lasers and coherent light sources

354
583 Lasers and Co 11. Lasers and Coherent Light Sources This chapter describes lasers and other sources of coherent light that operate in a wide wave- length range. First, the general principles for the generation of coherent continuous-wave and pulsed radiation are treated including the inter- action of radiation with matter, the properties of optical resonators and their modes as well as such processes as Q-switching and mode- locking. The general introduction is followed by sections on numerous types of lasers, the em- phasis being on today’s most important sources of coherent light, in particular on solid-state lasers and several types of gas lasers. An im- portant part of the chapter is devoted to the generation of coherent radiation by nonlin- ear processes with optical parametric oscillators, difference- and sum-frequency generation, and high-order harmonics. Radiation in the extended ultraviolet (EUV) and X-ray ranges can be gener- ated by free electron lasers (FEL) and advanced X-ray sources. Ultrahigh light intensities up to 10 21 W/ cm 2 open the door to studies of relativistic laser–matter interaction and laser particle accel- eration. The chapter closes with a section on laser stabilization. 11.1 Principles of Lasers ............................... 584 11.1.1 General Principles ....................... 584 11.1.2 Interaction of Radiation with Atoms ................................. 590 11.1.3 Laser Resonators and Modes ......... 595 11.1.4 Laser Rate Equations and Continuous-Wave Operation .. 602 11.1.5 Pulsed Laser Behavior .................. 605 11.2 Solid-State Lasers ................................. 614 11.2.1 Basics ........................................ 614 11.2.2 UV and Visible Rare-Earth Ion Lasers ........................................ 619 11.2.3 Near-Infrared Rare Earth Lasers .... 636 11.2.4 Mid-Infrared Lasers ..................... 660 11.2.5 Transition-Metal-Ion Lasers ......... 674 11.2.6 Overview of the most Important Laser Ions in Solid-State Lasers ..... 694 11.3 Semiconductor Lasers ............................ 695 11.3.1 Overview .................................... 695 11.3.2 Resonator Types and Modern Active Layer Materials: Quantum Effects and Strain .......... 698 11.3.3 Edge-Emitting Laser Diodes with Horizontal Resonators .......... 703 11.3.4 Basics of Surface-Emitting Lasers with Vertical Resonators (VCSELs) ... 720 11.3.5 Edge-Emitting Lasers and VCSELs with Low-Dimensional Active Regions ...................................... 725 11.3.6 Lasers with External Resonators .... 725 11.4 The CO 2 Laser ........................................ 726 11.4.1 Physical Principles ....................... 726 11.4.2 Typical Technical Designs.............. 737 11.5 Ion Lasers ............................................ 746 11.5.1 Ion-Laser Physics ........................ 747 11.5.2 Plasma Tube Design ..................... 749 11.5.3 Ion-Laser Resonators ................... 751 11.5.4 Electronics .................................. 753 11.5.5 Ion-Laser Applications ................. 755 11.6 The HeNe Laser ..................................... 756 11.6.1 The Active Medium ...................... 756 11.6.2 Construction and Design Principles 758 11.6.3 Stabilization ............................... 762 11.6.4 Manufacturing ............................ 763 11.6.5 Applications ............................... 764 11.7 Ultraviolet Lasers: Excimers, Fluorine (F 2 ), Nitrogen (N 2 ) ...... 764 11.7.1 The Unique Properties of Excimer Laser Radiation ........... 765 11.7.2 Technology of Current Excimer Lasers and the N 2 Laser ......................... 765 11.7.3 Applications ............................... 770 11.7.4 Outlook: Radiation in the EUV ....... 775 11.8 Dye Lasers ............................................ 777 11.8.1 Overview .................................... 777 11.8.2 General Description ..................... 777 11.8.3 Flashlamp-Pumped Dye Lasers ..... 777 11.8.4 Tunable Dye Lasers Pumped by High-Power Short-Wavelength Lasers ........................................ 778 Part C 11

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Page 1: Lasers and Coherent Light Sources

583

Lasers and Co11. Lasers and Coherent Light Sources

This chapter describes lasers and other sourcesof coherent light that operate in a wide wave-length range. First, the general principles forthe generation of coherent continuous-wave andpulsed radiation are treated including the inter-action of radiation with matter, the propertiesof optical resonators and their modes as wellas such processes as Q-switching and mode-locking. The general introduction is followed bysections on numerous types of lasers, the em-phasis being on today’s most important sourcesof coherent light, in particular on solid-statelasers and several types of gas lasers. An im-portant part of the chapter is devoted to thegeneration of coherent radiation by nonlin-ear processes with optical parametric oscillators,difference- and sum-frequency generation, andhigh-order harmonics. Radiation in the extendedultraviolet (EUV) and X-ray ranges can be gener-ated by free electron lasers (FEL) and advancedX-ray sources. Ultrahigh light intensities up to1021 W/cm2 open the door to studies of relativisticlaser–matter interaction and laser particle accel-eration. The chapter closes with a section on laserstabilization.

11.1 Principles of Lasers ............................... 58411.1.1 General Principles ....................... 58411.1.2 Interaction of Radiation

with Atoms................................. 59011.1.3 Laser Resonators and Modes ......... 59511.1.4 Laser Rate Equations

and Continuous-Wave Operation .. 60211.1.5 Pulsed Laser Behavior .................. 605

11.2 Solid-State Lasers ................................. 61411.2.1 Basics ........................................ 61411.2.2 UV and Visible Rare-Earth Ion

Lasers ........................................ 61911.2.3 Near-Infrared Rare Earth Lasers .... 63611.2.4 Mid-Infrared Lasers ..................... 66011.2.5 Transition-Metal-Ion Lasers ......... 67411.2.6 Overview of the most Important

Laser Ions in Solid-State Lasers ..... 694

11.3 Semiconductor Lasers............................ 69511.3.1 Overview .................................... 69511.3.2 Resonator Types

and Modern Active Layer Materials:Quantum Effects and Strain .......... 698

11.3.3 Edge-Emitting Laser Diodeswith Horizontal Resonators .......... 703

11.3.4 Basics of Surface-Emitting Laserswith Vertical Resonators (VCSELs) ... 720

11.3.5 Edge-Emitting Lasers and VCSELswith Low-Dimensional ActiveRegions...................................... 725

11.3.6 Lasers with External Resonators .... 725

11.4 The CO2 Laser........................................ 72611.4.1 Physical Principles ....................... 72611.4.2 Typical Technical Designs.............. 737

11.5 Ion Lasers ............................................ 74611.5.1 Ion-Laser Physics ........................ 74711.5.2 Plasma Tube Design ..................... 74911.5.3 Ion-Laser Resonators ................... 75111.5.4 Electronics.................................. 75311.5.5 Ion-Laser Applications ................. 755

11.6 The HeNe Laser ..................................... 75611.6.1 The Active Medium ...................... 75611.6.2 Construction and Design Principles 75811.6.3 Stabilization ............................... 76211.6.4 Manufacturing ............................ 76311.6.5 Applications ............................... 764

11.7 Ultraviolet Lasers:Excimers, Fluorine (F2), Nitrogen (N2) ...... 76411.7.1 The Unique Properties

of Excimer Laser Radiation ........... 76511.7.2 Technology of Current Excimer Lasers

and the N2 Laser ......................... 76511.7.3 Applications ............................... 77011.7.4 Outlook: Radiation in the EUV ....... 775

11.8 Dye Lasers ............................................ 77711.8.1 Overview .................................... 77711.8.2 General Description ..................... 77711.8.3 Flashlamp-Pumped Dye Lasers ..... 77711.8.4 Tunable Dye Lasers Pumped by

High-Power Short-WavelengthLasers ........................................ 778

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584 Part C Coherent and Incoherent Light Sources

11.8.5 Colliding-Pulse Mode-LockedDye Lasers .................................. 778

11.8.6 Tunable Continuous-WaveDye Lasers .................................. 779

11.8.7 Advanced Solid-State Dye Lasers ... 779

11.9 Optical Parametric Oscillators................. 78511.9.1 Optical Parametric Generation ...... 78611.9.2 Phase Matching .......................... 78711.9.3 Optical Parametric Oscillators ........ 79011.9.4 Design and Performance

of Optical Parametric Oscillators .... 790

11.10 Generation of Coherent Mid-Infrared Ra-diation by Difference-Frequency Mixing . 80111.10.1 Difference-Frequency Generation

(DFG) ......................................... 80211.10.2 DFG Laser Sources ........................ 80911.10.3 Outlook ...................................... 813

11.11 Free-Electron Lasers ............................. 81411.11.1 Principle of Operation .................. 81411.11.2 Current Status and Perspective

Applications of Free-ElectronLasers ........................................ 815

11.11.3 Suggested further reading ............ 819

11.12 X-ray and EUV Sources........................... 81911.12.1 X-Ray Lasers ............................... 81911.12.2 High-Order Harmonics ................. 822

11.13 Generation of Ultrahigh Light Intensitiesand Relativistic Laser–Matter Interaction 82711.13.1 Laser Systems for the Generation

of Ultrahigh Intensities ................ 82711.13.2 Relativistic Optics

and Laser Particle Acceleration...... 834

11.14 Frequency Stabilization of Lasers ........... 84111.14.1 Characterization

of Noise, Stability, Line Width,Reproducibility, and Uncertaintyof the Laser Frequency ................. 842

11.14.2 Basicsof Laser Frequency Stabilization .... 845

11.14.3 Examplesof Frequency-Stabilized Lasers...... 852

11.14.4 Measurementof Optical Frequencies ................. 863

11.14.5 Conclusion and Outlook ............... 864

References .................................................. 864

11.1 Principles of Lasers

11.1.1 General Principles

A laser (an acronym for light amplification by stim-ulated emission of radiation) is a device that producesand amplifies an intense beam of highly coherent, highlydirectional light. The original proposal of extending themaser (microwave amplification by stimulated emissionof radiation) idea to the infrared or visible region ofthe electromagnetic (EM) spectrum, thus giving a laser,was however first made in 1960 by Maiman [11.1] us-ing a flash-pumped rod of ruby with polished ends(ruby laser) after a proposal in 1958 by Townes andSchawlow [11.2]. Nowadays laser devices range in sizefrom semiconductor lasers as small as a grain of salt tosolid-state lasers as large as a storage building. Lasersare widely used in industry for, e.g., cutting and weld-ing metals and other materials, in medicine for surgery,in optical communications, in optical metrology, andin scientific research. They are an integral part of suchfamiliar devices as bar-code scanners used in supermar-kets, laser printers, compact disc and digital versatiledisc (DVD) players. The output wavelength of a laser is

determined by the properties of its active medium. Alto-gether, several thousand lasing lines have been reportedranging from the soft-X-ray down to the far-infraredspectral region and new lines appear frequently in opticsand laser journals. Depending upon the type of laser andits operational regime, the corresponding output powermay vary from a fraction of a milliwatt to several hun-dred kilowatts in continuous-wave operation, and fromtens of kilowatts to petawatt peak power in pulsed-modeoperation.

A laser consists of at least three components(Fig. 11.1):

1. a gain medium that can amplify light by means ofthe basic process of stimulated emission;

2. a pump source, which creates a population inversionin the gain medium;

3. two mirrors that form a resonator or optical cavityin which light is trapped, traveling back and forthbetween the mirrors.

The laser beam is usually the fractional part of lighttrapped in the cavity that escapes from one of the two

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Lasers and Coherent Light Sources 11.1 Principles of Lasers 585

Fig. 11.1 Schematic of a laser oscillator

mirrors (the output coupler), which has a nonvanish-ing transmission at the laser oscillation wavelength. Thegain medium can be solid (including semiconductors),liquid, or gas, and the pump source can be an electricaldischarge, a lamp, or another laser. Other specific com-ponents of a laser vary depending on the gain mediumand whether the laser is operated continuously or pulsed.Lasers may be in fact divided into two broad groups:

1. continuous wave (CW) or quasi-CW,2. pulsed.

A CW laser exhibits a steady flow of coherent energyand its output power undergoes little or no change withtime. Many gas lasers, such as HeNe and Ar-ion lasers,operate CW; several solid-state lasers, such as Nd3+and Ti3+:Al2O3 lasers, are also often operated in CWmode. In pulsed lasers, the output beam power changeswith time so as to produce a short optical pulse, usu-ally in a repetitive way and with pulse duration usuallyranging from nanoseconds (1 ns = 10−9 s) to femtosec-onds (1 fs = 10−15 s). Typical examples of pulsed lasersare many solid-state and liquid lasers, such as Nd:YAG,Ti:Al2O3, and dye lasers.

Spontaneousand Stimulated Emission, Absorption

A laser exploits three fundamental phenomena that occurwhen an electromagnetic wave interacts with a mater-ial, namely the processes of spontaneous and stimulatedemission, and the process of absorption (Fig. 11.2).

Spontaneous emission and nonradiative decay. Letus consider the energy levels of some given material andindicate by E1 and E2 the energies of the ground level,1, and of an excited level, 2, of the medium. If the atomis initially raised in the excited state 2, it spontaneouslytends to decay into the stable ground state 1. If the avail-able energy for the transition, E2 − E1 is released in

Fig. 11.2a–c Fundamental interaction processes of a quan-tized atom with an EM wave. (a) Spontaneous emission.(b) Stimulated emission. (c) Absorption

the form of an EM wave, the process is referred to asspontaneous, or radiative, emission (Fig. 11.2a). Owingto energy conservation, the frequency ν0 of the emittedradiation is given by:

ν0 = (E2 − E1)

h, (11.1)

where h is the Planck’s constant. Since the previous re-lation can also be written as E2 − E1 = hν0, we can saythat a single photon of energy hν0 is emitted during eachspontaneous emission process. Note also that this pro-cess occurs even if the atom is isolated and no externalperturbation is applied. The probability of spontaneousemission can be characterized in the following way. Letus consider an ensemble of atoms and assume that, attime t, there are N2 atoms per unit volume (population)in level 2.

Quantum mechanical calculations show that the rateof decay of these atoms due to spontaneous emission,i. e., (dN2/dt)sp, is proportional to N2 according to:(

dN2

dt

)sp

= − N2

τs, (11.2)

where τs is referred to as the spontaneous emission life-time and depends on the particular transition involved.The direction, the polarization, and the phase of theemission event are random, so that the overall emittedlight by the different atoms of the given population issaid to be incoherent.

If the atom is not isolated but interacts, e.g., by col-lisions with other atoms, the decay from the excitedstate 2 to the ground state 1 may occur by a releaseof the internal energy (E2 − E1) into some form otherthan EM radiation (e.g., into kinetic or internal energyof surrounding atoms in a gas or lattice vibrations ina crystal). The phenomenon is referred to as nonra-diative decay. The corresponding rate of decay fromthe excited state 2 can usually be expressed in a sim-ilar manner as (11.2), namely (dN2/dt)nr = −N2/τnr,

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586 Part C Coherent and Incoherent Light Sources

where τnr is the nonradiative lifetime. When both radia-tive and nonradiative decay are taken into account, therate of decay of population N2 can be written as:(

dN2

dt

)decay

= − N2

τs− N2

τnr≡ − N2

τ, (11.3)

where the time constant τ , defined by 1/τ = 1/τs +1/τnr, is referred to as the lifetime of the excited state 2.

Stimulated emission. If an EM wave of frequency νclose to ν0 is incident onto the atom while in its excitedstate 2, the interaction of the wave with the atom maystimulate the atom to decay to level 1 with the simultane-ous emission of EM radiation. The process is referred toas stimulated emission. In this case, one photon, with thesame frequency ν of the incoming radiation and with thesame propagation direction, polarization state and phaseis emitted (Fig. 11.2b). This is a major distinction tospontaneous emission. It is the fundamental reason whya laser emits coherent light as compared to the incoher-ent light emitted by other light sources [such as lampsor light-emitting diodes (LEDs)], which exploit sponta-neous emission. Quantum mechanical calculations showthat the process of stimulated emission can be describedby the equation(

dN2

dt

)st

= −W21 N2 , (11.4)

where W21 is the rate of stimulated emission. This rate isproportional to the photon flux F = I/(hν) of the incom-ing wave, where I is the wave intensity. We can in factwrite W21 = σ21 F, where σ21 is a quantity having thedimension of an area (the stimulated emission cross sec-tion), which depends on the characteristics of the giventransition and on the frequency difference ∆ν = ν−ν0,i. e., σ21 = σ21(∆ν). Owing to energy conservation in the

Fig. 11.3a,b Coherent amplification of an EM wave. (a) Photonflux-balance diagram in an infinitesimal section dz of the amplifier.(b) Physical meaning of the transition cross section

process of stimulated emission, the function σ21(∆ν) isvery narrow at around ∆ν = 0.

Absorption. Lastly, let us consider the case of an atominitially lying in its ground state 1. In the absence ofperturbations, such as collisions with other atoms or withphotons, the atom stably remains in this state. However,if an EM radiation of frequency ν ν0 is incident ontothe atom, there is a finite probability that the atom will beraised to level 2. The energy difference E2 − E1 requiredby the atom to undergo the transition is obtained fromthe energy of the EM wave, namely one photon of theincoming wave is destroyed. This process is referred toas absorption (Fig. 11.2c). As for stimulated emission,the absorption process can be described by(

dN2

dt

)a= −W12 N2 , (11.5)

where W12 is the rate of absorption. Again one can showthat W12 = σ12 F, where σ12 is the absorption crosssection. Einstein showed at the beginning of the 20thcentury that, for nondegenerate levels, W12 = W21, andthus σ12 = σ21. If levels 1 and 2 are g1-fold and g2-folddegenerate, respectively, one then has g2σ21 = g1σ12.

Coherent Amplification of LightConsider a monochromatic EM plane wave at fre-quency ν which propagates along the z-direction insidea medium made of a collection of atoms. Let E1 and E2be the energies of two nondegenerate levels 1 and 2 ofthe atom (this time 1 is not necessarily the ground state).We assume that the resonance frequency of the transi-tion, ν0 = (E2 − E1)/h, is coincident (or very close) toν. If F(z) is the photon flux of the EM wave at plane zand N1, N2 are the populations in levels 1 and 2 (whichare assumed to be z-independent), the change dF of thephoton flux due to the processes of absorption and stim-ulated emission along the elemental length dz of thematerial is given by (Fig. 11.3a)

dF = σ(N2 − N1)F(z)dz , (11.6)

where σ ≡ σ21 = σ12 is the transition cross section. Notethat in writing (11.6) we did not consider radiative andnonradiative decays since nonradiative decay does notadd new photons whereas spontaneous emission cre-ates photons which are emitted in any direction andthus gives negligible contribution to the incoming pho-ton flux. Most importantly, note that for N2 > N1, onehas dF/dz > 0 and the EM wave is amplified duringpropagation, i. e., the medium acts as a coherent opticalamplifier. Conversely, for N2 < N1, one has dF/dz < 0

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Lasers and Coherent Light Sources 11.1 Principles of Lasers 587

and the medium behaves as an absorber. If we let l bethe length of the medium, the photon flux F(l) at the out-put plane is related to the one F(0) at the input plane bythe relation F(l) = F(0) exp(g), where g = (N2 − N1)σlis the gain coefficient. The condition for light amplifica-tion (g> 0) is therefore N2 > N1, which is often referredto as population inversion. We note that in the previouscalculation N1 and N2 have been assumed independentof the intensity I of propagating wave. This is fine how-ever provided that I is weak enough so as that changesof N1 and N2 due to absorption and stimulated emis-sions can be neglected; however, for strong intensitiesone needs to account for the phenomenon of saturation.

As a final comment, we note that an examination of(11.6) leads to a simple physical interpretation of thetransition cross section σ . First, let us suppose that allthe atoms of the medium are in the ground state and letus associate with each atom an effective absorption crosssection σa in the sense that, if a photon enters this crosssection, it is absorbed by the atom (Fig. 11.3b). If S isthe cross-sectional area of the EM wave, the number ofatoms in the element dz of the material is NtS dz (Nt isthe total atomic population, i. e., N1 = Nt and N2 = 0),thus giving a total absorption cross section of σa NtS dz.The fractional decrease of photon flux in the element dzof the material is therefore (dF/F) = −(σa NtS dz/S) =−σa Nt dz. If we compare this expression with (11.6),we conclude that σa = σ , so that the meaning we canattribute to σ is that of an effective absorption crosssection as just defined.

Population Inversion: The Pumping ProcessAt thermal equilibrium any material behaves as an ab-sorber. In fact, the distribution of populations N e

2 andN e

1 of levels 1 and 2 at thermal equilibrium is describedby the Boltzmann statistics

N e2

N e1

= exp

(− E2 − E1

kBT

), (11.7)

where kB is the Boltzmann constant and T is the absolutetemperature. Note that N e

2 < N e1 ; in particular N e

2 is neg-ligible as compared to N e

1 if kBT E2 − E1; at roomtemperature (T = 300 K) one has kBT 208 cm−1.[A reciprocal centimeter is a simple convenient unitfor measuring energies in spectroscopy and refers tothe inverse of the wavelength of a photon possess-ing the given energy E. The actual energy E in theSI unit (Joule) can be obtained by multiplying cm−1

by hc, with c = 3 × 1010 cm/s and h = 6.63 × 10−34 Js].Therefore the condition N e

2 N e1 is well satisfied for

transitions in the near-infrared and visible. To obtain op-

tical amplification instead of optical absorption, we needto create population inversion in the medium by meansof a pumping process [11.3, 4], which drives the popu-lation distribution far from thermal equilibrium. At firstsight one might think to achieve population inversionbetween levels 2 and 1 through the interaction of thematerial with some strong EM radiation at frequencyν0, such as that emitted by a flash or arc lamp. Sinceat thermal equilibrium N e

1 > N e2 , absorption will domi-

nate over stimulated emission, and this will produce anincrease of N2 and a decrease of N1 from their thermalequilibrium values. However, when N1 tends to reachthe same value of N2, absorption and stimulated emis-sion processes compensate each other, i. e., the mediumtends to become transparent. Such a situation is referredto as saturation of the two-level transition. Therefore,owing to saturation, it is impossible to produce popu-lation inversion in a two-level system (at least in thesteady state). This goal can be achieved, however, whenmore than two energy levels are considered. Typicallythree or four energy levels are involved (Fig. 11.4), andcorrespondingly one speaks of three-level or four-levellasers. In a three-level laser (Fig. 11.4a), atoms are raisedfrom the ground level 1 to the level 3 through a pump-ing mechanism. If the material is such that, once anatom excited into level 3 rapidly decays into a lowerlevel 2 (by, e.g., a fast nonradiative decay), then popu-lation inversion can be obtained between level 2 and 1.Note that, in a three-level laser scheme, to achieve pop-ulation inversion it is necessary to raise at least half ofthe atoms from the ground state 1 to state 3. In a four-level laser (Fig. 11.4b), atoms are again raised from theground level 0 to an excited state 3, with a rapid decayto the upper laser level 2; however this time laser ac-tion takes place from level 2 to an excited lower-lyinglevel 1. Once laser oscillation starts, level 1 is populatedby stimulated emission; therefore to maintain popula-

!"#

!"#

!"#

Fig. 11.4a,b Schematic of the energy-level diagram for (a) a three-level laser, and (b) a four-level laser

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588 Part C Coherent and Incoherent Light Sources

tion inversion in stationary conditions (continuous-waveoperation), the lower laser level 1 should rapidly depop-ulate through, e.g., a very fast nonradiative decay intothe ground state 0. Compared to a three-level laser, thefour-level laser offers the great advantage that populationinversion is ideally achieved when just one atom is raisedto the pumping level 3. Four-level lasers are thus moreused, whenever possible, than three-level lasers. Morerecently, the so-called quasi-three-level lasers have alsobecome a rather important laser category. In these lasersthe energy level scheme is similar to that of a four-levellaser, however levels 0 and 1 are now nondegeneratesublevels of the ground energy level. The population ofthe ground state is distributed in all the sublevels ac-cording to Boltzmann statistics, and therefore, at roomtemperature, some population is left in level 1.

The mechanism that allows atom excitation from theground energy level into the excited pump level 3 is re-ferred to as the pumping process. The rate of populationof the upper state, 2, due to the pumping, (dN2/dt)p,can be written as:(

dN2

dt

)p= Wp Ng , (11.8)

where Wp is the pumping rate and Ng is the popula-tion of the ground state. For a four-level laser, one canassume Ng to be constant (and much larger than N2).In this case one can write from (11.8) (dN2/dt)p = Rp,where Rp = Wp Ng is the pump rate per unit volume. Theenergy required for pumping is generally supplied eitheroptically or electrically. The minimum pump power Pmneeded to produce a given pump rate Rp is given byPm = (dN2/dt)pVhνmp, where (dN2/dt)p is the num-ber of atoms per unit volume and time raised to the upperlaser level by the pumping process, V is the volume ofthe active medium, and νmp is the minimum pump fre-quency, given by the difference between the ground leveland the upper laser level. For either electrical or opti-cal pumping, the actual pump power Pp turns out to belarger than the minimum value Pm, so that one can definea pump efficiency ηp = Pm/Pp. Therefore, the relationbetween the pump rate Rp and the actual pump powerPp is given by:

Rp = ηp

(Pp

Vhνmp

). (11.9)

Optical pumping. In optical pumping by an incoherentsource, light from a powerful lamp (usually medium- tohigh-pressure Xe and Kr flashlamps for pulsed lasers,or high-pressure Kr lamps for CW lasers) is absorbed

by the active medium. Solid-state gain media, such asNd:YAG, and liquid lasers are particularly well suitedto optical pumping: absorption lines are in fact verybroad and this makes the absorption of the (broadband)light of the lamps efficient. The flashlamps generate sig-nificant heat into the material that must be dissipatedby water cooling. The pumping source may, however,itself be a laser; in this case one speaks of laser pump-ing. Among the most used laser pump sources, we justmention semiconductor lasers (diode laser pumping),argon-ion laser and second-harmonic or third-harmonicgeneration of Nd lasers. With the advent of powerfuland reliable semiconductor lasers, diode-laser pumpingis nowadays commonplace for many solid-state and fiberlasers.

Electrical pumping. In gas lasers and semiconductorlasers, the excitation mechanism usually consists ofan electrical current flow through the active medium.Gas lasers generally need to be electrically pumped be-cause, due to the small width of their absorption lines,optical pumping would be very inefficient. In gas lasers,electrical pumping is achieved by passing an electriccurrent either CW, at radio frequency or pulsed, directlythrough the gas itself. During the discharge, ions andfree electrons are produced which acquire kinetic en-ergy from the applied field and are able to excite neutralatoms via collisions: A+e → A∗ +e, where A∗ denotesthe atomic species in an excited state. Since electron im-pact excitation is a nonresonant process, it is a ratherefficient pumping method for a gas. In some cases, thegas may contain two species, one of which is first excitedby the discharge and then undergoes resonant energytransfer with the other via collision (an example is theHeNe laser). Electrical pumping in semiconductor lasersis achieved by flowing a large current density in a p–nor p–i–n diode. Though optical pumping can be used forsemiconductor lasers, electrical pumping proves to bemuch more convenient.

Pumping processes different from optical or elec-trical pumping may also be employed in some speciallasers; we just mention chemical pumping in chemicallasers, where population inversion is produced directlyfrom an exothermic reaction.

Laser OscillationTo obtain laser oscillation, the amplifying medium isplaced between two mirrors, forming a laser cavity(Fig. 11.1). Light propagating along the cavity axis andpassing through the pumped laser medium is reflectedback through it, stimulating further emission in the same

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direction. This means that laser photons undergo mul-tiple reflections within the cavity, being amplified ateach pass through the laser medium. One mirror (to-tal reflector) reflects almost all the incident light backthrough the laser medium while the other (partial reflec-tor or output coupler) transmits a fraction T2 = 1− R2,which constitutes the output laser beam. The combina-tion of laser gain medium, pumping source and opticalcavity forms a simple laser oscillator: if the amplifi-cation is large enough to overcame losses, i. e., whena threshold condition is reached, a single photon (whichis always present due to quantum noise) can be ampli-fied by several orders of magnitude to produce a hugenumber of coherently generated photons trapped insidethe resonator. In addition, for an open resonator, only thephotons which propagate in the paraxial direction of theresonator axis can reach the threshold for oscillation, sothat an important property of the output laser beam is thatit is directional. We can obtain the threshold conditionfor laser action by a simple argument based on gain/lossbalance of light photons in one cavity round-trip. Infact, in one round-trip the photons pass twice throughthe gain medium and hence the round-trip gain of pho-tons is exp(2σNl), where N = N2 − N1 is the populationinversion, and l is the length of the active medium. Onthe other hand, the round-trip loss for photons can bewritten as (1− T1)(1− T2)(1− L i)2, where T1 = 1− R1and T2 = 1− R2 are the power transmission of the twomirrors whereas L i accounts for the one-way internalloss in the cavity (due to, e.g., scattering or diffractionloss for open resonators). The population inversion Ncneeded to reach threshold (also called critical inversion)is simply obtained by equating the round-trip gain andloss. After setting γ1 = −ln(1− T1), γ2 = −ln(1− T2)and γi = −ln(1− L i), the critical inversion turns out tobe given by

Nc = γ

σl, (11.10)

where

γ = γi + γ1 +γ2

2(11.11)

is referred to as the single-pass logarithmic loss of thecavity. Once the critical inversion is reached, oscillationbuilds up from spontaneous emission.

To calculate the pump rate Rcp needed to reach thethreshold condition (the critical pump rate), let us con-sider the four-level laser (Fig. 11.4b). In steady-stateconditions and in the absence of lasing, the popula-tion accumulated on the upper laser level can simply

be calculated from a balance between the number ofatoms pumped per unit volume and time in the upperlaser level and the number of atoms that decay via ra-diative and nonradiative ways. Assuming N1 0, oneobtains ∆N N2 = Rpτ . When N2 = Nc, from the pre-vious equation and with the help of (11.10) the criticalpump rate is obtained as Rcp = γ/(στl). Note that thecritical pump rate increases as στ decreases. Thereforethe product στ , which depends on the properties of thegiven transition, can be regarded as a figure of merit fora given laser.

Properties of Laser BeamsLaser radiation shows an extremely high degreeof monochromaticity, coherence, directionality andbrightness as compared to other noncoherent lightsources [11.3].

The monochromaticity of laser radiation results fromthe circumstance that light oscillation sets in at one res-onance frequency of the optical cavity, and owing to thebalance between gain and loss in CW operation the linewidth ∆νL of the oscillating mode is ultimately limitedby quantum noise. Conversely, light from incoherentsources which exploit spontaneous emission (includingLEDs) has a much lower degree of monochromaticity (asmuch as 11 orders of magnitude) since the spectral distri-bution of spontaneously emitted photons is broadened ataround the atomic resonance frequency ν0 owing to var-ious broadening mechanisms. For laser radiation in thevisible or near-infrared, line widths ∆νL as low as a fewHz may, in fact, be achieved in frequency-stabilized lasersources.

The coherence of laser radiation refers to either tem-poral or spatial coherence. To define spatial coherence,let us consider two points P1 and P2 that, at a time t = 0,belong to the same wavefront, i. e., the phase differencebetween their electric fields at time t = 0 is zero. If thedifference ϕ2(t)−ϕ1(t) of their phases also remains zeroat times t> 0, we say that there is perfect spatial coher-ence between the two points P1 and P2. In practice, forany point P1, in order to have some degree of phase cor-relation, point P2 must lie within some finite area aroundP1, which is called the coherence area. The high degreeof spatial coherence of laser radiation stems again fromthe fact that the spatial field distribution of the beam gen-erated by stimulated emission is a mode of the opticalresonator.

To define temporal coherence at a given point P, letus consider the phase difference ϕ(t + τ)−ϕ(t) for theelectric field at P at times t + τ and t. For a given de-lay τ , if the phase difference is independent of time t,

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we say that there is temporal coherence over a time τ .If this occurs for any value of τ , the EM wave is saidto have perfect temporal coherence. If, conversely, thisoccurs only for delays τ smaller than a given delay τ0,the wave is said to have partial temporal coherence,and τ0 is called the coherence time of the EM wave atpoint P. The concept of temporal coherence is closelyrelated to that of monochromaticity; in fact, for a CWlaser one can show that the coherence time τ0 is re-lated to the laser line width ∆νL by the simple relationτ0 ≈ 1/∆νL. The high degree of temporal coherence oflaser radiation is therefore due to its extreme degree ofmonochromaticity.

The directionality of the laser beam is due to the factthat the gain medium is placed inside an open opticalresonator and, as a consequence, stimulated emissionpreferentially occurs in the direction orthogonal to thetwo cavity mirrors (Fig. 11.1), where feedback from themirrors is most effective and diffraction losses are thesmallest. The laser beam emitted from the output cou-pler shows a divergence angle which, in the ideal case, islimited by diffraction. From diffraction theory, the diver-gence angle for a monochromatic beam of wavelengthλ turns out to be given by

θd = β(λ

2w

), (11.12)

where β is a numerical coefficient (of order 1) whosespecific value depends on the particular transverse fielddistribution and 2w is the diameter of the beam. Forexample, considering laser radiation in the visible (λ=500 nm), the divergence of a laser beam of transversediameter 2w≈ 1 cm is solely θd ≈ λ/2w 5 × 10−5 rad.This means that, after propagation for a length L = 1 km,the beam size is increased to solely w+ θdL ≈ 6 cm!

The brightness of laser radiation is closely related tothe directionality and stems from the capability of a laseroscillator to emit a high optical power in a small solidangle of space. For a given emitting source, of area ∆S,if P denotes the optical power delivered in a fractionalsolid angle ∆Ω of space, the brightness of the emitteris defined as B = P/(∆S∆Ω). The brightness of a lasersource, in which the solid angle of emission ∆Ω isdetermined by its divergence angle θd, is given by

B =(

2

βπλ

)2

P . (11.13)

A laser of moderate power (e.g., a few milliwatts) hasa brightness several orders of magnitude greater thanthat of the brightest conventional sources.

11.1.2 Interaction of Radiation with Atoms

Absorption and Stimulated Emission RatesConsider a monochromatic EM wave of frequency ν in-cident on a two-level atom with a transition frequencyν0 = (E2 − E1)/h close to ν. The calculation of the ab-sorption and stimulated emission rates W12 and W21 canbe done following a semiclassical approach, in whichthe atom is treated quantum mechanically whereas theEM wave classically [11.3,5–9]. If the atom is initially inits ground state (level 1), the incident wave may inducea transition to level 2 owing to the coupling of the elec-tromagnetic field with the electric and magnetic dipole(and multipole) moments of the atom. The strongest in-teraction is usually that arising between the electric fieldE(t) = E0 cos(2πνt) of the EM and the electric dipolemoment of the transition, which is defined by the ma-trix dipole element µ12 = ∫ u∗

2(r)eru1(r)dr, where r isthe distance of the electron of the atom, with chargee, from its nucleus, and where u1(r) and u2(r) are theelectronic eigenfunctions for the atomic energy levels 1and 2 respectively. In this electric-dipole approximation,a perturbation theory, which assumes that the interactionbetween the EM wave and the atom is not disturbed by,e.g., collisions or other phenomena (including sponta-neous emission), leads to the following expression forthe absorption and stimulated emission rate

W12 = W21 = 2π2

3n2ε0h2 |µ12|2ρδ(ν−ν0) , (11.14)

where ρ = n2ε0 E20/2 is the energy density of the EM

wave, n is the refractive index of the medium, and δ is theimpulse Dirac function. In particular, for a plane waveof intensity I = cρ/n, from (11.14) the following ex-pression for the cross section σ12 = W12/F = hνW12/Ican be derived:

σ12 = 2π2

3nε0ch|µ12|2νδ(ν−ν0) . (11.15)

Analogously, if initially the electron is on level 2, owingto the interaction with the EM wave there is a prob-ability that the electron decays into state 1 emittinga photon by stimulated emission. The semiclassical per-turbation calculation leads for the stimulated emissioncross section σ21 = hνW21/I the same expression as thatof σ12, i. e., σ21 = σ12, provided that the two levels arenot degenerate.

The expression for σ12 = σ21 as given by (11.15) isunphysical since it implies that the transition probabilityis zero for ν = ν0 and ∞ for ν = ν0. The inconsis-tency is removed by observing that the interaction of

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the monochromatic EM wave with the atom is not per-fectly coherent, but it is disturbed by, e.g., collisionswith other atoms or with lattice phonons, spontaneousemission and nonradiative decay of the atom. The effectof such disturbing interactions is to broaden the tran-sition line of each atom in the ensemble, in the sensethat in (11.15) the Dirac δ function δ(ν−ν0) should bereplaced by a new function g(ν−ν0), symmetric aboutν = ν0 with

∫g(ν−ν0)dν = 1, which is generally given

by the Lorentzian function (Fig. 11.5a):

g(ν−ν0) = 2

π∆ν0

1

1+ [2(ν−ν0)/∆ν0]2, (11.16)

where ∆ν0 depends on the particular broadening mecha-nism involved. Note that the full width at half-maximum(FWHM) of the Lorentzian function is simply ∆ν0.The resonant character of the stimulated emission andabsorption processes is maintained since the line broad-ening ∆ν0 is typically several orders of magnitudesmaller than ν0 (e.g., in a low-pressure gas, for a tran-sition in the visible one has ν0 ≈ 5 × 1014 Hz whereas∆ν0 ≈ 106 –108 Hz). Since the aforementioned broad-ening mechanism act in the same way on each atomof the ensemble, it is referred to as a homogeneousbroadening mechanism.

A somewhat different case occurs when the reso-nance frequencies ν′0 of the atoms in the ensemble aredistributed around a central frequency ν0 (inhomoge-neous broadening). This distribution is described by thefunction g∗(ν′0 −ν0) such that

∫g∗(ν′0 −ν0)dν′0 = 1 and

g∗(ν′0 −ν0)dν′0 is the fractional part of the atoms in theensemble whose resonance frequency lies in the inter-val (ν′0, ν′0 + dν′0). For the most common mechanismsof inhomogeneous line broadening (such as Dopplerbroadening in a gas or local-field effects in ionic crys-tals or glasses) the distribution g∗(ν′0 −ν0) is given bya Gaussian function (Fig. 11.5b):

g∗(ν′0 −ν0) = 2

∆ν∗0

(ln2

π

)1/2

× exp

(−4ln2(ν′0 −ν0)2

∆ν∗20

), (11.17)

where ∆ν∗0 is the transition line width (FWHM), whichdepends on the particular broadening mechanism.

Taking into account the simultaneous presenceof both homogeneous and inhomogeneous broaden-ing mechanisms, one can show that the cross sectionσ = σ12 = σ21 for stimulated emission and absorption

$%&

'

$((

'

Fig. 11.5 Lorentzian (left side) and Gaussian (right side) lineshapes corresponding to homogeneous and inhomogeneous tran-sition broadening, respectively

assumes the most general form:

σ = 2π2

3nε0ch|µ12|2νgt(ν−ν0) , (11.18)

where the total line shape gt is given by the convolution:

gt(ν−ν0) =∫

g∗(ν′0 −ν0)g(ν−ν′0)dν′0 . (11.19)

If we consider an ensemble of atoms and indicate byN1 and N2 the populations of atoms in the energylevels 1 and 2, as discussed in the previous sectiona small-signal EM wave of frequency ν which prop-agates inside the medium experiences amplificationfor N2 > N1 or absorption for N2 < N1. In the for-mer case, one can introduce the absorption coefficientper unit length α(ν) = σ(ν− ν0)(N1 − N2), whereasin the latter case one can introduce the gain coeffi-cient per unit length g(ν) = σ(ν− ν0)(N2 − N1). Fora weak-signal wave, so that saturation can be ne-glected, the intensity of the propagating wave is thusexponentially attenuated or amplified along the propa-gation direction according to I(z) = I(0) exp(−αz) orI(z) = I(0) exp(gz), respectively.

These considerations apply to atomic or molecu-lar transitions which are electric-dipole allowed, i. e.,for which the dipole matrix element µ12 of the tran-sition does not vanish. Transitions between atomic ormolecular energy levels with the same parity (e.g., be-tween s states in an atom) are electric-dipole forbidden.This does not mean however that the atom or moleculecannot pass from level 1 to level 2 when interactingwith an EM wave; in this case the transition can oc-cur owing to the interaction of the EM wave with, e.g.,the magnetic dipole moment (or the electric quadrupolemoment) of the transition, though the strength of thecross section describing this process is much smallerthan that of an allowed electric dipole transition. For an

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electric dipole transition the absorption transition rateWe is proportional to ∼ |µe E0|2, where µe is the elec-tric dipole moment of the transition, E0 the amplitude ofthe electric field of the EM wave and −E0µe is the clas-sical energy of an electric dipole in an external field E0.Likewise, for a magnetic dipole interaction, the transi-tion rate Wm is proportional to ∼ |µm B0|2, where µmis the magnetic dipole of the transition, B0 is the ampli-tude of the magnetic field of the EM wave, and −B0µmis the classical energy of a magnetic dipole in an exter-nal magnetic field B0. By approximating µe ≈ ea andµm ≈ β, where a = 0.529 × 10−10 m is the Bohr radiusand β = 9.27 × 10−24 Am2 is the Bohr magneton, onethen obtains:(

We

Wm

)=(

eaE0

βB0

)2

=(

eac

β

)2

105 . (11.20)

Spontaneous EmissionSpontaneous emission is the phenomenon by whichan atom in an excited energy level tends to decay to-ward the ground state (even in absence of any externalperturbation) by emitting an EM wave, i. e., one photon.A correct explanation of spontaneous emission requiresa quantum electrodynamic approach in which both theatom and the EM field are quantized. In practice, due toquantization of the EM field, the mean values of bothE2 and H2 fields are nonvanishing even in the absenceof a (classical) EM wave (zero-point field fluctuations).Such intrinsic fluctuations always perturb the atom in anexcited state and trigger its decay toward a lower energylevel with the emission of one photon with frequencyν close to the atomic transition frequency ν0. Atomicdecay due to spontaneous emission follows an exponen-tial law with a time constant τs, which is referred toas the spontaneous emission lifetime [see (11.2)]. Thequantum electrodynamic calculation of τs in the electricdipole approximation and for an atom placed in an op-tical cavity was done by Weisskopf and Wigner [11.10].A simple and rigorous calculation of τs may be derivedusing an elegant thermodynamic argument, which wasproposed by Einstein well before any development of thequantum electrodynamics [11.11]. Assume that the ma-terial is placed inside a black-body cavity whose wallsare kept at a constant temperature T . Once the thermo-dynamic equilibrium is reached, the spectral EM energydensity distribution ρ(ν) inside the cavity is given by thePlanck distribution

ρ(ν) =(

8πν2

c3

)hν

exp(hν/kT )−1(11.21)

where ρ(ν) is such that ρ(ν)dν is the EM energy perunit volume of the cavity associated to modes with fre-quencies in the interval (ν, ν+ dν). We note that in(11.21) the factor (8πν2/c3) represents the density ofEM modes per unit volume of the cavity, whereas theterm hν/[exp(hν/kBT )−1] is the energy per mode. Thepopulations N e

1 and N e2 in the atomic levels 1 and 2 at

thermal equilibrium are described by a Boltzmann statis-tics (11.7); on the other hand, at steady state the numberof excitation per unit time from level 1 to level 2 dueto the absorption of black-body radiation should bal-ance the number of decays per unit time from level 2 tolevel 1 due to both stimulated emission and spontaneousemission, i. e., W12 N e

1 = W21 N e2 + N e

2 /τs. One thenobtains 1/τs = W12 exp(hν0/kT ) − W21. For a spec-trally broad radiation (such as black-body radiation)one can write W12 = W21 = ∫ dνcσ(ν− ν0)ρ(ν)/nhν= 2π2|µ12|2ρ(ν0)/(3n2ε0h2), where we used (11.15).Using this expression for W12 = W21 and (11.21) forρ(ν0), one finally obtains

τs = 3hε0c3

16π3ν30n|µ12|2

. (11.22)

Using a similar thermodynamic argument, one can alsoshow that the spectrum of the photon emitted by spon-taneous emission is given by the line shape gt(ν−ν0) ofthe transition (11.19), i. e., the probability that the pho-ton emitted by spontaneous emission has a frequency inthe range (ν, ν+ dν) is given by gt(ν−ν0)dν. This prop-erty is very interesting since it enables one to measurethe transition line shape gt(ν−ν0) simply in an emissionexperiment by passing the spontaneously emitted lightthrough a spectrometer of sufficiently high resolution.

To estimate the radiative lifetime τs, let us con-sider for instance an electric-dipole-allowed transitionat a frequency corresponding to the middle of the visi-ble range (λ0 = c/ν0 = 500 nm). Assuming |µ12| ea(where a 0.1 nm is the atomic radius), from (11.22)one obtains τs 10 ns. For a magnetic dipole transition1/τs turns out to be 105 times smaller, i. e., τs 1 ms.Note that, according to (11.22), τs decreases as the cubeof the transition frequency, so that the importance ofspontaneous emission increases rapidly with frequency.This implies that, as spontaneous emission is often neg-ligible in the middle- to far-infrared, where nonradiativedecay dominates, in the X-ray region (e.g., λ0 < 5 nm)spontaneous emission is the dominant decay processand τs becomes exceedingly short (10–100 fs). Sucha short lifetime represents a major challenge for achiev-ing a population inversion in X-ray lasers. It shouldfinally be noted that the rate of spontaneous emission as

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given by (11.22) can be either enhanced or reduced whenthe atom radiates in a dielectric cavity whose density ofEM modes (i. e., the number of resonant cavity modesper unit frequency and per unit volume) is changed fromthe value (8πν2/c3) due to, e.g., tight spatial confine-ment, such as in microlasers. For instance, spontaneousdecay is fully inhibited for an atom placed inside a pho-tonic crystal whenever the atomic transition frequencyν0 falls inside a bandgap of the photonic crystal.

Line Broadening MechanismsHomogeneous broadening. A line broadening mech-anism is said to be homogeneous when it broadens theline of each atom in the same way. In this case the lineshape of the single-atom cross section and that of theoverall absorption cross section are identical. There aretwo main homogeneous broadening mechanisms: colli-sional broadening and natural broadening.

Collisional broadening occurs in a gas, where theatom may collide with other atoms, ions, free elec-trons, or with the walls of the container, as well as ina solid, where collisions are due to the interaction ofthe atom with the lattice phonons. During a process ofabsorption or stimulated emission of a two-level atomwith a monochromatic EM wave, the collisions inter-rupt the coherent interaction of the EM with the atom;if we write the electronic wave function ψ of the atomduring the transition as ψ = c1(t)u1(r) exp[−iE1t/]+c2(t)u2(r) exp[−iE2t/], assuming that the collisiondoes not induce a decay, it simply introduces a ran-dom and rather instantaneous relative phase jump of thecoefficients c1 and c2, and thus of the oscillating part ofthe atomic dipole µ= ∫ −er|ψ(r, t)|2 dr, which is pro-portional to c1c∗

2. Since in the electric-dipole interactionthe radiation–atom coupling is expressed by the energyterm −µe E, a different but equivalent picture is to as-sume that it is the phase of the electric field that showsrandom phase jumps rather than the atomic dipole mo-ment (Fig. 11.6). Therefore we can consider the case ofcoherent (i. e., not disturbed) interaction of the two-levelatom with an EM wave which is not monochromatic butwhose spectral content is broadened at around ν owingto the phase jumps. Let I(ν′) = Ig(ν′ −ν) be the spec-tral intensity distribution of the EM wave with randomphase jumps (Fig. 11.6), where I is the total field inten-sity and g its spectral shape (with

∫g(ν′ −ν)dν′ = 1).

For each fractional spectral component Ig(ν′ −ν)dν′ ofthe EM wave, we may introduce an elemental absorp-tion transition rate dW12 given, according to (11.14), bydW12 = [2π2|µ12|2/(3nε0ch2)]δ(ν′ − ν0)Ig(ν′ − ν)dν′.The total transition rate, W12 = ∫ dW12, is then given

Fig. 11.6 Schematic of the sinusoidal electric field of fre-quency ν showing random phase jumps at time intervals τ

by W12 = σ12(ν−ν0)I/hν, where the cross section σ12is given by (11.15) after replacing the delta functionδ(ν−ν0) with g(ν−ν0). To calculate the line shape g,let us assume that the distribution of the time interval τbetween two successive collisions be described by an ex-ponential probability density, p(τ) = [exp(−τ/τc)]/τc,where τc is the mean value of τ . According to theWiener–Kintchine theorem, the spectrum g can be cal-culated as the Fourier transform of the autocorrelationfunction of the sinusoidal field with phase jumps attime intervals τ (Fig. 11.6). This yields a Lorentzianline shape (11.16) with a FWHM of ∆ν0, related to themean value of collision time τc by the equation

∆ν0 = 1

πτc. (11.23)

For instance, for an atomic or molecular gas at pres-sure p and absolute temperature T , from kinetic theoryand employing the hard-sphere model of a gas, one hasτc = (2MkBT/3)1/2[1/(8π pa2)], where M is the atomicmass and a its radius. Note that τc is inversely propor-tional, and hence ∆ν0 is directly proportional, to the gaspressure p. As a rough rule of thumb, we can say that,for any atom in a gas, collisions contribute to the linebroadening by an amount (∆νcoll/p) ≈ 1 MHz/Torr.

For an atom or an ion in a crystal, collisions occurwith lattice phonons. Since the number of phonons ina given lattice vibration is strongly dependent on thelattice temperature T , the corresponding line broad-ening ∆ν0 increases with increasing values of T . Forinstance, in the Nd:YAG laser Nd3+ ions are hostedin the YAG crystal, and collision broadening increasesfrom ∆ν0 ≈ 126 GHz at room temperature (T = 300 K)to about 250 GHz at T = 400 K.

A second homogeneous line broadening mechanismoriginates from spontaneous emission and is referredto as natural (or intrinsic) line broadening. It can beshown that the natural broadening is again described by

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a Lorentzian function with a FWHM of ∆ν0 given by

∆ν0 = 1

2πτs, (11.24)

where τs is the spontaneous lifetime. For, e.g., anelectric dipole transition at the center of the vis-ible (λ ≈ 500 nm), one has ∆νnat 16 MHz. Sinceτs ≈ 1/ν3, the natural line width rapidly increases fortransitions at shorter wavelengths (to the UV or X-rayspectral regions).

We finally note that, when the two aforementionedhomogeneous line broadening mechanisms act simul-taneously, the overall line shape is obtained from theconvolution of the two corresponding Lorentzian func-tions. One then obtains a Lorentzian function witha FWHM given by ∆ν0 = ∆νcoll +∆νnat.

Inhomogeneous broadening. A line broadening mech-anism is said to be inhomogeneous when it distributesthe atomic resonance frequencies over some spectralrange. A first case of inhomogeneous broadening is thatof ions in ionic crystals or glasses, where the local crys-tal field induces, via the Stark effect, a local variation ofthe energy-level separation of the ion. For random lo-cal field variations, the corresponding distribution of thetransition frequencies g∗(ν−ν0) turns out to be given bya Gaussian function (11.17) with a line width ∆ν∗0 thatdepends on the amount of field inhomogeneity withinthe crystal or glass.

A second inhomogeneous mechanism, typical ofgases, arises from atomic motion and is referred to asDoppler broadening. In fact, due to the motion of theatom, the frequency ν′ of the EM wave as seen in therest frame of the atom is shifted as compared to the fre-quency ν of the wave in the laboratory reference frameaccording to the relation ν′ = ν[1− (vz/c)], where vzis the component of the atomic velocity in the propa-gation z-direction of the EM wave. From the point ofview of atom–radiation interaction, this shift is equiva-lent to a change of the resonance frequency of the atomrather than to a change of the EM frequency. Taking intoaccount the Maxwellian distribution of molecular ve-locities in a gas, one can show that the distribution ofthe transition frequencies g∗(ν′ −ν0) is given again bya Gaussian function (11.17) with a line width ∆ν∗0 givenby

∆ν∗0 = 2ν0

(2kBT ln 2

Mc2

)1/2

, (11.25)

where M is the atomic (or molecular) mass and T the gastemperature. As an example, for theλ= 632.8 nm line of

the HeNe laser, at T = 300 K and using the appropriatemass for Ne, one obtains ∆ν∗0 1.7 GHz. As a generalrule, Doppler broadening in a gas is usually larger thancollisional broadening for a gas pressure lower than theatmospheric pressure; collisional broadening, in turn, isusually larger than the natural broadening.

Nonradiative DecayBesides decaying via radiative emission, an atom in anexcited state may decay toward a lower-lying level ina nonradiative way. A first mechanism of nonradiativedecay arises from collisions and is called collisional de-activation. In this case, for a liquid or a gas, the transitionenergy is released as excitation and/or kinetic energyof the colliding species, or it is transferred to the con-tainer walls; for an ion in a crystal or in a glass thedeactivation occurs through the interaction with latticephonons or glass vibrational modes. When, e.g., the ex-citation energy of an atomic excited species B∗ in a gasis released as a kinetic energy of the species A, the col-lisional deactivation process occurs via the superelasticcollision B∗ + A → B + A +∆E, where ∆E, the exci-tation energy to be released, is left as kinetic energy ofthe colliding partners. When the electronic energy ofspecies B∗ is released in the form of the internal energyof species A, we have instead B∗ + A → B + A∗ +∆E,where now ∆E is the difference between the internalenergies of the two colliding species. In this latter case,deactivation process is efficient provided that ∆E isappreciably smaller than the thermal energy kBT of col-liding species. A simple description of the nonradiativedecay of a given species in an excited state is expressedby a nonradiative lifetime τnr such that N2/τnr is thenumber of atoms, per unit volume and time, that decayowing to the deactivation process.

Nonradiative decay acts in combination with spon-taneous emission and, according to (11.3), the overalllifetime τ of an excited state is given by: τ = (1/τnr +1/τs)−1.

Concluding RemarksFrom the preceding discussions, we can say that the mostimportant material parameters of interest for a laser arethe transition wavelength λ, the transition cross sec-tion at the peak σp, the lifetime of the upper laser levelτ , and the line width ∆ν0 of the transition line shape.These parameters, for the most common gas, liquid andsolid-state lasers, are summarized in Table 11.1. Wenote that, as compared to gas and liquid lasers, the crosssections for solid-state lasers (Nd:YAG, Nd:glass, andTi3+:Al2O3) are relatively small and, correspondingly,

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Table 11.1 Main parameters for transitions in most common gas, liquid and solid-state lasers

Transition λ (nm) σp (cm2) τ (µs) ∆ν0

HeNe 632.8 5.8 × 10−13 30 × 10−3 1.7 GHz

Ar+ 514.5 2.5 × 10−13 6 × 10−3 3.5 GHz

Nd:YAG 1064 2.8 × 10−19 230 120 GHz

Nd:glass 1054 4 × 10−20 300 5.4 THz

Rhodamine 6 G 570 3.2 × 10−16 5.5 × 10−3 46 THz

Ti3+:Al2O3 790 4 × 10−19 3.9 100 THz

the lifetimes relatively long because in these lasers thetransitions are electric-dipole forbidden (or weakly al-lowed). We note also that the line widths of gas lasersare much smaller than those of solid-state or dye lasers.

11.1.3 Laser Resonators and Modes

As discussed in the introductory section, in a laser os-cillator the inverted amplifying medium is placed insidea laser resonator or laser cavity, which can be viewedas a trapping box for light radiation capable of sustain-ing stationary (i. e., monochromatic) or weakly dampedelectromagnetic field configurations at some selectedoptical frequencies [11.3, 5]. Such EM field configu-rations and the corresponding optical frequencies arecalled cavity modes and resonance frequencies, respec-tively. The most widely used resonators for lasers areopen cavities, composed of at least by two plane or spher-ical mirrors of circular shape (spherical resonators),placed in front each other at some distance L . Ex-cept for microlasers, in ordinary lasers the resonatorlength L is typically much larger than the oscillat-ing wavelength λ, ranging from a few centimeters toa few tens of centimeters, whereas mirror dimensionsrange from a fraction of centimeter to a few centime-ters. The optical cavity is open to reduce drastically thenumber of modes that can oscillate with low loss. Infact, if the cavity were closed, the number N of res-onant modes that might oscillate, i. e., the number ofmodes whose resonance frequency falls within the gain-line of the active medium, is approximately given byN ≈ (8πν2/c3)V∆ν0, where (8πν2/c3) is the density ofmodes, ∆ν0 the line width of the gain medium, and Vis the cavity volume. Note that V is usually several or-ders of magnitude larger than λ3 at optical wavelengths.To estimate N , let us consider an active medium witha narrow line width, such as the λ= 633 nm transitionof the HeNe laser (∆ν∗0 = 1.7 GHz). Assume a res-onator length L = 50 cm closed by a lateral cylinderwith diameter 2a = 3 mm. The cavity volume is thenV = πa2L , and the number of cavity modes that fall

under the gainline of the HeNe laser is N ≈ 1.2 × 109.In the open optical resonant cavities only those modestraveling nearly parallel to the resonator axis experiencelow losses to allow for laser oscillation. Therefore, theoscillating modes are expected to show a field distri-bution mostly confined around the optical axis of theresonator and propagating paraxially along it, makingthe output laser beam highly directional. These cavitymodes and corresponding resonance frequencies dependupon three integer numbers n, m and l, which are re-ferred to as mode indices. The latter two indices m and l(transverse mode indices) determine the transverse fielddistribution (i. e., in a plane orthogonal to the paraxialresonator axis) of the mode, whereas the former indexn (the longitudinal mode index) determines the longitu-dinal field distribution (i. e., along the resonator axis) ofthe mode and gives, in particular, the number of longitu-dinal nodes of the standing wave. For spherical mirrorswith sufficiently wide apertures, transverse modes areexpressed by Gauss–Hermite or Gauss–Laguerre func-tions, depending on rectangular or circular boundaryconditions. In particular, the leading-order mode, cor-responding to transverse mode indices m = l = 0, isa Gaussian beam and represents the most common trans-verse field distribution of any output laser beam. For thisreason the study of laser modes is closely related to thepropagation properties of Gaussian (or Gauss–Hermite)beams.

Gaussian BeamsThe electric field of a monochromatic (and uniformlypolarized) light wave propagating at a small angle (i. e.,paraxially) along the z-direction of an xyz cartesiansystem of coordinates can be described as follows:

E(x, y, z, t) = E0u(x, y, z)ei(ωt−kz) + c.c. , (11.26)

where ω= 2πν, ν is the optical frequency, k = 2π/λ isthe wavenumber, λ is the optical wavelength, u(x, y, z)is the complex field envelope obeying the so-calledparaxial wave equation, which in case of free-space

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propagation reads

∂2u

∂x2 + ∂2u

∂y2 −2ik∂u

∂z= 0 . (11.27)

Among the solutions of (11.27) that retain theirfunctional form during propagation, the fundamentalGaussian beam solution turns out to be particularlysuited to describe laser beams both inside and outsidethe resonator [11.12]. A Gaussian beam is a solution ofthe paraxial equation (11.27) of the form:

u(x, y, z) = w0

w(z)exp

(− x2 + y2

w2(z)

)

× exp

(−ik

x2 + y2

2R(z)

)

× exp[iϕ(z)]. (11.28)

In the preceeding equation,w(z), R(z) and ϕ(z) are givenby

w(z) =w0

√1+(

z

zR

)2

, (11.29)

)

)

*

)

)

*

)

'' '

*

Fig. 11.7a–c The fundamental Gaussian beam in free-space prop-agation: (a) the Gaussian transverse amplitude profile, beingr = (x2 + y2)1/2; (b) the beam spot size w(z); (c) the wavefrontradius of curvature R(z)

R(z) = z

[1+( zR

z

)2], (11.30)

ϕ(z) = tan−1(

z

zR

), (11.31)

where zR = πw20/λ is a parameter referred to as the

Rayleigh range. Note that u(x, y, z) is given by the prod-uct of three terms: an amplitude factor with a transverseGaussian distribution, (w0/w) exp[−(x2 + y2)/w2] (seeFig. 11.7a); a transverse phase factor, exp[−ik(x2 +y2)/(2R)]; and a longitudinal phase factor exp(iϕ). Theamplitude factor in (11.28) shows that, while propagat-ing, the beam intensity distribution retains its shape, butits transverse size w, which is called the beam spot size,changes along the propagation direction z according to(11.29). Note thatw(z) is a symmetric function of z witha minimum spot size w= w0 at the plane z = 0, whichis hence referred to as the beam waist (Fig. 11.7b). Forz = zR, one has w= √

2w0 so that the Rayleigh rangezR represents the distance from the beam waist at whichthe beam spot size increases by a factor

√2. At large

distances (i. e., for z zR), w increases linearly with z,according tow≈ (w0/zR)z. Hence we can define a beamdivergence due to diffraction as θd = limz→∞w(z)/z,obtaining

θd = λ

πw0. (11.32)

The transverse phase factor in (11.28) has the same formas that for a spherical wave in the paraxial approxima-tion, R playing the role of the radius of curvature of thespherical wavefront. Thus, we can say that a Gaussianbeam has an approximately spherical wavefront witha radius of curvature varying along propagation accord-ing to (11.30). Note that R(z) is an odd function of z(Fig. 11.7c), showing a minimum Rmin = 2zR at z = zR;for z zR, R increases linearly with z, whereas one hasR → ∞ as z = 0. Thus the wavefront is flat at z = 0 and,at large distances, its radius increases linearly with z justas for a spherical wave. The longitudinal phase factor ϕprovides, in addition to the usual phase shift −kz ofplane waves, a longitudinal phase shift (sometimes re-ferred to as the Gouy phase shift), slowly varying with zfrom −(π/2) to (π/2) on going from z zR to z zR.

An important parameter of a Gaussian beam ata given propagation plane z is the so-called complex-qparameter, defined by the relation:

1

q= 1

R− i

λ

πw2 . (11.33)

It can be shown that, for a Gaussian beam propagating infree space, the q parameter changes along propagation

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according to

q(z) = z + izR . (11.34)

Equations (11.29) and (11.30) can indeed be obtainedupon inserting the parameter q from (11.33) into (11.34)and then separating the real and imaginary part of theresulting equation.

The fundamental Gaussian beam described abovebelongs to a more general set of eigensolutions of (11.27)which can be written as the product of an Hermite poly-nomial with a Gaussian function. These are known asHermite–Gaussian beams, and assume the followingform [11.3, 5]

ul,m(x, y, z) = w0

w(z)Hl

(√2x

w(z)

)Hm

(√2y

w(z)

)

× exp

(− x2 + y2

w2(z)

)

× exp

[−ik

x2 + y2

2R(z)

]

× exp[i(1+ l +m)ϕ(z)] , (11.35)

where w(z), R(z) and ϕ(z) are given by (11.29, 30, 31),respectively, and Hl , Hm are Hermite polynomials of or-der l and m. The lowest-order Hermite–Gaussian beamis obtained by setting l = m = 0 in (11.35). These solu-tions are often referred to as TEMlm beams, where TEMstands for transverse electric magnetic: within the parax-ial approximation, both electric and magnetic fields ofthe EM wave are, in fact, approximately transverse to thez-direction. Note that, for a TEMlm beam, the number ofzeros of the field along the x- and y-directions is givenby the subscripts l and m, respectively, and thereforethe intensity distribution of the TEMlm beam consists ofl +1 lobes in the horizontal direction and m +1 lobes inthe vertical direction (Fig. 11.8).

As a final remark, it should be noted that Gauss–Hermite beams maintain their functional shape asthey propagate along an arbitrary paraxial opticalsystem, described by a paraxial ray matrix ABCD.Beam propagation, in this case, is simply ruled byan algebraic relation for the complex q parameter ofthe Gaussian beam, which is known as the ABCDlaw. In fact, if z = z1 and z = z2 are the input andoutput planes of the ABCD paraxial optical sys-tem, a Gauss–Hermite field distribution u(x, y, z1) =Hl(

√2x/w1)Hm(

√2y/w1) exp[−ik(x2 + y2)/(2q1)] at

the input plane z = z1 is transformed, at the output plane

+,- +,+,

+, +,+,

+, +,+,

Fig. 11.8 Grey scale intensity patterns of a few low-orderHermite–Gaussian modes

z = z2, into the distribution

u(x, y, z2) =(

1

A + (B/q1)

)1+l+m

× Hl

(√2x

w2

)Hm

(√2y

w2

)

× exp[−ik(x2 + y2)/(2q2)] , (11.36)

where the values q1 and q2 of the complex-q parameter atthe input (z = z1) and output (z = z2) planes are relatedby the so-called ABCD law

q2 = Aq1 + B

Cq1 + D. (11.37)

As a particular case, note that for free-space propagationfrom z1 = 0 to z2 = z one has A = 1, B = z, C = 0, andD = 1, so that the ABCD law (11.37) yields (11.34).

Optical Resonators: Introductory ConceptsThe simplest optical resonator is the plane-parallel orFabry–Pérot resonator, consisting of two plane metallicor dielectric mirrors set parallel to one another [11.13].At first approximation, the modes of this resonator can beconsidered as the superposition of two plane EM wavespropagating in opposite directions along the cavity axis(Fig. 11.9a). In this approximation, resonance frequen-cies can be readily obtained by imposing the conditionthat the cavity length L must be an integer multiple ofhalf-wavelengths, i. e., L = n(λ/2), where n is a posi-tive integer. This is a necessary condition for the electric

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field of the EM standing wave to be zero on the two, e.g.metallic, mirrors. It follows that the resonant frequenciesare given by

νn = n( c

2L

). (11.38)

Note that (11.38) can also be obtained by imposing thecondition that the phase shift of a plane wave due toone round-trip through the cavity must equal an inte-gral number times 2π, i. e., 2kL = 2nπ. This conditionis readily obtained by a self-consistency argument: if thefrequency of the plane wave is equal to that of a cav-ity mode, the phase shift after one round-trip must bezero (apart from an integer multiple of 2π). Only in thiscase, in fact, do the amplitudes at any arbitrary point,due to successive reflections, add up in phase to givean appreciable total field. According to (11.38) the fre-quency difference between two consecutive modes, i. e.,differing by 1 in the longitudinal mode index n, is givenby

∆ν = c

2L. (11.39)

This difference is referred to as the frequency differencebetween two consecutive longitudinal (or axial) modes.Note that, since the number n indicates the number ofhalf-wavelengths of the mode along the resonator axis,the two consecutive modes have a different longitudinalpattern.

A more general class of laser resonators is the onemade of spherical resonators, which are formed by twospherical mirrors of radius of curvatures R1 and R2,either concave (R > 0) or convex (R < 0), placed at

Fig. 11.9 (a) Plane-parallel resonator; (b) Spherical two-mirror resonator; (c) An unstable resonator; (d) A unidirec-tional ring resonator

some arbitrary distance L (Fig. 11.9b). These resonatorscan be divided into two categories: stable resonatorsand unstable resonators. A resonator is said to be un-stable when, in bouncing back and forth between thetwo mirrors, an arbitrary paraxial ray diverges indefi-nitely away from the resonator axis, either radially orangularly. Conversely, a resonator whose paraxial raysremain bounded is said to be stable. An example of anunstable resonator is shown in Fig. 11.9c. Among stablespherical resonators, symmetric resonators (i. e., hav-ing R1 = R2) are of particular importance; the confocalresonator, in which the two spherical mirrors have the fo-cus in common (R1 = R2 = L), is a noteworthy exampleof a spherical symmetric resonator. Another importantscheme for laser cavities is that employing a ring res-onator, where the path of the optical rays is arrangedto form a closed loop (Fig. 11.9d). Also in this case theresonant frequencies can be obtained by imposing thecondition that the total phase shift along the ring pathmust equal an integer number of 2π. The expressionfor the resonance frequencies of longitudinal (or axial)modes is thus given by

νn = n

(c

Lp

)(11.40)

where Lp is the length of the closed loop path. Ingeneral, a standing-wave pattern may be formed inring resonators, because the beam can propagate eitherclockwise or counterclockwise along the loop. Anyway,unidirectional ring resonators can be realized by meansof optical diodes inserted along the beam path.

Stability ConditionIn general, a laser cavity can be viewed as made of twospherical mirrors comprising a set of intermediate op-tical elements like lenses, mirrors, prisms, and so on(Fig. 11.10a). If we define an arbitrary plane β orthog-onal to the cavity optical axis, it is intuitive (and canbe rigorously demonstrated) that the round-trip propa-

. #"

Fig. 11.10 (a) A general laser cavity, and (b) its equivalentrepresentation given by resonator unfolding with respect toan arbitrary β plane

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gation along the cavity is equivalent to the propagationin a suitable optical system S having β as input andoutput planes, a transformation known as the unfoldingof the resonator. Paraxial propagation in such a sys-tem can be described by means of the ABCD cavityround-trip matrix (Fig. 11.10b). Accordingly, if we letr0 and r ′

0 be, respectively, the transverse coordinate andangle that a ray make with the optical axis when in-tercepting plane β at time t = 0, and rn and r ′

n be thecoordinate of the same ray after n cavity round-trip, wehave∣∣∣∣∣

rn

r ′n

∣∣∣∣∣=∣∣∣∣∣

A B

C D

∣∣∣∣∣n ∣∣∣∣∣

r0

r ′0

∣∣∣∣∣= Mn

∣∣∣∣∣r0

r ′0

∣∣∣∣∣ . (11.41)

Therefore, the optical resonator is stable if and onlyif, for any initial set of coordinates (r0, r ′

0), the cor-responding coordinates (rn, r ′

n) do not diverge as nincreases. This condition is met provided that the eigen-values λ1,2 of M are, in modulus, not larger thanone. Since λ1,2 = exp(±iθ), where cos(θ) = (A + D)/2,the stability condition requires that θ be real, i. e.,that∣∣∣∣ A + D

2

∣∣∣∣ 1 . (11.42)

For the particular case of two-mirror resonators, wecan go one step further by explicitly calculating the cor-responding ABCD matrix. We recall that a given overallmatrix can be obtained by the product of matrices of in-dividual optical elements traversed by the beam, withthe matrices written in the reverse order compared toray propagation through the corresponding elements. Inour case, the ABCD matrix is then given by the or-dered product of the following matrices: reflection frommirror 1, free-space propagation from mirror 1 to 2,reflection from mirror 2, free-space propagation frommirror 2 to 1:∣∣∣∣∣

A B

C D

∣∣∣∣∣=∣∣∣∣∣

1 0

−2/R1 1

∣∣∣∣∣∣∣∣∣∣

1 L

0 1

∣∣∣∣∣∣∣∣∣∣

1 0

−2/R2 1

∣∣∣∣∣∣∣∣∣∣

1 L

0 1

∣∣∣∣∣ .

After performing matrix multiplication we obtain

A + D

2=(

1− L

R1

)(1− L

R2

)−1 . (11.43)

It is customary to define dimensionless quantities, re-ferred to as the g1 and g2 parameters, defined asg1 = 1− L/R1 and g2 = 1− L/R2. In terms of theseparameters, the stability condition transforms into thevery simple relation:

0< g1g2 < 1 . (11.44)

/

.

0

Fig. 11.11 Stability diagram for a general two-mirror spher-ical resonator. Stable resonators correspond to (g1, g2)points lying in the gray region of the plane

The stability condition given by (11.44) can be con-veniently displayed in the (g1, g2)-plane as reportedin Fig. 11.11. Stable resonators correspond to thosepoints in the gray region of the plane, excluding thosewhich lie on the boundaries, (i. e., satisfying the con-ditions g1g2 = 0 or g1g2 = 1), which are referred toas marginally stable resonators. Note that symmetricresonators (i. e., having mirrors of the same radius ofcurvature R1 = R2 = R) lie on the bisector line b. Asparticular examples of these symmetric resonators, onecan see that those corresponding to point A, B and C ofthe figure are the concentric (L = 2R), confocal (L = R)and plane (R = ∞) resonators, respectively. Since pointsA, B and C lie on the boundary of the stability region,the corresponding resonators are only marginally stable.

Laser ModesThe modes of an optical resonator are defined as the sta-tionary (i. e., monochromatic) or weakly damped fielddistributions that can be sustained inside the cavity andthat satisfy the boundary conditions imposed by the cav-ity mirrors. We note that, in open resonators, diffractionlosses due to the finite aperture of the mirrors make themodes always leaky. The electric field for a mode ina leaky resonator can then be generally represented as

E(x, y, z, t) = a(x, y, z) cos(ωt)

× exp

[−t

2τc

](t> 0) , (11.45)

where a(x, y, z) is the mode field distribution, ω is theresonance frequency, and τc describes field decay dueto cavity losses and is referred to as the cavity pho-

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ton lifetime. In order to determine the resonator modes,the corresponding resonance frequencies and diffractionlosses let us consider the rather general class of res-onators with an optical axis z, as shown in Fig. 11.10a.Propagation of an EM wave back and forth betweenthe two end mirrors of the cavity is equivalent to theunidirectional propagation of an EM wave in a peri-odic sequence of optical elements (e.g., a lens guide)which is obtained by unfolding the resonator, as shownin Fig. 11.10b. Note that, in the unfolding scheme, theend spherical mirrors should be replaced by thin spheri-cal lenses with focal length equal to the mirror radius ofcurvature.

Let us first consider the propagation of a monochro-matic EM field in a periodic lens guide. Bywriting the electric field along the lens guide asE(x, y, z, t) = E(x, y, z)eiωt + c.c., due to the linear-ity of the Huygens–Fresnel integral the complex fieldamplitude E after one period of the lens guide can gen-erally be written by an integral transformation, namely(Fig. 11.12)

E(x, y, 2L)

= exp(−i2kL)

×∫∫1

K (x, y; x1, y1)E(x1, y1, 0)dx1 dy1 , (11.46)

where K (x, y; x1, y1) is a function of the transverse co-ordinates of both input and output planes, known asthe propagation kernel. Note the phase term (−2kL),which represents the phase shift if the wave werea plane wave. The kernel K accounts for all theelements encountered during propagation from in-put plane 1 (z = 0) to output plane 2 (z = 2L) andrepresents, from a physical viewpoint, the field dis-tribution observed at the (x, y)-plane (i. e., at z = 2L)

122

122

23 24

Fig. 11.12 Field calculation at plane 2 (z = 2L) when the fieldE(x1, y1, 0) at plane 1 (z = 0) is known

when a point-like source at the (x′1, y′

1) point isplaced at the z = 0 input plane. Indeed, if E(x1, y1, 0)were a bidimensional Dirac δ-function centered atx1 = x′

1 and y1 = y′1, i. e., if E(x1, y1, 0) = δ(x1 −

x′1, y1 − y′

1), then from (11.46) one would readily getE(x, y, 2L) = K (x, y; x′

1, y′1) exp(−2ikL). For a gen-

eral optical system, the calculation of the kernel Kis usually rather complicated. However, within the as-sumption of infinite aperture of all optical elements, thekernel K is simply expressed in terms of the round-trip resonator-matrix elements ABCD by the relation(Huygens–Fresnel–Kirchhoff kernel)

K (x, y; x1, y1)

= i

λBexp

− ik

2B

×[

A(x21 + y2

1)+ D(x2 + y2)− (2xx1 +2yy1)]

(11.47)

and the integral in (11.46) can be extended from −∞to ∞.

Let us now define a mode of a periodic lensguide as a field distribution which reproduces itselfafter one guide period except for an overall ampli-tude reduction, due to lens-guide losses, and a phaseshift ∆ϕ accounting for field propagation. HenceE(x, y, 2L) = |σ | exp(i∆ϕ)E(x, y, 0), where |σ | < 1.It is now convenient to write the phase shift as∆ϕ = −2kL +ϕ, where −2kL is the shift of a planewave and ϕ is an additional phase term due to the factthat the lens-guide mode is not a plane wave. Hence, fora lens-guide mode we require the condition

E(x, y, 2L) = σ exp(−2ikL)E(x, y, 0) , (11.48)

where σ = |σ | exp(iϕ). Substitution of (11.48) into(11.46) yields∫∫

1

K (x, y; x1, y1)E(x1, y1, 0)dx1 dy1

= σ E(x, y, 0) . (11.49)

Note that the mode distribution E(x, y, 0) is an eigen-function of a Fredholm homogeneous integral equationof the second kind corresponding to the eigenvalue σ . Asa rather general property, it turns out that the Fredholmintegral equation admits a doubly infinite discrete set ofconfined eigensolutions, which can be distinguished bya pair of integer positive numbers l and m. Accordingly,the corresponding eigenvalues will generally be indi-cated as σlm = |σlm | exp(iϕlm), with |σlm |< 1 owing to

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the overall losses (namely, diffraction losses, scatteringlosses from optical elements, and so on) encountered inone period of propagation along the lens guide.

Now, let us return to the problem of cavity modecalculation. In this case, the mode E of the lens guidecorresponds to a mode of the resonator of Fig. 11.10aprovided that, after one cavity round-trip, the overallphase shift ∆ϕ accumulated in one cavity round-tripis zero apart from an integer number of 2π, i. e.,∆ϕ = −2kL +ϕlm = −2πn. From this condition andusing the relation k = 2πν/c between the wavenumberk and frequency ν of the mode, one readily obtains thecavity resonance frequencies as

νnlm = c

2L

(n + ϕlm

). (11.50)

Note that we have indicated explicitly that these frequen-cies depend on the three integer numbers l, m, and n. Theintegers l and m define the transverse field profiles of themode (Fig. 11.8), i. e., they represent the number of zerosof the field along the x- and y-coordinates, respectively.The integer n defines the longitudinal field configura-tion, i. e., the number of zeros of the EM standing wave(nodes) as previously discussed for the plane-parallelresonator (11.38).

For stable resonators with infinite apertures theeigenmodes of the Fredholm equation are given bythe Gauss–Hermite functions, and their resonance fre-quencies can be calculated from (11.50) using thepropagation law (11.36). For instance, for the impor-tant case of two-mirror spherical resonators, it turns outthat ϕlm = 2(1+ l +m) cos−1(±√

g1g2), where the plusor minus sign depends on whether g2 (and hence g1) ispositive or negative. The resonance frequencies of thetwo-mirror spherical resonator are thus given by:

νnlm = c

2L

[n + 1+ l +m

πcos−1(±√

g1g2)

].

(11.51)

For, e.g., a confocal resonator, one has g1 = g2 = 0,and hence νnlm = [c/(4L)](2n +1+ l +m). Note thatmodes with the same value of (2n + l +m) have the sameresonance frequency, and they are said to be frequencydegenerate. Note also that, for a confocal resonator, themode spacing is c/(4L).

Photon Lifetime and Cavity QThe modes of an optical resonator are always leaky andtherefore show a finite photon cavity lifetime τc. In fact,besides diffraction losses due to finite aperture effectsof mirrors or intracavity optical elements, some other

loss mechanisms are always present in a laser resonator.For instance, the mirror reflectivity of the output coupleris always smaller than 100%, which means that a frac-tional part of the photons φ stored in the cavity escapesfrom the resonator at each round-trip. Scattering or ab-sorption losses of intracavity optical elements are alsoanother common cause of photon leakage. To calculatethe rate of energy decay in a given cavity mode, let I0 bethe initial intensity corresponding to the field amplitudeat a fixed point within the cavity, and let R1 and R2 bethe (power) reflectivities of the two mirrors and L i thefractional internal loss per pass, which accounts for scat-tering, absorption and diffraction losses. The intensityat the same point, after a round-trip time τR = 2L e/c,is I(τR) = R1 R2(1− L i)2 I0 = I0 exp(−2γ ), where L e isthe cavity optical length and γ is the logarithmic lossper pass, defined by the relation

γ = γ1 +γ2

2+γi (11.52)

with γ1 = −ln(R1), γ2 = −ln(R2), and γi = −ln(1−L i). In view of the exponential decay law introducedin (11.45), after a round-trip time we must haveI(τR) = I0 exp(−τR/τc), and thus we conclude that thephoton lifetime is given by

τc = τR

2γ= L e

cγ. (11.53)

Having calculated the photon lifetime, the timebehavior of the electric field at any point inside the res-onator can be written as E(t) = E0 exp(−t/2τc + iωt)+c.c., where ω is the angular frequency of the mode. Thesame time behavior then applies for the field of the waveleaving the resonator through the output mirror. Takingthe Fourier transform of this field (for t> 0) we find thatthe power spectrum of the emitted light has a Lorentzianline shape with line width (FWHM) given by

∆νc = 1

2πτc. (11.54)

We can now introduce an important quality factor whichis strictly related to the photon lifetime. This is the cavityQ-factor which is defined, for any resonant system, as2π times the ratio between the energy stored in the res-onator and the energy lost in one oscillation cycle. Thusa high cavity Q-factor implies low losses in the resonantsystem. Since in our case the energy stored is φhν andthe energy lost per cycle is (−dφ/dt)h we obtain

Q = − 2πνφ

dφ/dt= ν

∆νc, (11.55)

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602 Part C Coherent and Incoherent Light Sources

where the exponential decay law φ(t) = φ0 exp(−t/τc)for the stored cavity photons φ has been used and wherethe expression for ∆νc given by (11.54) has also beenutilized.

As an example, consider a two-mirror spherical res-onator with R1 = R2 = R = 0.98 and assume L i 0.From (11.53) we obtain τc = τT/[−ln(R)] = 49.5 τT,where τT = L/c is the transit time of the photon fora single pass in the cavity. Note that the photon life-time is much longer than the transit time, a typicalresult of low-loss cavities. If we now assume L = 90 cm,we get τT = 3 ns and τc 150 ns. From (11.54) wecan then calculate ∆νc 1.1 MHz. Finally, assuminga laser wavelength λ 630 nm corresponding to an op-tical frequency ν = 5 × 1014 Hz, from (11.55) we haveQ = 4.7 × 108. Thus laser resonators have a remark-ably high Q-value, which means that a very smallfraction of the energy is lost during one oscillationcycle.

11.1.4 Laser Rate Equationsand Continuous-Wave Operation

A simple and powerful approach for understanding thebasic dynamical behavior of a laser is based on a rate-equation model, in which simple balance equations forthe total number of atoms undergoing a transition andthe total number of photons created or annihilated arewritten [11.3]. For a more-refined treatment of laser dy-namics based on either a semiclassical or a full-quantumelectrodynamic approach, which may account for certainphenomena such as dynamical laser instabilities, lasercoherence and photon statistics, we refer the reader tomore-specialized literature [11.6, 8].

Laser Rate EquationsLet us consider a four-level laser scheme (Fig. 11.4b)and make the following assumptions:

1. the laser transition is homogeneously broadened,2. the lifetime τ1 of the lower laser level 1 is short

enough that we may neglect the population in level 1,3. a single longitudinal and transverse mode is oscil-

lating in the cavity,4. we neglect the precise transverse and longitudinal

spatial variation of the cavity mode,5. we assume uniform pumping of the active medium.

Under these assumptions, we can write the fol-lowing rate equations for the population inversionN = N2 − N1 N2 in the active medium and the number

of photons φ of the oscillating mode stored in the cavity:

dN

dt= Rp − BφN − N

τ(11.56)

dt= − φ

τc+ Va BφN , (11.57)

where Rp is the pump rate per unit volume, τ is thelifetime of the upper laser level 2, and τc is the photonlifetime for the oscillating mode. In (11.56), the termsRp, N/τ and BφN = W21 N account for the pumpingprocess, radiative and nonradiative decay, and stimulatedemission, respectively. The constant B, which representsthe stimulated transition rate per photon, per mode is re-lated to the transition cross section σ by the simpleequation B = σc/V , where V is the mode volume inthe laser cavity [11.3]. The first term on the right-handside in (11.57), φ/τc, represents the number of cavityphotons that are lost per unit time due to internal loss,diffraction loss and output coupling through the mirrors.Finally, the second term in (11.57) represents the numberof photons (per unit time) that are created in the oscil-lating mode owing to stimulated emission: since BφNrepresents the number of atoms per unit volume and perunit time that decay creating a photon in the oscillat-ing cavity mode, the total number of photons createdper unit time can be expressed as the product of BφNwith the volume Va occupied by the cavity mode insidethe gain medium. Spontaneous emission in not includedin the balance equation (11.57) since only a negligi-ble fraction of spontaneously emitted photons belongsto the oscillating mode. However, spontaneous emissionphotons are important to allow laser action starting.

The laser output power Pout is related to the photonnumber φ by the simple relation

Pout = γ2

2γ(hν)

φ

τc= γ2c

2L ehνφ . (11.58)

In fact, (hν)(φ/τc) is the total EM energy lost in thecavity per unit time, and solely a fraction γ2/(2γ ) ofthis power is available due to transmission throughthe output mirror. For a typical CW laser operated inthe continuous-wave regime, the number of photons φstored in the cavity may vary from about 1010 photonsfor low-power lasers (such as a HeNe laser deliveringPout = 10 mW power at λ= 632.8 nm) to 1017 photonsfor high-power lasers (such as a CO2 laser delivering anoutput power Pout = 10 kW at λ= 10.6 µm).

Threshold ConditionThe population inversion needed to reach the thresholdfor laser oscillation is simply obtained from (11.57) by

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Lasers and Coherent Light Sources 11.1 Principles of Lasers 603

imposing dφ/dt = 0. An initially small number of pho-tons turns out to be exponentially damped for N < Ncor exponentially amplified for N > Nc, where:

Nc ≡ 1

τc BVa= γ

σl(11.59)

is referred to as the critical inversion (or thresholdinversion). In this equation, l is the length of the ac-tive medium and the threshold condition σNcl = γis reached when the gain in the inverted medium,g = σNl, equals the logarithmic loss γ of the cavity.The pump rate corresponding to the threshold condi-tion is Rcp = Nc/τ = γ/(σlτ); the corresponding pumppower at threshold Pth is then obtained using (11.9). Theperturbation that starts the laser action when the pumprate Rp reaches the critical value Rcp is provided byspontaneous emission.

Output Power and Slope EfficiencyFor a pump rate Rp > Rcp, the rate equations (11.56, 57)admit the solution N0 = Nc and φ0 = [1/(Bτ)](x −1), corresponding to laser being above threshold(Fig. 11.13). Here x = Rp/Rcp = Pp/Pth > 1 is theabove-threshold pump behavior parameter, where Pp isthe pump power and Pth is its threshold value. The cor-responding output laser power can be then calculatedfrom (11.58) and can be cast in the form

Pout = ηs(Pp − Pth) , (11.60)

where

ηs = ηpηcηqηt . (11.61)

Equation (11.60) shows that, within the approximationmade, a linear relation is obtained between the outputpower and the pump power. One can then define the slopeefficiency of the laser as ηs = dPout/dPp. According to(11.61), ηs is given by the product of four contributions:

1. the pump efficiency ηp2. the output coupling efficiency ηc = γ2/(2γ )3. the laser quantum efficiency ηq = (hν)/(hνmp)4. the transverse efficiency ηt = Ab/A

where Ab = Va/l is the transverse mode area in the activemedium and A the transverse pumping area. The slopeefficiency of a laser may typically vary from less than1% in low-efficiency lasers (such as in the HeNe laser)to 20–50% or even higher in high-efficiency lasers.

Relaxation OscillationsOne can show that the solution, given above, for lasersabove threshold is stable, i. e., that any initial pertur-

566

2

"

" 66

Fig. 11.13 Behavior of population inversion N and photonnumber φ in the oscillating mode versus the pump rate Rp

for a four-level laser. Rcp is the critical pump rate abovewhich laser action takes place

bation of the system (e.g., of cavity losses) is damped.When the ratio τ/τc between the upper laser level life-time and cavity photon lifetime is larger than 1 (or muchlarger, such as in laser transitions which are electric-dipole forbidden), relaxation to the steady state occursthrough damped oscillations in both photon number andpopulation inversion. This results in damped oscillationsof the output power referred to as relaxation oscilla-tions. In solid-state lasers, the frequency of relaxationoscillations typically fall in the 10 kHz–10 MHz region,whereas, in semiconductor lasers, it falls in the GHz re-gion. Relaxation oscillations in slow-gain media (suchas in solid-state lasers) triggered by technical noise orby pump power fluctuations are one of the major causeof amplitude noise in the output laser power. Whenevera high degree of intensity stability is required, laser am-plitude stabilization may be provided by a suitable activefeedback loop.

Laser TuningThe gain line width of some lasers (e.g., dye lasers orvibronic solid-state lasers) is very wide and, for sev-eral applications, one needs to tune the laser outputwavelength away from the line center and across theentire available line width. In other cases, different las-ing transitions may compete or may potentially be used,and one needs to select one of them. In both circum-stances, one can employ a wavelength-selective elementinside the laser cavity, which is often referred to aslaser tuner. For lasers in the middle infrared (such asCO2 lasers), one generally uses a diffraction gratingin the so-called Littrow configuration (Fig. 11.14a) asone of the cavity mirror. Wavelength tuning is simply

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604 Part C Coherent and Incoherent Light Sources

Fig. 11.14 (a) Laser tuner based on a diffraction grating inthe Littrow configuration. (b) Laser tuner based on the useof a dispersive prism

achieved by grating rotation. In the visible or near-infrared spectral regions (such as for the Ar3+ laser),a dispersive prism is more commonly used and wave-length tuning is simply achieved by prism, or mirror,rotation (Fig. 11.14b). To reduce insertion losses, thetwo prisms facets are approximately inclined at Brew-ster angle. A third wavelength-selective element, whichis becoming increasingly popular in the visible or near-infrared, uses a birefringent filter inside the cavity.This device exploits a birefringent plate, inclined atthe Brewster angle, which generally changes the po-larization state of the intracavity laser beam. In thepresence of an intracavity polarizer, or just exploitingthe polarizing properties of the Brewster-angle filter, thebirefringence filter thus generally produces additionalcavity losses. However, at certain wavelengths, λ, the

7

8

/9

9

/5

Fig. 11.15 Schematic of a birefringent laser tuner made ofan intracavity birefringent plate and polarizer

birefringent plate does not change the polarization stateof the beam and thus allows for laser oscillation withlow loss. Upon rotation of the plate around the axis or-thogonal to its faces (Fig. 11.15) the direction of thebirefringence axes of the plates is changed and this cor-respondingly changes the wavelength at which low lossoccurs.

Single Mode Selectionand Limit to Laser Monochromaticity

Most often, lasers tend spontaneously to oscillate on sev-eral transverse and longitudinal modes, especially whenthe gainline is relatively broad. The reasons for mul-timode oscillations are rather involved and their studygoes beyond the scope of the present contribution. Formany applications, single-mode operation may be re-quired, and therefore one needs to force the laser tooscillate on a single transverse (usually the fundamen-tal TEM00 Gaussian mode) and longitudinal mode. Forstable resonators, single transverse mode oscillation iseasily achieved by placing an aperture inside the cav-ity of appropriate size in order to increase diffractionloss of higher-order modes with respect to the TEM00mode. In some cases, such as in longitudinally pumpedsolid-state lasers, limitation on the pump spot size lendsitself to TEM00 mode selection.

Even when a laser is oscillating on a single trans-verse mode, it can still oscillate on several longitudinalmodes. This usually occurs since the longitudinal-modeseparation ∆ν = c/(2L) is smaller (or much smaller)than the gain line width ∆ν0. For some gas lasers,where the gain line width is relatively small (up to a fewGHz), single-longitudinal-mode selection is simply ob-tained by making the cavity length short enough that thelongitudinal-mode separation ∆ν becomes larger than∆ν0/2. For, e.g., a HeNe laser (∆ν∗0 1.7 GHz), thiscondition implies L ≤ 17.5 cm. For solid-state laserswith a bandwidth up to a few hundred GHz, this condi-tion requires cavity lengths in the sub-millimeter region(microchip lasers). For lasers with much larger band-widths (e.g., dye lasers or tunable solid-state lasers),the required length is too small to make the tech-nique of practical relevance and to make the gain inthe medium large enough to reach threshold. In thesecases, different techniques are employed. For solid-state or dye lasers, the simplest method is to use one(or more) Fabry–Pérot etalons within the cavity, whichacts as a narrow frequency spectral-selective element(Fig. 11.16). The etalon thickness and finesse should bedesigned to ensure single mode selection. This impliesthat:

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Lasers and Coherent Light Sources 11.1 Principles of Lasers 605

1. the half width ∆νFP/2 of the transmission peaksof the Fabry–Pérot must be smaller than thelongitudinal-mode separation ∆ν = c/(2L);

2. the etalon free spectral range ∆νFSR must belarger than the half width of the gain line ∆ν0/2(Fig. 11.16).

For semiconductor lasers, single mode selection isachieved by using a distributed feedback (DFB) struc-ture, in which a longitudinal corrugation of therefractive index in the semiconductor induces fre-quency mode selection according to Bragg scatteringtheory.

A special case of single mode selection that deservesmention is that of a laser with a homogeneously broad-ened line (e.g., Nd:YAG and dye lasers). In this case,multimode oscillation is mainly due to the standing-wave character of the laser mode arising from theinterference of counter-propagating waves establishedbetween the two cavity mirrors. The use of a ringresonator instead of a linear cavity, in which unidirec-tional operation is forced by, e.g., an intracavity opticaldiode, may be enough in this case to achieve singlelongitudinal-mode operation.

We finally discuss the limit of monochromaticity(and hence of temporal coherence) of a laser. The linewidth ∆νL of a laser that oscillates on a single longi-tudinal mode is ultimately established by spontaneousemission noise. The quantum theory of a laser showsthat the spectral shape of emitted light is Lorentzian witha FWHM given by the well-known Schawlow–Townesformula:

∆νL = N2

N2 − N1

2πhνL(∆νC)2

P, (11.62)

where P is the output power, ∆νC = 1/(2πτc) is the linewidth of the cold cavity mode, N2 and N1 are the steady-state populations in the upper and lower laser levels,respectively, and νL is the emission frequency. Typi-cally the line width predicted by the Schawlow–Townesformula is negligibly small compared to that producedby other cavity disturbances (e.g., fluctuations of cavitylength or technical noise), except for the very importantclass of semiconductor lasers. For, e.g., a typical HeNelaser oscillating on its red transition (λ= 632.8 nm), τcis of the order of tens of µs, so that ∆νL is of the or-der of 1 mHz, which turns out to be much smallerthan the line broadening due to technical noise. For in-stance, a small cavity length change ∆L , due to technicalnoise, contributes to the frequency broadening ∆νL byan amount given by |∆νL| = (∆L/L)νL; for L = 1 mand νL = 4.7 × 1014 Hz (visible transition), a change of

!4

!

!#:

Fig. 11.16 Schematic of single longitudinal mode selection by useof an intracavity Fabry–Pérot etalon

∆L by just ≈ 10−8 times the atomic dimension leads toa contribution to ∆νL comparable to the quantum limit.Conversely, in semiconductor lasers the quantum limitof ∆νL is considerably larger and typically falls in theMHz range, owing to the much short photon lifetime (τcis of the order of few ps). Therefore, the laser line widthof a typical semiconductor laser arises from quantumnoise.

11.1.5 Pulsed Laser Behavior

In lasers operating in the CW or quasi-CW regimes, themaximum achievable optical output power is limited bythe maximum available pump power. For high-powerCW lasers (such as CO2 lasers), power levels up to≈ 100 kW can be reached; however larger power lev-els, which can be of interest for many applications, areprevented in the CW regime. Transient laser behavior al-lows one to obtain higher peak powers by concentratingthe available energy in a single, short optical pulse or ina periodic sequence of optical pulses [11.3,5]. Addition-ally, transient laser behavior is a powerful tool for thegeneration of ultrashort optical pulses, with durationsdown to ≈ 10 fs in lasers with a broad gainline (notablythe Ti3+:Al2O3 laser). From a dynamical viewpoint,pulsed laser behavior can be divided into two ratherdistinct categories:

1. Laser transients occurring on a time scale of the or-der of the cavity photon lifetime τc, i. e., appreciablylarger than the cavity round-trip time. This in-cludes the so-called Q-switching and gain-switchingregimes, which enable the generation of opticalpulses as short as a few nanoseconds with opticalpeak powers typically in the megawatt range. Theseare basically single-longitudinal-mode regimes andcan be described by means of a rate equation model(11.56, 57).

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606 Part C Coherent and Incoherent Light Sources

2. Laser transients occurring on a time scale appre-ciably shorter (and often much shorter) that thecavity round-trip time. These are basically multi-longitudinal-mode regimes, i. e., they involve thesimultaneous oscillation of many longitudinal lasermodes, and include the so-called mode-lockingregime, which enables the generation of trains ofultrashort laser pulses with durations down to a fewfemtoseconds.

Laser Q-switching: Dynamical AspectsQ-switching is a technique that enables the generationof a short optical pulse (of the order of the cavity photonlifetime τc) by a sudden switching of the cavity Q factor,i. e., of the cavity loss γ . The change of Q is produced, inprinciple, upon placing inside the laser cavity an opaqueshutter which can be opened or closed. When the shut-ter is closed (i. e., the cavity Q is low), laser action isprevented and the population inversion N can reach a rel-atively large value (well above the critical value Nc) due

2

"

9

#

Fig. 11.17a,b Dynamics of fast switching in a four-levellaser. (a) Temporal behavior of pump rate and populationinversion when the cavity Q is low. (b) Temporal behaviorof population inversion and cavity photons when the cav-ity Q is suddenly switched to a high value, showing theformation of the Q-switching pulse

to pumping. When the shutter is opened, the Q factoris suddenly switched to a high value and the laser ex-hibits a gain g = σlN that greatly exceeds the loss γ .Light emission then occurs via the generation of a shortand intense laser pulse. The duration of a Q-switchingpulse typically ranges from a few nanoseconds to a fewtens of nanoseconds, whereas its peak power is in themegawatt range. To achieve sufficient population inver-sion when laser action is prevented, a long lifetime τ ofthe upper laser level is required. Thus Q-switching canbe effectively used for electric-dipole-forbidden lasertransitions, where τ generally falls in the millisecondrange. This is the case of most solid-state lasers (e.g.,Nd, Yb, Er in different host materials; Cr-doped mater-ials, such as alexandrite, Cr:LiSAF, and ruby) and somegas lasers (e.g., CO2 or iodine).

To understand the basic dynamics of Q-switching,let us consider a four-level laser and assume that a steppump pulse is applied at time t = 0, i. e., Rp(t) = 0 fort< 0 and Rp = constant for 0< t< tp, where tp is theduration of the pump pulse; meanwhile the shutter isclosed (Fig. 11.17a) and laser action is prevented. From(11.56) with φ = 0, one then obtains that the transientpopulation inversion increases accordingly to the rela-tion N(t) = N∞[1− exp(−t/τ)], where the asymptoticvalue N∞ is given by N∞ = Rpτ (Fig. 11.17a). Aftera pump time of about 2τ , the population inversion al-ready reaches a value close to its asymptotic limit N∞,and therefore the pump pulse can be switched off andthe shutter opened. In fact, for tp larger than ≈ 2τ theenergy supplied to the medium is not useful for increas-ing the population inversion any further but is lost asradiative and nonradiative decay. Suppose now that theshutter is opened very rapidly at time tp (fast switching),and take the origin of time at the instant when switchingoccurs (Fig. 11.17b). From this time on, the evolutionof both population inversion and number of photons inthe cavity can be numerically computed by solving therate equations (11.56) and (11.57) with the initial con-ditionsφ(0) 1 and N(0) = Rpτ[1−exp(−tp/τ)] ≡ Ni,where the initial condition φ(0) 1 accounts for the factthat the laser action is started by spontaneous emission(the so-called extra photon). The qualitative transientbehavior of both N and φ is shown in Fig. 11.17b andcan be simply understood by observing that, just afterthe switching time t = 0, the gain g = σNl in the gainmedium greatly exceeds the single-pass cavity loss γ ;therefore the number of photons, which increases nearlyexponentially with time starting from the extra photondue to spontaneous emission, typically takes from sev-eral hundreds to a few thousand cavity round-trips to

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Lasers and Coherent Light Sources 11.1 Principles of Lasers 607

reach a sufficient value to saturate the laser transition,thus producing a reduction of the available inversion(Fig. 11.17b). This means that the time delay for thepeak of the laser pulse, τdelay typically ranges from sev-eral tens to a few hundred of nanoseconds. Note that,on such a time scale, the radiative and nonradiativedecays of the population N (which typically occur onthe millisecond time scale) are negligible, and thereforepopulation decay just occurs by stimulated emission.The decrease of N as φ increases leads, in turn, to a de-crease of the gain g = σNl. The peak of the pulse occursat the delay time, τdelay, such that the population inver-sion N decreases to its critical value, Nc. In this case, infact, laser gain g equals cavity loss γ and one has from(11.57) dφ/dt = 0. For t> τdelay, one has N < Nc anddφ/dt< 0. This means that the number of photons nowdecreases towards zero. Meanwhile, due to stimulatedemission, the population inversion also keeps decreas-ing until the photon pulse has decreased to zero. Atthis time some population inversion, say Nf , is gener-ally left in the medium after Q-switching (Fig. 11.17b).Note that the quantity ηE = (Ni − Nf )/Ni represents thefraction of the energy initially stored in the material thatgoes into stimulated emitted photons, usually referredto as the inversion, or energy, utilization factor. The du-ration ∆τp of the Q-switched pulse, the output energyE = (1/τc)[∫ φ(t)dt](hν)(γ2/(2γ )), the energy utiliza-tion factor ηE, as well as the time τdelay needed for pulseformation can be derived in a closed form by an analysisof the rate equations (11.56) and (11.57) in which pump-ing and radiative and nonradiative decay are neglected.One obtains:

∆τp = τc (Ni/Nc)ηE

(Ni/Nc)− ln(Ni/Nc)−1, (11.63)

E =(γ2

)(NiηEVa)(hν) , (11.64)

τdelay ≈ τc

(Ni/Nc)−1ln(φp/10) , (11.65)

where Ni/Nc is the ratio between the initially storedpopulation inversion and critical inversion; φp is thepeak photon number of the Q-switching pulse, given byφp = Va(Ni − Nc)− NcValn(Ni/Nc); and ηE is the en-ergy utilization factor. The value of ηE can be calculatedfrom the implicit equation (Ni/Nc)ηE = −ln(1−ηE)from which a plot of (Ni/Nc) vs. ηE can readily beobtained (Fig. 11.18). Note that (11.64) can readily beobtained by a simple energy-balance argument: thestored energy in the medium released as EM wave isin fact equal to hν(Ni − Nf )Va = hν(ηE Ni)Va, and, out

)"

,

$- $- $- *

$

$*

$%

$;

Fig. 11.18 Behavior of the energy utilization factor ηE ver-sus the normalized initial inversion Ni/Nc

of this energy, only the fraction [γ2/(2γ )] goes into theoutput beam.

Q-switched lasers may generally be operated in twodistinct regimes. In so-called pulsed Q-switching, thepump rate Rp(t) generally consists, as explained above,of a pulse with duration comparable to the upper-statelifetime τ . Of course, the pulsed operation can be pe-riodically repeated upon repeating the pulsed pump(generally up to a rate of a few tens of Hz). In so-called CW repetitive Q-switching, the pump rate Rp isheld constant and cavity losses are periodically switchedfrom a high to a low value (generally with a rate froma few to a few tens of kHz).

So far we have considered the dynamic behaviorcorresponding to fast switching, where the switchingof the cavity loss is treated as instantaneous (in prac-tice, much shorter than the time τdelay). In the case ofslow switching, the dynamic behavior is somewhat morecomplicated and multiple pulses may result, as shown

22

Fig. 11.19 Gain and photon dynamics in a slow-switchinglaser, showing the formation of multiple pulses

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608 Part C Coherent and Incoherent Light Sources

in Fig. 11.19. In the figure, the behavior of the cavityloss γ and laser gain g = σNl is depicted. The multipleintersections of the slowly decaying curve for the lossγ (t) with the gain curve g(t) explain the formation ofmultiple Q-switching pulses.

Methods of Q-SwitchingSeveral methods have been developed to achieve switch-ing of the cavity Q; the most common are [11.3, 4]:

1. Electrooptic Q-switching;2. Rotating prism;3. Acoustooptical Q-switching;4. Saturable absorber Q-switching.

Electrooptic Q-switching. In this case the shutter placedinside the laser cavity is made of a Pockels cell anda polarizer in the configuration shown in Fig. 11.20(electrooptic shutter). The Pockels cell consists of a suit-able nonlinear electrooptic crystal (such as KD*P orlithium niobate for the visible-to-near-infrared region orcadmium telluride for middle-infrared), in which an ap-plied dc voltage induces a change in the crystal refractiveindices. The induced birefringence turns out to be pro-portional to the applied voltage. The transmission axisof the polarizer is set at 45o with respect to the birefrin-gence axes of the crystal. When no dc field is appliedto the crystal, no polarization losses are introduced inthe cavity by the electrooptic shutter, i. e., cavity lossesare low. However, when a dc field is applied such thatthe phase difference ∆ϕ between the ordinary and ex-traordinary waves in the birefringent-induced crystal isequal to π/2, the Pockels cell operates as a λ/4 bire-fringent plate. Therefore, the linearly polarized light,coming from the polarizer, is rotated by 90 after a dou-ble pass through the cell and it is then fully reflected outof the cavity by the polarizer. The electrooptic shutter isnow closed and cavity Q is zero. The dc voltage to the

7

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,"=""#

Fig. 11.20 Schematic of a Q-switched laser using a Pockelscell

crystal required to produce the phase shift ∆ϕ = π/2is referred to as the quarter-wave voltage and typicallyranges from 1 to 5 kV. To avoid multiple pulses, thisvoltage must be switched off in a time typically smallerthan 20 ns.

Rotating prism. In this simple Q-switching technique,one of the cavity mirror is generally made of a roof-top prism and rotation is made through the axis parallelto the other mirror and orthogonal to the prism edge(Fig. 11.21). The high-Q condition is achieved whenthe prism edge passes through a position parallel to theother cavity mirror. Although rotating prisms are simpleand inexpensive devices which can be used at any wave-length, they suffer the limitation arising from the limitedrotation speed (≈ 400 Hz). The Q-switching time is thenrather long (typically ≈ 400 ns) which often results inthe production of multiple pulses (slow switching).

Acoustooptic Q-switching. In this case the shutterconsists of an acoustooptic modulator, driven by a radio-frequency (RF) oscillator, which is placed inside thelaser cavity. The modulator consists of a transparentblock of material (usually fused quartz in the visi-ble to near-infrared or cadmium selenide in the mid-to far-infrared) bonded on one side to a piezoelectrictransducer and to an acoustic absorber on the other(Fig. 11.22). When the transducer is on, traveling soundwaves are then produced in the material in the direc-tion orthogonal to the plane of the transducer. Due to thephotoelastic effect, the resulting strain in the materialresults in local changes of the material refractive index,i. e., in the generation of an index grating which is travel-ing along the material itself. Bragg scattering of the laser

8

Fig. 11.21 Schematic of a Q-switched laser using a rotatingprism

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7"" "!

0

>99"0"

"

5

/

/

Fig. 11.22 Schematic of an acoustooptic modulator used forlaser Q-switching (θB is the Bragg angle)

beam propagating across such a grating (Fig. 11.22) thusproduces a diffracted beam and hence additional lossesin the cavity (low-Q condition). Maximum diffractionefficiency is achieved when the incident light angle θBsatisfies the Bragg condition θB = λ/(2λa), where λand λa are the optical and acoustical wavelengths, re-spectively. The high-Q condition is simply obtainedby switching off the transducer voltage. Acoustoop-tic modulators have the advantages of introducing lowvalues of optical insertion losses and can be drivenat high repetition rates (several kHz). Therefore theyare used mainly for repetitive Q-switching of low-gainCW-pumped lasers (e.g., Nd:YAG or Ar-ion lasers).

Saturable absorber Q-switching. The Q-switchingtechniques discussed so far use active Q-switches, inthe sense that they need an external control source.A notable passive Q-switching technique, in which noexternal driving control is required, consists of placinga suitable saturable absorber inside the laser cav-ity. This absorber is basically an unpumped two-levelmedium which absorbs at the laser wavelength witha comparatively low saturation intensity. Thus, due tothe phenomenon of saturation, the absorption coeffi-cient of the absorber decreases as the intensity of theintracavity laser beam increases. To model this phe-nomenon we write the rate equation for the absorber as(dN2/dt) = σa(N1 − N2)I/(hν)− N2/τ , where N2 andτ are the population and lifetime of the excited level 2,N1 is the population in the lower (ground) level 1, andI = I(t) is the intracavity laser intensity. In CW opera-tion or if τ is short enough as compared to the changesof I(t), we may assume that (dN2/dt) ≈ 0 and therefore

obtain N1 − N2 Nt/(1+ I/Is), where Nt = N1 + N2 isthe total absorber population and Is ≡ hν/(2στ) is thesaturation intensity of the absorber. Since now the ab-sorption coefficient is α = σa(N1 − N2), from the lasttwo equations we obtain

α= α0

1+ I/Is, (11.66)

where α0 = Ntσa is the unsaturated absorption coef-ficient of the absorber. Equation (11.66) then showsthat, as the intracavity laser intensity I increases, thelosses introduced by the saturable absorber decreaseand the cavity Q correspondingly increases, i. e., Q-switching is achieved. The detailed dynamics underlyingthe formation of a Q-switching pulse with the saturable-absorber Q-switch, is however more involved than theone previously discussed. Here we just say that thesaturable absorber should have a low value of the sat-uration intensity so that, when laser action starts, theabsorber is bleached earlier (i. e., at lower intensity)than the time when population inversion in the gainmedium starts to decrease appreciably owing to stim-ulated emission.

Typical absorbers used for passive Q-switching con-sist of dyes in an appropriate solvent; the main drawbackof these absorbers is their photochemical degradation,i. e., poor chemical stability, and inadequate thermalproperties. Recently, the advent of solid-state absorbers(notably absorbers based on chromium doped into vari-ous crystalline hosts) are replacing dye absorbers, thusavoiding the problem of degradation.

Laser Mode-Locking: Dynamical AspectsMode-locking is a laser operation regime in which manylongitudinal modes of the cavity are simultaneouslyforced to oscillate with a precise phase relation so thatthe output laser beam shows a repetitive train of ultra-short optical pulses [11.3, 5]. To achieve mode-lockingoperation, a suitable device, which is usually referredto as the mode locker, must be placed inside the cavity.For a given laser medium, the lower limit to the achiev-able pulse duration ∆τp is set by the gain line width(∆τp 1/∆ν0), whereas the pulse repetition rate 1/τpis usually equal to the difference frequency between twoconsecutive longitudinal modes ∆ν (or an integer mul-tiple of ∆ν, as for harmonic mode-locking). Therefore,pulse duration, depending upon the gain line width, usu-ally ranges from about 1 ns, in gas lasers, down to 10 fs inbroad-bandwidth solid-state lasers. Pulse repetition ratedepends, of course, upon the cavity length and usuallyranges from about 100 MHz to a few GHz.

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610 Part C Coherent and Incoherent Light Sources

Frequency-domain description. The basic principle oflaser mode-locking can be explained as follows. Let usconsider, for simplicity, a traveling-wave ring laser ofoptical length L e, and assume that the separation of thelongitudinal cavity modes ∆ν = c/L e be smaller (usu-ally much smaller) than the gain line width ∆ν0. Inthis case, even in the absence of the mode locker, thelaser tends spontaneously to oscillate on several longi-tudinal modes (free-running regime). Let νl = ν0 + l∆νbe the frequency and El = Al exp(iϕl) the complex am-plitude of the l-th oscillating longitudinal cavity mode,where l is an integer number and l = 0 corresponds to thelongitudinal mode closest to the center of the gainline(Fig. 11.23). The electric field E(z, t) inside the lasercavity is then given by the superposition of the oscillat-ing longitudinal modes and can be written in the formE(z, t) = A(t − z/c) exp[2πiν0(t − z/c)], where z is thelongitudinal coordinate measured along the perimetricaxis of the ring and the envelope A(t′) is given by

A(t′) =∑

l

Al exp(2πil∆νt′ + iϕl) , (11.67)

where t′ = t − z/c is a retarded time. Note that, owing tothe dependence of the field on t − z/c, the field distribu-tion in the cavity is a traveling wave which propagateswith the speed of light; therefore we may limit the dis-cussion to the field behavior at a given reference planez = 0, e.g., at the output coupler of the laser. The outputlaser power, averaged over the rapidly varying opticalcycle, will be then proportional to |A(t)|2. Note that, ifthe mode amplitudes Al and phases ϕl are constant orslowly varying in time compared to the cavity round-triptime τR = L e/c = 1/∆ν, the signal A(t) is basically pe-riodic in time with a period equal to τR. However, thespecific form of the signal in one period depends on theprecise distribution of the mode amplitudes Al and, mostimportantly, on their phases ϕl .

'''

Fig. 11.23 Schematic of a laser oscillating on several longi-tudinal modes in the frequency domain

In a free-running laser, the phases ϕl do not havea precise relation to each other; they may also fluctuate intime. The superposition of N modes with, e.g., the sameamplitude Al = A0 but with randomly distributed phasestypically leads to a spiking signal |A(t)|2 made of a pe-riodic sequence of irregular pulses (Fig. 11.24a), eachwith duration approximately equal to ∆τp ≈ 1/∆νL,where ∆νL = N∆ν is the oscillating bandwidth. Notethat, since the response time of a conventional photode-tector is usually much longer than a few picoseconds,the complex temporal behavior shown in Fig. 11.24a isusually not resolved for free-running multimode lasers,instead its average value – proportional to NA2

0 – ismonitored.

In a mode-locked laser, the role of the mode lockeris to lock the phases of oscillating modes in a precisemanner. The most common and interesting case is thatof a mode locker that imposes a linear phase lockingcondition, i. e., ϕl = lϕ, where ϕ is a constant. In thiscase, one has:

A(t′′) =∞∑

l=−∞Al exp(2πil∆νt′′) , (11.68)

)

??

$- $-

)

??

$- $-

Fig. 11.24a,b Behavior of output power in a laser oscil-lating on many longitudinal modes with equal amplitude(a) Free-running regime, corresponding to random phasesφl . (b) Mode-locking regime, corresponding to linear phaselocking φl = lφ

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where t′′ is a translated time given by t′′ = t′ +ϕ/(2π∆ν). Usually, in a mode-locked laser the en-velope of the mode amplitudes Al follows the shapeof the gainline, i. e., Al is maximum at the center ofthe gainline (l = 0) and decreases toward zero as |l|increases; for instance, for active mode-locking in a ho-mogeneously broadened medium, one can show thatAl follows a Gaussian distribution. However, to cal-culate the series in (11.68) simply, we assume thatAl = A0 = const. for |l| ≤ N and Al = 0 for |l|> N ,i. e., we assume an odd number 2N +1 of oscillatingmodes with the same amplitude. In this case one ob-tains in (11.68) a geometric progression which can becalculated in a closed form, yielding

A(t′′) = A0sin[(2N +1)π∆νt′′

]sin(π∆νt′′)

. (11.69)

A plot of the squared amplitude of the electric field|A(t′′)|2 is reported in Fig. 11.24b. Note that a pulsetrain, at a repetition rate τp = 1/∆ν equal to the cavityround-trip time τR, is obtained. The squared ampli-tude of the electric field at the peak pulse is given by(2N +1)2 A2

0, whereas the FWHM pulse duration ∆τpis approximately given by

∆τp 1

(2N +1)∆ν= 1

∆νL, (11.70)

where ∆νL = (2N +1)∆ν is again the oscillating band-width. Therefore, for broad oscillating bandwidths,phase locking among longitudinal modes leads to thegeneration of short laser pulses with high peak power.The physical limit to the maximum number of phase-locked modes that can be forced to oscillate is ultimatelydetermined by the gain bandwidth of the active material,i. e., ∆τp is longer than or equal to 1/∆ν0.

In general, when the actual shape (e.g., Gaus-sian) of the mode amplitudes Al is taken intoaccount, the overall field amplitude A(t′′) can beobtained approximately from (11.68) upon trans-forming the sum over all modes into an integral,namely A(t′′) ∫ +∞

−∞ A(l) exp(2πil∆νt′′)dl. From thislast equation one then sees that the pulse amplitudeA(t′′) is the Fourier transform of the spectral modeenvelope A(l). Therefore, in this case, i. e., under lin-ear phase-locking conditions, the pulse amplitude issaid to be transform limited. Note, however, that un-der phase-locking conditions different from the linearcase (e.g., ϕl = ϕ1l +ϕ2l2, as in so-called frequencymode-locking) the mode-locked pulses are no longertransform-limited, i. e., their temporal duration is largerthan that predicted by the Fourier limit.

Time-domain description. The previous analysis ofmode-locking is often referred to as the frequency-domain description since the onset of the periodic pulsetrain is viewed as due to the coherent superposition oflongitudinal modes of the laser cavity. A different de-scription of the mode-locking regime, complementaryto the frequency-domain approach, is also possible inthe time domain. In fact, according to the results shownin Fig. 11.24b and since τR is the time for a round-trip,a single pulse of duration ∆τp is circulating within thelaser cavity (Fig. 11.25a). Note that the spatial extensionof the pulse ∆z = c∆τp, according to (11.70) is givenby ∆z = L e/(2N +1) where L e is the length of the ringperimeter. For a sufficiently large number (2N +1) of os-cillating modes, ∆z is then much smaller than the cavitylength L e. The temporal periodicity of the output laserbeam then simply results from the successive transitsof the intracavity circulating pulse at the output mirror,which occur at intervals τp = L e/c equal to the cavitytransit time. According to this picture, we readily under-

)

":

)

":

":

Fig. 11.25a–c Schematic of the mode-locking regime in thetime domain. (a) Mode-locking in a ring cavity. (b) Mode-locking in a linear cavity. (c) Harmonic mode-locking ina linear cavity (harmonic order n = 3)

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612 Part C Coherent and Incoherent Light Sources

stand that the mode-locking regime can be achieved byplacing a suitable fast shutter inside the cavity. In fact,if an initially non-mode-locked beam is present withinthe cavity, its spatial amplitude distribution can be rep-resented as in Fig. 11.24a, with the time t replaced byz/c. By periodically opening the shutter for a short timeinterval (of the order ∆τp) with period τp = L e/c, pos-sibly at the time when the most intense noisy pulse inFig. 11.24a reaches the shutter, then only this pulse willsurvive in the laser cavity, producing the mode-lockingcondition of Fig. 11.24b. After a transient leading fromthe temporal pattern of Fig. 11.24a to that of Fig. 11.24b,the mode-locked pulse will consistently reproduce itselfafter each transit in the cavity.

It should be noted that all considerations made abovefor a ring laser resonator apply mutatis mutandis to a lin-ear (i. e., Fabry–Pérot) laser cavity. In this case, however,self-consistent propagation of the mode-locked pulse inone cavity round-trip requires that the shutter must beplaced close to one end mirror of the cavity (Fig. 11.25b).Note that, if the shutter is placed at a distance L/2, L/3,· · · , L/n from one end mirror and is opened at inter-vals τR/2, τR/3, · · · , τR/n, where L is the cavity length,multiple pulses (precisely 2, 3, · · · , n pulses) may besimultaneously generated and the repetition rate of thepulse train is correspondingly increased by a factor of2, 3, · · · , n (see, e.g., Fig. 11.25c for the case n = 3).Such a mode-locking regime is referred to as harmonicmode-locking. Harmonic mode-locking is typically em-ployed in active mode-locked fiber lasers to increase thepulse repetition rate (≈ 1–40 GHz); owing to the rel-atively long cavity length (≈ 1–10 m), to reach highrepetition rates harmonic orders n up to ≈ 1000 areusually needed.

Mode-Locking MethodsThe methods to achieve mode-locking can generally bedivided into two categories:

1. Active mode locking, in which the mode-locker isdriven by an external source.

2. Passive mode locking, in which the mode-locker isnot externally driven but exploits some nonlinearoptical effect, such as the saturation of a saturableabsorber or the nonlinear change of the refractiveindex in a Kerr medium.

Active mode locking. Active mode locking is usu-ally achieved by placing, inside the laser cavity, eitheran amplitude modulator, which produces a periodicmodulation in time of the cavity loss [amplitude-

modulation (AM) mode locking], or a phase modulator,which periodically varies the optical length of the res-onator [frequency-modulation (FM) mode locking]. Inlasers with upper-state lifetimes shorter than the cavityround-trip time (e.g., dye lasers), active mode lock-ing can also be achieved by periodic modulation ofthe laser gain at a repetition rate equal to the longi-tudinal mode separation ∆ν (synchronous pumping).We limit here to describe the basic principle of AMmode locking, since it is the most common amongthe three mentioned techniques. In AM mode locking,the mode locker is usually a Pockels-cell electroop-tic modulator for pulsed and high-gain lasers, or anacoustooptic modulator for low-gain lasers. The elec-trooptic or acoustooptic modulator sinusoidally variesthe cavity loss γ (t) at a given modulation frequencyνm (Fig. 11.26). It is thus expected that the mode-locked pulse circulating inside the cavity (Fig. 11.25b)will pass through the modulator at the time t1 of thecycle where the cavity loss γ (t) is minimum. Sincethe pulse propagating inside the cavity passes againthrough the modulator at times t2 = t1 +τR, t3 = t2 +τR,etc., where τR = 1/∆ν is the cavity round-trip time,a steady mode-locking regime can be reached providedthat the synchronization condition ∆ν = νm is satisfied.It should be noted that the steady-state pulse dura-tion, ∆τp, is given by the inverse of the oscillationbandwidth, (2N +1)∆ν, and, thus, it is ultimately es-tablished by the gain bandwidth ∆ν0. However, thefinite bandwidth of the gain medium influences thesteady-state pulse duration in a quite different wayfor homogeneous or inhomogeneous lines. For an in-homogeneously broadened line, and for a laser wellabove threshold, the oscillating bandwidth tends tocover the whole gain bandwidth ∆ν∗0 even in the ab-sence of AM modulation, and the main role of themode-locker is just to lock the phases of these os-cillating modes. The resulting temporal duration ofthe mode-locked pulse is therefore given approxi-

)

??

Fig. 11.26 Schematic of AM active mode locking

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mately by

∆τp 0.44

∆ν∗0. (11.71)

Conversely, for a homogeneously broadened gainmedium the number of longitudinal modes oscillatingin the free-running regime is usually rather modest andthe role of the mode-locker is both to enlarge the oscil-lating bandwidth of the laser (by power transfer fromthe central to lateral longitudinal modes) and to lockthe phases of the oscillating modes. Under steady-stateconditions, band enlargement due to the modulator iscounteracted by band reduction due to the gain mediumand the pulse duration is given by [11.14]

∆τp 0.45√νm∆ν0

. (11.72)

As an example, we will consider a mode-locked Nd:YAGlaser oscillating on its homogeneously broadened line atλ= 1064 nm. Assuming ∆ν0 126 GHz (T = 300 K),a linear cavity of length L e = 1.5 m and an AM mode-locker placed close to one cavity mirror, the modulatorloss must be driven at a frequency νm = ∆ν= c/(2L e) 100 MHz, and the expected mode-locked pulse duration,according to (11.72), is ∆τp 125 ps.

It should be noted that in AM mode-locked lasers(as well as in the other active mode locking techniques)even small detunings of the modulation frequency νmfrom the cavity axial mode separation ∆ν may result inthe destruction of the mode-locking operation. In prac-tice a detuning |νm −∆ν|/νm of the order of ≈ 10−4

is enough to destroy mode-locking. To obtain stableAM mode-locking, active control of the cavity lengthis sometimes required, especially when relatively longcavities are employed (such as for AM mode-lockedfiber lasers).

Passive mode-locking. There are two main types ofpassive mode-locking (ML):

1. Fast saturable-absorber ML, which uses the satura-tion properties of a suitable absorber (e.g., a dyeor a semiconductor) with a very short upper-statelifetime;

2. Kerr-lens mode-locking (KLM), which exploits theself-focusing property of a suitable transparent Kerrmedium.

Fast saturable-absorber ML [11.15]. Consider a sat-urable absorber with a low saturation intensity anda relaxation time shorter than the duration of the mode-locked pulses. According to (11.66), the absorption

2

Fig. 11.27 Mode-locking with a fast saturable absorber

loss experienced by a pulse crossing the absorber isdependent on the instantaneous pulse intensity I(t),and decreases as the intensity increases. Thus, start-ing from the random sequence of light bursts occurringin the unlocked case (Fig. 11.24a), the gain–loss bal-ance will favor the growth and stabilization of the noisepulse with the highest intensity. The steady-state sit-uation, occurring in this case, can be described withthe help of Fig. 11.27. Here, gain saturation drives thegain g below the cavity losses γ (t) except at those in-stants where, due to the arrival of the pulse I(t) atthe saturable absorber, the losses are reduced due toabsorber saturation. During the time intervals corre-sponding to the hatched regions in Fig. 11.27, the gainis then larger than the instantaneous loss γ (t). A so-called window of net gain is thus produced, whichtends to increase the peak of the pulse and decreaseits wings, i. e., it tends to narrow the pulse. This nar-rowing is again counteracted by pulse broadening dueto the finite amplifier bandwidth until a steady-statepulse duration, whose duration again depends on the in-verse of the gain bandwidth, is eventually reached. Goodcandidates for saturable absorbers must have a short re-laxation time τ (∼ a few picoseconds or shorter) anda small saturation intensity, given by Is = hν/(2σaτ).Thus, very large values for the absorption cross sectionσa (≈ 10−16 cm2 or larger) are needed. Ideal absorbersare therefore dye molecules (e.g., cyanine dyes) or,even better, semiconductors. A particularly interest-ing saturable-absorber geometry consists of integratinga multiple-quantum-well absorber between two mirrorswhose spacing is such that the resulting Fabry–Pérotetalon operates in antiresonance. Such a device hasbeen convincingly proven to generate both picosecondand femtosecond laser pulses from several broadbandsolid-state lasers.

Kerr-lens mode-locking. This technique is based onthe use of a nonlinear loss element simply consist-ing of a nonlinear Kerr medium placed in front of

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614 Part C Coherent and Incoherent Light Sources

an aperture (Fig. 11.28). The nonlinear medium shows,via the optical Kerr effect, an intensity-dependent re-fractive index n = n0 +n2 I , where n0 is the linearrefractive index of the medium, I is the local lightintensity and n2 is a positive coefficient (for a self-focusing medium) which depends on the strengthof the nonlinearity (e.g., n2 4.5 × 10−16 cm2/Wfor fused quartz and n2 3.45 × 10−16 cm2/W forsapphire). A light beam with, e.g., a transverseGaussian intensity distribution I(r) = Ip exp[−2(r/w)2]that crosses a thin slice of the Kerr medium oflength l thus experiences a transversely varying phaseshift δϕ = 2πln2 I(r)/λ= (2πln2/λ)Ip exp[−2(r/w)2].Close to the beam center r = 0, one can writeδϕ ≈ (2πln2 Ip/λ)[1−2(r/w)2], i. e., the thin mediumintroduces a quadratic phase change of the field andthus acts, for n2 > 0, as a positive lens (called a Kerrlens) of dioptric power 1/ f = 4n2lIp/(n0w

2), whichincreases as the beam intensity Ip increases. If an aper-ture is placed at some suitable distance from the Kerrmedium, a beam with higher intensity will be focusedtighter and a higher fraction of the beam will be trans-

@6#

5#

A

0

Fig. 11.28 Schematic of a Kerr lens with an aperture usedin KLM

mitted through the aperture. Therefore, the Kerr mediumwith the aperture, like a fast saturable absorber, intro-duces losses that decrease when the instantaneous pulseintensity is increased, thus leading to mode-locking.Note that, by appropriately controlling the cavity dis-persion, the shortest mode-locked pulses (≈ 6 fs) havebeen achieved by this technique for Ti3+:Al2O3 lasers.

11.2 Solid-State Lasers

11.2.1 Basics

Solid-State LasersBased on Dielectric Insulators

The demonstration of the ruby laser [11.16](Cr3+:Al2O3) in 1960 led to a decade with the realizationof a number of crystalline and glass lasers. EspeciallyNd:YAG [11.17], Nd-doped yttrium aluminum garnet(Nd:Y3Al5O12), rapidly emerged as one of the most im-portant crystalline dielectric lasers. The first glass laserwas a fiber laser [11.18], which decades later led tothe development of highly efficient Er-doped fiber am-plifiers (EFDA) with important applications in opticalcommunications.

In the last two decades progress in the area ofdiode-pumped lasers has contributed very much toa renaissance in the field of solid-state lasers. Withdiode laser pumping it is possible to obtain higherefficiencies and to build rigid all-solid-state deviceswith simpler and more compact design. Besides Nd3+various efficient diode-pumped rare-earth lasers havebeen operated with Er3+, Tm3+, Ho3+ [11.19–21],and Yb3+ [11.22, 23]. In addition, the successful op-eration of Cr3+-doped [11.24–27] and Ti3+-doped

crystals [11.28, 29] as tunable room-temperature lasershas stimulated further research in transition-metal ions.Interesting new results have been obtained with the Cr4+ion [11.30–33] and recently also with the divalent Cr2+ion [11.34].

Nowadays, average and CW output powers forNd3+- and Yb3+-doped crystals in the kW range arecommercially available. Fiber lasers reach output pow-ers in the 100 W range with nearly diffraction-limitedbeam quality.

Compact solid-state lasers in the visible spectral re-gion (see for instance [11.35]) are of potential interest,especially for display and high-density optical data stor-age applications. Optical efficiencies of more than 20%with respect to the pump power can be obtained in Ndlasers by internal frequency doubling with nonlinearcrystals. An alternative approach enables the genera-tion of visible laser radiation by upconversion schemes,which incorporate energy transfer processes or two-step pump processes as ground-state and excited-stateabsorption.

In a most general sense lasers based on solids includeboth dielectric insulators and semiconductors as gainmedia. However, the development in the past yielded

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a subdivision into two laser classes based on solids:solid-state lasers (the topic of this chapter) and semi-conductor lasers (Sect. 11.3).

Nearly all modern, important solid-state lasers arebased on impurity-doped crystals or glasses. Typically,the impurity ions have unfilled electronic shells. So far,only laser ions from the iron, rare-earth and actinidegroup are known. The most important laser lines of theseions correspond to 4f–4f, 4f–5d, and 3d–3d transitions.In special cases the laser active ions can also be fullysubstituted into the lattice (stoichiometric laser mater-ials). It should be noted that in the past lattice defectsalso have been used as laser-active centers (color cen-ter lasers); these lasers are however not included in thischapter.

Spectra of Rare-Earthand Transition-Metal Ions in Solids

4f–4f transitions in rare-earth ions. In the free rare-earth ion the electrostatic interaction between the 4felectrons creates a splitting of the energy levels of the4fn configuration into different L S terms. The resultantwave functions are characterized by the quantum num-bers L , S, ML , and MS. The electrostatic energy splittingof the 2S+1L terms is typically 104 cm−1, and each levelis (2L +1)(2S +1)-fold degenerate with respect to MLand MS.

In addition, the energy levels are further split byspin–orbit coupling. If the energetic separation betweendifferent L S terms is large compared with the spin–orbit coupling energy there is only a small mixing ofthe L S terms. If this mixing is very small, the Russel–Saunders approximation holds and the wave functionsare characterized by the quantum numbers L , S, J , MJwith a degeneration of (2J +1). The typical splitting of2S+1L J terms is of the order of 1000 cm−1. AlthoughL S coupling is not strictly valid for rare-earth ions, it isusual to describe the 4f energy states with the Russel–Saunders approximation [11.36].

The interaction of the 4f electrons with the crystalfield, i. e., the electrostatic field of the surrounding lig-ands, results in a Stark splitting of the free ion 2S+1L Jterms. This interaction can be treated as a perturbation ofthe free ion levels. The crystal field splitting and the re-maining degeneracy depend on the symmetry of the localcrystal field. Lower symmetries increase the number ofsplit levels. However, according to Kramers theorem anodd number of electrons always yields at least twofolddegeneracy. The Stark splitting is typically in the energyrange of several 100 cm−1. The ground state of a special4fn configuration (the number of 4f electrons n = 1–14)

can be determined according to Hund’s rule. Fig. 11.29shows the 4fn energy levels of rare-earth ions (Diekediagram [11.37]).

Electric dipole transitions within the 4f-shell are par-ity forbidden in the free ion. When doped in a solid,acentric perturbations of the crystal field can howevercreate admixtures of wave functions with opposite parity(for instance 4fn−15d1 states) yielding so-called forcedelectric dipole transitions. Due to the screening of theouter filled 5s2 and 5p6 orbitals, the crystal field per-turbation is small and electron–phonon coupling is veryweak. So, for rare-earth ions at acentric sites one ob-serves electric-dipole zero-phonon transitions with veryweak vibronic sidebands. When doping occurs at cen-tric sites, parity remains a good quantum number and all4f–4f transitions remain electric dipole forbidden. Themagnetic-dipole emission cross sections are then verysmall and are not useful for laser applications.

Further selection rules for electric dipole transitionsbetween 4fn states are:

• ∆J ≤ 6; ∆S = 0, ∆L ≤ 6 (Russel–Saunders approx-imation)• J = 0 ⇔ J ′ = 0 is forbidden.

3d–3d transitions in transition-metal ions. The 3delectrons of transition-metal ions are not shielded andexperience a strong perturbation from the crystal fieldof the surrounding ligand ions. Therefore, the energy-level scheme and also the spectroscopic characteristicsof a transition metal ion depend strongly on the strengthand the symmetry of the crystalline field originatingfrom the surrounding ions. The energy-level schemesof the transition-metal ions in crystalline hosts are de-scribed by so-called Tanabe–Sugano diagrams [11.38].These diagrams are distinguished by the number of elec-trons within the 3d electron shell. In these diagrams theenergy levels of the transition-metal ion are presented asa function of the crystal field strength (for some exam-ples, see Sect. 11.2.3). For detailed reading appropriateliterature can be found in [11.36, 39–42].

Due to the strong interaction with the surroundingions of the lattice, transition-metal ions mostly exhibitbroadband emission because of the electron–phononcoupling between the electronic 3d levels and latticevibrations. As in the case of 4f–4f transitions, onlyacentric perturbations can induce electric dipole transi-tions. Generally the spectra consist of a purely electroniczero-phonon line with vibronic sidebands. In contrastto the situation in rare-earth ions, transition-metal ionsat centric sites may also have reasonable transition

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616 Part C Coherent and Incoherent Light Sources

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Lasers and Coherent Light Sources 11.2 Solid-State Lasers 617

probabilities due to dynamical admixtures of wave func-tions with different parity by acentric phonons. On theother hand, strong electron–phonon coupling can yieldhigher temperature-dependent nonradiative decay ratescompared to 4f–4f transitions. The main interest intransition-metal ions is based on their broad tunability.

Interconfigurational 4f–5d transitions in rare-earthions. Typically 4f↔5d interconfigurational transitionsof rare-earth ions are located in the ultraviolet spec-tral range. In some cases 4f↔5d transitions are alsoobserved in the visible spectral region. In contrast to3d↔3d and 4f↔4f transitions, 4f↔5d transitions areelectric-dipole allowed, because they obey the parityselection rule. Thus high transition probabilities andhence large absorption and emission cross sections(≈ 10−18 –10−17 cm2) are observed.

Similar to 3d↔3d transitions strong electron–phonon coupling yields broad absorption and emissionspectra with spectral half-widths of more than1000 cm−1. Therefore 4f↔5d interconfigurational tran-sitions are in principle suitable for the generation oftunable laser oscillation.

However, the difficulty in finding suitable pumpsources and often also excited-state absorption processesare major drawbacks with respect to the realization of5d–4f lasers.

Basic Spectroscopic Propertiesand Laser Parameters

Ground-state absorption. The ground-state absorptionof an ion is characterized by the absorption coefficientα and the ground-state absorption cross section σGSA.These values are derived using the Lambert–Beer law:

I (λ)= I0 (λ) e−α(λ)d = I0 (λ) e−nionσGSA(λ)d ,(11.73)

where I(λ) is the intensity transmitted through the crys-tal at wavelength λ, I0(λ) is the intensity in front of thecrystal, nion is the ion concentration, α(λ) is the absorp-tion coefficient,σGSA(λ) is the absorption cross section,and d is the crystal thickness.

The ground-state absorption spectrum contains in-formation about the energy-level structure of the ion,the cross sections σGSA(λ) and the oscillator strengthsof the observed transitions. Note, that the calculationof cross sections needs a second independent measure-ment of the ion concentration, which is usually done byX-ray microprobe analysis. The relation between the ab-sorption coefficient and the transition-matrix elementscan be derived using Fermi’s golden rule [11.36]. The

energy-level structure of transition-metal ions is theo-retically described by the ligand field theory and theangular overlap model (AOM), see for instance [11.43].

Spontaneous emission and emission cross section.The spontaneous emission is characterized by the Ein-stein coefficient A. In an emission measurement, thephoton flux (the number of photons per area per time)is usually measured. The normalization process withrespect to the instrument response function can be per-formed either with respect to photon flux or to thespectral intensity distribution Iλ(λ) (energy per area pertime). The relation between the spectral intensity dis-tribution Iλ(λ), the Einstein coefficient A (A = τ−1

rad )and the emission cross section σem(λ) is given by theFüchtbauer–Ladenburg equation [11.44]:

σem(λ) = λ5 Iλ(λ)A

8πn2c∫

Iλ(λ)λdλ, (11.74)

where n is the refractive index and c is the speed of light.The emission cross sections can also be calculated

from the absorption cross sections by the reciprocitymethod

σem (λ)= σGSA (λ)Zl

Zuexp

(Ezl −hc/λ

kT

),

(11.75)

where Zl and Zu are the partition functions of the lowerand upper energy levels, Ezl is the energy of the zero-phonon line of the corresponding transition, k is theBoltzmann constant, and T is the temperature.

For Gaussian band shapes, which are often observedfor emission spectra of transition-metal ions, the Mc-Cumber formula can be used for the determination ofthe peak emission cross section [11.45, 46]

σem =√

ln 2

π

A

4πcn2

λ40

∆λ, (11.76)

where λ0 is the peak emission wavelength and ∆λ is thefull width at half-maximum (FWHM).

Excitation spectra. The excitation spectra allow the de-termination of those absorption transitions, which yielda specific emission. These measurements are interestingin the case of different absorbing and emitting centersand also for finding energy transfer channels betweendifferent optical active centers.

Emission lifetime. The decay time of the metastablelevel of an ion in a crystal is usually measured afterexcitation with a short pulse. The measured decay rate

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618 Part C Coherent and Incoherent Light Sources

(measured as the number of transitions per unit time)is the sum of radiative and nonradiative decay rates. Ingeneral, the nonradiative decay of an ion can consist ofintra-ionic multiphonon processes and intra-ionic nonra-diative transfer processes (for details of energy transferprocesses see [11.36]). The equation for the decay is

1

τ= A + 1

τnr= 1

τr+ 1

τnror

W = A + Wnr = Wr + Wnr , (11.77)

where τ , τr, and τnr are the total, radiative, and nonra-diative decay times, respectively, and W , Wr, and Wnrare the total, radiative, and nonradiative decay rates,respectively.

The determination of the Einstein coefficient A is notan easy task, because in a simple decay measurementthe combined lifetime is always measured. An addi-tional measurement of the emission quantum efficiencyηQE would allow the determination of the radiative andnonradiative decay rate

ηQE = τ

τr= Wr

W= Wr

Wr + Wnr

⇒ τr = τ

ηQEand τnr = τrτ

τr − τ . (11.78)

However, a direct measurement of the quantum effi-ciency is difficult and usually contains a high error.

Another approach is the indirect determination ofthe quantum efficiency by analysis of the temperaturedependence of the emission lifetime. The two most com-monly used models are the simple Mott model of theactivation energy [11.47] and the more-sophisticatedmodel of Struck and Fonger [11.48] using the so-called single-configurational-coordinate model, whichdescribes in a simplified way the interaction betweenthe electronic center and the vibrating crystalline envi-ronment.

Excited-state absorption (ESA). Measurements of theexcited-state absorption (ESA) spectrum give furtherinsight into the energy-level structure of an ion. Inthe ground state absorption measurements spin-allowedtransitions are mainly observed and often spin-flip tran-sitions are hidden under these absorption bands. Thusthe determination of the energy of these spin-flip transi-tions is often not possible. If the metastable level of theion has a different spin to the ground state (e.g., Cr3+in strong crystal fields, Mn5+, Fe6+), the ESA spec-trum reveals the energetic positions of these states withdifferent spin. By using these data the crystal field pa-rameters can be determined with higher accuracy. The

knowledge of ESA processes is also very useful forthe determination of their influence on the efficiency ofa laser material. ESA may also inhibit gain and hencelaser action.

The laser aspect. In this section only the case of steady-state conditions for laser oscillation will be discussed.For more details and for the case of pulsed excitation,see e.g. [11.49–52].

The efficiency of a laser system can be describedby the laser threshold Pthr and the slope efficiencyη= dPout/dPabs, where Pout and Pabs are the laser out-put power and the absorbed pump power, respectively.Under the assumptions of only one metastable level (i. e.,the upper laser level), ideal overlap between pump beamand resonator mode, homogeneous pump profile, lowmirror transmission and low passive losses as well as theabsence of excited-state absorption, the following equa-tions hold for continuous-wave operation [11.49, 50]

Pthr = hνp

ηpσse (λl) τ

×[T + L +2dσGSA (λl) (nion −nthr)

] V

2d(3-level system) , (11.79)

Pthr = hνp

ηpσse (λl) τ(T + L)

V

2d

(4-level system) , (11.80)

η= ηpλp

λl

T

T + L, (11.81)

where hνp is the energy of pump photon, ηp is thepumping efficiency, i. e., the fraction of absorbed pumpphotons that are converted into excited ions in the upperlaser level, σse is the stimulated emission cross section,τ is the lifetime of the upper laser level, T is the mir-ror transmission, L is the passive losses, d is the lengthof the laser crystal, σgsa is the ground-state absorptioncross section, n is the concentration of laser ions, nthris the threshold inversion density, λp is the pump wave-length, λl is the laser wavelength, and V is the pumpvolume.

In the following, several factors affecting the pumppower threshold and the slope efficiency will be dis-cussed.

Influence of the emission quantum efficiency ηQE.A quantum efficiency less than unity leads to lifetimeshortening and therefore to an increase in the laserthreshold, as can be seen from (11.79) and (11.80) after

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Lasers and Coherent Light Sources 11.2 Solid-State Lasers 619

replacing τ by ηqeτr

Pthr = hνp

ηpσse (λl) ηqeτr(11.82)

×[T + L +2dσgsa (λl) (nion −nthr)

V

2d∝ 1

ηqe, (3-level) (11.83)

Pthr = hνp

ηpηqeσseτr(T + L)

V

2d∝ 1

ηqe(4-level) .

(11.84)

The quantum efficiency does not influence the slopeefficiency directly (11.81). However in practice a cor-responding, significant contribution of nonradiativetransitions increases the temperature in the pump vol-ume and usually leads to a lower slope efficiency dueto further lifetime reduction and other problems such asthermal lensing.

Influence of passive losses L. Passive losses in a lasersystem are due to imperfect optical components in thelaser resonator. This can be due to, e.g., stray centersand residual absorptions. Passive losses influence boththe threshold and the slope efficiency (11.79, 80, 81). Ina four-level system, the passive losses can be determinedby the Findlay–Clay method [11.53]. For such an anal-ysis, the laser threshold is measured as a function ofthe output mirror transmission. The following equationholds:

Pthr = hνp

ηpσseτ

V

2dT + hνp

ηpσseτ

V

2dL = mT +b ,

(11.85)

with slope m = hνpηpσseτ

V2d and axis parameter b = mL .

From a linear fit to Pthr = Pthr(T ) both m and b aredetermined and thus the value of the passive losses L .An alternative way to determine the passive losses is torewrite (11.81) for a so-called Caird plot [11.54]:

1

η= λ

ηpλp

L

T+ λ

ηpλp= m′ 1

T+b′ , (11.86)

with slope m′ = ληpλp

L = b′L . From a linear fit to 1η

=1η

(1T

)both m′ and b′ can be determined and thus the

value of the passive losses L .

Influence of excited-state absorption. ESA at thepump wavelength reduces the number of pump pho-tons converted into excited ions in the metastable laser

level according to

ηp = ηp,0

(1− n1σESA

(λp)

n1σESA(λp)+ (nion −n1) σgsa

(λp)),

(11.87)

whereηp,0 is the pumping efficiency without ESA, nion isthe concentration of active ions, and n1 is the populationdensity in the upper laser level. Therefore the pumpthreshold is increased and the laser slope efficiency isreduced according to (11.79, 80, 81).

ESA on the laser wavelength also affects the laserthreshold and the slope efficiency. The stimulated emis-sion cross section σse(λl) in (11.79) and (11.80) isreplaced by σEFF(λl) = σse(λl)−σESA(λl) and (11.81)has to be extended [11.54]

η= ηpλp

λl

σse (λl)−σESA (λl)

σse (λl)

T

T + L∝ σEFF

σse.

(11.88)

If σESA(λl)> σse(λl) laser oscillation is not possi-ble.

11.2.2 UV and Visible Rare-Earth Ion Lasers

Lasers based on 5d ↔ 4f transitions of trivalentand divalent rare-earth ions

In this section an overview of lasers oscillating in theultraviolet and visible spectral range based on transi-tions of rare-earth ions is given. The first part deals withlaser and possible laser systems based on interconfigu-rational transitions, i. e., 4fn−15d → 4fn . In the secondpart, visible and UV lasers based on intraconfigurational(4fn →4fn) transitions will be discussed. Both parts dealwith crystals as host materials. Finally, in the third partan overview of fiber lasers in the visible spectral rangeis given.

The 4f↔5d interconfigurational transitions of somedivalent (RE2+) and trivalent (RE3+) rare-earth ionsare located in the visible and ultraviolet spectralrange. They are in principle suitable for the realiza-tion of (tunable) laser oscillation. The transitions areelectric-dipole allowed and have high transition proba-bilities and large absorption and emission cross sectionsof the order of 10−17 to 10−18 cm2. Due to theirstrong electron–phonon coupling, the observed absorp-tion and emission transitions are broad (> 1000 cm−1).Difficulties with respect to laser operation are the non-availability of efficient and simple excitation sourcesand the high probability of excited-state absorption andsolarization (photoionization), e.g., Ce3+:LiCaAlF6 andCe3+:LiSrAlF6 depicted in Fig. 11.30.

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620 Part C Coherent and Incoherent Light Sources

.

-

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<"96

.!-)*9

;

B #

. "96

Fig. 11.30 Scheme of the energy levels in Ce3+-dopedLiCaAlF6 and LiSrAlF6, where sABS, sEM, sESA, and sSOL,are the absorption, emission, ESA, and solarization crosssections, respectively. From [11.55]

In the following the trivalent and divalent rare-earth ions exhibiting short-wavelength emission willbe briefly discussed together with a summary of theobtained laser results.

Ce3+ Lasers. The Ce3+ 4f1 ground-state configura-tion splits into the 2F5/2 ground state and the 2F7/2excited-state multiplet with an energy separation of ≈2000 cm−1. The 5d excited configuration consists in anideal octahedral or cubic symmetry of a threefold orbital

$-

$-

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Fig. 11.31 Left: Absorption (solid line) and emission (dotted line) spectra of Ce3+:LiYF4. The inset shows the schematicenergy level diagram. From [11.56]. Right: Input-output characteristic of a 2% Ce3+, 2% Na:LiSrAlF6

degenerated 2T2 state and of a twofold orbital degener-ated 2E level, from which the 2T2(2E) state is at lowerenergy in octahedral (cubic) symmetry. In Fig. 11.31 theroom-temperature absorption and emission spectrum ofCe3+:LiYF4 is shown. In general, in the absorption spec-trum two to five broad bands are expected, dependingon the crystal field symmetry. In emission two bandscorresponding to the 5d→ 2F5/2 and 5d→ 2F7/2 tran-sitions with an energetic separation of approximately2000 cm−1 are observed. The high cross sections ofthe electric-dipole- and spin-allowed 4f1 ↔5d1 transi-tions correspond to an emission lifetime in the ns range.Ce3+-doped materials were therefore thoroughly inves-tigated for application as tunable solid-state lasers andscintillators, see e.g., the overview articles by Couttset al. [11.57] and Dorenbos [11.58]. Laser oscillationwas thus far achieved in YLiF4 [11.56], LuLiF4 [11.59],LiCaAlF6 [11.60], LiSrAlF6 [11.55, 61], LaF3 [11.62],and BaY2F8 [11.63] crystals. The laser data are sum-marized in Table 11.2. For Ce3+-doped materials themain obstacle with respect to laser oscillation is excited-state absorption assigned to transitions to the conductionband, e.g., as is the case for Ce3+:YAG [11.64, 65]. ForCe3+-doped LiCaAlF6 and LiSrAlF6 the ESA transi-tion leads to a solarization, i. e., impurity traps withlong lifetimes are populated (Fig. 11.30). With the helpof an anti-solarent pump, these traps are depleted andhigher efficiencies are obtained [11.66, 67].

Pr3+ (4f 2). The energy-level scheme of Pr3+ is shownschematically in Fig. 11.32. Pr3+ has a 4f2 ground-state

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Lasers and Coherent Light Sources 11.2 Solid-State Lasers 621

Table 11.2 Overview of lasers based on the 5d→4f transitions of trivalent and divalent rare-earth ions (CVL: copper vapor laser,RS: Raman-shifted)

Laser material λlas (nm) Eout , Pout ηsl Tuning range (nm) Pump source Ref.

Ce3+:LiYF4 325.5 KrF, 249 nm [11.56]

Ce3+:LiLuF4 309 27 mJ 17% 307.8–311.7 & 323.5–326.5 KrF, 249 nm [11.75]

2.1 mJ 55% 307.6–313.5 & 324–328.5 Ce:LiSAF, 290 nm [11.76, 77]309 77 µJ 309.5–312.3 & 324.5–327.7 Nd:YAG, 5ω(213 nm) [11.78]

309.5 300 mW 38% 305.5–316 & 323–331 CVL, 2ω(289 nm), 7 kHz [11.79]309 67 mW 62% RS Nd:YAG, 4ω(289 nm), [11.80]

10 kHz

Ce3+:LiCaAlF6 290 21% Nd:YAG, 4ω(266 nm) [11.55]289 60 mJ 26% Nd:YAG, 4ω(266 nm) [11.81]

289 30 mJ 39% 284–294 Nd:YAG, 4ω(266 nm) [11.82]

289 550 mW 27% 280–311 Nd:YAG, 4ω(266 nm), 1 kHz [11.83]

288.5 530 mW 32% 280.5–316 2ω CVL, 271 nm, 7 kHz [11.79]

289 0.53 µJ 31% 283–314 Nd:YVO4, 4ω(266 nm), 1 kHz [11.84]

289 230 µJ 49% 280–317 Nd:YLiF4, 4ω(263.3 nm), [11.85]0.1–4.3 kHz

Ce3+:LiSrAlF6 290 29% Nd:YAG, 4ω(266 nm) [11.55]290 47% Nd:YAG, 4ω(266 nm) [11.66]

+ 2ω(532 nm)

281–315 nm Nd:YAG, 4ω(266 nm) [11.86]

Ce3+:LaF3 286 ≈ 5 µJ KrF, 249 nm [11.62]

Ce3+:BaY2F8 345 XeCl, 308 nm [11.63]

Nd3+:LaF3 172 Kr, 146 nm [11.87, 88]F2 Laser, 157 nm [11.89, 90]

Sm2+:CaF2 708.5 708.5–745 nm (20–210 K) Xe flash lamp [11.91, 92](20 K)

configuration. In this configuration 91 energy levelsexist, distributed over 11 manifolds, from which the3H4 level is the ground state and 1S0 is the highestlevel at approximately 46 500 cm−1. In the 4f15d1 con-figuration, 140 energy levels exist. Depending on thecrystal field strength and splitting, either the lowest4f15d1 level or the 1S0 state is lower in energy. Thusafter UV excitation, emission occurs either from the4f15d1 level as a 4f15d1 →4f2 transition or from the 1S0level as a 4f2 →4f2 transition. In any case, the emis-sion spectrum consists of several bands, because several4f2 terminal levels exist. Spectroscopic investigationsof the 4f↔5d transitions have so far been carried outfor a number of materials, e.g., Y3Al5O12 [11.68, 69],YAlO3 [11.68, 70, 71], CaF2, LiYF4 [11.71–73], andK5PrLi2F10 [11.71]. Thus far laser oscillation basedon the 4f15d1 →4f2 or on the 1S0 →4f2 emission ofPr3+ has not achieved. The main reason seems to beexcited-state absorption into the conduction band, e.g.,for YLiF4 [11.73], and into higher-lying 4f15d1 levels,e.g., for Y3Al5O12 [11.74].

Other RE3+ ions. The lowest 4fn−15d1 energy levels forions from Nd3+ to Yb3+ are at even higher energiesand therefore in the vacuum ultraviolet (VUV) spec-tral region. Thus the corresponding transitions are noteasy to access spectroscopically. A systematic study anddetailed analysis of the VUV spectroscopy of trivalentrare-earth ions were performed by Wegh et al. [11.93].The exploitation of these VUV transitions for laser os-cillation still requires much experimental efforts. Laseroscillation on a 5d→4f transition besides for Ce3+ hasthus far only been realized with Nd3+:LaF3 at 172 nmwith a Kr2 laser at 146 nm as the pump source [11.87].

Divalent rare-earth ions. Rare-earth ions have a strongtendency to be incorporated into crystals in the trivalentstate. In order to obtain the divalent state, the RE3+ ionneeds to be reduced. Eu2+, Yb2+, Sm2+ and, with somelimitations, Tm2+ can already be obtained in significantamounts during crystal growth by using suitable latticescontaining divalent cation sites, appropriate codopants,and a reducing growth atmosphere. The other RE2+ ions

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622 Part C Coherent and Incoherent Light Sources

M"'

*

-

%

&

;

(

222B%>

*!*22@%2-2*

4

*9-

,8"

,

ND!*

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Fig. 11.32a,b Energy-level schemes of Pr3+ (a) Pr3+:LiYF4, E(4f15d1)< E(1S0), (b) Pr3+:YF3, E(4f15d1)> E(1S0).(After [11.94])

are rather difficult to obtain in crystals; usually specialgrowth conditions as well as pre- and post-treatmentsare necessary, e.g., the use of sealed Ta ampoules dur-ing crystal growth, pre-synthesis of REF2 and RECl2,electron-beam irradiation etc.

5d→4f transitions are expected to occur for systemsin which the 4fn−15d level is energetically located withina large energy gap between 4fn multiplets. These ionsare Sm2+, Eu2+, Tm2+, and Yb2+.

Sm2+ (4f6). The 4f6 energy-level scheme of Sm2+ isspectroscopically similar to that of Eu3+ (Fig. 11.29).For most crystals, Sm2+ emission occurs as a 4f6 →4f6

transition from the metastable 5D0 level, which is locatedat about 15 000 cm−1. In CaF2, the lowest 4f55d1 levelis located just below the 5D0 level and broadband emis-sion occurs. Laser oscillation based on a 4f55d1 →4f6

transition was realized with Sm2+:CaF2 [11.91,92]. Thelaser wavelength is 708.5 nm at 20 K and increases withtemperature to 745 nm at 210 K, which is the highest op-eration temperature of the Sm2+:CaF2 laser. This laseris affected by excited-state absorption assigned to tran-sitions to the conduction band and to high-lying 4f6

levels [11.95, 96].

Eu2+ (4f7). The 4f7 energy-level scheme of Eu2+ isspectroscopically similar to that of Gd3+ (Fig. 11.29).A large energy gap between the 8S7/2 ground state andthe 6P7/2 first excited state exists. Depending on thecrystal field strength, the lowest 4f65d1 energy level islocated above or below the 6P7/2 level. Therefore, eithernarrow line emission due to the 6P7/2 → 8S7/2 transi-tion in the UV region or broadband emission due to the4f65d1 → 8S7/2 transition in the blue to yellow spectralrange occurs. Laser oscillation based on a 4f65d1 →4f7

transition of Eu2+ has not been realized thus far due toexcited-state absorption from the lowest 4f65d1 level tothe conduction band and to high-lying 4f7 levels [11.96].

Tm2+ (4f13). The 4f13 energy-level scheme of Tm2+ isspectroscopically similar to that of Yb3+ (Fig. 11.29).Laser oscillation of Tm2+ was realized in CaF2 on the2F5/2 → 2F7/2 transition at 1116 nm at temperatures be-low 27 K [11.97,98] (Table 11.9). Laser oscillation basedon the 4f125d1 → 4f13 transition has not been obtainedthus far. However, Tm2+ is an interesting candidate forlaser oscillation on a 4f125d1 → 4f13 transition, becausethere are no high-lying 4f13 energy levels, which couldinterfere with the 4f125d1 energy levels and thus act as

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Lasers and Coherent Light Sources 11.2 Solid-State Lasers 623

possible terminal levels for excited-state absorption tran-sitions or quenching processes. The transition betweenthe lowest 4f125d1 level and the ground state is parityallowed, but spin forbidden. Thus the emission crosssections are expected to be about one to two orders ofmagnitude smaller compared to, e.g., those of the 5d→4ftransitions of Ce3+. The Tm2+ → Tm3+ conversion ob-served under UV/VIS lamp excitation [11.98] indicatesthe strong tendency of the Tm ion towards the triva-lent state and has to be considered as a possible majordrawback for laser applications.

Yb2+ (4f14). The ground-state configuration of the Yb2+ion is 4f14. This completely filled shell leads to a 1S0ground state of the free ion and other 4f14 levelsdo not exist. In an octahedral crystal field, this statetransforms like the 1A1 irreducible representation. Theexcited 4f135d1 configuration consists in total of 140energy levels. The 4f135d level splits in a highly sym-metric crystal field into a threefold degenerated T2and a twofold degenerated E level. The 4f13 elec-trons can be considered as a Yb3+-ion configurationwith the two manifolds 2F7/2 and 2F5/2 separated byabout ∆E4 f ≈ 10 000 cm−1. The spin of the 5d elec-tron can either be parallel or antiparallel to that ofthe 4f13 core, thus the whole energy-level schemeexists for singlet and for triplet states, from which,according to Hunds’ rule, the triplet states are ener-getically lower. Note furthermore that for the Yb2+free ion the 6s level is energetically lower than the5d level. Therefore it might be the case that insome materials the 4f136s level is also the lowestexcited state. This would lead to a parity- and spin-forbidden transition requiring ∆L = 3 and ∆S = 1. Theemission spectra of Yb2+-doped materials consists ofa broad band (≈ 6000 cm−1) with the peak emissionwavelength strongly dependent on the host material(λpeak ≈ 390–575 nm) [11.99–103]. Laser oscillationhas not been obtained thus far. The main reason isexcited-state absorption. In Yb2+:MgF2 strong ESAtransitions in the whole spectral range of absorptionand emission are observed, preventing laser oscillation.The ESA cross section is about one order of magnitudelarger than the absorption cross section and three ordersof magnitude larger than the stimulated emission crosssection [11.102, 103].

Lasers based on 4f ↔ 4f transitions of trivalentand divalent rare-earth ions

This section deals with UV and visible lasers based onthe 4f↔4f transitions of trivalent and divalent rare-earth

%

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Fig. 11.33 Energy-level scheme of the Pr3+ ion. The ob-served laser transitions in the visible spectral range areindicated by arrows

ions. Laser oscillation has been realized under flashlamppumping, direct laser pumping into the upper laser levelor higher-lying energy levels and under upconversionpumping.

Pr3+ lasers. The Pr3+ ion is a very interesting andpromising ion for obtaining efficient visible laser os-cillation. Its energy-level scheme is shown in Fig. 11.33.

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Fig. 11.34 Emission spectra of Pr3+:BaY2F8 at room temperaturefor different polarizations

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624 Part C Coherent and Incoherent Light Sources

Laser transitions in the visible spectral range occur fromthe 3P0 and the thermally populated 3P1 and 3P2 levels.In Fig. 11.34 the emission spectrum of the Pr3+:BaY2F8as an example is shown. The peak cross sections are inthe range of ≈ 1 to 5 × 10−19 cm2 and thus comparableto the values of the Nd3+ 4F3/2 → 4I11/2 transition. Thehighest cross sections are in the orange and red spectralrange, therefore the most efficient lasers are expectedthere. The first Pr3+ laser (λ= 1047 nm) was realizedin CaWO4 as early as 1962 by Yariv et al. [11.104].Since then, laser oscillation has been obtained in morethan 20 materials on several transitions and under differ-ent pumping conditions, see e.g., the review article byKaminskii [11.105]. The major drawback of Pr3+ lasersis their excitation; see the discussion in section on Pr3+laser.

Directly pumped Pr3+ lasers. In Fig. 11.35 the absorp-tion spectrum of Pr3+:BaY2F8 for the spectral rangebetween 420 nm and 500 nm is shown. Direct pumpingof the 3P0 upper laser level is possible for several wave-lengths in the blue spectral range corresponding to the3H4 → 3P2, 3P1, 3P0, and 1I6 transitions. In principlethe following pumping schemes for direct excitation arepossible:

1. Ar+-ion laser pumping. Ar+-ion lasers offer thepossibility of continuous-wave pumping with a high-quality pump beam. Therefore it allows the

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Fig. 11.35 Absorption cross section spectrum of Pr3+,Yb3+:BaY2F8 at room temperature. The relevant transi-tions are assigned. The possible pump laser wavelengthsare also shown

characterization of a Pr3+-doped laser material. ForPr3+:LiYF4, laser oscillation on several transitionshas been obtained with slope efficiencies up to≈ 26% and output powers up to ≈ 270 mW [11.106](Fig. 11.36). However, the wavelength match be-tween the Ar+ pump and Pr3+ absorption is bad(Fig. 11.35). Furthermore Ar+-ion laser pumpingitself is very inefficient, therefore the overall ef-ficiency of a gas-laser-pumped Pr3+ laser is verylow.

2. Pumping with a frequency-doubled Nd3+ laser op-erating on the 4F3/2 → 4I9/2 ground-state transition.These ground-state lasers operate, depending on thehost material, between 910 nm and 960 nm, i. e.,the frequency doubling yields wavelengths between455 nm and 480 nm. A Pr3+ laser directly pumped bya frequency-doubled Nd3+:YAG ground-state laseroperating at 473 nm has been successfully demon-strated by Heumann et al. [11.107]. Pr3+ laseroscillation occurred at 639.5 nm on the 3P0 → 3F2transition with an output power of nearly 100 mWand a slope efficiency of 12%. Also for frequency-doubled Nd3+ ground-state lasers the match betweenthe pump and absorption wavelength is crucial.

G

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-

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%($-

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Fig. 11.36 Input–output diagram of the CW Pr3+:LiYF4

laser at different laser wavelengths pumped by an Ar+-ionlaser at 457.9 nm. (After [11.106])

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Lasers and Coherent Light Sources 11.2 Solid-State Lasers 625

3. Frequency-doubled optically pumped semiconduc-tor lasers (OPS). These lasers are commerciallyavailable with powers up to 200 mW [11.108]. Inprinciple, the laser wavelength is adjustable (bychoosing the material parameters of the semiconduc-tor) and can be tuned to match the absorption lineof the Pr3+ ion. Output powers of approximately35 mW with slope efficiencies of approximately30% have been obtained [11.109]. Furthermore,Richter et al. used an OPS operating at 480 nm asa direct pump source [11.110]. A slope efficiencyof 40% and a maximum output power of 75 mWwere obtained for Pr3+:YLiF4. For Pr3+:BaY2F8,the corresponding data are 30% and 51 mW. It isworth noting that intracavity frequency doublingleads to continuous-wave UV radiation of about19 mW for Pr3+:YLiF4 and Pr3+:BaY2F8. Due tothe multi-wavelength operation of Pr3+ lasers, UVgeneration at 360 nm, 303 nm and even 261 nmseems possible. These pump lasers are still expen-sive; however power scalability of OPS lasers can beexpected making these pump sources very attractivefor Pr3+.

4. Blue and UV GaN laser diodes. These diodes operatethus far in the spectral region below 450 nm and withoutput powers in the mW range. Recently, Richteret al. obtained laser oscillation at room temperaturewith Pr3+:YLiF4 under pumping with a 442 nm GaNlaser diode [11.35]. The Pr3+ laser emits 1.8 mWat 639.7 nm. The threshold pump power and slopeefficiency were 5.5 mW and 24%, respectively. If thedevelopment of these diodes towards higher outputpowers proceeds, laser diodes will certainly be oneof the best choices for the direct pumping of Pr3+lasers.

5. Flashlamp pumping [11.111]. Under flashlamp exci-tation only a small fraction of the emitted radiationof the lamp can be used for the excitation of thePr3+ 3P0 level, because of the narrow line widthof the Pr3+ absorption lines and of the small spec-tral range of these lines. Output energies up to87 mJ and slope efficiencies of 0.3% have beenobtained [11.111].

6. Dye-laser pumping. The pump laser wavelength istunable and thus can be adjusted to match the absorp-tion lines of the Pr3+ ion. Therefore, the obtainedlasers exhibit a high efficiency. For Pr3+:LiGdF4lasers, oscillation on several wavelengths and tran-sitions with slope efficiencies up to 37% have beenreported [11.112]. However, dye lasers are limitedin their practical use.

Table 11.3 lists Pr3+ lasers operating in the visiblespectral range under direct excitation. In Table 11.4 anoverview of the room-temperature laser data for visiblePr3+ lasers is given.

Upconversion pumped Pr3+ lasers. Because of the dif-ficulties of direct excitation, other pumping schemesusing upconversion processes are under investigationfor Pr3+ lasers. Upconversion describes a process inwhich the photon energy of the excitation light (pumplight) is converted via interaction with the active ionsinside the optical material into higher-energy pho-tons [11.113–116]. For Pr3+ with its high-lying energylevels upconversion is a suitable way to obtain visiblelaser oscillation under infrared pumping. Especially thephoton-avalanche pumping scheme, which is a com-bination of different upconversion and energy transferprocesses, has been efficiently exploited for the Pr,Ybcodoped system. For crystals the Ti:Al2O3 laser ismainly used as the excitation source, but investigationsare being performed on the use of commercially avail-able laser diodes. In fibers, efficient laser operation hasalready been achieved under laser-diode pumping (seepage 633, Visible fiber lasers).

The principle scheme of the photon-avalanche pro-cess is shown in Fig. 11.37. A weak ground-stateabsorption yields the excitation of a few ions into theintermediate (reservoir) level. The strong ESA processefficiently brings these ions to the emitting level. Aneffective feedback mechanism for the reservoir level(in our system a cross-relaxation process) is necessary,which couples the emitting level, the reservoir level andthe ground-state level. After these two steps there aretwo ions in the reservoir level. This cycle repeats andthus the population of the emitting (upper laser) levelincreases like an avalanche. If the threshold populationis reached, the laser oscillation on any transition from theupper laser level may start. The general characteristicsare described in detail in [11.113, 114, 117–119]; hereonly the main points are given for the example of Pr3+,Yb3+:BaY2F8 [11.120, 121]. The pump-power depen-dence of the emission intensity exhibits a threshold-likebehavior. At this threshold, the slope increases sig-nificantly. A second characteristic of the avalanchemechanism is the temporal S-shaped evolution of theupconverted emission (Fig. 11.38).

The photon-avalanche process has already been ob-served in a variety of materials, see e.g. [11.113, 114].However, laser oscillation has only been obtained ina few materials, see Table 11.5 for Pr3+ systems andTable 11.6 for other rare-earth-ion-doped systems. The

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626 Part C Coherent and Incoherent Light Sources

Table 11.3 Pr3+ lasers in the visible spectral range under direct excitation (bold: laser oscillation on some transitionsobtained at room temperature)

Crystal Transition Wavelength (nm) Ref.

LiYF43P0 →3 H4 479.0 [11.35, 106, 107, 110, 111, 122–126]3P1 →3 H5 522.03P0 →3 H5 537.8, 545.03P0 →3 H6 604.4, 607.2, 609.2, 613.03P1 →3 F2 615.8, 618, 620.13P0 →3 F2 638.8, 639.5, 644.43P1 →3 F3 670.33P0 →3 F3 695.4, 697.7, 705.53P1 →3 F4 699.41I6 →3 F4 708.23P0 →3 F4 719.5, 720.9, 722.2

LiLuF43P0 →3 H5 538 [11.127, 128]3P0 →3 H6 604.2, 607.13P0 →3 F2 639.9, 640.13P0 →3 F3 695.8, 697.73P0 →3 F4 719.2, 721.5

LiGdF43P1 →3 H5 522 [11.106, 112]3P0 →3 H5 5453P0 →3 H6 604.5, 6073P0 →3 F2 6393P0 →3 F3 6973P0 →3 F4 720

KYF43P0 →3 F2 642.5 [11.106, 112]

BaY2F83P0 →3 H6 607.1 [11.110, 125, 129–131]3P0 →3 F2 638.83P0 →3 F3 693.5–693.83P0 →3 F4 719.1

LaF33P0 →3 H6 598.5, 600.1 [11.132–135]3P0 →3 F4 719.4, 719.8

PrF33P0 →3 H6 598.4 [11.136]

LaCl3 3P0 →3 H4 489.2 [11.137–139]3P1 →3 H5 529.83P0 →3 H6 616.4, 619.03P0 →3 F2 645.1

PrCl3 3P0 →3 H4 489.2 [11.137, 140]3P1 →3 H5 529.8, 5313P0 →3 H6 617, 620, 6223P0 →3 F2 645.2, 647

LaBr33P1 →3 H5 532.0 [11.140]3P0 →3 H6 621.03P2 →3 F3 632.03P0 →3 F2 647.0

PrBr33P0 →3 H6 622 [11.137, 140]3P0 →3 F2 645.1, 649

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Table 11.3 (continued)

Crystal Transition Wavelength (nm) Ref.

Y3Al5O123P0 →3 H4 487.2 [11.141, 142]3P0 →3 H6 6163P0 →3 F4 747

YAlO33P0 →3 H6 613.9, 621.3, 621.6 [11.143–147]3P0 →3 F2 6623P0 →3 F3 719.5, 719.7, 7223P0 →3 F4 746.91D2 →3 F3 743.7, 753.7

LuAlO33P0 →3 H6 615.5 [11.144, 145, 147]3P0 →3 F3 722.03P0 →3 F4 749.6

SrLaGa3O73P0 →3 H4 488 [11.148]3P0 →3 F2 645

CaWO43P0 →3 F2 649.7 [11.149]

Ca(NbO3)23P0 →3 H6 610.5 [11.149]

LiPrP4O143P0 →3 H6 604.8, 608.5 [11.150]3P0 →3 F2 639.63P0 →3 F4 720.4

LaP5O143P0 →3 F2 637 [11.151]3P0 →3 F4 717

PrP5O143P0 →3 F2 637.4 [11.152–154]

LaP5O143P0 →3 F2 637.0 [11.151]

Table 11.4 Overview of room temperature laser data of directly pumped Pr3+ lasers in the visible spectral range

Host Transition λlaser (nm) Pump Pthr/Ethr Pout/Eout η (%) Ref.

LiYF43P1 →3 H5 522.0 457.9 nm 163 mW 144 mW 14.5 [11.106]3P0 →3 H5 545.0 CW, Ar+-ion laser 19 mW3P0 →3 H6 607 110 mW 7 mW 1.23P0 →3 F2 639.5 8 mW 266 mW 25.93P0 →3 F3 697 105 mW 71 mW 10.33P0 →3 F4 720 98 mW 40 mW 7.23P0 →1 G4 907.4 280 mW 23 mW 7.3

LiYF43P0 →3 F2 639.5 473 nm CW, SHG Nd:YAG 40 mW ≈ 100 mW 12 [11.107]

LiYF43P0 →3 H6 613 476 nm CW, Ar+ − ionlaser 45 mW ≈ 400 fs (ML) [11.124]

LiYF43P0 →3 F2 639.5 442 nm CW, GaN laser diode 5.5 mW ≈ 1.8 mW 24 [11.35]

LiYF43P0 →3 F2 639.5 480 nm CW OPS 37 mW 72 mW 40 [11.110]

LiYF43P0 →3 F2 639.5 Xe flashlamp (60 µs) ≈ 7 J 87 mJ ≈ 0.3 [11.111]

LiGdF43P1 →3 H5 522 468 nm 197 µJ 83 µJ 27 [11.112]3P0 →3 H5 545 pulsed dye laser 49 µJ 2 µJ3P0 →3 H6 604.5 144 µJ 32 µJ 373P0 →3 H6 607 50 µJ 2 µJ3P0 →3 F2 639 4 µJ 98 µJ 323P0 →3 F3 697 73 µJ 31 µJ 263P0 →3 F4 720 6 µJ 80 µJ 35

KYF43P0 →3 F2 642.5 457.9 nm CW, Ar+-ion laser 15 mW [11.106]

KYF43P0 →3 F2 642.5 465 nm pulsed dye laser [11.112]

YAlO33P0 →3 F4 746.9 476.5 nm CW, Ar+-ion laser 25 mW 130 mW 24.6 [11.146, 155]

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628 Part C Coherent and Incoherent Light Sources

40

,#

,

-

-

-

.

.

,40

Fig. 11.37 Scheme of the photon-avalanche process (GSA:ground-state absorption, ESA: excited-state absorption,CR: cross-relaxation)

reason is that there exist special requirements for an effi-cient avalanche process; its efficiency depends stronglyon the transfer rates between the ions, the ground-and excited-state absorption cross sections, the ionicconcentration, the emission branching ratios and thelifetimes of the energy levels involved. In principle itis possible to describe the photon-avalanche excitationmechanism for Pr,Yb-doped systems with rate equationsystems [11.113, 114, 117, 118, 120]. However, due toits complexity, it is not possible to predict the overallefficiency solely from knowledge of the spectroscopicparameters.

In Table 11.5 an overview of photon-avalanchepumped Pr3+ laser systems is given. Room-temperatureavalanche-pumped laser oscillation in bulk crystals was

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Fig. 11.38 Characteristics of the photon-avalanche excitation mechanisms. Left: pump-power dependence of the upcon-verted emission intensity (experiment, open circles) and 3P0 population (calculated, filled squares). Right: temporalevolution of the upconverted emission intensity (solid line: experiment, squares: rate equation model). (After [11.120])

obtained for the Pr3, Yb3+ codoped system, a so-calledsensitized photon avalanche, in YLiF4 (YLF) [11.156–158] and BaY2F8 (BYF) [11.121]. The principle schemeof the Pr-Yb system is shown in Fig. 11.39 [11.159–163]. The excitation around 840 nm corresponds toa very weak ground-state absorption process (veryprobable into a phonon tail of the Yb3+ absorption), fol-lowed by energy transfer (process r in Fig. 11.39) fromYb3+ to Pr3+ (2F5/2,

3H4) → (2F7/2,1G4). The ESA

process (1G4 → (1I6,3P1)), followed by fast phonon de-

excitation [(1I6,3P1) → 3P0] feeds the 3P0 level (i. e.,

the emitting level). The cross-relaxation process s(3P0,

2F7/2) → (1G4,2F5/2) followed again by the trans-

fer process r re-feeds the reservoir level. In this way,at each step, an increase of the population in the 1G4level is obtained and consequently a strong populationwill be build up in the 3P0 level (because of the strong1G4 → (1I6,

3P1) ESA process). The upconverted emis-sion excitation spectrum and the excited-state absorptionspectrum are shown in Fig. 11.40 indicating the goodmatch between these two spectra. The peak excited-state absorption cross section of the 1G4 → (1I6,

3P1)transition is about 1.5 × 10−19 cm2.

The laser input–output curves for Pr,Yb:YLF andPr,Yb:BYF under Ti:sapphire pumping are shown inFig. 11.41; the results are summarized in Table 11.5.The realization of diode pumping using commerciallyavailable infrared laser diodes is under investigation inseveral research groups. However, at the moment ion-doped fibers as materials for upconversion and avalanchepumped lasers are more efficient, because they allowwaveguiding of both the pump and the laser beam.

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Table 11.5 Photon-avalanche pumped crystalline CW Pr3+ lasers in the visible spectral range (T : temperature, η: slopeefficiency)

Dopant ions Host λlaser (nm) Transition λpump (nm) T Output (mW) η (%) Ref.

Pr3+ LaCl3 644 3P0 →3 F2 677 80–210 K 240 25 [11.164]Pr3+ / Yb3+ YLiF4 522 3P1 →3 H5 830 RT 143 7.5 [11.156–158]

639.5 3P0 →3 F2 830 276 15720 3P0 →3 F3 830

Pr3+/Yb3+ BaY2F8 607.5 3P0 →3 H6 822, 841 RT 98 30 [11.121]638.5 3P0 →3 F2 822, 841 60 15720.5 3P0 →3 F3 822 45 16

Table 11.6 Other rare-earth-doped solid-state upconversion lasers; for details see the overview in [11.114, 116] (ETU: energytransfer upconversion, STPA: sequential two-photon absorption, CT: cooperative transfer, PA: photon avalanche, CW: continuouswave, p: pulsed, SP: self pulsed, QS: Q-switched, ML: mode-locked, IR-fl: infrared flashlamp, RT: room temperature). ∗ Firstobservation of laser oscillation based on an upconversion process [11.169]

Dopant ions Host λlaser (nm) λpump (nm) Pump T (K) Output η (%) Ref.mechanism

Nd3+ LaF3 380 788 + 591 STPA ≤ 90 12 mW, CW 3 [11.165,166]

Nd3+ LaF3 380 578 STPA ≤ 20 4 mW, CW 0.7 [11.165,166]

Nd3+ LiYF4 730 603.6 PA ≤ 40 11 [11.165,167]

Nd3+ LiYF4 413 603.6 PA ≤ 40 10 µW, CW 4.3 [11.165,165, 167]

Ho3+ / Yb3+ KYF4 551 960 ETU 77 CW [11.168]Ho3+ / Yb3+ BaY2F8 551.5 IR-fl ETU 77 [11.169]∗Tm3+ YLiF4 450.2, 453 781 + 647.9 STPA 77 - RT 0.2 mJ, p 1.3 [11.170]

(pulsed lasers)Tm3+ YLiF4 450.2 784.5 + 648 STPA ≤ 70 9 mW, SP 2 [11.171,

172]Tm3+ YLiF4 483 628 PA ≤ 160 30 mW 7.5 [11.171,

172]Tm3+ YLiF4 483 647.9 PA ≤ 160 30 mW, SP 8 [11.171,

172]Tm3+ Y3Al5O12 486 785 + 638 STPA ≤ 3 0.07 mW, SP 0.01 [11.173]Tm3+ / Yb3+ BaY2F8 455, 510, 649, 799 960 ETU RT [11.174]Tm3+ / Yb3+ BaY2F8 348 960 ETU 77 CW [11.175]Tm3+ / Yb3+ BaY2F8 348 960 ETU RT SP [11.175]Tm3+ / Yb3+ BaY2F8 649 1054 ETU RT 1 [11.176]Tm3+ / Yb3+ YLiF4 810, 792 969 ETU RT 80 mW, CW [11.177]Tm3+ / Yb3+ YLiF4 650 969 ETU RT 5 mW, CW 0.2 [11.177]

Table 11.7 Visible Er3+ lasers with direct pumping (p = pulsed operation)

Crystal Laser transition λlaser (µm) Pump T Output mode Ref.

Ba(Y,Er)2F84S3/2 →4 I15/2 0.5540 Xe lamp 77 p [11.178]2H9/2 →4 I13/2 0.5617 Xe lamp 77 p [11.169, 178]4F9/2 →4 I15/2 0.6709 Xe lamp 77 p [11.178]2H9/2 →4 I11/2 0.7037 Xe lamp 77 p [11.178]

Ba(Y,Yb)2F84F9/2 →4 I15/2 0.6700 Xe lamp 77 p [11.169]

BaYb2F84F9/2 →4 I15/2 0.6700 Xe lamp 110 p [11.169, 179]

LiYF44S3/2 →4 I11/2 0.551 Dye laser 300 p [11.180]

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630 Part C Coherent and Incoherent Light Sources

40

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Fig. 11.39 Scheme of the avalanche mechanism in Yb–Pr-dopedsystems

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Fig. 11.40 Upconverted emission excitation spectrum andexcited-state absorption spectrum of Pr3+,Yb3+:BaY2F8.(After [11.120])

Therefore high pump intensities over a long distanceare possible, increasing the overall efficiency of theavalanche pumping mechanism, see the section on visi-ble fiber lasers in Sect. 11.2.2.

Er3+ lasers. The problem for the realization of directlypumped visible Er3+ lasers is the lack of efficient pumpsources that have a good spectral match to the ab-sorption lines of the Er3+ ion. Therefore most visibleEr3+ laser schemes have been realized by upconversionpumping in the near-infrared spectral range, as shownin Fig. 11.42. Table 11.7 summarizes a few results of di-rect pumping of Er3+ obtained with Xe lamps and a dyelaser.

The energy-level schemes of Erbium-doped crystalssuch as LiYF4 and LiLuF4 [11.127, 128] offer the fea-sibility of realizing an upconversion laser emitting inthe green spectral region (4S3/2 → 4I15/2 transition). Asshown in Fig. 11.42 the upconversion excitation of theupper laser level 4S3/2 requires sequential two-photonabsorption (STPA) either at 810 nm or 970 nm. Witha pump wavelength of 810 nm, the ground-state absorp-tion (GSA) 4I15/2 → 4I9/2 is followed by nonradiativedecay to the 4I11/2 level. From there, population is takento the 4F5/2 level by excited-state absorption (ESA).Finally, the 4S3/2 level is populated by nonradiativedecay. With a pump wavelength around 970 nm, the up-conversion excitation scheme of the upper laser levelis very similar, however, it involves the GSA process4I15/2 → 4I11/2 and the ESA process 4I11/2 → 4F7/2.

Using a Ti:sapphire laser for excitation, vari-ous rare-earth-doped fluoride crystals have shownroom-temperature CW upconversion lasing in the vis-ible spectral range, e.g., Er3+:LiYF4 [11.181] andEr3+:LiLuF4 [11.182]. Diode pumping of a green er-bium-doped upconversion laser in Er3+:LiLuF4 [11.183]and in the mixed fluoride crystal Er3+:LiKYF5 [11.184]has been demonstrated. In the latter case, however,laser operation could only be achieved under choppedexcitation with a duty cycle of 20%.

Table 11.8 presents an overview Er-based upcon-version lasers in different crystals and at variouswavelengths.

Besides YAlO3 most of the interesting candidatesare fluorides, due to their relatively small phonon ener-gies and correlated long lifetimes of intermediate states.Rare-earth-doped LiLuF4 exhibits a larger splitting ofthe manifolds by the crystal field compared to LiYF4and LiGdF4, which tends to produce more-favorablethermal occupation factors of both the upper and lowerlaser levels [11.185, 186]. A reasonable overlap be-tween GSA and ESA enables two-step excitation to the4S3/2 level of Er3+ using just a single pump wave-length. The emission cross section σem at 552 nm (πpolarization) is σem = 3.5 × 10−21 cm2. This value aswell as the GSA cross sections around 970 nm areslightly larger than the corresponding cross sections inEr3+(1%):LiYF4. The lifetime τ of the upper laser level4S3/2

[τ(4S3/2) = 400 µs

]is slightly longer in LiLuF4

than in LiYF4.A multipass pumping setup has been used in order

to increase the absorbed pump power (Fig. 11.43). Bothend faces of the 1.6 mm-long Er3+(1%):LiLuF4 crys-tal have been prepared with directly coated dielectricmirrors [11.183]. One of these mirrors is highly trans-

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Lasers and Coherent Light Sources 11.2 Solid-State Lasers 631

B" 5G

5G

%&$-2%6;-G2

%;$&2$;6-%G2-

&26%G2%

* - %

-

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-

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%($--

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%

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0 5G G

Fig. 11.41 Input–output curves of Pr,Yb:YLF [11.157, 158] (left) and Pr,Yb:BYF (right) under avalanche pumping witha Ti:sapphire laser

mitting for the pump wavelength and highly reflectingfor the laser wavelength, while the other mirror is highlytransmitting for the laser wavelength and highly reflect-ing for the pump wavelength (Fig. 11.43, coatings Land P, respectively). An additional concave mirror witha hole drilled slightly off axis is used to realize up tofour passes of the pump radiation through the activevolume of the laser crystal. The collimated pump beam

;

,#"'*!-)

*!&)@)

-

-(&

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--

*4)

*!()

*B()

*B)

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Fig. 11.42 Energy-level scheme of Er3+:LiLuF4 [11.127,128] with two possible upconversion excitation routes ofsequential two-photon absorption (STPA) at 810 nm or970 nm [11.182, 183] and emission at 552 nm

is focused through the mirror hole into the crystal. Withrespect to single-pass pumping significant improvementof the performance of the Er3+:LiLuF4 upconversionlaser could be achieved under Ti:sapphire laser excita-tion by using this setup. The maximum CW output powerwas 213 mW at an incident pump power of 2.6 W. Theslope efficiencies with respect to incident and absorbedpump power were 12% and 35%, respectively. Whenreplacing the Ti:sapphire laser by a 3 W diode, it waspossible to realize laser-diode-pumped CW operation ofthis erbium-doped upconversion laser at room temper-ature for the first time [11.183]. The maximum outputpower was 8 mW at an incident pump power of 2.5 W.The absorbed power under four-pass pumping was smalland estimated to be between 10% and 12%. The slopeefficiency with respect to the absorbed pump power was14%. Pumped with an optically pumped semiconductorlaser operating at 970 nm, continuous-wave laser oscil-lation at 550 nm with Er3+:LiLuF4 has been achievedwith an output power of 500 mW and a slope efficiencyof about 30% [11.127, 128, 182, 183].

Other 4f–4f divalent and trivalent rare-earth ionlasers. Besides visible lasers based on 4f–4f transitionsof Pr3+ and Er3+ several other lasers with 4f–4f tran-sitions in the visible spectral region have been realizedas shown in Table 11.6. When compared to Pr3+ andEr3+ lasers, the laser performance of all listed systemsin Table 11.6 is worse. However, at least the upconver-sion pumped Tm3+ and Ho3+ lasers seem to have somepotential for improvement of laser performance in thevisible spectral region.

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Table 11.8 Visible Er3+ lasers (ETU: energy transfer upconversion, STPA: sequential two-photon absorption, CT: cooperativetransfer, PA: photon avalanche, CW: continuous wave, p: pulsed, SP: self-pulsed, QS: Q-switched, ML: mode-locked, IR-fl:infrared flashlamp, OPS: optically pumped semiconductor, RT: room temperature)

Host λlaser (nm) λpump (nm) Pump T (K) Output η (%) Ref.

mechanism

BaY2F8 670 IR-fl ETU 77 [11.169]

YAlO3 550 792 + 840 STPA ≤ 77 0.8 mW, CW 0.2 [11.187]

YAlO3 550 785 + 840 STPA 34 8 mW, CW 1.8 [11.188, 189]

YAlO3 550 807 ETU 7–63 166 mW, CW 13 [11.188]

YAlO3 550 791.3 STPA 7–34 33 mW, CW 3.3 [11.188, 190]

+ looping

Y3Al5O12 561 647 + 810 STPA RT [11.191]

CaF2 855 1510 CT 77 64 mW, CW 18 [11.192]

LiYF4 551 797 or 791 (diode) STPA ≤ 90 0.2 [11.193–196]

LiYF4 551 791 (diode) ETU / STPA ≤ 90 0.1 mW, SP 0.03 [11.195]

LiYF4 551 802 (diode) ETU / STPA ≤ 77 2.3 mW, SP [11.197]

LiYF4 551 797 (diode) CT 48 100 mW, SP 5.5 [11.198]

LiYF4 702 1500 CT 10 360 µW, CW 0.06 [11.199]

LiYF4 551 810 STPA RT 40 mW, CW 1.4 [11.181]

LiYF4 551 974 STPA RT 45 mW, CW 2 [11.200, 201]

LiYF4 551 1500 ETU 80 10 mW, SP 2.9 [11.202]

LiYF4 561 1500 ETU 80 12 mW, SP 3.4 [11.202]

LiYF4 468 1500 ETU 80 0.7 mW, SP 0.2 [11.202]

LiYF4 561, 551, 544 797 ETU 49 467 mW, CW 11 [11.203]

LiYF4 551, 544 1550 CT 9–95 34 mW, CW 8.5 [11.204, 205]

LiYF4 551, 544 1500 CT ≤ 95 0.6 µJ (50 ns), QS 2 mW, ML [11.206]

LiYF4 551 647 + 810 STPA RT 0.95 mJ, p 8.5 [11.191]

LiYF4 467 1550 CT 70 [11.116]

LiYF4 469.7 969.3 ETU ≤ 35 2 mW, CW 0.3 [11.207]

LiYF4 469.7 653.2 ETU ≤ 35 6 mW, CW 4.8 [11.207]

LiYF4 560.6 969.3 ETU ≤ 35 2 [11.207]

LiYF4 551 802 ETU ≤ 90 5 mW, SP 2 [11.165]

LiYF4 1230, 850 1530 ETU 110 [11.208]

LiLuF4 552 970 STPA RT 213 mW 35 [11.183]

968 (diode) 8 mW 14

LiLuF4 552 970 (OPS) STPA RT 500 mW (CW) 30 [11.209]

800 mW (DC: 50)

KYF4 562 647 + 810 STPA RT 0.95 mJ, p 0.5 [11.191]

LiKYF5 550 488 (Ar+) STPA RT 40 mW 18 [11.184]

651 (diode) 50 mW (DC: 20) 6

808 (diode) 150 mW (DC: 20) 12

BaYb2F8 670 1540 + 1054 or 1054 ETU RT [11.210]

BaY2F8 470, 554, 555 792.4 CT 10 [11.211]

BaY2F8 552 792.4 CT 40 [11.211]

BaY2F8 617, 669 792.4 CT 20 [11.211]

BaY2F8 552, 470 ≈ 790 or ≈ 970 STPA 10 CW [11.212]

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Lasers and Coherent Light Sources 11.2 Solid-State Lasers 633

Blue upconversion laser emission in Tm-doped crys-tals can be achieved with STPA pumping. The energylevels and upconversion pump mechanisms in Tm3+ areillustrated in Fig. 11.44.

Figure 11.44a shows a pure STPA-pumped up-conversion laser scheme, whereas in Fig. 11.44b,upconversion pumping also requires an additionalcross-relaxation step between two Tm3+ ions. Thiscross-relaxation process populates the intermediate 3F4state which is the starting level for the second step of theSTPA process.

In sensitized upconversion lasers a donor ion D (thesensitizer) absorbs the pump light and transfers its exci-tation energy to an acceptor ion A. In many cases Yb3+has been used as the donor D for trivalent rare-earth-ionactivators A. Through an upconversion energy transfer(ETU), two donor ions transfer their excitation energysuccessively to an acceptor. Finally the excitation en-ergy of the donor is higher than the energy of an excitedYb3+ ion.

Figure 11.45 shows a system with D = Yb3+ andA = Tm3+. In this case a three-step STPA process isalso possible, yielding a variety of laser transitions inthe visible region.

Figure 11.46 illustrates the STPA and ETU upcon-version mechanisms in Ho3+ and in the donor–acceptorsystem Yb3+–Ho3+. Laser emission can be gener-ated near 750 nm and 550 nm, originating from themetastable 5S2,

3F4 states.It can be seen in Table 11.6 that, besides STPA and

ETU pumping, photon avalanche (PA) pumping has alsobeen used in a few cases (Nd3+ and Tm3+).

In Table 11.9 other rare-earth-ion lasers in the visiblespectral range based on 4f–4f transitions are listed. Thedata are taken from [11.213].

Visible fiber lasersA different and very promising approach to realize ef-ficient upconversion room-temperature laser oscillationis the use of rare-earth-ion-doped fibers. The geome-try of the fiber provides waveguiding of both the pumpradiation and the stimulated emission, thus long inter-action lengths can be realized, yielding high intensitiesover a long distance, a critical requirement for upcon-version lasers. This is one advantage of the fiber concept(for a detailed description of the fiber concept, seethe description of Yb fiber lasers in Sect. 11.2.2) com-pared to bulk materials. Another requirement for visibleupconversion lasers is the existence of metastable inter-mediate levels to act as initial levels for an excited-stateabsorption or an energy transfer process. Like in crys-

"

N@P--20P(&N@P(&20P--

" *

.#

+N6

Fig. 11.43 Experimental setup for fourfold-pass pumping [11.183](HR: high reflection, AR: antireflection)

tals, fluoride materials are preferred because of theirlow phonon energies and generally larger bandgaps.The fiber material of choice is the fluorocirconate glassZBLAN (ZrF4 −BaF2 −LaF3 −AlF3 −NaF). Anotheradvantage of fibers compared to bulk crystals is thatthe transitions in glass are broadened and therefore thepossibility for resonant transitions or energy transfer isenhanced. An overview of visible fluoride fibers is givenin [11.214].

Nearly all rare-earth-ion-doped fluoride fiber laserscan be pumped in the 0.63–1.2 µm region andthus take advantage of the mature semiconductortechnology, e.g., AlGaInP (0.63–0.69 µm), GaAlAs(0.78–0.88 µm) and InGaAs (0.90–1.2 µm) and ofthe highly developed solid-state laser technology, e.g.,

!*

-

@%

&;

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*

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-

-

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-

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&;-

@*

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-

,#"' ,#"'

Fig. 11.44a,b Upconversion pump mechanisms for Tm3+. (a) Se-quential two-photon absorption, (b) sequential two-photon absorp-tion with additional cross-relaxation (dashed line)

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634 Part C Coherent and Incoherent Light Sources

Table 11.9 Visible and UV lasers based on divalent and trivalent rare-earth ions

Dopant ions Host Transition λlaser (nm) T (K) Ref.

Sm3+ TbF34G5/2 →6 H7/2 593.2 116 [11.215]

Sm2+ SrF25D0 →7 F1 696.9 4.2 [11.216]

Eu3+ Y2O35D0 →7 F2 611.3 220 [11.217]

Eu3+ YVO45D0 →7 F2 619.3 90 [11.218]

Gd3+ Y3Al5O126P7/2 →8 S7/2 314.5 300 [11.219]

Tb3+ LiYF45D4 →7 F5 544.5 300 [11.220]

Ho3+ CaF25S2 →5 I8 551.2 77 [11.221]

Tm2+ CaF22F5/2 →2 F7/2 1116 < 27 [11.97, 98]

Ag+ KI, RbBr, CsBr 335 5 [11.222]

Nd3+ and Yb3+ lasers in the near-infrared spectralregion.

Visible Pr3+ fiber lasers. As in crystals, thepraseodymium ion is very attractive for visible fiberlasers because of its energy-level scheme (see Fig. 11.33)and the possibility of upconversion pumping by two-step absorption, photon avalanche or energy transferprocesses. Laser oscillation at room temperature hasbeen achieved in the red, orange, green and blue spec-tral ranges. Some of these transitions have even beenoperated simultaneously [11.223]. Laser oscillation atroom temperature originates from the 3P0, 3P1 and 1I6

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!*

!

%*(

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*;

*;

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(%

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Fig. 11.45 Sensitized upconversion in Yb3+,Tm3+-dopedBaY2F8 after Trash and Johnson [11.174] with two andthree steps in the STPA process

levels, which are thermally coupled. In Pr3+:ZBLAN,the lifetime of these coupled multiplets is in the range40–50 µs [11.224,225]. Recently, Richter et al. obtained

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-

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--

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,#"'

-

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(%

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,#"'

-B%

-B&

-42 -!*

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Fig. 11.46a,b STPA (a) and ETU (b) pumping of Ho3+upconversion lasers

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Lasers and Coherent Light Sources 11.2 Solid-State Lasers 635

Table 11.10 Visible Pr3+-doped ZBLAN fiber lasers (ESA: excited-state absorption (i. e., sequential two-step absorption), ETU:energy transfer upconversion, PA: photon avalanche, RT: room temperature)

Dopant ions λlaser (nm) Transition λpump (nm) Pump T (K) Output η (%) Ref.mechanism

Pr3+ 635 3P0 →3 F2 1010 + 835 ESA RT 180 mW 10 [11.227]

605 3P0 →3 H6 1010 + 835 ESA RT 30 mW 3.3

520 3P1 →3 H5 1010 + 835 ESA RT 1 mW

491 3P0 →3 H4 1010 + 835 ESA RT 1 mW

Pr3+ 635 3P0 →3 F2 1020 + 840 ESA RT 54 mW 14 [11.228]

520 3P1 →3 H5 1020 + 840 ESA RT 20 mW 5

491 3P0 →3 H4 1020 + 840 ESA RT 7 mW 1.5

Pr3+, Yb3+ 635 3P0 →3 F2 849 ETU RT 20 mW [11.229]

Pr3+, Yb3+ 635 3P0 →3 F2 1016 diode ETU RT 6.2 mW 3.2 [11.230]

+ 833 diode

532 3P1 →3 H5 ETU RT 0.7 mW 0.3

Pr3+, Yb3+ 635 3P0 →3 F2 860 diode ETU RT 4 mW 2.2 [11.231]

602 3P0 →3 H6 860 diode ETU 0.2 mW

Pr3+, Yb3+ 520 + 490 3P1 →3 H5 856 diode RT 1.4 mW [11.223]

+3P0 →3 H4

Pr3+, Yb3+ 492 3P0 →3 H4 1017 diode RT 1.2 mW 8.5 [11.232]

+ 835 diode

Pr3+, Yb3+ 635–637 3P0 →3 F2 780–880 PA RT 300 mW 16.8 [11.161]

605–622 3P0 →3 H6 780–880 PA RT 45 mW 4.6

517–540 3P1 →3 H5 780–880 PA RT 20 mW 5

491–493 3P0 →3 H4 780–880 PA RT 4 mW 1.2

Pr3+, Yb3+ 635 3P0 →3 F2 850 PA RT 1020 mW 19 [11.162]

Pr3+, Yb3+ 635 3P0 →3 F2 850 diode PA RT 440 mW 17 [11.233]

520 3P1 →3 H5 850 diode PA RT 100 mW ≈ 4

Pr3+, Yb3+ 635 3P0 →3 F2 850 diode PA RT 2 W 45 [11.234, 235]

520 3P1 →3 H5 850 diode PA RT 0.3 W 17

Pr3+, Yb3+ 491 3P0 →3 H4 850 PA RT 165 mW 12.1 [11.236]

491 + 520 3P0 →3 H4 850 PA RT 230 mW 14.3

+3P1 →3 H5

491 3P0 →3 H4 840 diode PA RT 8 mW ≈ 6

Pr3+, Yb3+ 635 3P0 →3 F2 838 diode PA RT ML: 550 ps [11.237]

(239 MHz)

Pr3+, Yb3+ 603 (tunable) 3P0 →3 H6 840, Ti:Sapphire PA RT 55 mW 19 [11.238]

634 3P0 →3 F2 100 mW

Pr3+ 635 3P0 →3 F2 480, diode Direct RT 94 mW 41.5 [11.226]

laser oscillation of a Pr3+-doped ZBLAN fiber under di-rect pumping with a blue semiconductor laser [11.226].Output powers of 94 mW and a slope efficiency of 41.5%at 635 nm were obtained.

In Table 11.10 the results to date with Pr3+-dopedZBLAN fibers are summarized. The best results ob-tained thus far, at the most efficient wavelength of635 nm, are output powers up to 2 W and slope effi-ciencies up to 45% [11.234]. The efficiencies in the

green and blue wavelength regions are about one orderof magnitude smaller.

Generally, a double-clad fiber can be used to couplehighly divergent pump radiation of near-infrared high-power laser diodes into the large-numerical-apertureinner cladding and propagate it along the fiber axis.On its way down the fiber, the pump radiation is gradu-ally absorbed in the embedded small-numerical-apertureactive core. Since the number of transverse radiation

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636 Part C Coherent and Incoherent Light Sources

+N0

2

.

! .

Fig. 11.47 Experimental setup of the Pr3+,Yb3+-doped up-conversion ZBLAN fiber laser [11.162]. M1 denotes thedichroic mirror, C1 an aspheric lens, M2 the dielectricresonator mirror, and C2 a collimator

modes supported in a cylindrical fiber core depends onlyon the wavelength of the radiation, the core diameter,and its numerical aperture, the geometry and mater-ial of the active fiber core can be chosen to supportjust one or a few transverse modes of the radiationgenerated in the active core by near-infrared pumping.Thus, an upconversion double-clad fiber with a suit-ably chosen rare-earth-ion-doped active core can beused to convert the highly multimode radiation of high-power near-infrared laser-diode arrays into single-modeor few-mode visible laser emission with excellent beamquality. The best results in terms of upconversion laseroutput power, however, have been achieved when thepump radiation of a high-beam-quality laser source waslaunched directly into the active core of the describedfiber. This is due to the high pump intensities requiredlocally for photon-avalanche upconversion, which can

B" 5G

5 G

$

$*

$%

$;

+N6P;-+N6P;- +N6P;%

$- $- $- $- * *$- - -$-

28G6G"(

Fig. 11.48 Input–output characteristics of the Pr3+,Yb3+-doped up-conversion ZBLAN fiber laser [11.162]

be best provided by diffraction-limited pump radia-tion focused onto the fiber end face and guided in therare-earth-ion-doped active core.

For example, Fig. 11.47 shows a demonstration ofhigh-power upconversion laser operation under pump-ing with two dichroic coupled Ti3+:Al2O3 laserstuned to pumping wavelengths near 850 nm [11.162].The maximum upconversion-laser output powerwas 1020 mW at an incident near-infrared total pumppower of 5.51 W (Fig. 11.48). The overall slope ef-ficiency with respect to the incident pump powerwas 19% [11.162].

Also, near-infrared high-power diode-laser barswith beam-shaping optics have been used to pumpa Pr3+,Yb3+-doped ZBLAN fiber with the commonsingle-clad structure and a large-area multimode activecore. In this case, 4.5 W of pump power were launchedinto the fiber core, generating red output powers in ex-cess of 2 W at 635 nm [11.234]. In the blue spectralrange, a Pr3+,Yb3+-doped ZBLAN-based upconver-sion fiber laser has been demonstrated at an emissionwavelength of 491 nm (3P0 → 3H4 transition of Pr3+).The pump source was a single-mode diode laser emit-ting at a wavelength of 840 nm. The maximum blueoutput power was 8 mW at an incident pump powerof 200 mW [11.236].

Other visible fiber lasers. Laser oscillation in the visiblespectral range has also been obtained with other rare-earth ions in ZBLAN. In Table 11.11 an overview ofthese laser systems is given. A detailed description ofthese systems and their prospects is given in [11.214].

11.2.3 Near-Infrared Rare Earth Lasers

Nd lasersmost intensively investigated class of solid-state lasersis based on Nd3+-ion-doped materials. The Nd3+-ionoffers various groups of laser lines in the near-IR spectralregion.

Energy-level scheme. Figure 11.49 shows transitionsfrom the 4F3/2 upper laser level into the 4I13/2, 4I11/2,and 4I9/2 manifolds. The specific absorption and emis-sion wavelengths depend on the crystal field, whichinfluences the splitting within a manifold and betweendifferent manifolds. As shown in Fig. 11.49 the splittingbetween the 4I (and also between 4F) manifolds is dom-inated by L S coupling of the 4f electrons and is onlyaffected to second order by covalency effects caused bythe crystal field. The energetic splitting between the 4F

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Lasers and Coherent Light Sources 11.2 Solid-State Lasers 637

Table 11.11 Overview of room temperature rare earth ion doped upconversion fiber lasers. (in): incident, (l) launched,(abs) absorbed

Active ion λlaser (nm) Transition λpump (nm) Pout (mW) η (%) Ref.

Nd 381 4D3/2 →4 I11/2 590 0.076 0.25 (l) [11.239]

Nd 412 2P3/2 →4 I11/2 590 0.470 1.7 (l) [11.240]

Tm 455 1D2 →3 F4 645 + 1064 3 1.5 [11.241]

Tm 480 1G4 →3 H4 1130 33 34.6 (abs) [11.242]

1123 230 25 (in) [11.243]

680 + ≈ 1100 14.8 18.9 (abs) [11.244–249]

Tm / Yb 480 1G4 →3 H4 1070 375 [11.250]

1065 106 6.6 (in) [11.251]

1120, 1140 116 15 [11.252]

Dy 478 4F9/2 →6 H15/2 457 (Ar+) 2.3 0.9 [11.253]

575 4F9/2 →6 H13/2 10 1.5

Er 540 4S3/2 →4 I15/2 801 23 16 [11.254, 255]

Er 540 4S3/2 →4 I15/2 970 50 51 [11.256–258]

Ho 544 / 549 5F4 →5 I8 / 5S2 →5 I8 ≈ 640 40 22.4 (l) [11.259–263]

Ho 753 647 0.54 3.3 [11.259]

Tm 810 1064 1200 37 [11.264]

and 4I levels is dominated by Coulomb interaction of the4f electrons, which is also only effected to second or-der by covalency effects of the crystal field. Therefore,the transition wavelengths of Nd-doped crystals vary ina certain range near the values given for Nd:Y3Al5O12(Nd:YAG) in Fig. 11.49. The strongest and most com-monly used laser transition 4F3/2 → 4I11/2 emits near1060 nm (Fig. 11.49).

Figures 11.50 and 11.51 show as an example the ab-sorption and emission spectra of Nd:YAG, measured at300 K. The strongest absorption is located near 808 nmand the strongest laser transition at 1064 nm.

Longitudinal and transversal diode pumping. Ndlasers are usually pumped by lamps or diode lasers.The first generation of Nd-doped solid-state lasers werepumped with CW krypton or pulsed xenon lamps.These lamps have high electrical-to-optical efficienciesof about 70% and are available at reasonable costs. Un-fortunately, the overlap of their emission spectra withthe narrow 4f–4f absorption spectra of Nd-doped solid-state laser materials is usually poor; thus the electrical tooptical pump efficiency is low – typically a few percent.

In modern Nd lasers, diode lasers at 808 nm are em-ployed as pump sources. Laser diodes typically haveefficiencies of about 50% and offer several advantagesover lamp pumping. Diodes emit spectrally narrow-bandradiation (≈ 1–5 nm). Therefore, a much better overlapwith the 4f–4f absorption spectra of Nd can be achieved.

Because of the good spectral match, the total pump effi-ciency of diode pumping is much higher than for lamppumping. In addition the transfer efficiency of pumplight into the crystal is much better in the case of diodepumping, so that a total efficiency of 10% to 30% forthe conversion of electrical power into laser power isachievable.

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49*B*! #4"

4:###69"#9

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*B)

*B)

%*(*%(;$-

Fig. 11.49 Level scheme, pump and laser transitions of Nd3+ inYAG

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638 Part C Coherent and Incoherent Light Sources

&;

G 6

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; ; ;* ;% ;; (

*

%

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%

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Fig. 11.50 Absorption spectrum of Nd(1%):YAG and absorptioncross sections

The selective spectral pumping with diodes alsoleads to reduced heat deposition in the laser crystal withreduced thermal lensing effects, giving rise to improvedbeam quality. Diffraction-limited beams can be realizedeasier.

However, there are also disadvantages of diodelasers. Because of their small active laser cross-sectional

;%

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Fig. 11.51 Emission cross section spectra of Nd:YAG

area and their relatively large refractive index, laserdiode beams have a large divergence. Their beams alsopossess large beam-quality differences in the transversaland sagittal directions since the active area dimensionsare usually ≈ 1 µm × 100 µm and confinement is givenonly in one dimension. For power scaling, bars or ar-rays of diodes must be used. This can lead to outputpowers of up to several tens of watts. Unfortunately,combination of many single diode stripes also broad-ens the emission spectrum, while the beam quality ofthe combined radiation is further decreased. So, sophis-ticated pump optics such as beam shaping must often beemployed [11.265].

Several pump geometries can be used with diodelaser excitation. The common geometry to pump lasers,which are capable of generating several watts of out-put power with excellent beam quality, is end pumping(Fig. 11.52). In this geometry, pump radiation is fo-cused into the laser crystal along the laser resonatoraxis. A very good overlap of pump and laser modes canbe provided. Gain media of small volume and with shortabsorption lengths can be used, and high population in-versions at low pump power levels can be achieved.Because most of the pump power is deposited within thevolume of the TEM00 laser mode, higher-order modesusually cannot oscillate. Excellent beam quality is there-fore inherent. However, scaling of end-pumped lasers tooutput powers above 20 W is difficult to achieve due to

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Lasers and Coherent Light Sources 11.2 Solid-State Lasers 639

possible thermally induced fracture of the laser crystalwithin the small volume of pump absorption.

An alternative diode pumping method is side pump-ing (Fig. 11.53, [11.266]). In this approach, the pumpinggeometry is principally similar to the arrangement forlamp pumping. One or several linear pump diode barsare positioned around the side surface of the laser crystal.Pump radiation is then usually imaged into the crystalperpendicular to the laser resonator axis. The crystal isrelatively homogeneously pumped throughout its totalvolume, and a smaller excitation density is usually real-ized than in diode laser end pumped systems. Because ofthe use of larger gain media, however, more energy canbe deposited into the laser crystal and therefore higheroutput powers can be achieved. Since a larger laser modeis present inside the crystal, higher-order modes alsousually oscillate in these lasers. Therefore beam qualityis often rather low.

Nd-doped fiber lasers can also be operated in specificconfigurations, see for example [11.267–278], and areused in many applications. For high-power laser gener-ation, double-clad fibers are usually used. Basic aspectsare described below in this chapter (page 650/651).

The most important Nd lasers. The most importantNd-doped laser materials feature relatively high emis-sion cross sections, a relatively long upper-state lifetimeof Nd3+, a high damage threshold, high mechanicaland chemical stability, good thermal conductivity andvery good optical quality, see Table 11.12. Many hostcrystals have been investigated, including yttrium alu-minium garnet (YAG), yttrium aluminium perovskite(YAP or YALO), yttrium lithium fluoride (YLF) andyttrium vanadate (YVO). Very efficient and compactdiode-pumped lasers with slope efficiencies of over60% have been demonstrated in many Nd-doped lasermaterials.

Nd:YAG is still regarded as the most important solid-state laser. Due to their very good optical and mechanicalproperties, diode-pumped Nd:YAG lasers are robust andreliable. They are in use in many applications. Withinthe last few years Nd:YAG have also become availableas ceramics with high optical quality.

The vanadates Nd:YVO and Nd:GVO emit polarizedradiation and exhibit large cross sections and gain.

Nd laser wavelength and materials. A variety of Ndlasers have been reported so far. Tables 11.13, 14, 15 listlaser materials and their laser wavelengths which havebeen observed at 300 K on the transitions 4F3/2 → 4I9/2,4I11/2 and 4I13/2. The tables also list Nd-doped ceram-

"

Fig. 11.52 Diode end-pumping scheme

ics, which have been developed with very good opticalquality and low scattering losses. So, the efficiency ofceramic lasers can be as high as for crystals.

If not specified otherwise, the references for the lasermaterials can be found in [11.213].

Yb lasersCoherent oscillation of Yb3+ was first observed in YAGat 77 K [11.279]. In 1991 the first realization of a diode-pumped Yb:YAG laser [11.280] at room temperatureinitialized an intensive renaissance of research on Yb-doped laser materials for laser diode pumping at 300 K.Yb3+-doped solid-state lasers feature several importantadvantages compared with other rare-earth lasers:

1. Yb3+ ions have only two states, the ground state2F7/2 and the excited state 2F5/2, which are separatedby an energy of about 10 000 cm−1. Thus, there isno excited-state absorption of the pump and laserradiation (Fig. 11.54).

2. The quantum efficiency of Yb3+ is close to unity.

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Fig. 11.53 Diode side (transversal) pumping scheme[11.266]

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640 Part C Coherent and Incoherent Light Sources

Table 11.12 Important host materials for Nd3+ [11.285]

Host YAG YAlO (FAP) YVO GVO YLF

Chemical formula Y3Al5O12 YAlO3 YVO4 GdVO4 YLiF4

Lattice symmetry Cubic Orthorhombic Tetragonal Tetragonal TetragonalSpace group 1a3d Pnma 141/amd 141/amd 141/aLattice constant (Å) 12.00 a = 5.33 a = 7.120 a = 7.123 a = 5.18

b = 7.37 c = 6.289 b = 6.291 c = 10.74c = 5.18

Density of Nd sites (1% doping) 1.39 1.96 1.255 1.25 1.39Heat conductivity (Wm−1K−1) 11–13 11 ∼ 5–12 a ∼ 8–12 a 6dni/dT (10−6 K−1) 9.9 14.5(a) 8.5(a) −0.9(a)

9.7(b) 3.0(c) −2.9(c)dL/dT (10−6 K−1) 8.2 4.4(a) 3.1(a) 1.6(a) 13(a)

10.8(b) 7.2(c) 7.3(c) 8(c)9.5(c)

Max. phonon energy (cm−1) 700 550 850 490Refractive indices n = 1.822 na = 1.9260 no = 1.958 no = 1.972 no = 1.454

nb = 1.9118 ne = 1.2168 ne = 1.2192 ne = 1.477nc = 1.9346

τ(4F3/2) (µs) 250 160 97 100 500λabs (nm) 808 813 (E ‖ a) 808 808 792σabs (10−20 cm2) 7.9 7.2 (E ‖ a) 60 (π) 54 (π) 14 (π)

12 (σ) 12(σ) 1.2(σ)λlaser(4I11/2) (nm) 1064 1080 (E ‖ a) 1064 1063 1047 (π)

1054 (σ)σem(4I11/2) (10−20 cm2) 29 25 (E ‖ a) 123 (π) 125 (π) 1.5 (π)

52 (σ) 61 (σ) 1.5 (σ)∆λ(4I11/2) (nm) 0.8 2.5 (E ‖ a) 1.0 (π) 1.2 (π) 1.5 (π)

1.5 (σ) 1.3 (σ) 1.5( σ)λlaser(4I9/2) (nm) 946 930 914 (π) 912 (π) 904 (π)

915 (σ) 912 (σ) 909 (σ)σem(4I9/2) (10−20 cm2) 3.9 4.1 (E ‖ a) 4.8 (π) 6.6 (π) 1.2 (π)

4.3(σ) 5.6 (σ) 1.3 (σ)∆λ (nm) 1.0 2.5 (E ‖ a) 2.8 (π) 2.5 (π) 3.0(π)

3.4 (σ) 3.3 (σ) 3.0 (σ)a The published data of heat conductivities of the vanadates YVO and GVO differ considerably [11.281–284]

3. Due to the small Stokes shift and the related smallquantum defect (typically 500 cm−1) heat generationin the lasing process of Yb3+ is small and makes ita suitable ion for high-average-power lasers.

4. The small ionic radius of Yb3+ compared to thatof other rare-earth ions favors its incorporation intoY-based host crystals such as YAG, allowing higherdopant concentration and thus shorter gain elementssuch as discs.

5. Yb3+ ions exhibit a relatively broad emission bandwhich leads to tunability and the generation of ultra-short pulses.

6. The radiative lifetime of the laser level ranges in dif-ferent crystals from a few hundred microseconds

to a few milliseconds, which implies greater en-ergy storage efficiency, especially for Q-switchedoperation with diode pumping.

One disadvantage of Yb3+-doped lasers is that theyoperate in a quasi-three-level scheme with temperature-dependent reabsorption at the laser wavelength. Thisleads to an increased threshold compared to a four-levelscheme, because the pump must bleach the reabsorptionlosses (see the section on basic spectroscopic propertiesin Sect. 11.2.1).

Figures 11.55 and 11.56 show as an examplethe absorption and emission cross section spectra ofYb:YAG [11.286]. When pumped near the absorption

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Table 11.13 Laser wavelengths of the 4F3/2 → 4I9/2 transi-tion at 300 K [11.213]

Wavelength (nm) Material

0.8910 Y3Al5O12

0.8999 Y3Al3O12

0.901 Sr1−xLaxMgxAl12−xO19 [11.287]

0.9106 Ba3LaNb3O12

0.910 LiLuF4 [11.285]

0.912 Y2SiO5

0.912 YVO4 [11.288]

0.912 GdVO4 [11.289–291]

0.914 YVO4 [11.292]

0.916 LuVO4 [11.293, 294]

0.930 YAlO3 [11.213, 295]

0.9312 YAlO3

0.936 Gd3Sc2Ga3O12

0.9385 Y3Al5O12

≈ 0.94 Y3Al5O12 [11.296]

0.941 CaY2Mg2Ge3O12

0.9458–0.9464 Y3Al5O12 [11.297]

0.946 Y3Al5O12 [11.213, 298–306]

0.946 Y3Al5O12 ceramic [11.307]

0.9660 YAlO3

0.966 Sc2O3 [11.308]

peaks at 940 nm or 970 nm, the Stokes shift to the lasingwavelength at 1030 nm corresponds to less than 10%,which allows in principle a slope efficiency of morethan 90%.

Yb3+ thin-disc lasers. In connection with the powerscaling of diode-pumped Yb lasers Giesen et al. [11.310]invented the thin-disc laser concept as a pump andresonator design for high-power lasers (Fig. 11.57).

The active medium is a thin circular disc, which iscoated for both the pump and laser wavelengths witha highly reflecting (HR) dielectric mirror on the rearside and with an antireflection coating on the front side.The disc is then bonded with the rear HR side to a heatsink. The resonator is formed by the coated crystal andan output coupler. The pump light is provided by fiber-coupled laser diodes which are focused onto the crystal.Since the thin crystal disc only absorbs a small fractionof the pump light during one pass through the crystal, thepump light is reflected back into the disc several timesby a parabolic mirror and the folding prism [11.311]. Upto a total of 32 pump light passes may be used.

Due to the geometrical setup a nearly one-dimensional thermal gradient along the laser axis can

Table 11.14 Laser wavelengths of the 4F3/2 → 4I11/2 tran-sition at 300 K [11.213]

Wavelength Material

0.97 (transition RbPb2Br5 [11.309]4F5/2,

2 H9/2 →4 I11/2)

1.0369 CaF2 −SrF2

1.0370 CaF2

1.0370–1.0395 SrF2

1.04065–1.0410 LaF3

1.0410 CeF3

1.0412 KYF4

1.042–1.075 Na0.4Y0.6F2.2

1.0445 SrF2

1.046–1.064 LiNdP4O12

1.0461 CaF2 −YF3

1.0461–1.0468 CaF2

1.047 LiGdF4

1.047 LiNdP4O12

1.047 LiYF4

1.047–1.078 NdP5O14

1.0471 LiYF4

1.0472 LiLuF4

1.0475 LaBGeO5

1.0477 Li(Nd,La)P4O12

1.0477 Li(Nd,Gd)P4O12

1.048 Li(Bi,Nd)P4O12

1.048 Li(Nd,La)P4O12

1.048 Li(Nd,Gd)P4O12

1.048 K5(Nd,Ce)Li2F10

1.0481 LiKYF5

1.0482 LaBGeO5

1.0482 NaLa(MoO4)2

1.0486 LaF3 −SrF2

1.049–1.077 NaNdP4O12

1.0491 SrAl12O19

1.0493 Sr2Y5Fl9

1.0495 BaY2F8

1.0495 GdF3 −CaF2

1.0497 SrAl2O4

1.0498 Ca2Y5F19

1.0498 SrAl12O19

1.05 KNdP4O12

1.05 NdP5O14

1.05 LaP5O14

1.05 LiLuF4

1.0500 CaF2 −ScF3

1.0505 (Nd,La)P5O14

1.0505 5NaF−9YF3

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Table 11.14 (continued)

Wavelength Material

1.0506 5NaF−9YF3

1.0507 CdF2 −ScF3

1.051 NaNdP4O12

1.051 YP5Ol4

1.051 (La,Nd)P5O14

1.051 CeP5O14

1.051 GdP5O14

1.051 NdP5O14

1.0511 (Nd,La)P5O14

1.0512 NdP5O14

1.0512 (Nd,La)P5O14

1.0512 (Y,Nd)P5O14

1.0513 NdP5O14

1.0515 YP5O14

1.0515 NdP5O14

1.052 (Nd,La)P5O14

1.052 KNdP4O12

1.052 K5NdLi2F10

1.052 Y3Al5O12

1.0521 BaF2 −YF3

1.0521 NdP5O14

1.0521 YF3

1.0525 YP5O14

1.0526 BaF2 −GdF3

1.0528 SrF2 −GdF3

1.0529 NdP5O14

1.053 LiYF4

1.053 (La,Nd)P5O14

1.053–1.062 Ca3(Nb,Ga)2Ga3O12

1.0530 LiYF4

1.0530 BaY2F8

1.0530 CaF2 −LuF3

1.0530–1.059 LaMgAl11O19

1.0531 LiLuF4

1.0532 LiKYF5

1.0534–1.0563 BaF2 −LaF3

1.0535 Lu3Al5O12

1.0535–1.0547 CaF2 −SrF2 −BaF2 −YF3 −LaF3

1.0537 BaF2 −CeF3

1.0539–1.0549 α−NaCaYF6

1.054 Gd3Ga5O12

1.054 LaAl11MgO19

1.054–1.086 LaAl11MgO19

1.0540 CaF2 −YF3

1.0540 BaF2

1.0540 CaF2 −YF3

Table 11.14 (continued)

Wavelength Material

1.0543 BaF2 −CeF3

1.0543 SrF2 −ScF3

1.05436 Ba2MgGe2O7

1.05437 Ba2ZnGe2O7

1.0547 LaMgAl11O19

1.05499 CsY2F7

1.055 Na3Nd(PO4)2

1.055 K3(La,Nd)(PO4)2

1.055 Na3(La,Nd)(PO4)2

1.0551 Pb5(PO4)3F

1.0552 LaMgAl11O19

1.0554 LiNdP4O12

1.0554 KY3F10

1.0555 CsGd2F7

1.0555 Ba5(PO4)3F

1.0556 SrF2 −LuF3

1.0560 SrF2 −LuF3

1.0566 La2Si2O7

1.0567 SrF2YF3

1.0569 NdGaGe2O7

1.0570 GdGaGe2O7

1.0572 LaSr2Ga11O20

1.0573 CaMoO4

1.0575 CsLa(WO4)2

1.0576 SrAl4O7

1.0576 SrMoO4

1.0576 La2Si2O7

1.058 Gd3Sc2Ga3O12

1.0582 Ca3Ga4O9

1.0582–1.0597 CaWO4

1.0583 Y3Sc2Ga3O12

1.0584 Y3Sc2Ga3O12 : Cr

1.0584 Y3Sc2Ga3O12

1.0584 CaY2Mg2Ge3O12

1.0585 YAlO3

1.0585 LiLa(MoO4)2

1.0585 Sr5(PO4)3F

1.0585 CaF2 −SrF2 −BaF2 −YF3 −LaF3

1.0585 KLa(MoO4)2

1.0586 Sr5(PO4)3F

1.0586 PbMoO4

1.0587 KLa(MoO4)2

1.0587 CaWO4

1.0588 Ca3(Nb,Ga)2Ga3O12

1.0589 SrF2 −CeF3 −GdF3

1.0589 Y3Ga5O12

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Table 11.14 (continued)

Wavelength Material

1.05896 CaMg2Y2Ge3O12

1.059 (La, Sr)(Al,Ta)O3

1.059 BaLaGa3O7

1.059 NaY(WO4)2

1.059 Na1+xMgxAl11−xO17

1.059 Na2Nd2Pb6(PO4)6Cl21.059 Sr5(PO4)3F [11.312]

1.0590 SrF2 −CeF3

1.0590 Gd3Ga5O12 waveguide [11.313]

1.0591 Gd3Ga5O12

1.0591 Lu3Sc2Al3O12

1.0591 LaGaGe2O7

1.0593 Sr4Ca(PO4)3F

1.0594 Lu3Ga5O12

1.0595 5NaF−9YF3

1.0595 Y3Sc2Al3O12

1.0595 NaLa(MoO4)2

1.0595 BaLaGa3O7

1.0596 CaAl4O7

1.0596 Ca3Ga2Ge3O12

1.0596 SrF2

1.0597 Ca3Ga2Ge3O12

1.0597–1.0583 SrF2 −LaF3

1.0597–1.0629 α−NaCaYF6

1.0599 Lu3Sc2Al3O12

1.0599 LiGd(MoO4)2

1.05995 Gd3Sc2Al3O12

≈ 1.06 (Gd,Ca)3(Ga,Mg,Zr)5O12 : Cr

≈ 1.06 CaGd4(SiO4)3O

≈ 1.06 Ca4YO(BO3)3 [11.314]

≈ 1.06 Y3Al5O12 [11.296]

≈ 1.06 Y3Sc1.0Al4.0O12 ceramic [11.315]

≈ 1.06 Gd3Sc2Al3O12

≈ 1.06 GdAl3(BO3)4 [11.316]

≈ 1.06 NaLa(MoO4)2

≈ 1.06 NaGd(WO4)2

≈ 1.06 NdAl3(BO3)4

≈ 1.06 YAl3(BO3)4 [11.213, 317]

≈ 1.06 Gd3Ga5O12

≈ 1.06 GdVO4 [11.213, 318–320]

≈ 1.06 YVO4 [11.319, 320]

≈ 1.06 La0.2Gd0.8VO4 [11.319, 321]

1.060 Gd3Ga5O12 : Cr

1.060 Ca4GdO(BO3)3

[11.213, 322–324]

1.060 Ca4YO(BO3)3 [11.325]

Table 11.14 (continued)

Wavelength Material

1.060 LaB3O6 [11.326]

1.0600 Gd3Ga5O12

1.0601 GdGaGe2O7

1.0603 Y3Ga5O12

1.0603 Gd2(WO4)3

1.0603–1.0632 CaF2 −YF3

1.0604 HfO2 −Y2O3

1.0604 NaLuGeO4

1.0605 SrF2 −ScF3

1.0606 Gd2(MoO4)3

1.0606 Gd3Ga5O12

1.0606 Gd3Ga5O12 waveguide [11.313]

1.0607 Sr3Ca2(PO4)3

1.0607 CaF2 −ScF3

1.0608 ZrO2 −Y2O3

1.0608 Nd3Ga5O12

1.0608 NaGaGe2O7

1.0609 Lu3Ga5O12

1.0609 NaYGeO4

1.061 Ca2Al2SiO7

1.061 BaGd2(MoO4)4

1.061 CaMoO4

1.061 YVO4 [11.327]

1.061 CaLa4(SiO4)3O

1.0610 Ca2Ga2SiO7

1.0610 7La2O3 −9SiO2

1.0610–1.0627 Y3Al5O12

1.0612 Gd3Sc2Ga3O12

1.0612 CaLa4(SiO4)3O

1.0612 Ca3(Nb,Ga)2Ga3O12

1.0613 Ca4La(PO4)3O

1.0613 Ba2NaNb5O15

1.0613 Gd3Sc2Ga3O12

1.0615 Ca(NbO3)2

1.0615 Y3Al5O12

1.0615 Lu3Al5OI21.0615 Y3Sc2Ga3O12

1.0615 Ba0.25Mg2.75Y2Ge3O12

1.0615 NaGdGeO4

1.0615 Y3Al5O12

1.0615–1.0625 Ca(NbO3)2

1.0618 Sr2Ca3(PO4)3F

1.0618 SrAl12O19

1.0618 CaF2 −ScF3

1.0618 LaNbO4

1.062 LaSc3(BO3)4 [11.213, 328]

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Table 11.14 (continued)

Wavelength Material

1.0620 Gd3Sc2Al3O12

1.0620 Lu3Sc2Al3O12

1.0621 SrAl12O19

1.0621 Gd3Ga5O12

1.0622 Y3Sc2Al3O12

1.0623–1.10585 CaF2 −SrF2 −BaF2 −YF3 −LaF3

1.0623 Lu3Ga5O12

1.0623 CaF2 −LuF3

1.0623–1.0628 CaF2

1.0624 LaNbO4

1.0625 Y3Ga5O12

1.0625 YVO4

1.0628 SrWO4

1.0629 Ca5(PO4)3F

1.0629 α−NaCaYF6

1.0629 Bi4Si3O12

1.0629–1.0656 CdF2 −YF3

1.063 GdVO4 [11.329]

1.063 Gd3Sc2Ga3O12 : Cr

1.063 SrWO4

1.063 Na5(Nd,La)(WO4)4

1.063 NdAl3(BO3)4

1.063 (La,Nd)P5O14

1.063 NdAl3(BO3)4

1.0630 Ca5(PO4)3F

1.0632 CaF2 −YF3 −NdF3

1.0632 CaF2 −YF3

1.0633–1.0653 α−NaCaCeF6

1.06335–1.0638 LaF3

1.0634 YVO4

1.0635 LaF3 −SrF2

1.0635 NdAl3(BO3)4

1.0635 NaLa(WO4)2

1.0635 (Nd,Gd)Al3(BO3)4

1.0635 Bi4(Si,Ge)3O12

1.0635 CaF2 −LuF3

1.0637–1.0670 Y3Al5O12

1.06375–1.0672 Lu3Al5O12

1.0638 CeF3

1.0638 CaAl4O7

1.0638 NaBi(WO4)2

1.0638 NdP5O14

1.0638 Ca3Ga2Ge3O12

1.0638–1.0644 (Y,Ce)3Al5O12

1.0639 Ca3Ga2Ge3O12

1.064 Y3Al5O12 [11.213, 301, 330–332]

Table 11.14 (continued)

Wavelength Material

1.064 Y3Al5O12 ceramic [11.315, 333–346]

1.064 Y3Al5O12 : Fe

1.064 Y3Al5O12 : Ti

1.064 Y3Al5O12 : Cr,Ce

1.064 Y3Al5O12 : Ho

1.064 Y3Al5O12 : Er

1.064 YVO4 [11.213, 347–350]

1.064 YVO4 single crystal fiber [11.351]

1.064 LaF3

1.064 KGd(WO4)2 [11.352, 353]

1.064 La3Ga5.5Ta0.5O14 [11.354]

1.0640 La3Ga5SiO14

1.0640–1.0657 CaF2 −CeF3

1.06405–1.0654 YAlO3

1.0641 Y3Al5O12 : Cr

1.0641 YVO4

1.0641 La3Ga5.5Ta0.5O14

1.06415 Y3Al5O12

1.06415 Y3Al5O12 [11.355]

1.0642 Ca3Ga2Ge3O12

1.0642 NaBi(WO4)2

1.06425 Lu3Al5O12

1.0643 SrMO4

1.0644 Bi4Ge3O12

1.0645 CaF2 −LaF3

1.0645 La3Ga5.5Nb0.5O14

1.0645 La3Ga5SiO14

1.0645 YAlO3

1.0645 YAlO : Cr

1.0646 Y3Al5O12

1.0646 KLa(MoO4)2

1.0647 CeCl31.0648 YVO4

1.0649 CaY2Mg2Ge3O12

1.065 GdVO4

1.065 (Nd,Gd)Al3(BO3)4

1.065 Sr5(VO4)3Cl

1.065 Sr5(VO4)3F

1.0650 La3Ga5GeO14

1.0650 RbNd(WO4)2

1.0652 CaWO4

1.0652 CdF2 −LuF3

1.0653–1.0633 α−NaCaCeF6

1.0653 NaLa(MoO4)2

1.0653–1.0665 NaLa(MoO4)2

1.0654 CaF2 −GdF3

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Lasers and Coherent Light Sources 11.2 Solid-State Lasers 645

Table 11.14 (continued)

Wavelength Material

1.0654 NdGaGe2O7

1.0656 CdF2 −YF3

1.0657–1.0640 CaF2 −CeF3

1.0657 CaF2

1.0658 LiLa(MoO4)2

1.0658 CsNd(MoO4)2

1.0658 LuVO4 [11.356]

1.0659 GdGaGe2O7

1.066 Nd(Ga,Cr)3(BO3)4

1.066 K5Nd(MoO4)4

1.066 K5Bi(MoO4)4

1.0661 CaF2

1.0664–1.0672 YVO4

≈ 1.0665 CdF2 −LaF3

1.0666 CdF2

1.0667 CdF2 −GeF3

1.0667 NaGd(MoO4)2

1.0668 CdF2 −LaF3

1.0669 KY(MoO4)2

1.067 Ca3(VO4)2

1.0670 La3Ga5SiO14

1.0672 CaY4(SiO4)3O

1.0672 CdF2 −GdF3

1.0672 KGd(WO4)2

1.0672 La3Ga5SiO14

1.0673 GaMoO4

1.0673 La3Ga5SiO14

1.0674 NaY(MoO4)2

1.0675 LuAlO3

1.0675 Na2Nd2Pb6(PO4)6Cl21.0675 Nd3Ga5SiO14

1.0675 Nd3Ga5GeO14

1.068 Na2Nd2Pb6(PO4)6Cl21.0680 Nd3Ga5GeO14

1.0682 Y3Al5O12

1.0687–1.0690 KY(WO4)2

1.0688 Ga3Ga2SiO7

1.0688 Ca2Ga2Ge4O14

1.0688 KY(WO4)2

1.0689 NdGaGe2O7

1.0690 GdAlO3

1.0690 Ca3Ga2Ge4O14

1.0694 Sr3Ga2Ge4O14

1.0698 La2Be2O5

1.07 KPb2Br5 [11.309]

1.07 RbPb2Br5 [11.309]

Table 11.14 (continued)

Wavelength Material

1.070 La2Be2O5

1.0701 Gd2(MoO4)3

1.0701–1.0706 KLu(WO4)2

1.0706 KY(WO4)2

1.0706 LaSr2Ga11O20

1.0711 Y2SiO5

1.0714 KLu(WO4)2

1.0714–1.0716 KLu(WO4)2

1.0715 Y2SiO5

1.0716–1.0721 KLu(WO4)2

1.0720 CaSc2O4

1.0721 KLu(WO4)2

1.07255–1.0730 YAlO3

1.0726 YAlO3

1.0729 YAlO3

1.073 KLu(WO4)2 [11.357]

1.0737 Y3Al5O12

≈ 1.074 Y2O3 −ThO2 −Nd2O3

1.074 SrAl12O19

1.0741 Gd2O3

1.0741 Y2SiO5

1.0742 Y2SiO5

1.0746 Y2O3

1.0746 Y2O3 ceramic [11.358]

1.075 La2O2S

1.0757 Sr3Ga2Ge4O14

1.0759 LuAlO3

1.0759 Lu2O3 ceramic [11.359]

1.0760 GdAlO3

1.0775–1.0845 CaYAlO4

1.0780 Y3Al5O12

1.0780–1.086 LaMgAl11O19

1.0782 Y2SiO5

1.0782–1.0815 YAlO3

1.0785 LuScO3

1.0786 CaAl4O7

1.0786 Y2O3 ceramic [11.360]

1.0788 Ca2Ga2SiO7

1.0789 Gd2O3

1.079 La2Be2O5

1.0790 La2Be2O5

1.0790 Lu2SiO5

1.07925 Lu2SiO5

1.0795 YAlO3

1.0795–1.0802 YAlO3

1.0796 YAlO3

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646 Part C Coherent and Incoherent Light Sources

Table 11.14 (continued)

Wavelength Material

1.08 Y2O3

1.08 YAlO3 [11.360]

1.080 Lu2O3 ceramic [11.359]

1.0804 LaAlO3

1.0806 CaYAlO4

1.0812 Sc2SiO5

1.08145 Sc2SiO5

1.0817 LaMgAl11O19

1.0824 LaMgAl11O19

1.082 Sc2O3 [11.308]

1.082–1.084 LaMgAl11O19

1.0828 SrAl4O7

1.0829–1.0859 LiNbO3

1.083 YAlO3

1.0832 LuAlO3

1.0832–1.0855 YAlO3

1.0843 YScO3

1.0845 YAlO3

1.0846 LiNbO3

1.085 LiNbO3 : Mg

1.08515 GdScO3

1.0868 CaSc2O4

≈ 1.0885 CaF2 −CeO2

1.0885–1.0889 CaF2

1.0909 YAlO3

1.091 Ca4GdO(BO3)3 [11.322]

1.0921 YAlO3

1.0922–1.0933 LiNbO3

1.093 LiNbO3

1.0933 LiNbO3

≈ 1.094 LiNbO3 : MgO

1.0989 YAlO3

1.1054 Y3Al5O12

1.1119 Y3Al5O12

1.1158 Y3Al5O12

1.1225 Y3Al5O12

be reached within the crystal. This setup minimizes theformation of thermal lenses, and therefore yields bet-ter beam quality at high powers compared to a rod laser.Multiple passes of the pump radiation through the crystalincrease the absorption efficiency and also the effectivepump power density in the crystal. Therefore, the thindisc design is suitable for quasi-three-level systems suchas Yb3+. Figure 11.58 shows the input versus outputpower of a thin-disc Yb:YAG laser [11.311]. CW output

Table 11.15 Laser wavelengths of the 4F3/2 → 4I13/2 tran-sition at 300 K [11.213]

Wavelength Material

1.18 (transition RbPb2Br5 [11.309]4F5/2,

2 H9/2 →4 I13/2)

1.3 KNdP4O12

1.3 NdAl3(BO3)4

1.302 KYF4

1.304–1.372 (La,Nd)P5O14

1.3065 SrAl12O19

1.307 KYF4

1.3070 5NaF−9YF3

1.311–1.334 NaNdP4O12

1.313 LiYF4

1.3133 LiLuF4

1.3150 Ca3Ga2Ge3O12

1.316–1.340 LiNdP4O12

1.317 Li(La,Nd)P4O12

1.317 LiNdP4O12

1.3170 CeF3

1.3175 BaF2

1.318 Y3Al5O12

1.318 BaY2F8

1.3185 CaF2 −GdF3

1.3185 BaF2 −LaF3

1.3185 KY3F10

1.3187 Y3Al5O12

1.3188 Y3Al5O12

1.319 Y3Al5O12 [11.330, 361]

1.319 LiNdP4O12

1.319 Y3Ga5O12

1.319 Y3Al5O12 ceramic [11.362]

1.319–1.325 (Y,Nd)P5O14

1.3190 Ca2Y5F19

1.3190 CaF2 −LaF3

1.3190 CaF2 −CeF3

1.3190 Sr2Y5F19

1.3190 α−NaCaCeF6

≈ 1.32 Y3Al5O12 ceramic [11.334]

1.32 Gd3Sc2Ga3O12 : Cr

1.32 NdP5O14

1.32 (La,Nd)P5O14

1.32 K(Nd,Gd)P4O12

1.32 YLiF4 [11.363]

1.320 NaNdP4O12

1.3200 SrF2 −LuF3

1.3200 BaF2 −YF3

1.3200 Y3Al5O12

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Lasers and Coherent Light Sources 11.2 Solid-State Lasers 647

Table 11.15 (continued)

Wavelength Material

1.3200 YP5O14

1.3208 LiLuF4

1.3209 Lu3Al5O12

1.3209 Ba5(PO4)3F

1.3212 LiYF4

1.3225 CaF2

1.323 NdP5O14

1.323 (La,Nd)P5O14

1.324 (La,Nd)P5O14

1.3245 CdF2 −YF3

1.3250 SrF2 −LaF3

1.3250 SrF2

1.3255 SrF2 −CeF3

1.3260 SrF2 −GdF3

1.3270 CaF2 −YF3

1.3270 BaF2

1.3270 Ca3(Nb,Ga)2Ga3O12

1.328 Sr5(PO4)3F [11.213, 364]

1.3280 BaF2 −LaF3

1.3285 α−NaCaYF6

1.3285 SrF2 −ScF3

1.3298 CsLa(WO4)2

1.3298 GdGaGe2O7

1.330 CaF2 −LuF3

1.3300 Gd3Ga5O12

1.3300 CdF2 −ScF3

1.3303 NdGaGe2O7

1.3305 HfO2 −Y2O3

1.3305 Y3Ga5O12

1.3310 LaF3

1.3310 Y3Sc2Ga3O12

1.3310 NaLuGeO4

1.3315 LaF3 −SrF2

1.3315 Lu3Ga5O12

1.3315 Gd3Ga5O12

1.3315 Ca3Ga2Ge3O12

1.3317 Ca3Ga2Ge3O12

1.3320 CeF3

1.3320 ZrO2 −Y2O3

1.3320 Ca3Ga4O9

1.3325 SrMoO4

1.3325 NaYGeO4

1.3326 Lu3Al5O12

1.3334 NaGdGeO4

1.3338 Y3Al5O12

1.3340 CaWO4

Table 11.15 (continued)

Wavelength Material

1.3340 PbMoO4

1.3342 KLa(MoO4)2

1.3342 Lu3Al5O12

1.3342 NaBi(WO4)2

1.3345 SrAl4O7

1.3347 Ca5(PO4)3F

1.3347 SrWO4

1.3350 KLa(MoO4)2

1.3350 Y3Al5O12

1.3354 CaLa4(SiO4)3O

1.3355 NaLa(WO4)2

1.3360 Y3Sc2Al3O12

1.3360 Gd3Sc2Al3O12

1.3360 Lu3Sc2Al3O12

13360 Y3Sc2Al3O12

1.3360 CdF2 −CeF3

1.3365 Ca2Ga2SiO7

1.3365 CdF2 −GaF3

1.3365 CdF2 −LaF3

1.3370 CaF2 −YF3

1.3370 CaWO4

1.3370 Gd3Ga5O12

1.3370 LiLa(MoO4)2

1.3375 α−NaCaYF6

1.338 Y3Al5O12 [11.365]

1.338 Y3Ga5O12

1.338 Sr5(PO4)3F [11.366]

1.3380 Ca(NbO3)2

1.3380 NaLa(MoO4)2

1.3381 Y3Al5O12

1.3382 Y3Al5O12

1.3385 NaGd(MoO4)2

1.3387 Lu3Al5O12

1.339 YF3

1.3390 CaWO4

≈ 1.34 Y3Al5O12 [11.296, 367]

≈ 1.34 YAlO3 [11.360]

≈ 1.34 GdVO4

[11.213, 319, 320, 368–372]

≈ 1.34 YVO4 [11.213, 319, 320]

≈ 1.34 La0.2Gd0.8VO4 [11.319, 321]

≈ 1.34 LuVO4 [11.373]

≈ 1.34 GdAl3(BO3)4 [11.316]

1.3400 LiGd(MoO4)2

1.3400 YAlO3

1.3407 Bi4Si3O12

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648 Part C Coherent and Incoherent Light Sources

Table 11.15 (continued)

Wavelength Material

1.341 NdAl3(BO3)4

1.3410 Y3Al5O12

1.3410 Lu3Al5O12

1.3410 YAlO3

1.3413 YAlO3

1.3414 YAlO3 [11.213, 374, 375]

1.3416 YAlO3

1.3418 Bi4Ge3O12

1.342 YVO4 [11.347, 376–383]

1.3420 CaAl4O7

1.3425 Ca(NbO3)2

1.3425 YVO4

1.3425 PbMoO4

1.3425 CdF2 −YF3

1.3437 LuAlO3

1.3440 NaLa(MoO4)2

1.345 NdAl3(BO3)4

1.3475 CaWO4

1.3482 KLu(WO4)2

1.3485 KY(MoO4)2

1.3493 Ca3Ga2Ge4O14

1.35 KGd(WO4)2 [11.384, 385]

1.3500 CdF2 −LuF3

1.3505 CaF2 −ScF3

1.351 La2Be2O5

1.351 KGd(WO4)2 [11.353]

1.3510 KGd(WO4)2

1.3510 La2Be2O5

1.3510 Sr3Ga2Ge4O14

1.3512 YAlO3

1.3514 YAlO3

1.3520 CdF2 −GdF3

1.3525 Ca2Y5F19

1.3525 KY(WO4)2

1.3525 Lu3Al5O12

1.3532 Lu3Al5O12

1.3533 Y3Al5O12

1.3533 KLa(MoO4)2

1.3533 KLu(WO4)2

1.354 La2Be2O5

1.3545 KY(WO4)2

1.3550 LiNbO3

1.3550 KLu(WO4)2

1.3565 CaSc2O4

1.3572 Y3Al5O12

≈ 1.358 Y2O3

Table 11.15 (continued)

Wavelength Material

1.3585 CaF2 −YF3

1.3585 Y2SiO5

1.3585 Lu2SiO5

1.3595 LaF3

1.3600 α−NaCaYF6

1.3628 LaSr2Ga11O20

1.3630 KLa(MoO4)2

1.3630–2 Sc2SiO5

1.365 La2Be2O5

1.3657 KLa(MoO4)2

1.3665 SrAl4O7

1.3675 LaF3

1.3680 SrAl4O7

1.3690 CeF3

1.3707 La3Ga5.5Nb0.5O14

1.3710 CaAl4O7

1.3730 La3Ga5GeO14

1.3730 La3Ga5SiO14

1.3730 La3Ga5.5Ta0.5O14

1.3745 LiNbO3

1.3760 LaMgAll1O19

1.386 YVO4 [11.377, 386]

1.3868 LaBGeO5

1.3870 LiNbO3

1.3885 CaWO4

1.4150 Y3Al5O12

1.430 YAlO3 [11.387]

1.44 SrGd4(SiO4)3O

1.4444 Y3Al5O12 [11.213, 387]

1.486 Sc2O3 [11.308]

powers in the kilowatt range from one disc are possible.Further power scaling with several discs in the cavity isalso possible [11.311].

Yb fiber laser. Fiber lasers present another approach forthe generation of high powers with diffraction-limitedbeam quality. Since the introduction of the double-cladfiber more than two decades ago and with the recenttechnological advances in the fields of fiber fabricationand beam-shaped high-power diode lasers, the perfor-mance of diode-pumped fiber lasers has dramaticallyimproved. Today, fiber lasers can compete with theircorresponding bulk crystalline systems in certain appli-cations, especially when fundamental-transverse-modecontinuous-wave (CW) laser operation at output powersin the milliwatt to kilowatt range is required.

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Lasers and Coherent Light Sources 11.2 Solid-State Lasers 649

Basic aspects of fiber lasers. The invention of thedouble-clad fiber geometry and the holey fiber concepthas accelerated the scaling of the output power and hencethe success of high-power Er, Nd and Yb fiber lasers.For more-detailed reading a comprehensive introductionto the field of rare-earth-doped fiber lasers can be foundin [11.388].

The choice of the fiber material involves a numberof considerations: The maximum phonon energy, envi-ronmental durability, the drawability and the rare-earthsolubility. The maximum phonon energy of the glasssets the overall infrared transparency range of the fiberand the multiphonon relaxation rates that influence thequantum efficiency of radiative electronic transitions bynonradiative decay. The important physical propertiesof the popular glasses used for optical fibers are shownin Table 11.16.

• Silicate glass. This glass is the most important ma-terial used for optical fiber production [11.388,389].However, the maximum phonon energy is high(≈ 1100 cm−1) and has so far limited the emissionwavelength for infrared fiber lasers using this mater-ial to approximately 2.2 µm [11.390]. Silica is robustand fibers fabricated from this material involve thevery effective modified chemical vapor deposition(MCVD) technique. Reducing the OH− content inthe glass, which has two main absorption peaksin the range 1.3–2.0 µm, reduces the backgroundabsorption of fibers [11.391].• Fluoride glass. These glasses, especially theheavy-metal fluorides [11.392, 393], are usedas host materials for mid-infrared fiber lasers.The most widespread fluoride fiber material isZBLAN [11.394] with a mixture of 53 mol % ZrF4,20 mol % BaF2, 4 mol % LaF3, 3 mol % AlF3, and20 mol % NaF. Since it can be readily drawn intosingle-mode optical fiber [11.395] it is particu-larly important to mid-infrared fiber lasers [11.396]and allows for high infrared transparency up to≈ 6 µm. Nonradiative decay by multiphonon relax-ation, however, becomes significant for transitions atwavelengths longer than ≈ 3 µm. In addition to mid-infrared applications, ZBLAN is mostly also usedfor upconversion fiber lasers which need metastableintermediate pump levels with low multiphonon re-laxation rates. An overview of the spectroscopicproperties of rare-earth ions doped into ZBLAN isgiven in [11.397].• Chalcogenide glasses. Chalcogenides are composedof the chalcogen elements S, Se and Te [11.398,

!-)% !&)

!-)

4)

!**96

!&);

@4/@ @"

Fig. 11.54 Level scheme of Yb3+: free-ion state (one 4f hole 4f1h),

electron–electron interaction (Hee), splitting by spin–orbit inter-action HSB, and crystal field Hc. Pump and laser transitions areindicated by arrows

399]. When rare-earth ions are doped into theseglasses [11.400], the radiative transition probabili-ties, and therefore the absorption and emission crosssections, are high as a result of the high (≈ 2.6) re-fractive index and the high degree of covalency ofthe glass. Small phonon energies of 300–450 cm−1

produce low rates of multiphonon relaxation formid-infrared transitions. The low thermal conduc-tivity (Table 11.16) is, however, an important factor

;-

G 6

'"

DND00A

-

( (- -

Fig. 11.55 Absorption spectrum of Yb:YAG at 300 K

(

G 6

'"

DND0,A

(- -

*

Fig. 11.56 Emission spectrum of Yb:YAG at 300 K

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650 Part C Coherent and Incoherent Light Sources

+6"9"

9 @:

4" 4"

4"!

4"

"

+6"566:

.

.#9"

"

*

-

;

%

&

Fig. 11.57 (a) Thin-disc laser setup, (b) multiple-pass pump opticsfor 16 passes

to be considered in the design of chalcogenide-based lasers. So far, the most important glassesare the sulfide glasses GaLaS (GLS) [11.401] andGeGaS [11.402] because of the reasonably highrare-earth solubility.

Fiber, pump and resonator geometries. As bulk lasers,fiber lasers can be operated continuous wave, pulsed(including Q-switching) and mode-locked. These op-eration modes have been investigated intensively forthe common laser transitions near 1 µm in Nd3+ andYb3+, and near 1.5 µm in Er3+. However, the smallfiber diameter limits the peak power by the dam-age threshold intensity and, hence, crystalline lasersin bulk geometries are mostly preferred when high-energy short pulses are needed. In an analogous wayto the optical excitation of bulk gain media (seethe section on longitudinal and transversal pump-ing in this Sect. 11.2.2), doped optical fibers can beeither end pumped (core pumped) or side pumped

Table 11.16 Properties of popular fiber materials

Fiber material Max. phonon energy Infrared transparency Propagation losses Thermal conductivity(cm−1) (µm) (λ at minimum) (dB/km) (W/Km)

Silica 1100 [11.403] < 2.5 0.2 (1.55 µm) 1.38 [11.404]

ZBLAN 550 [11.397] < 6.0 0.05 (2.55 µm) 0.7–0.8 [11.405]

GaLaS 425 [11.406] < 8.0 0.5 (3.50 µm) 0.43–0.5 [11.407]

G

G

*

-

%

&

*

-

%

* % ; *

"$-

Fig. 11.58 Output power and optical efficiency for a thin-disc Yb:YAG laser using a single disc. (Pp = pump power,Pl = laser output power, ηopt = optical-optica efficiency)

(cladding pumped). The former method is less scalablesince it relies on the use of expensive high-beam-quality pump sources because core areas are usually< 100 µm2. On the other hand, the larger claddingarea (> 104 µm2) allows for high-power diode-arraypumping [11.408–411]. We will here describe thecladding-pumping technique, which is one of themost important developments in high-power fiber-lasertechnology.

• Fiber designs for cladding pumping. In the de-sign of cladding pumping, the core of the fiber isgenerally made to guide a single transverse LP01mode. The shape of the multimode pump cladding(Fig. 11.59), however, can be shaped with a num-ber of geometries. The pump cladding, which inturn is surrounded by a low-refractive-index trans-parent polymer or glass, provides a high numericalaperture (NA) of typically 0.3–0.55. Photonic crys-tal structures can be also used to improve claddingpumping [11.412] for the operation of fiber lasersat the multi-hundred-watt level. There are threemain double-clad-fiber layouts: circular, circularwith offset core, and rectangular, as shown schemat-

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Lasers and Coherent Light Sources 11.2 Solid-State Lasers 651

ically in Fig. 11.59. In the case of circular pumpcladding [11.53] a portion of the launched pumplight is skew to the fiber axis and produces an in-ner pump beam caustic that never crosses the core.Asymmetric configurations significantly improvethe pump beam absorption in the core [11.413,414].Double-clad pump schemes have been demonstratedwith holey or photonic crystal fibers [11.415]. Insuch fibers single-mode guiding and very large modeareas are possible [11.416].• Fiber-laser resonators. Typical free-running fiber-laser resonators are shown schematically inFig. 11.60. In the simplest resonator (Fig. 11.60a),the pump light passes through a dichroic mirror thatis highly reflective for the oscillating laser light.Fresnel reflection at the cleaved output end facetof the fiber can provide sufficient feedback for laseroscillation, although with an output-coupler mirrorat the output end of the fiber the optical efficiencycan be maximized. In an alternative arrangement, thepump light can be launched into the output end of thefiber (Fig. 11.60b). In order to scale the output power,each end of the fiber can be pumped (Fig. 11.60c).Due to its geometry, the fiber provides potentiallyhigh pump- and signal-beam intensities without the

Table 11.17 Material and laser parameters of Yb:YAG

Growth method Czochralski [11.417]

Temperature (C) 1930 [11.417]

Crucible Ir (Re) [11.417]

Yb-distribution coefficient 1.0 [11.417]

Max. doping level (%) ≤ 100 [11.417]

Structure Cubic [11.417]

Space group Ia3d-O10h [11.417]

Heat conductivity (Wm−1K−1)

Undoped YAG 11.0 [11.417]

Yb-doped (5%) 6.8 [11.417]

RE-density (1021 cm−3) 14 [11.417]

λlaser (nm) 1030 [11.417]

1050

σem (cm2) at 1030 nm 19 × 10−21 [11.286]

20 × 10−21 [11.418]

21 × 10−21 [11.419]

σem (cm2) at 1050 nm 3 × 10−21 [11.286]

σabs (cm2) at 969 nm 8.3 × 10−21 [11.286]

7.7 × 10−21 [11.419]

σabs (cm2) at 941 nm 8.2 × 10−21 [11.286]

τrad (µs) 1040 [11.417]

951 [11.420]

J":

.

"

"

Fig. 11.59a–d Principal double-clad fiber geometries which in-clude (a) circular-shaped pump cladding with axially positionedcore, (b) circular-shaped pump cladding with off-axially positionedcore, (c) rectangular-shaped pump cladding and (d) D-shaped pumpcladding

drawbacks of significant thermal and thermo-opticaleffects. Its large surface-area-to-volume ratio meansthat the heat generated in the core is dissipated ef-fectively by radiation and convection from the outersurface of the fiber.

High-power Yb fiber lasers. Yb-doped fiber-lasershave been operated at CW and average output pow-ers in excess of 10 W [11.421–426], in excess of100 W [11.427–433] and above 1 kW [11.434]. Also,a 30 µm Yb-doped large-mode-area fiber has been usedfor the amplification of Q-switched Nd:YAG pulses atrepetition rates between 3 and 50 kHz with average out-put powers up to 100 W. Pulse energies as high as

!

!

!

Fig. 11.60a–c Schematic diagram of resonators used forfree-running fiber lasers with (a) a single-end co-propagating pump, (b) a single-end counter-propagatingpump and (c) dual-end pumps. M represents a mirror

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652 Part C Coherent and Incoherent Light Sources

4 mJ with diffraction-limited beam quality have beenobtained in this case [11.435].

Yb3+ crystalline lasers. The properties and laser pa-rameters of Yb:YAG are listed in Table 11.17. In themeantime, besides Yb:YAG, a lot of Yb-doped lasermaterials have been explored and tested in CW, Q-switched and mode-locked operation. Table 11.18 showsan overview of Yb-doped laser crystals.

Er lasers at 1.5 µm (4I13/2 → 4I15/2)For many years Er3+-doped materials have beenwidely investigated for laser applications in the spec-tral range around 1.6 µm [11.436]. This laser transition(4I13/2 → 4I15/2, see Er3+ energy-level scheme depictedin Fig. 11.61) is used for eye-safe lasers for medicine,telecommunication, remote sensing and light detectingand ranging (LIDAR). Suitable erbium-doped mater-ials for the 1.6 µm laser transition should be at firstcharacterized by a high phonon energy, which en-ables fast depopulation of the pump level 4I11/2 vianonradiative decay in order to prevent excited-stateabsorption (ESA, 4I11/2 → 2H11/2,

4S3/2) and upcon-version

[UC1, (4I11/2,

4I11/2) → (4I15/2,4F7/2)

]from

the 4I11/2 level and to populate efficiently the 4I13/2upper laser level (Fig. 11.61). The second importantcondition is that the ESA transition 4I13/2 → 4I9/2should not spectrally overlap with the range ofstimulated emission around 1.6 µm and that the upcon-version process

[UC2, (4I13/2,

4I13/2) → (4I9/2,4I15/2)

]is weak. Additionally, a significant splitting of theground-state multiplet is advantageous to achievea quasi-four-level system. These conditions are bestfulfilled by Er3+-doped glasses and fibers, whichare to date the most efficient lasers at this tran-sition. However, glasses suffer from poor thermaland mechanical stability, thus Er3+-doped crys-talline matrices are still being intensively investigatedin order to find suitable crystals for this lasertransition.

For most applications of Er3+-doped laser mater-ials, laser diodes operating in the wavelength rangearound 975 nm (the 4I15/2 → 4I11/2 transition) are usedas a pump source, thus enabling all-solid-state laser sys-tems. With Ti:sapphire pump lasers in general betterlaser results are obtained, however, the overall effi-ciency is low. Another possible pump wavelength liesaround 1.5 µm, i. e., directly into the upper laser mul-tiplet. However, for this wavelength region high-powerlaser diodes are not yet available, therefore usually Er–glass lasers as pump sources are used. In any case,

the Er absorption at these wavelengths is rather small,because the absorption cross sections are in the or-der of 1 to 2 × 10−20 cm2 (Figs. 11.62, 63). Dopinglevels higher than 1 to 2% are critical, because ofincreased reabsorption losses and higher rates for en-ergy transfer processes, which depopulates the upperlaser level (4I13/2,

4I13/2) → (4I15/2,4I9/2). Therefore,

to keep the Er3+ concentration low but achieve higherabsorption, the usual approach is to codope the Er3+-doped laser material with Yb3+, which can be veryefficiently pumped around 975–980 nm. Then, the en-ergy transfer process (2F5/2,

4I15/2) → (2F7/2,4I11/2)

is exploited (Fig. 11.61). The main task for theoptimization of Er lasers around 1.55 µm is thusto find the optimum concentration for both dopantions.

Crystals. The absorption spectra of Er3+:YVO4 forthe 4I15/2 → 4I11/2 and 4I15/2 → 4I13/2 transitionsare shown in Figures 11.62 and 11.63, respec-tively [11.437]. The peak absorption cross section forthe 4I15/2 → 4I11/2 transition around 970 nm is up to2 × 10−20 cm2, for the 4I15/2 → 4I13/2 transition around1500 nm cross sections are higher, approximately upto 4 × 10−20 cm2. The emission spectrum is shownin Fig. 11.63. The peak emission cross sections ofthe 4I13/2 → 4I15/2 emission in Er3+:YVO4 are upto 2 × 10−20 cm2, in the long-wavelength tail, wherethe laser oscillation occurs, the peak cross sectionsare around 0.5 × 10−20 cm2. These values are typi-cal for Er3+-doped crystals, e.g., the peak emissioncross section around 1550 nm is 0.31 × 10−20 cm2 for

*B-)

*!&)

"'

(;

, D

-

-,40

,#9

I.

*B)

*B)

*B()

*!()

@)*4)

(&&

I.

,40

$%Q

Fig. 11.61 Energy-level diagram of Er3+ and Yb3+ ina YVO4 crystal

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Lasers and Coherent Light Sources 11.2 Solid-State Lasers 653

Table 11.18 Yb3+-doped laser crystals and laser transitions 2F5/2 → 2F7/2

Crystal λlaser (µm) Pump T (K) Output mode Ref.

BaCaBO3F 1.034 TiS laser 300 p [11.440]

CaF2 : Na ≈ 1.046–1.048 Laser diode 300 ML [11.441]

1.051 Laser diode 300 Q-switch [11.441]

CaF2 : Nd 1.0336 Xe lamp 120 p [11.442]

1.030–1.055 TiS laser 300 CW [11.443]

Ca4GdO(BO3)3 - Laser diode 300 CW [11.444]

1.032 TiS laser 300 CW [11.445]

1.035–1.088 Laser diode 300 CW [11.446]

1.050 Laser diode 300 CW [11.446, 447]

1.050 TiS laser 300 CW [11.447]

1.082 TiS laser 300 CW [11.445]

1.082 Laser diode 300 CW [11.447]

Ca5(PO4)3F 1.043 TiS laser 300 CW [11.448, 449]

Ca3Sr2(PO4)3F 1.046 TiS laser 300 CW [11.450]

Ca4Sr(PO4)3F 0.985 TiS laser 300 CW [11.450]

1.046 TiS laser 300 CW [11.450]

1.110 TiS laser 300 CW [11.450]

Ca4YO(BO3)3 1.018–1.087 TiS laser 300 CW [11.451]

1.032 Laser diode 300 CW [11.452]

1.060 TiS laser 300 CW [11.453]

1.084–1.096 TiS laser 300 CW [11.454]

Gd3Ga5O12 1.039 Laser diode 300 Q-switch [11.455]

Gd3Ga5O12 : Nd 1.0232 Xe lamp 77 p [11.456]

Gd3Sc2Al3O12 : Nd 1.0299 Xe lamp 77 p [11.457]

Gd2SiO5 1.028–1.093 Laser diode 300 CW [11.458]

1.030–1.039 Laser diode 300 CW [11.459]

1.045–1.070 Laser diode 300 CW [11.459]

1.081–1.097 Laser diode 399 CW [11.459]

1.089 Laser diode 300 CW [11.460, 461]

1.090 Laser diode 300 CW [11.462, 463]

1.091–1.105 Laser diode 300 p [11.459]

1.094 Laser diode 300 CW [11.461]

GdVO4 1.015 Laser diode 300 CW [11.464]

1.015 TiS laser 300 CW [11.465]

1.015–1.019 Laser diode 300 CW [11.466]

1.026–1.031 Laser diode 300 CW [11.466]

1.029 Laser diode 300 CW [11.464]

1.029 TiS laser 300 CW [11.465]

1.040 Laser diode 300 CW [11.466]

1.045 Laser diode 300 CW [11.466]

(Gd,Y)2SiO5 1.030–1.089 Laser diode 300 CW [11.458]

KGd(WO4)2 1.026–1.044 Laser diode 300 CW [11.467]

1.030–1.051 Laser diode 300 CW [11.468]

1.031–1.0374 Laser diode 300 ML [11.469]

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654 Part C Coherent and Incoherent Light Sources

Table 11.18 (continued)

Crystal λlaser (µm) Pump T (K) Output mode Ref.

KLu(WO4)2 1.030 Laser diode 300 CW [11.470]

1.030 TiS laser 300 CW [11.470]

1.044 TiS laser 300 CW [11.471]

1.0435 Laser diode 300 CW [11.471]

KY(WO4)2 1.026 Laser diode 300 CW [11.417]

1.026–1.042 Laser diode 300 CW [11.467]

1.030 TiS laser 300 CW [11.472]

1.048 Laser diode 300 ML [11.473]

≈ 1.028 Laser diode 300 ML [11.474]

1.030 Laser diode 300 CW [11.475]

0.987–1.051 Laser diode 300 CW [11.476]

KYb(WO4)2 1.068 TiS laser 300 CW [11.477]

1.074 TiS laser 300 qCW [11.478]

LaSc3(BO3)4 1.044 TiS laser 300 CW [11.417]

1.045 TiS laser 300 CW [11.479]

0.995–1.087 Laser diode 300 CW [11.480]

LiGd(MoO4)2 1.027–1.0335 Laser diode 300 CW [11.481]

LiNbO3-waveguide 1.008 TiS laser 300 CW [11.482]

1.030 TiS laser 300 CW [11.482]

1.060 TiS laser 300 CW [11.482]

LiNbO3 : MgO 1.063 Laser diode 300 CW [11.483]

Li6Y(BO3)3 1.040 Laser diode 300 CW [11.484]

Lu3Al5O12 1.0297 Xe lamp 77 p [11.456]

1.03 Laser diode 175 CW [11.485]

Lu3Al5O12 : Nd,Cr 1.0294 Xe lamp 77 p [11.456]

Lu3Ga5O12 : Nd 1.0230 Xe lamp 77 p [11.456]

Lu2O3 ≈ 1.029–1.038 TiS laser 300 ML [11.486]

1.032 Laser diode 300 CW [11.417]

ceramics 1.035 Laser diode 300 CW [11.487]

ceramics 1.079 Laser diode 300 CW [11.487]

Laser diode 300 CW [11.475]

Lu3Sc2Al3O12 : Nd 1.0299 Xe lamp 77 p [11.457]

LuVO4 1.0347 Laser diode 300 CW [11.488]

1.041 Ti laser 300 CW [11.488]

1.0444 Laser diode 300 CW [11.488]

1.0527 Laser diode 300 CW [11.488]

NaGd(WO4)2 1.016–1.049 TiS laser 300 CW [11.489]

1.023 TiS laser 300 CW [11.490]

1.033 Laser diode 300 CW [11.490]

NaLa(MoO4)2 1.016–1.064 TiS laser 300 CW [11.491]

1.017 Laser diode 300 CW [11.492]

≈ 1.020 Laser diode 300 Q-switch [11.492]

1.023 Laser diode 300 CW [11.492]

1.035 Laser diode 300 CW [11.491]

NaLa(WO4)2 1.017–1.057 TiS laser 300 CW [11.493]

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Lasers and Coherent Light Sources 11.2 Solid-State Lasers 655

Table 11.18 (continued)

Crystal λlaser (µm) Pump T (K) Output mode Ref.

Sc2O3 1.041 TiS laser 300 CW [11.494]

1.0416 TiS laser 300 CW [11.495]

1.0946 TiS laser 300 CW [11.495]

ceramic 1.041 Laser diode 300 CW [11.496]

ceramic 1.094 Laser diode 300 CW [11.496]

Sr5(PO4)3F 0.985 Cr:LiSAF 300 p [11.497]

0.985 TiS laser 300 CW [11.498, 499]

1.047 TiS laser 300 CW, qCW [11.500, 501]

Laser diode 300 p [11.502]

Sr5−xBax(PO4)3F ≈ 1.048 TiS laser 300 CW [11.500]

(Sr0.7Ca0.3)3Y(BO3)3 Laser diode 300 CW, qCW [11.503]

Sr5(VO4)3F 1.044 TiS laser 300 p [11.450]

Sr3Y(BO3)3 Laser diode 300 CW, qCW [11.494]

SrY4(SiO4)3 1.020–1.095 Laser diode 300 CW [11.504]

≈ 1.068 Laser diode 300 ML [11.504]

Y3Al5O12 1.016–1.095 TiS laser 300 CW [11.505]

1.0293 Xe lamp 77 p [11.456]

1.0296 Xe lamp 77 p [11.279]

1.023–1.052 Laser diode 300 CW [11.506]

1.029 Laser diode 300 CW, qCW [11.507]

waveguide 1.03 TiS laser 300 CW [11.508]

1.030 TiS laser 300 CW [11.509]

1.03 Laser diode 300 CW [11.280, 417,507, 510]

1.03 Laser diode 300 Q-switch [11.511, 512]

1.03 Laser diode 300 ML [11.513]

1.03 TiS laser 300 Q-switch [11.514, 515]

waveguide 1.03 Laser diode 300 CW [11.516]

waveguide 1.03 Laser diode 300 CW [11.517]

waveguide 1.03 Laser diode 300 Q-switch [11.517]

1.031 Laser diode 300 p [11.501]

1.031 Laser diode 300 CW [11.518, 519]

≈ 1.0312 Laser diode 300 ML [11.519]

1.0494–1.0504 Laser diode 300 CW [11.520]

1.070 Laser diode 300 CW [11.311]

Y3Al5O12 : Nd 1.0297 Xe lamp 200 p [11.456]

waveguide 1.03 TiS laser 300 CW [11.521]

Y3Al5O12 : Nd,Cr 1.0298 Xe lamp 210 P [11.456]

YAl3(BO3)4 ≈ 1.040 Laser diode 300 CW [11.522]

1.120–1.140 Laser diode 300 CW [11.523, 524]

Y3Ga5O12 : Nd 1.0233 Xe lamp 77 P [11.456]

YLiF4 ≈ 0.991–1.022 Laser diode 77 P [11.525]

YLuSiO5 1.014–1.091 Laser diode 300 CW [11.526]

Y2O3 ceramic ≈ 1.076 Laser diode 300 CW [11.527]

ceramic ≈ 1.076 Laser diode 300 ML [11.527]

ceramic 1.0767–1.0784 Laser diode 300 CW [11.520]

ceramic 1.078 Laser diode 300 CW [11.528]

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656 Part C Coherent and Incoherent Light Sources

Table 11.18 (continued)

Crystal λlaser (µm) Pump T (K) Output mode Ref.

Y2SiO5 1.000–1.010 Laser diode 300 CW [11.529]

1.082 Laser diode 300 CW [11.529]

Y3Sc1.0Al4.1O12 ceramic ≈ 1.060 Laser diode 300 CW, ML [11.530]

YVO4 1.020–1.027 Laser diode 300 CW [11.531]

1.037 TiS laser 300 CW [11.532]

1.039 Laser diode 300 CW [11.532]

Er:YAlO3 [11.436], 0.33 × 10−20 cm2 for Er:Y2SiO5[11.438], 0.45 × 10−20 cm2 for Er:YAG [11.436],0.42 × 10−20 cm2 for Er:YLF [11.436], and 0.59 ×10−20 cm2 for Er:LaGaO3 [11.439].

From the absorption and emission spectra the gaincoefficient curves (Fig. 11.64) can be calculated by

g = N[Pσem − (1− P)σabs] ,where N is the ion concentration and P is the inversioncoefficient, defined as the ratio between the populationsin the 4I13/2 and the 4I15/2 levels. It can be seen that forEr3+:YVO4 already for an inversion coefficient of P ≈0.2 laser oscillation should be possible within a spectralrange from approximately 1530 nm to 1610 nm.

In Fig. 11.65a the input–output characteristics ofTi:sapphire-pumped Er3+:YVO4 crystals with 0.5% Erand 1% Er doping, operating at 1604 nm, are shown.The slope efficiency with respect to the incident poweris about 7–8%. It should be noted that with respectto the absorbed power the 0.5% Er:YVO4 crystal ex-hibits a higher slope efficiency. This indicates that forthe more highly doped sample (1% Er) the aforemen-

G 6

"'

$-

2-

(& (; ((

" "

*B-)*B)

Fig. 11.62 Polarized absorption spectra at room tempera-ture of an Er3+:YVO4 crystal in the spectral range of the4I15/2 → 4I11/2 transition. (After [11.437])

tioned loss mechanism of reabsorption and upconversionalready take place. The laser threshold is below 200 mWof incident power and below 100 mW of absorbed pumppower.

These values for the slope efficiency, the laser thresh-old and the output power are typical for well-performingEr3+-doped crystals, such as for Y3Al5O12 [11.533] andLaSc3(BO3)4 [11.534].

Recently, high-power and ultra-efficient laser op-eration of an Er:YAG laser near 1645 nm withapproximately 60 W [11.535–538] of output powerwas demonstrated using a high-power fibre laser at1532 nm as a pump source (so-called in-band pump-ing (Fig. 11.65b). This work demonstrates that the smallStokes shift between the pump and laser radiation(1532 nm/1645 nm) yields very efficient Er lasers withslope efficiencies up to 80%.

In Table 11.19 an overview of crystalline room-temperature Er3+ lasers is given. Note that lasers onthe 4S3/2 → 4I9/2 transition are also listed.

G 6

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*- &- -- % %-

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Fig. 11.63 Spectra of absorption cross sections due to the4I15/2 → 4I13/2 transition of Er3+ in YVO4 crystal (dottedlines) and emission cross sections (4I13/2 → 4I15/2) (solidlines) for σ and π polarization. (After [11.437])

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Lasers and Coherent Light Sources 11.2 Solid-State Lasers 657

Glass. Laser oscillation in Er3+-doped and Yb3+,Er3+-codoped silicate and phosphate glasses was investigatedas early as the mid-1960s [11.539–541]. The three-levelbehavior of the Er3+ system and the weak absorp-tion of pump radiation caused by the requirementof low dopant concentration (usually approximately2 to 5 × 1019 cm−3) makes it difficult to obtain ef-ficient laser operation in singly Er3+-doped glasses.Therefore, Yb3+ codoping is necessary in order toabsorb the pump light efficiently at wavelengthsaround 1 µm. Continuous-wave laser oscillation, Q-switched [11.542, 543] and quasi-CW operation withpulse energies up to 35 J and average output pow-ers up to 20 W have been obtained [11.544–548].The pulse duration of the Xe flashlamps is sev-eral milliseconds, thus matching the upper-laser-levellifetime of the Er3+ 4I13/2 level. Also laser diodescan be efficiently applied as pump sources. Wuet al. [11.549] used pulsed laser diodes in a transver-sal excitation scheme. Using a repetition rate of50 Hz, a pump pulse duration of 2.5 ms and a peakpump power of ≈ 1.5 kW, an average output powerof 8.5 W was obtained, corresponding to an out-put energy of 170 mJ per pulse. A comparison anddiscussion of different pumping schemes is givenin [11.550]. In all experiments, the main problemof glass materials compared to crystals is the lowheat conductivity (for Kigre QE7 0.82 W/mK [11.547]versus 13 W/mK for YAG). Thus, the introducedheating power and therefore also the extractableoutput power and/or repetition rate is limited andhigh-power continuous-wave excitation of Er3+-dopedglasses is difficult. Despite this, the overall perfor-mance of the Kigre QE7 and QX glasses is better thanthat of Er3+-doped crystals. Obaton et al. [11.551]obtained a slope efficiency of 21% in a diode end-pumped setup for a QX glass. Diening [11.552]compared Kigre QE7 glass with Er3+:LaSc3(BO3)4and Y3Al5O12 in the same setup (Fig. 11.66). Theachieved output power and slope efficiency forthe QE7 glass are about twice as high as forEr3+:LaSc3(BO3)4.

Fig. 11.65 (a) Input–output curve of the CW laser oscil-lation of Er3+ (0.5%):YVO4 crystal (λlaser = 1604 nm,output mirror transmission = 1%). The slope efficienciesare given with respect to the absorbed (ηabs) and incident(ηin) power. (Results from [11.437]). (b) In-band pump-ing of Er:YAG with a fiber laser at 1532 nm and lasing atλlaser = 1645 nm [11.535]

'

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$3$3$3$*

$3$3$3$*

*; - - -* -% -; % %

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';'%'*'*

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Fig. 11.64 Gain coefficient curves derived for both polarizations forEr3+:YVO4 in the spectral range of the 4I15/2 → 4I11/2 laser tran-sition for four values of the inversion parameter P (larger gainvalues for larger P). Arrows denote the wavelengths for which laseroscillation was realized. (After [11.437])

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658 Part C Coherent and Incoherent Light Sources

Table 11.19 Overview of room-temperature Er3+-doped laser operating around 1.6 µm on the 4I13/2 → 4I15/2 and on the4S3/2 → 4I9/2 transition

Crystal λlaser (nm) Transition Slope Mode of Pump source/Remarks Ref.efficiency (%) operation

Ca2Al2SiO7 1530 nm 4I13/2 →4 I15/2 1.1 CW Ti:Sapphire 940 nm, 975 nm [11.553]

1550 nm 4I13/2 →4 I15/2 1.5 CW Ti:Sapphire 940 nm, 975 nm [11.553]

1555 nm 4I13/2 →4 I15/2 5 CW Ti:Sapphire 940 nm, 975 nm [11.553]

LiNbO3 : Ti 1532 nm 4I13/2 →4 I15/2 6 CW Tl:KCl, 1477 nm [11.554]

1563 nm 4I13/2 →4 I15/2 3 CW, pulsed Tl:KCl 1479 nm, 1484 nm [11.555]

1576 nm 4I13/2 →4 I15/2 CW, pulsed Tl:KCl 1479 nm, 1484 nm [11.555]

SrY4(SiO4)3O 1554 nm 4I13/2 →4 I15/2 0.4 CW Laser diode 980 nm [11.556]

YAlO3 1662 nm 4S3/2 →4 I9/2 10.1 CW Kr+ [11.557]

1663 nm 4S3/2 →4 I9/2 0.07 Pulsed Xe-flash lamp [11.558]

1663 nm 4S3/2 →4 I9/2 Pulsed Xe-flash lamp [11.559,560]

1663.2 nm 4S3/2 →4 I9/2 Pulsed Xe-flash lamp [11.561–563]

1663.2 nm 4S3/2 →4 I9/2 CW Ar+, 488 nm [11.564]

1667 nm 4S3/2 →4 I9/2 0.02 Pulsed Xe-flash lamp [11.565]

1677.6 nm 4S3/2 →4 I9/2 2.2 CW Ar+, 488 nm [11.564]

1706 nm 4S3/2 →4 I9/2 0.02 Pulsed Xe-flash lamp [11.565]

1706.1 nm 4S3/2 →4 I9/2 CW Ar+, 488 nm [11.564]

1729 nm 4S3/2 →4 I9/2 0.02 Pulsed Xe-flash lamp [11.565]

1729.6 nm 4S3/2 →4 I9/2 CW Ar+, 488 nm [11.564]

LiYF4 1620 nm 4I13/2 →4 I15/2 CW Kr+, 647 nm [11.566]

1640 nm 4I13/2 →4 I15/2 CW Kr+, 647 nm [11.566]

1664.0 nm 4S3/2 →4 I9/2 Pulsed Xe flashlamp [11.567]

1730 nm 4S3/2 →4 I9/2 0.6 Pulsed Xe flashlamp [11.568]

1732.0 nm 4S3/2 →4 I9/2 Pulsed Xe flashlamp [11.569]

Y3Al5O12 1617 nm 4I13/2 →4 I15/2 10 Q-switched (4 kHz) Er-fiber laser, 1543 nm [11.570]

1632 nm 4I13/2 →4 I15/2 Pulsed Xe flashlamp [11.571]

1634 nm 4I13/2 →4 I15/2 Pulsed, intracavity Er:glass 1549 nm [11.572]

1640 nm 4I13/2 →4 I15/2 12.7 CW Kr+, 647 nm [11.533]

1640 nm 4I13/2 →4 I15/2 0,5 Q-switched Er:glass 1534 nm [11.573]

1644 nm 4I13/2 →4 I15/2 7 Pulsed Er:glass 1535 nm [11.574]

1644.9 nm 4I13/2 →4 I15/2 Pulsed Xe flashlamp [11.575]

1645 nm 4I13/2 →4 I15/2 40 Pulsed Er:glass 1532 nm [11.576]

1645 nm 4I13/2 →4 I15/2 40 Q-switched Yb, Er-doped fiber 1530 nm [11.577]

1645 nm 4I13/2 →4 I15/2 46 Pulsed 1.5 µm laser diodes [11.578]

1645 nm 4I13/2 →4 I15/2 40 CW Er-fiber laser, 1543 nm [11.570]

1645.3 nm 4I13/2 →4 I15/2 81 CW, Q-switched Yb, Er-doped fiber 1530 nm [11.535]

1645.9 nm 4I13/2 →4 I15/2 Pulsed Xe flashlamp [11.579]

1646 nm 4I13/2 →4 I15/2 7 CW laser diode [11.580]

1775.7 nm 4S3/2 →4 I9/2 Pulsed Xe flashlamp [11.563,571, 581]

Y3Ga5O12 1640 nm 4I13/2 →4 I15/2 0.9 CW Kr+, 647 nm [11.533]

Y3Sc2Ga3O12 1643 nm 4I13/2 →4 I15/2 10 Pulsed Er:glass, 1532 nm [11.582]

Lu3Al5O12 1776.2 nm 4S3/2 →4 I9/2 Pulsed Xe flashlamp [11.563]

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Table 11.19 (continued)

Crystal λlaser (nm) Transition Slope Mode of Pump source/Remarks Ref.efficiency (%) operation

KGd(WO4)2 1715.5 nm 4S3/2 →4 I9/2 Pulsed Xe flashlamp [11.583]

1732.5 nm 4S3/2 →4 I9/2 Pulsed Xe flashlamp [11.583]

1733.0 nm 4S3/2 →4 I9/2 Pulsed Xe flashlamp [11.584]

KY(WO4)2 1540 nm 4I13/2 →4 I15/2 1 CW Ti:Sapphire [11.585]

1737.2 nm 4S3/2 →4 I9/2 Pulsed Xe flashlamp [11.584]

KLa(MoO4)2 1730 nm 4S3/2 →4 I9/2 Pulsed Xe flashlamp [11.586]

LiLuF4 1734.5 nm 4S3/2 →4 I9/2 Pulsed Xe flashlamp [11.587,588]

KLu(WO4)2 1739.0 nm 4S3/2 →4 I9/2 Pulsed Xe flashlamp [11.584]

KEr(WO4)2 1737.2 nm 4S3/2 →4 I9/2 Pulsed Xe flashlamp [11.584]

YVO4 1604 nm 4I13/2 →4 I15/2 19 CW Ti:Sapphire [11.437]

(1531 nm,

1553 nm,

1564 nm,

1580 nm)

Y2SiO5 1617 nm 4I13/2 →4 I15/2 5.6 CW laser diode [11.580]

(1545 nm,

1567 nm,

1576 nm)

Sc2SiO5 1558 nm 4I13/2 →4 I15/2 CW Ti:Sapphire 979 nm [11.589]

1551 nm 4I13/2 →4 I15/2 1.8 CW Ti:Sapphire 920 nm [11.589]

1551 nm 4I13/2 →4 I15/2 2.4 CW laser diode 968 nm [11.589]

Sc2Si2O7 1545 nm 4I13/2 →4 I15/2 2.6 CW Ti:Sapphire 980 nm [11.589]

1556 nm 4I13/2 →4 I15/2 2.3 CW Ti:Sapphire 978 nm [11.589]

LaSc3(BO3)4 1563 nm 4I13/2 →4 I15/2 6 CW laser diode 975 nm [11.534]

Ca4YO(BO3)3 1.5–1.6 µm 4I13/2 →4 I15/2 26.8 CW laser diode [11.590,591]

Ca4GdO(BO3)3 1.54 µm 4I13/2 →4 I15/2 15 CW laser diode 975 nm [11.592]

7 Ti:Sapphire 902 nm

Fibers. Erbium-doped fiber lasers have been exten-sively studied for their potential use as sources incommunication systems operation in the third com-munication window around 1.55 µm. All these lasersoscillate on the 4I13/2 → 4I15/2 transition, either incontinuous-wave or pulsed mode. In singly Er3+-dopedfibers, suitable pump wavelengths for using laser diodesare 810 nm (4I15/2 → 4I9/2), 980 nm (4I15/2 → 4I11/2)and 1480 nm (4I15/2 → 4I13/2). Other possible pumpbands are around 660 nm (4I15/2 → 4F9/2), 532 nm and514.5 nm (4I15/2 → 4H11/2). The pump wavelengths at810 nm and 514.5 nm suffer from strong excited-stateabsorption, yielding a loss of pump photons [11.593].The gain coefficient of Er-doped fibers is rather high(11 dB/mW [11.594]) due to the fairly high peak emis-sion cross section of 4–7 × 10−21 and the long lifetime

of the 4I13/2 level (8–10 ms) in silica fibers, despitethe three-level laser character causing ground-state ab-sorption at this wavelength. Like in Er3+-doped crystalsand glasses, concentration quenching occurs. In orderto enhance the absorption efficiency without increas-ing the Er3+ concentration and/or fiber length, Yb3+codoping is used, especially when diode pumping be-tween 900 nm and 1000 nm is used. The requirement forefficient operation is – as in the crystals and glasses – ef-ficient energy transfer from the Yb3+ 2F5/2 level to theEr3+ 4I13/2 level (Fig. 11.61). In Table 11.20, some ofthe Er-fiber lasers around 1.55 µm are summarized. Fora very detailed discussion of Er3+-doped fiber lasersand amplifier see [11.595]. In summary, Er3+-dopedfiber lasers operating around 1.55 mm are extremely ef-ficient. Output powers in the watt range are possible.

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660 Part C Coherent and Incoherent Light Sources

G

D2,N4"/*

AH,&

D2,ND0-

G

-

-

-

- - -

Fig. 11.66 Results of the laser experiments with Kigre QE7glass, Yb3+ (10%), Er3+ (0.5%): LaSc3(BO3)4 and Yb3+,Er3+ (0.5%): Y3AL5O12. (After [11.552])

Nowadays up to 100 W of output power are commer-cially available [11.596]. Most efficient pumping occursat 1480 nm, here slope efficiencies close to the quantumlimit of 95% are possible. Pumping at 980 nm is lessefficient due to the higher quantum defect, however, atthis wavelength highly efficient and reliable high-powerlaser diodes are available.

Other near-infrared Er3+ lasers. Room-temperaturelaser oscillation in Er3+-doped crystals in the near-infrared spectral range has also been observed atother wavelengths. In Yb3+-codoped Er3+:YLiF4 anupconversion pumping scheme was realized for the4S3/2 → 4I11/2 transition at 1234 nm, which allowedcontinuous-wave room-temperature Ti:sapphire- anddiode-pumped laser oscillation [11.597, 598]. UnderTi:sapphire excitation around 966 nm output powersof 160 mW and slope efficiencies of up to 22% wereobtained. Note, that without the Yb codoping, the out-put powers were an order of magnitude lower. Underdiode laser pumping at 966 nm, Yb3+,Er3+:YLiF4 ex-hibited output powers of 80 mW and slope efficienciesof 7%.

For this transition Xe flashlamp excitation isalso possible allowing room-temperature laser oscil-lation in YLiF4 [11.599–601], LuLiF4 [11.602] andYAlO3 [11.603]. In BaYb2F8, the 4F9/2 → 4I13/2 transi-tion at 1260 nm exhibited laser oscillation under pulsedNd laser or Xe flashlamp excitation [11.604]. Laser os-cillation on the 4F9/2 → 4I11/2 transition was obtainedaround 1.96 µm in BaYb2F8 [11.605–607] under pulsedNd laser and Xe flashlamp excitation.

11.2.4 Mid-Infrared Lasers

BasicsThe mid-infrared wavelength range ≈ 1.9–5.0 µm is ofinterest for a number of applications. Mid-infrared solid-state lasers serve as light sources for spectroscopy, e.g.,in remote sensing of the atmosphere, as the frequenciesof internal vibrational motion of many molecules can befound in this spectral region. Other applications includemedicine, e.g., microsurgery and dentistry in the regionof high water absorption around 2.7–3 µm. Laser wave-lengths near 2 µm are suitable for tissue welding andlithotripsy.

The first mid-infrared laser was operated in 1960,shortly after the invention of the laser itself. It was op-erated at 2.6 µm in a calcium fluoride crystal dopedwith trivalent uranium [11.608]. Pulsed excitation andcooling to low temperatures were typically required fornovel laser transitions in the early years. Two years later,a Dy2+-doped CaF2 laser at 2.36 µm was also demon-strated in continuous-wave operation [11.609, 610].Among the lasers based on trivalent rare-earth ions, thetransitions near 2 µm in Tm3+ and Ho3+ were operatedin CaWO4 in 1962 [11.611,612]. The first observation ofcoherent emission near 3 µm from erbium ions was re-ported in 1967 [11.613]. Since those early years, a largenumber of new host materials has been developed andvarious new laser transitions in the mid-infrared spectralregion have been demonstrated (for a comprehen-sive overview of ion–host combinations, see [11.614]).Around 1990, Tm3+- and Ho3+-doped solid-state lasersystems in Y3Al5O12 (YAG), YLiF4 (YLF), YVO4, andY3Sc2Ga3O12 (YSGG) were shown to operate between1.86 and 2.46 µm [11.615–618], and Er3+-doped lasersin similar host systems cover the wavelength range of2.66–2.94 µm. Short-pulse lasers at these wavelengthshave been demonstrated [11.619, 620].

Nowadays, mid-infrared laser transitions rangingfrom 1.8 µm up to 7.2 µm [11.621] are known in di-valent Dy, trivalent Tm, Ho, Er, Dy, Pr, Tb, and Nd, aswell as in trivalent U. The current state of the art in solid-state lasers occurring in rare-earth ions such as Tm3+,Ho3+, Er3+ and others and their population mecha-nisms are discussed in this section on the basis of thespectroscopic properties of these ions. Continuous-wavefundamental-mode power levels ranging from a few mWnear 4 µm up to ≈ 100 W near 2 µm have been demon-strated in recent years. Power-scaling methods and theirlimitations, the possibilities to optimize the populationmechanisms and increase the efficiencies of these lasers,as well as the prospects of future mid-infrared lasers in

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Table 11.20 Overview over Er3+ and Yb3+:Er3+-doped silica fiber lasers. λlaser: laser wavelength, λpump: pump wavelength, lfiber:fiber length, Pthr: laser threshold, η: slope efficiency, Pout: output power, Ppump: pump power, Ref.: reference, (l): launched, (inc):incident, (abs): absorbed, NA: not available. (After [11.595])

λlaser λPump Er Yb lfiber Remarks Pthr η Pout max. Ppump Ref.

(nm) (nm) concen- concen- (m) (mW) (%) (mW) (mW)tration tration

1566 514.5 35 ppm Er - 13 Ar+ pump 44 (l) 10 (l) 56 600 (l) [11.622]

1560 532 150 ppm - 1 Ring laser 10 (l) 5.1 (l) 1.8 45 (l) [11.623]

Er2O3

1535 532 100 ppm Er - 15 Doubled NA 28 1000 3600 [11.624]

Nd:YAG

1560 806 500 ppm Er - 3.7 Laser diode 10 (l) 16 (l) 8 56 (l) [11.625]

1620 808 300 ppm Er - 1.5 Laser diode 3 (abs) 3.3 (abs) 0.13 7 (abs) [11.626]

1560 980 0.08 wt % - 0.9 Dye laser 2.5 (abs) 58 (abs) 4.7 11.3 (abs) [11.627]

1540 980 1100 ppm Er - 9.5 Ti:Sapphire > 10 (l) > 49 (l) 260 540 (l) [11.628]

1552 1460 1370 ppm Er - 5 2 laser diodes 37 (l) 14 (l) 8 93 (l) [11.629]

1552 1470 1370 ppm Er - 7 laser diode 44 (abs) 6.3 (abs) 1 60 (abs) [11.630]

1555 1480 45 ppm Er - 60 laser diode 6.5 (l) 38.8 (l) 3.3 15 (l) [11.631]

1560 1480 110 ppm - 42.6 laser diode 4.8 (abs) 58.6 (abs) 14.2 29 (abs) [11.632]

Er2O3

1570 810 0.06 wt % 1.3 wt % 1.45 2 laser diodes 12.7 (l) 15.4 (l) 2.3 28 (l) [11.633]

1560 820 0.08% 1.7% 0.7 Dye laser 3.7 (abs) 7 (abs) NA NA [11.634]

1560 832 0.08% 1.7% 0.7 Dye laser 5 (abs) 8.5 (abs) NA NA [11.635]

1537 962 900 ppm 1.1% 1.6 laser diode 130 (l) 19 (l) 96 620 (l) [11.636]

1545 980 NA NA 0.07 laser diode 1 (abs) 25 (inc) 18.6 95 (inc) [11.637]

1535 1047 0.06% 1.8% 4 Nd:YLF 20 (l) 23 (l) 285 640 (l) [11.638]

1560 1064 0.08% 1.7% 0.91 Nd:YAG 8 (abs) 4.2 (abs) 1.3 (abs) 80 (abs) [11.634]

1535 1064 880 ppm 7500 NA Nd:YAG 37 (abs) 27 (abs) NA NA [11.639]

ppm

1545.6 980/1480 NA NA 0.07 Laser diode 10 (l) 50 (l) 166 340 (l) [11.640]

a number of rare-earth ions at transitions in the wave-length range beyond 3 µm and extending to 5 µm aredescribed.

The aspects relevant to rare-earth-ion-doped mid-infrared solid-state lasers, such as the competitionbetween radiative and multiphonon decay and the con-sequent choice of host materials for these wavelengthswill be introduced next. The performance of the mostimportant mid-infrared laser transitions in the wave-length range 2–3 µm will then be discussed in detail:Tm3+-doped lasers at 1.9 µm and 2.3 µm, Ho3+-dopedlasers at 2.1 µm and 2.9 µm, Er3+-doped lasers at2.7–2.9 µm, and Dy3+-doped lasers at 2.9–3.4 µm.At wavelengths beyond 3 µm, it becomes increasinglydifficult to find suitable host materials for activelydoped laser systems. This statement holds true for glassfibers in the same way as for crystalline materials. Theprospects for future mid-infrared solid-state lasers in

this wavelength range will be discussed at the end of thesection.

Introductions to the fields of mid-infrared solid-statecrystalline and fiber lasers can be found in [11.641]and [11.642], respectively.

Decay mechanisms, host materials,and thermal issues

Here a few fundamental aspects of lasers are discussedwith emphasis on their impact on mid-infrared solid-state lasers.

Radiative versus multiphonon decay. The choice ofhost material for mid-infrared solid-state lasers involvesa number of considerations. The maximum phononenergy is the most important aspect. The optical trans-parency range relates to both the size of the bandgapand also the infrared absorption cut-off, hence to the vi-

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662 Part C Coherent and Incoherent Light Sources

brational frequency ν of the anion–cation bonds of thematerial. For an ordered structure,

ν =(

1

)√k

M, (11.89)

where M = m1m2/(m1 +m2) is the reduced mass fortwo bodies m1 and m2 vibrating with an elastic restoringforce k. The relative cation–anion bond strength is inti-mated by the field strength Z/r2, where Z is the valencestate of the cation or anion and r is the ionic radius. Gen-erally, materials composed of large anions and cationswith low field strengths display high transparency in themid-infrared spectral region.

Radiative decay of excited states is in competitionwith nonradiative multiphonon decay. The maximumphonon energy of the material sets the multiphonon re-laxation rates, which influence the quantum efficiency.The rate constant of a multiphonon relaxation processdecreases exponentially with the energy gap to the nextlower-lying state and with the order of the process, i. e.,the number of phonons required to bridge the energygap [11.643, 646]. As an example, the multiphonon re-laxation rates for the common fiber glasses as a functionof the energy gap between energy levels are shown inFig. 11.67.

The influence of multiphonon decay is stronger inoxides than in fluorides because of the smaller atomicmass m2 of the anion and the larger elastic restoringforce k (11.89), due to the stronger covalent bonds in ox-ides [11.644], both resulting in larger maximum phononenergies in oxides. Typically, nonradiative decay be-comes dominant if five or fewer phonons are required tobridge the energy gap [11.645]. Since an energy gap of≈ 3300 cm−1 corresponds to a transition wavelength of3 µm, radiative decay prevails for phonon energies below≈ 600 cm−1, which is roughly the maximum phononenergy of fluorides. Fluorides are, therefore, preferredover oxides as host materials for most mid-infrared lasertransitions.

In the example of Fig. 11.68, dominant laser (solidlines) and multiphonon (dotted lines) transitions fromthe three lowest-energy levels of Er3+ are indicated,together with the corresponding lifetimes of the levelsin different classes of host materials. In high-phonon-energy oxide host materials, only the 1.5 µm lasertransition possesses sufficiently high frequency andlarge energy gap, resulting in a long 4I13/2 lifetime.On the other hand, the 4I11/2 lifetime is significantlyquenched by multiphonon relaxation and the 2.8 µmlaser originating in this level is more easily operatedin fluoride host materials. Finally, low-phonon host ma-

,#"'

68'

* - % &

*-%&;(

E/0-"'

/*"'

+ &"'4-"'

("'4""'

6"'

Fig. 11.67 Calculated and measured multiphonon relax-ation rates as a function of the energy gap between energylevels for glasses with different maximum phonon energies.(After [11.643, 644])

terials such as chlorides are required to ensure a long4I9/2 lifetime in order to operate the 4.5 µm laser.

Host materials for mid-infrared lasers. Crystallineoxide materials have been the laser host materials ofchoice for several decades, as many oxide crystals arecomparatively easy to grow, environmentally stable,and possess high heat conductivities [11.647], frac-ture limits, and refractive indices, the latter resultingin large absorption and emission cross sections. Mostprominent is Y3Al5O12 (YAG), together with other

,

*$-Q

*B)

'2!'2.'

*B()

'2!'2.'

$Q2 Q2 -

Q2 -2

-2 2 -

$;Q

$-Q

*B)

*B-)

Fig. 11.68 Radiative and laser (solid lines) versus nonradia-tive (dotted lines) decay of the lowest three excited states ofEr3+ in oxide, fluoride, and chloride host materials [11.645]

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Lasers and Coherent Light Sources 11.2 Solid-State Lasers 663

garnets, but in recent years materials such as mixedgarnets, the vanadates YVO4 and GdVO4, the doubletungstates KY(WO4)2 and KGd(WO4)2, the sesquiox-ides Y2O3, Sc2O3, and Lu2O3 and others have showngreat promise for efficient lasing when doped withvarious rare-earth ions. Their performance decreasessignificantly when lasers are operated at wavelengthsabove ≈ 2.5–3 µm because of their high maximumphonon energies (≈ 700–900 cm−1). Silicate glass isperhaps the most important material used for opticalfiber production [11.644,648], however, in this materialthe maximum phonon energy of ≈ 1100 cm−1 [11.649]is even higher and has so far limited the emission wave-length of mid-infrared fiber lasers using this material to≈ 2.2 µm [11.650]. Silica is robust and involves the veryeffective modified chemical vapor deposition (MCVD)technique for fiber fabrication. Reducing the OH− con-tent in the glass, which has two main absorption peaksin the range 1.3–2.0 µm [11.651], improves the near-to-mid-infrared utility. Rare-earth ions such as Nd3+ andEr3+ which have high field strengths have low solubil-ity in silicate glass, which can lead to clustering andmicroscale phase separation.

The use of fluoride crystals and glasses as host ma-terials for mid-infrared solid-state lasers has found wideacceptance. The heavy-metal fluorides [11.652,653] arepreferred as fiber materials, especially ZBLAN [11.654,655], a mixture of 53 mol % ZrF4, 20 mol % BaF2,4 mol % LaF3, 3 mol % AlF3, and 20 mol % NaF.Since it can be readily drawn into single-mode opticalfiber [11.656] it is particularly important to mid-infraredfiber lasers [11.657]. The large atomic weight of thezirconium atom combined with relatively weak bond-ing provides a maximum phonon energy for ZBLANof ≈ 550 cm−1 [11.658] and allows for high infraredtransparency up to ≈ 6 µm. Multiphonon relaxation,however, becomes significant for transitions at wave-lengths longer than ≈ 3–3.5 µm. Compared to silica,ZBLAN has a lower damage threshold. The crystal fieldstrength is also weaker [11.659]. An overview of thespectroscopic properties of rare-earth ions doped intoZBLAN is given in [11.658]. On the crystalline side,the host materials LiYF4, BaY2F8, and their respectiveisostructural relatives have become the workhorses formany mid-infrared laser transitions [11.660, 661]. Be-cause of their fluoride content, these materials have tobe grown under an atmosphere which excludes oxygen.

Among the low-phonon host materials, many com-pounds naturally possess low heat conductivity and arehygroscopic. This accounts for most of the halides,with increasing hygroscopicity from chloride to iodide.

On the other hand, these materials provide phononenergies in the range of 350–150 cm−1 [11.662]. Re-cently, KPb2Cl5 and related compounds have emergedas non-hygroscopic, hence promising candidates formid-infrared lasers [11.663–665]. Chalcogenide glassesare composed of the chalcogen elements S, Se andTe [11.666–668]. They are environmentally durable andhave reasonably large glass-forming regions. When therare-earth ions are doped into these glasses [11.669],the radiative transition probabilities and, therefore, theabsorption and emission cross sections are high as a re-sult of the high refractive index (≈ 2.6) of the glassand the high degree of covalency of the rare-earth ionwith the surrounding medium. Maximum phonon en-ergies of 300–450 cm−1 [11.670] produce low ratesof multiphonon relaxation (Fig. 11.67), and thereforehigh quantum efficiencies. The most important glassesare the sulfide glasses GaLaS (GLS) [11.671] andGeGaS [11.672] because of the reasonably high rare-earth solubility.

Studies into the use of ceramics as host mater-ials for the rare-earths have recently made a lot ofprogress [11.673]. These ceramics are composed ofnanocrystallites of materials such as YAG and canbe produced in a simple cost-efficient process at rel-atively low temperatures. This allows the fabricationof materials with very high melting points [11.674]that are difficult to grow by other techniques such asthe Czochralski method. This class of materials is alsoavailable in a fiber geometry [11.675]. Ceramic fiberscombine the characteristics of crystalline materials suchas high absorption and emission cross sections, largethermal conductivity, and even the possibility of dopingwith transition-metal ions [11.675] with the convenienceof guiding the pump and signal light in a fiber. Whilebulk ceramics have already matured as laser host mater-ials, the losses of ceramic fibers are still comparativelyhigh.

Specific aspects of operating mid-infrared lasers.As higher pump powers become available from laser-diode systems, it is generally recognized that thermaland thermo-optical issues set limitations to the powerscalability of end-pumped bulk laser systems. Owing tothe unfavorable temperature dependence of thermal andthermo-optical parameters [11.647], the large heat loadin the crystal leads, firstly, to a significant temperatureincrease in the rod, secondly, to strong thermal lensingwith pronounced spherical aberrations, and ultimately,to rod fracture in a high-average-power end-pumpedsystem. Thermal management will be required when

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664 Part C Coherent and Incoherent Light Sources

very high output powers are desired. In particular, forhigh-power mid-infrared operation, thermal manage-ment may be very important because of the decreasedquantum efficiency and the consequently higher amountof heat dissipation [11.676].

Due to its geometry, the fiber provides potentiallyhigh pump and signal beam intensities without the draw-backs of significant thermal and thermo-optical effects.Its large surface-area-to-volume ratio means that the heatgenerated from multiphonon relaxation in the core isdissipated effectively by radiation and convection fromthe outer surface of the fiber. This is especially truefor single-clad, core-pumped single-mode fibers wherethis ratio is highest [11.678]. The larger cladding area(> 104 µm2) of double-clad fiber lasers allows for high-power diode-array pumping [11.408, 679–682]. On theother hand, double-clad fibers have a smaller surface-area-to-volume ratio and thermal issues need to be takeninto account [11.683–685].

While bulk crystalline mid-infrared lasers maturedalready during the 1990s, the high costs of fabricatingfibers with sufficiently low losses in the mid-infraredregion of the spectrum has impeded the necessary re-search efforts in the field of mid-infrared fiber lasers.However, with the introduction of the double-clad fiberand recent technological advances in the fields offiber fabrication and beam-shaped high-power diodelasers, the performance of diode-pumped fiber lasershas dramatically improved. Today, mid-infrared fiberlasers can compete with the corresponding bulk crys-talline systems in certain applications, especially whenfundamental-transverse-mode, CW laser operation atoutput powers in the milliwatt to the hundred watt rangeis required.

A large number of techniques for pulsed operationincluding Q-switching and mode locking of fiber lasershave been explored. These techniques have been inves-tigated intensively for the common laser transitions at1 µm in Nd3+ and Yb3+, and at 1.5 µm in Er3+, andare usually described in combination with these lasers.The small fiber size limits the peak power through thedamage-threshold intensity (propagating power per corearea) and, hence, crystalline lasers in bulk geometries oroptical parametric processes are often preferred whenhigh-energy short pulses are needed. This argument ac-counts especially for mid-infrared ZBLAN-based fiberlasers, because these fibers possess a lower damagethreshold compared to silica fibers. The description ofmid-infrared fiber lasers is, therefore, confined to CWoperation and specific techniques for pulsed operationof fiber lasers are not discussed in this chapter.

Thulium-doped solid-state lasersat 1.9–2.0 µm and 2.3–2.5 µm

The use of the Tm3+ ion for mid-infrared solid-statelaser applications has been widespread, partly as a re-sult of the convenient absorption band near 0.79 µm,which allows for direct AlGaAs diode-laser pumping.The primary luminescent transitions of Tm3+ relevantto mid-infrared laser emission are the 3F4 → 3H5 tran-sition at ≈ 2.3 µm and the 3H4 → 3H6 ground-statetransition at ≈ 1.9 µm; see the energy-level scheme inFig. 11.69. The 3F4 level is excited by the 0.79 µm pumpwavelength.

Three-level lasers at 1.9–2.0µm. The first laseremission from Tm3+ ions was reported on thephonon-terminated 2 µm transition 3H4 → 3H6 inCaWO4:Tm3+ in 1962 [11.611]. In 1975, pulsedoperation at room temperature was demonstratedin Cr3+-codoped YAG and YAlO3 [11.686]. Cr3+codoping allowed the experimentalist to improve theabsorption of flashlamp or ion-laser pump light inthe visible spectral range by the active mediumand subsequent energy transfer from Cr3+ to theTm3+ lasing ions [11.687, 688]. The cross-relaxationprocess (3F4,

3H6) → (3H4,3H4) can transform one

pump photon absorbed in the 3F4 or a higher-lyingenergy level into two excitations in the 3H4 up-per laser level of the 2 µm transition [11.616, 689](Fig. 11.69), thereby enhancing the quantum efficiencyof this laser by a factor of 2. Laser emission underdiode pumping of the 3F4 level at 780–790 nm wasachieved in YAG in the late 1980s [11.690]. A single-

+

$Q

@*

$-!*

Q

@-

40

.

@%

%$;

Fig. 11.69 Partial energy-level scheme of Tm3+ display-ing the measured lifetimes when doped into fluorideglass [11.677], NR and CR represent nonradiative decayand cross-relaxation, respectively

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Lasers and Coherent Light Sources 11.2 Solid-State Lasers 665

frequency monolithic laser has also been reported inYAG:Tm3+ [11.691].

Recently, 14 W, 18 W, and 36 W of output powerwith high beam quality has been achieved with diodepumping at room temperature in YAG:Tm3+ [11.692]and in LiYF4:Tm3+ in slab [11.693] and bulk [11.694]geometries, respectively. The latter approach couldbe scaled to 70 W, however currently with lowbeam quality [11.695]. Two research groups reportedabout 115 W and 120 W, respectively, of multi-mode output power from diode-pumped YAG:Tm3+lasers [11.696, 697]. A microchip laser was demon-strated in GdVO4:Tm3+ [11.698]. A thin-disc laser hasalso been demonstrated in YAG [11.699].

The large degree of Stark splitting of the 3H6ground state, combined with vibronic broadening ofthe spectrum [11.700], provides the 3H4 → 3H6 tran-sition with a very broad emission, spanning ≈ 400 nmin many hosts, which represents one of the broadestluminescent transitions available from any rare-earthion. Accordingly, its tunability is rather large, rangingfrom 1.87–2.16 µm in YAG [11.701], 1.84–2.14 µmin YSGG [11.701], 1.93–2.00 µm in YAlO3 [11.702],1.93–2.09 µm in Y2O3 [11.703], 1.93–2.16 µm inSc2O3 [11.703], 1.83–1.97 µm in CaF2 [11.704],1.91–2.07 µm in LiYF4 [11.693], 1.85–2.06 µm inBaY2F8 [11.705], 1.86–1.99 µm in GdVO4 [11.706],1.84–1.95 µm in LuVO4 [11.707], 1.79–2.04 µmin KGd(WO4)2 [11.708], and 1.81–2.03 µm inNaGd(WO4)2 [11.709]. As for many other transitions,shifts in the center emission wavelength can be achievedby substitution of host ions, e.g., from Y3Al5O12 toLu3Al5O12 [11.710]. The vanadate crystals GdVO4 andYVO4 as well as double tungstates possess compara-tively high absorption coefficients [11.698, 711, 712],allowing pumping also at 805–810 nm, where cheaperand more-reliable pump diodes than at 780–790 nmare available. Exploiting the large gain bandwidth,mode-locked operation of Tm3+ 2 µm lasers with pulsedurations of 35 ps and 41 ps has been reported inYAG:Tm3+ [11.713] and YAG:Cr3+,Tm3+ [11.714], re-spectively. Actively [11.715, 716] or passively [11.717]Q-switched laser operation is useful in micro-surgery [11.718].

Recent progress in the fields of crystalline epitax-ial growth and in-bulk refractive-index modificationprocesses in glasses and crystals has enabled novelsolid-state lasers in the waveguide geometry [11.719].As one of the results, Tm3+ waveguide lasers at2 µm have also been demonstrated in lead germanateglass [11.720], YAG [11.721] with up to 15 W of output

power under high-power diode side pumping [11.722],and KY(WO4)2 [11.723]. The latter laser has, as yet, notshown a performance as good as for the same materialin bulk geometry [11.724] or in a waveguide geometrybut doped with Yb3+ and lasing at 1 µm [11.725]. Epi-taxial layers of Tm:KLu(WO4)2/KLu(WO4)2 have alsobeen operated with the laser cavity perpendicular to thelayer in the 2 µm spectral range [11.726].

The first explorations into fiber lasers utilizingthe 1.9 µm ground-state transition related to the dye-laser pumping at 797 nm of a Tm3+-doped silicafiber laser [11.727]. Overlap of the main absorp-tion band with the emission wavelength of AlGaAsdiode lasers quickly resulted in diode-laser pumpingof these fiber lasers based on either silica [11.728]or fluoride [11.729] glass hosts. The cross-relaxationprocess (3F4,

3H6) → (3H4,3H4) and enhancement of

excited ions in the 3H4 upper laser level of the2 µm transition (Fig. 11.69), is highly dependent onthe overall concentration of Tm3+ ions and competi-tion from multiphonon relaxation from the 3F4 level.Although generally high concentrations of Tm3+ inlow-phonon-energy glasses enable full exploitation ofthis beneficial process, it has been shown recently thatthis cross-relaxation process is resonant in a silica hostand hence only moderate (2–3 wt %) Tm3+ ion con-centrations are required to maximize the benefits ofcross-relaxation [11.730].

Also in fibers, the broad emission spectrumallows a large degree of wavelength tunabil-ity [11.731]. Recently, tuning ranges of 230 nmfrom 1.86–2.09 µm [11.732] and 250 nm from1.72–1.97 µm [11.733] have been demonstrated. Sincethe Tm3+ 1.9 µm transition can be favorably oper-ated in silica fiber (with its higher peak-power damagethreshold compared to ZBLAN fiber), pulses in therange of 190–500 fs have been obtained in additive-pulse [11.734] or passive [11.735] mode-lockingarrangements using this broad emission spectrum. Thesmaller emission cross section and the three-level natureof the laser transition resulted in higher pump thresholdscompared to standard Nd3+-doped silica fiber lasers. Re-absorption from the ground state of the Tm3+ ion hasto be overcome because the ground-state multiplet isthe lower laser level. Reducing the population of thehigher Stark levels of the ground state by cooling thefiber causes emission at shorter wavelengths. Tunabil-ity to longer wavelengths can be obtained by variationof the fiber length because of the increased level of re-absorption by the ground state with longer lengths offiber [11.736].

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666 Part C Coherent and Incoherent Light Sources

Early power-scaling experiments involved the useof the convenient 1.064 µm YAG:Nd3+ laser whichcore-pumped the short-wavelength side of the 3H5level [11.737]. Pumping the long-wavelength side of the3H5 level with a high-power 1.319 µm YAG:Nd3+ laseralso yielded efficient output [11.738]. In-band pump-ing of the transition at 1.57 µm in silica [11.739] andat 1.58–1.60 µm in fluoride glass [11.740, 741] hasalso been demonstrated. Whilst theoretical modeling ofTm3+-doped silica fiber lasers [11.742] indicates thatin-band pumping is the most efficient pump methodfor silica-based fiber lasers because of the high Stokesefficiency, nevertheless, the wide availability of high-power AlGaAs diode lasers at 790–800 nm and thestrong level of cross-relaxation in Tm3+-doped silicameans that such diode-cladding-pumped systems in bothstanding-wave [11.736,743] and ring-resonator [11.744]arrangements are perhaps the most practical ways ofproducing high output power from this ion (Fig. 11.70).With Yb3+ sensitization and pumping at 975 nm, 75 Wof output power has been demonstrated [11.745]. Cur-rently, the Tm3+-doped silica fiber laser is the mostmature of the mid-infrared fiber-laser systems pri-marily because of the robustness and convenienceoffered by the silica glass host. The maximum out-put power from high-power Tm3+-doped fiber lasersis now ≈ 85 W [11.743], which is comparable to theequivalent diode-pumped Tm3+-doped crystalline lasersystems [11.697].

B" 5G

$;5+

5G

$%+2$*@

$5+

*

*

%

;

- *

Fig. 11.70 Measured output powers from diode-cladding-pumped fiber lasers using 1.8 wt % Tm3+-doped sil-ica [11.735], 2.2 wt % Tm3+-doped silica [11.746],and 3.6 mol % Tm3+, 0.4 mol % Ho3+-doped fluorideglass [11.747]

Four-level lasers at 2.3–2.5µm. The mid-infraredfour-level CW laser at ≈ 2.3 µm on the transition3F4 → 3H5 has been operated in GSGG:Tm3+ andLiYF4:Tm3+, with wavelength tunability ranging from2.2–2.37 µm [11.748] and 2.2–2.46 µm [11.749], re-spectively. This laser operates best at low Tm3+concentrations of < 2 at % in order to avoid the afore-mentioned cross-relaxation, which in this case woulddepopulate the upper laser level (Fig. 11.69). The life-time of the lower laser level of the 3F4 → 3H5 transitionis quite short and leads to a low pump threshold.

Doping Tm3+ ions into a ZBLAN fiber offers anincreased quantum efficiency of the 3F4 level [11.750–752]. Deliberately designing the fiber to have a relativelylow Tm3+-ion concentration reduces cross-relaxationand hence severe lifetime quenching of the 3F4 level.The tunability extends from 2.25 µm to 2.5 µm [11.677].Simultaneous lasing on the 3H4 → 3H6 transition at1.9 µm produces a two-color fiber laser [11.753].Applications requiring highly efficient output or multi-mid-infrared-wavelength output will benefit from theuse of Tm3+-doped ZBLAN fibers.

Holmium-doped solid-state lasers at 2.1 µmand 2.9 µm

The use of the Ho3+ ion as the active dopant forsolid-state lasers opens up a number of very usefulmid-infrared transitions. In this section, we will con-centrate on the 5I7 → 5I8 ground-state transition at≈ 2.1 µm and the 5I6 → 5I7 transition at ≈ 2.9 µm; seethe energy-level scheme in Fig. 11.71. One of the sig-nificant shortcomings of Ho3+, however, is the lack of

+

$Q

@*

!*

@-

40

.

@%

$(Q

,+

@

$Q

$(Q

-B&

-B%

-B;

-B-

-B*

-!-

-4

Fig. 11.71 Partial energy-level scheme of Ho3+ with a Tm3+sensitizer. ET represents energy transfer

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Lasers and Coherent Light Sources 11.2 Solid-State Lasers 667

ground-state-absorption (GSA) transitions [11.754] thatoverlap with convenient high-power pump sources. Asa result, many of the early demonstrations of Ho3+-doped room-temperature crystalline CW lasers [11.616]involved sensitizing with Tm3+ in order to access theconvenient absorption bands and the practical cross-relaxation process Tm3+ provides, as we have discussedin Sect. 11.3.1. Energy migration amongst the Tm3+ions and a suitable Tm3+:Ho3+ concentration ratioensures that efficient energy transfer to Ho3+ takesplace [11.755, 756] (Fig. 11.71).

Three-level lasers at 2.1µm. The 2 µm transition5I7 → 5I8 in Ho3+ ions was first demonstrated as a laserin CaWO4:Ho3+ in 1962 [11.612] and in Tm3+-codopedCaWO4:Ho3+ in 1963 [11.757]. In 1971, pulsed op-eration at room temperature on this three-level lasertransition was demonstrated in LiYF4 [11.758]. CWlaser emission was achieved under Kr+-laser pumping inCr3+,Tm3+-codoped YSAG:Ho3+ and YSGG:Ho3+ in1986 [11.759]. Like in 2 µm Tm3+ lasers (Sect. 11.3.1),the Cr3+ codopant served as a sensitizer for the ab-sorption of pump light in the visible spectral range andexcitation of Tm3+ ions by energy transfer. The subse-quent excitation of Ho3+ ions by energy transfer fromTm3+ [11.760, 761] profits from the same Tm3+-Tm3+cross-relaxation as described for 2 µm Tm3+ lasers inSect. 11.3.1 (Fig. 11.71). In the 1980s, laser emissionat 2.1 µm in Ho3+ under diode pumping of the Tm3+3F4 level at 780–790 nm with pump thresholds as lowas 5 mW was achieved in YAG [11.762–764]. Com-pact, monolithic, low-threshold laser devices can beachieved in this way [11.765]. Besides many differ-ent garnet crystal systems, LiYF4 also regained interestas a host material for CW diode-pumped 2 µm Ho3+lasers around 1990 [11.766, 767]. An Yb3+-codoped,diode-pumped Ho3+ laser at 2.1 µm has also beendemonstrated [11.768]. Noise suppression [11.769,770],amplitude and frequency stabilization [11.771–773] of2 µm Ho3+ lasers have been investigated.

Attempts to exploit the rather large gain band-width of Ho3+ near 2.0–2.1 µm by tuning the emissionwavelength were reported in the early 1990s [11.774,775]. Nowadays, tuning ranges of > 80 nm areachieved in host materials such as mixed YSGG:GSAG[11.776], BaY2F8:Ho3+ [11.777], and KYF4 [11.778].Mode-locking experiments have resulted in 800 ps,370 ps, and 70 ps pulse durations, obtained inYAG:Cr3+,Tm3+,Ho3+ [11.714], LiYF4:Tm3+,Ho3+[11.779], and BaY2F8:Tm3+,Ho3+ [11.780], respective-ly. In mixed crystals of YSGG:GSAG:Cr3+,Tm3+,Ho3+,

which provide an inhomogeneously broadened, andtherefore smoother, gain shape, a pulse duration asshort as 25 ps could be achieved [11.776]. Q-switchedlaser operation [11.781–783] has been investigated andapplied for microsurgery [11.784].

Besides the cross-relaxation and energy-transferprocesses shown in Fig. 11.71, several other energy-transfer processes can occur in Tm3+,Ho3+-codopedmaterials [11.785–789], thereby making the systemrather complex and introducing parasitic processeswhich can deplete the Ho3+ 5I7 upper laser level,increase the laser threshold, and diminish the laser ef-ficiency. Rather than codoping the host with Tm3+ions and exciting the Ho3+ ions via nonradiative en-ergy transfer from the Tm3+ ions, one can directlypump the Ho3+ 5I7 upper laser level at 1.9 µm usinglaser diodes [11.790], the output from a 1.9 µm Tm3+laser [11.791], which provided up to 15 W [11.792] and19 W [11.694] output power from Ho3+, or a MgF2:Colaser [11.793]. This approach ensures a low quantumdefect and, hence, low heat generation in the laser crys-tal. This scheme was proven very successful by the useof a high-power Tm3+ fiber laser as the pump source,providing 6.4 W of output power and a slope efficiencyof 80% versus incident pump power at 1.9 µm [11.794].Also an efficient 2 µm Ho3+ single-frequency ring laserhas been demonstrated in this way [11.795].

The first fiber laser configuration making use ofthis transition employed ZBLANP glass (a variant ofZBLAN) and argon-ion pumping [11.796]. A year later,this was followed by the demonstration of an argon-ion-pumped Ho3+-doped silica fiber laser [11.797]. In bothcases, the fiber was singly doped with Ho3+, the out-put power < 1 mW, and each needed a relatively highpump power to reach laser threshold. Improvements inthe output power and efficiency have been made recentlywith Yb3+-doped silica fiber laser pumping of the 5I6level [11.798]; however, the output power had only in-creased to 280 mW because the pump fiber laser wasoperated at the low-efficiency 1150 nm wavelength. Us-ing the pump wavelength of 1100 nm, which is moreoptimal with respect to the operation of an Yb3+-dopedsilica fiber laser, the output power has been increased byapproximately an order of magnitude [11.799].

As mentioned above, a practical method of ef-ficiently generating laser emission on the 5I7 → 5I8transition is to codope Ho3+ laser ions with Tm3+ sen-sitizer ions. The first demonstration of a fiber laseroperating with the Tm3+,Ho3+ system occurred in1991 [11.800] when 250 mW was generated at a slopeefficiency of 52% from a Ti:sapphire-pumped fluoride

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668 Part C Coherent and Incoherent Light Sources

fiber laser. A year later [11.801], this work was followedby an increase in the Tm3+ concentration to improvecross-relaxation and resulted in a higher slope efficiencybeing obtained. Demonstration of a Tm3+,Ho3+-dopedsilica fiber laser soon followed [11.802, 803]; how-ever, owing to lower Tm3+ concentrations which forcesweaker cross-relaxation, significantly lower slope ef-ficiencies were measured, especially when pumped at1.064 µm [11.804]. When the concentrations are in-creased and the double-clad pump arrangement used,a significant augmentation of the output power to5.4 W has been demonstrated [11.805]. To date, thehighest output power of 8.8 W from a fiber laser operat-ing on the 5I7 → 5I8 transition has been produced bya diode-cladding-pumped Tm3+,Ho3+-doped fluoridefiber laser [11.747] (Fig. 11.70). In an analogous wayto recent demonstrations in bulk laser systems, tandem-pumping Ho3+ with a separate Tm3+ laser operatingat 1.9 µm may also prove effective in fibers, because itsimilarly exploits the cross-relaxation process betweenTm3+ ions but avoids any ETU between Ho3+ ions inthe 5I7 upper laser level and excited Tm3+ ions [11.789].When sensitizing with Yb3+ ions, a Ho3+-doped silicafiber laser at 2.1 µm has been shown [11.806] to operateat moderate efficiency levels despite the fact that the en-ergy transfer from Yb3+ to Ho3+ is quite nonresonant(Fig. 11.72).

Four-level lasers at 2.9µm. Laser emission at thetransition 5I6 → 5I7 near 3 µm in an Ho3+-doped crys-tal was demonstrated in 1976 [11.808]. SensitizingHo3+ with Yb3+ ions (see the energy-level schemein Fig. 11.72) in order to exploit the more-favorableabsorption features of Yb3+ has been used in diode-

D

!-)

40

,+

@

$Q

$(Q

-B&

-B;

-B%

-B-

$-

!&)

Fig. 11.72 Partial energy-level scheme of Ho3+ with Yb3+sensitizer displaying the measured lifetimes of Ho3+ whendoped into fluoride glass host [11.807]

pumped crystalline lasers for the generation of 2.9 µmoutput [11.809]. Recent reports on this transition includelaser investigations of YAlO3:Ho3+ [11.810], diode-pumped Yb3+-codoped YGSS:Ho3+ with 10 mJ outputenergy [11.811], and Cr3+,Yb3+-codoped YGSS:Ho3+tunable in the range 2.84–3.05 µm [11.812] and in Q-switched operation [11.813]. Since the 5I7 lower laserlevel is a metastable excited state with a longer lifetimethan the 5I6 upper laser level, it is difficult to achieveCW inversion on this transition. Cascade lasing on the5I6 → 5I7 and 5I7 → 5I8 transitions at 3 and 2 µm, re-spectively [11.814, 815], may help to deplete the 5I7level radiatively, i. e., without significant heat genera-tion. Passive Q-switching of this transition has also beendemonstrated [11.813].

The combined effect of the infrared absorptioncut-off wavelength of ≈ 2.5 µm for pure silica glassand the strong multiphonon relaxation quenching ofmid-infrared transitions of rare-earth ions in this hostmeans that four-level fiber lasers operating on the5I6 → 5I7 transition at ≈ 2.9 µm have only involvedfluoride glass as the host material. The first demon-stration of a fiber laser using this transition [11.816]produced only ≈ 13 mW when pumped at a wavelengthof 640 nm. High-power cascade lasing at 2.9 µm and2.1 µm has been employed to extend the output powerto 1.3 W [11.817] by removing bottlenecking at the 5I7level [11.807] via the second laser transition at 2.1 µm.In an analogous manner to the Er3+-doped fluorideglass system discussed below, the most successful ar-rangement to date for extracting high power from thistransition has involved the use of Pr3+ as a desensi-tizer for the 5I7 energy level: a maximum output powerof 2.5 W was produced when the pump wavelengthof 1100 nm from a Yb3+-doped silica fiber laser wasused [11.818].

An Yb3+-sensitized Ho3+-doped ZBLAN fibercould be directly pumped with diode lasers and mayefficiently provide high-power 2.9 µm output withoutthe costly requirement of an intermediate laser system.Initial spectroscopic results look encouraging [11.819],however, the many excited ion interactions that a fluoridehost provides may be problematic. A recent demonstra-tion has shown that ion–ion interactions (specificallyETU) in Ho3+-doped fluoride glass are critical to theproduction of 2.9 µm output from singly Ho3+-dopedfluoride glass fiber lasers [11.820].

Erbium-doped solid-state lasers at 2.7–2.9 µmFor a long time, the development of erbium lasers op-erating on the 4I11/2 → 4I13/2 transition near 3 µm was

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dominated by crystalline systems. The early success ofthe erbium 3 µm crystal laser has given rise to a signifi-cant amount of spectroscopic investigations. This has ledto a deep understanding of the rather complex populationmechanisms of this laser system and to the developmentof a large number of suitable host materials.

Crystalline lasers. The first observation of coherentemission near 3 µm from erbium ions was reportedin 1967 [11.613]. Yttrium aluminum garnet (YAG)was demonstrated as a host for the erbium 3 µmlaser [11.822] in 1975. In 1983, the first CW lasing near3 µm was obtained in this material [11.823]. At aboutthe same time, it was established [11.824–827] that en-ergy transfer processes [11.828] between neighboringerbium ions in the host lattice play an important rolein this laser system. Energy transfer processes can be-come very efficient at high excitation density [11.829]and govern the population mechanisms of the 3 µmlaser at high erbium concentration. In the energy-levelscheme of Fig. 11.73a, the important ETU and cross-relaxation processes are introduced. The ETU process(4I13/2,

4I13/2) → (4I15/2,4I9/2) leads to a fast deple-

tion of the lower laser level and enables CW operationof a laser transition which, otherwise, could be self-terminating owing to the unfavorable lifetime ratio ofthe upper compared to the lower laser level. The ETUprocess from 4I13/2 can be so dominant that even underdirect pumping of the 4I13/2 lower laser level and subse-quent excitation of the 4I11/2 upper laser level by ETU,3 µm laser operation was demonstrated in several hostmaterials [11.830].

This ETU process offers another great advantage.Half of the ions that undergo this process are upcon-verted to the 4I9/2 level and, by subsequent multiphononrelaxation, are recycled to the 4I11/2 upper laser levelfrom where they can each emit a second laser photon,for a single pump-photon absorption. For a large numberof ions participating in this process, a slope efficiencyηsl of twice the Stokes efficiency ηSt = λpump/λlaseris obtained [11.831], because the quantum efficiencyηQE = nlaser/npump of pump photons converted to laserphotons increases from 1 to 2 (λ and n are thewavelengths and photon numbers of laser and pumptransitions, respectively):

ηsl = ηQEηSt = 2ηSt . (11.90)

This is illustrated in Fig. 11.74.In a simple rate-equation system which includes the

processes shown in Fig. 11.73a the slope efficiency is

, """'

,

.

*B)

*B()

*B-)

,+I

,+I

*B)

*!()

*!&)@)

,+I

,+I .

- -

*-%&;(

'&"'2

*4)

Fig. 11.73 (a) Partial energy-level scheme of erbium indicating thepump and laser transitions, ETU1 from 4I13/2, ETU2 from 4I11/2, andcross-relaxation (CR) from the thermally coupled 4S3/2 and 2H11/2

levels. (b) Macroscopic parameters of ETU1 from 4I13/2(W11) andETU2 from 4I11/2(W22) and the ratio W11/W22 in ZBLAN:Er3+bulk glasses. (After [11.821])

given by [11.831]

ηsl = ηStln(1− T )

ln[(1− T )(1− L)]

(2− b2

1

b22

W22

W11

),

(11.91)

with T the transmission of the out-coupling mirror, Lthe internal resonator losses, and bi and Wii , the Boltz-mann factors and ETU parameters of the upper (i = 2)and lower (i = 1) laser levels, respectively. If ETU oc-curs only from the lower laser level, i. e., W22 = 0, weobtain the predicted factor of two increase in slope ef-ficiency from (11.91). The slope efficiency is reduced,however, by the resonator losses, the imperfect modeoverlap, and the ETU process from the upper laser levelin the case of W22 > 0. In the investigated host mater-ials, the parameters Wii of both ETU processes increasewith increasing erbium concentration because of the in-fluence of energy migration within the erbium 4I11/2 and4I13/2 levels on ETU. The slope efficiency of (11.91) isoptimum for a maximum ratio W1/W2. Spectroscopy ofthese processes in crystal hosts and laser experimentsrevealed that the maximum ratio is obtained at dopantconcentrations of ≈ 12–15% in BaY2F8 [11.832, 833],≈ 15% in LiYF4 [11.661], ≈ 30% in YSGG [11.834],

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670 Part C Coherent and Incoherent Light Sources

,

*B)

*B ()

,+I40

*B-)

- -

*!()

*B)

,+I ,+I

$ $- $&- $;$$$H,

Fig. 11.74 Partial energy-level scheme of erbium illustrat-ing the process of energy recycling from the lower tothe upper laser level by ETU. Indicated are the relativepump rate ηp of the upper laser level and the quantum effi-ciency ηQE which increases from 1 to 2 if a large numberof ions participate in the process. (After [11.836])

and ≈ 50% in Y3Al5O12 [11.835]. A trend in this se-ries is the increase of the optimum erbium concentrationwith phonon energy of the host material.

Energy recycling by ETU is the most efficientway to operate a CW erbium laser near 3 µm. Thehighest slope efficiency obtained experimentally is cur-rently 50% in LiYF4:15% Er3+ [11.837]. The pumpwavelength that provides the highest Stokes efficiencyof ηSt = λpump/λlaser = 35% is 980 nm, which cor-responds to pumping directly into the upper laserlevel [11.838] (Fig. 11.73a). The highest slope efficiencyobtained experimentally [11.837] is currently ηsl = 50%in LiYF4:15% Er3+. This result shows that energy re-cycling is indeed efficient and that slope efficienciesfar above the Stokes efficiency can be obtained underCW pumping. Under quasi-CW excitation, the slope ef-ficiency is strongly reduced [11.839], because the lowerlaser level is much less populated than in the steady-state regime and ETU is less efficient [11.840, 841].Other operational regimes which deplete the lower laserlevel without recycling the energy to the upper laserlevel are less efficient. Consequently, neither co-lasingat the 1.6 µm transition from the lower laser level to theground state [11.842] nor energy transfer from the er-bium lower laser level to a rare-earth codopant [11.660]have reached the efficiency of the recycling regime.

Lifetime quenching of the 4I11/2 upper laser levelby multiphonon relaxation is stronger in oxide com-

pared to fluoride host materials because of the largermaximum phonon energies in oxides. With an energygap between the 4I11/2 upper and 4I13/2 lower laser lev-els of ≈ 3400–3500 cm−1, the radiative decay becomesdominant for phonon energies below ≈ 550 cm−1. Sincea long lifetime of the 4I11/2 upper laser level providesa small pump threshold, fluorides are preferable hostmaterials [11.843] for this laser transition if the pumppower is not many times above threshold.

In the 1980s and 1990s, numerous host materialswere investigated for CW and pulsed laser operation ofEr3+ in the 2.7–2.9 µm region. A major role was playedby the family of garnet crystals YAG, YSGG, YSAG,YGG, and GGG [11.615, 844, 845]. In the early years,Cr3+ codoping was used in order to improve the absorp-tion of broadband flashlamp pump light in the visiblespectral range and transfer the absorbed energy fromCr3+ to Er3+. Typical output characteristics obtainedwere 2.7 W average power at a pump energy of 5 J witha repetition rate of 10 Hz [11.845]. Doping levels up to100% substitution of Y3+ by Er3+ were tested [11.846].A new class of host materials, fluoride crystals such asLiYF4, BaY2F8, and others, became of importance inthe late 1980s [11.847–850]. Laser thresholds as lowas 5 mW were obtained under CW excitation by diodelasers [11.849].

In recent years, researchers have obtained CW andquasi-CW diode-pumped output power levels exceeding1 W at 3 µm from fluoride [11.661] and oxide [11.851–853] crystalline host materials. A significant problem inthe energy-recycling regime is increased heat genera-tion due to the multiphonon relaxation 4I9/2 → 4I11/2that follows each ETU process from the lower laserlevel [11.676]. Glass bulk materials [11.854] suffer fromthe same thermal and thermo-optical drawbacks as thecrystalline bulk materials, with even decreased thermalconductivity in the glass. A possible solution is diodeside pumping, which leads to lower excitation den-sities and correspondingly weaker ETU processes, aswell as better heat removal in the slab geometry. Thehighest output powers of 1.8 W and later 4 W from anerbium 3 µm crystal laser [11.855, 856] have been ob-tained in this way. A reduced erbium concentration withcorrespondingly smaller parameters of the ETU pro-cesses may aid this approach. However, the efficiencyof the energy-recycling regime cannot be reached in thisapproach.

Other special configurations include the operationof Er3+-doped YAG, GGG, and YSGG lasers in mono-lithic cavities with output powers up to 0.5 W andtunable single-frequency output [11.857] as well as mi-

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crolasers in YSGG [11.858]. Pulsed output at 3 µm hasbeen generated from erbium-doped crystalline mater-ials in many configurations and regimes, e.g., underquasi-CW pumping [11.839, 841], active [11.859–868]and passive [11.869–872] Q-switching, and mode-locking [11.619, 869, 873].

Fiber lasers. The erbium-doped fluoride fiber representsa promising alternative for the construction of a com-pact and efficient all-solid-state laser emitting at thetransition at 3 µm. Due to its geometry, the fiber pro-vides large flexibility and potentially high pump andsignal beam intensities without the drawbacks of ther-mal and thermo-optical effects. The first erbium 3 µmfiber laser was demonstrated in 1988 [11.875]. Single-mode [11.876] and diode-pumped [11.877] operationwere demonstrated shortly afterwards. Although the life-time of the 4I13/2 lower laser level exceeds that of the4I11/2 upper laser level, CW lasing can be obtainedon this four-level-laser transition in ZBLAN (but alsoin fluoride crystals, see the paragraph on crystallinelasers) without employing special techniques to depop-ulate the 4I13/2 lower laser level, because the lowerlaser level is not fed significantly by luminescent de-cay or multiphonon relaxation from the upper laserlevel [11.878]. In addition, the Stark splitting of the laserlevels contributes to population inversion, because thelaser transition occurs between a low-lying Stark com-ponent of the upper and a high-lying Stark componentof the lower laser level [11.836]. During the relaxationoscillations at the onset of lasing, a red-shift of the las-ing wavelength is often observed in erbium 3 µm lasersystems [11.879–882], because the excitation energy isaccumulated in the long-lived 4I13/2 lower laser level andthe character of the lasing process changes from four-level to three-level lasing [11.836]. For the same reason,the tunability range of a 3 µm CW laser [11.883] isnarrowed and red-shifted with increasing pump power.

Pump excited-state absorption (ESA), which ispresent in Er3+ at almost all available GSA wave-lengths [11.884], has a major influence on theperformance of low-doped, core-pumped erbium 2.7 µmZBLAN fiber lasers because of the significant amountof ground-state bleaching and excitation of the laser lev-els under these conditions [11.885]. Pumping at 980 nmdirectly into the upper laser level provides the highestStokes efficiency of ηSt = λpump/λlaser = 35% [11.838].However, ESA at 980 nm from the 4I11/2 upper laserlevel [11.886] is detrimental to lasing. Experimentally,the best pump wavelength [11.885] is near 792 nm, at thepeak of ESA from the 4I13/2 lower laser level [11.874];

see the measured GSA and ESA cross sections inFig. 11.75a. Depletion of the lower laser level by ESAfavorably results in a redistribution of its population den-sity and overcomes the bottleneck that results from thelong lower-level lifetime. However, slope efficienciesobtained in this way were < 15%. Moreover, satura-tion of the output power at 2.7 µm was observed andthe highest reported output powers were in the 20 mWregion [11.887, 888]. The excitation of the metastable4S3/2 level (lifetime ≈ 580 µs [11.821]) led to inversionwith respect to the 4I13/2 level. A second laser transi-tion at 850 nm repopulated the 4I13/2 lower laser level ofthe 2.7 µm transition (Fig. 11.75b), causing the 2.7 µmlaser to saturate at low output powers [11.885]. Sig-nificant improvement in the performance of this lasersystem was obtained by deliberately operating a thirdlaser transition 4S3/2 → 4I9/2 at 1.7 µm, thereby sup-pressing the competitive laser at 850 nm and recyclingthe excitation energy accumulated in the 4S3/2 level intothe upper laser level; see the energy-level scheme inFig. 11.75b. The slope efficiency of the 2.7 µm transi-tion increased significantly to 23% [11.889], close to theStokes efficiency limit of 29% under 800 nm pumping.An output power of 150 mW was demonstrated ex-perimentally [11.889]. Also a three-transition-cascade

&;

G 6

,

$&Q

*B)

,40*B()

*B-)

40

*B)

*@)

(

,40'"

*4)

*!&)

*!()

;-

$&Q

%$(

- -;Q*B)

*B-)

*B)

&( ; ; ; ; ;*

*

;

%

*

Fig. 11.75 (a) Absorption cross sections in ZBLAN:Er3+ near800 nm: GSA 4I15/2 → 4I9/2 and ESA 4I13/2 → 2H11/2,

4I11/2 →4F3/2, and 4I11/2 → 4F5/2 (After [11.874]). (b) Partial energy-level scheme of erbium indicating the processes relevant to theZBLAN:Er3+ cascade laser: lower loop with GSA to 4I9/2, mul-tiphonon relaxation, laser transition at 2.7 µm, luminescent decay,and upper loop with ESA to 2H11/2, thermal relaxation, laser tran-sition at 1.7 µm, multiphonon relaxation, laser transition at 2.7 µm.Competitive lasing at 850 nm is suppressed in the cascade regime

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672 Part C Coherent and Incoherent Light Sources

lasing regime with additional lasing at the transition4I13/2 → 4I15/2 near 1.6 µm was demonstrated [11.890].

In ZBLAN fibers with higher dopant concentra-tions of typically 1–5 mol % (≈ 1.6–8 × 1020 cm−3) andwith the double-clad geometry, ESA is much lessimportant, because the reduced pump intensity withlow-brightness diode lasers leads to smaller excita-tion densities. Currently, the most successful approachtowards a high-power erbium 2.7 µm fiber laser iscodoping of the fiber with Pr3+ [11.891,892]. This ideawas reported already in [11.887, 893, 894] and was pro-posed for the double-clad fiber laser in [11.895]. Inthis approach, the Er3+ 2.7 µm transition is operatedas a simple four-level laser; see the energy-level schemein Fig. 11.76a. The 4I13/2 lower laser level is depopu-lated by the energy transfer process ET1 to the Pr3+codopant and fast decay to the ground state by mul-tiphonon relaxation within Pr3+. The energy-transferprocess ET2 from the 4I11/2 upper laser level to the Pr3+codopant is weak [11.821]. The strong lifetime quench-ing of the 4I13/2 lower laser level significantly reducesground-state bleaching and excitation of the laser lev-els, thus making the influence of ESA negligible, but

"6 5G

,

*B)

*B()

*B-)

40

*B)

*!()

*!&)@)

,+

5G

* % ;

!*

*

!!@%

@-

@*

>

,+

*4)

Fig. 11.76 (a) Partial energy-level scheme of erbium indicating theprocesses relevant to the ZBLAN:Er3+ lifetime-quenching laser:GSA at 980 nm to the 4I11/2 upper laser level (or at 790 nm to the4I9/2 pump level and subsequent multiphonon relaxation to 4I11/2),laser transition to the 4I13/2 lower laser level, and relaxation tothe ground state via energy transfer ET1 to the Pr3+ codopant.The energy transfer ET2 from the 4I11/2 upper laser level to thePr3+ codopant is weak. (b) Output power at 2.7 µm under 792 nmpumping. (After [11.891])

similarly preventing energy recycling by ETU [11.896].Each pump photon can at best produce one laser pho-ton in the Er3+,Pr3+-codoped system. The theoreticallimit of the slope efficiency is given by the Stokesefficiency, which is 29% under 800 nm pumping. Ex-perimentally, a slope efficiency of 17% and an outputpower of 1.7 W were obtained [11.891] (Fig. 11.76b).Other researchers [11.897] reported output powers of660 mW. Since ESA from both laser levels is negligible,the system can alternatively be pumped near 980 nm,which provides a Stokes efficiency of 35%. In this way,the experimental slope efficiency could be increased to25% [11.898]. With improvements in diode-laser tech-nology and an optimized fiber design, an output powerof 5.4 W at 2.7 µm with a slope efficiency of 21% couldrecently be demonstrated from an Er3+,Pr3+-codopedZBLAN fiber laser [11.892, 899].

The first steps toward pulsed output from erbium3 µm ZBLAN lasers [11.882,900,901] were unsatisfac-tory in terms of output energies and average powers.There have also been attempts to operate the ZBLANfiber laser in the energy-recycling regime. The pa-rameters Wii of both ETU processes in ZBLAN bulkglasses [11.821] versus Er3+ concentration are shown inFig. 11.73b. The criterion for optimization of the slopeefficiency in (11.90) is maximizing the ratio W11/W22.For Er3+ concentrations of > 2–3 mol % at which ETUprocesses become important, this ratio is ≈ 3, see thedashed line in Fig. 11.73b, a more favorable value thanreported for LiYF4:Er3+ [11.851]. Energy recycling byETU at high Er3+ concentrations [11.902] might leadto output powers at 3 µm on the order of 10 W. In earlyattempts, two research groups tried to exploit energyrecycling [11.903, 904], however the slope efficienciesin these experiments did not exceed the slope efficien-cies obtained in Er3+,Pr3+-codoped fibers pumped atcorresponding pump wavelengths [11.891,892,898]. Infibers with relatively large core diameters and there-fore transverse multimode operation, singly Er-dopedwith concentrations smaller than those required forefficient energy recycling, 3 W of output power wasobtained [11.905]. With further increase of the corediameter to 90 µm, the highest pulsed output of> 0.5 mJcould be demonstrated by the same researchers. Re-cently, 8 W of output power with a slope efficiencyof 24.4% were demonstrated from a highly Er-doped(60 000 ppm) ZBLAN fiber when dual-end-pumping thefiber with 25 W of launched pump power at 975 nm.This result represents the highest output obtained by theend of 2005 from a fiber laser near 3 µm. Again, theachieved slope efficiency of 24.4% is very similar to

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that of 25% obtained with the same pump wavelength ina Er3+,Pr3+-codoped fiber [11.898]. Therefore, it mustbe concluded that efficient energy recycling with cor-respondingly improved slope efficiency similar to theresult of 50% in a crystalline host material [11.837] isstill lacking.

Dysprosium-doped solid-state lasers at2.3–2.4 µm and 2.9–3.4 µm

The search for new mid-infrared laser transitions de-pends entirely on the structure of the energy-leveldiagram of the rare-earth ions. The Dy3+ ion offersa rather dense energy-level scheme in the infrared spec-tral region, resulting in a range of absorption peaks, aswell as a four-level laser transition at 2.3–2.4 µm anda phonon-terminated 3 µm laser transition. The transi-tion at 2.3–2.4 µm was among the first laser transitionsreported and the first to be demonstrated in continuous-wave operation [11.609, 610]. Because of the denseenergy-level scheme of Dy3+, multiphonon relaxationis an issue of concern in this ion, hence low-phonon hostmaterials may improve its laser performance. Recently,room-temperature laser emission at 2.43 µm was re-ported in the low-phonon host materials CaGa2S4:Dy3+and KPb2Cl5:Dy3+. At the 3.4 µm transition, room-temperature laser oscillation from Dy3+ in BaYb2F8has been reported [11.906]. The 1100 nm output froma Yb3+-doped silica fiber laser has been successfullyused to pump a Dy3+-doped fluoride fiber laser [11.907].In this case, a maximum output power of 275 mW wasgenerated with a slope efficiency of only ≈ 5%, however,when the pump wavelength was increased to ≈ 1.3 µmusing a YAG:Nd3+ laser, the slope efficiency is approx-imately quadrupled to ≈ 20% [11.908]. Reduced levelsof pump ESA are believed to cause this augmentation inthe slope efficiency. Future Dy3+-doped fluoride fiberlasers may benefit from further increases in the pumpwavelength to 1.7 µm or 2.8 µm.

Before turning our attention to rare-earth-ion-dopedsolid-state lasers in the wavelength range beyond 3 µm,it should be noted that solid-state lasers based on theactinide ion U3+ have also attracted attention in therecent past [11.909, 910].

Solid-state lasers at wavelengths beyond 3 µmLaser wavelengths longer than typically 3 µm are gen-erally difficult to achieve in solid-state host materialsby direct generation from rare-earth or transition-metalions, because the energy gap between the upper andlower laser level is necessarily small and all the com-mon oxide and fluoride host materials possess maximum

phonon energies, which lead to fast multiphonon relax-ation of the excitation of the upper laser level. Therefore,many of the laser transitions reported in the literature re-quire cooling of the active device. On the other hand, theattractiveness of this wavelength range for a number ofapplications has inspired the search for host materialswith maximum phonon energies below ≈ 300 cm−1.

In crystalline hosts, a flashlamp-pumped, cooledEr3+ laser at 4.75 µm [11.614], a CW diode-pumped, cooled Er3+ laser at 3.41 µm [11.911], anda pulsed diode-pumped room-temperature Er3+ laserat 4.6 µm [11.912] have been operated. A room-temperature BaY2F8:Ho3+ laser was operated at3.9 µm [11.913]. Due to its dense energy-level scheme,Dy3+ offers a large range of possible mid-infraredtransitions. Recently, room-temperature lasing of Dy3+transitions in the low-phonon host materials CaGaS2and KPb2Cl5 was demonstrated [11.914], in the for-mer material even at 4.31 µm. The longest-wavelengthlasers shown to operate in a solid-state mater-ial are room-temperature 5 µm and 7 µm lasers inLaCl3:Pr3+ [11.915, 916].

The operation of lasers at wavelengths of3.22 µm [11.917] and 3.95 µm [11.918] has beenobtained from Ho3+-doped ZBLAN fiber and at3.45 µm [11.919] from Er3+-doped ZBLAN fiber. Itwas, however, necessary to cool the ZBLAN fiber forthe 3.45 µm and 3.95 µm transitions. These two lasertransitions span five or six maximum phonon ener-gies in ZBLAN, therefore the lifetime of the upperlaser level for each of these transitions is short and en-genders an increase in the pump threshold comparedto other ZBLAN fiber lasers operating at the shortermid-infrared wavelengths. In addition, the lower laserlevels of these transitions possess quite long lifetimesand some saturation of the output power has been ob-served [11.920]. This problem (while it can be mitigatedwith cascaded lasing), combined with the use of incon-venient pump sources has impeded the full utilizationof these laser transitions. The 3.95 µm wavelength emit-ted from the cooled ZBLAN fiber laser is currently thelongest laser wavelength that has been generated froma fiber laser.

Generating wavelengths longer than 3 µm from fiberlasers is a task that tests the limits of current glasstechnology. The need for lower phonon energies has tobe balanced with acceptable mechanical, chemical, andthermal properties. Since the highly developed ZBLANglass is only useful for laser transitions up to 3–3.5 µm,glasses such as the chalcogenides [11.921] will needto fill the gap. It is because these glasses have to

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be drawn into low-loss fiber that has prevented long-wavelength emission to the extent that is possible incrystalline-based solid-state lasers. Creating efficient,high-power mid-infrared fiber lasers with output wave-lengths > 3 µm is at the forefront of current fiber-laserresearch efforts.

As mentioned above, fiber lasers operating on lasertransitions that have wavelengths > 3 µm will need touse glasses, which have very low phonon energies. Whilerare-earth-ion-doped heavy-metal oxides [11.925] havebeen studied for 2–3 µm mid-infrared emission; to date,there has been no report of laser action for a fiber lasercomprised of such a glass. Heavy-metal oxides do notseem to be suitable for lasers at wavelengths beyond3 µm, because their maximum phonon energies are com-parable to fluoride glasses and are too high for lasertransitions beyond 3 µm.

The chalcogenide glasses have been doped witha number of rare-earth ions including Ho3+ [11.924],Tm3+ [11.923], Tb3+ [11.923], Dy3+ [11.922],Pr3+ [11.926], and Er3+ [11.927, 928] for studies into> 3 µm mid-infrared luminescence (Table 11.21). Fiber-laser action has been reported, however, only for anNd3+-doped GLS glass operating at a wavelength of≈ 1 µm [11.929]. Recent demonstrations of fabricatingBragg gratings [11.930], single-mode fibers [11.931]and holey fibers [11.932] with chalcogenide glass high-light the utility of this glass for fiber-based applications;however, the purity and toxicity of the starting materialsand the difficulty of making ultralow-loss fiber currentlyimpede the widespread use of chalcogenide glass formid-infrared fiber-laser applications. Once these obsta-cles have been overcome, future > 3 µm fiber laserswill most likely involve the rare-earth ions Pr3+, Nd3+,Dy3+, and Ho3+ doped into chalcogenide glass, becausemost of the important mid-infrared transitions relevant tothese ions can be accessed with pump-photon wavenum-bers < 10 000 cm−1. Judicious choice of the overalldopant-ion concentration and the use of particular sen-

Table 11.21 Examples of luminescent transitions investi-gated as candidates for mid-infrared lasers in sulfide glasses

Ion λlaser (µm) Transition Ref.

Dy3+ 3.2 6H13/2 →6 H15/2 [11.922]

Tm3+ 3.8 3H5 →3 H4 [11.923]

Ho3+ 3.9 5I5 →5 H6 [11.924]

Dy3+ 4.3 6H11/2 →6 H13/2 [11.922]

Tb3+ 4.8 7F5 →7 F6 [11.923]

Ho3+ 4.9 5I4 →5 I5 [11.924]

sitizer and quenching ions will enable the production ofefficient > 3 µm output some time in the future.

ConclusionsIn the roughly four decades since the demonstrationof the first mid-infrared solid-state lasers, thousands ofscientific papers have been published which have re-ported on lasing in novel host materials, replacementof flashlamps by ion-laser and later diode-laser pumpsources, ever-growing output powers, higher efficien-cies, larger tunability ranges, shorter pulse durationsand the like. A general tendency is that, the shorterthe wavelength, the better the laser performance. Whenwe approach longer wavelengths in the mid-infraredspectrum, we find that the quality and durability ofthe required low-phonon-energy host materials decline,Stokes and slope efficiencies decrease, whereas the ther-mal problems increase. While many crystalline hostmaterials and the corresponding laser techniques havematured during the 1990s, the fast development of high-power, fundamental-mode fiber lasers, which could bewitnessed in the 1 µm spectral range, has now alsoreached the mid-infrared spectral region. However, theassumption that, due to its large surface-to-volume ra-tio, the fiber geometry might avoid all thermal problemshas been questioned by several recent high-power fiber-laser experiments in the near- and mid-infrared spectralregion. These phenomena are, in principle, not muchdifferent from the situation found in crystalline lasers.Nevertheless, there remain distinct differences betweenthese two host categories. When flexibility of the res-onator design, short pulses, and high peak powers arerequired, crystalline lasers have advantages. On the otherhand, fiber lasers are preferred when high beam qualityor low pump threshold combined with medium CW out-put power are desired. The low pump threshold of fiberlasers is an invaluable advantage when cascade-laseroperation is required to depopulate the long-lived ter-minating level of one laser transition by a second lasertransition. The comparatively low dopant concentrationsthat are useful in fiber lasers due to the long interactionlengths can minimize energy dissipation by interionicprocesses but, equally, limit the exploitation of these pro-cesses as a tool to optimize the population mechanismsof a certain laser system, as has been done successfullyin several of the mid-infrared crystalline laser systemsdiscussed above. Although still a great challenge with re-spect to fabrication process and durability, low-phononcrystalline and fiber host materials have the potential torevolutionize CW mid-infrared lasers in the wavelengthrange between 3–5 µm.

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11.2.5 Transition-Metal-Ion Lasers

BasicsIn this chapter an overview of transition-metal-ion laserswill be given. The main focus lies on the laser charac-teristics and results, a more spectroscopically orientedoverview is given in [11.933]. Transition metals are el-ements of the third, fourth and fifth row of elementsin the periodic table. Laser oscillation has so far onlybeen obtained with ions of the transition metals of thethird row (the Fe row; Ti to Cu). Due to the strongcoupling between the electronic levels of the transition-metal ion with the surrounding field established by thecrystalline environment, transition-metal-ion lasers areusually tunable over a wide spectral range up to sev-eral hundred nanometers. These lasers are of interest fora wide field of applications, e.g., in scientific research,in medicine, for measurement and testing techniques,ultrashort-pulse generation, and communication. Theycan also be used as coherent light sources for second-harmonic generation, for optical parametric oscillators,and for sum- and difference-frequency generation.

The energy-level scheme and thus also the spectro-scopic and laser characteristics of a transition-metal ionin a crystalline field is strongly dependent on the valencestate of the ion, the number of ligands (i. e., the coor-dination number) and the strength and symmetry of thesurrounding crystalline field. Therefore it is not possibleto draw some kind of Dieke diagram for the transition-metal ions, as it is possible for the trivalent rare-earthions (see the section on 4f–4f transitions in Sect. 11.2.1).The energy-level schemes of the transition-metal ions incrystalline hosts are in principle described by the so-called Tanabe–Sugano diagrams [11.934–936]. Thesediagrams are distinguished by the number of electronwithin the 3d electron shell. In these diagrams the en-ergy of a specific level of the transition-metal ion isdepicted as a function of the crystal field strength. Wewill not describe the quantum-mechanical backgroundto obtain these diagrams, which would be beyond theframe of this handbook. The reader is referred to theappropriate literature [11.937–942].

In comparison to laser systems based on the 4f↔4ftransitions of trivalent rare-earth ions, lasers based on3d↔3d transitions are in general more affected bya higher probability of excited-state absorption, a higherprobability of nonradiative decay, and a higher saturationintensity, leading to higher laser thresholds. Often laseroscillation cannot be obtained at all. In the following sec-tion, the focus is on specific transition-metal-ion lasers,ordered according to their laser wavelength from visible

G 6

.*N4*

+N/0*

* -

.*ND4"80-=8

.*N.*

.*ND0-

-N4-<*!2-N4<*2-N/-<*

.N0

.N-4*

.N/0*

.N.0!%

.N/04%

.N40!%

+N0

+ND0

.NE4

.N.4

.NE4

.N.+

<N!

<N..!

CN

CN-

CN!

.N!

!NE4

Fig. 11.77 Overview of tunable solid-state lasers based on transition-metal-ion-doped crystals

over the near-infrared to the mid-infrared. This overviewincludes Ti3+ lasers, especially Ti3+:Al2O3, Cr3+lasers, Cr4+ lasers and finally Cr2+ lasers. At the endof this section, other transition-metal-ion lasers are pre-sented, including Co2+ and Ni2+ lasers. Finally, somegeneral comments about transition-metal-ion lasers willbe given in the last section. In Fig. 11.77 an overviewof lasers based on 3d↔3d transitions of transition-metalions is given. It can be seen that almost the whole spectralrange between 650 nm and 4500 nm is covered.

Overview of Transition-Metal-Ion LasersTi3+ lasers. Ti3+-doped Al2O3 (Ti:sapphire) has beenintensively investigated as a tunable laser material sincethe first laser operation was reported [11.943, 944].Efficient laser oscillation was obtained in Al2O3 inpulsed [11.943,945,946] and continuous-wave [11.945,947, 948] operation, for further references see [11.213].The tuning range covers more than 400 nm and spansapproximately 670–1100 nm [11.949]. It has a rela-tively high emission cross section of approximately

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,#

."66

, +

,+

Fig. 11.78 Basic energy-level schemes of 3d1 ions in octa-hedral and tetrahedral coordination

4.1 × 10−19 cm2 [11.950] at the maximum of the emis-sion spectrum. A slope efficiency of 62% has beenreported [11.951], which is close to the quantum limitin that experiment of 78% (pump wavelength 589 nm,emission wavelength 750 nm). This indicates the lowintrinsic losses of the system.

The Ti3+ ion belongs to the 3d1 configuration,which is very favorable with respect to laser appli-cation because of its simple energy-level scheme (seeFigs. 11.78, 11.79). There are only two 3d1 energy lev-els, which diminishes the possibility of excited-stateabsorption of the laser radiation, a process that limits

.9 "H

,#

+,.+

,40

40

J+,

J++

Fig. 11.79 Schematic diagram of the Jahn–Teller effectfor octahedrally coordinated d1-systems. 2E, 2T2 and CT(charge transfer) are the energy levels, GSA, ESA, and emare ground-state absorption, excited-state absorption, andemission, respectively. nr represents the nonradiative de-cay via tunneling between the excited state and the groundstate. ∆EJT is the Jahn–Teller stabilization energy

the tuning range and efficiency of other transition-metal-ion-doped lasers. The orbital degeneracy of thed levels is removed due to the Jahn–Teller effect, whichyields the large absorption and emission bandwidths.The room-temperature absorption and emission spec-tra are shown in Fig. 11.80. Excited-state absorptiontransitions occurs to energy levels correlated to chargetransfer and conduction-band levels and between thetwo Jahn–Teller-split components of the upper (2E) dlevel. In Ti:sapphire, excited-state absorption was notobserved in the spectral range of emission [11.952]. InTable 11.22 the laser-relevant parameters of Ti:sapphireare listed. Besides the favorable spectroscopic and lasercharacteristic, the Al2O3 host material offers a varietyof advantageous properties: a high thermal conductivity,as well as mechanical and chemical hardness.

To date, Ti3+:Al2O3 is the most common and com-mercially available tunable solid-state laser. Nowadays,it can be pumped with frequency-doubled Nd lasersat wavelengths around 532 nm, thus efficient all-solid-state laser operation is possible. Formerly, Ar+-ion laserpumping was applied. In commercial systems, overallefficiencies as high as 30% are obtained.

Besides the broad tuning capability of theTi:sapphire laser, its ability for ultrashort-pulse gen-eration and amplification is especially exploited. Inmode-locked operation, pulses as short as 5 fs [11.953–957] and octave-spanning spectra (e.g., 600 nm to1200 nm [11.957]) have been obtained.

Laser oscillation with reasonable efficiency hasalso been reported for Ti3+:BeAl2O4 [11.958–961]

Table 11.22 Overview of Ti3+:Al2O3 laser-relevant param-eters [11.948, 950]

Index of refraction 1.76

Absorption cross section 6.5 × 10−20 cm2 (E ‖ c)

Fluorescence lifetime 3.2 µs

Fluorescence bandwidth ≈ 200 nm

(FWHM)

Peak emission wavelength 790 nm

Peak stimulated emission 4.1 × 10−19 cm2 (E ‖ c)

cross section

2.0 × 10−19 cm2 (E⊥c)

Quantum efficiency ≈ 0.9–1

Saturation fluence 0.9 J/cm2

Dopant concentration 0.1% (weight)

Growth Czochralski, heat exchange

Tm 2050 C

Thermal conductivity 28 W/mK

Thermal lens (dn/dT ) 12 × 10−19 K−1

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Lasers and Coherent Light Sources 11.2 Solid-State Lasers 677

(Table 11.23). However, the efficiency as well as the tun-ing range is smaller than Ti3+:Al2O3, thus this laser hasno commercial application. In Y3Al5O12 [11.962] andYAlO3 [11.963, 964] crystals doped with Ti3+ the ob-served efficiencies are very low (Table 11.23). The mainreason for this is excited-state absorption in the spec-tral region of emission and absorption [11.952, 965]. InTi3+-doped systems nonradiative decay processes due tophonon-assisted tunneling between the Jahn–Teller-splitexcited and ground states also occur, which prevent effi-cient laser oscillation, e.g., in Ti3+:Y3Al5O12 [11.966,967].

Cr3+ lasers.Basics. The Cr3+ ion is almost always found in oc-tahedral coordination and its energy-level scheme isdescribed by the Tanabe–Sugano diagram shown inFig. 11.81. In low crystal fields, the first excited stateis the 4T2 level, whereas in strong crystal fields the2E level is the first excited state. This means, that ei-ther broadband emission (4T2 → 4A2) or narrow-lineemission (2E → 4A2) occurs. The absorption spectraare dominated by quartet–quartet transitions, whereasthe excited-state absorption spectra are dominated ei-ther by quartet–quartet or doublet–doublet transitionsdepending on the total spin of the lowest excited state.

Laser characteristics. The first laser was realized in1960 with ruby, i. e., Cr3+-doped Al2O3 [11.968]. Inruby – due to the strong crystal field experienced by theCr3+ ion – the laser oscillation occurs on the 2E → 4A2transition. Thus, ruby is a three-level laser and nottunable over a wide range. Ruby is still a commer-cially available pulsed laser system with peak outputpowers in the MW range used for applications inmeasurement and pulsed holography. Ruby has remark-able thermo-mechanical properties which allows highpeak power operation, especially in the Q-switchedregime. In 1976, Morris et al. [11.969] realized withCr3+-doped BeAl2O4 (alexandrite) the first tunablelaser based on the Cr3+ ion. The laser transition is4T2 → 4A2 and terminates in the higher vibronics of

Table 11.23 Overview of other Ti3+-laser materials

Crystal λlaser Tuning range Slope Mode of Pump source Ref.(nm) (nm) efficiency (%) operation

BeAl2O4 810 730–950 15 Pulsed SHG Q-switched Nd:YAG (532 nm) [11.959]

- 753–949 0.013 Pulsed flashlamp (10 µs) [11.961]

Y3Al5O12 No details given in the reference [11.962]

YAlO3 615 - 0.3 Pulsed SHG Q-switched Nd:YAlO3 (540 nm) [11.963, 964]

0"99""'

-

% * -

A BB

%

! ""#$

A

M

-(-

0$

%-% %- %

*A BBR

%

! ""#$

%- % %-

*AR

& ; (

Fig. 11.80a,b Absorption (a) and emission (b) spectra at roomtemperature and at 4 K of Ti:sapphire. (After [11.948])

the ground state. Therefore, a four-level laser system isrealized. To date, the alexandrite laser has found signifi-cant commercial applications in industry (e.g., marking,writing and printing), scientific research (e.g., fluores-cence dynamics, fluorescence imaging, LIDAR), and

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)

)

+

*

;

*

%

,

*0

+

*+

*+

0

*+

!

>

>@2

*

*!

Fig. 11.81 Tanabe–Sugano diagram for octahedrally coor-dinated 3d3 ions, e.g., Cr3+ with C/B = 5.5. The dashedvertical line marks the border between low and high crystalfield strength

medicine (e.g., hair and tattoo removal,). Its advan-tages are its tunability between 700 nm and 820 nm, thehigh heat conductivity of 23 W/mK, which allows high-power pulsed operation, and its high slope efficiencyof up to 51% [11.970, 971]. Even broader laser tun-ing ranges with similar efficiencies were achieved withthe Cr3+-doped colquirite crystals LiCaAlF6, LiSrAlF6,and LiSrGaF6 [11.972] (Table 11.25). However, thesecrystals suffer from poor thermo-mechanical properties(Table 11.24), which allow only relatively low pump andlaser powers. The broad tuning range is exploited formode-locked operation with pulse lengths down to 9 fsfor Cr3+:LiCAF [11.973]. The crystals also offer thepossibility of efficient laser-diode pumping with laserdiodes at wavelengths around 670–690 nm. Thus com-pact, mode-locked laser systems in the low-power rangeare possible [11.974–976]. It is interesting to note thatthe highest slope efficiency ever obtained with a Cr3+-based laser was achieved with Cr3+:Be3Al2(SiO3)6(emerald). Also here, however, the thermo-mechanicalproperties are poor and, furthermore, the emerald crys-tal is very difficult to grow in laser quality. For themost efficient Cr3+ laser materials the relevant laser andmaterials parameters are listed in Table 11.24. Laser os-cillation with Cr3+-doped systems has been realized in

more than 30 materials. In Table 11.25 an overview ofreported laser systems is given [11.103,977]. In generalthe tuning ranges for Cr3+ lasers are not as broad as forTi3+:Al2O3 lasers, however, they exhibit the advantageof direct diode-laser pumping around 670 nm, i. e., inthe spectral region of the 4A2 → 4T2 absorption.

For Cr3+ laser systems excited-state absorption(ESA) plays a very important role and is the mainreason for the observed large differences in the laser ef-ficiencies. Due to its energy-level scheme (Fig. 11.81),spin-allowed ESA transitions are expected either be-tween the quartet states (for low-crystal-field hosts) orbetween the doublet states (in strong-field hosts). Thesetransitions cover a wide spectral range, due to the strongelectron–phonon coupling. Thus, in general, these ESAtransitions overlap with the absorption and emissionbands. The influence of the ESA is strongly dependenton the host lattice, i. e., on the crystal field experiencedby the Cr3+ ion. In general, it is observed that crystalswith a medium crystal field for the Cr3+ ion are favor-able. Furthermore, in some crystals (e.g., the colquirites,alexandrite, emerald etc.) the polarization-dependent se-lection rules can be exploited. Cubic hosts are generallymore strongly affected.

Summary and perspectives for Cr3+ lasers. Cr3+lasers are generally interesting for laser applications.High-efficiency diode-pumped and broadly tunablecontinuous-wave and mode-locked laser oscillation havebeen obtained in a variety of crystals. Some laser sys-tems are commercially available. However, all of thesecrystals have specific drawbacks: they have poor thermalmechanical properties, the growing process is difficultor the tuning range is small. Therefore Cr3+ lasers are atthis time in general not competitive with the Ti3+:Al2O3laser, which offers a broader tuning range, allows shorterpulses in the mode-locked regime and has better thermo-mechanical properties. Ti3+:Al2O3 cannot be pumpeddirectly with diode lasers; however, advances in thefrequency-doubling technique of neodymium lasers hasled to efficient all-solid-state pump lasers which re-placed the argon-ion laser as pump. Additionally, pumplaser diodes in the spectral region between 630 nm and700 nm with high beam quality and output power arecurrently not available in a satisfactory manner. Con-cerning the research for new Cr3+ laser systems one hasto take into account that the Cr3+ ion has already beeninvestigated in a large number of systems. In princi-ple, new host materials with the perspective of efficientlaser operation should have a medium crystal field forthe Cr3+ ion and polarization-dependent optical prop-

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Table 11.24 Materials, spectroscopic and laser parameters of the most important Cr3+ lasers

LiCaAlF6 LiSrAlF6 Alexandrite Ruby

Structure Trigonal Trigonal Orthorhombic Hexagonal

P31c P31c Pnma R3c

Lattice parameters (Å) 5.007 (a) 5.084 (a) 9.404 (a) 4.759 (a)

9.641 (c) 10.21 (c) 5.476 (b) 12.989 (c)

4.425 (c)

Typical Cr3+ concentration (cm−3) ≈ 1019 –1020 ≈ 1019 –1020 ≈ 1019 –1020 ≈ 1.6 × 1019

Growth Czochralski Czochralski Czochralski Czochralski

Tm (C) 810±10 766±10 1870 2050

Density (g/cm3) 2.99 3.45 3.69 3.98

Thermal conductivity (W/mK) 4.58 (|| a) 3.0 (|| a) 20 33 (|| a)

5.14 (|| c) 3.3 (|| c) 35 (|| c)

Thermal expansion (10−6 C−1) 22 (|| a) 25 (|| a) 6 (|| a) 6.65 (|| a)

3.6 (|| c) -10 (|| c) 6 (|| b) 7.15 (|| c)

7 (|| c)

n 1.390 (a) 1.405 (a) na = 1.7381 ( 800 nm) 1.763 (o)

1.389 (c) 1.407 (c) nb = 1.7436 ( 800 nm) 1.755 (e)

nc = 1.7361 ( 800 nm)

dn/dT (10−6 /K) -4.2 (|| a) -2.5 (|| a) 13.6 (o)

-4.6 (|| c) -4 (|| c) 14.7 (e)

σem (10−20 cm2) 1.3 (π) 4.8 (π) 0.7 2.5

τem(300 K) (µs) 170 67 260 3000

σemτ(10−24 cm−2s−1) 2.2 3.2 1.8 75

λpeak,em (nm) 763 846 697 694.3

∆λ (nm) ≈ 120 ≈ 200 ≈ 75 -

∆λ/λpeak,em ≈ 0.16 ≈ 0.24 ≈ 0.11 -

erties that could help to avoid or at least significantlyreduce excited-state absorption in the spectral regionof the emission. Furthermore, good thermo-mechanicalproperties are required.

Cr4+ lasers.Basics. Cr4+-doped crystals have been of interest astunable, room-temperature laser materials since the late1980s. In a variety of materials laser oscillation indifferent operation schemes has been achieved, seeTable 11.26 and Table 11.28, where an overview ofthe obtained laser results is given. The energy-levelscheme of the Cr4+ ion in crystals can be describedwith the Tanabe–Sugano diagram shown in Fig. 11.82.The absorption spectra are dominated by the three spin-allowed transitions between the 3A2 ground state andthe 3T2(3F), 3T1(3F), and 3T1(3P) excited states. Usu-ally, the energy levels are strongly crystal field dependentand split, thus the absorption spectra of different mater-ials differ significantly. In all materials investigated thusfar, broadband emission due to the transition between the

3T2 excited state and the 3A2 ground state is observed.Referring to the Tanabe–Sugano diagram and the indi-cated area of crystal field values for the Cr4+ ion, onewould expect – at least in some materials – narrow-lineemission. However, due to lattice relaxation and crys-tal field splitting of the excited state, the 3T2 or one ofits crystal field components becomes lower than the 1Elevel.

The most efficient laser oscillation of Cr4+-dopedmaterials was realized in Mg2SiO4 and Y3Al5O12(YAG). The laser data and main spectroscopic data ofthese materials are listed in Table 11.26 and Table 11.27.A variety of further host materials for the Cr4+ ionwere also investigated, but with either low efficiency(Table 11.28) or without realization of laser oscillation.A detailed overview of Cr4+-doped systems is givenin [11.933].

In all Cr4+-doped materials investigated thus far,there are two main drawbacks for efficient laser oscil-lation or laser oscillation at all: excited-state absorptionand nonradiative decay. Investigation of excited-state

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Table 11.25 Free-running laser wavelengths, tuning ranges, laser temperatures, slope efficiencies, operation modes and outputpower/output energy of Cr3+-laser materials. (CW: continuous wave, p: pulsed, SHG: second harmonic, fl: flash lamp, dc: dutycycle, QS: Q-switched, g-sw: gain-switched, ML: mode locked, * discontinuously)

Host material Wavelength Tuning T (K) η (%) Mode Pout/Eout Ref.(nm) range (nm)

Be3Al2(SiO3)6 684.8 300 p (SHG QS Nd:YAG) [11.988]757.4 751–759.2 300 p (fl) 6.8 mJ [11.989]765 728.8–809.0 300 34 CW (Kr+) ≈ 330 mW [11.990]768 720–840 300 64 q-CW (3% DC) 1.6 W [11.991,

992]LiCaAlF6 780 720–840 52.4 CW (5% DC) Kr+ 850 mW [11.972]

780 720–840 300 52 CW (5% DC) Kr+ [11.993]780 1.55 p – fl 1.8 J [11.993]

LiSrGaF6 820 - 300 51 CW, Kr+ 1 W [11.994]800–900 300 QS (10 kHz), 12 µJ [11.995]

laser diode [11.996]BeAl2O4 679.9 77 [11.997]

680.3 77 p (fl) [11.998,999]

680.3 700–800 300 p (fl) [11.1000]750 701–794 300 p (fl) 500 mJ [11.1000,

1001]765 744–788 300 ≈ 0.7 q-CW (ac Hg-lamp) 6.5 W [11.1002]

745–785 300 2.3 CW 20 W [11.1003](DC Hg-,Xe-lamp)

750 700–820 300 p (fl 125 Hz) 150 W [11.1004]755 300 1.2 CW 60 W [11.1004]750 300 ML (38 ps) [11.1004]

752–790 300–583 p (fl), temp. tuning [11.1005]700–820 300–583 [11.970,

971, 997,1006–1011]

701–818 300 [11.1000]744–788 300 [11.1002]701–818 300 2.5 fl 35 W / 5 J [11.970]

680.4 300 0.15 fl-QS (20 ns) 500 mJ [11.970,1012]

680.4 300 0.15 fl 400 mJ [11.970]752 726–802 300 51 CW, Kr+ 600 mW [11.971]752 700–820 [11.969]753 300 63.8 dye, 645 nm 150 mW [11.1013]753 300 laser diode, ≈ 640 nm 25 mW [11.1013]765 300 28 QS SHG Nd: GVO, 150 mW [11.1014]

671 nm (80 kHz)LiSrAlF6 825 780–920 300 36 CW, Kr+ 650 mW [11.993,

1015]865 815–915 300 a-ML (30 ps, 160 fs) 3.5 mW [11.1016]845 780–1010 300 5 p (fl) 2.7 J [11.1017]834 300 CW (laser diode) 20 mW [11.1018]870 858–920 300 laser diode, 4.3 mW [11.1019]

electronically tuned849 810–860 300 laser diode 43 mW [11.1020]

300 p (fl, 5 Hz) 44 W/ 8.8 J [11.1021]

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Table 11.25 (continued)

Host material Wavelength Tuning T (K) η (%) Mode Pout/Eout Ref.(nm) range (nm)

ScBO3 843 787–892 300 25 CW (3%), Kr+ 250 mW [11.1022]300 26 CW (10%), Kr+ 275 mW [11.972]

Gd3Sc2Ga3O12 777 745–805 300 11 CW (1:50 DC) 60 mW [11.1023]785 742–842 300 28 quasi-CW 200 mW [11.1024–

1026]790 300 1 p (10 µs, dye) 10 µJ [11.1027]

766–820 300 0.06 p – fl 20 mJ [11.1028]300 0.02 p – fl 10 mJ [11.1029]

0.57 p – fl ≈ 70 mJ [11.1030]

Na3Ga2Li3F12 791 741–841 300 18.4 CW [11.54]

Y3Sc2Al3O12 769 9 q-CW, Kr+ 50 mW [11.1031]

Gd3Sc2Al3O12 780 750–800 300 0.24 p – fl 200 mJ [11.1029]784 765–801 300 0.12 p – fl 110 mJ [11.1032]784 300 18.5 CW Kr+ 90 mW [11.1025,

1026,1033]

780 750–810 300 0.38 p – fl 260 mJ [11.1034]300 QS (p – fl) 30 mJ [11.1034]

784 735–820 300 19 CW, Kr+ 200 mW [11.1031]780 750–803 300 0.24 p – fl 206 mJ [11.1031]

SrAlF5 921, 935 852–947 300 3.6 q-CW (DC 3%), Kr+ 35 mW [11.1035]910 15 q-CW (DC 2%), Kr+ [11.1036]932 825–1011 [11.1037]930 825–1010 300 10 Kr+ [11.1038]

KZnF3 810, 826 300 1 p – dye (0.5 µs) [11.1039,1040]

790–826 775–825 20–260 0.1 CW, Kr+ [11.1039,1040]

785–865 300 14 CW, Kr+ 85 mW [11.1041]785–865 p – ruby [11.1042]780–845 300 3 CW, Kr+ 55 mW [11.1043]

820 766–865 300 14 [11.1044]

ZnWO4 980–1090 77 13 CW 110 mW [11.1026,1045]

300 p – dye [11.1045]

La3Ga5SiO14 960 862–1107 300 7.6 CW (3% DC) 80 mW [11.1046]815–1110 300 10 p 10 mJ [11.1047]

968 [11.1048]

Gd3Ga5O12 769 - 300 10 quasi-CW [11.1023,1024,1026]

La3Ga5.5Nb0.5O14 1040 900–1250 300 5 p 10 mJ [11.1047,1049]

Y3Ga5O12 740 - 5 quasi-CW [11.1023,1026]

Y3Sc2Ga3O12 750 - 5 quasi-CW [11.1023,1026]

La3Lu2Ga3O12 830 790–850 3 quasi-CW [11.1024,1026,1048]

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Table 11.25 (continued)

Host material Wavelength Tuning T (K) η (%) Mode Pout/Eout Ref.(nm) range (nm)

MgO 830 824–878* 2.3 CW, Ar+ 48 mW [11.1050–1052]

Al2(WO4)3 800 q-CW, Kr+ [11.1053]

BeAl6O10 820 780–920 300 ≈ 0.03 p – fl 6 mJ [11.1054]

834 795–874 300 - p – SHG Nd:YAG [11.1055]

Al2O3 692.9 (R2) 300 p – fl [11.1056]

693.4 300 p – fl [11.1057]

693.4 77 CW, Hg-lamp 4 mW [11.1058]

77 12 CW, Ar+ 42 mW [11.1059]

694.3 (R1) 300 [11.968,1060–1065]

700.9 (N2) 77 [11.1066]

704.1 (N1) 77 [11.1066]

767 300 [11.1067]

Y3Al5O12 687.4 not tunable ≈ 77 [11.1068]

LiSr0.8Ca0.2AlF6 835 750–950 300 1.25 p – fl 1.2 J [11.1069]

847 783–945 300 CW, Kr+ 300 mW [11.1070]

LiSrCrF6 890 300 33 q-CW, TiSa (DC: 2%) 200 mW [11.1071]

ScBeAlO4 792 740–828 300 31 CW, Kr+ [11.1072]

La3Ga5GeO14 880–1220 300 5 p 8 mJ [11.1047,1073]

Sr3Ga2Ge4O14 895–1150 300 3 p 4 mJ [11.1047,1073]

Ca3Ga2Ge4O14 870–1210 300 6 p 16 mJ [11.1047,1049]

La3Ga5.5Ta0.5O14 925–1240 300 5 p 6 mJ [11.1047,1049]

(Ca,Gd)3(Ga,Mn,Zr)5O12 774–814 300 12 p – ruby 170 mJ [11.1074]

≈ 777 300 0.08 p – fl 40 mJ [11.1074]

LaSc3(BO3)4 934 300 0.65 q-CW, Kr+ 3 mW [11.1075]

(DC: 10%)

LiInGeO4 1150–1480 300 p (g-sw. Ti:Al2O3) [11.1076]

LiScGeO4 1220–1380 300 p (g-sw. Ti:Al2O3) [11.1076]

Li:Mg2SiO4 1121 1030–1180 300 p (QS Cr3+:BeAl2O4) [11.1077]

1120, 1130, 285 1.27 CW (Ar+) 5.5 mW [11.1077]

1140

absorption has been performed on garnets [11.978–980], forsterite [11.981–984], cunyite [11.985], sili-cates [11.982] and Wurtzite-type crystals [11.984, 986,987]. The nonradiative decay via multiphonon relaxationleads to quantum efficiencies far below 100% at roomtemperature; see the overview given in [11.933]. Thepreparation of Cr4+-doped crystals requires for mostmaterials special conditions before, during or after thegrowth process. The Cr4+ ion is not as stable in its va-

lence state as Cr3+. Therefore, there is the tendencyfor incorporation of chromium ions in different valen-cies in the crystals, especially for materials that do notexhibit an appropriate tetravalent tetrahedrally coordi-nated lattice site. This is, e.g., the case for Y3Al5O12,therefore here additional codoping with divalent cations(Mg, Ca) is necessary. But also in Mg2SiO4, Cr3+ is in-corporated into the Mg lattice. No traces of Cr3+ wereobserved in these materials, which do not exhibit an ap-

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Table 11.26 Overview of laser results obtained with Cr4+:YAG and Cr4+:Mg2SiO4. (CW: continuous wave, lp: longpulse pumped, g-sw: gain switched, DP: diode pumped, ML: mode locked, CF: crystalline fiber)

Crystal λlaser (nm) Output ηsl (%) Tuning range Mode of operation Ref.

Y3Al5O12 1430 7.5 mJ 22 1350–1530 g-sw (55 ns) [11.1078–1083]

1450 1900 mW 42 1340–1570 CW (T = 3 C) [11.1081, 1083–1085]

1420 58 mJ 28 1309–1596 lp (200 µs) [11.1082, 1083,

1086, 1087]

ML [11.1088, 1089]

1440 20 mW 5 1396–1482 CW-ML (26 ps) [11.1090]

1520 360 mW 8 1510–1530 CW-ML (120 fs) [11.1091]

1510 50 mW < 1 1490–1580 CW-ML (70 fs) [11.1092]

1540 - - - CW-ML (53 fs) [11.1093]

1569 30 mW DP-ML (65 fs) [11.1094, 1095]

1450 400 mW Nd−YVO4, ML (20 fs) [11.1096]

1470 80 mW 5.5 1420–1530 DP [11.1097]

1420 150 mW 1.9 - DP-CF [11.1098]

1440 Intracavity Nd:YAG [11.1099]

Mg2SiO4 CW [11.1100]

1242 38 - CW (duty cycle 1:15) [11.1101, 1102]

1.1 W 26 - CW [11.1103]

DP [11.1104, 1105]

lp [11.1106]

flashlamp pumped [11.1107]

4.95 mJ - 1206–1250 flashlamp pumped [11.1108]

g-sw [11.1106, 1109–1113]

1235 1170–1370 g-sw [11.1114]

370 mW 13 1173–1338 g-sw (1.5 kHz/10kHz) [11.1115]

ML [11.1116–1122]

300 mW - - ML (25 fs) [11.1122]

1300 Nd:YAG, ML (14 fs) [11.1123]

1260 10 pJ - - DP-ML (1.5 ps) [11.1124]

1260 10 mW 5 1236–1300 CW-DP (T = −10 C) [11.1104]

propriate lattice site for the Cr3+ ion, e.g., Ca2GeO4and Y2SiO5. However, the laser results obtained withY3Al5O12 and Mg2SiO4 indicate that the laser efficiencyis not necessarily affected by the presence of Cr3+.

Laser characteristics. In Table 11.26 an overviewof the laser results obtained with Cr4+:YAG andCr4+:Mg2SiO4, the materials in which the most effi-cient laser operation has been obtained, is given. InYAG, the highest slope efficiency obtained thus far inthe continuous-wave regime is 42% [11.1085]. The cor-responding input–output curve is shown in Fig. 11.83a.Attempts to improve the laser efficiency by changing thecrystals composition, i. e., by substituting Lu for Y onthe dodecahedral site or Sc for Al on the octahedral sitewere not successful [11.1086, 1087]. The efficiency de-

creased significantly. The reason is mainly excited-stateabsorption, but the lower crystal quality and increasednonradiative rate also contribute [11.1125]. A remark-able characteristic – on first sight – is that the Cr4+:YAGlaser oscillates polarized parallel to one of its maincrystallographic axes. The laser output was highestwhen the pump beam of a Nd:YAG laser operatingat 1064 nm and propagating along the [001]-axis ofthe Cr4+:YAG crystal is polarized parallel to one ofthe crystallographic < 100>-axes of the Cr4+:YAGcrystal, and was lowest when its polarization was par-allel to one of the < 110>-axes (Fig. 11.83b). TheCr4+:YAG laser output is polarized and maintains itspolarization while rotating the pump beam polarization.When the pump beam polarization is parallel to oneof the < 110>-axes, the Cr4+:YAG laser polarization

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Table 11.27 Parameters of Cr4+-doped Y3Al5O12 and Mg2SiO4

Y3Al5O12 (YAG) Mg2SiO4 (Forsterite)

Structure Ia3d (O10h ) Pbnm (D16

2h)

Hardness 8.25–8.5 7

Site symmetry S4 m

Growth Czochralski, divalent codopant required Czochralski

Tm (1930±20) C (1890±20) C

Cr4+ concentration ≈ 1017 –1018 cm−3 ≈ 1018 –1019 cm−3

Thermal conductivity 0.13 W/cmK 0.08 W/cmK

Refractive index (λpeak) 1.81 1.669 (a)

1.651 (b)

1.636 (c)

dn/dT (undoped) 7.7–8.2 × 10−6 /K 9.5 × 10−6 /K

Density 4.56 g/cm3 3.22 g/cm3

σabs (1064 nm) ≈ 6.5 × 10−18 cm2 ≈ 5.0 × 10−19 cm2

σem (λpeak) ≈ 3.3 × 10−19 cm2 ≈ 2.0 × 10−19 cm2

σESA (λpeak) < 0.3 × 10−19 cm2 < 0.2 × 10−19 cm2

τem (300 K) 4.1 µs 3.0 µs

σemτ 1.35 × 10−24 cm−2s−1 0.6 × 10−24 cm−2s−1

Quantum efficiency ≈ 0.2 ≈ 0.16

λpeak,em 1380 nm 1140 nm

∆λ ≈ 300 nm ≈ 250 nm

∆λ/λpeak,em ≈ 0.22 ≈ 0.22

switches [11.1084]. This characteristic can be explainedby the crystal structure, the location of the Cr4+ ionsand the local symmetry they experience. For details,see [11.1126, 1127].

Table 11.28 Overview of other Cr4+-doped laser materials. (CW: continuous wave, lp: long pulse pumped, g-sw: gainswitched, DP: diode pumped. * laser-active center assigned to Cr3+ in [11.1076], Table 11.24)

Crystal λlaser (nm) Output ηsl (%) Tuning range Mode of operation Ref.

Y3ScxAl5−xO12 1498 (x = 0.5) 23 mJ 10 1394–1628 lp (100 µs) [11.1087](YSAG)

1548 (x = 1.0) 4.5 mJ 3 1464–1604 lp (100 µs) [11.1087]1584 (x = 1.5) 0.9 mJ 0.5 lp (100 µs) [11.1087]

Lu3Al5O12 Not given 50 mW 1.5 - quasi-CW [11.1128, 1129]

Ca2GeO4 [11.1104, 1130–1132]

1400 0.4 mJ 6.1 1348–1482 g-sw (T = 0 C) [11.1130, 1133]1410 20 mW 8.5 1390–1475 CW-DP (T = −10 C) [11.1104]

LiScGeO∗4 1300 0.1 mJ 3 1220–1380 g-sw [11.1134]

Y2SiO5 [11.1135, 1136]1304 20 mW 0.4 - quasi-CW (1:8) [11.1084, 1137]1348 0.55 mJ 0.4 - lp (200 µs) [11.1084, 1137]

LiNbGeO5 - - - 1320–1430 g-sw (110 K) [11.1138–1140]

CaGd4(SiO4)3O 1370 37 µJ ≈ 1 g-sw [11.1141]

SrGd4(SiO4)3O 1440 g-sw [11.1141]

With the Cr4+:Mg2SiO4 laser, slope efficienciesof up to 38% [11.1102] and output powers around1.1 W [11.1103] in continuous-wave operation at roomtemperature have been realized (Fig. 11.83c. For crys-

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)

)

*

-

,

0

+

0

+

>

!

.* -

++

+2, +

*

Fig. 11.82 Tanabe–Sugano diagram for tetrahedrally coor-dinated 3d2 ions. The marked area correspond roughly tothe regions of Dq/B values of Cr4+ and Mn5+. For sim-plicity, C/B was set to 5.6, although it differs for differentsystems

tals from the forsterite group the attempts to obtain betterlaser results by substituting the constituent ions of thehost lattice were also not successful. A variety of crys-tals were investigated [11.1142–1144] (see the overviewgiven in [11.933]) but only for Ca2GeO4 was laser oscil-lation obtained. For Ca2GeO4 excited-state absorptionon the laser wavelength is detrimental [11.985].

Laser oscillation was also obtained for Cr4+-doped Y2SiO5, LiGeNbO5, CaGd4(SiO4)3O, andSrGd4(SiO4)3O. The low efficiencies are probably dueto excited-state absorption and the high nonradiativedecay rate.

Summary and outlook for Cr4+ lasers. Cr4+:YAGand Cr4+:Mg2SiO4 are efficient and broadly tun-able laser systems for the infrared spectral rangeincluding the very interesting region for telecom-munication applications around 1.3 µm and 1.55 µm.Mode-locked operation with pulse lengths as shortas 20 fs for Cr4+:YAG [11.1096] and 14 fs forCr4+:Mg2SiO4 [11.1123] were obtained. Also directdiode-pumped laser operation was realized, however,with lower efficiencies [11.1094, 1097]. For both sys-tems, power handling is a problem and the ion

0 5G

5G

$%$

* - % &

-

-

-

S.$*-Q

"

0OTU=82 7V

$

%

0 5G

5G

%$(-*$($$-

*- ( - ; - & -

$

$*

$%

$;

$

* - % &

Fig. 11.83 (a) Input–output characteristic of a Cr4+:YAG-laser (after [11.1085]), (b) Polarization dependence ofa Cr4+:YAG laser (after [11.933, 1084]), (c) Input–outputcharacteristic of a Cr4+:Mg2SiO4 laser (after [11.1103])

concentration in both materials is rather low, yieldinga low absorption efficiency of the pump light.

The main obstacle for Cr4+ laser materials is excited-state absorption at the emission wavelength, which

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686 Part C Coherent and Incoherent Light Sources

)

)

0

+

->

@

0

-,

-+

! +

-+

+

H +4

Fig. 11.84 Tanabe–Sugano diagram for tetrahedrally coor-dinated ions with 3d4 configuration, C/B = 4

is present in practically all crystals. Its influence issmall in systems with an advantageous energy-levelscheme and/or a crystal structure that supports strongpolarization-dependent selection rules. A second but less

G 6

'("

- -

;

*

'*

404,',40

4,',40

40

4,

Fig. 11.85 σGSA (dashed line), σSE (dotted line), (σGSA +σSE −σESA) (thin solid line), and (σSE −σESA) spectrum (thick solid line)of Cr2+:ZnSe at room temperature. The sharp structure beyond2500 nm is attributed to the water absorption in air and the followingnormalization process. (After [11.1145])

important point concerning Cr4+ laser materials is thenonradiative decay rate. To decrease this rate, crystalswith low-energy phonons and/or small electron–phononcoupling should be used as host materials, e.g., thosein which silicon is substituted by germanium or alu-minum is substituted by gallium. However, attempts inthis direction have not been successful thus far.

Cr2+ lasers.Basics. The energy-level scheme of tetrahedrally co-ordinated Cr2+ is shown in Fig. 11.84. The Cr2+ ionexperiences a low crystal field, e.g., in ZnSe the value ofDq/B is about 0.9. Thus, the 5T2 level is the ground stateand the 5E level is the first excited state, whilst all higher-lying levels are triplet and singlet states. In consequence,the 5E → 5T2 emission is a spin-allowed transition,while all interionic excited-state transitions are spin-forbidden. Such systems are in general very promisingfor the realization of efficient tunable laser oscillation,because even in the case of a spectral overlap betweenstimulated emission and excited-state absorption, thetransition probabilities for the latter are expected to beabout a factor of 10 smaller.

Cr2+-doped chalcogenide crystals have been shownto be efficient and broadband tunable solid-state lasersfor the infrared spectral range between 2 µm and 3 µm.Pulsed, continuous-wave, mode-locked and diode-pumped laser operation have been demonstrated inrecent years. Possible applications of these mid-infraredlasers include scientific research, remote sensing, trace-gas analysis, medicine, biology, materials processing,and ultrashort-pulse generation.

Materials. The choice of host materials for the Cr2+laser ion is limited due to special conditions. First, thematerials have to exhibit a tetrahedrally coordinatedlattice site. A divalent lattice site is also preferably,because otherwise a charge-compensation mechanismwould have to be established in the lattice. Furthermore,host crystals with low phonon frequencies have to bechosen in order to decrease the possibility of nonra-diative decay via multiphonon relaxation. Chalcogenidecrystals, with their tetrahedrally coordinated divalentcation lattice sites and with phonon energies lower than400 cm−1, are thus very suitable with respect to therealization of efficient broadband emission in this mid-infrared spectral range. In Table 11.29 some materialparameters of the investigated chalcogenide crystals aregiven in comparison to the data for Al2O3. The ther-mal conductivities of the chalcogenides are rather highand comparable to the values for Al2O3. However, the

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Table 11.29 Materials parameter of some chalcogenide crystals suitable for Cr2+. The data for Y3Al5O12 are given for comparison.(*: for CdTe). (After [11.1146–1148]) w: Wurtzite, z: Zincblende

ZnS ZnSe Cd1−xMnxTe CdSe Y3Al5O12

Structure Wurtzite: hexagonal Zincblende: cubic Zincblende: cubic Wurtzite: hexagonal Garnet: cubic

Zincblende: cubic

Site symmetry C3v (hex.) Td Td C3v D2, C3i, S4

Td (cubic)

Growth Vertical Brigdeman Vertical Bridgeman Vertical Brigdeman Vapor transport Czochralski

Tm (C) w: 1700 1525 1070–1092 1250–1350 1930

z: 1020

Hardness (Knoop) w: 210–240 130 45∗ 70 1250

z: 150–160

Refractive index 2.29 2.48 2.75 2.57 1.8

Thermal conductivity w: 17 18 ≈ 2 6.2 10

(W/mK) z: 27 (6.2∗)

dn/dT (10−6 /K) w: 46 70 107∗ 9

Transmission range w: 0.4–17 0.5–18 1–28∗ 0.8–18 0.2–10

(µm) z: 0.4–14

main disadvantages of the chalcogenide crystals are thegrowth techniques, which are Bridgman or vapor trans-port, which usually lead to a lower crystal quality thanin the case of growth by the Czochralski method, andthe high values for dn/dt, which lead to strong thermallensing during laser operation, especially in the case ofhigh-power operation.

Spectroscopy. The spectroscopy of Cr2+ ions in chalco-genide crystals has been thoroughly investigated in thepast [11.1154–1162]. Tetrahedrally coordinated Cr2+ions exhibit a broad band in the absorption spectrum dueto the 5T2 → 5E transition in the infrared spectral rangewith a maximum around 1.7–1.9 µm. The emission

Table 11.30 Overview about the spectroscopic characteristics of Cr2+ doped chalcogenide crystals. Data for Ti3+:Al2O3

are given for comparison

ZnS ZnSe ZnTe Cd0.85Mn0.15Te Cd0.55Mn0.45Te CdSe Ti3+ : Al2O3

[11.1149–1151]

[11.1149–1151]

[11.1149–1151]

[11.1152] [11.1153] [11.1154] [11.948]

σabs (10−20 cm2) 52 87 123 ≈ 270 ≈ 170 300 6.5

σem (10−20 cm2) 75 90 188 270 170 200 45

τem (300 K) (µs) 8 9 3 1.4 4.8 6 3

σemτ 6.0 8.1 5.6 3.8 8.2 12.0 1.4

(10−22 cm−2s−1)

η ≈ 0.73 ≈ 1 ≈ 1 ≈ 0.38 ≈ 1 ≈ 1 ≈ 0.9

λpeak,em (nm) 2300 2300 2400 2250 2480 2200 800

∆λ (nm) ≈ 780 1000 ≈ 900 ≈ 500 770 ≈ 550 300

∆λ/λpeak,em ≈ 0.34 ≈ 0.43 ≈ 0.38 ≈ 0.22 0.31 ≈ 0.25 0.38

spectrum due to the 5E → 5T2 transition is also broadand occurs between 2 µm and 3 µm. The absorptionand emission cross sections for the Cr2+-doped chalco-genides are on the order of ≈ 10−18 cm2; these are valuesexpected for tetrahedrally coordinated transition-metalions and are larger than those for Ti3+:Al2O3 [11.948].The room-temperature absorption and emission spec-tra are shown in Fig. 11.85. The emission lifetimes atroom temperature are on the order of several µs and theemission quantum efficiencies are close to unity. Theσemτ product gives an indication of the expected laserthreshold, because Pthr ∝ (σemτ)−1. The values for theCr2+-doped chalcogenides are higher than in the caseof Ti3+:Al2O3, so that generally lower laser thresholds

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for the Cr2+-doped materials are expected. The ratio be-tween the emission bandwidth and the central emissionwavelength, ∆λ/λpeak,em, is a measure for the principleability to generate ultrashort pulses in the mode-lockedregime; the higher this value, the shorter the pulses. Thevalues for Cr2+-doped crystals are comparable to thoseof Ti3+:Al2O3, for which laser pulses shorter than 5 fshave been realized. However, one has to keep in mindthat bandwidth is not the only important parameter forultrashort-pulse generation. Other important parametersare the nonlinearities of the material and the thermallensing introduced by the high peak power, which oc-curs in the mode-locked regime. Only recently Sorokinaet al. [11.1163] obtained mode-locking in the fs regime,after a lot of work was spent in order to understandthe mechanisms behind the pulse-forming processes andhow they can be controlled. In Table 11.30 the mainspectroscopic data for the Cr2+-doped chalcogenides aresummarized in comparison to the data for Ti3+:Al2O3.

For the tetrahedrally coordinated Cr2+ ion strongESA transitions due to inner-shell 3d transitions arenot expected, because all possible transitions are sup-posed to be spin-forbidden. This assumption was provenby ESA measurements [11.1145, 1164] (Fig. 11.85).Neither in the spectral region of the ground-state ab-sorption nor in the emission region is ESA observed.Tunability up to and beyond 3 µm was predicted forCr2+:ZnSe [11.1145]. This prediction was later provenwith laser experiments, where laser oscillation was ob-served up to 3100 nm [11.1165].

Laser results. Laser materials based on the Cr2+ ionas the active ion have been investigated since the mid1990s. Nowadays, Cr2+ lasers are operating in a vari-ety of different operation schemes and under differentexcitation sources. In Table 11.31 an overview of theobtained laser results is given.

The best laser results were thus far obtained forCr2+:ZnSe. In different setups using different pumpsources (Tm3+ lasers, Co2+:MgF2 lasers, diode lasersbetween 1.54 µm and 2.0 µm, erbium-doped fiber am-plifiers) slope efficiencies up to 73%, output powers upto 7 W, thresholds lower than 100 mW, a tuning range of2000–3100 nm and mode locking with pulse durationsas short as ≈ 100 fs were obtained (Table 11.31). McKayand coworkers [11.1166] reported results of a thin-disclaser setup for a Cr2+:ZnSe laser, a setup that was suc-cessfully applied to Yb-doped laser materials [11.1167].This setup appears to be favorable also for Cr2+:ZnSe,because Wagner et al. [11.1168] reported on thermalrollover in the case of high-power pumping. In McKay’s

experiment, a Q-switched Tm,Ho:YLF laser operatingat 2.05 µm with a repetition rate of 10 kHz was usedas the pump source. An output power of 4.27 W witha slope efficiency of 47% with respect to the absorbedpump power was obtained. The other possibility is touse a rather large pump beam radius of 260 µm (1/e2

radius), as was done by Alford et al. [11.1169]. Usinga 35 W Tm3+:YAlO3 as the pump laser, a continuous-wave output power of 7 W at 2.51 µm was achieved.

Besides ZnSe, other chalcogenide and mixed chalco-genide host materials for the Cr2+ ion are also suitablefor efficient laser oscillation. For Cr2+-doped ZnS, thespectroscopic characteristics are very similar to thoseof Cr2+:ZnSe. From the material point of view, ZnSeven seems to have some advantages over ZnSe as ZnShas a higher bandgap energy (3.84 eV for ZnS, 2.83 eVfor ZnSe), a higher hardness, a higher thermal con-ductivity [27 W/mK for ZnS (cubic phase), 19 W/mKfor ZnSe] and a lower dn/dT (46 × 10−6 1/K for ZnS,70 × 10−6 1/K for ZnSe) (Table 11.29). However, ZnSis much more difficult to grow and many differentstructure types exist. Thus far, the obtained laser re-sults for Cr2+:ZnS are not as good as the results forCr2+:ZnSe (Table 11.31). The highest output power ob-tained was about 700 mW at an absorbed pump powerof 2.65 W from an Er-doped fiber laser [11.1170,1171].The thresholds are around 100 mW and are thus com-parable to those observed for Cr2+:ZnSe. The widesttuning range obtained so far is 2110–2840 nm. Thehighest slope efficiency in CW regime is about 40%.Direct diode pumping has also been realized, with anoutput power of 25 mW at an absorbed pump powerof 570 mW [11.1170]. However, in the same setupa Cr2+:ZnSe laser showed better results. Investigationsrevealed that the passive losses of the Cr2+:ZnS crys-tal (14%/cm) were much higher than the losses from theCr2+:ZnSe crystal (4%/cm), indicating the larger prob-lems with crystal growth and crystal quality in the caseof Cr2+:ZnS compared to Cr2+:ZnSe.

Compared to ZnSe and ZnS, the thermal propertiesand material parameters of Cd0.55Mn0.45Te are muchworse, i. e., the dn/dT is higher and the thermal con-ductivity is lower (Table 11.29). Therefore, efficientlaser operation was only achieved under pulsed pump-ing [11.1153]. An output power of 170 mW with a slopeefficiency of 64% was realized for a repetition rate of2 Hz; the tuning range was 2.17–3.01 µm [11.1153].Mond et al. [11.1172] reported diode-pumped CW op-eration with an output power of 6 mW and a slopeefficiency of 4%. A thermal rollover at higher pumppowers was observed, indicating the strong thermal lens-

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Table 11.31 Overview of the laser results obtained for Cr2+-doped materials

ZnS ZnSe Cd0.85Mn0.15Te Cd0.55Mn0.45Te CdSe CdTe

λlaser (nm) 2350 [11.1149,1173] 2350 [11.1149] 2515 nm [11.1152] 2550 [11.1153] 2600 [11.1174] 2535

2500 [11.1145] 2660 nm [11.1175] [11.1176]

2600 [11.1164]

3000 [11.1164]

ηsl (%) 40 [11.1170] 73 [11.1145] 44 [11.1175] 64 [11.1153] 50 [11.1177] 1 [11.1176]

Tuning (nm) 2050–2400 2000–3100 2300–2600 2170–3010 2400–3400

[11.1178] [11.1165] [11.1175] [11.1153] [11.1179]

2110–2840

[11.1170]

Pout or Eout 0.1 mJ 7 W 0.6 mJ 170 mW (2 Hz) 0.5 mJ pulsed

[11.1149, 1173] [11.1169] [11.1175] [11.1153] mode

Pulse length (ML) 4.4 ps [11.1180,1181]

≈ 4 ps [11.1182]

≈ 100 fs[11.1163]

Other references

Pulsed [11.1149–1151,1170, 1171, 1173,1178]

[11.1149–1151,1173]

[11.1152, 1175, 1183] [11.1153, 1183] [11.1174, 1177,1179, 1184]

CW [11.1145, 1164,1168, 1185–1189]

Gain-switched [11.1190]

Diode pumped [11.1145, 1165,1172, 1191–1198]

Mode-locked [11.1180, 1182]

Thin disc [11.1166]

Multi-wavelength [11.1199]

ing problems. The threshold pump power was only120 mW. This low value is expected from the spec-troscopic parameters. Under diode pumping at a dutycycle of 1:4, the thermal rollover is not observed at thepump powers available. Then the highest output poweris 15 mW with a slope efficiency of 5% and a thresholdof ≈ 100 mW.

Cr2+:CdSe exhibits a similar behavior toCr2+:Cd0.55Mn0.45Te, as expected from the materialparameters and spectroscopic characteristics. Thus alsostrong thermal lensing and power-handling problems areencountered in the laser experiments. CW laser opera-tion has not yet been reported, however, the laser resultsunder pulsed pumping are very promising. Using a Q-switched Tm,Ho:YAG laser operating at 2.05 µm witha repetition rate of 1 kHz, a maximum output energyof 0.5 mJ per pulse and a slope efficiency of 50% wasobtained [11.1177]. The highest average output power

was 815 mW [11.1184] while the largest tuning rangeachieved thus far is 2.4–3.4 µm [11.1179].

Outlook for Cr2+ systems. Cr2+-doped materials arehighly efficient lasers in a very interesting wavelengthrange for application; see the overview in Table 11.31.In all materials investigated, the material parameters arestill a large problem, i. e., the quality of the crystals, thestrong thermal lensing, the high nonlinearity, yieldinga strong tendency for self-focusing, and the relativelylow damage threshold. All these problems are most se-vere for CdMnTe and CdSe. Therefore, the thin-discsetup may be a way to overcome some of these prob-lems. Another alternative is the use of larger pump andlaser modes. As far as ultrashort-pulse generation is con-cerned, Sorokina et al. recently realized mode locking inthe fs regime [11.1163]. Thus far, only a few materialshave been investigated for laser applications. Therefore,

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there is the possibility to look for different host materialsfor the tetrahedral Cr2+ ion, e.g., ZnGa2S4, ZnGa2Se4,CaGa2S4, and CaGa2Se4.

Other transition-metal-ion lasers.V2+ lasers. The V2+ ion is isoelectronic to the Cr3+ion. Therefore, its energy-level scheme can also bedescribed with the Tanabe–Sugano diagram shown inFig. 11.81. The absorption spectra of octahedrally coor-dinated V2+ is similar to that of Cr3+, but red-shifteddue to the lower valence of the V2+ ion. The spectraare dominated by the three broad spin-allowed bandsdue to transitions between the 4A2(4F) ground stateand the 4T2(4F), 4T1(4F), and 4T1(4P) excited states.The emission spectra consist of a broad band due tothe 4T2(4F) → 4A2(4F) transition, also shifted to longerwavelengths compared to the emission spectra of Cr3+.Laser oscillation with V2+ on the 4T2(4F) → 4A2(4F)transition was realized only in MgF2 [11.1200–1202]and CsCaF3 [11.1203, 1204]. An overview of the ob-tained laser results is given in Table 11.31. The laserefficiencies are very low. Payne et al. [11.44, 1205] andMoncorgé et al. [11.1206] found that ESA is the domi-nant loss mechanism for laser oscillation. In some mater-ials nonradiative decay also competes with the emission,leading to small emission quantum efficiencies.

Ni2+ lasers. The energy-level scheme of octahedrallycoordinated Ni2+ in crystals can be described with theTanabe–Sugano diagram shown in Fig. 11.86. In theabsorption spectra, three bands according to the spin-allowed transitions from the 3A2(3F) ground state to the3T2(3F), 3T1a(3F), and 3T1b(3P) excited states are ob-served. The laser transition of Ni2+ lies in the infraredspectral range due to the 3T2(3F) → 3A2(3F) transition.Its spectral position is strongly wavelength dependent(Table 11.33). The emission lifetime is typically on theorder of ms and the emission quantum efficiency is nearunity at room temperature in most materials.

Despite these advantageous spectroscopic data, laseroscillation with Ni2+ was obtained only at temperaturesbelow 240 K and only in a few materials (Table 11.33).The absence of laser oscillation at room temperature canbe explained by excited-state absorption (ESA), whichoverlaps with the spectral range of emission. Detailed

Table 11.32 Overview of V2+-laser materials

Host material λlaser (nm) Tuning range (nm) T (K) Mode of operation Pout/Eout η (%) Ref.

MgF2 1121 1070–1150 77 pulsed [11.1200–1202]

CsCaF3 1280 1240–1330 80 CW ≈ 15 µW 0.06 [11.1203, 1204]

)

)

*

-

,

0

+

0

+

>

!

C

++

+2, +

*

Fig. 11.86 Tanabe–Sugano diagram for the d8 electron con-figuration. The dashed area corresponds to the Dq/B valuesof Ni2+

measurements of the ESA for several Ni2+-doped crys-tals were performed by Koetke et al. [11.1207, 1208].With increasing temperature the excited-state absorp-tion and ground-state absorption bands become broaderand thus overlap to a larger extent with the stimu-lated emission. As a result, the spectral region whereσeff = σse −σESA > 0 becomes narrower. Furthermore,the losses due to ground-state absorption increase. How-ever, at low temperatures output powers up to 10 W andslope efficiencies up to 57% were obtained [11.1202].

Mn5+ lasers. The Mn5+ ion incorporates into crys-tals mainly at tetrahedrally coordinated lattice sites. Itsenergy-level scheme can be described with the Tanabe–Sugano diagram shown in Fig. 11.82. Compared to Cr4+,the crystal field is higher due to the higher valencestate, thus the Mn5+ ion exhibits narrow-line emis-sion due to the 1E(1D) → 3A2(3F) transition. Opticalproperties of Mn5+ ions in solids have been underinvestigation for more than 30 years [11.1209–1217].

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Table 11.33 Overview of the results of Ni2+ lasers

Host material λlaser (nm) Tuning range T (K) Mode of operation Pout/Eout η (%) Ref.(nm)

MgO 1314.4 77 Pulsed [11.1200]

1318 80 CW 10 W 57 [11.1202,1223]

≈ 1320 [11.1202]

≈ 1410 [11.1202]

MgF2 1610–1740 89 CW, CW-Q-qw 1.85 W 28 [11.1202]

1608–1730 80 CW 185 mW 10 [11.1224]

1610–1740 80 CW ≈ 100 mW [11.1225,1226]

1670 80 QS (480 ns) 25 mW (1 kHz) [11.1225]

1610–1730 ML (23 ps) ≈ 100 mW [11.1227]

1623 77 Pulsed [11.1200,1228, 1229]

1630 20–90 CW 1.74 W 37 [11.1230]

1730–1750 100–200 CW ≈ 0.5 W [11.1230]

1636 77–82 Pulsed [11.1200]

1674–1676 82–100 Pulsed, CW [11.1200]

1731–1756 100–192 Pulsed, CW [11.1200]

1785–1797 198–240 Pulsed [11.1200]

MnF2 1865 20 Pulsed [11.1200]

1915 77 Pulsed [11.1200]

1922 77 Pulsed [11.1200]

1929 85 CW (exc.) [11.1200]

1939 85 CW (exc.) [11.1200]

KMgF3 1591 77 Pulsed [11.1231]

CaY2Mg2Ge3O12 1460 80 Pulsed ≈ 3.8 mJ 0.7 [11.1202,1229]

Gd3Ga5O12 1434–1520 100 Pulsed 6 [11.1207,1232]

Mn5+ laser operation at room temperature was demon-strated by Merkle et al. in Ba3(VO4)2, Sr3(VO4)2,and in Sr5(VO4)3F [11.1218–1220]. The laser tran-sition is realized between the 1E(1D) excited stateand the 3A2(3F) ground state, thus these lasers arethree-level systems. The efficiency is rather low (laseroutput energy ∼ µJ, ηsl ≤ 1.6%) and laser oscillationfrom other Mn5+ systems has not been reported. Themajor drawback for these lasers is the excited-stateabsorption at the stimulated-emission wavelength. In-vestigations of the excited-state absorption and gainwere performed in detail by Verdun [11.1217], Merkleet al. [11.1218], Manaa et al. [11.1221] and Kücket al. [11.933, 1222].

To obtain a four-level system, one should searchfor crystals with low crystal field strengths or withlarge energy-level splittings. Then the 3T2(3F) or oneof its crystal field components would be the lowest-

energy level. However, thus far Mn5+ systems exhibitingbroadband emission are not known.

Co2+ lasers. The Co2+ ion in octahedral coordination ex-hibits laser oscillation in the mid-infrared spectral regionbetween 1.5 µm and 2.5 µm. Its energy-level schemecan be described with the Tanabe–Sugano diagram forthe 3d7 electron configuration shown in Fig. 11.87.Three broad and spin-allowed transitions between the4T1a ground state and the 4T2, 4A2 and 4T1b ex-cited states exist. In MgF2, these transitions are locatedaround 7000 cm−1, 15 000 cm−1, and 20 000 cm−1, re-spectively [11.1233]. The emission occurs between1.5 µm and 2.5 µm, according to the 4T2 → 4T1a transi-tion. At low temperatures, the lifetimes are on the orderof several ms, while at room temperature the emis-sion is strongly quenched due to nonradiative decayvia multiphonon relaxation, yielding very low quantum

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)

)

*

&

*

*!

!

*+

++

*+

,

*+

,*+

*+

0*0

0

-

Fig. 11.87 Tanabe–Sugano diagram for the d7 configuration(after [11.36])

efficiencies [11.1234]. Excited-state absorption over-laps with the 4T1a → 4T2 absorption band, which isthe main pump band for laser operation. In the regionof stimulated emission, the excited-state absorption isnegligible [11.1233].

Laser oscillation was obtained in the continuous-wave regime only at cryogenic temperatures, whilepulsed laser operation was also realized at room temper-ature for Co2+-doped MgF2 and KZnF3. An overviewof the laser results for Co2+ laser systems is given inTable 11.35. The best laser results were obtained for

Table 11.34 Overview of Mn5+-laser materials. The data are from the literature stated in the text [11.1209, 1222]

Ba3(VO4)2 Sr3(VO4)2 Sr5(VO4)3F

Structure Hexagonal, R-3m Hexagonal, R-3m Hexagonal, P63/m

Site symmetry C3v (V-site) C3v (V-site) Cs (V-site)

Growth Czochralski/LHPG Czochralski/LHPG Czochralski

Tm (C) 1560 1923

σabs (10−20 cm2) ≈ 300 ( 800 nm) ≈ 300 ( 800 nm)

σem (10−20 cm2) 10–20 27 (E ‖ c), 13 (E⊥c)

σESA (10−20 cm2) 14 (E ‖ c), 24 (E⊥c)

τem (300 K) (µs) 430–480 525 475–500

σemτ (10−22 cm−2s−1) ≈ 0.7 ≈ 1.3 (E ‖ c), ≈ 0.6 (E⊥c)

λlaser (nm) 1181.0 1168.0 1163.7

ηsl (%) 0.21 0.08 1.6

Pout or Eout ≈ 2 µJ ≈ 1 µJ

Co2+:MgF2, which is also a commercial laser system.With Co2+:MgF2 an overall tuning range of 1.5–2.5 µmwas realized with output powers up to 4.2 W, outputenergies up to 1.6 J and slope efficiencies as high as65%. As the pump source, Nd:YAG or Nd:glass lasersaround 1.3 µm are usually used, however, laser oscilla-tion has also been realized under flashlamp pumping,argon-ion laser excitation and oxygen–iodine laser exci-tation. Operation regimes are CW, pulsed, Q-switchedand mode-locked.

Fe2+ lasers. The Fe2+ ion has a 3d6 electron configu-ration that is complementary to that of Cr2+. The 5Dfree-ion state also splits into a 5T2 and a 5E state,however, for the Fe2+ ion 5E is the ground state and5T2 is the first excited state. Consequently, there is justone spin-allowed absorption (5E → 5T2) and emission(5T2 → 5E) transition. As for the Cr2+ ion, all excited-state transitions are spin-forbidden. Thus, one mightexpect similar laser characteristics for Fe2+-doped lasersas for Cr2+ lasers. However, the energy gap between the5T2 excited state and the 5E ground state is smaller, thusthe emission is at longer wavelengths and the nonradia-tive decay rate is higher, leading to lifetime shorteningand low quantum efficiencies at elevated temperatures.For Fe2+:ZnSe, the emission lifetime first increases from12 to 120 K from 33 µs to 105 µs and then decreases toabout 5 µs at 250 K due to thermally activated multi-phonon decay [11.1235, 1236]. At 14 K, the 5E → 5T2absorption band is between 2.5 µm and 3.75 µm, and the5T2 → 5E emission band is between 3.7 µm and 4.8 µm.

Laser oscillation of Fe2+ was obtained inZnSe [11.1235–1239] and in n-InP [11.1240]. ForFe2+:ZnSe, room-temperature tunable laser oscil-lation was recently reported in a gain-switched

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Table 11.35 Overview of Co2+-laser materials. (*: versus input power, QS: Q-switched operation, ML: mode-lockedoperation)

Host material λlaser (nm) λlaser (nm) T (K) Mode of operation Pout/Eout η (%) Ref.

MgF2 1860 1500–2000 80 CW 1 W 31 [11.1241]

1920 1600–2300 80 Pulsed 150 mJ 65 [11.1202, 1242,1243]

1920 80 Pulsed 7.3 W (50 Hz) ≈ 23∗ [11.1242,1243]

1920 QS (220 ns) 25 mJ [11.1242,1243]

1940 225 Pulsed ≈ 11 mJ 14∗ [11.1243]

1920 80 4.2 W 28∗ [11.1202,1243]

2100 1750–2500 RT Pulsed 70 mJ 46 [11.1244]

1650–2010 80 ML (34 ps) [11.1227]

1500–2000 QS-ML (200 ps) 400 mW (ave) [11.1245]

1600–1900 77 QS (600 ns) 15 mJ [11.1246]

1890 1600–2150 77 CW 2 W 32 [11.1247]

2050 RT Pulsed 0.2 mJ 29 [11.1248]

2050 77 Pulsed 1.6 J 44 [11.1249]

2040 1960–2180 RT Pulsed 20 mJ 25 [11.1233]

1630–2080 80 CW ≈ 100 mW 5 [11.1226,1250]

1750 77 Pulsed [11.1200,1251]

2060 1800–2450 282 Pulsed 900 mJ 33∗ [11.1252]

1803.5 77 Pulsed [11.1200,1251]

1990 77 Pulsed [11.1200,1251]

2050 77 Pulsed [11.1200,1251]

KMgF3 1620–1900 80 CW 20 [11.1253]

1821 77 Pulsed [11.1200]

KZnF3 1650–2110 80 30 [11.1253]

1770 1650–2070 80 120 mW [11.1254]

1950 1850–2050 85 CW ≈ 20 mW ≈ 2 [11.1255]

1700–2150 98 CW ≈ 55 mW 8 [11.1256]

2024 300 Pulsed 3 mJ 8 [11.1233]

ZnF2 2165 77 Pulsed [11.1200,1251]

mode [11.1237]. The pump source was the secondStokes output of a Nd:YAG laser at 2.92 µm. The out-put pulse energy was about 1 µJ. The highest outputpower and slope efficiency of 12 µJ and 8.2%, respec-tively, were obtained at lower temperatures. In this case,the Fe2+:ZnSe laser was pumped by a pulsed Er3+:YAGlaser operating at 2.698 µm [11.1235, 1236]. The laserwavelength is tunable with temperature from 3.98 µm at15 K to 4.54 µm at 180 K. In n-InP, Fe2+ laser oscillationwas obtained at 2 K at 3.53 µm, i. e., at the zero-phonontransition [11.1240].

SummaryIn this chapter an overview of transition-metal-ion-doped crystals as solid-state laser materials is given.

It was shown that they are efficient (usually tunable)laser sources covering a wide spectral range (Fig. 11.77).However, compared to lasers based on 4fn →4fn transi-tions of trivalent rare-earth ions, they play only a smallrole as far as commercialization is concerned. In prin-ciple only the Ti3+:Al2O3 and – with limitations – theCr3+:BeAl2O4 laser can be mentioned. These lasers aremainly used in the field of scientific research. The mainreason for this is that their advantages, i. e., their tunabil-ity and their capability to generate ultrashort pulses, arenot relevant for most industrial applications. They ex-hibit lower output power than lasers based on 4fn →4fn

transitions of trivalent rare-earth ions (e.g., Nd3+:YAG,Yb3+:YAG) and are more expensive and less efficientthan diode lasers. Direct diode-laser pumping is either

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Table 11.36 Overview about transition-metal ions sorted according to the corresponding Tanabe–Sugano diagram dn

(TSD-dn). Normal: octahedral coordination; italic: tetrahedral coordination. Light brown: laser oscillation in octahedralcoordination; Dark brown: laser oscillation in tetrahedral coordination. (After [11.933])

Ion TSD-d1 TSD-d2 TSD-d3 TSD-d4 TSD-d5 TSD-d6 TSD-d7 TSD-d8 TSD-d9

Ti Ti3+ Ti2+

V V4+ V3+ V2+ V3+ V4+

Cr Cr4+ Cr3+ Cr2+ Cr2+ Cr4+ Cr5+

Mn Mn5+ Mn4+ Mn3+ Mn2+, Mn2+ Mn3+ Mn5+ Mn6+

Fe Fe2+ Fe3+, Fe3+ Fe2+ Fe6+

Co Co3+ Co2+ Co3+ Co2+

Ni Ni2+ Ni3+ Ni3+ Ni2+

Cu Cu2+ Cu3+ Cu2+

inefficient or – in the case of Cr3+ and Cr2+ lasers– requires laser diodes that are not yet available ata satisfactory price and quality on the market.

Transition-metal-ion and especially tunable lasersystems exhibit a stronger coupling of the electroniclevels to the vibrating lattice of the crystal. This leads tohigher possibilities for ESA and nonradiative decay pro-cesses compared to the 4fn →4fn transitions of trivalentrare-earth ions in crystals.

The main problem in realizing efficient laser oscilla-tion is the excited-state absorption from the metastableupper laser level. From (11.87) and (11.88) its ef-fect on laser threshold and slope efficiency becomesclear. It occurs in principle for every electron configura-tion, either as intra- or interconfigurational transitionsor as a transition to charge transfer or conduction-band-related levels. The electron configurations are ingeneral favorable, when intraconfigurational excited-state absorption transitions are not possible, e.g., inthe d1 and d9 configuration, or are less strong due toselection rules, e.g., in the d4 and d6 configurations.Other configurations exhibit more-complex energy-levelschemes, therefore intraconfigurational ESA transitionsoccur with high probability in the spectral regions ofemission and excitation. The influence of ESA, how-ever, can even in this case be reduced, e.g., by takingadvantage of polarization-dependent transition rules,which result in higher stimulated-emission cross sec-tions than ESA cross sections, as it is the case, e.g., forCr3+:LiSrAlF6 and Cr4+:Y3Al5O12.

The role of nonradiative decay for the realizationof efficient laser operation is less important, althoughnot negligible. It affects to a first approximation onlythe laser threshold, which is increased (11.84). How-ever, a nonradiative rate leads to a temperature increase

in the pump channel, which also affects the overalllaser characteristics. Thus, its general influence dependsstrongly on the material parameters of the laser sys-tem, i. e., mainly of the host material. Materials witha high thermal conductivity and mechanical strength arefavored. As an example, Cr4+:Y3Al5O12 can be given.Here, the quantum efficiency is less than 20%, whereaslaser operation with a slope efficiency close to 40% wasrealized.

Table 11.36 gives an overview of the transition-metal ions in octahedral and tetrahedral coordinationinvestigated to date. They are listed according totheir corresponding energy-level diagram (i. e., Tanabe–Sugano diagram). The laser-active transition-metal ionsare indicated. Almost all transition-metal ions with dif-ferent valence states and ligand coordinations have beeninvestigated thus far. Efficient room-temperature laseroscillation was only obtained for Ti3+ and Cr3+ inoctahedral and for Cr2+ and Cr4+ in tetrahedral co-ordination. Whether efficient laser operation will alsobe obtained for other ions depends very much on thehost material chosen. For example, the Ti3+ ion ex-hibits efficient laser operation only in Al2O3. Therefore,other ions cannot a priori be excluded as efficient laserions.

11.2.6 Overview of the most ImportantLaser Ions in Solid-State Lasers

Research on laser materials has created a number ofcompact, efficient solid-state laser sources for a largevariety of applications. Laser materials have been de-veloped for various wavelengths (near-infrared, visible,UV) and power regimes (mW to multi-kW). Spe-cial geometries of the active material (microchip, rod,

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disc, fiber) strongly correlate with the active ion con-centrations and cross sections of absorption and gaintransitions.

The values of the cross sections depend on the quan-tum numbers of the final and initial states as well as onthe local environment of the active-ion site in the mater-ial. The lifetimes of the states are influenced by radiativeand nonradiative transitions. So, the microscopic crystalproperties play an important role for static and dynamicprocesses in the laser crystal.

In the near-infrared spectral region high efficien-cies have been achieved with diode-pumped oxide-and fluoride-based laser materials doped with the rare-earth ions Nd3+, Tm3+, Ho3+, Er3+, and Yb3+. Forhigh-average-power operation Nd3+- and Yb3+-dopedcrystals are of greatest interest. Especially the Yb3+ion exhibits very small Stokes losses and minimum heatgeneration, which reduces thermal lensing and improvesbeam quality. Transition-metal-doped crystals based onthe ions Ti3+, Cr2+, Cr3+, and Cr4+ offer broadly tun-able radiation within the spectral region 680–3000 nm.In the visible region Er3+-, Tm3+-, and Pr3+-dopedlaser materials operate at several red, green, and bluetransitions with laser diode upconversion and/or directpumping. So far Ce3+ is the only ion with reasonabledirect laser performance in the UV.

The following wavelength data provide a roughguide to the spectral range of the various laser ions.

Near-IR rare-earth lasers.

Nd3+ 0.9, 1.06, 1.3 µmYb3+ 1–1.1 µmTm3+ 2 µm, 1.5 µm (upconversion)Ho3+ 2 µmEr3+ 1.6, 3 µm, 0.85 µm (upconversion)

Visible rare-earth lasers.

Pr3+ 0.64 µm (diode-pumped)Pr3+,Yb3+ 0.52, 0.63 µm (upconversion)Er3+ 0.55 µm (upconversion)Tm3+ 0.45, 0.48, 0.51, 0.65, 0.79 µm (up-

conversion)

Frequency doubling of near IR rare-earth lasers (Nd, Yb)

UV rare-earth lasers.

Ce3+ 0.3 µm

Frequency doubling of visible rare-earth lasersFrequency tripling/quadrupling of near-IR lasers

Transition-metal lasers.

Ti3+ 0.68–1.1 µmCr3+ 0.7–1.1 µmCr4+ 1.2–1.6 µmCr2+ 2–3 µm

11.3 Semiconductor Lasers

11.3.1 Overview

In crystalline solids the interaction between atomicenergy levels generates energy bands. A quantum-mechanical treatment provides, in the single-electronapproximation, energy bands which overlap which eachother or are separated by bandgaps from each other.In semiconductors, we find between the energeticallyhighest band, which is fully occupied by electrons atT = 0 K, the valence band (VB), and the energeticallylowest band, which is completely empty at T = 0 K theconduction band (CB) an energy range, in which wefind no allowed energy states, disregarding energy statesoriginating from dopants. This bandgap, Eg, is the en-ergy difference between the lower conduction-band edgeEC and the upper valence-band edge EV. For laser op-eration an extreme deviation from thermal equilibrium(nonthermal carrier distribution) is required: carrier in-version. In the CB close to EC carrier inversion means

a much higher density of electrons than holes and in theVB close to EV a much higher density of holes than elec-trons. This extremely nonthermal condition is generatedby strong electrical or optical pumping of the laser-activematerial. The electrical carrier injection in most cases isobtained by a p–i–n heterojunction made of III/V semi-conductor materials. Choosing InP as a typical example,Fig. 11.88 displays a planarization (projection into 2-D space for simplification) of the real 3-D zinkblendecrystal structure with the corresponding band structuresbelow. From the left to the right p-InP, intrinsic i-InPand p-InP are shown.

In the highly p-doped InP bulk regions, at room tem-perature nearly all acceptors have released their holesinto the valence band. Well away from the pn junc-tion the hole field current (drift current) dominates thecarrier transport. At the left edge of the device, holesare generated under the p contact. At the right edge ofthe device, electrons are injected through the n contact.

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Well away from the pn junction the electron field cur-rent dominates the carrier transport there. The bandgapEg nearly equals the energy of the photons generated inthe central intrinsic region. However, such a homojunc-tion p–i–n structure (Fig. 11.88) suffers from enormouscarrier leakage and the lack of a medium guiding thegenerated light. To enable operation at room temperatureand considerably reduce the threshold currents at leasta double heterostructure (heterojunction p–i–n structure)is required (Fig. 11.89). This Nobel-prize-winning ideaby Koemer and Alferov [11.1257, 1258] enables carrierconfinement and optical confinement by a single impor-tant modification: using higher-bandgap Eg materialsfor the p and n regions compared to the central intrin-sic active layer. Thus, for semiconductor lasers at leasttwo different materials are required, as shown in spacez(x, y) (Fig. 11.89a) and in the band structure E(y, z) inspace (Fig. 11.89b) for an undoped and unbiased struc-ture. The central layer (material 1) has a lower Eg thanthe embedding bulk layers (materials 2) and therefore

electrically confines electrons in the CB and holes inthe VB, which is visualized by the band edge. Sincefor semiconductors decreasing Eg in nearly all casesincreases the refractive index n we fortunately obtainan optical waveguide: a central higher-n material em-bedded in a lower-index material (Fig. 11.89c). Thus,the laser-active layer is the core layer of the waveguideat the same time. By a proper design, essentially ofthe refractive index differences and dimensions, we canefficiently guide the generated light in a fundamentalmode (see the profile in Fig. 11.89c) similar to a single-mode optical fibre. Considering an edge-emitting laser,the structure is now p–i–n doped and forward-biased(Fig. 11.89d), enabling the carrier transport describedabove: hole injection from the left and electron injec-tion from right. The band structure (Fig. 11.89e) of thedoped and biased laser structure visualizes the operationof the p–i heterointerface acting as a border (electricalconfinement) for the electrons. The small potential bar-rier at the i–n heterointerface is no real obstacle. The

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electrons can tunnel through or thermally jump across.In analogy, the p–i heterointerface is no real obstaclefor the holes, whereas the n–i heterointerface providesthe required border (electrical confinement). Modern

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semiconductor lasers, however, include quantum wells(QWs, a 2-D carrier system) [11.1259] or quantum dots(QDs, a 0-D carrier system) [11.1260] as the laser-activemedium, instead of a 3-D laser-active region in bulk

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Fig. 11.90 Schematic of a semiconductor laser with 3-Dlaser-active layer and buried waveguide. Insets: refractiveindex profiles and mode profiles in the x- and y-directionsintersecting the active layer at A and B, respectively

lasers (Fig. 11.89e). In Fig. 11.89f two QWs are depictedas an example, providing the emission energy as the dif-ference between the energetically lowest bound states inthe wells (note that the ordinate indicates the electronenergy).

Finally, electrical and optical confinement in the y-direction (Fig. 11.89c) has to be completed by electricaland optical confinement in the x-direction (Fig. 11.90).Note that this figure is rotated by 90 with respect toFig. 11.89. In Fig. 11.90 the active material is embed-ded in the x-direction by semi-insulating (si) materialto force the current to move mainly in the active layers(lateral electrical confinement). The si material is cho-sen to have a higher bandgap, and thus a lower refractiveindex, than the active material (lateral optical confine-ment). Also in this case, wave guiding is applied in the x-and z-directions. This is shown in the two insets by thetwo cross sections A and B, showing the refractive-indexprofiles and light-intensity profiles. Here, waveguidingin fundamental mode is shown for the case of bulk 3-Dactive layers.

11.3.2 Resonator Typesand Modern Active Layer Materials:Quantum Effects and Strain

Modern optoelectronic semiconductor devices are basedon a sequence of materials of different composition. Em-bedding a film of lower Eg between a material of largerEg, we obtain quantization effects if the film thicknessL y is on the order of the electron wavelength or below(QW) [11.1259]. Using the Schrödinger equation, weobtain at least one bound state in the CB and VB poten-tial wells, respectively. The electrons and holes are stillfully mobile in the x- and z-directions. This means thata wavevector k|| parallel to the heterointerfaces exists.The motion in the y-direction is considerably restrictedand statistically described by quantum mechanics. Fig-ure 11.91 displays the VB structure in k-space [E(k‖)]obtained by a theoretical model calculation [11.1261]based on the Schrödinger equation for four differentGa(In)As/(Al)GaAs QWs (k‖ is oriented parallel to theheterointerfaces).

In most cases in addition the QWs and the barriers inthe active layer are additionally strained [11.1262]. Usu-ally, the wells are compressively strained, which means

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that the lattice constant of that layer is compressivelyreduced in the x- and z-directions to match that of thesubstrate ao. In many cases, the barriers are in tension,which indicates that the lattice constant of the barriersis increased in the x- and z-directions by tensile strainto be identical to the substrate lattice constant ao.

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Strain and quantization effects in the active layersof semiconductor lasers are used to improve the deviceproperties. For the laser it is beneficial to make the effec-tive masses of electrons and holes as similar as possible,to increase material gain, to reduce threshold, to tailorthe density of states and to increase differential gain.This can be obtained by applying strain and/or quantiza-tion. In Fig. 11.92, the energy dependence of the densityof states is shown for 3-D material to be root-like, andto be constant for 2-D material.

Figure 11.93 depicts examples of InAs [11.1263]and CdSe [11.1264] QDs. Schematically, Fig. 11.93ashows the QD formation in a cross section of the crys-tal structure. We see the three upper monolayers of theGaAs substrate. During the self-organized QD formation

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the intentionally integrated strain (detailed explanationbelow) of the InAs versus GaAs plays an importantrole. For thermodynamical and elastomechanical rea-sons, two monolayers of strained InAs are first formed.Continuing the growth, for total-energy reasons, it is ofconsiderable advantage for the semiconductor surface tocontinue by a localized island-like growth. In this casea possible geometric shape is a pyramid with a baseplane directly joined to the top of the InAs monolay-ers. Experimentally, this phase can be directly studiedusing an atomic force microscope (AFM) [11.1260].Figure 11.93b displays in a top view a typical AFMsurface profile. In the next process step the QDs areovergrown by GaAs and thereby embedded. Experi-mentally the final layer sequence can be studied bytransmission electron microscopy (TEM) after cleaving.In order to increase the number of QDs in the direc-tion perpendicular to the substrate interface, the wholeInAs process is repeated as often as desired after a de-fined GaAs spacer layer thickness. This provides layeredQD arrangements. Figures 11.93c, d show TEM micro-graphs for two different spacer-layer widths: 40 nm ad20 nm. We observe a vertical correlation of the dotsfor the smaller separation. At very high magnificationFig. 11.93e shows QD formations in the II/VI semicon-ductor system CdSe/ZnSe; note the different scales inthe subfigures. This high TEM magnification resolvesthe individual crystal layers. The CdSe QD is contrasteddark against the surrounding brighter ZnSe.

However, quantization is not limited to a single di-rection. If we limit the carrier movement in anotherdirection (e.g., in the z- and y-direction as in Fig. 11.92),

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we obtain a 1-D carrier system and, thus, a quantumwire [11.1265] with hyperbolically shaped density ofstates branches. Limitation of carrier movement in allthree space directions results in quantum dots [11.1260,1261, 1263–1272], i. e., a zero-dimensional (0-D) car-rier system with a δ-like density of states (Fig. 11.92d).For many physical properties of the semiconductor (e.g.,carrier mobilities, carrier capture in quantized states andspontaneous or stimulated emission of light), the densityof states plays an important role. Using dimensionalityand strain efficiently during the design of lasers en-ables one to enhance desired and suppress undesiredproperties. Here, modern epitaxy [e.g., metal organicchemical vapour deposition (MOCVD) and molecularbeam epitaxy (MBE)] are powerful tools. To date 2-Dand 0-D structures have been used successfully for laserdevices. 1-D structures have not yet had a breakthroughdue to their disadvantageous dynamic properties. QDsare grown today using self-organization in the Stranski–Krastanow growth mode [11.1263,1265,1267,1271]. Bygrowing a large number of QDs with identical quantizedenergy levels in the laser-active layer, we would bene-fit from a strongly reduced temperature sensitivity anda very high differential gain and, thus, extremely highbit rates in optical communication. Although many QDlasers have been implemented, the pyramid-like QDssuffer from strong fluctuations in size and thus in theirenergy levels.

On the basis of these QDs, however, the aforemen-tioned improvements in device properties have not yetbe demonstrated. On the other hand, the strong fluctua-tion causes a strongly spectrally broadened gain profile,

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which is beneficial for other applications, e.g., short-pulse generation by mode locking or wide spectraltuning of the laser device. However, we should alsoremember that 10 years were necessary after implement-ing the first QW lasers to demonstrate that QW laserswere superior to bulk lasers. Similarly, QD lasers willrequire time before their final breakthrough. However,we believe that QW and QD lasers will share the ap-plication area in the future, depending on the specificrequirements.

Figure 11.94 shows the dependence of the bandgapon the lattice constant for various III/V and II/VIsemiconductors. The brown area corresponds to the qua-ternary AlzGa1−x−zInxN. The grey area describes thequaternary Ga1−xInxAs1−yPy. For high laser efficien-cies in interband lasers, active materials with a directbandgap are required, disregarding for the moment someintraband lasers. Note that simple bulk Si has an indirectband structure and does not provide efficient radiative re-combination. Modified Si structures have demonstratedstrong luminescence, LED operation or laser oscillation,e.g., from Si/Ge superlattices, QDs in Si and Si-based Raman laser structures [11.1273]. All compoundsemiconductors, located on one of the grey verticallines (Fig. 11.94) that indicate the lattice constant ofimportant semiconductor substrate materials, can belattice-matched to the respective substrate. Accord-ing to Fig. 11.94 the ternary compounds Al0.48In0.52As

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(Eg = 1.43 eV) and Ga0.47In0.53As (Eg = 0.75 eV) havethe same lattice constant as InP (Eg = 1.34 eV), whichis available in wafers today up to a diameter of 150 mm.

QW widths (or QD dimensions), effective masses[more precisely the E(k) functions], and bandgaps ofwell and barrier materials energetically determine thequantized states and, thus, the spectral gain profile ofthe laser, i. e., the possible range of laser emission. Fig-ure 11.94 shows that a very large wavelength range canbe covered for different applications. Some examplesare red and blue lasers which are used for data stor-age in digital versatile and blue-ray discs, respectively.Emission at about 850 nm is used for short-range opti-cal fibre communication and CD devices, 980 nm lasersfor pumping Er-doped fibre amplifiers, while between1.25 and 1.65 µm the lasers are applied in ultrahigh-bit-rate long-haul optical fibre communication. Visible andinfrared wavelengths are very attractive for optical sens-ing. The range between 0.8 and 1 µm is used for directlaser applications (welding, drilling, cutting and solder-ing) since it includes lasers with extremely high outputpower, highest wall-plug efficiencies and lowest priceper watt of optical power.

Stimulated emission and optical gain (Fig. 11.95)are essentially determined by the product of the re-duced electronic density of states and the Fermi factor( fc − fv). This Fermi factor originates from fc(1− fv) –fv(1− fc), namely the probability of processes photon-

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generating photons (emission), i. e., the probability thatthe CB states are occupied by electrons and VB states arenot occupied by electrons fc(1− fv) minus the probabil-ity of processes destroying photons (reabsorption), i. e.,the probability of VB states occupied by electrons andCB states not occupied by electrons fv(1− fc). The fulllines in Fig. 11.95a, b schematically show the spectralmaterial gain profile of 3-D and 2-D semiconductors,respectively. Gain ranges spectrally from the bandgapEg to the difference between the two quasi-Fermi lev-els ∆EF. In reality there are additional effects such asFermi-level filling, profile broadening [11.1261] (boldbroken line) or many-body effects. Material gain minusloss provides net gain (Fig. 11.95c, d). If these laser-active materials are placed in a Fabry–Pérot (FP) laser

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the net gain can support the modes (vertical lines) as indi-cated. In Fig. 11.95 the spontaneous emission is shownby the thin broken lines, revealing a different spectralprofile, which is important, e.g., for LEDs.

Figure 11.96 depicts a classification summary of themost important semiconductor laser geometries. Mostof them will be treated in detail below. Generally, theheterointerfaces are located horizontally. We distinguishbetween horizontal (left) and vertical (right) resonatorstructures, which is indicated by the orientation of thebroad brown double arrow. Thus, we have horizontalcavity lasers (in-plane lasers) and vertical cavity (VC)lasers.

First case: Fabry–Pérot (FP) structures. Here the opti-cal reflection (feedback) is provided by the borders ofthe resonator (facets). In many cases, the high refractiveindex difference between the semiconductor and the airis already sufficient, providing an optical reflection co-efficient of about 30%. By additional facet coatings thiscoefficient can be tailored continuously between 0% (an-tireflection) and 100% (perfect reflection). The resonatormodes are given by

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Intuitively, the resonator length L has to be a positiveinteger multiple of half of the wavelength of light inthe medium, where neff is the effective refractive index(eigenvalue of the Helmholtz equation, see below) of thewaveguide. In the second case, in so-called structureswith distributed feedback (DFB), the optical reflectionis extended over the whole resonator. A very efficientfeedback occurs at the Bragg wavelength λB, which iscorrelated with the DFB grating periodΛ via the Braggcondition:

mDFB(λB/neff ) = 2Λ . (11.93)

mDFB is a positive integer and describes the gratingorder. An example: for a DFB laser emitting at λB =1.55 µm having a first-order grating (mDFB = 1) and aneffective refractive index neff = 3.27, a grating period ofΛ= 237 nm is required. Intuitively, an integer multipleof the wavelength in the medium has to correspond,according to (11.93), to the double grating period. Notethat the mathematical structure of (11.92) and (11.93) isidentical. In lasers with a typical length of 200 µm weobatin for FP laser modes within the gain profile a verylarge mFP (on the order of 1000) while for DFB lasersin most cases mDFB is 1 (for a first-order grating).

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As expected, light is emitted in horizontal directionsfor edge-emitting lasers (Fig. 11.96). Surface emitterscan be implemented by etching an out-coupling mirrorinclined at 45 or by a second-order DFB grating (notethe doubled grating period in the figure). In a second-order grating the light field is reflected back horizontally(180) and coupled out vertically. If the Bragg con-dition is exactly fulfilled, the emission occurs exactlyperpendicular to the surface (90). However, the largerthe deviation in (11.93) in the (>)-direction or (<)-direction, the larger or smaller the angle, respectively. Ifthe grating is interrupted we have a distributed Bragg re-flector (DBR) structure. Typically, a central grating-freesection is embedded between two DBR sections. In theDFB and DBR structures the horizontally propagatinglight field alternatively passes two virtual quasi-layers ofslightly different refractive index. For DBR resonatorswith real layers (see Sect. 11.3.4 or the next section)these operation principles become more evident.

Second case: VCSELs. They are also based on DBRstructures, which in contrast to the first case consist ofreal multiple layers having a very high refractive-indexdifference. The central cavity has no grating and is em-bedded between two DBR mirrors, thus also forming anFP-like structure. However, feedback is distributed overthe two DBR structures. In most cases the thicknessof a single period (equivalent to a pair of neighbor-ing layers) is chosen to equal half the wavelength oflight in the medium for the design wavelength. This cor-responds to a first-order grating according to (11.93).Since the resonator is oriented perpendicularly and theemission occurs perpendicular to the main chip surface(oriented parallel to the substrate area), this laser is calleda vertical-cavity surface-emitting laser (VCSEL). Thetwo mirrors must have a very high reflectivity to reachthe laser threshold, since the laser-active layer is rela-tively thin and has a weak overlap with the light field inthe resonator.

Edge-emitting lasers as well as VCSELs will betreated in detail in the following sections. On the ba-sis of both cases (A and B), it is possible to implementlasers with external resonators, as indicated in the lastrow of Fig. 11.96.

11.3.3 Edge-Emitting Laser Diodeswith Horizontal Resonators

As already mentioned, the population inversion betweenthe valence and conduction bands, which is necessaryfor the coherent amplification of radiation [11.1274],

can be obtained by an injection of electrons and holesvia a forward-biased pn junction (see Figs. 11.89, 90).The electrons injected into the n semiconductor as wellas the holes injected into the p semiconductor diffuse tothe pn junction and can recombine there radiatively andgenerate a photon with energy ω (Fig. 11.88). If theexternal voltage is increased and the carrier density ex-ceeds a critical value in the range of 1018 cm−3, the rateof photon emission becomes higher than the absorptionrate, so that an incident wave can be amplified coher-ently due to stimulated emission. The condition for thispopulation inversion is that the separation between thequasi-Fermi levels E fc and E fv , which describe the fill-ing of the conduction and valence bands, is larger thanthe bandgap Eg (Bernard–Durafourg condition):

E fc − E fv ≥ ω≥ Eg . (11.94)

In this case the semiconductor material is transpar-ent for the generated wave with the wavelength definedby the bandgap. The resulting gain values are very high(in the range of 103 cm−1) due to the high carrier den-sity. Lasing occurs if the additional losses of the opticalresonator providing the feedback are compensated.

Double-Heterostructure LasersThe active region, where population inversion isachieved, is very thin in homojunction lasers and thethreshold current is very high, since only a small portionof the injected carriers is utilized for the lasing pro-cess. Lower threshold currents and continuous operationat room temperature can be achieved with double-heterostructure lasers, where the low-bandgap activelayer is sandwiched between n- and p-doped claddinglayers with a higher bandgap epitaxially grown on a sub-strate (Fig. 11.89). The technical realization of thesestructures is possible if the lattice mismatch between thedifferent layers of the material system does not exceeda critical value. Double heterostructures [11.1257] havethree main advantages with respect to laser operation:

• The bandgap difference between the layers is dis-tributed between the valence and conduction bandsand creates potential barriers for the injected elec-trons and holes. For an appropriate choice of dopingand applied voltage, a nearly rectangular-shapedpotential well can be achieved (Fig. 11.89), whichefficiently confines the carriers in the low-bandgapactive layer if the potential difference is higher thanthe thermal activation energy kBT . The width of theactive layer dact is defined by the heterostructure ge-ometry. The carriers injected via the pn junction are

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captured in the potential well and confined to a smallvolume, thus decreasing the injection current, whichis necessary to achieve inversion.• There is no reabsorption of the radiation emittedfrom the active layer in the surrounding layers dueto the difference in bandgap energies.• In the material systems used for semiconductorlasers, the low-bandgap active layer has a higherrefractive index than the surrounding cladding lay-ers. Therefore, the double heterostructure acts asa dielectric planar slab waveguide, confining thegenerated optical field to the active region due tothe refractive-index difference between the layers.The number of modes supported by this opticalwaveguide for a given wavelength depends on thethicknesses and the refractive indices of the lay-ers. By a proper choice of the design parameters,a single transverse (perpendicular to the pn junc-tion) mode can be selected, which concentrates thephoton density for stimulated emission in the gainregion.

Thus, a double heterostructure enables the confinementof carriers and generated photons in the active layer. Forbulk active layers, the de Broglie wavelength of the car-riers is small compared to the thickness of the activelayer, leading to a high degree of carrier confinement.The wavelength of the photons, however, is comparableto the dimension of the structure, so that only a portion ofthe optical intensity is confined to the active region. Thetransversal distribution of the light intensity in the dou-ble heterostructure is given by the solution of the waveequation of a planar slab waveguide supporting TE andTM modes with an effective index neff as the respectiveeigenvalue. Figure 11.89c shows the optical intensity ofthe fundamental TE mode in a double heterostructure asa function of the transversal coordinate y, revealing theincomplete confinement of the optical field in the activeregion. The fraction of the mode intensity within the ac-tive layer is called the optical confinement or the fillingfactor Γact, which is an important design parameter fora semiconductor laser:

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, (11.95)

where E(y) is the electric field. If the light is also con-fined in the lateral direction of the laser, the definitionhas to be modified accordingly.

The dependence of the optical confinement factor ofthe guided modes on the thickness d for a layer struc-ture is shown in Fig. 11.97. For increasing thickness of

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The correction factor nact/neff takes into account thewaveguiding effect on the modal gain [11.1275].

Laser StructuresUsing modern epitaxial growth methods it is possibleto realize semiconductor multilayer structures very pre-cisely so that stable, transverse single-mode operationcan be obtained using dielectric waveguiding [11.1276].For most applications, however, lateral (parallel to thepn junction) patterning of the laser structure is also re-quired to obtain lateral carrier and photon confinement,which is important to obtain stable and efficient laseroperation with high spectral purity and good couplingefficiency into a fibre. In addition, lateral confinement ofthe injection current is necessary to avoid leakage cur-rents bypassing the active region. The confinement ofphotons, carriers and current has been implemented inmany ways, reflecting the specific purpose of the de-vice. Semiconductor lasers can be classified accordingto the mechanism of lateral waveguiding as gain-guidedor index-guided, depending on whether it is the lateralvariation of the optical gain or the refractive index thatconfines the mode. Index-guided lasers can further besubclassified as weakly or strongly index-guided, de-pending on the magnitude of the lateral refractive-index

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706 Part C Coherent and Incoherent Light Sources

step. In a gain-guided laser structure (oxide stripe laser,Fig. 11.98, left) the current is injected via a stripe con-tact (width w≈ 5 µm) in a laterally unpatterned activelayer. The optical field is mainly guided by the result-ing variation of the gain. The lateral waveguiding isweak, so that even small variations of the refractiveindex, e.g., due to temperature changes or carrier in-jection lead to unstable operation. In an index-guidedlaser structure (buried laser structure, Fig. 11.98, right)the active layer (w≈ 2 µm) is laterally embedded intoa material with a lower refractive index (n< nact), higherbandgap and higher electrical resistivity to facilitate sta-ble lateral waveguiding, carrier confinement and currentconfinement. The typical features of index- and gain-guided laser structures are summarized in Fig. 11.98.The power–current characteristic (P–I curve) of gain-guided lasers is characterized by a high threshold current(typical 50–100 mA) and kinks originating from theunstable lateral waveguiding. The optical spectrum ismultimodal due to the enhanced spontaneous emission.The phase-fronts are curved in the resonator and the farfield shows the characteristic twin lobes caused by thelaterally inhomogeneous gain distribution. The main ad-vantage of gain-guided lasers is the simplicity of theirfabrication. For very large stripe widths (50–100 µm)a so-called broad-area laser results, in which the cur-rent is injected laterally and homogeneously via a largecontact. Since there is no lateral waveguiding in broad-area lasers, the threshold current is very high (typicala few A) and the multimode emission cannot be coupledefficiently into a fibre. Due to the high output power,broad-area lasers are used, e.g., for optical pumping ofsolid-state lasers.

Index-guided lasers show stable, lateral single-modeemission with a low threshold current (typical 10 mA)and a P–I curve without kinks. The spontaneous emis-sion in each mode is significantly smaller, resulting ina spectrum with a few dominating modes. The phase-fronts are planar and the far field has a smooth shape,enabling a high coupling efficiency into a single-modefibre.

Weakly index-guided laser structures. In weaklyindex-guided lasers the thickness of the waveguidinglayer is varied, thus resulting in a lateral waveguidingstructure. Lateral single-mode emission can be ob-tained by a proper choice of the thickness and widthof this variation. The lateral index step has to exceedthe carrier-induced reduction of the refractive index(∆n ≈ 5 × 10−3) so that index-guiding is dominating.Weakly index-guided laser structures can be divided into

two categories: ridge waveguide lasers and channeledsubstrate lasers.

In ridge-waveguide lasers, a rib waveguide is de-fined by etching a narrow stripe (≈ 3–5 µm) down nearto the active layer (at a typical distance of 200 nm). In thecase of a metal-clad-ridge waveguide (MCRW) laser in

.=0 "

="=5 .G

5

.6= .4

*Q

=0

=00

=00

0"

=0

0 C:

""

*Q

=B0

=B

=B

B0"

=B

""

4

0 "

Q

0 E"

=0

=0$%$*0=0$-$;-0: =0$%$*00

$-Q$ Q

$- Q

=E99

Fig. 11.99 Weakly index-guided laser structures

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Lasers and Coherent Light Sources 11.3 Semiconductor Lasers 707

the material system GaAlAs (Fig. 11.99a) a CrAu elec-trode is deposited, which gives an electrically conductivecontact on the p+ top layer of the rib and an insulat-ing contact on the p-GaAlAs layer with lower dopingalong the rib. Thus, the current injection is focused inthe region of the active layer below the rib. This currentconfinement can also be realized by additional isolatinglayers, e.g., SiO2 (Fig. 11.99b). The lateral waveguid-ing is accomplished by the higher refractive index ofthe semiconductor material compared to the surround-ing SiO2 and air. The etch depth has to be carefullycontrolled (e.g., by using etch-stop layers) in order toselect one lateral mode and minimize current bypass.

The first step in the processing of channeled sub-strate lasers is the etch of a 2–3 µm-wide and 1 µm-deepchannel into the substrate (Fig. 11.99c). In the subse-quent epitaxy (e.g., liquid-phase epitaxy) this channel is

/ 6 " /@ ,"6= 6 " ,/@

> ="6 6 " >./@

6

=00

=0=

0

=00

=00

*Q

40 E

=0=00

0"

=B

""

Q

""

=B0

=B

=B

=B

=B

B0"

=B

0

Q

40 E

=B0

=B

=B=B

B0"

=B

0 E""

4

=B

=B

Q=B

$$=B

B0"

%Q

0 )0

=B =B

=B

Fig. 11.100 Buried heterostructure lasers

nearly planarized, thus generating a lateral waveguidingstructure, since the active layer has a higher refractiveindex. Current bypass can be reduced by guiding theinjection current using, e.g., localized Zn diffusion.

Weakly index-guided lasers are suitable to obtainlow threshold currents of typically 20–40 mA and highoutput powers with lateral single-mode emission. Theweak waveguiding allows for broader active layers com-pared to buried laser structures, which has a positiveeffect, e.g., on the series resistance. The low currentleakage in ridge waveguide lasers usually results in goodlinearity of the P–I curve. The emission is more com-plex than in buried lasers, since index- and gain-guidingare important and even small changes of the refractive in-dex by temperature or current injection can influence theperformance. The active layer in weakly index-guidedlasers is not affected during processing, so that lateral

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708 Part C Coherent and Incoherent Light Sources

carrier diffusion in the active layer cannot be avoided, buton the other hand, the unpatterned active layer is advan-tageous for the reliability of the device. Since only oneepitaxial growth step is needed, the fabrication of theselasers is significantly easier than buried semiconductorlasers.

Strongly index-guided laser structures. Strong index-guiding can be accomplished by buried laser structures,where a small stripe of the active layer material witha high refractive index is embedded into semiconductormaterial with lower refractive index and larger bandgap.For this purpose, the active layer has to be patternedand epitaxially regrown afterwards. Lateral single-modeoperation is possible, if the resulting stripe width of theactive layer does not exceed a critical value defined bythe cut-off for the higher lateral modes [11.1277].

Figure 11.100 shows some examples for stronglyindex-guided laser structures. In the case of a buried-heterostructure (BH) laser structure in the materialsystem GaAs/GaAlAs (Fig. 11.100a) the GaAs activelayer is surrounded by AlGaAs, which has a largerbandgap and a lower refractive index. The active layeris grown in the first epitaxial step and structured into1–2 µm-wide stripes. Using liquid-phase epitaxy (LPE),which planarizes the structure, the regions alongsidethe stripe are refilled with alternating n- and p-dopedAlGaAs layers. This reverse-biased diode structure pre-vents the current from bypassing the active region. Ina similar way, a BH structure can be realized in thelong-wavelength material system InGaAsP/InP (Etched-mesa buried heterostructure laser, Fig. 11.100b). Thereduction of leakage current using alternating n- andp-doped layers, however, leads to a significant increaseof the parasitic electrical capacity of the laser, whichdegrades the high-frequency response of the device. Al-ternatively, electrically isolating regions for the currentconfinement can be fabricated using semi-insulating ma-terial (e.g., Fe-doped InP) or by proton implantation. InDouble-channel substrate planar buried heterostructure(DCPBH) lasers (Fig. 11.100c) the first epitaxial growthstep produces the active InGaAsP-layer and a p-InP caplayer on the n-InP substrate. Then, a double-stripe struc-ture embedding the active region is created by etching.In the subsequent LPE step the generated channels arerefilled with p-InP/n-InP and the epitaxial growth of thelaser structure is finished up to the InGaAsP top layer.The processing of the mushroom laser (Fig. 11.100d)begins with the etching of 6 µm-wide mesa in the layerstructure, which cuts through the active layer grown inthe first epitaxy [11.1278]. Using a selective wet chem-

ical etching process the width of the active region isreduced up to 1–2 µm to obtain lateral single-mode op-eration. The resulting undercut area is epitaxially refilledafterwards, e.g., using vapor phase epitaxy (VPE) withsemi-insulating InP [11.1279].

In addition to this strongly index-guided laser struc-tures with a planar active layer, which are useful forintegration with a DFB grating, there also exist buriedlaser structures utilizing nonplanar active layers basedon the regrowth of, e.g., V-grooves or mesas.

Using strongly index-guided structures, a very sta-ble and lateral single-mode laser operation with very lowthreshold currents (< 10 mA) and excellent high-speedcharacteristics can be obtained, since stable opticalwaveguiding, carrier confinement and current confine-ment are combined. The fabrication of these devices,however, is complicated due to the additional epitaxystep.

Edge-emitting Fabry–Pérot laser diodesIn Fabry–Pérot (FP) lasers, the cleaved facets of thesemiconductor crystal form the optical resonator, whichenables laser operation by providing the optical feedbackof the stimulated amplified radiation. This resonator se-lects the photons generated by stimulated emission withrespect to direction and wavelength. The light wave trav-eling perpendicularly to the facets is amplified if thewavelength matches a longitudinal mode of the res-onator [(11.92), Figs. 11.90, 95, 96, 103a]. The lasingprocess starts if the gain experienced during one round-trip in the cavity equals the losses caused by absorption,scattering and the light output through the facets.

Lasing condition. The length L of the resonator istypically a few hundred micrometers. The intensity re-flection and transmission coefficients of the end facetscan be estimated using the Fresnel equations, neglect-ing the transversal and lateral structure of the waveguidewith the air (n = 1) assumed to be outside the cavity:

R = (neff −1)2

(neff +1)2, T = 4neff

(neff +1)2, (11.97)

where neff is the effective refractive index of the wave-guide mode considered.

In this one-dimensional model plane waves withelectric field amplitude E(z) travel in the longitudi-nal direction (z) of the FP resonator, experiencing themodal intensity gain g due to stimulated emission. Thethreshold for laser operation is defined by a round-tripcondition requiring that in a stationary state the opticalwave remains unchanged after one complete round-trip

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Lasers and Coherent Light Sources 11.3 Semiconductor Lasers 709

6

$

6

4

4

Fig. 11.101 Dependence of the output power (P) and carrierdensity (N) on the injected current

in the cavity. This round-trip condition gives the mirrorlosses of the FP resonator

g = αm = 1

2Lln(R1 R2) , (11.98)

where R1 and R2 denote the intensity reflection coeffi-cients of the end facets. Since the net gain g in the cavityis composed of the material gain of the semiconductorgeff and the waveguide losses αs, the lasing conditioncan be written as

gthr = nact

neffΓactgact(Nthr) = αs +αm (11.99)

if the contribution of spontaneous emission is neglectedand Nthr denotes the carrier density at threshold. Theoptical losses of the waveguide are caused by opticalscattering from imperfections in the bulk media or atinterfaces and free-carrier absorption in the active andcladding layers.

The longitudinal distribution of the photon densitys(z) in the active FP resonator is given by the sum of theforward and backward traveling photon densities, whichgrow exponentially due to the gain g. For a symmetricresonator (R = R1 = R2) the total photon density s(z) isgiven by a cosh function (Fig. 11.103a) with a minimumin the middle of the resonator. For a laser diode withas-cleaved facets (R = 0.28), the intensity distributionis relatively flat, whereas for antireflection-coated facetsa strongly inhomogeneous photon distribution results.

The round-trip condition also gives the optical fre-quencies of the longitudinal modes of the FP resonatoraccording to νq = qc/(2Lneff), where q = 1, 2, 3 · · ·and c denotes the vacuum speed of light. The frequency

%&

"

"

Fig. 11.102 Scheme of a DFB laser structure with an integrated rect-angular Bragg grating (duty cyclew/Λ) including the correspondingtransversal intensity distribution I (x)

separation ∆ν of adjacent modes is influenced by the dis-persion of the waveguide, which is taken into accountby the group refractive index ng

∆ν = c

2Lng(11.100)

with

ng = neff +ν dneff

dν. (11.101)

For a typical FP laser diode the group indexof ng ≈ 3.5–5 is higher than the effective indexneff ≈ 3–3.5. The separation of the equidistant modefrequencies is about 150 GHz for a 300 µm-long cavity,which is small compared to the width of the gain curve of

!

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>!/

<.4,

4

4

#

Fig. 11.103 Schematic structural cross sections (left) and the corre-sponding emission spectra (right) for various diode lasers

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710 Part C Coherent and Incoherent Light Sources

the semiconductor (≈ 5 THz). Thus, longitudinal multi-mode operation can be expected above threshold for anFP laser diode.

For many applications the degree of single-modeemission is important; it can be quantified by the side-mode suppression ratio (SMSR):

SMSR = 10 log10(P1/P2) , (11.102)

where P1 and P2 ≤ P1 denote the output power of thetwo strongest modes in the optical spectrum. TypicalFP laser diodes achieve a maximum SMSR of approxi-mately 20 dB.

Rate equations. The fundamental static and dynamicproperties of semiconductor laser diodes can be mod-eled using a set of rate equations [11.1280] describingthe interaction of electron–hole pairs and photons inthe active layer. We consider a strongly index-guidingdouble heterostructure supporting a single optical modetraveling as a plane wave axially in the cavity. Thecurrent I is assumed to be uniformly injected into theactive layer with volume V and recombines there, com-pletely neglecting leakage currents. The carrier densitywithin the active layer is treated as homogeneous in thetransversal and lateral direction, since the correspond-ing inhomogeneities of photon density are small and theresulting gradients in carrier density are smoothed outby diffusion. In FP lasers with a sufficiently high mirrorreflectivity, the axial variations of the photon density sand the carrier density N can be neglected and the rateequations can be written as:

dN

dt= I

eV− N

τnr− BN2 −CN3

−vgnact

neffgact(N, s)s + FN (t)

V, (11.103)

ds

dt=vg

[Γact

nact

neffgact(N, s)− gthr

]s

+ Γact[Rsp + Fs(t)

]V

, (11.104)

dt=1

2αHvgΓact

nact

neffgact(N, s)+ FΦ(t) , (11.105)

where vg = c/ng is the group velocity of the waveguide,e the electron charge, gact is the material gain of theactive layer, Γact is the optical confinement factor ofthe active layer and 1/τnr, B and C are the parametersdescribing nonradiative, bimolecular and Auger recom-bination, respectively. Φ is the phase of the complexelectric field E, which is connected with the photon

number S = sV/Γact via

E(t) =√S(t) exp[iΦ(t)] , (11.106)

Rsp is the time-averaged rate of spontaneous emissioninto the lasing mode; Fs(t), FN (t) and FΦ(t) representLangevin noise sources taking into account the statisticnature of the spontaneous emission and the shot-noisecharacter of the carrier recombination and generation.The Langevin forces leading to fluctuations of carrierdensity and photon density are correlated and have zeromean [11.1280].

The longitudinal excess factor Kz accounts for theenhancement of the spontaneous-emission noise due tothe axial dependence of the complex electric field E(z, t)

Kz(t) = | ∫ L0 |E(z, t)|2 dz|2

| ∫ L0 E2(z, t)dz|2

. (11.107)

In the case of a transversely single-mode index-guidedFP laser the factor Kz is given by [11.1281]

KFPz =

((√

R1 +√R2)(1−√

R1 R2)√R1 R2 ln(1/R1 R2)

)2

.

(11.108)

The first rate equation (11.103) can be formallyderived from the quantum-mechanical density-matrixformalism. It can be interpreted as a balance of carri-ers that are injected as a current I and contribute tostimulated emission or are lost for the lasing process viathe different recombination processes. The second andthird equations (11.104, 105) can be derived from theMaxwell equations with the rotating-wave and slowlyvarying amplitude approximations.

For semiconductor lasers with a bulk active layer, thedependence of the gain on the carrier density gact(N) canbe approximated as linear [11.1282]

gact(N) = dg

dN(N − Ntr) , (11.109)

where dg/dN is the differential gain and Ntr is thetransparency carrier density. In quantum well structures(Figs. 11.91, 92) this dependence is usually described bya logarithmic function [11.1283, 1284]

gact(N) = dg

dNN ln

N

Ntrfor gact ≥ 0 . (11.110)

The influence of the photon density on the gain is takeninto account by introducing a nonlinear gain coefficientε according to:

gact(N, s) = gact(N)

1+ εs . (11.111)

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Lasers and Coherent Light Sources 11.3 Semiconductor Lasers 711

This nonlinear gain compression is caused by spectralhole burning and carrier heating, which is significantif the time constant of stimulated emission becomescomparable to the intraband relaxation time.

Refractive index change. In semiconductors the realpart of the refractive index depends on the carrier densitybecause of various physical mechanisms. With increas-ing injection the band-to-band absorption is reduced dueto band-filling effects. In addition, the absorption in-creases due to the reduction of the bandgap (bandgaprenormalization resulting from many-body effects) andabsorption also increases due to the increasing absorp-tion of free carriers. The resulting total change of the realpart of the refractive index, which is related to the gainspectrum via the Kramers–Kronig relation, depends onthe wavelength relative to the gain maximum. In the caseof 1.5 µm InGaAsP the refractive index decreases withinjection. The dependence of the refractive index on thecarrier density is theoretically described by the effectiveline width enhancement or Henry factor α

α= ∂neff

∂γeff, (11.112)

where the complex effective refractive index is definedby neff − iγeff . The change of the effective index withcarrier density can be written as

δneff ∼= Γactnact

neffδnact = −Γact

nact

neff

αλ

∂gact(N)

∂NδN .

(11.113)

The line width enhancement factor is important forthe treatment of line width and frequency chirp un-der modulation [11.1281]. Typically it ranges from 3to 5, decreasing from the long-wavelength to the short-wavelength side of the gain curve [11.1282].

Steady-state characteristics. The single-mode rateequations can be used to analyze the steady-state behav-ior of a semiconductor laser. Setting the time derivativeto zero in (11.104), we obtain an implicit expression forthe photon number S in the case of continuous-wave(CW) operation:

S = Rsp

vg[gthr −Γactgact(N, s)] . (11.114)

The number of photons increases as the gain valueasymptotically approaches the losses gthr. The small gaindifference is compensated by spontaneous emission,which provides the noise input amplified by stimu-lated emission. Below threshold the photon density is

small and (11.103) gives a linear increase of the car-rier density according to N ∝ I/eV . Above thresholdthe gain is approximately clamped at g(Nthr) = gthr andthe corresponding threshold current, which is definedin the limiting case of vanishing spontaneous emission(Rsp = 0), becomes

Ithr = eV

(Nthr

τnr+ BN2

thr +CN3thr

). (11.115)

Using (11.103) the photon number above thresh-old can then be written as S = (I − Ithr)/evggthr. Sincethe carrier density is clamped at threshold, all injectedcarriers in excess of the threshold current contributeto stimulated emission and the number of photons in-creases proportionally to (I − Ithr). The total outputpower P = vgωαmS emitted from both facets becomes

P = ω

eηiαm

gthr(I − Ithr) , (11.116)

where we assume that only a fraction ηi of the ex-ternal drive current reaches the active region and theremaining fraction (1 −ηi) is lost via leakage cur-rent or nonradiative recombination. Thus, the P–Icurve of a semiconductor laser diode is a straight line(Fig. 11.101) above threshold with a slope defined bythe external quantum efficiency ηext:

ηext = dP

dI

e

ω= ηi

αm

gthr, (11.117)

which can be interpreted as the ratio of the numberof emitted photons to the number of injected elec-trons per time. The sharpness of the transition fromthe spontaneous emission below threshold to the stimu-lated emission above threshold depends on the amountof spontaneous emission into the lasing mode. Leak-age currents, thermal effects and spectral hole-burningneglected so far lead to bending of the P–I curve.

Characteristic temperature. The threshold current ofa semiconductor laser depends on the temperature T ,which can be described phenomenologically by

Ithr(T ) = I0 expT

T0, (11.118)

where T0 is the characteristic temperature, which typi-cally ranges between 40 K and 90 K for semiconductorlasers emitting around 1550 nm.

Single-Mode Laser StructuresTransmitters used, e.g., in optical-fibre communicationsystems should emit light predominantly in a single

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712 Part C Coherent and Incoherent Light Sources

longitudinal mode since the presence of side modes lim-its the transmission capacity due to pulse broadeningcaused by the chromatic dispersion of the fibre. Semi-conductor lasers with an FP resonator usually exhibitmultimode operation since the gain spectrum is widerthan the longitudinal mode spacing and the broadeningof the gain profile, which due to spectral hole burning isnot perfectly homogeneous, offers several modes withsufficient gain to oscillate. The techniques to achievereliable longitudinal mode control even under high-bit-rate modulation can be categorized into two maingroups:

Short lasers. The discrimination against side modes inFP resonators can be enhanced by reducing the cavitylength L . If the mode spacing ∆ν ∝ L−1 becomes com-parable to the width of the gain curve, only one modewill oscillate near the gain peak. To obtain stable single-mode operation, however, the lasers must be extremelyshort. This requires a very good reflectivity of the endfacets to overcome the high mirror losses αm ∝ L−1,leading to high threshold current densities. The prob-lems of fabricating very short semiconductor devicescan be solved using a vertical-cavity surface-emittinglaser (VCSEL) structure (Fig. 11.96).

Frequency-selective feedback. The second methodto obtain single-mode operation is to incorporatea frequency-selective element in the resonator struc-ture. This can be realized by using coupled cavities,an external grating or a Bragg grating:

Coupled cavities. If one or more additional mirrors areintroduced in the FP resonator, the boundary conditionsadded due to the reflections at each interface severelylimit the number of longitudinal modes. To achievesingle-mode operation, however, it is often necessary totune the resonator by changing the drive current or thetemperature. Usually the single-mode regime is small sothat such structures can only be modulated over a limitedcurrent range without mode jumps. In addition, the re-producible fabrication of nearly identical devices turnsout to be difficult since the spectral properties stronglydepend on the exact lengths of the sections.

External grating. Frequency selection can also be real-ized by an external grating outside the resonator. Themechanical stability of such lasers, however, is a crit-ical point since the grating is not integrated on thewafer. Consequently, lasers with external gratings areexpensive devices (Fig. 11.110a).

Bragg grating. The method most frequently used toachieve single-mode emission is to incorporate a Bragggrating, which creates a periodic variation of thecomplex refractive index and distributes the feedbackthroughout the cavity. Dynamic single-mode operationis achieved if the threshold gain for the oscillatingmode is significantly smaller than the threshold gainfor the other modes. Devices employing Bragg grat-ings can be classified roughly into three categories:distributed Bragg reflector (DBR), distributed feedback(DFB) (Figs. 11.96, 102, 103, section on Fabry–Pérotstructures in Sect. 11.3.2, and Sect. 11.3.6) and gain-coupled (GC) lasers.

In DBR lasers the Bragg grating is etched into pas-sive regions near the cavity ends. The index grating (thevariation of the real part of the refractive index) acts asan effective mirror with wavelength-dependent reflec-tivity and surrounds the central part of the cavity whichis active and remains uncorrugated. The longitudinalmode with a wavelength located near the reflectivitymaximum of the grating is selected. Since a DBR laseris formed by replacing the mirrors by passive gratings,the properties can be described by an effective mirrormodel. The transition between the active section andthe passive gratings usually complicates the technolog-ical realization of in-plane DBR lasers. An importantadvantage of DBR lasers is that the wavelength can bechanged if the grating regions are equipped with sep-arate electrodes that can tune the Bragg frequency viathe carrier-induced refractive index change. Thus, DBRgratings are often used in tunable lasers.

In DFB lasers the index grating covers the entireresonator length. At the wavelength corresponding tothe corrugation period of the grating, the forward- andbackward-traveling waves created by the Bragg scat-tering are confined in the central part of the cavity sothat the mirror losses become a function of the wave-length. The longitudinal mode with the lowest mirrorlosses corresponding to the most effective concentrationof photons in the resonator is selected.

In gain-coupled devices a periodic variation of gainor loss is used to favor a longitudinal mode of theFP resonator. In the ideal case, there is no Braggscattering at the gain grating, and the longitudinalphoton distribution as well as the mirror losses are un-changed compared to the FP cavity. The overlap withthe loss or gain grating, however, varies between thedifferent longitudinal modes of the FP resonator. Themode experiencing the largest overlap with the gaingrating (or minimum overlap with a loss grating) isselected.

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Lasers and Coherent Light Sources 11.3 Semiconductor Lasers 713

DFB lasers. The spectral properties of DFBlasers [11.1285] are essentially determined by the inte-grated Bragg grating. The waveguding in such periodicstructures can be analyzed by the coupled mode the-ory [11.1286], which yields approximate analyticalsolutions describing the light propagation in waveguideswith a periodic variation of the complex refractive indexby counter-propagating modes exchanging energy byscattering. The strength of the interaction and the amountof feedback in the grating structure are determined bythe complex coupling coefficient

κ = π∆n/λ0 + i∆g/4 , (11.119)

which is proportional to the variation of the refractiveindex step ∆n, the gain variation ∆g and the number ofcorrugations per length in the grating. The Bragg wave-length λB is given by the effective index, the corrugationperiodΛ and the order mDFB of the grating according to(11.93). For an axially homogeneous first-order grating(mDFB = 1) with length L and a perfect antireflectioncoating on the facets (R1 = R2 = 0) the theory revealsthe following solutions

• In the case of pure index coupling (∆neff =0, ∆g = 0) the transmission spectrum turns outto be symmetric with respect to the Bragg wave-length λB = 2neffΛ where oscillation is forbidden.For a small coupling coefficient, the mode spacingapproximately takes the value of an FP resonator∆λ= λ2/2neff L but, in contrast to an FP cavity, thethreshold gain of the modes is wavelength dependentand increases with growing distance from the Braggwavelength. A strong coupling produces a trans-mission stop-band with width ∆λ∼= κλ2

B/(πneff )centered at the Bragg wavelength, in which trans-mission is strongly damped. The two modes withthe lowest threshold gain g ∼= 2π2/(κ2L3) are lo-cated at the edges of the stop-band symmetrical tothe Bragg wavelength.• In the case of pure gain coupling (∆g = 0, ∆neff =0) the mode degeneracy is removed, which meansthat the mode with the lowest threshold gain os-cillates at the Bragg wavelength symmetricallysurrounded by the other modes with higher thresholdgain values. The mode spacing is ∆λ= λ2/2neff Land no stop-band occurs since there is no backscat-tering at index steps in the grating. The modeselection is due to the different overlap of the stand-ing waves in the FP resonator with the gain grating.

In second-order gratings (mDFB = 2) additional scatter-ing occurs in the transversal direction, leading to higher

losses. In addition, the coupling coefficient of second-order gratings depends more sensitively on the exactshape of the grating so that it becomes more difficult tocontrol. That is why first-order gratings predominate al-though the corrugation periods are smaller (Λ∼= 240 nmfor λ= 1.55 µm). Due to the scattering perpendicularto the optical axis (Fig. 11.96), second-order gratingscan be utilized for the vertical emission of light fromedge-emitting laser diodes [11.1287, 1288].

Basic properties of index-coupled DFB lasers. Thespectrum of an index-coupled DFB laser mainly con-sists of two degenerate lasing modes at the edges ofthe stop-band (Fig. 11.103b). This mode degeneracy inindex-coupled DFB laser structures is usually removedby incorporating a λ/4 phase shift in the grating. Thiscan technologically be implemented by inserting an ad-ditional section of lengthλ0/(4neff ) =Λ/2 in the middleof the grating. The introduction of the λ/4 phase shiftselects the Bragg mode in the middle of the stop-band,revealing the lowest mirror losses so that single-modeoperation with an SMSR of > 40 dB can be obtained(Fig. 11.103c).

The axial distribution of the light intensity in the cav-ity is connected to the mirror lossesαm via the round-tripcondition. Thus, a decrease of the mirror losses for thelongitudinal modes in a DFB grating is equivalent to in-creasing longitudinal optical confinement, which meansthat the photons are concentrated inside the cavity andonly a small fraction of the light intensity leaves the res-onator through the end facets. The typical longitudinalphoton distribution for a DFB laser with and withouta λ/4 phase shift is shown in Fig. 11.103, revealing thepronounced maximum in the middle of the resonatoreven for moderate coupling coefficients. This strong in-homogeneity of the photon density distribution in DFBlasers leads to an inhomogeneous carrier density dis-tribution above threshold due to the recombination bystimulated emission. With increasing injection the car-rier density is depleted in places with a high photondensity. This phenomenon is called longitudinal spatialhole burning (LSHB) and has several important conse-quences for the static and dynamic behavior of DFBlasers above threshold. First, the mode discrimination isinfluenced since a variation of the carrier density distri-bution changes the round-trip gain of the various modeshaving different photon density distributions. Thus, theside-mode suppression can degrade with increasing out-put power due to LSHB. Second, the mode wavelengthschange, even above threshold, since the inhomogeneityof the carrier density caused by LSHB leads to an inho-

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mogeneous axial distribution of the effective refractiveindex. This effect is utilized in several types of tunableDFB lasers.

Various DFB lasers with a more complex gratingstructure have been developed in order to obtain a highyield of single-mode devices and a flat axial photondistribution with reduced LSHB. Some examples are:

• 2 ×λ/8 phase shifts. The distance between the phaseshifts, however, must be optimized to achieve a highyield [11.1289].• Corrugation pitch modulation (CPM). The DFBgrating is divided into three sections. The cor-rugation period in the central section is slightlyhigher than in the outer sections so that the phaseshift is quasi-continuously distributed along the cav-ity [11.1290].• Axial variation of duty cycle by using, e.g., a holo-graphic double-exposure technique [11.1291].• Axial variation of coupling coefficient by vari-ation of the etch depth [11.1292] or sampledgratings [11.1293].• Bent waveguides superimposed on homogeneousgrating fields can be used to obtain quasi-continuously and arbitrarily chirped gratings withhigh spatial resolution [11.1294].• Axially inhomogeneous injection using a three-electrode structure. Spatial hole burning can be

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compensated for if the injected current density ishigher in the central section near the peak of thephoton density, thus reducing the gain of the side-modes suffering from the lower current in the outersection [11.1295].

The phase relation between the grating and the endfacets is difficult to control during the cleaving pro-cess of DFB lasers, since the corrugation period fora first-order grating (λ= 1.55 µm) is typically 240 nm.Experimental and theoretical investigations show thatall static and dynamic optical properties of as-cleavedDFB lasers are strongly influenced by this phase rela-tion between the grating and facet. The mirror lossesof the various longitudinal modes, the mode discrimi-nation, the intensity distribution, the optical spectra andthe dynamic and noise characteristics vary considerablyas a function of the end facet phases [11.1296]. Sincethese end facet phases are distributed randomly after thecleaving process, the yield of good DFB devices is lim-ited. The problem of the uncertain end facet phases canbe reduced by appropriate antireflection coatings.

Gain-coupled lasers. In gain-coupled laser structures,a longitudinal mode of the FP resonator is selected byimplementing an axial gain or loss grating. In contrastto index-coupled DFB lasers, there is almost no reflec-tion of the light wave in a gain grating. Therefore, thelongitudinal intensity distribution and the spectral posi-tions of the various modes are equal to those of an FPresonator. The mode selection is caused by the differentoverlap of the longitudinal modes with the gain grating.The mode, whose longitudinal field distribution exhibitsthe largest overlap with the gain grating or the smallestoverlap with the loss grating, is selected.

Gain coupling can be realized in different ways:

• The grating structure can be etched directly into theactive layer and regrown afterwards with a semi-conductor material [11.1297] that is transparent tothe laser emission (Fig. 11.104b). The resulting gaincoupling in this grating structure is accompanied bya strong index coupling due to the high refractive-index difference between the active layer and the re-grown material. With respect to the phase differencebetween the index and gain grating one distinguishesbetween in-phase and anti-phase gratings.• A current blocking pnp layer structure above theactive layer is corrugated and regrown, so that a pe-riodic variation of the current density injected intothe active layer is achieved [11.1298]. The parasiticindex coupling can be kept very small in this type of

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gain grating. Due to the Kramers–Kronig relations,however, the gain grating is inevitably accompaniedby an index grating.

Loss coupling is obtained if the grating is etched intoan additional layer that is separated from the activelayer and is absorbing for the emission (Fig. 11.104c).This grating acts as a saturable periodic absorber for thelaser emission from the active layer [11.1299]. In con-trast to index coupling (Fig. 11.104a) the bandgap Egof the grating is smaller than the bandgap of the ac-tive layer Eg,act. Usually, the absorbing grating layerhas a different refractive index than the material usedfor the regrowth, so that considerable index couplingoccurs. Alternatively, the loss grating can be integratedinto a metallization at the surface of the device, so thatno epitaxial regrowth is required [11.1300].

There are significant differences between the charac-teristics of gain- and loss-coupled semiconductor laserscompared to index-coupled DFB lasers:

• In contrast to index-coupled devices, there is almostno reflection of the light wave in an ideal gain grat-ing. Thus, the interference of the reflected waveinside the grating and at the facets is avoided, sothat the influence of the end facet phases is stronglyreduced compared with index-coupled DFB lasers.Therefore, a high single-mode yield can be obtainedwithout the necessity to use an antireflection coatingof the end facets.• The longitudinal photon distribution is, in the caseof a small parasitic index coupling, similar to that ofan FP resonator. Thus, the strongly inhomogeneousphoton distribution in DFB lasers and the result-ing problem of longitudinal spatial hole burning isreduced in gain-coupled devices.

Basics of Laserswith High Modulation Bandwidth

For the highest-bit-rate fibre-optic data communication(1.26–1.68 µm), the ultrafast conversion of the datafrom the electronic bit sequence into the appropriate op-tical bit pattern takes place by means of semiconductorlasers. This is done either with a CW laser combined witha subsequent ultrafast optical modulator or via directmodulation of the laser (intensity or frequency modula-tion). As the simplest example the intensity modulationof the laser is described. The bit pattern available as a se-quence of ultrashort current pulses is translated by thelaser into the appropriate sequence of ultrashort lightpulses, which then propagate e.g. via a fibre towards

the receiver. The photodetector retranslates the lightpulses back to the electronic bit sequence. Seen in termsof a two-language dictionary, the laser translates ultra-fast from electronics into optics and the photodetectorultrafast from optics into electronics.

For this purpose, the electrons injected throughthe n contact in the semiconductor laser must arriveas fast as possible at the most deeply bound energystates of the conduction-band QWs and the holes in-jected through the p contact as fast as possible into theground states of the valence-band QWs (Fig. 11.105).Several retarding physical transport and relaxation pro-cesses are involved, whose combined time delay effectcan however be minimized [11.1301, 1302]. Owing tothe very high doping in the long conduction paths,which start at the contacts, very short dielectric re-

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716 Part C Coherent and Incoherent Light Sources

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laxation times arise there. This can be visualized bymeans of a pipe completely filled with ping pong balls(high doping throughout the length of the conductionpath). A ball injected at one end of the pipe, causes an-other to be ejected from the other end of the pipe. Inthis manner, a current pulse is transferred to the otherend of the pipe almost without delay in heavily dopedsemiconductors.

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Fig. 11.107a–f Schematic cross-sectional views in the y–z-plane of different laser types: (a) three-section DFB, (b) three-section DBR, (c) TTG, (d) GAC, (e) SG, which is related to the SSG laser (not displayed) and (f) GCSR

In order to reduce optical losses through reabsorp-tion, the confinement layers should be undoped oronly slightly doped. Retarding effects arise as a con-sequence of the charge-carrier transport (number 1 inFig. 11.106). Using the analogy of the balls and pipeabove, a ball in an empty pipe (undoped) must gothrough the entire length of pipe (the length of a con-finement layer) before finally exiting from the otherend. Further retarding effects arise from the charge car-rier capture (2), relaxation (3) in the respective QWground states, by the reconciliation of charge carrierinhomogeneities between the individual pots, throughtunneling (4) and thermal reemission (5). Since the mo-bility of the electrons is substantially higher than thatof the holes, in asymmetrical laser structures [11.1302]the p-side confinement layers are reduced in thicknessin favor of the n side. This favors the transfer of theless-mobile holes. In addition, this asymmetric laserstructure design takes into account the different cap-ture probabilities: the capture of holes into the QWsis much more efficient than that of electrons. Thus,a smaller p-sided confinement layer (reservoir of uncap-tured electrons giving rise to bit interfering) is beneficial.In today’s fastest laser diodes (largest modulation band-width) −3 dB modulation frequencies of up to 40 GHzcan be obtained.

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Fig. 11.108a–d Schematic perspective view of (a) a Y-laser [11.1303] and (b-d) three-section bent-waveguide chirpedDFB lasers [11.1304], (b) the principle of generating chirped DFB gratings by specifically bent waveguides on homoge-neous grating fields, cross section of two independent lasers originating from the bending displayed in (b), (d) perspectiveschematic of a device shown in (c). A scanning electron micrograph (SEM) of such a tunable three-section chirped DFBlaser is displayed in Fig. 11.116a

Tunable LasersWavelength tuning of an edge-emitting DFB semicon-ductor laser can be obtained according to (11.93), inthe simplest case, by varying the effective refractive in-dex via the injection currents in different sections, eitherthermally or by using plasma effects (carrier densityvariations). In these cases rising injection currents lead

1. to increasing emission wavelengths (red shift) viathermal effects and

2. to a blue shift via the plasma effect.

A superposition of both effects is found in multi-sectionDFB lasers [11.1305] since gain, tuning and DFB gratingexist in all sections (Fig. 11.107a). In order to increasethe total tuning range it is necessary to separate gainand tuning [11.1306, 1307]. In the three-section DBRlaser (Fig. 11.107b) this is done longitudinally by us-ing a gain section (right, the active region is shownin black), a grating-section (left) and a phase section

(center) [11.1308]. The phase section only includes thebrown waveguide layer(s) and is used to match thephase of the standing light wave after one round-trip.The gain is controlled, e.g., via the current betweenthe top right contact and the bottom contact. In con-trast, a vertical separation is used in the tunable twinguide (TTG) laser [11.1306,1307,1309] (Fig. 11.107c).A current from the left-side contact to the top contactonly controls the carrier density in the brown waveguid-ing layer. The gain is controlled via the current betweenthe left side contact and the bottom contact. In the three-section DBR and the TTG laser the tuning is based onthe plasma effect, if only forward biasing is used.

Another very important tuning principle is based ontwo mode combs of slightly different mode spacing. Inanalogy to a vernier based on two scales [11.1306,1307],a small detuning of the combs can address individualmodes separated widely in the spectrum: only thosemodes which exactly occur in both combs can oscillatesince only this situation provides sufficient reflectivity

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Table 11.37 Tuning characteristics of different 1.55 µm laser types, DFB: distributed feedback, DBR: distributed Bragg reflector,TTG: tunable twin guide, SSG: superstructure grating, SG: sampled grating, GAC: grating-assisted coupler, GCSR: grating-assisted co-directional coupler laser with rear sampled grating reflector. The tuning occurs either thermally (t), via the plasmaeffect (p), via the co-directional coupler (CC) grating or via Vernier effect (V). TR: tuning range

Laser type Control Tuning Tuning Tuning Miscellaneous Ref.

currents (exp.) (max. con- principles

(∆λ/nm) tinuous)

3 sect. DFB 2 3 3 t, p Continuous TR [11.1305]

3 sect. DFB 2 5.5 3 t, p Bent waveguide, quasi contiunuous TR [11.1294, 1304,1312]

3 sect. DBR 3 10 7 p, (t) Quasi continuous TR [11.1313]

3 sect. DBR 3 22 7 p, t Incl. both polarities, quasi continuous TR [11.1314]

3 sect. DBR 3 7 p Mode-hop-free, really continuous TR [11.1315]

TTG 2 7 13 p, (t) Continuous TR [11.1306,1309]

TTG 2 13 13 p, t Incl. both polarities, continuous TR [11.1306]

Y 4 51 V, p No continuous TR [11.1303]

SSG/SG 4 95 V, p Quasi contiunuous TR, gaps in the total TR [11.1310,1311]

SSG/SG 11 38-50 V, p Quasi contiunuous TR, [11.1316]

without gaps in the total TR

GAC 3 50-70 CC, p No continuous TR [11.1262, 1317,1318]

GCSR 4 100 CC, p Quasi continuous TR, [11.1319]

without gaps in the total TR

GCSR 4 50-114 CC, p Quasi contiunuous TR, gaps in the total TR [11.1320]

SSG/SG TTG 3 30-50 V, p Quasi contiunuous TR [11.1321]

MG-Y 4-5 46 V, p Quasi contiunuous TR [11.1322]

and, thus, sufficient net gain. This principle is appliedin the (C3) laser, Y laser, Mach–Zehnder interferome-ter laser, superstructure grating (SSG) DBR laser andsampled grating (SG) laser. Two FP cavities of differ-ent length and, thus, two FP mode combs are used in theY laser (Fig. 11.108a) and the C3 laser (two coupled FPlasers of slightly different length, not displayed). Two su-perstructure mode combs are shifted against each otherin the SSG laser [11.1310] and SG DBR laser [11.1311](Fig. 11.107e). The codirectional coupling in the gratingassisted coupler (GAC) laser between two waveguidesrealizes an additional spectral filter which can be shiftedspectrally in a controlled way against an FP mode spec-trum. The two waveguides can be laterally or vertically(Fig. 11.107d) coupled. Depending on the size of thegrating period, a grating can reflect light (inversion ofthe propagation wave vector, contra-directional couplinggrating, short grating period, e.g., Fig. 11.107a,b,c,e) orchange the magnitude of the propagation vector (main-taining propagation direction, codirectional couplinggrating, long period, shown in Fig. 11.107d below thecenter contact). Table 11.37 includes a comparison of

different tunable laser types with respect to importantdevice characteristics and properties.

Generally the types discussed up to now cannotprovide excellent characteristics in all features suchas simple tuning (low number of control parameters),high SMSR, wide tuning, continuous tuning and highefficiency. This motivates the search for promising com-binations of the above principles. Using a GAC to selecta mode from an SG yields the grating-assisted codirec-tional coupler laser with rear sampled grating reflector(GCSR) [11.1319, 1320] (Fig. 11.107f). Another laserwith 11 sections combines a SSG or a SG with manyDBR sections of different grating period each [11.1316].SSG or the SG laser and the TTG laser has been pro-posed [11.1321]. Combining the SSG laser and the Ylaser yields the modulated grating Y laser [11.1322].For comprehensive details of tunable lasers we referto [11.1306, 1307]. Finally, coupled Mach–Zehnder in-terferometers or arrayed waveguide (AWG) structurescan also be used as filter structures in lasers, e.g., inthe digitally tunable ring laser using a ladder of ring-resonator filters [11.1323, 1324].

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DFB lasers with an axially varying grating periodare very attractive to tailor specific device properties.Using bent waveguides on homogeneous DFB gratings(Λz = const.) enables to generate a local effective pitchlength Λ(z) that exceeds Λz the more, the larger thelocal tilt angle of the bent waveguide deviates froma vertical intersection of the pitches i. e., the largerthe local tilt angle ϑ(z) is. Figure 11.108b schemati-cally shows a bent waveguide covering a homogeneousgrating. Choosing appropriate bending functions x(z)generates chirped DFB gratings with a varying effectivepitch lengthΛ(z) =Λz/ cos(ϑ(z)), as shown in the insetto Fig. 11.108b [11.1294]. The fabrication method is in-dicated in Fig. 11.108c, which shows two chirped DFBlasers formed after cleaving the structure in Fig. 11.108bin the x-direction in the center of the side of the deviceoriented in the z-direction. Since the light is stronglyguided along the bent w-direction, the light sequen-tially passes quasi continuously varying pitch lengthsΛ(w), as displayed in Fig. 11.108c. Since ϑ is largestin the center of Fig. 11.108b, the two chirped DFBlasers have the largest pitch at the center facets. Var-ious applications of chirped DFB gratings have beendemonstrated [11.1294,1304,1312,1325], including ax-ially distributed phase shifts, higher SMSR, enhancedsingle-mode stability and lower line widths. In addition,chirped three-section DFB lasers (Fig. 11.108d) havedemonstrated enhanced tuning ranges [11.1294, 1304,1325] (Table 11.37). The mushroom-type laser struc-ture (Fig. 11.100d) is applied in the chirped three-sectionDFB laser (Fig. 11.108d), which shows a modulationbandwidth of up to 26 GHz [11.1302].

Further Laser Types with Horizontal ResonatorsIn the bipolar lasers considered so far the light emissionin the active layer takes place through the recombina-tion of an electron in the conduction band with a hole inthe valence band (band–band transition or inter-bandtransition). In unipolar lasers the radiative recombi-nation takes place within a single band between twobound states of a potential well (intra-band transition,in Fig. 11.109 shown for the conduction band). Throughso-called electron recycling, this process is cascadedin stages (explaining the name quantum cascade laser,Fig. 11.109). The band edges are accordingly tilted bythe applied voltage (electric field). The multiple-QWstructure [11.1326] shown here is based on a uniformQW composition and a different uniform barrier com-position. However, there are in each case six differentQW widths (the lower sequence of numbers) and barrierwidths (the upper sequence of numbers) involved.

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Through tailored dimensioning, we ensure that inthe case of resonance (for a certain applied voltage)each of the four narrowest QWs exhibits a bound state,and that these four states are energetically identical(i. e., they line up exactly horizontally). Furthermore,the widest QW is dimensioned in such a way that thehighest state of the three bound states, marked witha 3 in Fig. 11.109, deviates only insignificantly fromthe four identical levels mentioned above. Due to thethin barriers and the resulting tunneling processes, anelectron injected from the left tunnels horizontally intothe excited level 3. After a radiative transition it fallsdown to level 2. Due to the very thin barriers, the wavefunctions are generally strongly delocalized. By opti-mizing the QW geometries the relaxation time fromstate 2 to state 1 is designed to be very small, so thatthe maxima of the wave function of state 2 lies onthe left and that of state 1 on the right of both of thewider QWs.

This tunneling process proceeds in a stair-like cas-cade, which is illustrated with just two stages inFig. 11.109. This example also illustrates very success-ful recent complex QW structures, which are based ona variety of different quantum cascade structures. Quan-tum cascade lasers are very successful to produce laseremissions in the mid infrared (3.5–10 µm, in some caseseven up to 60 µm), where there are very few suitablesemiconductor materials with extremely small bandgapsfor the construction of bipolar lasers.

A very elegant way to obtain a longitudinal single-mode oscillation of an FP laser is to use one highlyreflecting (HR) and one antireflection (AR) coated facet,with a piece of fibre optically coupled to the latter(Fig. 11.110a) including a DBR grating with a period((11.93)) such that it filters exactly one single oscillat-ing mode from the spectral amplification profile. Thislaser design is called a (semiconductor-)fibre laser. In

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720 Part C Coherent and Incoherent Light Sources

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Fig. 11.110a–d Various designs of lasers with horizon-tal resonator. (a) FP laser with an external fibre mirror,(b) double ring resonators coupled to a straight waveguide,(c) standing wave in a single ring resonator coupled to 2waveguides, (d) DFB microdisc laser [11.1327]; inset: FPmicrodisc laser with whispering-gallery mode

contrast, the active medium for optically pumped fibrelasers is a doped fibre.

Laser designs with straight aligned resonators havebeen considered so far. However, in the ring laser(Fig. 11.110b, c) and the microdisk laser (Fig. 11.110d),

the beginning and end of the resonator merge into oneanother. In the case of waveguiding (Fig. 11.110b) modalsolutions result if the length of the circular optical axisis close to an integer multiple of the wavelength inthe medium. Similar selection rules apply in micro-disc lasers, which can also be illustrated in the raymodel with multiple reflections (eight in Fig. 11.110d)at the exterior surfaces. These modes are also known aswhispering-gallery modes after the acoustic phenomenain the gallery of the St. Paul’s cathedral in London andthe Gol Gumgaz mausoleum in Bijapur city in India.There, a whispered word is discernible, after multiplereflections from the inner walls of the cupola, back atthe ear of the speaker.

11.3.4 Basics of Surface-Emitting Laserswith Vertical Resonators (VCSELs)

The VCSEL presents a large technological challenge interms of the implementation of DBR mirrors with ex-tremely high reflectivity. In contrast to the edge-emittingDFB or DBR laser, the individual layers of the DBRmirrors in the VCSEL are successively deposited (e.g.,by epitaxy). In this way, layer A and the neighboringlayer B together make up one period. Since the reflec-tivity of the DBR mirrors rises with increasing numberof periods and refractive index contrast, materials withlarge refractive index contrast are preferable in orderto reduce the total number of periods and thus devicecost. The insets in Fig. 11.111 show a DBR mirror (leftinset) and the light intensity reflected by the mirroras a function of the wavelength (reflection spectrum,right inset). For λ= 1.55 µm, the maximum reflectiv-ity Rmax in the stop-band is presented in the mainpicture as a function of the number of periods for dif-ferent material systems. The data result from theoreticalmodel calculations with consideration of the spectralvariations of the refractive indices and absorption coef-ficients [11.1328,1329]. Large differences appear in thefigure due to different absorption and different refractiveindex contrast 2(nA −nB)/(nA +nB). For semiconduc-tor epitaxial layers, absorption depends on the dopingconcentration and is well controllable. Absorption in di-electric layers is more difficult to control and dependsstrongly on the technological process. Unfortunately, nolarge refractive index contrasts can be realized in theGaInAsP material system. To achieve a reflectivity of99.8%, 50 periods, a number that is in practice too high,are needed. In contrast, this is achieved in AlAs/GaAsDBRs with just 20 periods. If the two dielectrics Si3N4and SiO2 are combined, then even 13 periods are suf-

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ficient due to the very high relative refractive indexcontrast.

This enormously high reflectivity is achievedthrough constructive interference of the partial wavesreflected at all heterointerfaces. Note that with a reflec-tion at the interfaces from the optically thinner to theoptically denser medium a phase change of π occurs,but not vice versa [11.1330, 1331]. The condition forconstructive interference is fulfilled, e.g., by the com-bination of λ/4-thick layers A and B, as is the casein Fig. 11.111. Here, dA and dB are the physical thick-nesses and λ/4nA and λ/4nB are the optical thicknesses.In general, the following relation applies to achieve highreflectivity using constructive interference

nAdA +nBdA = mλ

2(with m = 1, 3, 5 . . . ) .

(11.120)

On GaAs substrates the almost lattice-matchedAlAs/(Al)GaAs combination is ideal for lasers in thewavelength range 800–1300 nm. Although, as withthe edge emitters, trials have been made to imple-ment long-wavelength VCSELs on GaAs substrates,1.55 µm VCSELs are still based on InP. This is dueto the fact that the active laser layers (e.g., GaInAsPand AlGaInAs) for the spectral range close to 1.55 µmcan be best implemented on InP substrates. There aredifferent possibilities for implementing highly reflect-ing 1.55 µm DBR mirrors on InP substrates: a kind ofpressure bonding technique (wafer fusion) ofAlAs/GaAs DBRs [11.1332, 1333] to embed theGaInAsP active region, strongly lattice-mismatched so-called pseudomorphic AlAs/GaAs DBRs [11.1334],lattice-matched AlGaInAs, AlAsSb/AlGaAsSb,AlGaAsSb/InP DBRs [11.1335, 1336] or fusing ofa low-period GaInAsP/InP DBR with active layers ontoa AlAs/GaAs DBR [11.1337].

A resonator results from the combination of twoDBR mirrors, where the volume between the DBRmirror-ends towards the center is called the cavity. Ifthe cavity material is passive, then the setup is an opticalfilter, if it is active then a VCSEL results. Figure 11.112shows schematically the structure of a VCSEL with twomultiple-layer semiconductor DBR mirrors. In the illus-tration the upper mirror is highly reflective, so the laserlight is emitted essentially downwards (brown arrow).The envelope of the vertical intensity distribution dis-cussed below is indicated by the lateral expansion ofthe brown arrow (Fig. 11.96, depicted in grey-tone vari-ations). The holes are injected from above through theupper p-doped DBR mirror into the active laser zone;

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the electrons are injected from below through the lowern-doped DBR mirror. The outer curved black arrows in-dicate the current flow. Here, it is challenging to reducethe electrical resistance of the p-doped DBR mirror bysophisticated doping (impurity concentration profiles) inorder to reduce the operating voltages and the buildup ofJoule heating. Figure 11.112a schematically depicts thestructure of a VCSEL with two dielectric DBR mirrors.The holes and electrons are injected by means of ringcontacts, whereby the current flow bypasses the elec-trically insulating DBR mirrors. Furthermore, of crucialimportance is the lateral electrical confinement to enablethe production of high electron–hole-pair densities in thecentral part of the active zone, and thereby very smallthreshold current densities. Therefore, an electrically in-sulating ring is built surrounding the lateral laser mode.The realization of this is technologically demanding andis achieved, e.g., by ion implantation or selective oxi-dation. A sufficiently high thermal conductivity of theDBR mirrors is of further importance in enabling effi-cient heat dissipation from the active region to the heatsink and, if necessary, of that heat generated inside theDBR mirrors.

For the VCSEL with two dielectric DBR mirrorsand a cavity length of 3λ/2 shown in Fig. 11.112a,the electric field of the standing light wave is com-puted [11.1328, 1329] and presented in Fig. 11.113together with the entire multilayer structure. In order

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722 Part C Coherent and Incoherent Light Sources

""

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Fig. 11.112a,b Schematic setup of a VCSEL with elec-trically insulating (a) and electrically conducting DBRmirrors (b)

to achieve a high gain in the laser design, the QWs arepositioned at the maxima (antinodes) of the half-waves.Ideally, three QWs, three double QWs or three tripleQWs should be placed in the 3λ/2 cavity shown. Theenvelope of the standing light wave field gradually dropsoutwards from center due to the distributed reflecting ef-fect of the DBR mirrors. We note again that exactly onequarter of a wavelength is allotted to each mirror layer(λ/4 layers) and that the nodes for the design wavelengthlie exactly at the interfaces.

In the following, the VCSEL spectra are intuitivelyconstructed. To derive this, we consider a semiconduc-tor cavity between two mirrors indicated in the top leftof Fig. 11.114. Based on the fixed cavity length, (11.92)leads to the three modes displayed as lines in the spec-trum on the right of the figure, which become sharper

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" #

Fig. 11.113 Layered structure of a VCSEL including thestanding wave of the electrical field

for higher mirror reflectivities. This means that a cavityof this length is suitable for each of these (laser) wave-lengths if the standing wave experiences a sufficientlylarge optical gain at its peak (antinodes). On the con-trary case, for a desired (fixed) laser wavelength λ thecavity length can be selected straight away as λ/2, 2λ/2or 3λ/2. If the two mirrors are implement as DBRs (bot-tom left of the figure), then the line spectrum at the topright is superposed with the known reflection spectrum(Fig. 11.114 bottom right) of the DBR mirrors, resultingin the laser spectrum already shown in Fig. 11.103d. Thestop-band of the VCSEL is substantially larger becauseof the higher refractive index contrast in the DBR mir-rors compared to that of the DFB laser with the relativelysmall refractive index contrast between the quasi-layersof the DFB grating. Following from Fig. 11.103c, d theλ/4-phase-shifted DFB laser has many similarities toa VCSEL with a λ/2 cavity and two very long DBRmirrors with a small refractive index contrast.

Note that the following numerical values are stronglydependent on wavelength and material and that a thor-oughly comprehensive treatment would exceed theframework of this volume. For VCSELs, optical poweroutputs of < 1 mW (lateral single-mode operation) aretypical. With a laterally expanded active region, i. e.,laterally multimode VCSELs, achieve up to 120 mW.Typical threshold currents lie around 1 mA, with recordvalues of about 0.06 mA (corresponding to 350 A/cm2).The highest power outputs and the lowest thresholdsare achieved in the spectral range 850–1000 nm. Com-pared to edge emitters the thresholds are amazinglylow, but with rather moderate maximum power out-puts. In broad-area edge emitters for example, over6 W [11.1338] and external quantum efficiencies of

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Lasers and Coherent Light Sources 11.3 Semiconductor Lasers 723

over 56% [11.1338] have been achieved. While theVCSEL has enormous advantages in the laser–fibercoupling due to the small beam divergence and thesymmetric mode profile, the edge emitter incurs sub-stantial overhead costs for light coupling due to thehigher divergence and the elliptical mode profile. Thisis achieved in practice with aspherical lenses or withcomplicated tapered waveguide structures in the laseror fibre. The VCSEL offers further substantial ad-vantages as it enables simple optical on-wafer testingsimilar to the testing of integrated electronic circuits.In contrary, edge emitters must be individually iso-lated for characterization, i. e., at least cleaved into bars,or additionally for surface out-coupling, a 45 mirror(Fig. 11.96) must be implement in combination withetched mirrors.

The modulation bandwidths of VCSELs are typ-ically around 1 GHz (the record being 10 GHz),compared with typical values of 15 GHz (the recordbeing 40 GHz) for edge emitters. If one compares thelaser line widths of devices without external resonators,then VCSELs typically attain 200 MHz (the record be-ing 50 MHz) while edge emitters typically attain 1 MHz(the record being 10 kHz). Edge-emitting semiconduc-tor lasers exist in the range between 350 nm and 12 µm,while in the green and mid-IR spectral regions there areas yet either no devices exist or the component lifetimesare not sufficient for practical applications. Electricallypumped VCSELs exist so far within a spectrally muchsmaller range (420 nm to 2.05 µm) [11.1339–1341]. InVCSELs wavelength tuning is particularly difficult us-ing charge-carrier-induced (current-induced) or thermalrefractive index changes. A typical tuning range of 1 nmis attained in VCSELs compared to 80 nm in single-mode edge emitters. Finally we present a tuning conceptfor optoelectronic devices with vertical resonators whichpermits ultrawide wavelength tuning on the basis of onlyone control parameter.

Micromechanical Tunable Filters and VCSELsA perpendicular light wave (wavelength λD) incident ona highly reflecting DBR mirror is up to 99.9% reflected.The period is normally selected in such a way thatλD liesat the center of the stop-band (Fig. 11.114 bottom right).When a second identical mirror is positioned parallel ata distance, say, 1.5 ×λD from the first one, as shown inFig. 11.114, it is experimentally observed that the firstmirror does not reflect the wavelength λD any more.Although at first sight astonishing, the arrangement ofthe two DBR mirrors is now almost 100% transparentfor λD.

9"" 9>/

4"9" #

Q

"

4" "

" #

>/

>/

HGHGHG

$- $-

"

Fig. 11.114 Characteristic spectral components of a VC resonator.Three standing waves in the cavity ((a–c) above left) cause, asmarked by the bracket, the build up of the FP modes (above right),the DBR mirrors (bottom left) are characterized by the reflectionspectrum (bottom right), as indicated by the two brackets. Witha very short cavity length only one FP mode is located in the stopband (along the broken vertical lines)

ForλD in a DBR mirror, all reflected partial waves in-terfere constructively (99.9% reflection) on the incidentside of the mirror and destructively on the opposite side.This is the result of complicated multiple reflections andcomplex zigzag paths (in the simple ray model) por-trayed in a simplified manner occur. The light wavespenetrate the mirror, but due to the perfect destruc-tive interference, behind the mirrors, they transfer noenergy. By the targeted positioning of the second mir-ror (and adding further boundary surfaces) all partialwaves now interfere destructively on the incident sideand constructively on the opposite side (behind the sec-ond DBR mirror). Furthermore, a standing wave formsin the cavity for λD (Fig. 11.114).

Within the stop-band, except within the range of thevery sharp filter line (centered at λD) all wavelengthsare up to 99.9% reflected. In this way extremely high-quality optical filters can be implement, e.g., for fibre-optic telecommunications based on dense wavelengthdivision multiplexing.

Figure 11.111 contains a further material system,so far not discussed, with an extremely high refractiveindex contrast (n

InP= 3.2, nair = 1). With only four pe-

riods, a reflectivity of over 99.8% can be obtained. Thisunusual structure can be made, e.g., from a semicon-ductor multilayered structure with alternating InP andGa0.43In0.57As layers by selective etching of the GaInAs

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724 Part C Coherent and Incoherent Light Sources

'%0" <

G 6

'*

'

'

';

'%

'*

'

'$-'$- ' '$- ' '$- '

* %-

* %-

>/$-B0)

>/$-B0)

B

)" #

4

4

Fig. 11.115a–c Micromachined filter device with vertical resonator based on semiconductor multiple air gaps [11.1342,1343]. Electron micrograph of the cross section of a filter (a), details in (b), corresponding experimental wavelengthtuning as function of the actuation voltage in (c)

layers (micromechanical sacrificial-layer technology).Lattice-matching already exists through epitaxy on anInP substrate and, if necessary, an outstanding com-patibility with 1.55 µm GaInAsP laser-active layersis available. Figures 11.115a,b show a filter structurewith six InP membranes, each of which is fixed tothe supporting posts at four points. If a reverse-biasvoltage is applied between the upper p-doped and thelower n-doped DBR mirrors the membranes includ-ing the sandwiched air cavity can be electrostaticallyactuated. The cavity length and thus the filter wave-length can be varied in this way with only one controlparameter (voltage). With such an optical filter, an enor-mous continuous wavelength tuning range of 142 nm(Fig. 11.115c) was experimentally achieved with only3.2 V [11.1328,1329,1342,1343]. In other structures ofthe same type, a continuous tuning range of as much as221 nm has been obtained [11.1342]. The same tuningprinciple occurs in Figures 11.116c,d showing a VCSELwith InP/multiple-air-gap DBR mirrors, GaInAsP laser-active QW layers and an InP substrate.

In conclusion, based on Fig. 11.116 a veryinteresting parallel between quantum electronics

(Fig. 11.116a,b) and quantum photonics (Fig. 11.116c,d)can be drawn. In order to tailor the electronic levelsfor electrons and holes, semiconductor heterostructuresof predetermined composition and thickness are imple-mented. If the layer thickness of the material with thesmaller bandgap is on the order of the magnitude of theelectron wavelength, a quantization occurs that leads todefined quantized energy levels in the QWs. These quan-tized energy levels (eigenvalues) and the correspondingelectronic wave functions (eigenfunctions, i. e., modes)are solutions of the Schrödinger equation. The materials,stress and layer thicknesses are set up in this examplewith the 10 AlGaInAs QWs (Fig. 11.116b) in such a waythat the optical emission is at 1.55 µm. These QWs serveas the laser-active medium of an ultrafast semiconductordiode laser (Fig. 11.116a) [11.1302,1344]. In a very sim-plified depiction, the multiple QWs define a resonatorfor electron waves.

The electron microscope micrographs on the rightshow the exact analogy for photons. In order to selecta defined mode in the resonator of a VCSEL, takinginto account the refractive indices, layer thicknessesin the order of magnitude of the photon wavelength

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Lasers and Coherent Light Sources 11.3 Semiconductor Lasers 725

are chosen. A kind of quantization also arises in thiscase. The effective refractive indices (eigenvalues) andthe corresponding photon wave functions (eigenfunc-tions, i. e., modes) are solutions of the Helmholtzequation. From the mathematical point of view, thespace-dependent parts of the Schrödinger and Helmholtzequations have equivalent behavior. The strong re-fractive index contrast of InP/air-gap multi-membranestructures is again used in Fig. 11.116d to implementhighly reflective DBR mirrors for a 1.55 µm VCSEL(Fig. 11.116c) [11.1342, 1343]. Analogously the peri-odic refractive index variation defines a resonator forphoton waves.

Further analogies in this area are offered byperiodic 2-D quantum-dot fields and 2-D photonic crys-tals, or photonic band structures and electronic bandstructures.

11.3.5 Edge-Emitting Lasers and VCSELswith Low-Dimensional ActiveRegions

In commercial semiconductor laser diodes, bulk semi-conductor materials (3-D) were exclusively used forthe active zones for 25 years. Starting from 1990 edgeemitters with QWs (2-D) were implement in researchlabs and continuously improved. Only after a furtherfive years QW lasers were commercialized and subse-quently surpassed the 3-D lasers with lower thresholdcurrent densities, higher power outputs, higher char-acteristic temperatures T0 and higher bit rates due tomore-favorable gain profiles and better electronic con-finement. Although quantum wire (1-D) lasers shouldexhibit even better characteristics, they have so far notbeen very successful due to technological and geo-metrical constraints. In particular long carrier-transportand carrier-capture times substantially limit the bitrates in quantum wire lasers. Quantum dot QD (0-D) laser structures [11.1263–1272] are a lot morepromising. However, a problem at the moment is thestrongly varying size of QDs (Fig. 11.93). The theoret-ically very sharp density-of-state profiles are stronglyinhomogeneously spread, so that the predicted highdifferential gain dg/dn does not arise in practice.Compared with the 3-D and 2-D structures so far nospectrally narrower and higher-gain profile has beenobtained. Particularly broad gain profiles can be of ad-vantage however for extremely widely tunable lasers(see the section on tunable lasers in Sect. 11.3.3 and thaton micromechanical tunable VCSELs in Sect. 11.3.4).However, as already mentioned, additional research and

$--Q>!/ $--Q<.4,

0B0)0B0HG"5

B0)=>/65

Q

Q

Fig. 11.116 (a) Tunable, edge-emitting three-section laser withaxially varying DFB grating known as a chirped DFB grat-ing [11.1294], (b) strain-compensated multiple QW structure withtensile-strained AlGaInAs barriers and compressively strainedAlGaInAs wells[11.1344], (c) VCSEL based on multiple InP/airmembranes (centre), each of which is supported by four suspensionsconnected to the square supporting posts [11.1343], (d) cross sec-tion of the vertical resonator, consisting of the laser-active GaInAsPQW region, which is embedded between two InP/air-gap DBRmirrors [11.1342]

development effort is expected to provide excellentQD lasers.

11.3.6 Lasers with External Resonators

Both edge emitters and VCSELs can be implemented, asshown in Fig. 11.96, with an external resonator mirror,in contrast to Figs. 11.115 and 11.116c, which involvea relatively large air gap. This can serve several purposes:

1. the extension of the resonator length to achievenarrower line widths,

2. wavelength tuning by inserting a wavelength-selective, rotating element (prism, etalon, grating),

3. mode coupling for the generation of periodic se-quences of ultrashort pulses and

4. wavelength conversion by insertion of an opticallynonlinear crystal.

Examples of the last of these are lasers in whicha frequency-doubling crystal in the air gap of the res-

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726 Part C Coherent and Incoherent Light Sources

onator transforms the IR light, which is emitted from anedge emitter with one side antireflection-coated or a half-

cavity VCSEL (active region plus one DBR mirror), intothe yellow, green and blue range.

11.4 The CO2 Laser

The CO2 laser is one of the most important lasersfor industrial, medical and scientific applications. Ap-plications include high-precision material processing,cutting and welding of sheet metal, marking of plas-tics, cutting of paper and fabrics, surface treatments forsteel hardening or medical applications such as tissuecoagulation.

CO2 lasers use a gas mixture of helium, nitro-gen and carbon dioxide as the active medium, whichis usually excited by an electrical gas discharge. Thekey characteristics are emission in the mid-infraredat wavelengths around 10 µm and a continuous-waveoutput power ranging from a few watts for sealed-offminiature laser modules to more than 10 kW for high-power CO2 lasers with fast gas flow. While most CO2lasers operate in a CW mode with quite fast powermodulation enabled by electrical excitation, there arealso pulsed systems based on Q-switching or the so-called tranversal-excited atmospheric-pressure (TEA)laser. The beam quality is often excellent and in manycases nearly diffraction-limited. This is important for ap-plications such as precision cutting or remote welding.The mid-infrared emission implies that standard glasslenses and glass fiber-optic waveguides cannot be usedbecause of the high infrared absorption of silica-basedglasses. Transmitting optics such as focussing lensesor partially reflecting mirror substrates are commonlymade of zinc selenide, silicon or germanium. For beamguidance moving-mirror systems mounted on linear andswivel axes are used. Besides the most common emis-

G"G"

.Z":9"

'

+>.

0

/555A

/

4" "

Fig. 11.117 Basic longitudinally DC-excited CO2 laser

sion wavelength of 10.6 µm, special tunable CO2 laserscan generate laser light at dozens of distinct emissionlines between roughly 9.2 µm and 11 µm. Such lasersare used for scientific applications such as molecularspectroscopy and optically pumping of far-infrared gaslasers.

The physical principles and the technical realizationsof typical CO2 lasers are described in the followingsections.

11.4.1 Physical Principles

Basic CO2 Laser Tube PrinciplesLaser action in a CO2 gas was first described by Patelin 1964 [11.1345, 1346]. The basic physical principleshave changed little since then. An excellent and de-tailed discussion of CO2 laser fundamentals is givenin [11.1347] while additional general laser fundamentalsare described in [11.1348] and [11.1349]. Figure 11.117shows a sketch of a basic longitudinally direct current(DC)-excited CO2 laser. The laser gas is usually a mix-ture of CO2, N2 and He in a ratio of 1:2:8, for example.The total gas pressure in such a DC-excited laser isa few tens of hPa. The gas is excited by an electricalgas discharge, for example by a steady longitudinal DCglow discharge inside a quartz glass tube. Typical re-quired voltages are 15 kV per meter of discharge length,with a DC current of a few tens of mA. The voltagesneeded to start the discharge can be considerably higher.To maintain a stable α-type glow discharge [11.1350]

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Lasers and Coherent Light Sources 11.4 The CO2 Laser 727

an external ballast resistor Rb is required in series withthe high-voltage power supply. This compensates for thedifferential negative impedance characteristic of the gasdischarge and limits the discharge current in order toprevent arcing. Alternatively, power supplies with fastvoltage- and current-regulating electronics can be used.Typical lengths of the gas discharge are on the order ofone meter, depending on the desired output power.

Such laser tubes can be filled once with the lasergas mixture and are then sealed-off for the lifetime ofthe laser tube. Other lasers are provided with a gas in-let and outlet to maintain a gas flow through the lasertube. This improves the laser power as fresh laser gas isprovided to the laser but also requires additional periph-erals such as gas reservoirs and vacuum pumps. Opticalwindows to seal off the tube can be realized by planetransparent plates at the Brewster angle with respectto the beam axis. One polarization state of the laserlight can pass through the Brewster window withoutunwanted reflections. Thus, the Brewster window alsoacts as a polarization-selecting element inside the laserresonator.

The CO2–N2 molecule mixture can be excitedinside the gas discharge by collisions with free elec-trons [11.1351]. The excited roto-vibrational statesprovide optical gain at certain wavelengths in the in-frared. A simple plano-concave stable optical resonatorwith two mirrors located at each ends of the dischargetube can be used to provide continuous laser oscillation.The totally reflecting mirror is often made of copper,optionally with additional gold or protective dielectriclayers to enhance lifetime and reflectivity. The outputcoupler can be made of zinc selenide (ZnSe), germaniumor silicon, as these materials have good transparency inthe mid-infrared, in contrast to common silica-basedglasses. The required reflectivity is provided by dielec-tric coatings on the mirror substrate. As the opticalsmall-signal gain of the active CO2 gas medium is quitehigh compared to a helium–neon (HeNe) gas laser andis in the range of 1 m−1, reflectivities of output cou-plers may range from 20% to 90%. Optimum outputcoupler reflectivities for a maximum laser power are of-ten determined experimentally. Alternatively, they canbe calculated by a so-called Rigrod analysis if param-eters such as the small-signal gain and the saturationintensity of the medium are known [11.1352, 1353].

Typical efficiencies ηE, defined as the ratio of the ex-tracted optical laser power PL to the applied electricalpower PE into the discharge, are in the range of 10%.The largest part of the electrical excitation power is dissi-pated as thermal energy inside the gas volume. To avoid

a high thermal population of the lower laser energy lev-els, the CO2 laser gas must be kept cool at temperaturesof 400–500 K, depending largely on the different CO2laser types. A simple method for cooling is a secondglass tube, which acts as a cooling jacket coaxial to thelaser gas tube with a continuous water flow, as shown inFig. 11.117.

A rough estimate for these diffusion-cooled DC ex-cited lasers is an output power per discharge length of80 W/m. This cannot be scaled by using larger crosssections of the laser tube since this would reduce theamount of thermal energy that can be removed from themiddle of the gas discharge by diffusion cooling to thewalls. The overheating of the gas and the resulting ther-mal population of the lower energy level of the lasertransition would reduce the optical gain. In fact, for op-timized cooling the glass tube should be as small aspossible without introducing significant aperture lossesby truncating the free-space laser beam. Typical diam-eters are 5–10 mm, depending on the tube length andthe optical resonator design [11.1354].

Vibration and Rotation of the CO2 MoleculeThe emission wavelengths of the CO2 laser are deter-mined by the vibrational and rotational energy levels ofthe CO2 molecule, which are discussed in more detail in[11.1355, 1356] or [11.1357]. All data in the followingsections are given for the naturally most abundant iso-topologue 16O12C16O. Figure 11.118 shows the threefundamental vibrational modes: the symmetric stretchmode v1 along the molecule axis, the bending mode v2with motion of the C atom in a plane perpendicular to themolecule axis, and the asymmetric stretch mode v3. Thebending mode is twofold degenerate because of the pos-

0#""6

.

/

4#""6

.

.

Fig. 11.118 Vi-brational modesof the CO2 mol-ecule

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728 Part C Coherent and Incoherent Light Sources

.

,#

**

**

Fig. 11.119 Some of the lowest vibrational energy levels of the CO2

molecule. The Fermi resonance couples energy levels as indicatedby the dashed lines

sible bending motions in the two orthogonal coordinatesof the plane.

In the classical model of the unperturbed harmonicoscillator each mode has an associated vibrational fre-quency:

f1 = 40.51 THz, f2 = 20.15 THz,

f3 = 71.84 THz . (11.121)

In the quantum-mechanical model the energy of anyvibrational state of the molecule can only have dis-crete values called the vibrational energy levels. Usingthe integer vibrational quantum numbers n1, n2 and n3that describe the degree of excitation of the vibrationalmodes, the total vibrational energy Wv of the moleculecan be written as a sum over the single vibrations:

Wv = h f1

(n1 + 1

2

)+h f2 (n2 +1)

+h f3

(n3 + 1

2

). (11.122)

Table 11.38 Labeling, symmetry/parity classification and energies of some low-lying vibrational states of the CO2

molecule [11.1358]

State Frequency and energy

Herzberg AFGL Type ν (cm−1) f (THz) W (meV)

(0000) 00001 Σ+g 0 0 0

(0110) 01101 Πu 667.380 20.008 82.745

(0200) 10002 Σ+g 1285.408 38.536 159.370

(0220) 02201 ∆g 1335.132 40.026 165.535

(1000) 10001 Σ+g 1388.184 41.617 172.113

(0310) 11102 Πu 1932.470 57.934 239.596

(0330) 03301 Φu 2003.246 60.056 248.371

(1110) 11101 Πu 2076.856 62.263 257.497

(0001) 00011 Σ+u 2349.143 70.426 291.257

Even if there is no vibrational excitation present (allni = 0) and the molecule is in its ground state there isenergy stored inside the molecule that cannot be ex-changed. Thus, this zero-point energy is neglected in allfurther energy formulas and diagrams.

To characterize the vibrational state completelyanother quantum number l is required to define the distri-bution and the phase of the bending vibration quanta inv2 into the two orthogonal coordinates. For each numbern2 there are n2 +1 values for the angular momentum l:

|l| =⎧⎨⎩

n2, n2 −2, n2 −4, . . . , 0 ; n2 even

n2, n2 −2, n2 −4, . . . , 1 ; n2 odd .

(11.123)

For example for n2 = 2 the value l = 0 describes a linearbending motion, and l = −2 or l = 2 describes circularmotions of the C atom in opposite directions in the planeperpendicular to the molecule axis. A vibrational state ofthe CO2 molecule is labeled with the common Herzbergnotation (n1nl

2n3), for example (0000) for the groundstate or (1220) for a state with simultaneous excitationof several vibrational modes [11.1355].

Figure 11.119 shows the energies of some of thelowest vibrational states. The energy levels observed byemission and absorption of radiation, as shown in Ta-ble 11.38, are slightly different than those calculatedusing the simple-harmonic-oscillator theory (11.122).More-precise calculations using anharmonic correctionsand considering the mutual coupling of the three vibra-tional modes are therefore required.

The so-called Fermi resonance is of particular impor-tance for the laser process in the CO2 laser. Vibrationalstates with equal values of the term 2n1 +n2 and l = 0have almost the same energy and are strongly coupled.Due to this Fermi resonance the (1000) state is shifted up

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Lasers and Coherent Light Sources 11.4 The CO2 Laser 729

.

Fig. 11.120 Rota-tion of the CO2

molecule

and the (0200) state is shifted down in energy. The popu-lation densities of these states are quickly exchanged bycollisions with each other. Also the quantum-mechanicalwave functions and the vibrational motion of the two re-sulting levels are a strong mixture of the unperturbedvibrational states. In the CO2 laser literature the re-sulting levels are sometimes labeled (I) for the more(1000)-like state and (II) for the more (0200)-like state.Spectroscopic databases, which are an excellent refer-ence for accurate absorption and emission wavelengthsof the CO2 molecule, often use a different labelingscheme than the simple Herzberg scheme, for examplethe so-called AFGL (Air Force Geophysics Lab, USA)notation, which gives a better treatment of the Fermiresonance [11.1359, 1360].

Superimposed onto the vibration is the rotationalmotion of the CO2 molecule, as shown in Fig. 11.120.Only the rotation around an axis perpendicular to themolecule axis has a significant moment of inertia andis considered further. The discrete energy levels in therigid-rotator model are given by

Wr = B · J(J +1) , (11.124)

where B is the rotational constant and J is the quantumnumber for the rotational state. The higher the value ofJ the faster would be the rotation of the molecule in theclassical mechanical model. For the (0001) vibrationalstate, for example, B has a value of

B = 48.0 meV = h fr = h 11.6 GHz . (11.125)

More precise calculations show that the rotational con-stant B is slightly dependent on the vibrational statebecause of the anharmonicity and also dependent on therotational quantum number J itself because of centrifu-gal forces.

The total internal energy W = Wv + Wr of the CO2molecule is the sum of the vibrational and the rotationalenergies. The energy stored in the rotational motionis usually much smaller than in the vibration motion.Thus, the rotational energy levels can be thought of assuperimposed on each vibrational level.

Emission Lines of the CO2 LaserAbsorption, spontaneous and stimulated emission oflight can occur when the energy of a photon Wp = h fp

equals the energy difference ∆W = W2 − W1 betweentwo given energy levels of the respective atom ormolecule. Additionally, selection rules based on the con-servation of the spin or angular momentum and dueto symmetry properties of the wave functions haveto be considered. Based on the formalism of vibra-tional matrix dipole elements only certain transitionsare allowed [11.1357]. The strongest allowed CO2laser emission lines result from the vibrational transi-tions (0001) → (1000) centered at wavelengths around10.4 µm (10 µm band) and (0001) → (0200) around9.4 µm (9 µm band). For these regular transitions thefollowing selection rule for the rotational quantum num-bers of upper and lower energy levels apply:

∆J = J2 − J1 = ±1 . (11.126)

For a given lower rotational state J1 there are twodifferent possibilities for the emission of photons, asshown in Fig. 11.121. Transitions are named after therotational quantum number of the lower state. Addi-tionally, the transitions are labeled P (the P branch)for ∆J = −1 and R (the R branch) for ∆J = +1. For

,

,

;

,

,

;

,

,

(

-

(

((;

Fig. 11.121 Examples of names for allowed roto-vibrationaltransitions of the regular bands. The rotational distributionis calculated for T = 400 K

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730 Part C Coherent and Incoherent Light Sources

(G 6Q

($- $- $-

![ "##+@7 ( ; &

(

(

* *

*

*

Fig. 11.122 Calculated relative small-signal gain for the regular bands ofthe CO2 laser at T = 500 K

example one of the strongest emission lines of theCO2 laser is labeled 10P(20), indicating a transitionfrom the 10 µm band with the lower rotational stateJ = 20 and ∆J = −1. Different transitions have differ-ent energy differences ∆W and thus different emissionfrequencies f or wavelengths λ because of the non-equidistant rotational energies in (11.124). This is theorigin of the multitude of possible emission lines of theCO2 laser.

Figure 11.122 shows the calculated relative gain ofthe regular bands of a typical CO2 laser. Both vibrationaltransitions have characteristic R and P branches due tothe superimposed rotational energy levels and the selec-tion rules. Exact line positions are listed in [11.1347]based on [11.1361], or in spectral databases suchas [11.1359], which also lists transition dipole momentsthat are useful for the calculation of the absorption andgain of a transition [11.1362]. Note that in Fig. 11.121only rotational states of even order J are present forthe lower states (1000) and (0200). The odd states aremissing because of the symmetry properties and parityrules (Σ+

g ) of these states in the 16O12C16O isotopo-logue [11.1356]. Similar considerations are valid for theupper state of the regular band (0001) (Σ+

u state), whichonly has odd rotational quantum numbers J .

Table 11.39 Possible vibrational lasing transitions

Vibrational band Comment

(0001) → (1000) Regular band, 10 µm

(0001) → (0200) Regular band, 9 µm

(0111) → (1110) Hot band

(0111) → (0310) Hot band

(0002) → (1001) Sequence band

(0002) → (0201) Sequence band

Besides of the regular bands, there are other vibra-tional transitions with slightly shifted wavelengths thatcan be operated as lasers: the so-called hot band andsequence band. Table 11.39 shows a summary of vi-brational transitions. For laser emission in the sequenceor hot bands, lasing in the regular bands must be sup-pressed, for example by passive absorption cells filledwith hot CO2 gas inside the resonator [11.1347]. Almostall technical lasers operate in the regular bands.

The optical small-signal intensity gain per unitlength g0 of a given transition is proportional to thepopulation density difference of the upper and the lowerstates, weighted with the respective degeneracy factors:

g0 ∼ (Nn2 J2 − 2J2 −1

2J1 −1Nn1 J1 ) . (11.127)

The population densities Nn J are the numbers ofmolecules per unit volume in the roto-vibrational energylevel n and J . Within each vibrational state n with a totalpopulation of Nn the distribution of the molecules in thedifferent rotational states Nn J is described by a thermalBoltzmann distribution, weighted by the total rotationalpartition sum and the degeneracy factor for each J :

Nn J = Nn

(2B

kT

)(2J +1) exp

(− Wr

kT

), (11.128)

with the translational gas temperature T and the Boltz-mann constant k. The rotational level at the maximum ofthis distribution at a given temperature can be approxi-mately calculated as

Jmax ≈√

kT

2B− 1

2. (11.129)

For typical gas temperatures of T = 400 K the levelsaround J = 19 have the strongest population. In CO2

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Lasers and Coherent Light Sources 11.4 The CO2 Laser 731

lasers without any further wavelength-selective elementsnormally only one transition around the 10P(20) lineat a wavelength of 10.59 µm will lase, as this one hasthe strongest gain (Fig. 11.122). Typical values for thesmall signal gain g0 are in the range from 0.5 m−1

to 1.5 m−1.Even under lasing conditions the rotational distribu-

tion will maintain its thermal Boltzmann shape becauseof fast population exchange between the rotational lev-els by molecule collisions. Thus, almost all rotationalstates within the upper vibrational level can contributeto a single lasing transition rather than just the popula-tion of the particular upper level, for example the J = 19state of the (0001) vibrational level. This affects mainlythe saturation intensity Is of the optical gain, which ismuch larger than expected form the population Nn J ofa single rotational level J itself. The saturation inten-sity Is is in the range from 100 W cm−2 for sealed-offlarge-bore DC-excited lasers to over 1000 W cm−2 forfast-flow systems.

Electrical Excitation and Gas CompositionWithout any excitation and in thermal equilibriumthe population of the vibrational states is given bya Boltzmann distribution. As the energies of the lowervibrational levels are almost of the same order as thethermal energy kT = 25.5 meV at 296 K, the thermalpopulation of levels above the ground state cannot be ne-glected. The population Nn of a given vibrational leveln is given by:

Nn = N0

exp(− dn Wn

kT

)Qvib

, (11.130)

where N0 is the total density of CO2 molecules, dn isthe degeneracy factor and Wn the energy of level n, andQvib is the vibrational total partition sum. Without ex-

7

<'<

7

7 7

<'+

.

'

,

Q

2 2 2 2

7 7

.2.

7

'

(Q

'

Fig. 11.123 Energy-level diagram ofthe CO2 laser process with excitationand decay paths

citation the population of the upper laser state (0001)is always smaller than the lower laser levels (1000)or (0200). This is the normal case for passive media,which show absorption lines on the respective transitionwavelengths.

To achieve an inversion, and thus optical gain, inan active medium an electrical gas discharge is usu-ally used to excite the CO2 laser. Accelerated electronsinside the discharge collide with the molecules andlose part of their kinetic energy. This energy can betransformed into vibrational excitation energy of themolecules or into kinetic energy by a pure transla-tory motion. How effective a certain vibrational stateof the molecule is excited is described by the excita-tion cross section, which is dependent on the energy ofthe electrons. For inversion of the regular CO2 laserlines, a selective population of the upper laser state(0001) is required. This is barely possible in a pureCO2 gas discharge as the cross sections for excitation ofthe different vibrational states are of the same order ofmagnitude.

Therefore, nitrogen (N2) is added to the lasergas [11.1351]. As a molecule with two atoms it onlyhas one vibrational mode. As a homonuclear moleculeit has no electric dipole moment and hence, no radiativedecay of excited vibrational states. Thus, nitrogen canefficiently store energy in its vibrational excited states.More than 50% of all N2 molecules can be excited tohigher vibrational levels inside a gas discharge. The firstexcited vibrational state, n = 1, has nearly the same en-ergy (289 meV) as the upper laser level (0001). Due toresonant collisions between excited nitrogen atoms vi-brational energy is readily transferred to the desired CO2state

N2(n)+CO2(0000)

N2(n −1)+CO2(0001)+∆W . (11.131)

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732 Part C Coherent and Incoherent Light Sources

The energy difference ∆W = −2.2 meV kT is eas-ily available from the thermal kinetic energy of themolecules.

Figure 11.123 shows the energy-level diagram of theCO2 laser process. Accelerated electrons from the gasdischarge excite nitrogen molecules to higher vibrationallevels by collisions. These excited nitrogen moleculestransfer their energy to the upper laser level (0001) ofthe CO2 molecule and provide a significant populationinversion compared to the lower laser levels. From there,two strong lasing transitions by stimulated emission ofphotons are possible, the 9 µm band with (0200) as thelower laser level and the 10 µm band with (1000).

To maintain a large inversion thermal population ofthe lower laser levels must be avoided by efficient cool-ing of the laser gas. Therefore, helium (He) is added tothe laser gas. Helium is the gas with the largest ther-mal conductivity. It also plays an important role for thepopulation decay of the (0200) and (0110) levels byvibrational–translational (V–T) relaxation. Thereby, vi-brational energy of the CO2 molecule is transferred tokinetic energy, the translatory motion of the molecules.Radiative decay by spontaneous emission is also pos-sible for the (0200) and (0110) levels, but not for the(1000) level, which shows no change in the dipole mo-ment during vibration. It can reduce its population byV–T relaxation directly to the (0110) level, but also bythe Fermi resonance via collisions and energy exchangewith the (0200) state. In fact, the degree of vibrationalexcitation of these two vibrational modes is almost thesame due to this Fermi coupling. After relaxation to theground state (0000) the CO2 molecule is available forexcitation and the laser process again.

Within one vibrational mode, for example v1 of theCO2 molecule, there is a strong interaction of the vibra-tional levels by vibrational–vibrational (V–V) relaxationand excitation processes during collisions:

CO2(n′1)+CO2(n′′

1)

CO2(n′1 +1)+CO2(n′′

1 −1)+∆W , (11.132)

where the small energy difference ∆W coming fromthe anharmonic forces can almost be neglected. By thisfast thermalization the population distribution of the vi-brational levels within one mode vi can be describedby a Boltzmann distribution similar to (11.130) witha specific vibrational temperature Ti for this particu-lar mode. Three vibrational temperatures T1, T2, andT3 are required to describe the three vibrational modesof the CO2 molecule. Together with the vibrationaltemperature of the nitrogen TN and the physical gas

temperature T they form the so-called five-temperaturemodel of the CO2 laser. This model can be used toquantify the degree of vibrational excitation in a givenlaser system and to calculate the optical gain of differenttransitions [11.1347, 1363, 1364].

Note that the vibrational temperatures inside a gasdischarge can be significantly higher than the usual phys-ical gas temperature T which describes the translatorykinetic energy. For typical laser conditions the temper-atures T3 and TN can be in excess of 1000 K, even ifthe physical gas temperature T is only 450 K, for exam-ple. This describes the desired high level of excitationof these modes for a strong inversion, especially if thereis no laser action to reduce the population in (0001)by stimulated emission. The vibrational temperaturesT1 and T2 are almost equal because of their couplingand are slightly larger than the physical temperature, be-cause their population decay to the ground state takesplace with a finite rate. Here, as low as possible vibra-tional temperatures are desired for a low population ofthe lower laser levels. Of course, no vibrational temper-ature can be lower than the physical gas temperature T .It is also the temperature T that is valid for the rotationalpopulation distribution.

Thus, a gas mixture for a CO2 laser could beHe : N2 : CO2 = 8 : 2 : 1, with large individual variationsbetween specific laser designs, which are often exper-imentally defined for maximum laser output. The gaspressure p is typically a few tens of hPa for longitudi-nally DC-excited lasers, in the range of 100 hPa for laserswith a transversal RF discharge or can equal the atmo-spheric pressure in so-called TEA (transversaly excitedatmospheric pressure) lasers with a pulsed transversaldischarge. The gas pressure influences the stability ofthe gas discharge and the electric field strengths requiredto start and maintain a stable discharge at the dif-ferent excitation frequencies and geometries. Togetherwith the applied electric field E inside the dischargethe gas pressure p affects the electron energy distri-bution function (EEDF). For efficient excitation of thedesired vibrational nitrogen states electrons with ener-gies of 2–3 eV are required, where the effective crosssection for vibrational excitation of N2 has a maxi-mum [11.1347].

Dissociation and Gas AdditivesThe complex-shaped electron energy distribution func-tion (EEDF) in such a laser gas discharge always hasa tail extending to quite high electron energies. Theseelectrons can start unwanted chemical dissociation pro-cesses of the CO2 molecule inside the gas discharge, for

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Lasers and Coherent Light Sources 11.4 The CO2 Laser 733

example

CO2 + e− CO+O− ,∆W = −3.85 eV ,(11.133)

CO2 + e− CO+O+ e− ,∆W = −5.5 eV .(11.134)

This is the origin of the complex gas chemistry thatleads to a reduced CO2 concentration and the forma-tion of a significant CO concentration inside the lasergas tube [11.1367]. By the law of mass action a typicalequilibrium ratio of 1 : 1 between CO2 and CO is estab-lished if no means to reduce the dissociation are present.Clearly this reduces the overall laser output power andefficiency.

In lasers with a steady gas exchange by a gas flowthe laser gas can be replaced by fresh gas from a premixbottle or can be partially regenerated by forced circu-lation through catalysts to prevent too high a degreeof dissociation. Naturally, the gas dissociation is morecrucial for sealed-off lasers without any gas exchange.To decrease dissociation processes and maintain a highoutput power, many different gas additives to the basicmixture have been studied [11.1347]. Table 11.40 showsthe composition of some commercial laser gas premixesfor high-power CO2 lasers.

Xenon (Xe) is without doubt the most effi-cient and most widely used additive for sealed CO2lasers [11.1368–1370]. Xe has a first ionizing poten-tial of 12.1 eV, which is some eV less than the othergases. Hence, the same current density in the dischargecan be maintained at a lower applied electric field. Thisin turn shifts the EEDF towards lower energies, whichare more efficient for vibrational excitation, and thereare fewer fast electrons to cause dissociation processes.On the other hand, Xe with its heavy atomic weight hasa very low thermal conductivity. Also it is a very rareand expensive noble gas. Sealed laser mixtures typicallycontain a few percent of Xe.

Water vapor (H2O) and molecular hydrogen (H2)have also been extensively studied [11.1347,1371]. Both

Table 11.40 Some commercial gas premixes for different industrial CO2 laser types [11.1365, 1366]

Premix name / Composition Laser type,

(Maker) He N2 CO2 CO O2 Xe H2 model series

Lasermix 322 65.5 29 5.5 Fast axial-flow,

(Linde AG) Trumpf TLF Series

Lasermix 690 65 19 4 6 3 3 Diffusion-cooled sealed-off,

(Linde AG) Rofin Slab DC 0XX

LasalTM 81 80.8 15 4 0.2 DC excited slow-flow,

(Air Liquide) FEHA SM Series

gases react inside the gas discharge, for example withoxygen radicals to form hydroxyl radicals OH. Thishydroxyl radical is an efficient catalyst for oxidizingCO back to CO2. On the other hand, water vapor withits multitude of vibrational states rapidly deexcites theupper laser state (0001) and quenches the desired pop-ulation densities of this level. The reported water vaporconcentrations for optimum catalysis and yet low vi-brational quenching are quite low [11.1347] and werefound to be hard to control in real laser systems. Inother laser systems no clear proof of an optimum wa-ter vapor content was found at all [11.1372, 1373]. Asa rule of thump the dew point of the laser gas shouldbe below −40 C, equivalent to a water vapor partialpressure of 0.13 hPa. Creeping of water from the cool-ing system has to be avoided. However, some laser gaspremixes may contain small amounts of hydrogen, asdeemed appropriate by the manufacturer of the lasersystem.

Other gas premixes include carbon monoxide (CO)or oxygen (O2). By the law of mass action both additivescan shift the equilibrium of the dissociation reactions(11.133) and (11.134) towards the left. Too high anO2 concentration has a detrimental influence on thedischarge stability and the EEDF, as it is a stronglyelectronegative gas. CO as a diatomic molecule has sim-ilar vibrational energy levels to nitrogen and can alsotransfer vibrational energy by resonant collisions to theupper laser state of the CO2 molecule (Fig. 11.123). An-other benefit of adding CO and O2 initially to the gaspremix for filling the laser is that the gas is alreadynear its equilibrium composition for the dissociationprocess, so no overshoot or undershoot of the laserpower occurs when the laser is operated for the firsttime after filling. However, CO is not as effective as ni-trogen for vibrational excitation. The energy differencefrom the CO:(n = 1) vibrational level to the CO2:(0001)level is 25.5 meV and is thus larger than with nitro-gen. Additionally, CO has allowed dipole transitionsto the ground state by spontaneous emission, which

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734 Part C Coherent and Incoherent Light Sources

decreases vibrational excitation compared to nitrogen.Also CO is the only toxic gas of the species discussedhere. Care has to be taken when handling such gasmixtures.

Materials for CO2 LasersMaterials for the laser tube have to be chosen properly.All common rules for building and preparing vacuumsystems apply. Materials with low out-gasing and lowvapor pressure must be used. This normally inhibits theapplication of plastics and polymer-based glues in CO2laser tubes. Additionally, the UV radiation of the gasdischarge will degrade most organic compounds rapidly.There are only a few epoxy-based adhesives that may beused.

Many low-power sealed-off lasers and even high-power lasers with slow or fast gas circulation use quartzglass tubes to confine the laser gas. Quartz (SiO2) glassis thermally stable, chemically inert, robust against UVradiation and has low dielectric losses. However, silica-based glasses have high infrared absorption, even forgrazing incidence of light to the surface. Thus, the diam-eter of the quartz glass tube needs to be considerablylarger than the laser beam formed by the resonator mir-rors. For waveguide lasers (see Sect. 11.4.2) tubes madefrom other materials must be used. Alumina (polycrys-talline Al2O3 ceramics) is often used. It is a dense andvacuum-tight ceramic material, which is also dimen-sionally and thermally stable. Its infrared absorption islower than quartz glass, making it suitable for hollowdielectric waveguides. Also it has very low dielectriclosses at radio frequencies, making it an ideal mater-ial for transversal capacitively coupled RF dischargeexcitation.

Good material choices to confine the gas vol-ume are also inert passivated metals such as stainlesssteel [11.1374] and passivated aluminium. Steel caneasily be welded, both with conventional techniques orwith lasers, to built a vacuum-tight laser tube. Weld-ing of aluminum is more difficult. On the other hand,aluminum has a far better thermal and electrical con-ductivity and a higher infrared reflectivity. This makesaluminum a very good material for waveguide plates andelectrodes for transversal RF discharges [11.1375].

Electrolytic copper or oxygen-free high-conductivity(OFHC) copper can also be used because of their ex-cellent thermal and electrical conductivity and infraredreflectivity. On the other hand, copper can be oxidizedby oxygen radicals from the dissociation process in-side the gas discharge. Copper does not form stable,inert oxides on its surface but rather absorbs and binds

the oxygen in the volume. This shifts the equilibriumof the dissociation towards lower CO2 content in thetube. Therefore, uncoated copper is not well suitedas a material in direct contact with the gas discharge.Away from the discharge, however, copper is an excel-lent material for mirrors. Copper mirrors can be cooledeffectively by water channels from the rear side. Thisis important for resonator and deflecting mirrors in-side high-power lasers. For example, with an outputcoupler reflectivity of 50% the circulating light powerinside the resonator is twice the rated laser power ofthe system. Even if the infrared reflectivity is very high,this can lead to considerable warming and mechanicalstress if the mirrors are not cooled. Copper can be elec-troplated with nickel (Ni) and then coated with gold(Au). This makes the copper surface more stable againstoxidation.

Catalysts and Gas AnalysisMuch research was carried out to find proper cata-lysts to reduce the dissociation of CO2 and to maintaina high output power. In lasers with a gas flow, thelaser gas can be circulated through external catalysts.Common catalysts such as hot platinum [11.1376] orplatinum on metallic oxides such as tin oxide with a largesurface area [11.1377,1378] can be used. However, fast-flow systems require gas replenishment of a few litersper hour to compensate for residual vacuum leaks ormaterial contamination. As the price of the laser gasmixture is rather low compared to the costs of installinga catalyst, they are usually not applied in fast-flowsystems.

For sealed lasers without any gas flow the sit-uation is quite different. The dissociation reactionsinside the gas discharge are fast and equilibriumis reached in a tenth of a second within the dis-charge. On the other hand, diffusion time constantsto a nearby gas reservoir are in the range of min-utes. Thus, any catalytic surfaces need to be in closecontact with the discharge itself. For example, cat-alytic active platinum cathodes [11.1371] and distributedplatinum [11.1379] or sputtered gold coatings onthe inside of a longitudinal discharge tube [11.1380]have been reported. For waveguide lasers, gold-coatedelectrodes were found to be effective in some re-ports [11.1381, 1382]. The catalytic activity and theadvantages for laser output power of these approachesdepend on the preparation of the catalytic surfaces, thegas composition and the laser geometry itself, leavingopen many questions and an impression of alchemy.Some commercial laser designs embody some sort of

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Lasers and Coherent Light Sources 11.4 The CO2 Laser 735

these techniques, as deemed best for the given laserdesign.

To study the effects of catalysts on the gas com-position or, more generally, to have more data aboutthe active medium such as rotational and vibrationaltemperatures various methods can be applied. Massspectrometers are commonly used for gas analysis.However, short-lived species from the discharge can-not be easily measured. The required pressure reductionfrom some tens of hPa down to 10 × 10−7 hPa tendsto change the original gas compositions. Special tech-niques in experimental laser systems have been usedto measure gas compositions directly from the gas dis-charge, and have given insight into the complex gaschemistry [11.1367, 1384].

Another method is to study the visible and UVspontaneous emission from electronically excited statesinside the gas discharge [11.1385]. This can even bedone with the naked eye. Figure 11.124 shows an ex-ample spectrum. A gas discharge without much COappears pink or redish-purple because of vibronic nitro-gen emission lines in the red and blue visible spectrum.With a significant carbon monoxide content there areadditionally some CO emission lines spread over thevisible and a strong emission band in the blue–UVspectral region. Thus, a gas discharge with a higherdegree of dissociation looks more blueish-white. Thecolor change can be observed in slow-flow DC-excitedgas lasers with glass tubes from the gas inlet to thegas outlet, for example. For quantitative studies fiber-coupled spectrometers with fast readout can be used togain time-resolved concentration data [11.1383]. Withhigh-resolution spectroscopy, for example of the UV ni-trogen emission band, the rotational temperatures of thegas can be measured. Using an imaging system, spatiallyresolved data for these parameters have even been meas-ured in the discharge gap of an RF-excited high-powerslab laser [11.1386].

Absorption spectroscopy with tunable diode lasersis also very effective for measuring species concentra-tions and roto-vibrational temperatures directly insidethe laser tube or gas discharge. To do this, a wavelength-tunable diode laser is adjusted to the various absorptiontransitions of the relevant molecules (i. e., CO2, CO,N2O, NO). By aiming the diode-laser beam throughthe gas volume and measuring the relative intensi-ties of several rotational and vibrational transitionsan accurate diagnostic of the active medium is possi-ble [11.1387–1391]. Recently, the catalytic activity ofspecial gold coatings on electrodes for RF-excited slablasers was verified using this technique [11.1392].

G 6

; * - % &B#

)"

@

F FF

@ @@@@ @@@@

CCCCN. /

.N

'

'

'

'

'*

'-

.N/0

'

'

' ' '*' '

," $

Fig. 11.124 Visible spectrum the sidewards spontaneous emission ofa CO2 laser gas discharge (after [11.1373, 1383])

Output Spectra and Line BroadeningDue to the fast population relaxation between the rota-tional levels only the rotational transition with the largestgain is usually observed. A similar competition as forthe rotational lines is also given for the two regular vi-brational transitions. As both have the same upper laserlevel (0001), and the population levels of the lower states(1000) and (0200) are coupled by the Fermi resonanceand have similar decay rates, only the vibrational tran-sition with the largest gain will survive. Typically this isthe 10P(20) transition or one of its neighbors for manylasers.

Every roto-vibrational transmission line is broad-ened by several broadening mechanisms. At gas pres-sures above 10 hPa collisional or pressure broadeningdominates over Doppler broadening. The collision-broadened line shape is described by a Lorentzianfunction having a line width (FWHM) ∆ fL proportionalto the total gas pressure p:

∆ fL = 2p

CO2b

CO2+ψ

N2b

N2+ψ

Heb

He

)

×

(300

T

)n

(11.135)

where the ψi are the fractions of the respective gasesand T is the gas temperature. The temperature exponentn is 0.58 for constant pressure [11.1393], with valuesranging from 0.5 to 0.7 also found in the literature. The biare the pressure-broadening coefficients resulting fromcollisions with species i and are given in [11.1393] to

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736 Part C Coherent and Incoherent Light Sources

9[ "#

C7

C79[ "##= #-#

9[ "#

"#

4=

4

#

Fig. 11.125 Homogeneous saturation of a Lorentzian gainprofile leads to lasing of a single longitudinal resonatoreigenfrequency

be:

bCO2

= (3.40−|m| ·0.0272) MHz hPa−1 (11.136)

bN2

= (2.35−|m| ·0.0127) MHz hPa−1 (11.137)

bHe

= (1.77−|m| ·0.00083) MHz hPa−1 ,

(11.138)

where m = −J for the P branch and m = J +1 for theR branch. For example, the most common 10P(20) lineat 100 hPa and 500 K in Lasermix 322 from Table 11.40has a Lorentzian line width (FWHM) of 284 MHz. TheDoppler line width at this temperature would be ∆ fD =36.2 MHz and can be considered if a Voigt line shapeinstead of a pure Lorentzian is used [11.1394].

Within this line width one or more longitudi-nal resonator eigenfrequencies fq may be located.For an example length for a CO2 laser resonatorof Lres = 2 m a spacing of the eigenfrequencies∆ fq = c/2L = 75 MHz results. As the collision-broadened line saturates homogeneously, only theeigenfrequency closest to the center frequency of thetransition can lase in the steady state after saturation,as shown in Fig. 11.125. Thus, CO2 lasers are typicallylongitudinally single-mode lasers.

Efficiency, Output Power and CoolingThe theoretical limit for the efficiency of a laser is theinternal quantum efficiency ηq. For the CO2 laser thisis the ratio of the energy of the laser photons to the en-ergy of the upper laser level (0001) and is roughly 40%.This is quite high for a gas laser. Real technical lasers

have significantly lower efficiencies for several reasons.The electrons inside the gas discharge do not only excitethe required vibrational states of the molecules but alsoother states. Also a significant part of the kinetic elec-tron energy is transferred into pure translatory motionof the molecules and thus transferred into heat. Addi-tional energy is lost in the ionizing processes requiredto maintain the gas discharge and in chemical disso-ciation processes. Also not every vibrationally excitedmolecule contributes to the laser process. Vibrationalenergy decays by vibrational–translational relaxation,which produces heat again and by radiative decay byspontaneous emission of excited states. Finally, opticalenergy is dissipated in absorption losses of the mirrorsand waveguides or by limited aperture diameters.

Typical electrical efficiencies ηE, defined as the ratioof the extracted laser power PL to the electrical power PEapplied directly to the discharge tube, are in the range of10% for slow-flow DC-excited laser, up to 15% for RF-excited waveguide lasers and up to 20% for fast-flowingsystems. The overall wall-plug efficiencies ηtot as a ratioof laser power PL to mains supply power Pmains arealways lower than ηE. This is due to conversion lossesin the electronic power supplies to provide the requiredDC high-voltage or RF power. A considerable amountof energy is also dissipated in the gas circulating systemof fast-flow lasers.

The output power of a given CO2 laser can beroughly estimated by simple thermodynamics. If allaspects of optical design, electrical excitation and gascomposition are optimized, electrical efficiencies ηE asdiscussed above can be assumed. From this point, thelaser power is finally limited by the cooling and heat re-moval capacity of the laser system. The power dissipatedby the electric excitation Pdiss should not heat the lasergas above a certain temperature, for example 450 K.

In diffusion-cooled lasers no forced gas flow con-tributes to the heat removal. The thermal conductivityof the gas mixture and the diffusion of excited vibra-tional states to the housing walls determine the thermaltransport processes. For a given tolerable temperaturerise ∆T of the laser gas in the discharge area, the max-imum applicable dissipated thermal power Pdiss can becalculated by solving the differential heat-transfer equa-tion for the given geometry. For fast-flowing systemsthe amount of heat removal is given by the specific heatcapacity cP of the gas and the mass flow m.

With the assumption that almost all electrical exci-tation power PE is finally dissipated into heat Pdiss, themaximum permissable excitation power PE,max for thegiven temperature rise ∆T is thus known. The maximum

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Lasers and Coherent Light Sources 11.4 The CO2 Laser 737

possible output power PL,max of this laser system canthen be calculated with the assumed electrical efficiencyηE. This exaggerates the importance of the cooling meth-ods for high-power lasers. Different approaches will bediscussed in the next section on technical lasers.

11.4.2 Typical Technical Designs

Although CO2 lasers with a longitudinal DC dischargeinside a glass tube as described in Sect. 11.4.1 are stillbeing used and sold because of their simplicity, manyother types and realizations of the CO2 laser have beendeveloped over the last decades. They can by systemat-ically categorized according to various characteristics,as below.

Gas flow inside the laser.• No gas flow, sealed-off• Quasi-sealed-off, periodic gas exchange• Slow gas flow axial to the laser beam• Fast gas flow axial to the laser beam• Fast gas flow transversal to the laser beam

Gas cooling.• Diffusion-cooled, cooled walls of the gas discharge• Fast gas flow with external heat exchanger

Electrical excitation.• Longitudinal DC discharge, continuous• Transversal DC discharge at high gas pressure,pulsed• Capacitively coupled transversal RF discharge• Inductively coupled RF discharge• Microwave-excited gas discharge

Optical resonator.• Stable optical resonator• Unstable optical resonator• Free-space propagation between the mirrors• Optical waveguide between the mirrors• Combinations of these in different planes

In principle, almost any of these characteristics couldbe combined to give specific advantages. The three axesof the optical laser beam, the gas flow and the excitingelectrical field can be mutually parallel or orthogonal ingeneral. Some of the most important designs with theirspecific technical characteristics are discussed furtherin the next sections. Commercial manufacturers of CO2lasers often have quite different approaches, also de-pending on their own intellectual property and patents.

General design aspects are: as efficient a heat removalas possible, a compact optical resonator for a small foot-print even at very high power levels and a rugged androbust design for maintenance-free long-term industrialoperation.

More-exotic laser types such as optically pumpedCO2 lasers, black-body radiation-pumped lasers, laserswith electron-beam sustained gas discharges and gas-dynamic CO2 lasers are not further considered in detailhere.

DC-Excited Fast-Axial-Gas-Flow LasersLongitudinally DC-excited lasers have the advantage ofa rather simple and cost-effective design. Modern DC-excited lasers use sophisticated electronically regulatedhigh-voltage (HV) supply circuits with both voltage andcurrent control to omit the ballast resistor shown inFig. 11.117. This reduces ohmic losses in the HV circuit.Vacuum-tube-based or more modern semiconductor-based current regulators and switched-mode voltageconverters are applied.

For high-power lasers a fast axial gas flow can beused to cool the gas. Lasers with a power in the kWrange [11.1396, 1397] were demonstrated in the late1970s. Figure 11.126 shows an example of a modernDC-excited fast-flow high-power laser. The gas dis-charge in the gas tube is divided into four independentsections to keep the required high-voltage levels mod-erate. Two of the discharge and flow tubes shown canbe used in parallel, with a U-shaped folded light pathand resonator to double the length of the active medium.This is because the output power scales with the lengthof the gain medium.

="

95

@8"6

00

"

.6

>"6

0

="

.5 + 5

>.@< #

Fig. 11.126 Longitudinally DC-excited high-power CO2 laser withfast axial gas flow [11.1395]

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738 Part C Coherent and Incoherent Light Sources

As discussed in Sect. 11.4.1, the maximum outputpower of a CO2 laser is also largely determined byits heat removal capacity. A fast gas circulation can bedriven by a turbo blower. From thermodynamic consid-erations, the maximum laser power PL,max of a fast-flowlaser can be calculated with a given electrical efficiencyηE:

PL,max = ηE PE,max =(ηE

1−ηE

)Pth

=(ηE

1−ηE

)m cp ∆T , (11.139)

where PE,max is the maximum applicable electricalpower, Pth is the thermal power dissipated in andremoved from the discharge, m is the mass flow,cp is the specific heat of the gas mixture, typically2500 J kg−1 m−3, and ∆T is the maximum gas temper-ature rise that can be tolerated, typically 250 K. It canbe seen clearly that the laser power scales with the massthroughput and thus with the flow velocity, which is lim-ited by the onset of turbulence and the speed of sound.For an assumed efficiency of PE = 15%, one yields anapproximate required mass flow of 0.1 kg/s per kW oflaser power. The heat dissipated in the gas is removed bya heat exchanger after the gas has passed the discharge

,"

+!

>"6

/ ##

B

!

!

>"6

,"

!

>"6

B

.

Fig. 11.127a–c Geometries for RF excitation with current IRF andelectric field E: (a) capacitively coupled RF discharge, insulatedelectrode, (b) capacitively coupled RF discharge, non-insulatedelectrodes, (c) inductively coupled RF discharge

sections. The additional heat resulting from the gas com-pression by the turbo blower is removed by pre-coolersbefore the gas enters the discharge tube again.

Besides the fast gas flow there is also a permanentslow gas exchange with a fresh gas mixture, for exampleat a rate of 37 l (standard pressure and temperature) perhour in a ratio He : N2 : CO2 = 25 : 12 : 2 for a particularcommercial system. Longitudinally DC-excited lasersup to several kW laser power are available. Depending onthe laser power and the application, such laser systemscan emit a mixture of an ideal TEM00 mode and a hybridTEM∗

10 donut-shaped mode.

RF-Excited Gas DischargesRadio-frequency (RF) excitation of the gas dischargeis widely used in CO2 lasers. In this context RF meansa frequency range of roughly 1–500 MHz. Figure 11.127shows cross sections of typical geometries found in CO2lasers. In capacitively coupled RF (CCRF) dischargesthe electric field is applied by two metallic electrodeswith the gas volume in between. The electric field Ein the volume of the gas is sufficiently high to startand maintain a self-sustained glow discharge [11.1350].The RF voltage URF and the RF electric field E aretypically applied transversally to the laser beam. Thus,much lower voltages URF are required compared tolongitudinally excited discharges. URF is roughly 100 Vper mm discharge gap width d, depending largely on thegas pressure.

The electrodes can be electrically insulated from thedischarge by a dielectric material, for example a glass oralumina tube as shown in Fig. 11.127a. No direct contactof the gas with the electrodes is necessary. The currentflow is closed by the displacement current, which is∼ ∂E/∂t according to Maxwell’s laws.

Alternatively, the metallic electrodes can be placedin direct contact with the gas volume (Fig. 11.127b). Inthis case the surface regions on both electrode bound-aries have reduced free-electron densities and are calledion sheaths. In these ion sheaths no efficient excitationof vibrational states takes place. Power dissipated in thesheaths does not contribute to the laser power. The widthof the sheaths ds is inversely proportional to the excita-tion frequency ds ∼ 1/ f and is, for example, 0.35 mmat f = 125 MHz at a pressure of p = 90 hPa [11.1398].The electrode separation must be significantly largerthan this value to ensure sufficient excitation of the gasvolume. On the other hand, this reduces the efficiencyof diffusion cooling for heat removal from the gas to theelectrodes. For small gaps and efficient cooling, excita-tion frequencies in the 100 MHz range are often chosen.

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An optimized value for the electrode separation d istypically 2 mm under these conditions.

With insulated electrodes a so-called dielectric bar-rier discharge has been reported, which allows efficientexcitation at much lower frequencies around 1 MHzwhile still maintaining a stable discharge with effi-cient vibrational excitation and good cooling of thegas [11.1400].

The α-type RF discharge is a volume discharge withall ionizing and electron-collision processes taking placethroughout the entire gas volume. No ionizing processesat the electrodes are required. Such RF discharges arehence called electrodeless discharges. However, in thecase of non-isolated electrodes (Fig. 11.127b) there isalso an onset of a γ -type discharge at higher currentdensities, where surface ionizing processes play an im-portant role [11.1350]. This is usually not the desireddischarge type for laser excitation.

RF gas discharges can also be inductively cou-pled (Fig. 11.127c). The RF current IRF through a coilgenerates a time-varying magnetic field B which isaccompanied by an electric field E according tocurl E = −∂B/∂t. Inductively coupled discharges arenot commonly used for CO2 lasers because the gas cool-ing and electrical field geometry are not as favorable asfor CCRFs.

RF-excited discharges have several advantages. Oneadvantage is that such an electrodeless discharge shows

!9B""6

!+9 .-

.2

.2

*

.2 .2*

. .

+

+

+8

Fig. 11.128 Homogenization of the RF voltage distribution URF(z) on long electrodes with parallel inductors [11.1399]

no material sputtering from the cathode or anode andno resulting contamination of the laser gas or degra-dation by material ablation. Very high energy densitiescan be applied to the gas volume while still maintain-ing a stable glow discharge without arcing. Higher gaspressures can be used, typically around 100 hPa. Thus,more active medium and more laser power is availableper volume. The required voltages are lower than for DCexcitation. Insulation of electric wires and componentson the air-pressure side is easily done by sufficient largeair gaps or common dielectrics. The impedance of thegas discharge, which can be modeled as a strongly lossycapacitor [11.1401], can be matched to the RF genera-tor impedance with almost lossless reactive components(LC matching circuit). No ohmic ballast resistors arerequired to stabilize the discharge.

However, there are also some issues to be properlyconsidered in the design of RF-excited lasers. One ofthose is voltage and discharge homogeneity along theRF electrodes. At a frequency of 100 MHz the vac-uum wavelength λ0 is 3 m. Wave propagation effectsand waveguide theory must be applied if structureswith one dimension larger than λ0/10 are used. Thisis often the case for the length of typical RF-excitedwaveguide lasers. Therefore, regularly spaced inductorsparallel to the electrodes are used [11.1369,1402,1403].Figure 11.128 shows an example for an electric cir-cuit model of the electrodes with a gas discharge.

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The electrical wave propagation along the electrodesis described by a characteristic impedance Z and thepropagation constant of a lossy transmission line γ .Parallel inductors L i are used to compensate for thecapacity per unit length of the electrodes to homogenizethe voltage distribution. The higher the RF frequency,the more inductors in a shorter distance li must beused. Also shown is an impedance-matching networkto match the complex impedance of the laser Zlaserto the system impedance Z0 of 50 Ω of common RFgenerators.

Alternatively, an array of segmented electrodesmuch smaller than the wavelength can be used. Thiscan be problematic if the electrodes also act as wave-guides for the laser light. Mechanical misalignment ofthe electrode segments may then cause excessive opticallosses.

RF power generators are somewhat more com-plex than DC high-voltage supplies. They mustprovide roughly ten times the nominal laser poweras electrical power at RF frequencies. For low- andmedium-power CO2 lasers up to 500 W, solid-stateRF power generators based on semiconductor tran-sistors [mostly metal–oxide–semiconductor field-effecttransistors (MOSFETs)] are available at the required fre-quencies. Such RF generators can be integrated directlyinto the laser system housing with air or water coolingof the power supply.

For an RF power beyond this level electron-tube-based generators are required. Coaxial power tetrodes,as installed in amplitude-modulation (AM) short-waveradio transmitter stations, are used. This is a matureand reliable technique to generate RF powers of tensof kilowatts in the desired frequency range. Tubes cantolerate large reflected power levels from badly matchedloads without permanent damage. Electron tubes aremore robust against overvoltage and current transientsthan transistors. A drawback is that the tubes itself needa high-voltage DC power supply and that they dissipate

/8

!!

>""

!

Fig. 11.129a–c Cross sections ofsome waveguides for CO2 laserswith RF electrodes: (a) circular di-electric waveguide, (b) all-dielectricrectangular waveguide, (c) hybridmetal–dielectric waveguide

additional power for cathode heating. The lifetime ofsuch tubes is on the order of 10 000 hours. However,costs for replacement tubes are moderate.

Finally, issues of electromagnetic compatibility(EMC) have to be considered. Therefore, many RF-excited lasers operate in the ISM (Industrial Scientificand Medical) bands of 13.6 MHz, 27 MHz or 40.6 MHz,where less-stringent regulations apply. Systems at differ-ent frequencies, especially in the frequency-modulated(FM) radio bands around 100 MHz need to have prop-erly shielded housings, best made of all metal and withcontact strips for all service flaps of the laser housing.

Microwave excitation at a frequency of 2.45 GHzhas been studied intensively [11.1404–1406]. Power-ful magnetrons are very efficient microwave-generatingtubes at this frequency and are available at low cost,because of their use in mass-produced products suchas kitchen microwave ovens. However, the short wave-length makes it difficult to realize large-area dischargeswith good uniformity. Additionally, the power densityrequired to maintain a microwave-excited gas dischargeis larger than at common RF frequencies and tends tooverheat the gas. Thus, microwave-excited CO2 lasershave been realized mostly in pulsed operation [11.1407]or with a fast gas flow [11.1408]. Diffusion-cooledlaser systems have also been demonstrated [11.1409].In spite of the cost advantages of microwave gener-ation with magnetrons, the mentioned disadvantagesand the progress of RF-excited lasers has hindered thecommercial success of microwave excitation to date.

Waveguide LasersFor the laser design in Fig. 11.117 it would be desir-able to reduce the diameter of the glass tube to achievebetter cooling of the laser gas. This can introduce exces-sive optical losses by obstructing the laser beam insidethe resonator. Losses from such limiting apertures witha radius of a for common Gaussian TEM modes intwo-mirror stable resonators of length l are calculated

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in [11.1410]. These losses rise quickly if the Fresnelnumber N = a2/λl gets smaller than 1.

This problem can be overcome if the gas tube haswaveguide properties at the optical laser frequency.A comprehensive overview of waveguide laser designis given in [11.1402]. Figure 11.129 shows the princi-ple geometries of waveguides used in CO2 lasers. Thesimplest form is a circular hollow dielectric waveguide,which is theoretically described in [11.1411]. Circularmetallic waveguides are not an option because they willobviously shorten any electrical field required for thedischarge excitation. Rectangular waveguides can havedielectric walls on all four sides or one pair of dielec-tric and one pair of metallic walls [11.1412,1413]. Suchwaveguides support hybrid EHmn modes at optical fre-quencies with very low losses. These modes can bethought of as a superposition of electromagnetic wavespropagating at different angles to the waveguide axis,reflected by the walls at grazing incidence. Therefore,the materials used should have low absorption losses inthe infrared. Alumina or beryllium oxide ceramics area good choice for the dielectrics because of their highmechanical, chemical and thermal stability and theirhigh thermal conductivity for heat removal. Aluminumor gold-plated copper surfaces can be used as metallicwaveguide boundaries.

The lowest circular mode EH11 is very similar to theGaussian free-space mode TEM00, with a power overlapof 98%. Thus, waveguide lasers operating only in thelowest possible mode have excellent beam quality.

To build a waveguide laser, such waveguides arefilled with laser gas. The excitation of the gas dischargeis therefore often realized by a transversal RF discharge.A pair of metallic electrodes are placed on oppositesides of the dielectric waveguide (Figure 11.129a,b) orare part of the waveguide itself (Figure 11.129c). Theoptical resonator is realized by mirrors at both endsof the waveguide. Design rules for low-loss couplingand good mode discrimination are given in [11.1414].A good solution is a mirror in a small distance d fromthe waveguide, which is half the radius of curvature ρMof the mirror. Alternatively, plane mirrors close to thewaveguide can be used.

The power of a waveguide laser scales almost lin-early with the length. Up to 110 W of laser power permeter length have been reported [11.1415].

Many commercial CO2 lasers use waveguide tech-nology. Figure 11.130 shows the cross section ofa particular commercial waveguide laser [11.1416]. Theresonator light path is folded in a zigzag shape threetimes through multiple waveguide channels formed by

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Fig. 11.130 Cross section of a commercial RF excited waveguidelaser with a folded resonator light path

a ceramic plate and an aluminum profile. This resultsin a compact laser head design with high power, as thelaser power scales with the length of the waveguide. Thealuminum profile serves as the RF ground electrode, asone side of the optical waveguide, as a cooling plateand as an RF- and gas-tight housing for sealed opera-tion. The second RF electrode is on top of the ceramicplate. Regularly spaced spiral inductors homogenize theRF voltage distribution along the structure. Air or wa-ter cooling can be realized by the cooling channels onthe profile. An output power of 100 W is available froma compact package size [11.1417]. Q-switching can op-tionally be applied to realize pulsed lasers with highpeak power, especially for marking applications.

Another interesting waveguide laser structure isshown in Fig. 11.131 [11.1418]. The waveguide isformed by two pairs of metallic aluminum electrodesand two ridges from the metallic housing profile. RFpower is applied to both electrodes in a push–pull mode

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Fig. 11.131 Cross section of an all-metal waveguide laser

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with respect to the grounded housing. That means thatthe RF voltage amplitude on both electrodes is the samebut with a 180 phase shift relative to one other. Theelectrodes are separated from the ridges by small gapswhich enable gas circulation in the whole structure forcooling and heat removal. The electrodes are anodizedsuch that the oxide layer provides sufficient insulationto the grounded ridges of the housing. Additional in-sulators are used to hold the electrodes in the housing.A homogeneous gas discharge is formed in the middleof the structure with lateral dimensions of a few mm. Inspite of the small gaps, the structure can act as an opti-cal waveguide in all directions to form a laser beam withgood beam quality.

The advantage of this all-metal structure is thatno ceramic–metal joints are required. Such joints canpotentially cause failure because of the mismatched ther-mal expansion coefficients. Both the electrodes and thehousing can economically be manufactured by extrudedaluminum profiles. Heat is easily removed by conductionor passive thermal convection. Without ceramics withtheir high dielectric constants between the electrodes,the capacitive loading of the electrical waveguide is low.Fewer inductors are thus required for voltage homoge-nization. The metal housing can be sealed with metalbrazing to build long-living sealed-off lasers. A rangeof medium-power compact lasers up to 240 W is avail-able in this technology [11.1420], partially using foldedresonators for power scaling with the waveguide length.

The Slab LaserFigure 11.132 shows the structure of an RF-excitedslab laser. This design has been pioneered by severalgroups [11.1421–1424], accompanied by the well-known Tulip patent [11.1425]. It is formed by two

>"6

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Fig. 11.132 RF-excited slab laser with hybrid waveguide-unstableresonator [11.1419]

large-area metallic electrodes separated by a small gap.The length of the electrodes can be 1 m with a width ofsome tens of centimeters. A typical gap height is 2 mm.An RF voltage at a frequency in the range of 100 MHzis applied between the electrodes to start a gas dischargein the laser gas mixture between the electrodes. Theelectrodes are provided with channels for cooling water.Thermal energy dissipated in the gas discharge is effec-tively removed by conductive or diffusion cooling. Gasdischarges with very large areas can be operated stablywith this geometry.

In a plane perpendicular to the large-area surfacesthe electrode pair acts as a waveguide for optical fre-quencies. A low-order mode is easily obtained in thiswaveguide direction. In the plane parallel to the elec-trode surfaces free-space conditions are valid as thereare no limiting boundaries on the sides of the elec-trodes. With such broad dimensions this would resultin a strongly multimode beam profile in this direction, ifcommon stable resonators were applied. Therefore, anunstable resonator is used which has advantages for theextraction of high-quality beams from active laser me-dia with large transversal dimensions [11.1348, 1426].An unstable resonator is characterized by two mirrorswith radii of curvature ρ1 and ρ2 and a mirror separa-tion L that do noes fulfill the stability criterion for stableGaussian beam resonators:

0 ≤(

1− L

ρ1

)(1− L

ρ2

)≤ 1 (11.140)

Such a resonator does not reflect a Gaussian beambackwards into itself but rather uses the partial trans-mission of the beam on one mirror side as theoutput coupling. It can be shown that the beamprofile can have an almost diffraction-limited beamquality in the unstable-resonator plane. In the wave-guide plane, the mirrors can be almost flat. Bothmirrors can be made of solid copper and canbe cooled to withstand very high power densities.By a proper design, the coupling losses betweenthe mirrors and the slab waveguide can be mini-mized [11.1427].

The laser beam coming directly from this type ofresonator has an elliptical shape and has different diver-gence angles in the waveguide and the free-space plane.This is corrected by beam-shaping optics with at leastone cylindrical mirror. Additionally, a spatial mode fil-ter is used to eliminate shadows on the beam wings inthe free-space unstable-resonator direction. After that,the beam has a nearly circular shape with an excellentbeam quality of M2 = 1.1.

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The main advantage of this structure is that the laserpower can be scaled by the area Ae of the gas discharge,which extends in two dimensions. This is in contrast totube-type or waveguide lasers where the power scalesonly over one dimension, namely the tube length L .From thermodynamic calculations and based on ex-tending results from conventional waveguide lasers, themaximum laser power PL of a slab with an electrodearea Ae and an electrode separation de is approximatelygiven by:

PL = CAe

de, C ≈ 3

W mm

cm2 . (11.141)

By this area scaling over Ae, very compact and yet pow-erful lasers can be made. As with all large RF-excitedlaser systems, parallel inductors are used for homo-geneous RF voltage distribution along the electrodes.The efficient diffusion cooling requires no gas flow,which saves system and operating costs. Medium-powerlasers can be operated completely sealed and are avail-able at power levels up to 500 W [11.1429], partiallywith transistor-based RF generators. High-power slablasers operate typically with a periodic gas exchange,for example after 72 h of continuous operation, to com-pensate for residual gas leaks from the large structure.Commercial slab lasers with powers up to 8 kW in-tegrate the slab structure and the electron-tube-basedself-oscillating RF generator into one compact housingwith an integrated premix gas supply which lasts forapproximately 12 months of continuous use [11.1419].

RF-Excited Fast-Flow CO2 LasersThe basic principle of a fast-axial-gas-flow laser asdescribed in Sect. 11.4.2 can be combined with the ad-vantages of RF excitation. Figure 11.133 shows sucha laser system. The gas circulates through glass tubesdriven by a turbo radial blower in the center of the struc-ture. The optical resonator is quadratically folded. For aneven longer discharge length and more output power, theresonator can be folded in two planes with 16 dischargesections (Fig. 11.134). Optically stable resonator con-figurations are used. The RF discharge is capacitivelycoupled with eight pairs of electrodes, two on each side.The electrodes are mutually rotated by an angle of 45to ensure a homogeneous gain in the cross section, aver-aged over one resonator cycle. A heat exchanger coolsthe gas coming from the discharge section and a pre-cooler removes compression heat after the blower beforethe gas enters the discharge sections again. Typical gaspressures are in the range of 150 hPa. The power scalingwith mass flow according to (11.139) is also valid for this

/

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="+ 595

@8"6

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Fig. 11.133 RF-excited fast axial gas flow with a folded res-onator [11.1428]

system. Volume flow rates are up to 500 m3/h per kWextracted laser power with gas velocities of 100 m/s.Lasers with output powers up to 20 kW are commer-cially available while laser powers above 30 kW havebeen demonstrated experimentally [11.1430]. The beamquality is near a perfect diffraction-limited TEM00 modewith an M2 of 1.1 for high-quality series up to 4 kW.At the high-power end the 20 kW system still has anM2 value as low as 5. The wall-plug efficiency of thesesystems is 10% including the losses in the RF power sup-ply and the gas-circulating system. Electron-tube-basedhigh-power RF oscillators at an industrial ISM standardfrequency of 13.6 MHz are used.

Other RF-Excited CO2 Laser SystemsThere are many other commercial laser designs basedon RF excitation and with specific resonator or coolinggeometries. Some interesting designs will be discussedbriefly here.

*

" *"6

*

/

Fig. 11.134 Quadratically folded resonator in two planes[11.1428]

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744 Part C Coherent and Incoherent Light Sources

,"

/

46"

," " 9"

+6=9

Fig. 11.135 Folded free-space resonator between large-area slabelectrodes [11.1431]

Figure 11.135 shows a folded free-space resonatorbetween two large-area RF electrodes. Only three mir-rors are required for beam folding. The output couplerand the plane total reflector can even be integratedin one mirror with inhomogeneously reflecting coat-ings. Compact and robust sealed original equipmentmanufacturer (OEM) laser modules up to 120 W areavailable [11.1431].

A combination of large-area electrodes with a hy-brid stable–unstable resonator is described in [11.1433]and [11.1434]. In contrast to the slab laser with itsplane-parallel electrodes, the electrodes do not act asa waveguide here. Instead, they have a slightly V-shaped surface geometry along the beam propagationdirection between the mirrors to select the lowest-orderfree-space mode in a plane perpendicular to the elec-trodes, whereas an unstable resonator is used in the

!

!"

5

@8"6

95"

+5

!

55

Fig. 11.136 RF-excited laser with fast transversal gasflow [11.1432]

free-space plane again. Sealed-off OEM modules in anall-metal design and a maximum power of 400 W canbe realized [11.1435].

Large-area RF-excited gas discharges can also becombined with a fast gas flow between the electrodeplates transversally to the optical resonator and the laserbeam (Fig. 11.136). The gas flow is driven by a tangentialblower integrated into the large-diameter cylindrical gasvessel. A folded resonator is used to extract the energyfrom the large active medium. Such lasers can produce8 kW of laser power from a very compact footprint. Thebeam quality with M2 ≈ 5 is still well suited for weldingand surface-treatment applications.

Coaxial laser systems with an annular gas dischargeas shown in Fig. 11.137 can be thought of as slab-typelasers with an electrode pair rolled up to form coax-ial tubes [11.1436]. RF power is applied to the innertube, while the outer tube is grounded. This is similarto a coaxial cable, but with an inner-conductor diameterof approximately 10 cm and a discharge gap width of7 mm, for example. In this design, the electrodes haveno optical waveguiding properties, thus such lasers areless susceptible to power and beam-quality variationsunder mechanical and thermal stress. The resonator isformed by a helical mirror and an axicon mirror, bendingthe beam to the opposite side of the tube while slightlyshifting the azimuthal angle after each pass [11.1437].In the radial direction the optical resonator configu-ration is stable, while in the azimuthal direction anunstable resonator is used. The laser is diffusion-cooledby water-cooled electrode tubes. The laser can be op-erated quasi-sealed-off with a regularly scheduled gasexchange. Compact and robust lasers with 2 kW powercan be realized [11.1438]. They are well suited formounting on moving systems such as robots withoutbending mirrors for beam steering.

Pulsed TEA LasersAll the laser designs discussed so far are continuous-wave (CW) lasers by principle. Their power can bemodulated by the electrical excitation power with fre-quencies of a few kHz for DC excitation and up to100 kHz for some RF-excited lasers. In contrast, thetranversal-excited atmospheric-pressure (TEA) laser isa CO2 laser that can inherently operate only in pulsedmode, but is capable of generation very high pulse en-ergies and peak powers that are not possible with CWCO2 lasers. This is achieved by a high gas pressure, atatmospheric pressure or above. A homogeneous DC dis-charge can only be operated for a short time of 1µs underthese conditions before filamentation and arcing would

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start. The required voltages at this pressure are in theorder of magnitude of 100 kV per meter. Longitudinaldischarges would need voltages that are not practical andwould not produce stable discharges. Therefore, pulsedtransversal discharges are used. Figure 11.138 showsa sketch of a simple TEA laser circuit. A high-voltagepower supply (HV) charges a storage capacitor C slowlyvia the resistors RV1 and RV2. When the maximum volt-age at the capacitor is reached, a fast high-voltage switch,which is also capable of carrying high currents, is trig-gered to close the circuit quickly. The capacitor voltageis applied to the electrode pair, where a short but intensedischarge transversal to the resonator axis and the laserbeam starts. The discharge self-terminates when all theenergy stored in the capacitor is dissipated. Afterwardsthe high-voltage switch resets itself to an open state andcharging of the capacitor starts again before the nextpulse can be triggered.

With TEA lasers pulse energies in the range of10 J per liter of active volume and per bar of pres-sure can be achieved, with peak intensities in the MWrange [11.1440, 1441]. Pulse repetition rates are a few10 Hz for high-power systems and kHz for mini-TEAlasers [11.1442].

Critical design issues are the electrode profile tomaintain a stable and homogeneous discharge in thecross section and also over the length of the elec-trodes. Often pre-ionization techniques by additionalelectrodes, UV light from corona discharges or evenelectron beams are therefore applied. The high-voltageswitch is also a crucial part. Typically, triggered sparkgaps or thyratrons are used, which must be replacedafter a given number of pulses. Recently, progressin semiconductor technology has made it possible toreplace these vacuum tubes by solid-state switches.A device called a silicon controlled rectifier (SCR) iscapable of switching high currents very fast. In com-bination with a step-up transformer and a saturableinductor/capacitor ladder network the high-voltage leveland the fast rise time required for the electrode voltagecan be achieved.

Much research was done on TEA lasers in the1980s, especially in the fields of plasma physics, LIDARand military applications. Partially due to the suc-cess of high-power fast-flow and RF-excited systems,only a few commercial TEA lasers are available. Sys-tems with pulse energies of 2.4 J at a repetition rateof 125 Hz are available, for example. Applications in-clude non-metal processing such as marking and paintor rubber-compound stripping from surfaces in the au-tomotive industry [11.1443].

E455

@"

08"

B 2"0 "6 2"

Fig. 11.137 Coaxial laser with an annular gas discharge [11.1439]

Wavelength Selection and Tunable CO2 LasersWithout any wavelength-selective elements, a CO2 laseremits at wavelengths around 10.6 µm in the vicinity ofthe 10P(20) line. Some commercial lasers are optionallyavailable at the wavelengths of the strongest lines of theother bands, for example at 9.3 µm, 9.6 µm or 10.3 µm(Fig. 11.122). This is done by weakly wavelength-selective elements such as mirror coatings to select oneof the four regular emission bands where the strongestline will start to oscillate. For example, this is interest-ing for plastics machining where materials can showa strong variation of absorption over wavelength.

For precise selection of a particular roto-vibrationaltransition, a diffraction grating is used. Figure 11.139shows a DC-excited tunable laser in the Littrow config-uration. Depending on the angle α of the incoming beamwith respect to the surface normal of the grating, onlya light beam at a specific wavelength λL is reflected:

λL = 2Λ sinα , (11.142)

withΛ being the grating period. For this particular wave-length λL the grating acts as a plane reflector on axis tothe second resonator mirror. This wavelength is selected

,"

,"

>"6

/8

45"6

+

/

/@<

Fig. 11.138 Excitation circuit for a basic TEA laser

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746 Part C Coherent and Incoherent Light Sources

+>.

>"6

/5

>99"

46" "

/

Fig. 11.139 Tunable CO2 laser inLittrow configuration

Table 11.41 Emission range of a tunable CO2 laser [11.1444]

Band Wavelengths (µm) Linesmin – center – max min – center – max

9R 9.158 – 9.271 – 9.367 9R(44) – 9R(20) – 9R(4)

9P 9.443 – 9.552 – 9.836 9P(6) – 9P(20) – 9P(50)

10R 10.095 – 10.247 – 10.365 10R(46) – 10R(20) – 10R(4)

10P 10.441 – 10.591 – 10.936 10P(4) – 10P(20) – 10R(50)

to lase. Other wavelengths λx , if present, would be re-flected at a different angle not parallel to the resonatoraxis. They would suffer significant losses from the aper-ture of the laser tube, which would prevent them fromlasing. By rotation of the grating the angle α can be var-ied to select different roto-vibrational transitions one byone. Such gratings are mostly made of copper or steelwith precision-ruled grooves and a gold coating to en-hance IR reflectivity. The grating period Λ is chosen tobe slightly smaller than the laser wavelength in order tosuppress higher diffraction orders. The groove geometrycan be adjusted such that the reflectivity in the desiredminus-first diffraction order at a given wavelength ismaximal (blazed gratings, blazing wavelength).

Table 11.41 shows for example the tuning range ofa commercial tunable CO2 laser. Note that, because ofthe separation of the rotational lines in the range of30–90 GHz and the pressure-broadened line width ofsome 100 MHz, continuous tuning over the entire rangeis not possible. Within one line width, fine-tuning can bedone by changing the resonator length with a piezoelec-tric element. An output power of 180 W can be achievedfor the strongest lines and more than 20 W on the weaklines at the band edges, for example.

Applications include scientific spectroscopy, indus-trial trace-gas monitoring by absorption spectroscopy oroptically pumping of far-infrared molecular gas lasers.By using gas mixtures enriched with special isotopo-logues, for example 13CO2, emission lines at additionalwavelengths can be realized. Tables of such wavelengthsare given, for example, in [11.1347].

Besides longitudinally DC-excited lasers, RF-excited waveguide lasers are also good candidatesfor being tuned by a standard Littrow grating. Tun-able slab-type lasers with their broad active mediumrequire special grating structures for the extrac-tion of a low-order mode. This can be realizedby apodized Littrow gratings where the splittingratio between the diffraction orders m = 0 andm = −1 is variable over direction of the grat-ing [11.1445]. Such gratings can act as an outputcoupler with spatially variably reflectivity for beamforming and as line-selective element for wavelengthtuning at the same time. They have been realizedon copper mirror substrates with photolithographyand a microgalvanic process as well as on siliconsubstrates with photolithography and anisotropic etch-ing [11.1446].

11.5 Ion Lasers

Ion lasers became a success very soon after theirinvention in the 1960s. In the following decades,low-power air-cooled argon lasers were made in thetens of thousands for use in printing, color separa-

tion, and medical technology. Mid-power ion laserswere a staple of the entertainment industry, used inlight shows, special effects, and for holography, aswell as in disc mastering. Large-frame ion lasers

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were ubiquitous in well-equipped chemistry and bi-ology laboratories around the world. Ion lasers werethe flagship products of the major commercial lasermanufacturers.

But even in their heyday, alternatives to ion laserswere eagerly sought. There were several reasons for this,including early problems with reliability and lifetime, aswell as high cost, but the major dissatisfactions resultedfrom the fundamental low efficiency of the devices. Thispoor efficiency meant ion lasers were large, of consid-erable weight, and needed substantial power suppliesdrawing large utility loads. They also had to be cooledaggressively either by air or water.

With the development of solid-state lasers inthe 1990s, in particular diode-pumped lasers withfrequency-doubled green output at 532 nm, many ex-pected ion-laser technology to become obsolete. Thishas not happened. While not nearly at the levels theyexperienced in the 1980s, sales continue to be steady,ion lasers are still found in their traditional applications,and are a significant contributor to the bottom line ofa number of companies, both as new systems and in thereplacement market. While there are fewer types of newion lasers available now, these systems still vary con-siderably in performance, size, utility requirements, andcost.

An ion-laser system consists of a power supply, con-trol electronics, and the laser head. The laser head itselfcontains the plasma tube placed within a resonator (orcavity) structure. Control of the laser is typically throughan interface included in the power supply, which itselfcan vary considerably in size, complexity, and coolingrequirements.

Small ion lasers are air cooled, require only standardwall-plug electrical service, have a laser head roughlythe size of a toaster, and are most commonly used toproduce only a single blue–green output at 488 nm inthe range 5–100 mW. When operated in the low-currentrange, these lasers can be expected to last for years ina stable environment. Tube replacement when necessarycan be performed by the user and is relatively quick andinexpensive.

The greatest variety of product offerings is in themid-range of performance of 1–5 W of laser out-put. These systems range from simpler models basedon the original ion technology, with air cooling andceramic plasma tubes, to specialized systems withsingle-wavelength output intended for specific indus-trial applications, to only slightly smaller versions ofthe sophisticated large-frame systems used in advancedlaboratories.

A high-performance ion system has a laser headabout a meter and half long and can weigh 100 kg. Itmay require 70 A per phase of three-phase electricalservice at 480 V and in addition a cooling-water flowof several liters per minute. But in compensation, theselasers are capable of producing many watts of continu-ous laser power in a range of selectable wavelengths notavailable from any other commercial laser system.

Wavelengths available from ion lasers extend fromthe infrared well into the ultraviolet, with argon, the mostcommon type of ion laser, emitting light primarily in theblue–green to ultraviolet range [11.1447,1448]. Specificwavelength output is determined either by the choice ofthe coating for the laser output mirror, or by tuning to thedesired wavelength using a Littrow prism placed in thelaser resonator. The section below discussing the laserresonator will further describe control of the spectralcharacteristics of the ion-laser output.

Krypton is not nearly as popular an ion laser as ar-gon, in part because of its lower power, but it remainscommercially viable because, like argon, it offers signif-icant levels of continuous output power at wavelengthsnot easily achieved by other means. For example, morethan 1 W CW is available from a single line in the yellow,a color region where laser sources are scarce, and linesin the violet and ultraviolet have important applicationsas well.

The output power of an ion laser depends on twosets of variables. One set derives from the design of thelaser resonator, and includes, for example, the cross-sectional area of the laser beam. This set of variableswill be discussed in the section which describes the res-onator. The other set of variables that determines laserpower results from the physics of the laser process inthe plasma created from the noble gas. These variablesinclude the total light amplification available from theplasma.

11.5.1 Ion-Laser Physics

An energy-level diagram is used to show the wave-lengths available from a laser material that has beenexcited into a condition of population inversion. A sim-plified version illustrating the transitions in singlyionized argon that produce visible laser light is shownin Fig. 11.140 [11.1449]. The transitions shown resultfrom energetic electrons that have decayed to the 4p up-per laser level from a still higher level not shown inthe figure. From the upper laser level, the electrons arestimulated to fall into the 4s states. The most importantindividual transitions, which result in the highest output

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Fig. 11.140 The visible laser transitions of ionized argon(wavelengths in nanometers)

power, are those for 488.0 nm, a blue–green color, and514.5 nm in the green.

Having many upper laser levels transitioning to onlytwo lower levels raises the possibility of competitionfor the lower states, which can limit laser output. Inthe case of argon, however, the lower states depopulateto the ground state very rapidly, avoiding this potentialbottleneck. The lifetime of an electron in one of theupper 4p states is about 10 ns, while the lower 4s statesmake their transition to the ground state in a favorable0.5 ns. With a proper choice of mirror coating, all-linesvisible operation can produce laser power from most ofthese visible lines simultaneously, and the power presentin any one of the lines will not be much less than thatobtained from single-line operation.

An energy-level diagram can also illustrate prob-lems or requirements in the design of a specific lasersystem. Figure 11.141 shows the 4p and 4s laser lev-els of ionized singly argon in relation to the argonground state. The energy needed to ionize the argonto start the population inversion is about 16 eV, andan additional 20 eV is required to reach the states thatdecay to the upper laser level. Comparing this total en-ergy input of 36 eV to the energy emitted by the lasertransition itself, about 2 eV, it is evident that the effi-ciency of an argon ion laser will be low. In addition,the temperature of the gas discharge needed to produceelectrons with sufficient kinetic energy to achieve therequired level of ionization is in the vicinity of 3000 K.This, combined with the low efficiency, indicates thatheat management is a primary concern in ion-laserdesign [11.1450].

Ion lasers use an electric discharge to energize thegas. As can be seen in Fig. 11.141, two collisions withfree electrons are required to excite an electron in an

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Fig. 11.141 The two-step ion-laser excitation. One electroncollision ionizes neutral argon and a second pumps the ionto an excited state from which it decays to the upper laserlevel

argon ion to the upper level of visible lasing transitions.The gain available from the ionized gas therefore variesas the square of the current density, up to the point wherethe gain saturates (i. e., other gas-phase processes beginto limit the gain) [11.1451].

The optimum gas pressure that produces the max-imum laser output is a balance between competingfactors. High pressure provides a greater number ofpotential ions to stimulate laser emission, while lowpressure allows the voltage drop along the plasma moretime to accelerate the free electrons to a greater energybetween collisions. Different wavelengths have differ-ent optimum pressures, and the final choice for a laserdesigned to operate at multiple wavelengths is a com-promise. In general the optimum is in the low-pressureregime. Argon plasma tubes have a fill pressure on theorder of 1 Torr [11.1452].

Performance is improved by a magnetic field di-rected along the length of the plasma discharge, whichhelps confine the discharge towards the center of theplasma tube [11.1450]. The benefit is greatest for lasersoperating at high current. Again, different wavelengthshave different optima, and a compromise value ofmagnetic field strength is employed. Air-cooled laserstypically do not use magnets, as the benefit of a magneticfield for low powers does not justify the added expense,

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Lasers and Coherent Light Sources 11.5 Ion Lasers 749

and the addition of magnets would complicate the flowof cooling air.

If the argon is doubly ionized (i. e., stripped of twoelectrons), a new set of transitions become available thatproduce ultraviolet laser light [11.1453]. The groundstate of Ar2+ is about 43 eV above that of the neutralatom, compared to the Ar+ ground-state difference of16 eV mentioned, so operating in the ultraviolet can beexpected to be even less efficient. Once above a higherthreshold current, UV output rises more rapidly than thevisible output does.

The energy diagram for krypton is similar to thatfor argon, with laser transitions shifted to somewhatlower photon energies in the visible from the upper5p to the lower 5s bands [11.1454]. Important linesare at 647.1 nm in the red, corresponding to the green514.5 nm argon line, and at 568.2 nm in the yellow, cor-responding to the blue–green line at 488 nm in argon.Unlike argon, however, in krypton these strong lines doexperience competition for the same lower state, andso all-lines operation in krypton does not produce thepower that might be expected judging from single-lineoutput.

Argon and krypton can be mixed together in the samelaser tube to produce a laser offering a dazzling variety ofvisible wavelengths. A compromise must be reached inthe parameters such as fill pressure and magnetic fieldthat allows for the differences in the optimization foreach type of laser. Such a mixed-gas laser using broad-band mirror coatings has even been used to producea white-light laser, but the argon–krypton mixture be-comes unbalanced fairly quickly due to the difference ingas sputtering rates, and the white-light balance is lost as

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Fig. 11.142 Design of an air-cooled plasma tube

the component wavelengths change in relative strengthof output.

11.5.2 Plasma Tube Design

The heart of the ion laser is the plasma tube, which am-plifies laser wavelengths available from a low-pressurenoble gas by confining a high-current discharge withina narrow bore. The demands placed on the tube struc-ture are extreme, resulting from the need to confine andsustain a discharge as high as 70 A for hours at a time.The bore of the tube must withstand the bombardmentof high-energy ions (sputtering) and the heat from thedischarge, which can exceed 50 kW under some oper-ating conditions. This heat must be efficiently removedfrom the tube or tube components will melt or rupture.

Early tube designs relied on cylinders of the ce-ramic beryllia, or BeO, about 1–2 cm in diameter witha centrally drilled hole of about 1 mm to form the bore.Beryllia cylinders about 10 to 15 cm long are still inuse today in lower-power ion lasers, particularly for air-cooled argon. Figure 11.142 shows a cross section ofan air-cooled plasma tube. However beryllia proved in-capable of sufficiently withstanding the sputtering fromthe plasmas of high-power or ultraviolet ion lasers, andalternative designs were sought.

Years of intensive research and development haveresulted in sophisticated designs for high-current tubesthat replace beryllia with a tube structured of compos-ite materials [11.1452,1455]. These complex (and moreexpensive) designs can withstand high current densitiesfor an extended period, thus making practical the op-eration of ion lasers at higher powers or less-efficient

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Fig. 11.143 Design of a water-cooled plasma tube

wavelengths. The description that follows is based onthe segmented-bore technology used by Spectra-Physicsin the manufacture of its water-cooled ion lasers. Theoverall design is illustrated in Fig. 11.143.

The exterior of the plasma tube is a long, hollowthin-walled ceramic pipe or sleeve a few centimeters indiameter. Inside this sleeve, spaced about one centimeterapart, are many thin discs brazed to the interior of theceramic. The discs are copper with a central tungstenring with a hole a few millimeters in diameter. The boreof the plasma tube is defined by the diameter of the holein the tungsten segments (Fig. 11.144).

The tungsten bore segments are extremely resistantto sputtering action and can withstand the high tempera-tures of the discharge. The copper discs to which they arebonded conduct the heat of the discharge from the tung-sten to the thin ceramic wall of the exterior sleeve, whichis bathed outside by a cooling water flow of several litersper minute. The flow of this cooling water is designedto prevent local thermal stresses affecting the tube.

The plasma tube must ensure uniform gas pressurealong the length of the bore. In arc discharges in low-pressure gases, the ions lose their momentum in frequentcollisions with the tube walls. In contrast, the electronslose little momentum, so an imbalance is created in thenet momentum in the gas. As a result, the neutral gasatoms are driven to the anode end of the plasma tube. Toequalize the gas pressure along the length of the tube,holes in the copper discs provide a return path to thecathode end, which also allows the copper discs to helpcool the gas in the return flow.

The design of the bore where it opens to the cath-ode, the so-called throat region, is critically importantto achieve good laser lifetime. Damage to the bore fromsputtering is maximum in the throat. The diameter of thebore segments (or, in lower-power tubes, the conicallyshaped opening drilled in the beryllia to form the throat)is tapered to match the contour of the electrical field asit enters the bore.

The design of the cathode element itself is also crit-ical. The cathode is helical and is made from a spongymaterial formed of tungsten powder. It may also in-clude some admixture of calcium, aluminum, barium,and strontium. These additives reduce the work functionof the cathode surface so that the temperature needed toproduce electron emission is reduced [11.1455].

The efficiency of laser amplification is a function ofthe gas pressure in the discharge. However ionic sputter-ing not only erodes the tube materials, but also over timeembeds a significant amount of gas in the tube walls andcomponents. Low-power lasers compensate for this ef-fect by widening the body of the plasma tube to forma reservoir, so that the gas lost from sputtering action isa small fraction of the total volume in the tube. Mid- andhigh-power designs incorporate an active fill system toreplenish the gas as it is lost.

To determine when the gas must be replenished,a microprocessor compares the operating voltage of thelaser to values stored in a look-up table, taking intoaccount the current, magnetic field and warm-up time.When the voltage indicates that the tube pressure is low,a high-pressure reservoir automatically injects a smallquantity of gas into a plenum. A separate valve then

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Fig. 11.144 Seg-mented boredesign

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Lasers and Coherent Light Sources 11.5 Ion Lasers 751

opens to allow the plenum to add a precise quantity ofgas to the plasma tube. Since this fill process brieflydisturbs laser operation, the fill system may be man-ually disabled for short periods to ensure that criticalexperiments are not effected.

For air-cooled lasers, which do not employ a mag-netic field, copper cooling fins are brazed onto theexterior of the beryllia cylinder that forms the plasmatube. Cooling air is forced over the fins at a rate ofa few hundred cubic feet per minute (about 10 m3 perminute). Care must be taken that heat removal is evenlydistributed so that the tube does not warp and move thelaser beam (the laser output mirror is attached directlyto the tube), and the air flow should not induce vibra-tions in the fins which might translate into motion of thebeam.

For larger lasers, the entire plasma tube is encasedin a large electromagnet which also forms the confiningwall for the water flowing next to the tube. The field cho-sen for the magnet depends on the diameter of the boreand the details of how the laser is operated, but is usuallyin the neighborhood of 1000 G. The magnetic field is par-ticularly important for ultraviolet output, which requiresstronger magnetic fields for optimum performance.

The ends of the plasma tube are sealed with opticsthat allow the laser light to exit the tube. These opticsare either laser mirrors with dielectric coatings, used inessentially all low-power and some mid-power lasers,or quartz windows oriented at Brewster’s angle. Tubesare constructed with Brewster windows to polarize theiroutput, generally oriented vertically. Mirror-sealed tubeswill have an internal optic to achieve polarization.

The light emitted from the discharge of high-currenttubes poses a challenge for optics. The transitions fromthe lower laser levels back to the ground state of neu-tral argon shown in Fig. 11.141 emits a highly energeticphoton around 80 nm. This vacuum UV radiation is very

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Fig. 11.145 Open resonator design

deleterious to optical materials. The coatings on the mir-rors that seal low-power and some mid-power tubes aremade of many alternating layers of dielectric materials,such as silica and alumina, that are capable of with-standing this radiation. Brewster windows used to sealmid-power lasers are typically made of fused silica.

The levels of the vacuum ultraviolet of high-currentdischarges present a greater problem for the Brewsterwindow seals of these tubes [11.1456,1457]. Fused silicacannot be used because, when exposed to these radi-ation levels, it forms color-center defects which thenabsorb even more radiation in a runaway effect that willdegrade laser action. To meet the challenge of sealinghigh-current plasma tubes, manufacturers use crystallinequartz, carefully cut and oriented on the tube so that thecrystalline axis of the two windows on each end of thetube are in alignment.

11.5.3 Ion-Laser Resonators

To complete the ion-laser structure, the plasma tubemust be placed between suitable mirrors that providelight feedback for amplification by the discharge. Thesemirrors are a high reflector at one end of the laser,and a partial transmitter or output coupler that allowsa portion of the circulating light to exit the laser. Thesemirrors, together with the structure that holds them inposition, form the laser resonator [11.1458]. For smallion lasers and some mid-frame systems, this resonatorstructure is very simple: the mirrors are bonded onto theplasma tube itself.

Air-cooled ion lasers sacrifice flexibility for sim-plicity. Except for the power level, their outputcharacteristics cannot be changed. A mirror position isfixed at manufacture by the plastic deformation of thethin metal tube to which the mirror is bonded. The out-put wavelength, mode structure of the beam, and so on

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remain fixed. For many applications this is acceptable,even desirable, but other applications benefit greatly byretaining the flexibility of output that is a hallmark ofthe ion laser.

The external or open resonator commonly employedwith mid- and large-frame systems retains this flexibil-ity. This design requires the use of Brewster-windowseals for the plasma tube, as discussed in the preced-ing section. The resonator mirrors are held in place bya rigid structure that frames the plasma tube (hence theuse of mid-frame or large frame as an indication oflaser size). While it outwardly appears simple, the de-sign of the resonator is critical to the performance of thelaser.

The resonator frame, as shown in Fig. 11.145, mustkeep the mirrors aligned to each other and pointingthrough the laser bore within a very tight tolerance overthe entire range of operating and environmental condi-tions. The choice of materials affects the stability of theresonator. The ideal material has both a low thermal ex-pansion coefficient and a high ability to distribute heatevenly. Graphite and low-expansion compounds such asiron–nickel alloys are typically used for the long rodsthat form the length of the frame.

Stability also depends on the rigidity of the res-onator. Jitter impressed on the laser output due to themicrophonic movement of the mirrors can be caused bycooling-water flow, vibration of the resonator structure,and acoustic noise. Isolation of the resonator from boththe plasma tube, the magnet, and the cover of the laserhead helps reduce jitter.

The mechanical design of the structure is also criticalto stability. The most stable configuration is an arrange-ment of three resonator rods in an equilateral triangle.As a practical matter, this ideal structure does not leavesufficient room for the plasma tube and magnet. As oneof the angles of the resonator triangle increases, the re-sistance to flexure is reduced. The closer the design canbe to the ideal equilateral triangle, the better will be the

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Fig. 11.146 Active mirror positioning allows resonators to extractmore power from the discharge

mechanical stability provided for stable power and beampointing [11.1459].

A significant advantage that modern large-framelasers have over their forerunners is the use of ac-tive mirror positioning. The output coupler is mountedon three-point piezoelectric positioner that providessmall corrections to the mirror alignment in responseto changes in laser output power, most commonly ex-perienced when the laser is warming up. This activeresonator also allows for a more aggressive opticaldesign to extract power from the plasma tube.

With active mirror positioning, the cross-sectionalarea of the beam, and therefore the volume of the plasmadischarge used to provide laser amplification, can be in-creased by enlarging the bore and using a longer-radiusoutput coupler (Fig. 11.146). This is especially benefi-cial for wavelengths with lower gain. However activecontrol alone cannot provide all of the stability requiredby this design: the use of long-radius optics requires anextremely rigid resonator structure.

The open resonator allows the user to employ dif-ferent mirrors to obtain different wavelength output. Toobtain all-lines argon-ion output in the visible, for ex-ample, mirrors are used with coatings that reflect overabout 70 nm in the blue and green. A prism on a rotat-able mount placed before the high reflector allows thelaser to be tuned to single-line output throughout theall-lines range. (The dispersion of the prism directs onlyone wavelength at a time to the high reflector.) A simi-lar arrangement using different optics enables the sameoperation for all-lines ultraviolet output.

The open resonator also allows the spatial mode ofthe beam to be controlled. An adjustable aperture placedbetween the Brewster window and the output coupler,where the beam is large, provides a variable amount ofloss to the edges of the beam while it is inside the res-onator. The aperture, when opened to allow the highestpower output, will permit a number of transverse modes,or if reduced will result in TEM00 mode operation. Anaperture diameter between 1.5 and 2 times the TEM00mode diameter is used to achieve this [11.1460].

The spectral content of the laser output may also bechanged by insertion of an etalon into the laser resonator.An etalon is a thin optical cavity, such as a plate of glass.When inserted into the resonator, the internal reflectionsfrom its surfaces act to narrow the frequency content ofthe laser output, its line width. The resonator producesstanding waves of light between its mirrors, called longi-tudinal modes, many of which have frequencies that fallwithin the bandwidth of the laser amplifier. The etalonacts as a bandpass filter that introduces variable loss and

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13(4

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6

Fig. 11.147 Using an etalon to narrow the spectrum of thelaser output

favors oscillation on only one of the longitudinal modes(Fig. 11.147).

Active mirror positioning may also be used to stabi-lize the line width further, resulting in an even narrowerspectral output. The coherence length – the distance overwhich the output beam maintains a fixed phase relation-ship – is inversely proportional to the line width. Whenthe laser output is changed from single line to singlefrequency, the coherence length increases from about50 mm to 20 m.

A drawback of the open resonator is that the spacebetween the mirrors and the Brewster windows is notvacuum-sealed. Laser performance strongly depends onkeeping these optical surfaces clean. In addition, thevacuum UV light from the arc discharge, transmittedby the Brewster windows, converts oxygen to ozone,which is deleterious to the laser optics. This intracavityspace must be sealed and its atmosphere controlled tomaintain cleanliness and avoid optical damage. Lasersproducing short-wavelength UV may employ a nitrogenpurge to avoid ozone production. Longer-wavelengthlasers can use a simpler technique of placing a catalyst inthe enclosed space that converts ozone back into oxygen.

11.5.4 Electronics

A variety of electronic subsystems are required to op-erate an ion laser, as shown in the block diagram of

a large-frame system in Fig. 11.148. Typically packagedin the laser head along with the plasma tube are the startcircuit and the light pick-off. Lasers that are not soldas OEM components for a larger system are required tohave an interlock switch that will reliably terminate laseroperation if the head cover is opened. For water-cooledlasers, the laser head will usually also contain the flow-and temperature-control circuits for the cooling water,and the fill circuitry to monitor and replenish the gas inthe plasma tube.

The start circuit provides a high-voltage spark to ini-tiate the discharge. Typically, a pulse of several kV isinjected in series with the tube by a pulse transformer ora spark-gap circuit. The light pick-off is the sensor forthe feedback loop that measures and regulates the laseroutput power, and usually consists of a beam splitterwith a silicon photodetector. Since an ion laser may becapable of producing a wide spectrum of wavelengthsover a considerable range of powers, multiple pream-plifier ranges are usually used in the feedback loop, aswell as wavelength sensitivity correction color filters orelectronic gain modification.

The power supply that drives the plasma tube isalmost always packaged as a separate unit. This is be-cause the supply itself is roughly the same volume as thelaser head, and separating it from the laser allows muchgreater flexibility in positioning the laser system. Thedrive current for the plasma tube is delivered to the laserhead through a robust umbilical. Since the power supplyalso serves as the control interface for the laser, cables

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Fig. 11.148 Block diagram of a large-frame ion-laser system

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Table 11.42 Ion-laser wavelengths

Powers available in a type of laser (W) (not all wavelengths are available simultaneously)

Available Lasing species Large-frame Mid-frame Large air-cooled Small air-cooledwavelengths (nm) water cooled (W) water cooled (W) (W) (W)

275.4 Argon 0.375

300.3 Argon 0.5

302.4 Argon 0.5

305.5 Argon 0.17

333.6 Argon 0.4

334.5 Argon 0.8

335.8 Argon 0.8

350.7 Krypton 2.0 0.3

351.1 Argon 1.5 0.18

351.4 Argon 0.5 0.06

356.4 Krypton 0.5 0.12

363.8 Argon 2.0 0.25

406.7 Krypton 1.2 0.22

413.1 Krypton 2.5 0.3

415.4 Krypton 0.35

454.5 Argon 0.8 0.18 0.005 0.002

457.9 Argon 1.5 0.55 0.015 0.005

465.8 Argon 0.8 0.24 0.015 0.005

468.0 Krypton 0.5 0.1

472.7 Argon 1.3 0.32

476.2 Krypton 0.4 0.1

476.5 Argon 3.0 1.0 0.025 0.008

482.5 Krypton 0.4 0.05

488.0 Argon 8.0 2.5 0.1 0.03

496.5 Argon 3.0 1.0 0.03 0.012

501.7 Argon 1.8 0.64 0.015 0.005

514.5 Argon 10.0 3.2 0.1 0.025

528.7 Argon 1.8 0.55 0.01

530.9 Krypton 1.5 0.25

568.2 Krypton 0.6 0.25 0.02

631.2 Krypton 0.2

647.1 Krypton 3.0 0.8 0.015

676.4 Krypton 0.9 0.15

752.5 Krypton 1.2 0.15

793.1 Krypton 0.1

799.3 Krypton 0.2 0.03

that carry monitoring and control signals are typicallybundled in the umbilical as well.

Ion-laser power supplies show the same range ofvariation in size and design as the laser heads. The powersupplies of small air-cooled argon units can be as smallabout 15 cm on a side, while supplies that drive large-frame systems are roughly 0.7 m on a side and weigh

almost 100 kg. These larger units may require coolingwater. Table 11.43 displays typical service requirementsand loads for the three main classes of ion lasers.

The electrical characteristics of the arc dischargeare such that it essentially requires a constant-currentsupply in order to be sustained [11.1458]. The heart ofan ion-laser power supply is the plasma current regulator.

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Table 11.43 Power supply characteristics of typical ion-laser systems

Ion- Max- Electrical Plasma Dischargelaser imum input dis- powertype laser charge require-

power ment

Air- 20 mW 115 VAC 90 V 0.7 kW

cooled single phase 8 A

Small 5 W 208 VAC 240 V 13 kW

frame three phase 55 A

Large 25 W 480 VAC 550 V 36 kW

frame three phase 65 A

The plasma current needs to be well regulated in orderto prevent line ripple and other variations from beingimpressed on the laser output. The plasma current maybe regulated by a linear transistor passbank, or moreefficiently by various switching-type power supplies.

In the patented regulator used by Spectra-Physics,a switched resistor regulator consisting of a pulse-widthmodulator (PWM) controlling a switch transistor is con-nected to a low-impedance high-power water-cooledresistor. This resistor is connected to the plasma load.The PWM varies the percent of the time that the tran-sistor is turned on and the resistor is conducting current.A parallel capacitor provides a current path to the loadwhen the transistor is off. This regulator looks likea variable resistor that can vary from infinity down tothe minimum resistance of the circuit. A shunt regula-tor is used to provide ripple rejection and high-speedsmall-signal regulation.

The power supply must power the electromagnetwhen an axial magnetic field is employed. The electro-magnet requires about 10–25% of the power suppliedto the arc discharge. Thus a large-frame supply is reallytwo supplies: a 36 kW plasma supply and a regulated8 kW magnet supply. The power supply must also pro-vide power to heat the tungsten cathode in the plasmatube. Typically a filament transformer provides 15–25 Aat about 4 VAC for this purpose.

11.5.5 Ion-Laser Applications

The operator interface to control power and the basicon/off function is through the power supply. In inexpen-sive models of air-cooled systems, the control interfacetypically consists of hand-wired knobs and switches onthe front panel of the supply. The next step in sophis-tication is a hand-held system controller with a visualdisplay that attaches to the power supply with a long

flexible cable. Upon turn-on, all functions of the lasercan be accessed from the system controller. High-endmodels offer a fully functioning computer interface forautomated remote operation and monitoring of the laser.

Ion lasers operate in either current mode or powermode (also called light mode). Current mode holds theplasma discharge current at a fixed value and allows thelaser output to vary. Power mode, a much more com-mon method of operating the laser, adjusts the plasmacurrent as necessary to match the output power to a user-requested value. When the current can no longer beincreased to achieve the power set-point, it may be timeto replace the plasma tube.

Like many modern laser systems, most ion lasersare designed for hands-free operation. Air-cooled argonsystems in particular are intended to provide a con-stant laser source available from a simple flip ofa switch. Large-frame systems in laboratory applica-tions in contrast trade on the flexibility they provide theuser.

Small air-cooled systems continue to be economicaland reliable sources of coherent, polarized. TEM00 blue–green laser light. They are usually operated at a powersignificantly derated from their maximum, which canextend their useful lifetime for years. Biological andmedical applications for these lasers include cell sorting.Desoxyribonucleic acid (DNA) sequencing, bacterialanalysis, confocal microscopy, and hematology. Manyof the dyes used in these applications were originallydeveloped for argon-laser wavelengths.

These lasers are also used in many applications re-lated to producing text and images. The blue beam isvaluable for exposing printing plates for high-speedprinting, and to provide the color separation requiredfor full-color printing. Similar applications are found inphotoprocessing and other photographic sources.

Mid-frame systems are used in many of the sameapplications as lower-power lasers. In addition, they areused in entertainment, especially for laser light shows,and have laboratory applications in Raman spectroscopyor as a pump sources for tunable laser systems suchas a Ti:sapphire. In ophthalmology, the beam can befocused on the retina to repair diabetes – induced retinaldetachment, for instance.

The deep-blue and UV outputs of ion lasers havebeen used in the semiconductor industry for wafer in-spection and lithography. The ability of the output toexpose photoresist is also used in producing mastersfor compact discs, which then serve as the molds inthe injection-molding manufacture of high-volume CDproduction.

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756 Part C Coherent and Incoherent Light Sources

11.6 The HeNe Laser

The HeNe laser is an electrically pumped continuouslyemitting gas laser. Its basic principle is a gas dischargein a glass tube filled with a mixture of helium and neonunder low pressure. The gas discharge is set up by a cath-ode and an anode placed at opposite sides of the glasstube. The laser mirrors are usually fixed to the end ofthe tubes.

The HeNe laser was one of the first lasers to be real-ized. It was developed in 1960 as the first laser emittinglight continuously [11.1461]. However, this was not thewell-known bright red line (632.8 nm) that has been dis-covered, the first laser emitted light at 1.15 µm, whichis one of the strongest lines of the HeNe laser. Red laseremission was achieved shortly after, in 1962 [11.1462].Though other possible transitions delivering light in thevisible had been found theoretically, it took a certaintime to demonstrate this practically. Improvements inresonator design and the performance of the mirrorshad to be achieved. Especially the well-known 543.3 nmline, with its low gain, needed several attempts by re-searchers until it was demonstrated for the first time in1970 [11.1463].

Shortly after its first demonstration, the HeNe laserfound more and more applications and became the mostcommon laser worldwide with millions of units sold,until laser diodes appeared on the market. More than40 years after its first appearance the HeNe laser is still ofgreat importance in the worldwide laser market. Around44 000 units were sold in 2005 [11.1464].

The HeNe laser has been used for adjusting andpositioning but also, because of its excellent opticalproperties, in interferometers, sensors or spectrometers.It was applied in the first scanner tills and even the firstCD players were equipped with HeNe lasers. Althoughthese applications have been taken over completely bydiode lasers, HeNe lasers still find their use in manyfields of analytics, instrumentation, sensor technology,science and education.

Its advantages are excellent beam quality, longlifetime and an unbeatable price–performance ratio. Be-cause of the mature technology and the still large salesnumbers manufacturing costs have decreased continu-ously. Against the trend for shorter product lifetime theHeNe laser has asserted itself for 40 years and is stilla competitive product. Many OEM manufactures stillchoose the HeNe laser for there new products, irrespec-tive of the competition from diode lasers and solid-statelasers. Considering especially the laser diodes we seethat, in fact, the diode itself is a rather cheap compo-

nent. To achieve a beam quality comparable to the HeNelaser, however, requires considerable effort and addi-tional cost, which compensates for the higher costs ofa HeNe laser. For example, the strongly elliptical beamof the diode has to be made circular and, in order to avoidwavelength drift, temperature stabilization has to be pro-vided. Another advantage is the longer coherence lengthof a HeNe laser compared to that of standard diodes.

Further advantages of the HeNe laser are:

• Excellent mode purity, typical > 95% GaussianTEM00• A favorable relation between resonator length andresonator width (diameter)• A nearly diffraction-limited beam• High beam-pointing stability• High reproducibility in manufacturing

11.6.1 The Active Medium

Energy-Level DiagramThe most detailed energy-level diagram which we couldfind in the literature is given in [11.1465]. Nowadays,however, the green and yellow laser lines are of muchmore importance than the infrared ones, so it was nec-essary to add these new visible lines to the diagram. Itcan be seen that neutral helium is excited by electroncollisions and transfers its energy by nearly resonant in-elastic atomic collisions to the excited states of neon.This means that the lasing atom is the neon, helium isonly necessary for energy transfer from the gas dischargeto the upper neon levels. The energy transfer from thelower laser levels 3p and 2p ends at the 1s level due tospontaneous emission and the 1s level is depopulated bywall collisions to the ground state of neon. For this rea-son it is necessary to have a small-diameter dischargetube, in order to secure a quick emptying of the 1s level.It should be mentioned that the rare 3He isotope is usedand not 4He. The reason is that the lighter 3He isotopehas a higher velocity difference to neon, resulting inbetter energy transfer to the upper laser levels [11.1466]and consequently a higher gain and output power (about25% better at 633 nm).

It is known that electron collisions and stimulatedabsorption of visible light can cause transitions from the1s to the 2p level, resulting in a higher lifetime of thislower laser level (2p). This leads to a higher inversionfor the 3.39 µm laser line as for the other lines, since ithas the 3p level as the lower laser level.

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Lasers and Coherent Light Sources 11.6 The HeNe Laser 757

Table 11.44 Typical laser transitions

Wavelength Transition Gain Typical power

(in air) (nm) (m−1) (mW)

543.3 3s2–2p10 0.03 0.5–3.0

594.1 3s2–2p8 2.0

611.8 3s2–2p6 0.1 2.0

632.8 3s2–2p4 0.5 0.5–50

640.2 3s2–2p2

1152.3 2s2–2p4 2.0

1523.1 2s2–2p1 4 1

3391.3 3s2–3p4 100 10

Table 11.44 lists the commercially available laserlines and some typical parameters. The gain [11.1465,1467] depends strongly on the inversion and the notedvalues should only give an impression of how the gainchanges with wavelength (g0 ∼ ν−3

0 ). This illustratesthat the most difficult laser to build is the green one(543.3 nm).

Another difficulty in making a green laser is that,within each group of energy levels (2s, 3s, 2p, 3p, andso on), the energy distribution of the sublevels (e.g., 2p1to 2p10) is determined by thermic Ne–Ne collisions, re-sulting in a Boltzmann distribution, which means thatthe population rises with falling energy. Thus, the lowerlaser level for the 543.3 nm line (2p10) has a muchhigher population as the 2p4 level for the red line. Inorder to achieve population inversion for this laser lineit is necessary to reduce the number of Ne–Ne colli-sions by reducing the neon pressure. The consequenceis a lower output power at this line and a lower lifetimefor green lasers (about 10 000 h) compared to red lasers(> 20 000 h).

Table 11.44 lists only the most important laser lines.A good compilation of laser lines at different elements isgiven in [11.213], which also contains references to theoriginal literature. Sometimes in an experiment one mayhave unwanted light from scattering from the dischargetube itself (not laser light). Such discharge lines can befound in [11.1468].

Gas DischargeThe gas discharge is the most important process in thelaser, since it has to transform the electrical energy ofthe power supply into the laser light. This process de-termines the laser power, power stability, optical noiseand lifetime of the laser. Hence, the key know-how ofa laser manufacturer is not the resonator design but howto optimize the discharge.

,#<

&

;

(

@ C

.

8"#"" 6

G"

$(Q4

4

4

,"

"

-

-

$*Q

*

*

$-* Q

$-(*Q

$- Q$-( Q

$%; Q

-

Fig. 11.149 Energy-level diagram of the HeNe laser

As shown in Fig. 11.149 the HeNe laser tube is filledwith a helium–neon gas mixture, having a total pressureof about 4–7 mbar and a neon content of about 10%.The operating DC current is in the range of 3.5–11 mAand the corresponding tube voltage is about 1–5 kV.This is a cold-cathode glow discharge, where the activelaser medium is formed by the positive column of thedischarge, which is located in a capillary of diameters of0.5–2 mm. Since such a glow discharge has a decreasingvoltage–current characteristic curve (see Fig. 11.151),

+

Fig. 11.150 Electrical scheme of the laser

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758 Part C Coherent and Incoherent Light Sources

+

++ +B +

G:

/

Fig. 11.151 Characteristic voltage–current curve of a HeNelaser

a ballast resistor R of about 60–100 kΩ is used to bringthe whole system to an increasing voltage–current curve.

In Fig. 11.151 the black curve shows the electricalbehavior of the discharge tube and the brown line thecorresponding ballast resistor. After switching on thelaser, the power supply has to raise voltage to the ig-nition point (UI ≈ 10 kV). After the ignition, the powersupply switches to its normal fixed-current operatingmode. The corresponding voltage is determined by thecrossing point of the discharge curve and the resistorline (voltage at the tube UT). It is important to choosethe resistor in such a way that only one crossing pointexists.

Whereas in the tube design a small metal anode isused, the cathode is considerably larger, and is mostlymade of aluminium because the cathode has to deliverthe electrons for the discharge and to transfer the heat ofthe cathode fall, the region where the electrons are ac-celerated from zero to the colliding velocity, outside thelaser tube. The capability of the cathode to emit elec-

* % &

(;-

Fig. 11.152 Sectional view of a HeNe laser tube (LASOS Lasertechnik GmbH), see text for explanation

trons depends on the properties of the cathode surface.Normally the current density at the cathode surface isconstant, so the cathode surface must be large enoughfor the applied operating current, otherwise sputteringoccurs and the tube is quickly damaged. The typicalcathode size is in the range of some 10 cm2. In orderto avoid sputtering, at the surface a nanometer-scalelayer of Al2O3 is formed. Normally this oxide layerwithstands sputtering for some 10 000 h. If this layer isdestroyed, the laser fails within a few hundred hours.

11.6.2 Construction and Design Principles

Construction of a HeNe LaserFigure 11.152 shows a cross-sectional view of a mod-ern HeNe laser. The principal setup is very similar forall major manufactures of HeNe lasers and differs indetails only. The laser tube consists of a glass tube (8)which is melted at both ends to metal end caps (3, 10).To these end caps the laser mirrors (1, 12) are connectedusing a metal–glass soldering technique. These metal–glass connections provide long-term vacuum sealing ofthe laser tube. Inside the laser tube the cathode (6) andthe gas discharge capillary (7) are located. The glass ofthe capillary is melted to the glass of the outer tube (8),so the discharge is concentrated inside this capillary. Ifthe mirror (1) is a flat mirror and mirror (12) is concave,the end of the capillary (9) acts as a mode field aperture.If linear polarized laser radiation is required, a Brewsterwindow (2) is placed inside the laser. A favorable posi-tion for this window is near the flat mirror, because atthis position the beam diameter is the smallest in the tubeand there is no angular displacement between the laserbeam and the window (the beam is always perpendicularto the mirror surface). The tube is evacuated and filledusing the pipe (4), which is sealed after filling the tubeby means of a cold soldering process. The spring (13)centers the capillary end. At the mirror holders (3, 10)

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Lasers and Coherent Light Sources 11.6 The HeNe Laser 759

a region of reduced material thickness is formed (5, 11).This is used for the alignment of the mirrors by meansof plastic deformation of the metal.

The setup according to Fig. 11.152 represents thebasic design of commercial laser tubes up to about25 mW output power which has been used for aboutthree decades with minor changes only. It provides long-term gas stability, clean optics over the whole lifetimeof the laser and a long lifetime of the laser.

In contrast to the present design, early HeNe laserswere not sealed by the mirrors but by a Brewster win-dow at each end. Thus, they used an external resonator,mostly made of mirrors hold on invar rods. This olderdesign is used today only in high-power HeNe lasers(> 35 mW), because such lasers need a very long cavity(> 1 m). Here, the external resonator allows a better sta-bility of the mirror alignment, and the capillary (whichsuffers from bending due to their own weight) can bealigned too.

Resonator DesignThe design goal of a HeNe resonator is mainly to achievea Gaussian beam profile. The calculation of all beamparameters can be easily done by means of the ABCDmatrix formalism [11.1469]. In this case, every opticalelement in the laser is described by a matrix, as describedbelow.

The matrix of a mirror with curvature r

M =(

1 0

−2/r 1

). (11.143)

The matrix for the free-space propagation of dis-tance s

S =(

1 s

0 1

). (11.144)

Starting at the output mirror all single matrices for everyelement are multiplied for one complete round-trip (RT),resulting in the following calculation:

RT = S M2 S M1 =(

A B

C D

). (11.145)

In a last step, the 1/e2 beam radius (w) and the wavefrontcurvature (R) are calculated from this matrix:

The beam radius at the output mirror is

w=√λ

π

2|B|√4− (A + D)2

. (11.146)

Fig. 11.153 Laser tube with fixed mirrors (LASOS LasertechnikGmbH)

The wave-front curvature at the output mirror is

R = 2B

D − A. (11.147)

Using the well-known formalism for Gaussian beampropagation, the beam diameter can be calculated at anydesired position.

The complex parameter q is defined as:1

q= 1

R− i

λ

πw2 . (11.148)

Fig. 11.154 35 mW laser tube with external resonator (LASOSLasertechnik GmbH)

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The propagation of a Gaussian beam through anoptical system described by the ABCD matrix yields

qout = Aqin + B

Cqin + D. (11.149)

The beam diameter at the position of the mode field aper-ture is important for the laser design (Fig. 11.155). Theratio between this aperture and the beam diameter hasto be chosen in such a wa, that the losses for the TEM00mode are low but high enough for the first higher-ordertransversal modes, e.g., TEM01, in order to suppressthe higher modes. This ratio depends on the laser gainand can be calculated theoretically. The practical ex-perience for a common 633 nm laser leads to a ratiobetween the aperture diameter and beam radius of about3.5 : 1. For high-power HeNe lasers, this value must bereduced, whereas for the low-gain 543 nm laser it has tobe increased.

In practical designs the above formalism is used tobalance the resonator design between the beam diameter,required by the application (the beam waist is normallylocated at the surface of the output mirror), and the laserpower, which determines the laser length on one hand,and the available mirror curvatures as well the availableaperture diameters and aperture positions on the other.

Scaling RelationsScaling relations are very useful for the practical designof new lasers. The most important scaling relations aredescribed below. Firstly, the laser power P, which isproportional to the discharge length l (capillary):

P

l= const. (11.150)

For this reason, today HeNe lasers with a power above25 mW are very rare. These lasers are very long (> 1 m)and need sensitive and expensive external resonators.Thus, in this power region the costs for beam shaping oflaser diodes (aspheric lens systems with prisms, cylin-drical lenses and/or zoom telescopes) are comparableor lower than for a HeNe laser and the mechanical di-

Fig. 11.155 Principle setup of a HeNe resonator with rearand output mirror and a mode-field aperture

mensions of the diode-lasers systems are much smaller(about 15 cm).

The product of the pressure p and capillary diameterd is constant:

pd = const. (11.151)

Following this equation, two systems with the same pdproduct have comparable discharge properties and theexcitation of the upper laser levels is comparable in bothcases if the same current is applied. This rule is veryuseful if a well-running laser must be customized inorder to get more power or different beam diameter.If the new capillary diameter has been estimated usingthe formalism of Sect. 11.6.2 then the corresponding gaspressure for this new design can be calculated by meansof (11.151).

The current–density scaling relation is less importantbecause there is only a weak dependence on capillarydiameter

j 4√d = const. (11.152)

The scaling of the small signal gain g0 with capillarydiameter d

g0d = const. (11.153)

Equation (11.153) shows that a small capillary diameterd should be achieved in the design process because, withhigher gain, losses can be higher too and it is possibleto come closer to the theoretical maximum power.

Additionally two electrical relations [11.1470]should be noted: the current density jK at the cathode

jKp2

= const. (11.154)

and the ignition voltage UI

UI = f

(pl

T

). (11.155)

Equation (11.154) has the consequence that low-pressure laser tubes, like versions for 543 nm, need largercathodes. Equation (11.155) shows that the voltage nec-essary to ignite the laser increases with pressure andlength and decreases with temperature T .

Laser Line SelectionToday, the HeNe laser has been replaced in the infraredrange by laser diodes, but its visible lines, especiallythose at 632.8 nm, 594.1 nm and 543.3 nm, are widelyused in scientific applications such as laser scanningmicroscopy. Because of this fact, we will only consider

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the visible range. Except for some educational lasers,all modern HeNe lasers operate at one wavelength only.The reason is that it is only possible by this approachto obtain the maximum power of each laser line and toreduce the optical noise. The operating wavelength isselected from the possible lasing lines by means of thelaser mirrors. In the most common design the rear mirrorhas as high as possible a reflectivity (> 99.9%) for thedesired wavelength, and the output coupler will haveapprox. 1–2% transmission at this line and > 10% atthe other lines. So, the unwanted laser lines have lossesgreater than their gain and will not be amplified. Becauseof the low gain of the 543.3 nm line, lasers for thiswavelength must have output mirrors with transmissionsof about 0.05–0.15%.

3.39 µm SuppressionFrom Table 11.44 it can be seen that the 3.39 µm linemust be suppressed very carefully, because it has a gainthat is some orders of magnitude higher than all vis-ible lines. Insufficient suppression of this line causespower loss and power fluctuations at the desired lines.Therefore, both laser mirrors must have a reflectivity forthis line below 0.5%. Another approach for the 3.39 µmsuppression is to use the Zeeman effect. Applying aninhomogeneous transversal magnetic field to the lasersplits the laser levels into the Zeeman sublevels. Thegain of the sublevels is the gain of the unsplit level di-vided by the number of sublevels, so the gain may fallbelow the losses and this line is suppressed. The fre-quency difference of the sublevels is the same for alllaser lines. However, Doppler broadening of the laserlines depends on their frequencies.

The Doppler broadening of a laser line λ is given by

∂νD = 2

λ

√2kT ln 2

m, (11.156)

where k is Boltzmann’s constant, T is the gas tempera-ture and m is the mass of the Ne atom.

For visible wavelengths the frequency split causedby the Zeeman effect is lower than the Doppler-broadened line. Thus, the gain is not influenced by theZeeman effect, whereas for the 3.39 µm line the Zeemansplitting is larger than the Doppler-broadened line andthe gain decreases with the number of sublevels. Thismethod is normally used in high-power 633 nm lasers(> 35 mW) or in the 543 nm laser.

Line Width and Coherence LengthThe natural line width of a red HeNe laser is about20 MHz [11.1466] but this line is broadened by two

-;

G 6

+

*

-

%;-( % % % % %* %- %% %&

Fig. 11.156 Typical transmission curve of the output mirrorfor a 632.8 nm laser

different processes. The first process is collision broad-ening, leading to a line width of about 500 MHz at typicalpressures of 4–6 mbar [11.1466]. This line width is pro-portional to the gas pressure. The second process isDoppler broadening (11.156)), which results in a widthof 1–1.5 GHz for the visible laser lines.

A TEM00 laser usually runs at several longitudinalmodes, which are spaced by

∆ f = c

2L. (11.157)

Thus, the longer the resonator (mirror distance L), themore modes can oscillate simultaneously.

As can be seen in Fig. 11.157, the mode spac-ing is 257 MHz (seven modes) and the gain profile isabout 1.5 GHz. A Lamb dip does not occur becausethe collision broadening is larger than the mode spac-ing [11.1466]. For interferometric purposes tubes that

Fig. 11.157 Longitudinal modes (upper curve) and corre-sponding gain profile of a 584 mm-long 20 mW laser at633 nm (LASOS: LGK 7665 P), measured with a scanninginterferometer

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exhibit only one or two longitudinal modes are oftenused. The typical length of such a tube is about 140 mmwith a power of 0.5–1.0 mW. Whereas the gain curve isfixed in frequency space, the longitudinal mode patternshows a movement when the resonator length changesin the dimensions of the wavelength λ, which may re-sult from small ambient-temperature fluctuations. Sucha movement causes power fluctuations in the range ofabout 5% (depending on the laser length). Because inmost applications only the sum power of all modes isused, it is possible to stabilize the laser power by in-creasing the number of modes under the gain profile.This can be done by two methods. The first is to in-crease the laser length and the second uses a 1 : 1 mixtureof the neon isotopes 20Ne and 22Ne instead of naturalNe (90% 20Ne and 10% 22Ne). Because of the heaviernuclear core of the 22Ne isotope the radiated light hasa frequency about 800 MHz higher than the light from20Ne (isotopic shift, [11.1471] p. 333). This means thatthe difference between these two lines is smaller thanthe Doppler broadening. Therefore, using a mixture ofthese two isotopes gives a broader gain curve. The iso-topic mixture has no influence on other laser parameterssuch as the power or noise.

One of the most exciting applications of the HeNelaser is interferometry. For this purpose, the coherencelength of the laser is important. The coherence lengthcan by calculated from the line width ∆ν by

lc = c

∆ν.

The problem of a multimode laser is that it does not runat a single frequency but at several equidistant modes.

$

$*

$%

$;

* % ;

Fig. 11.158 Contrast ratio for the interference of two four-longitudinal-mode laser beams

Figure 11.158 shows the calculation of the meas-ured contrast ratio of a longitudinal multimode laserbeam after a Michelson interferometer over the opti-cal path difference. The beam was formed of four lasermodes with equal intensity. This picture is overlayed bythe decreasing contrast function of the single mode, butthe coherence length of the single mode is in the rangeof kilometers and cannot be seen here. Whereas the in-terference can be observed over very long distances, theperiodic structure makes it necessary to define the coher-ence length only in the region up to the first minimum.In the case of N longitudinal modes, this first minimumis located at the position

lmin = 2L

N. (11.158)

Thus, for the laser of Fig. 11.157 the coherence lengthis below 16 cm.

It should be mentioned that in a real laser the dif-ferent modes have different intensities, but this onlychanges the amplitudes in Fig. 11.158 and leads toa nonzero intensity at the minima.

11.6.3 Stabilization

In order to obtain a larger coherence length, a com-mon approach is it to use only one of the longitudinalmodes. In order to select a single mode, the fact that in anunpolarized laser (without internal Brewster windows)neighboring modes have perpendicular polarization di-rections can be used. Hence, using a short unpolarizedlaser with only two modes, it is possible to select onemode by means of an external polarizer.

In some applications the frequency of this mode mustbe fixed. This can be achieved by controlling the lengthof the laser tube by means of a heater around the glasstube. One way is to select the two perpendicular modesusing a polarizing beam splitter and measure the power

Fig. 11.159 Modes and gain profile of a 140 mm-long laser

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Fig. 11.160 Components of a laser tube

difference between these two modes. By means of theheater, this signal should be held at zero. In this case(Fig. 11.159) both modes are located at the sharp flanksof the gain profile. Here, small frequency changes re-sult in large power changes and exact control of thefrequency is possible. This principle is much easier thanstabilizing a laser diode, at least with an external res-onator with a grating, and the frequency depends onlyon the atomic properties of the neon. This first approachto frequency stabilization is widely used in the field ofmechanical engineering for measurement devices withan accuracy of about 1 nm. A detailed description offrequency stabilization is given in Sect. 11.14.

11.6.4 Manufacturing

Because the first HeNe lasers were built in the 1960s,today the manufacturing technology is a fully developed

process, where the processes of cleaning, connectingand the vacuum processes are the key know-how of themanufactures.

After an initial cleaning process of the differentcomponents, in a first step the metal parts are solderedtogether in a high-temperature process (about 1000 C).In the next step, the metal and glass parts are meltedtogether at temperatures of about 800 C. During thisstep, care must be taken that the capillary bore is atthe position of the subsequent optical axis. In order toavoid mechanical stress, the different materials must bechosen in such a manner that the thermal expansion co-efficients of the metal and glass parts are matched. Thethird step is soldering the mirrors to the mirror hold-ers (at about 500 C). After this preparation, the tubesare backed out under vacuum for many hours. Than,by means of an oxygen discharge, the inner surfacesof the tube are cleaned of residual organic traces. The

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oxygen discharge is also used to form an Al2O3 layeron the cathode surface, which is important for the life-time of the tube. The next steps are filling the tube withthe helium–neon gas mixture, carrying out a burn-inprocess and the final filling. In order to obtain laseremission, after these steps it is only necessary to alignthe mirrors. A pre-alignment may be done by meansof external lasers, using the back reflection of the lasermirrors from the tube, which should be adjusted. Thecriteria for final alignment is maximization of the outputpower.

In order to get a robust product and/or because thelong high-power tubes are very sensitive to air flow, theglass tube is usually connected to the ballast resistor andboth are placed in an outer aluminium tube.

11.6.5 Applications

Most applications make use of the visible wavelengths ofthe HeNe laser. Typical applications are flow cytometry,confocal microscopy, DNA sequencing, sensing, haema-tology and photo-finishing. Last but not least the HeNelaser is a very popular teaching apparatus in education,school or studies. A variety of applications use fluores-cence imaging and interferometry which are describedin more detail below.

Fluorescence ImagingFluorescence imaging makes use of the different wave-lengths to excite fluorescence of certain fluorophores.Particularly in biomedical applications this is one of thebasic methods. Different fluorophores bind to specific

cells and allow the detection of abnormal structures. Theworking principle is that each particular fluorophore isexcited by a special wavelength, emitting light at an-other wavelength, which is detected by highly sensitivesensors. Using various of these markers allows one todistinguish, label or sort cells. This method is applied inbasic research as well as in clinical diagnostics, whereit helps to detect serious diseases such as leukaemia oracquired immune-deficiency syndrome (AIDS).

Since most of these methods were developed in a pe-riod when the only lasers sources available were gaslasers, the fluorescing substances were made especiallyfor these wavelengths. Together with the common wave-lengths of the Ar-ion laser they have become a kind ofstandard for this application, although a variety of otherwavelengths and corresponding dyes are also used to-day. The emission lines of the HeNe lasers are foundin nearly all commercial devices. Common applicationsare flow cytometry and confocal microscopy.

InterferometryInterferometric methods are used for the precision mea-surement of various physical values. Examples are theposition or speed of particles, distances, stress or vi-bration. The optical properties and the long coherencelength of the HeNe laser make it ideally suited forthis kind of application. The narrow line width of thelaser allows its wavelength to be used as a measure-ment standard. For many interferometric applicationsthe frequency of the emission line has to be stabi-lized to achieve the maximum measurement precision(Sect. 11.6.3).

11.7 Ultraviolet Lasers: Excimers, Fluorine (F2), Nitrogen (N2)In 1984, Ch. K. Rhodes stated in the preface to theworld’s first book on excimer lasers [11.1472]: “The de-velopment of excimer laser systems marked a significantturning point in the development of coherent sources.The progress of the last years has been largely predicatedupon the combined knowledge of several disciplinesincluding atomic and molecular physics, optical technol-ogy, and pulsed-power technology.” This early statementwas primarily associated with electron-beam excitationof laser transitions of rare gases and rare-gas halogenmixtures. Present excimer lasers are, however, basedon precisely controlled electrical discharges, and relyon detailed knowledge of material chemistry to assurelong service life of the discharge tubes. Cost-effective

operation has therefore become possible, a prerequi-site for industrial applications. Together with laser-beammonitoring and control by electronic means, the multi-disciplinary approach has made excimer lasers importanttools for a large variety of industrial and medical appli-cations.

For semantic reasons, the term excimer originallyreferred to excited dimers. Today it is used for all kindof laser-active media that are characterized by boundexcited states and dissociative ground states, more gen-erally named exciplexes.

This section is organized as follows. An overviewof the unique properties of excimer lasers is followedby a guide to the physics and technology behind these

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laser systems, including beam characterization usingwavefront diagnostics. The section on applications fo-cuses on the various techniques of material modificationusing excimer lasers and includes the use of fluorine(F2) laser radiation at 157 nm. The promising appli-cations of femtosecond excimer laser pulses are alsoconsidered. At the leading edge of current research, thechapter on ultrahigh-intensity applications deals withthe generation of hollow atoms and their application forultrashort-wavelength X-ray generation.

Lastly, we present the new radiation sources at13.5 nm for next-generation lithography. Radiationsources in the extreme UV (EUV) can be realized byboth laser and discharge pumping and are expected totransform the present microlithography into nanolithog-raphy.

Since the field of UV lasers is very large and hasrecently been covered in a review book [11.1473] thisshort section just summarizes the various possibilitiesand requests the reader to browse in [11.1472, 1473],and in primary publications if more details are required.

11.7.1 The Unique Propertiesof Excimer Laser Radiation

The most impressive property of excimer laser radia-tion is the large variety of emission wavelengths, whichcover the entire ultraviolet spectral region (Fig. 11.161).The shorter the wavelength, the higher the resolutionthat can be achieved in microprojection and -imaging,thereby opening a wide field of applications. Concur-rent with the short wavelength is the high quantumenergy. Short-wavelength photons are strongly absorbedby most materials and they can supply sufficient quan-tum energy to induce photochemical reactions and causemolecules to dissociate. Together with the high peakpower available in a laser pulse, the bond-breakingcapability of excimer laser radiation allows ablativeevaporation that opens the door to microprocessing ofmany materials, ranging from soft biological tissue tohard diamond.

The pulse duration is an important parameter. Typ-ical excimer lasers emit pulses in the range of a fewnanoseconds so that material processing can be fre-quently performed on the fly, i. e., can be applied toa continuous flow of components to be processed. In ad-dition, thanks to the naturally broad line width, excimerlasers can be tailored to supply pulses in the femtosec-ond range with extremely high peak power. This allowsone to generate a plasma that consists of electrons andhollow atoms, i. e., inner-shell ionized atoms that recom-

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Fig. 11.161 Wavelength (lower scale) and corresponding photon en-ergy (upper scale) of the various excimer transitions. The filledsymbols indicate commercially important wavelengths

bine via highly energetic transitions, thereby deliveringradiation in the extreme ultraviolet or weak-X-ray re-gion. This is surely one of the most advanced promisingapplications.

Finally, excimer lasers are scalable to high pulseenergies in the joule range, to high repetition rates ofa few kHz, to high average powers up to 1 kW, so thatthey can be conveniently adapted to specific industrialtasks. Compared to these facts, their drawbacks are thefairly broad line widths, the low degree of coherence andthe non-continuity of the radiation. However, the lattercharacteristics are of minor importance considering thestrengths discussed above. This leaves some space forother lasers that might conquer the UV range by fre-quency conversion, until more-powerful semiconductoremitters may partly fill the gap. The low degree of coher-ence may be utilized for speckle-free imaging, and thebroad line width, which enables ultrashort pulses, maybe narrowed by some frequency-selecting means in theresonator setup. However, the beam, which somehow re-sembles the emission of a lighthouse (large divergence),requires some harnessing by beam-shaping optics.

Figure 11.161 summarizes the data on wavelengthand quantum energy of rare-gas halide lasers, togetherwith those of the fluorine laser and some other diatomicand triatomic species. The F2 laser is actually not anexcimer laser but can be excited in almost the sameway. These UV sources are most important for a widefield of applications outlined in the next sections. Theexotic types such as KrBr, rare-gas dimers and otherdimers as well as trimers are discussed in some detailin [11.1472, 1473].

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766 Part C Coherent and Incoherent Light Sources

11.7.2 Technology of Current Excimer Lasersand the N2 Laser

We start with the fundamentals of excimer lasers andthe fluorine laser. Due to its role as a forefather ofall gaseous UV lasers, the (molecular) nitrogen laser(N2 laser) will be briefly dealt with as well. Finally, thetechnology of discharge-pumped excimer lasers will bebriefly outlined.

Excimer Transitions:Unusual Four-Level Laser Systems

Excimer lasers derive their emission from moleculesthat are generated a priori in an electronically excitedstate, which decays by emission of (laser) radiation intoa repulsive or loosely bound ground state out of whichthe molecule dissociates. In usual test-tube chemistry therare gases are noble or inert, i. e., they avoid chemicalbinding, although special noble-gas compounds do existalso in the electronic ground state. The most importantmolecules for commercially available excimer lasers areArF∗, KrF∗, XeCl∗, XeF∗, where the asterisk refers toelectronic excitation.

The dissociative electronic ground state is respon-sible for the four-level character of these systems.Generally, four-level laser systems can be operated incontinuous manner. However, this is not possible withrare-gas halide excimer lasers because of physical andtechnical restrictions. A fundamental one is the impor-tant role that spontaneous emission plays with respectto the stimulated emission in the ultraviolet spectral re-gion; the ratio of the Einstein coefficient A, which isresponsible for spontaneous emission, and that respon-sible for the stimulated emission in the same transition,B, is proportional to the third power of the transitionfrequency ν [11.1475]:

A

B= 2hν3

c2 .

Hence stimulated emission can only compete with spon-taneous emission if the radiation intensity within thetransition I(ν) is so large that the product BI(ν) ex-ceeds A, that is, if the system is pumped extremelyhard. Electrical power densities of some 100 MW/l mustbe deposited in the gas volume. In the early days, thisexcitation density was achieved by using energetic elec-tron beams [11.1472]; nowadays short-pulse transverseelectrical discharges are being used. An extension ofthe pulse duration beyond about 1 µs is hampered bydischarge instabilities caused by arcing. The detrimen-tal reaction products and heat must be removed before

the next shot so that rapid exchange of the laser gas isrequired.

After the first experimental demonstration ofa bound–free laser system in liquid Xe2 [11.1476],a large number of other excimer or exciplex moleculeswere successfully investigated, including homonuclearas well as heteronuclear species [11.1472], but most ofthese did not gain importance for applications. The doorto the systems listed in Fig. 11.161 was opened wide in1975 [11.1477–1481], with convincing efficiency, wave-lengths, and after some technical effort, ease of handlingof the laser gases. The gases are primarily the dischargecarrier, which is the buffer gas (mostly helium), and, toa much lower extent, the reactants, which are the lasergases.

Molecular Potentials and Reactions KineticsThe rare-gas halide systems are characterized by twoelectronic ground states, correlating with the electronicground states of the rare gas and halogen atoms, whichcombine to a molecular Σ and a Π state, arising fromthe p-hole of the halogen atom (Fig. 11.162 [11.1474]).Whereas the Π state is strongly repulsive, the Σ stateshows a minimum, mostly with a depth of a few hundredcm−1 only, so that thermal energy allows the moleculeto dissociate within a few picoseconds. The first elec-tronically excited states correlate with the positive raregas and negative halogen ions, and hence show a deepminimum. The higher states correlate with the elec-

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Fig. 11.162 Typical potential energy diagram of a rare-gashalide molecule. M stands for the rare gas, X for the halogenatom [11.1474]

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Lasers and Coherent Light Sources 11.7 Ultraviolet Lasers: Excimers, Fluorine (F2), Nitrogen (N2) 767

tronic excitations of the neutrals, and their potential isshallower, as is typical for covalent binding. This gen-eral scheme holds for the rare-gas halides, except XeFwhere the potential of the ground state reaches a depthof 1065 cm−1 according to [11.1474].

In the buffer-gas-supported electrical discharge, theexcited and ionic species are formed as shown for theKrF∗ case in Fig. 11.163. The formation of the excitedrare-gas halide molecule follows both a neutral channel,a chemical exchange reaction, and an ionic recombina-tion channel, stabilized by third-body collisions with thebuffer-gas atoms. Hence the formation of exciplexes isfavored at high pressure, i. e., pressures of a few bars.The buffer gas also provides fast relaxation to the lowestvibrational level within the electronically excited state,followed by a radiative transition to the ground statewithin a few nanoseconds.

The laser transitions couple with the continuumstates above the ground state, showing a homogeneouslybroadened non-Lorentzian line shape [11.1484]. Hence,excimer lasers can be tuned within a certain bandwidthrange if frequency-selective optics are introduced – seethe tuning ranges in Fig. 11.161. The homogeneous linebroadening turns out to be favorable for the amplificationof short pulses.

In more detail, many more excited states that cor-relate to the excited atomic states exist and are beingpopulated. However, their buffer-gas-induced relaxationfeeds the population of the lowest electronically excitedstate. An overview of the reaction kinetics of both purerare gases and rare-gas atoms with halogens and theunderlying rate constants is given in [11.1474].

The fluorine laser is based on bound–bound tran-sitions. With 157 nm it shows the shortest wavelengthof all homonuclear halogen lasers [11.1485]. From thetwo p-holes of the F atoms, three levels belonging toground electronic configuration are obtained for theF2 molecule, two of which are bound, so that the sta-ble molecule can be directly electronically excited in anelectrical discharge. In addition, excitation transfer fromthe electronically excited He∗ and excited F∗ atoms, andionic recombination of F+ and F− ions, which havea large electron affinity, during interaction with a Heatom lead to the population of the upper 3Π laser levelthat decays to the weakly bound lower 3Π level. TheF2 laser transitions can be excited in discharges likethose in rare-gas halide lasers. Therefore the F2 laseris frequently mentioned among the commercial excimerlasers – in particular with respect to its short wave-length [11.1486]. Figure 11.164 shows details of theF2 laser emission together with a line-selected spectrum

>"6A2!

A2AW2A

!2! '

A!W

A2!

AW2! A2! '

C "6 B"6

*;

Fig. 11.163 Dominant reaction paths for the generation ofKrF∗ [11.1482]

with a half-width of only 1.034 pm (FWHM, [11.1473,Chap. 6, pp. 97]).

Technical Design Principles for Excimer LasersFrom the physical preconditions described in thetwo preceding sections it follows that the elec-trical energy of high density, on the order of10−2 J/cm3 [11.1487], must be discharged homoge-neously into the laser gas (Fig. 11.165a) on a time scaleof a few nanoseconds so that a high-pressure (up to0.5 MPa) glow discharge can be maintained for as longas possible.

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Fig. 11.164 Emission spectrum of the fluorine laser accord-ing [11.1483]. Inset shows spectral details of the stronger transitionafter line selection [11.1473]

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768 Part C Coherent and Incoherent Light Sources

"

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Fig. 11.165 (a) Schematic layout of an excimer lasertogether with gas discharge and laser resonator; (b,c) Ex-citation circuits employing magnetic pulse compression:(b) L–C inversion circuit with single-stage pulse compres-sion, (c) thyristor-switched circuit using double-stage pulsecompression [11.1487]

The electrical charge comes from a storage capacitoror, more effective, a low-impedance pulse-forming line.For a homogeneous ignition, an efficient pre-ionizationis required. Pre-ionization by ultraviolet radiation hasprevailed over X-rays. Table 11.45 summarizes the dif-ferent methods of UV pre-ionization. Glow discharges,especially in halogen-containing gases, tend to trans-form into an arc or sparking discharge due to electroncapture by the halogens so that laser emission breaksdown. Hence, considerable efforts have been made tostabilize the glow discharge [11.1488,1489]. In order tosupply the charge sufficiently fast, favorable use is madeof magnetic pulse compression (Fig. 11.165b,c). The en-ergy switches, formerly thyratrons, are mostly thyristors,gate-turn-off (GTO) thyristors or insulator gate bipolartransistors (IGBT).

To guarantee a long laser-tube lifetime, the choiceof electrode material is very important. Considerable

material research has taken place and led to differentmaterials for the fluorine and chlorine rare-gas lasers.Even different materials for either the cathode or theanode, or over the profile of one electrode have beenused [11.1487]. Rapid transverse gas circulation pro-vides cooling and supply of fresh laser gas for everyshot. In order to achieve stable laser output power,halogen gas consumption is compensated by processor-controlled gas injections so that the laser tube can beoperated with a single main fill typically for about 1billion pulses.

Excimer laser resonators (Fig. 11.165a) mostly con-sist simply of plane windows made of CaF2, allowinghigh out-coupling in the range 50–92%. For specialapplications, unstable resonators or resonators withfrequency-selective optics are employed, as well asoscillator-amplifier systems.

The number of technical papers on the technologyand construction of excimer lasers and their applica-tions has become numerous. Excimer laser parametersare scalable to a considerable extent. Special excimerlasers have been developed by putting in the fore-ground either high single-pulse energy (up to 10 J),or high repetition rate (6 kHz), or medium or highaverage power (up to 1 kW). The efficiencies, calcu-lated from the ratio of laser output energy to storedelectrical energy, are typically a few percent (< 5%).Since digital inter-publisher retrieval has become possi-ble [11.1490], nearly every combination of the above canbe electronically traced back, in part, over the last threedecades. As an example, Table 11.46 shows data spec-ified for an XeCl excimer laser designed for industrialapplications.

Beam Characterization of Excimer LasersFor various applications such as the production ofdiffractive structures via direct patterning [11.1491,1492], semiconductor microlithography or eye surgery,a detailed knowledge of the wavefront of the emitted ex-cimer laser radiation is necessary. In addition, all theseapplications strongly rely on the stability and the precisecontrol of the laser parameters, such as pulse energy,beam width, divergence, pointing stability, uniformityetc. Thus reliable, standardized methods for the evalua-tion of beam parameters as well as accurate diagnostictools for UV laser beam characterization are mandatory.While the output energy and the power can be moni-tored with the standard tools that are used also for otherhigh-power lasers, the recording of spatial beam pro-files and directional distributions (wavefronts) requiresspecific instrumentation adapted to the characteristics of

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Table 11.45 Pre-ionization techniques applied in excimer-laser discharges

Preionization of the laser gas by UV radiation emitted from spark dischargesbetween pin electrodes mounted in a row parallel to the laser channel. Sparksare being ignited some 10 ns before the main discharge starts so that at least108 electrons/cm3 are created as a seed for the main discharge.

A possible modification of a: spark preionization through one of the elec-trodes designed as a screen or mesh electrode.

." Dielectric(alumina)-surface-guided spark discharge for preionization. Be-cause the discharge spreads, erosion of the pins is reduced considerably, thusthe lifetime of the laser gas is increased, and the laser tube can be operatedup to 10 billion shots. This design is mostly used in high-average-powerexcimer lasers.

@$<$ Surface corona discharges on a dielectric sheet emit UV with high spatialuniformity. Consuming a lower amount of energy as compared to sparks,they supply an electron density sufficient for a narrow discharge volume asused in high-repetition-rate excimer lasers, and provide increased dischargeelectrode lifetime.

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Knife-edge-released discharges creeping on a dielectric surface (similar tothe well-known Lichtenberg discharges) cover a large area, thus can beconveniently applied for the preionization of large-aperture excimer lasers.The knife edge is connected to the ground electrode of the main discharge.

Table 11.46 Data from an XeCl excimer laser for industrial use. (Type: Steel 2000, Coherent Lambda Physik 2005)

Maximum Maximum Maximum Pulse FWHM beam Divergence Beam point.stabilized energy average power repetition rate duration dimensions V × H stability(mJ) (W) (Hz) (ns) V × H (mm) (mrad) V × H (mrad)

1050 315 300 29 37 × 13 4.5 × 1.5 0.45 × 0.15

excimer lasers. Besides sensitivity to the various wave-lengths in the deep-UV spectral range, such diagnosticsystems must also possess detector apertures adaptedto the large near-field beam cross section of excimerlasers, and, most important, must guarantee long-term

stability of the employed optics and sensors under pulsedhigh-power UV irradiation. The evaluation of relevantexcimer beam propagation parameters is now possiblewith camera-based profile and wavefront measurementsin accordance with current International Organization

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Fig. 11.166 Near-field (left) and far-field profile (right)of F2 laser (157 nm, Novaline F1030, Lambda Physik)recorded simultaneously with a camera-based measurementsystem. The evaluated second-moment beam widths in thehorizontal and vertical directions are also indicated

for Standardization (ISO) standards [11.1493, 1494].Figures 11.166 and 11.167 show results obtained witha Hartmann camera-based measurement system.

Brief Recollection of the N2 UV laserThe molecular nitrogen laser, realized as early as1963 with emission in the near infrared [11.1495]

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Fig. 11.167 Wavefront measurement of the emission of an F2 laser(157 nm) using a Hartmann camera (40 × 30 pinholes, diameter100 µm, pitch 1 mm). This technique provides comprehensive beamcharacterization and propagation analysis of excimer lasers froma single measurement

and shortly later between 300 and 400 nm with thestrongest line at 337 nm [11.1496] (described in detailin 1965 [11.1497,1498]) quickly became the workhorsefor laser spectroscopy. Its predominant use was, and stillis, optical pumping of pulsed dye lasers throughout thenear-UV, the visible and near-infrared spectrum. In addi-tion, it serves as an excitation source in photochemistryand biological research. The lasing is achieved in high-voltage (10–20 kV) high-speed (few-ns) transverse gasdischarges in a broad pressure range from some 10 mbarup to atmospheric pressure (TEA laser), in pure nitro-gen and also, with lower efficiency, in air. The 337 nmtransition is the most intense of the vibronic transitions(0–0) between the C3Πu and the B3Πg states. Sincethe B state is long-lived (about 10 µs) with respect tothe C state (about 10 ns), the laser action is quicklyself-terminating.

The pumping is achieved from the X state by di-rect electronic excitation of the C state, which happensto occur with a larger cross section than that for theB state, so that population inversion can be obtainedif the discharge current rises much faster than radia-tive increase of population of the B state. To emit pulseenergies in the mJ range, N2 lasers must be equippedwith low-impedance discharge circuits, such as for ex-ample can be realized by embedding the laser channelinto a Blümlein transmission line, which enables a risetime of a few nanoseconds. Since the gain is very large,the emission may be stimulated predominantly in onedirection when the excitation wave is made to reachthe discharge channel synchronous to the light propaga-tion; in this way a 10:1 ratio in favor of one directioncan be achieved [11.1499], and fairly short lasers withconsiderable peak output power (1.2 MW) are possi-ble [11.1500]. Transmission line circuits are thoroughlyanalyzed in [11.1501]. A pulse energy of 20 mJ hasbeen realized using a fast circuit with magnetic pulsecompression [11.1502]. On the other hand, just fordemonstration purpose, a nitrogen laser can be easilyrealized by adjusting two electrodes on a glass sheetand supplying the electrodes with a spark-gap-switchedhigh-voltage source. A home-built nitrogen laser, usingstandard ignition transformers, spark gaps, refrigeratorcirculating pumps, and some dielectrics and aluminiumfoil or copper-layered circuit boards, is precisely de-scribed on the web [11.1503].

Historically speaking, N2 lasers as well as thehomonuclear excimer lasers and metastable mercurycompounds was proposed as early as 1960 by Houter-mans [11.1504] – much earlier than the first ruby laserhad been realized.

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Lasers and Coherent Light Sources 11.7 Ultraviolet Lasers: Excimers, Fluorine (F2), Nitrogen (N2) 771

11.7.3 Applications

Physically, the applications of excimer lasers can begrouped according to the benefits from

1. short wavelength, and correspondingly,2. their high quantum energy,3. the high pulse energy density obtained by focusing,

and4. the extreme peak power when the pulse energy is

compressed into ultrashort pulses.

Property 1 allows exposing and patterning of sub-100 nm structures in photoresists for semiconductorlithography, property 2 allows photochemical near-surface modifications such as color change, as usedfor marking of plastics, and index-of-refraction changeused for the generation of fibre Bragg gratings. Prop-erty 3 allows the melting of thin silicon films, usedto induce large-grain crystallization in the production

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Fig. 11.168 Potential lithography exposure tool solutions according to the International Technology Roadmap forSemiconductors 2005 [11.1505]. Courtesy ITRS

line of thin-film transistor (TFT) displays. The prop-erties 1–3 enable microstructuring by material ablationincluding biological tissue, and property 4 opens thedoor for the generation of X-rays of extreme brilliance.Besides these technical applications, excimer laser radi-ation is still widely used for scientific purposes, such asoptical pumping of dye lasers. Here, we briefly refer toa few of the technical applications.

Excimer Lasers in the Electronics IndustryLithography for the generation of semiconductor circuitshas been based on excimer laser light sources for morethan a decade now, and will continue to do so for a coupleof years. Thus this is the economically most importantapplication of these lasers. Because the diffraction-limited minimum (half pitch) that is resolvable in opticalprojection is given by k1λ/NA, every effort is made todecrease the wavelength λ for the illumination togetherwith the process factor k1, and to increase the numericalaperture (NA) of the projection system. After extensive

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Fig. 11.169 30 nm-wide lines and equally sized spaces ob-tained at IBM Almaden Research Center (left) using 193 nminterferometric high-index immersion lithography, com-pared to current 90 nm features. Courtesy IBM

use of 248 nm lasers, 193 nm illumination is now state ofthe art, and increasing the NA by using immersion op-tics turned out to be a highly successful method to arriveconsiderably below the diffraction limit. In the Interna-tional Technology Roadmap for Semiconductors 2005(Fig. 11.168 [11.1505]) the 45 nm half-pitch for dynamicrandom-access memories (DRAMs) in production isscheduled for 2010, to be achieved with 193 nm im-mersion technology. Fluorine lasers (157 nm) are not indiscussion any more since immersion fluids with an in-

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dex of refraction higher than that of pure water are beinginvestigated. These could extend the use of 193 nm downto a 30 nm half-pitch (see Fig. 11.169 from [11.1506]).

For the lasers to be used in lithography, there areconsiderable requirements with respect to line width,wavelength and pulse-to-pulse stability, lifetime of thelaser tube and all the optics [11.1508]. Dual-chambersystems distribute the task of narrow-band generationin the oscillator and power generation in the amplifiertube; the latter can be designed as a regenerative ringamplifier [11.1508]. Recently, a 60 W system operatingat 193 nm was introduced (Fig. 11.170 [11.1507]) withthe benefit of improved energy stability of the 6 kHzpulses, so that the exposure dose of the resist can beprecisely controlled. This system aims for 45 nm half-pitch production.

Transistors in thin films are the key elements to con-trol the individual pixels in displays. To increase themobility of the electrons, large silicon crystal grains areproduced by applying excimer laser radiation to causemelting and controlled recrystallization, the beam be-ing scanned in lines across the surface [11.1509, 1510].This process, summarized as excimer laser annealing(ELA), is performed by applying 308 nm radiation thatis strongly absorbed by silicon so that thin films on glasscan be melted without damage to the substrate [11.1511].In this way, large-area TFT displays can be manufac-tured. Even doping imperfections due to incompleteannealing near source/drain junctions can be eliminatedby oblique-incidence ELA [11.1512].

Further excimer laser applications that are well es-tablished in electronics are printed circuit board (PCB)via drilling [11.1513], wire stripping and wire mark-ing [11.1473]. A rather new technology is the lift-offmethod to de-bond for example an electronic GaN LEDfrom its sapphire substrate, which is used for crys-tal growth, by shining the laser through the substrate,thereby ejecting the LED onto its heat-sink electricalinterconnect [11.1514]. For future optical coupling ofPCBs, attempts to integrate optical mirrors manufac-tured by ablation should be mentioned [11.1515, 1516].

Processing of Optical, Ceramic, Polymeric,and Biological Materials

Ablation is surely the best-known application of excimerlasers: drilling and microstructuring of glass, quartz,even diamond, ceramics and polymers [11.1517] allowshighly precise contouring with high reproducibility. Thissubject is covered in chapters 11–14 and 16 of [11.1473],which also includes the ophthalmological application in-cluding the laser-assisted in situ keratomileusis (LASIK)

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Lasers and Coherent Light Sources 11.7 Ultraviolet Lasers: Excimers, Fluorine (F2), Nitrogen (N2) 773

method ([11.1473, Chap. 19], [11.1518]). Nonablativeprocessing of glass and polymers allows one to generatefiber Bragg gratings in optical waveguides by changingthe index of refraction [11.1473,1519]. Microstructuringis going to play a role also in the upcoming technologyof organic light emitting devices (OLEDs) [11.1520].

Femtosecond Excimer Laser PulsesAnother interesting subject of today’s excimer tech-nology is their conversion into UV light sources withfemtosecond pulse duration. Basically a femtosecondseed pulse at 248 nm is injected into an appropriatelymodified standard excimer laser resonator to achievea powerful highly directed short pulse emission. Asexamples are presented material processing with thisradiation and high-intensity studies for the generationof tunable X-rays.

Conversion of a nanosecond excimer-laser systeminto a fs system is described in [11.1521–1523] andbriefly in Fig. 11.171a. The ultrafast UV laser systemconsists of a Ti:sapphire front laser, a frequency-triplingunit to convert the wavelength of the ultrashort pulsesinto the UV spectral range, and a specially designed KrFamplifier to boost the energy of the pulses to several mJ.The current laser arrangement, as shown in Fig. 11.171a,uses a commercial Ti:sapphire front-end system (Coher-ent) delivering pulses of 150 fs duration at a wavelengthof 745 nm. After frequency tripling, seed pulses are ob-tained for the KrF amplifier module, which is the keycomponent of the system. This module is a modifiedversion of a commercial excimer laser. In a three-passamplification scheme the pulses are amplified up to en-ergies of ≈ 30 mJ at repetition rates exceeding 300 Hz,resulting in an average power of 10 W at 248 nm.

UV femtosecond material processing. Nanoscale fab-rication of materials is more and more in demand inscientific and industrial applications. The general trendto reduce the size of optomechanical devices and thegrowing need for assemblies with feature sizes belowone micron generates new challenges for laser fabrica-tion techniques.

Short-pulse lasers with picosecond and femtosec-ond pulse duration offer material processing capabilitieswith highly decreased damaged area around the irradi-ated spot, and consequently smaller feature sizes. If theshort pulse duration is combined with short wavelengths,unprecedented results can be achieved due to the depen-dence of the spatial resolution on the wavelength. ThusUV femtosecond laser systems provide superior materialprocessing quality.

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Fig. 11.171 (a) Layout of the UV femtosecond hybrid laser systemshowing Ti:sapphire seed system and three-pass off-axis KrF am-plifier. (b) Comparison between calculated and measured surfacetextures for four-beam interference at a phase difference of 0π (left)and 0.5π (right). The surface contours top left and bottom right weremeasured via atomic force microscopy (AFM)

Such pulses are ideally suited for sub-micron ma-chining of solid surfaces. Applying various diffractiveoptical masks in combination with reflective imag-ing/focusing systems allows the generation of complex2-D and 3-D structures with feature sizes down to≈ 200 nm or lower on all materials, including metals,semiconductors and dielectrics. Figure 11.171 showsthe calculated and experimentally obtained surfacerelief structures fabricated with the technique of phase-controlled multiple-beam interference [11.1524].

High-intensity UV femtosecond studies. A leading ap-plication of KrF∗ (248 nm) excimer lasers has been thedemonstration of saturated amplification in the multi-

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774 Part C Coherent and Incoherent Light Sources

kilovolt X-ray regime that arises from the ability toproduce and controllably combine two new highly or-dered forms of excited matter:

1. hollow atoms and2. stable electronically hollow plasma channels.

The realization of this new X-ray source provides a peakbrightness that is sufficient for the implementation ofa new high-resolution technology for biological mi-croimaging.

Hollow atoms and the cluster concept for X-ray am-plification. Previous work [11.1526] on the nitrogenmolecule N2 suggested the possibility of designing a newclass of molecular materials optimized for the efficientproduction and amplification of X-rays [11.1527]. Thisidea was immediately tested with Xe clusters [11.1528]with the outcome that the copious production of Xehollow-atom states [11.1529] emitting both Xe(M) andXe(L) radiation in the kilovolt spectral region [11.1528]

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Fig. 11.172 Characteristic spontaneous emission profile ofthe Xe(L) 3d2p hollow-atom spectrum produced from Xeclusters with intense (≈ 1019 W/cm2) femtosecond 248 nmexcitation. The splitting between the major and minor lobesarises from the spin–orbit interaction of the 2p vacancy. Thefull width of the main feature is ≈ 200 eV. The positionsof selected charge-state transition arrays (Xe31+, Xe32+,Xe34+, Xe35+, and Xe36+) are indicated. The spectralresolution of these film data is ≈ 4 eV. (After [11.1525])

was observed. These data confirmed that the selectiveexcitation of the inner-shell states initially observedin N2 could be scaled into the multi-kilovolt spec-tral region. A good example [11.1528, 1529] is givenby the characteristic Xe(L) 3d2p hollow-atom emis-sion profile centered at ≈ 2.8 Å shown in Fig. 11.172.Hollow atoms are atoms (ions) that intrinsically pos-sess an inverted electronic configuration consisting ofdeeply bound inner-shell vacancies, perhaps multiple,with the simultaneous retention of several electrons inrelatively weakly bound outer orbitals. Accordingly,these states are optimally suited for the prompt emis-sion and amplification of X-rays. The demonstration ofsaturated amplification [11.1525,1530–1532] followed,together with the ability to form self-trapped plasmachannels [11.1533, 1534]] that are well matched to theconditions necessary for Xe(L) excitation in clusters.

Multi-kilovolt X-ray amplification with clusters inself-trapped plasma channels. The fundamentalpower compression and its spatial organization werefound in the alliance of the two basic phenomena men-tioned above. They are

1. the direct multiphoton excitation of hollow atomsfrom clusters with ultraviolet radiation and

2. a nonlinear mode of stable confined propagationin plasmas resulting from a relativistic/charge-displacement mechanism of self-channeling [11.1533,1534].

The spectrum typical of the X-ray beam amplifiedaxially in the Xe plasma channel and recorded onfilm with a von-Hámos-spectrograph is illustratedin Fig. 11.173.

In Fig. 11.174 the axially recorded spectrum pre-sented in Fig. 11.173 is compared to a transverselyobserved single-pulse spectrum that exhibits deep andbroad spectral hole burning [11.1535] that correspondswell with the axially amplified transition arrays. This re-sult demonstrates two key attributes of the Xe(L) system,namely,

1. a high efficiency of energy extraction and2. a very broad bandwidth (60 eV) for amplification.

In this case, the strongly enhanced lines observed cor-respond to the Xe34+, Xe35+, and Xe36+ charge statearrays of the major lobe shown in Fig. 11.172. Theseobservations, along with additional spectra showingcomparable results on several other transition arrays in

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Lasers and Coherent Light Sources 11.7 Ultraviolet Lasers: Excimers, Fluorine (F2), Nitrogen (N2) 775

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Fig. 11.173 Axially recorded spectrum showing sharp peaksat the positions of the Xe34+, Xe35+, and Xe36+ transitionarrays. On the basis of geometric considerations concern-ing the von-Hámos-spectrograph and the ratio of exposurescorresponding to the spontaneous signal and the amplifiedlines, the recorded enhancement of these features is esti-mated to be minimally ≈ 1.5–3.0 × 103 over the strengthof the spontaneous emission

the ≈ 2.71–2.93 Å region, indicate that the hollow-atomstates are strongly inverted and that the amplificationcan be tuned across a substantial fraction of the spec-tral profile illustrated in Fig. 11.172. An estimate of thepeak spectral brightness achieved in the initial experi-ments [11.1525] gave a value of ≈ 1031 –1032 photons(s mm2 mr2)−1 and (0.1% bandwidth)−1, a range ap-proaching that required for single-molecule imaging inliving biological material [11.1536–1538].

In summary, the study of high-intensity interactionswith ultraviolet excimer lasers on atoms, molecules,and plasmas over the last 20 years has culminatedin the ability to produce new forms of matter thatare both highly excited and highly ordered [11.1539].A consequence is the capacity to achieve saturatedX-ray amplification in the multi-kilovolt regime at peakbrightness figures sufficient for the implementation ofbiological microimaging. Basically, the amplificationproceeds through the creation of a highly ordered ex-cited state [11.1525, 1535] that is comprised of fourmutually coupled components: atomic (ionic) matter,plasma electrons, and the two coherent radiative fields,

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Fig. 11.174 Comparison of the axially detected film #2shown in Fig. 11.173 with a single pulse spectrum(#030226B) recorded transversely, illustrating deep spec-tral hole-burning corresponding to the region of the Xe34+,Xe35+, and Xe36+ arrays. Since the hole burning descendsto the noise level of the detector, efficient energy extractionis evident. The width of the spectral gap is ≈ 60 eV, a valuesufficient for amplification of a pulse with a duration of≈ 30 as

which are the ultraviolet and X-ray waves propagatingin the channel. The chief consequence is well-orderedand efficient energy flow conducted through radiation-dominated interactions that are confined to a smallphase-space volume. Finally, it is significant to notethat ordered energy flow was the principal character-istic of excimer systems that made [11.1472] them soabundantly useful over the last three decades. Hence,a key feature of that history appears prominently againin the X-ray range.

11.7.4 Outlook:Radiation in the EUV

The roadmap of the semiconductor industry (Fig. 11.168)shows in which way computer chips with criticaldimensions of 32 nm and below are planned to bemanufactured: light sources with emission in the ex-treme ultraviolet (EUV) at a wavelength of 13.5 nm willbe used. EUV lithography is considered as the next-generation lithography (NGL) to be established after

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Fig. 11.175 EUV microexposure tool from Exitech with anintegrated EUV source from XTREME technologies (left)

reaching the physical limits of ultraviolet (immersion)lithography based on ArF excimer lasers.

13.5 nm TechnologyThe first EUV microexposure tools are already underoperation for technology development and feasibilitystudies (Fig. 11.175).

One of the biggest challenges concerning the in-troduction of EUV lithography is the development ofhigh-power EUV sources at a reasonable cost. Mak-ing the operation of the photolithography manufacturingtools economically viable requires a source with powerin the kilowatt range at the 13.5 nm exposure wave-length. In addition, the optical design of the collectoroptics sets limits on the source size, i. e., an emissionvolume of a few cubic millimeters. Only the combina-tion of both the high power and small emission volumewill enable high optical efficiency and therefore highwafer throughput. The goal is the development andmanufacturing of these high-power EUV sources aswell as their integration into the optical system of thephotolithography tool (Fig. 11.176).

Plasmas are known as efficient emitters of 13.5 nmEUV radiation, if their temperature reaches about200 000 C, i. e., approximately 35 times higher thanat the surface of the sun. Plasmas can be generated ei-ther by an electrical discharge or by means of pulsedlaser excitation. Both methods are able to generate smallplasma volumes fulfilling the technical requirementsof photolithography [11.1540, 1541]. The spatially ho-

Fig. 11.176 EUV plasma source integrated into an opticalsystem

mogeneously emitted radiation from such plasmas ispulsed with lengths in the nanosecond range (typically).The distribution of the wavelength spectrum is narrowfrom elements with low atomic numbers and broad fromelements with high atomic numbers. In addition, the ra-diation is almost fully incoherent, as from a thermalPlanck emitter.

Today the highest EUV power is achieved in so-called pinch plasma sources. However, the heat load onthe static electrode configuration can lead to fast erosionand even melting of the surfaces, thereby limiting thepower scaling through increasing repetition rates. Alter-native technologies with moving electrodes have beeninvestigated and finally a potential solution with rotatingdisc electrodes (RDE) has been found. A new excitationscheme has been applied to this technology at XTREMEtechnologies, resulting in a world record (170 mJ/2π sr)of the achieved EUV pulse energy [11.1542]. This effectresulted in a reasonable repetition rate to fulfill the powerrequirements of high volume manufacturing tools. Fur-ther development efforts will be directed to combine thepower-scaling capability with the reliability goals forthis concept.

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Lasers and Coherent Light Sources 11.8 Dye Lasers 777

11.8 Dye Lasers

11.8.1 Overview

The distinctive feature of dyes as a lasing media is thebroad emission band with a typical bandwidth of 50 nm.Laser action from dye molecules was first observed byseveral research groups in 1966 [11.1543, 1544]. Com-pared to the gas and solid-state lasers of the 1960s,dye lasers easily excelled both in terms of broad spec-tral coverage and in versatility of output performance(viz. high per-pulse energy output, long or short pulseoperation, flashlamp or laser pumping) Researchersquickly recognized the wavelength tuning capability ofdye lasers, which is probably one of the most impor-tant operation characteristics in view of applications. Inthe following years, literally hundreds of organic dyeswith emission spectra from the near ultraviolet to thenear infrared (300–1200 nm) have been made to lase.Continuous-wave (CW) or ultrashort pulse (femtosec-onds) operation of dye laser was also demonstrated. Withtheir broad spectral coverage and narrow linewidth tun-ability, dye lasers have been the workhorse in scientificlaboratories for applications ranging from fundamen-tal physics to clinical medicine for many years. Despitethe bad reputation of being prone to various mishapsduring operation (e.g., spilling of dye solution beinga not infrequent occurrence), there are many loyal fansof liquid dye lasers in the scientific community. Thispopular support is a tribute to the versatile output per-formance of these liquid lasers and the simplicity ofthe technology involved in constructing a liquid dyelaser. Almost any research lab can put together a flash-lamp with power supply, two mirrors, some commondye (e.g., rhodamine 6 G), and nuts and bolts to builda pulsed liquid dye laser. There are of course numeroususer-friendly and powerful liquid dye lasers available.We will first give a short description of the charac-teristics of liquid dye lasers and then discuss severaltypes that are commonly found in research laborato-ries for photophysical, photochemical and spectroscopyapplications.

11.8.2 General Description

Dyes are organic compounds that contain conjugate dou-ble bonds. The presence of the conjugate double bondsrenders these compounds optically active. There are over200 laser dyes that can, in principle, provide spectralcoverage from 320 to 1200 nm [11.1545]. The tuningrange of each dye is 40–60 nm. When used sequen-

tially, continuous tunable laser action can be obtainedfrom the near ultraviolet to the near infrared. Typi-cally a strongly absorbing and strongly emitting dyeis dissolved in a suitable solvent (e.g., a polar solventsuch as ethanol or a nonpolar solvent such as chloro-form) at concentrations of 10−3 –10−4 molar to serve asthe gain medium. A flashlamp or another laser is usedas the pump. Powerful coaxial flashlamp-pumped rho-damine 6 G dye lasers can deliver laser energy up to400 J per pulse [11.1546]. Average output power of upto 1.2 kW in burst mode were obtained in a flashlamp-pumped dye laser by Morton and Dragoo [11.1547].Since the advent of high-power short-wavelength lasersin the 1970s, excimer, nitrogen or frequency-multipliedNd:YAG lasers are frequently used for pumping liquiddye lasers with output pulse around 10 ns and energyoutput of tens of millijoules. A saturable absorber (an-other dye) is used to passively mode-lock the dye laserto generate pulses of about 200 fs without dispersiveelements and down to tens of fs with dispersive op-tical elements [11.1548]. For applications that requiresingle-longitudinal-mode laser output, CW operation ofdye lasers is achieved by using a modified flow systemto remove the long-lived triplet state of the dye mol-ecule. The linewidth of a free-running jet-stream CWdye laser can be as low as to 2 MHz [11.1549]. In thefollowing sections, we discuss several common dye laserarrangements.

11.8.3 Flashlamp-Pumped Dye Lasers

A flashlamp was the pump source for Maiman’s epoch-making ruby laser. Today’s flashlamps are still usedto pump liquid and solid-state lasers. Linear flash-

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Fig. 11.177 Schematics of a coaxial flashlamp-pumped dyelaser

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lamps are often the choice for laser pumping andusually consist of a tubular quartz envelope sealed bytungsten electrodes at both ends. The tube is oftenfilled with a heavy rare gas (e.g., xenon or kryp-ton) for high electrical-to-optical conversion efficiency.Still close to 90% of the electrical power ends upas heat, which requires a cooling system for its re-moval. Often, an elliptical reflector is used to focus theoutput from the flashlamp into the dye cell. For op-timum coupling of the light from the flashlamp intothe dye, the dye cell must be placed close to the lamp.Coaxial lamps have emitting surfaces that completelysurround the flowing dye. Figure 11.177 shows a coax-ial flashlamp-pumped dye laser, a structure commonlyemployed in many commercial dye as well as solid-state lasers. The dye cell is set at the center of thecoaxial, cylindrically shaped xenon flashlamp for op-timal pumping. Two broadband mirrors provide theoptical feedback. Such a flashlamp-pumped dye lasertypically produces a broadband emission (about 10 nmlinewidth) centered at the peak of the gain profilewith a pulse duration of 1–2 µs. The output energy isabout 100 mJ for most commercial flashlamp-pump dyelasers.

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Fig. 11.178 Schematics of a Hänsch-type cavity dye laser. A tele-scopic lens is used to improve the spectral resolving power of thegrating

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Fig. 11.179 Schematic of a colliding-pulse dye laser for ultrashort-pulse generation

11.8.4 Tunable Dye Lasers Pumpedby High-Power Short-WavelengthLasers

By incorporating wavelength-selective elements insidethe resonator cavity, dye laser output can be tuned.Increasingly excimer or frequency-multiplied Nd:YAGlasers are used to pump dye lasers. Excimer, nitrogenand frequency-multiplied Nd:YAG lasers all providehigh pump power (peak power of several MW) atshort wavelengths, making them highly suitable forpumping dye lasers. Dye lasers pumped by high-powershort-wavelength lasers exhibit such high gain that lossintroduced by the additional wavelength-selective el-ements inside the resonator cavity can be overcome.Either prisms or diffraction gratings can be used forwavelength selection, and gratings are generally superiorin terms of large dispersion and wavelength-resolvingpower. Figure 11.178 shows a variant of the telescopicgrating resonator originally used by Hänsch [11.1550]for the generation of narrow-linewidth laser output.A nitrogen laser provided the pumping. The use of thetelescopic lens increases the number of grooves illumi-nated by the laser light and reduces the light intensityon the grating, preventing damage to the coating on thegrating surface. The linewidth of a telescopic-gratingcavity dye laser is in the range of 0.1 nm. To furtherreduce the linewidth to 0.01–0.05 nm, an intracavityetalon (e.g., a coated optical flat or air-spaced Fabry–Pérot cavity) may be introduced. Many variants of thegrating-cavity (e.g., the hybrid prism–grating cavity) dyelasers have been developed to achieve narrow-linewidthoutput [11.1551].

11.8.5 Colliding-Pulse Mode-Locked DyeLasers

The broad emission band of a dye can be used ef-fectively to generate ultrashort laser pulses, since thetheoretical limit of the ultrashort-pulse duration sup-ported by the gain medium is proportional to the inverseof the gain bandwidth. As the emission bandwidth ofmany laser dyes is 40–50 nm, pulse durations down totens of fs can be produced. The development of passivemode-locking is essential to the successful demonstra-tion of femtosecond pulse generation in dye lasers. Inpassive mode-locking, a saturable (nonlinear) absorber(an absorbing dye) with an absorption that matches theemission wavelength of the lasing dye is placed insidethe resonator cavity. Ideally, the leading and trailingedges of the optical pulse are removed by the absorber

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Lasers and Coherent Light Sources 11.8 Dye Lasers 779

whilst the peak of the pulse, unaffected by the absorber,experiences amplification. The efficiency of the absorberincreases considerably if two oppositely traveling pulsesinteract or collide in the saturable absorber at the sametime (colliding-pulse mode locking). This is a result ofthe two coherent pulses interfering constructively, lead-ing to the reduction of power required for the saturationof the absorber. The colliding-pulse mode-locking con-figuration can be realized in either a linear or ring cavity.Because of the ease of alignment, the ring cavity isoften preferred. Figure 11.179 shows the type of ringlaser used originally by Fork et al. to generate pulsesshorter than 100 fs [11.1552]. Both the laser dye (e.g.,rhodamine 6 G) and the absorbing dye (e.g., DODCI)are in the form of a free-flowing stream jet, and a CWargon-ion laser serves as the pump. The radii of curva-ture of the mirrors must be carefully chosen so that thespot size in the absorber is smaller than that in the gainmedium to assure pulse-forming stability.

11.8.6 Tunable Continuous-Wave Dye Lasers

The linewidth of tunable pulsed dye lasers with intracav-ity dispersive element for narrow-linewidth operationare of the order of 500 MHz. In contrast, a continuous-wave dye laser is capable of delivering laser output withlinewidths as narrow as tens of kHz. The high resolvingpower is ideal for spectroscopy. Such high performanceis connected with high cost, complicated design anda high-power pump source (often a CW argon-ion laserwith 10–20 W output power). Continuous-wave dyelasers have much lower gain than their pulsed counterpart. The loss reduction becomes critical and the re-moval of the long-lived triplet state is crucial for thestable operation of the laser. A free-flowing jet is usedto effect rapid circulation of the dye. Since the gain pro-file of most dyes may be regarded as homogeneouslybroadened, one might expect single longitudinal-modeoutput from a simple grating or prism-cavity CW dyelaser. However, single longitudinal-mode operation maynot be possible in such linear-cavity dye lasers with-out intracavity frequency-selecting elements becausespatial hole burning can arise gain saturation by thestanding wave. Multimode output then results. Addi-tional frequency-selecting elements such as an etalonmust be inserted inside the cavity for single-mode op-eration increasing losses and lowering output power.Figure 11.180 shows a linear (standing-wave) cavityarrangement for a CW dye laser end-pumped by anargon-ion laser for single-mode operation. The dye jetflows perpendicularly to the page. The etalons assure

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narrow-linewidth output. The birefringent filters are forwavelength tuning. A ring-cavity supports the propaga-tion of traveling waves and thus is ideal for single-modeoperation. Figure 11.181 shows a tunable single-modering laser supporting single-mode operation. The op-tical diode (a Faraday isolator) ensures unidirectionaltravelling waves thus avoiding spatial hole burning.

11.8.7 Advanced Solid-State Dye Lasers

The contribution of liquid dye lasers to the advance oflaser technology, particularly to ultrafast laser technol-ogy and the many applications of tunable lasers, cannotbe overstated. Interested readers should consult mono-graphs such as Dye Lasers by Schäfer [11.1553] or DyeLaser Principles by Duarte and Hillman [11.1554] andreferences therein for more detailed information on therange of applications of dye lasers. The excellent out-

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put performance (as mentioned in the last paragraph)notwithstanding, dye lasers earn skepticism among theirusers. The main misgivings for dye lasers in liquid form,may be the disposal of the used dye and the maintenanceproblems associated with the physically large circulationloop for the dye flow. With the emergence of transition-metal solid-state lasers in 1980s, the position of dyelasers as the tunable laser-light source of choice has beenfacing a rising challenge. These solid-state lasers arebased on the vibronic transitions of 3d transition-metalions such as Cr3+, Ti3+, and Co2+ doped into oxideor fluoride crystals that served as the host [11.1555].These transitional-metal ion-doped crystals show broademission bands (typically 200–300 nm) in the visible tonear-infrared spectral region and thus are good tunable-laser candidates. The most well known among them isthe Ti3+:Al2O3 (titanium:sapphire) laser. Tunable ra-diation in the blue–green and in the ultraviolet can inprinciple be obtained by frequency doubling or triplingthe fundamental output of Ti3+:Al2O3 lasers. In additionoptical parametric oscillators (OPO) based on nonlin-ear optical crystals such as ADP or BBO (BaB2O4)are capable of providing tunable radiation well intothe infrared [11.1556]. The leading role of dye lasersas a tunable coherent-light source is indeed somewhatdiminished.

Challenges and OpportunitiesSolid-state lasers are often preferred because of theirruggedness and easy maintenance. User-friendly dyelasers must do without the flow loop that circulates thedye solvent. Solid-state dye lasers (SSDL) incorporatingthe dye molecules in solid matrices appear to be able tocombine the cost-effectiveness of liquid dye lasers andthe ruggedness of inorganic solid-state lasers. In orderfor SSDL to be a serious competitor to inorganic solid-state lasers in various fields of applications, the problemof photodegradation of dye molecules must first be dealtwith.

Good progress has been made in the synthesis ofphotostable dyes. The recently synthesized perylenefamily [11.1557] and the pyrromethene family [11.1558]of laser dyes have been shown to outperform rhodamine-6G in efficiency, tunability and photostability. Equallyimportant to the development of SSDL are the solid-state matrices that serve as host materials where therehas also been impressive progress. In the 1980s Avniret al. [11.1559] and Gromov et al. [11.1560] demon-strated dye-doped sol-gel materials and dye-dopedpolymeric materials, respectively, as promising lasermedia. Both sol-gel materials and polymeric materials

were shown to have good chemical stability and wideoptical transparency for use as host materials for laserdyes. The pace of research activity has increased con-siderably since then. In the sections to follow, we willbring the reader up to date with the latest development ofsolid-state dye lasers. The state of the art of solid-statedye lasers based on pure and hybrid sol-gel materials andon advanced polymers is covered in the next section. Themost current development of solid-state dye lasers maybe polymeric [11.1561] or sol-gel [11.1562] waveguidelasers using a distributed feedback configuration. Thesecompact lasers produce tunable narrow-linewidth outputand appear to be readily integrable into planar opticalcircuits. This is covered in the penultimate section. Fi-nally the topic of tunable upconverted DFB dye lasers ispresented.

Solid-State Dye Lasers Based on a Polymer Host

In 1967, one year after the demonstration of thefirst liquid dye lasers, Soffer and McFarland observedlasing in rhodamine-doped poly(methyl methacrylate)(PMMA) [11.1563]. The first polymeric hosts sufferedfrom large thermal coefficients, stress birefringence, op-tical inhomogeneity and chemical reactivity with laserdyes. The most serious problem of dye-doped poly-mers is the tendency of aggregation of dye molecules,which effectively quenches fluorescence. As a result,the performance of the initial solid-state dye laserswas less than satisfactory. The properties of PMMAcan, however, be improved by purification of themonomer, by introducing alcohol additives [11.1560]

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Fig. 11.182 Experimental arrangement of a sol-gel DFBwaveguide dye laser

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Lasers and Coherent Light Sources 11.8 Dye Lasers 781

and can be modified through copolymerization witha low molecular-weight polymer [11.1564]. The result isan improved polymer: modified poly(methyl methacry-late) (MPMMA) with the desired optical homogeneityand chemical stability suitable for use as a solid-statedye laser host [11.1565]. Work on the developmentof new polymeric hosts has been very active. Kingand coworkers showed that the photostability of solid-state dye lasers can be enhanced by deoxygenation ofthe host [11.1566] and the addition of a triplet-statequencher [11.1567]. Following the idea that photodegra-dation of dye molecules can be reduced by increasing therigidity and hence the rate of heat dissipation of the hostmatrix, Costela et al. [11.1568] prepared pyrromethenedye-doped copolymers of methyl methacrylate (MMA)and different methacrylic and acrylic crosslinking poly-mers. The combined use of a photostable dye and newpolymeric hosts results in SSDLs that boost the life-time exceeding 106 shots at a repetition rate of a fewHz when pumped at 532 nm. Narrow-linewidth opera-tion with a linewidth of 1.12 GHz has also been achievedin dye-doped MPMMA using a multiple-prism gratingresonator cavity [11.1565]. Further improvement of thepolymer host will undoubtedly lead to SSDLs that wouldgain the acceptance of the common laser users.

Solid-State Dye Lasers Based on a Sol-Gel HostThe sol-gel method is a low-temperature glass-makingtechnique that enables the introduction of organic dyesinto inorganic glasses. Porous glass can be obtainedvia the sol-gel route by hydrolysis and polyconden-sation of metal alkoxides [11.1568]. Initial studies ofsol-gel silica doped with organic dyes indicated thatsol-gel materials held good promise as SSDL host ma-terials because of their wide transparency range and theapparently excellent optical and thermal properties ofsilica [11.1559, 1569]. An added advantage is the highconcentration without aggregation and the photostabil-ity of a dye when trapped in sol-gel silica, as a resultof the isolation of dye molecules in the silica cage.Soon tunable laser action from sulforhodamine-dopedsol-gel silica was demonstrated [11.1570]. Lasing andfluorescence properties of a large number of dyes insol-gel silica that cover the spectral range from the nearultraviolet to the near infrared were fabricated and ex-amined [11.1571]. Several variants of sol-gel materialshave been used as solid-state dye-laser host materialswith varying degree of success. The first sol-gel mater-ial used in dye-laser experiments was in fact a glassy gelobtained by the gelation of a solution and is sometimescalled a xerogel [11.1572]. Xerogel derived from inor-

ganic precursors is mechanically fragile and opticallylossy due to the presence of numerous pores. The useof organically modified precursors or organic modifiersduring the sol-gel process results in organically modifiedsilicates (ORMOSILs) that show improved mechanicalstrength and much reduced optical loss [11.1565,1566].The combined use of the improved sol-gel materials ashost and the deoxygenation procedures during samplepreparation has lead to sol-gel dye lasers with lifetimeexceeding one million shots [11.1573].

One of the attractions of sol-gel materials is thepotential that such glassy materials can have prop-erties similar to glasses made via the traditionalhigh-temperature approach. Another attraction is theability of sol-gel materials to trap both organic andinorganic dopants while showing exceptional chem-ical stability. Whilst very few polymeric dye laserswork in the blue to near-UV range due to problemsrelated to attenuation and photostability under UV ex-citation, several UV laser dyes have been doped intosol-gel materials [11.1574, 1575]. Laser action withoutput wavelengths as short as 340 nm have been ob-served [11.1575].

From the recent development of polymer or sol-gel dye lasers, one must rely on the role of materialsengineering in advancing solid-state dye lasers.

Distributed Feedback Waveguide Dye LasersDistributed feedback (DFB) lasers are compact tunablelaser sources that produce narrow-linewidth output. Thefirst DFB laser was in fact a solid-state dye laser (dye-doped gelatin film), but it did not operate as a waveguidelaser because of the high film thickness [11.1576]. Insubsequent developments, the majority of DFB laserwork has concentrated on semiconductor lasers thathave been of obvious industrial significance. Recentlythere has been a renewal of interest in organic DFBlasers, particularly those based on waveguide struc-tures. This renewal in interest was caused in part by theapplication of conjugated polymers as luminescent ma-terials [11.1577, 1578] and the subsequent conjugatedpolymer laser experiments [11.1579, 1580]. Dye-dopedpolymeric materials or sol-gel materials can be pre-pared in a planar waveguide structure by simply usingspin-coating or dip-coating. Sol–gel materials have theadditional advantage of a larger range of refractive in-dex variation [11.1581, 1582] which allows integratedoptics application on a large number of polymer or glasssubstrates. DFB configuration seems ideally suited forlaser output generation in these waveguide structures.Polymeric [11.1583] and sol-gel [11.1562] DFB wave-

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Fig. 11.183 Schematics of Nd:YAG-microchip-pumped DFB dyelasers [11.1578]

guide dye lasers with tunable narrow-linewidth outputhave indeed been demonstrated.

Figure 11.182 shows the typical setup for the gen-eration of a sol-gel DFB waveguide dye laser. Thedynamic grating responsible for the lasing effect wasproduced by the crossing beams, which also served asthe pump. Tuning was by varying the incident angleof the beams and hence the grating period. The pe-riodic perturbation necessary for DFB lasing can alsobe produced by permanent morphological modulation

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+

Fig. 11.184 Scans of TE TM waveguide modes for a sol-geltitania–zirconia organically modified silicate waveguide ofthickness 6.7 µm and a refractive index of 1.56 on a glasssubstrate using a prism coupler

of the substrate surface. One of the interesting featuresof DFB waveguide dye lasers is the possibility of theproduction of tunable multiple-wavelength output. Okiet al. [11.1584, 1585] fabricated a multi-striped plasticwaveguide laser array that could generate multiple out-put wavelengths in the range 575–945 nm (Fig. 11.183).Another approach to obtain multiple output wavelengthsis to take advantage of waveguide structures that supportmultiple propagation modes [11.1586]. Figure 11.184shows the scan traces of a prism coupler for a titania–zirconia organically modified silicate waveguide ona glass substrate [11.1587]. The titania–zirconia organ-ically modified silicate film has a thickness of 6.7 µmand a refractive index of 1.56. Eight TE modes andeight TM modes were observed. The action of the cross-ing beams produced DFB laser output with wavelengthsthat obey the Bragg condition, λL = ηλp/M sin θ, whereη is the refractive index of the gain medium at λL, λpis the pump-laser wavelength, and M is the Bragg re-flection order. For DFB laser action in waveguides, ηtakes on the values of the effective indices for TEimodes or TMi modes (i. e., i varies from 0 to 7 fora 6.7 µm film). Figure 11.185 shows a typical DFB laseroutput spectrum with a polarizer that blocks the TMmodes [11.1587]. Without the polarizer all eight pairs ofTE/TM modes could be observed. It was found that thecrossing s-polarized beam generated purely TE modes,whilst pairs of TE/TM modes were produced when p-polarized beams were used. Furthermore, the separationbetween the modes and the number of modes can becontrolled by varying the waveguide parameters such

-%G 6

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$-

-; % %

Fig. 11.185 Output emission spectrum of a multiple wave-length sol-gel DFB waveguide dye laser. The conditions ofthe waveguide are the same as in Fig. 11.184

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Lasers and Coherent Light Sources 11.8 Dye Lasers 783

as the index difference and guiding-film thickness. Si-multaneous tuning of multiple output modes was alsoachieved (Fig. 11.186). Such compact tunable multiple-wavelength lasers should find wide applications fromanalytical spectroscopy to optical communications.

Distributed Feedback Laser Actionby Polarization Modulation

The original DFB laser theory describes the laser actioncaused by periodic perturbation of gain or refractive in-dex [11.1576, 1588] both of which could be generatedby intensity modulation. A schematic of a typical cross-ing beam experiment is illustrated in Fig. 11.187. Thetwo beams are shown to have their polarization direc-tions at an angle Φ. The crossing beams must both bes-polarized (Φ = 0) for the formation of an intensityinterference pattern (intensity modulation). The inten-sity interference pattern in the gain medium producesa concentration grating of excited-state atoms/moleculeswhich will provide the periodic change in gain or re-fractive index necessary for DFB laser action. Thecrossing of an s-polarized beam with a p-polarized beam(Φ = 90), however, does not produce an intensity in-terference pattern. Instead, a periodic change of thepolarization (polarization modulation) of the resultantfield that changes from linear polarization to ellipticalpolarization to circular polarization and then back to el-liptical polarization after one period. The grating thatresults from polarization modulation is a polarizationgrating.

Figure 11.188 shows the DFB laser emission spec-tra of rhodamoine 6G-doped zirconia waveguides as Φchanges from 0 to 90 for θ ≈ 44 [11.1587]. The pumpenergy used was 10 µJ. The situation at Φ = 0 corre-sponds to the case of pure intensity modulation. Thevariation of Φ changes the amplitude of the s-polarizedcomponent of the beam. As Φ increases, the effective-ness of intensity modulation weakens as the disparityin amplitude of the s-polarized components of the twocrossing beams grows, resulting in a low degree of mod-ulation in the transient intensity grating. At Φ = 60,the amplitude of the electric field of the s-polarizedcomponent is 1/2 of the companion beam. The effectof intensity modulation is already so weakened thata substantial ASE (attenuated spontaneous emission)background (ratio of the DFB output intensity to the ASEintensity of 10:3) appears in the emission spectrum. AtΦ = 90, DFB lasing is completely extinguished sincethe s-polarized component of one of the beams has zeroamplitude. The pump energy was then gradually raisedto about 30 µJ. DFB lasing reappeared at 30 µJ, but this

-;

%

-(

%

+,+,&

+,%

+,-+,

+,

+,

+,*

*B"

G 6

*-* **

Fig. 11.186 Wavelength tuning of a multiple wavelength sol-gelDFB waveguide dye laser. The conditions of the waveguide arethe same as in Fig. 11.184

time the feedback mechanism was provided by polar-ization modulation. The distinguishing feature of DFBlasing induced by the polarization modulation is the ap-pearance of a pair of TE0/TM0 output modes, whereasonly the TE0 mode is observed in the case of intensitymodulation.

Our latest experiments on polarization modulationhave revealed that DFB lasing can also be induced inliquid dye solution. The pump threshold is drastically re-duced by increasing θ. For θ larger than 75, as is the casefor DFB lasing of liquid oxazine dye near 800 nm un-der the first-order Bragg condition, the threshold pump

>!/ >!/ 4

Fig. 11.187 Schematics for polarization modulation DFBlaser experiments

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784 Part C Coherent and Incoherent Light Sources

G 6

B#$

$-

-; % %

S+,

G 6

B#$

$-

-; % %

%S+,

G 6

B#$

$-

-; % %

;S+,

G 6

B#$

$-

-; % %

(S

G 6

B#$

$-

-; % %

(S

+,

+

Fig. 11.188 Output emission spectra of sol-gel DFB waveguide dye lasers as Φ varies

energy for DFB lasing by polarization modulation is thesame as that by intensity modulation. Further work onDFB lasing by polarization modulation is underway.

Two-Photon-Pumped Solid-State Dye LasersCompact visible lasers by direct upconversion of in-frared light are versatile light sources that hold promise

for many applications in optoelectronics. Many dyes(e.g., R6G, DCM) have been known to emit weaklyin the visible when pumped in the near infrared. In-deed, broadband laser emission in the visible wasobserved in two-photon pumped polymer waveguideand fiber [11.1589, 1590]. A number of dyes withlarge two-photon pump upconverted absorption cross

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Lasers and Coherent Light Sources 11.9 Optical Parametric Oscillators 785

G 6

B#$

* % ;

$-

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Fig. 11.189 Absorption, two-photon-pumped fluorescenceand ASE spectra for an HMASPS-doped zirconia wave-guide

section have been synthesized recently. In particular,a styrylpyridinium dye: trans-4-[p-(N-hydroxyethyl-N-methylamino) styryl]-N-methylpyridinium p-toluenesulfonate (abbreviated as HMASPS) that shows strongemission in the red [11.1591] was doped in sol-gelzirconia thin films on a glass substrate [11.1592]. Fig-ure 11.189 shows the absorption, two-photon-pumpedfluorescence and ASE spectra from the HMASPS thin-film waveguides. A strong emission peak centered at620 nm is seen. Using two crossing 1.06 µm beams as

*

%

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G 6

- % &

%*

%

0-

0%*

Fig. 11.190 Tuning curves for DFB lasing in HMASPS-doped zirconia waveguides

the pump, DFB lasing was observed in the red witha linewidth as narrow as 45 GHz. Subsequent experi-ments demonstrated tuning by varying the intersectionangle of the two 1.06 µm crossing beams [11.1592].Figure 11.190 shows the tuning curves for second(M = 2) and third (M = 3) Bragg orders. A tuning rangeof 25–30 nm was observed. Extension of the spectralcoverage and improved output performance of the two-photon-pumped DFB lasing are expected as more newtwo-photon dyes are explored for DFB lasing study.

11.9 Optical Parametric Oscillators

Optical parametric oscillators (OPO) are based on non-linear frequency conversion of laser radiation into twocoherent light waves with lower frequencies, called thesignal and idler waves. The first optical nonlinear con-version experiment was demonstrated soon after theinvention of the laser by the observation of second-harmonic generation of ruby laser radiation in a quartzcrystal [11.1593]. This first result initiated intense ex-perimental and theoretical investigations to increasethe conversion efficiency and the available wavelengthrange. In the following several years the physics of thenonlinear frequency conversion was theoretically wellunderstood and all important methods of maximizingthe conversion efficiency were elaborated, includingquasi-phase matching [11.1594, 1595], birefringencephase matching in crystals [11.1596, 1597] and the re-quirement for optimum focusing [11.1598]. The firstoptical parametric oscillator was proposed in 1962 by

Kroll [11.1599] and realized in 1965 by Giordmaineand Miller [11.1600], followed by detailed theoreticaland experimental investigations [11.1601–1603]. Sev-eral review articles display a survey about the state ofthe art of optical parametric oscillation until the mid1980s [11.1604–1607].

However, despite this initial efforts, the success ofOPOs as a versatile tunable light source fell short of ex-pectations for almost a decade. At that time the reasonswere unsuitable birefringence of the crystals to permitphase matching, low transparency and damage thresh-olds, difficulties in growing large homogenous crystalsand the lack of reliable laser sources with good spatialand spectral properties, which are important for efficientfrequency conversion.

This situation changed due to significant advancesin crystal growth techniques in the mid 1980s for bothlaser-active and nonlinear crystals [11.1608,1609], lead-

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786 Part C Coherent and Incoherent Light Sources

ing to a renaissance of solid-state lasers. The availabilityof reliable pump sources with high spatial and spectralcoherence (in continuous and pulsed operation mode),low-loss optical components and the appearance of newor optimized nonlinear materials (BBO, LiB3O5 (LBO),KTiOPO4 (KTP), KNbO3) started a new OPO evolu-tion. Optical parametric oscillators are powerful devicesfor the efficient generation of widely tunable coher-ent light. The intense research in OPOs has resultedin reliable OPO systems, which are commercially avail-able. The tunability and the high output power definethe optical parametric oscillator as an attractive sourcein many applications including high-resolution spec-troscopy, environmental monitoring, medical research,process control, remote sensing and precision frequencymeasurements.

Recently a new generation of conversion mediahave appeared with periodically structured ferroelec-tric materials for quasi-phase matching. Together withthe development of diode-pumped all-solid-state lasersand in particular high-brightness diode-laser systemsnew perspectives are opened towards modern, compact,efficient, powerful, tunable OPO systems.

11.9.1 Optical Parametric Generation

The theoretical treatment of optical parametric gener-ation has been presented in the early 1960s [11.1610,1611] and is found in many textbooks and review arti-cles [11.1604–1606,1612–1614]. Therefore, this sectionwill provide only a brief description of the most impor-tant equations. The interaction of intense laser fieldswith a dielectric material induces nonlinear susceptibili-ties and causes the polarization of the medium to developnew frequency components not present in the incidentradiation field.

The lowest-order nonlinear susceptibility χ(2) isresponsible for three-wave mixing processes suchas second-harmonic generation (SHG), sum- anddifference-frequency mixing (SFG, DFG), optical rec-tification and the optical parametric generation (OPG).Optical parametric generation with a feedback resonatorfor at least one of the waves is called optical paramet-ric oscillation (OPO). The relation between the electricfield E and the polarization density P in the medium isgiven for a second-order susceptibility as

P = ε0(χ(1) E+χ(2) EE)

P = Pl + Pnl (11.159)

with ε0 the vacuum permittivity and the susceptibilityχ(k), which are tensors of the rank k for an anisotropic

medium. Because χ(2) is zero in centrosymmetric sys-tems three-wave mixing is only possible in non-isotropicmedia. Generally, the χ(2) tensor possesses 27 elements,but many of the components vanish under certain sym-metry conditions. The tensor can be reduced even toa scalar if the interacting waves are linearly polarizedand monochromatic for a fixed crystal orientation. In thiscase, the effective nonlinearity is denoted by a coefficientχ

(2)eff . In the literature [11.1614] the second-order suscep-

tibility is often expressed by a coefficient d defined byd := χ(2)/2.

The dynamics of optical parametric generation orthree wave-mixing processes in general can be describedby a set of nonlinear coupled differential equations forthe field amplitudes of the optical waves interacting withthe nonlinear medium.

Considering the basic wave equation derived fromMaxwell’s equations for nonmagnetic dielectric mediawhere Pnl is regarded as source radiation

∇2 E = µ0εd2

dt2E+µ0

d2

dt2Pnl (11.160)

and assuming

∂E1(z)

∂z= iκ1 E3(z)E∗

2(z)ei∆kz , (11.161)

∂E2(z)

∂z= iκ2 E3(z)E∗

1(z)ei∆kz , (11.162)

∂E3(z)

∂z= iκ3 E1(z)E2(z)e−i∆kz , (11.163)

with the coefficients κi =ωideff/nic0 (i = 1, 2, 3), deffthe nonlinear coefficient, z the propagation direction and∆k = k3 − k2 − k1 the phase mismatch.

This set of equations, the coupled wave equationsrepresents the interaction of three plane waves in a χ(2)

nonlinear dielectric. They have been solved exactly byArmstrong et al. [11.1594] for various input conditions.Depending on the initial conditions for the three complexamplitudes, these equations describe the various typesof three wave-mixing processes generated by the single-pass propagation of the electromagnetic field through thecrystal, such as second-harmonic generation, sum- anddifference-frequency mixing and parametric generation.

The optical parametric generation process whereonly one wave with frequency ω3 enters the dielec-tric medium and generates two waves with the lowerfrequencies ω1 and ω2 is based on the optical para-metric fluorescence starting from the quantum noise.This pure quantum mechanical effect was first pro-posed and studied in 1961 by Louisell et al. [11.1615].A quantum-mechanical model [11.1616] and semiclas-

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sical descriptions of the optical parametric fluorescencecan be found in the literature [11.1612, 1613, 1617].

The process of parametric amplification starts witha small field amplitude of the signal (ω1 = ωs) or idlerwave (ω2 = ωi). The relation between the frequencies isalways ωs +ωi = ωp and per the definition ωs ≥ ωi.

Introducing the definition for the intensityI = 1/2ε0nc|E|2 the coupled field equations lead to theManley–Rowe equation [11.1618] which is equivalentto photon conservation.

dIs

ωs dz= dIi

ωi dz= − dIp

ωp dz(11.164)

The conversion of every pump photon generates a pairof signal and corresponding idler photons. In this sensethe relation ωp = ωs +ωi corresponds to energy con-servation and ∆k = kp − ks − ki = 0 to the impulseconservation of the parametric process.

Neglecting pump field depletion (dEp/dz = 0) andassuming the presence of only the signal field (Ei = 0)in the beginning of parametric process the coupled waveequations can be solved analytically [11.1605, 1606].The single-pass fractional gain in signal intensity isobtained as

Gs(l) = Is(z = l)

Is(z = 0)−1 = Γ 2l2 sinh2 gl

(gl)2, (11.165)

where l is the length of the nonlinear medium and g thetotal gain factor

g =√Γ 2 −

(∆k

2

)2

, (11.166)

and Γ the parametric gain factor

Γ 2 = 2ωsωi|deff |2 Ip

npnsniε0c3 . (11.167)

The material-dependent part of the gain coefficient|deff |2/(npnsni) is called the figure of merit (FOM)and classifies the nonlinear quality of the conversionmedium.

In the high-gain limit the single-pass gain becomes

Gs =[

1+(

∆k

2g

)2]

sinh2 gl (11.168)

which reduces to

Gs(l) ∼= 1

4e2Γ l (11.169)

when ∆k < g.

In the low-gain limit where Γ 2l2 < (∆k/2)2 the gainof a parametric amplifier is

Gs(l) = Γ 2l2sin2[(

∆k2

)2 −Γ 2] 1

2l

[(

∆k2

)2 −Γ 2]

l. (11.170)

Considering perfect phase matching ∆k = 0, the single-pass signal gain in the low-gain limit approximates to

Gs(l) ∼= Γ 2l2 ∝ d2effl

2 (11.171)

It can be seen that the magnitude of the parametricgain depends on the intensity of the incoming field, aswell as on the material parameters such as nonlinearcoefficient, refractive index and the interaction length.However, the main condition for an efficient parametricamplification is the phase-matching condition. As shownin Fig. 11.191 the parametric gain reaches a maximumfor ∆k = 0, and decreases symmetrically to zero for|∆kl| = π.

The gain bandwidth is defined by∣∣∣∣(

1

2∆k2)

−Γ 2∣∣∣∣1/2

l = π , (11.172)

which reduces for low gain to(1

2

)∆kl = π . (11.173)

The gain bandwidth increases with higher gain, butuntil reaching Γ 2l2 ∼= π the bandwidth broadening issmall.

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$;

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Fig. 11.191 Parametric gain of a nonlinear crystal of lengthl as a function of the phase mismatch ∆kl. The gain ismaximum by optimum phase matching ∆kl = 0. The fullwidth at half-maximum (FHWM) of the gain function is0.88π

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11.9.2 Phase Matching

As shown before effective three-wave-mixing in a non-linear medium is only achieved if the phase-matchingcondition (∆k = kp −ks −ki = 0) is fulfilled. Togetherwith the frequency-matching condition (ωp = ωs +ωi),this condition implies the spatial phase matching of thethree waves which is necessary for constructive inter-ference of the interacting waves within the nonlinearmedium.

This conservation of energy and momentum is notpossible in materials with isotropic dispersion. Due tothe different diffraction indices, the optical fields of thethree frequencies differ in phase velocity and the relativephase of the interactive waves varies along the medium.A measure of the phase mismatch is the coherence lengthLc = π/|∆k|, which defines the distance over whichthe relative phase of the interacting waves shifts by π.The propagation beyond the coherence length, which istypically several µm, leads to back-conversion from thegenerated waves into the pump wave. The oscillatingbehavior of the generated intensity depending on theinteraction length is shown in Fig. 11.192.

Today there are two different techniques to real-ize a phase-matched conversion process: birefringencephase matching and the quasi-phase matching.

Birefringence Phase MatchingThis method uses the dependence of the refractive in-dex on the propagation direction and polarization ofelectromagnetic waves in birefringence crystals. Thewaves are distinguished into ordinary (o) and extraordi-nary (e) waves depending on their polarization direction.In uniaxial crystals the refractive index of the ordi-nary wave is independent from the propagation directionwhereas the refractive index of the extraordinary wavedepends on the polar angle between the optical axisand the propagation direction. To achieve phase match-ing the polarization of the high-frequency pump waveshould be selected so that the index of refraction issmallest. The phase-matching condition can be realizedby choosing an appropriate angle between propaga-tion direction and optical axis. Angular phase-matchedwavelength tuning is then possible by changing thephase-matching angle with regard to the phase-matchingcondition npωp = nsωs +niωi. The process is calledtype I phase matching when both generated waves havethe same polarization perpendicular to the polarizationof the pump wave, for example (e–oo). For type II phasematching the polarization of the generated waves areoriented perpendicular to each other (e–oe). An analyti-

6"

B#

*

H/

C

Fig. 11.192 Conversion intensity as function of the prop-agation length in a nonlinear crystal. NPM: Non-phase-matched process with no significant total conversionintensity. BPM: birefringence phase matching. Intensity in-creases quadratically with z, with a slope given by d2

eff,BPM.QPM: Quasi-phase-matched process. Intensity increasesquadratically with z on average, with a slope given byd2

eff,QPM d2eff,BPM

cal solution of the phase matching angle is only possiblefor type I phase matching in uniaxial crystals, the otherprocesses have to be solved numerically [11.1609]. Theanalysis in biaxial crystals is more complicate and hasonly numerical solutions; only in their principle planescan the formalism be reduced to the uniaxial condi-tions (Table 11.47 [11.1619, 1620]. A consequence ofangular birefringence phase matching is the spatial walk-off generated by the birefringence, which reduces theinteraction length of the optical waves. Besides the an-gular phase matching the indices of refraction can bematched by changing the crystal temperature (temper-ature phase matching). Combined with a propagationdirection perpendicular to the principal axis this methodis called noncritical phase matching (NCPM) and re-sults in the absence of spatial walk-off and less angularsensitivity, which is important in applications with tightfocusing.

The vector equation of the phase matching condi-tion can be realized in a non-collinear or a collinearway. In contrast to the collinear phase matching is thenon-collinear phase matching a vector phase matching.Figure 11.193 shows the non-collinear angle includedby the wavevectors. With the choice of a suitable non-collinear angle the angular acceptance is enlarged orthe Poynting vector walk-off can be compensated. Theconditions for the walk-off compensation are shownin Fig. 11.193a; the Poynting vectors of the ordinarysignal and the extraordinary pump wave are paral-

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Table 11.47 Refraction indices of ordinary and extraordinary wave for uniaxial and biaxial crystals in the principle planes

Crystal Extraordinary wave (e) Ordinary wave (o)

Uniaxial n e(Θ) = non e√n2

e cos2 Θ+n2o sin2

no(Θ) = no

Biaxial, xy-plane nxye (Φ) = nx ny√

n2x cos2 Φ+n2

y sin2 Φno(Φ) = nz

Biaxial, xz-plane nxze (Θ) = nx nz√

n2x cos2 Θ+n2

z sin2Θno(Θ) = ny

Biaxial, yz-plane nyze (Θ) = nynz√

n2y cos2 Θ+n2

z sin2 Θno(Θ) = nx

lel. Such walk-off compensation is of advantage fornanosecond OPOs reaching higher conversion efficien-cies [11.1621] and necessary for critical-phase-matchedOPOs pumped by ultrashort pulses because of the smallbeam radii [11.1622, 1623]. The diagram of tangentialphase matching is shown in Fig. 11.193b. This specialcase of non-collinear phase matching produces a signif-icant enlarged angular acceptance for the extraordinarypump wave. In addition, group-velocity matching forultrashort pulses is realized via non-collinear phasematching [11.1624].

Quasi-Phase MatchingAn alternative method to achieve phase matchingvia phase correction is quasi-phase matching (QPM).Here the relative phase between the three waves iscorrected using a periodic change in the sign ofthe nonlinear susceptibility (Fig. 11.192) before theback conversion starts. Although the technique ofQPM was proposed even before birefringence phasematching [11.1594], difficulties in fabrication of the pe-riodic structure in the range of the coherence length(typically 1–100 µm) have prevented its realization

"8

C'"

"8

C'"

Fig. 11.193a,b Wavevector diagramfor different non-collinear phase-matching schemes. (a) Walk-offcompensation, (b) scheme for tan-gential phase matching

for a long time. Recent advances in the fabricationof structured ferroelectrics meanwhile have estab-lished this technique for efficient frequency-conversionapplications.

Quasi-phase matching through periodically pol-ing offers two major advantages over birefringencephase matching. First, the polarization of the in-teracting waves can be equal. This allows the useof the largest χ(2) coefficient of the crystal, whichreduces the pump threshold and increases the con-version efficiency. The second advantage is that thequasi-phase matching of any combination of pump,signal and idler wavelength can be realized in suchmaterials via periodic poling. In addition, QPM pro-vides design flexibility of nonlinear conversion devicesby engineered domain structures. Applications withnovel configurations such as chirped gratings forpulse compression [11.1625], fan-out gratings forbroad tunability [11.1626], quasi-periodic structuresfor simultaneous operation of two three-wave-mixingprocesses [11.1627] and two-dimensional gratings formultiple direction phase-matching processes [11.1628]have been demonstrated.

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Phase Mismatch and Acceptance BandwidthsThe ideal phase-matched interaction, where the pumpwave and idler and signal waves propagate with thesame velocity through the nonlinear crystal, ∆k = 0,is affected under real experimental conditions by sev-eral parameters such as the divergence and bandwidthof the pump radiation, the bandwidth of the generatedradiation and an inhomogeneous temperature distribu-tion. The tolerances of the nonlinear material concerningto these perturbations are expressed by the acceptancebandwidths. They are defined by the FWHM of the para-metric gain width (Fig. 11.191) and developed in Taylorseries of ∆k(0):

Angular acceptance

∆δ= 2 × 0.886πl−1(δ∆k/δδ)−1 ;Gain band width

∆ωs = 2 × 0.886πl−1(δ∆k/δωs)−1 ;

Spectral acceptance

∆ωp = 2 × 0.886πl−1(δ∆k/δωp)−1 ;Temperature acceptance

∆T = 2 × 0.886πl−1(δ∆k/δT )−1 . (11.174)

11.9.3 Optical Parametric Oscillators

An optical parametric oscillator consists of three basiccomponents, namely the pump source, the gain medium,and the feedback resonator. For a qualitative descriptionof the operation principle we consider the most basicOPO setup, as shown schematically in Fig. 11.194.

An intense coherent light field produced by a laserbeam propagates through an optically nonlinear crystalwith the frequency ωp. The crystal is placed inside anoptical resonator, which resonates for reasons of sim-plicity with one wave, for example the signal wave.As the nonlinear crystal provides a sufficient nonlin-earity, parametric generation takes place and a pump

"

C"#

Fig. 11.194 Schematic setup of an OPO. An optical wavewith frequency νp from the pump source is converted intotwo output waves with frequencies νs and νi via three-waveinteraction in a nonlinear crystal. M1 and M2 are mirrorsof a feedback resonator

photon is converted into a signal and idler photon fulfill-ing photon-wise energy conservation ωp = ωs +ωi. Theresonator feeds the signal wave back into the crystalwhere it is further amplified through the power transferfrom the pump wave, as expressed in (11.164). Opticalparametric oscillation starts, when the increased signalintensity surpasses threshold that means the amplifiedsignal wave compensates for the round-trip losses inthe resonator (due to mirror transmission, absorption,scattering and diffraction). Reaching this distinct pumpintensity, the threshold pump intensity, a significant por-tion of the pump intensity is converted into signal andidler intensity. This power transfer from the pump to thegenerated waves reduces the (spatially averaged) pumpintensity inside the nonlinear crystal and thus the sig-nal gain. This effect is called gain saturation. It leadsto steady-state operation of the OPO, where the signalpower generated inside the crystal exactly balances theresonator losses for the signal wave.

As any pair of signal and idler photon can be gener-ated from the initially random vacuum field fluctuationsin steady-state operation, the signal and idler frequencypair that has the minimum threshold pump intensityis generated. Thus the generated frequencies, and withthem the ratio of the signal and idler frequency, are de-termined by the frequency dependence of the parametricgain in the crystal and the frequency dependence of theresonator losses.

A detailed description of the behavior of opticalparametric oscillators has been published in severalreviews [11.1604–1606] and the physics of OPOs isdiscussed more fundamentally in a number of textbooksabout nonlinear optics [11.1612–1614].

In the literature, optical parametric oscillators aremostly classified into three groups: continuous-wave op-tical parametric oscillators, nanosecond pulsed opticalparametric oscillators and synchronously pumped pi-cosecond or femtosecond optical parametric oscillators,depending on the temporal characteristic of their pumplaser.

This distinction is not only a practical classificationbut is important for the theoretical description and thebasic performance, such as pump intensity at threshold,conversion efficiency and spectral performance.

11.9.4 Design and Performanceof Optical Parametric Oscillators

The performance of an optical parametric system con-cerning tunability, high output intensity, spectral andspatial coherence is directly related to key elements

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of the OPO’s architecture, which are pump source,nonlinear material and feedback resonator. In general,the physics of the optical parametric process imposesseveral requirements of potential OPO components.

A simultaneous optimization of all OPO featurescannot be fairly done because of material problems andphysical conflicts. Therefore, a general description ofan optimized OPO design is not possible and the deviceconfiguration is always a compromise depending on therequirements of the specific application.

The choice of the pump laser depends together withthe selection of the nonlinear material on the requiredwavelength range and time resolution for a given ap-plication. Obviously, the pump laser has to be powerfulenough to drive the OPO well above threshold and guar-antee stable OPO operation. Since good spatial andspectral beam quality reduces the threshold, preferredpump sources will be high-power lasers of several watts,which also provide an excellent spatial beam quality,and good power and frequency stability. If the availablepump power is restricted, low beam divergence is of par-ticular importance to allow focusing into the nonlinearcrystal. The deviations in spectral and spatial coherencetolerated by the parametric frequency conversion are de-termined by the acceptance bandwidths of the nonlinearmaterial. Wavelength tunability of the pump laser offersan additional attractive feature: tuning of the OPO viathe pump wavelength.

The selection of the nonlinear material dependson the nonlinear parametric process, the experimentaldesign and the properties of the pump source. Funda-mentally, the nonlinear crystal should be distinguishedby a broad transparency range, a high optical-damagethreshold and large effective nonlinearity. An importantparameter is the birefringence of the crystal that hasto be high enough to maintain phase matching. How-ever, this effect should not be too large because anotherconsequence of birefringence is spatial walk-off, whichreduces the interacting length of the coupled fields.Large spectral and angular acceptance bandwidths areadvantageous because real laser systems show finitebandwidths and divergence and in ultrashort opticalparametric oscillation, the temporal walk-off and group-velocity dispersion between pump, signal and idler waveshould be low. For stable operation of OPO systems withhigh energies the thermo-mechanical and thermo-opticalstability of the crystal should be sufficient. Finally, thecrystals should be available in appropriate dimensionsand excellent optical homogeneity.

The design of the feedback resonator is strongly con-nected to the output characteristic of the pump source. If

the available pump power is restricted the OPO designmay have to be modified for lower threshold, whichmight however restrict the tuning capabilities of theOPO. Three types of feedback resonators are distin-guished, depending on the number of resonating waves.The singly resonant OPO (SRO) has highly reflectingmirrors at the signal or the idler wave. In principle, thisconfiguration is of particular advantage because it shouldprovide continuous wavelength tunability. The disad-vantage of these systems are their high pump power atthreshold, which exceeds several watts [11.1629,1630].Therefore SRO devices are mainly found for pulsed OPOconfigurations.

Doubly resonant OPO configurations (DRO) whereboth signal and idler waves are resonated have receivedmuch attention, because the pump power at thresh-old is reduced by one to three orders of magnitude.However, such DRO OPOs do not provide mode-hop-free tunability and often suffer from frequencyand power instabilities [11.1631, 1632]. Continuouswavelength tuning requires active stabilization tech-niques [11.1633, 1634].

The doubly resonant configuration where pumpand signal (or idler) waves are resonated is calledpump-enhanced SRO (PESRO). Such devices are veryattractive because they provide a compromise betweenthe low pump power at threshold of DROs and thewide tunability of SROs when the length of the OPOis carefully locked to a frequency-stabilized pumplaser [11.1635]. In addition, a continuous tunability ispossible with a pump-enhanced SRO, by using a res-onator internal beam splitter to separate the pumpspatially from the resonated OPO wave [11.1636]. Thesewaves are then resonated on two separate cavity endmirrors (dual cavity) for independent control of theresonated OPO wave and the length of the pump cavity.

Triply resonant OPOs (TRO) suffer from the com-bined problems of both types of DROs, consequentlythey provide less tunability in comparison to SRO andDRO, but they are remarkable because of their lowthreshold pump power in the range of several mW orbelow [11.1637, 1638].

Continuous-Wave Optical Parametric OscillatorsContinuous-wave optical parametric oscillators (CW-OPOs) are efficient and highly coherent light sourcesin the near- and mid-infrared. Due to their tunability,these sources are of considerable interest for high-resolution spectroscopy of molecules and trace-gasdetection [11.1639]. Simplified and improved frequencystandards are currently being considered, based on the

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strong frequency and phase correlation of the OPO’ssignal and idler output waves [11.1640, 1641].

After the first demonstration of CW OPOs in 1968by Smith et al. [11.1601] and Byer et al. [11.1602] sig-nificant progress is this field was made in the beginningof the 1990s, supported by advances in crystal growthtechniques [11.1609], low-loss optical components andCW solid-state lasers.

As discussed before pump power at threshold de-creases with increasing cavity finesse for each of thewavelengths. Typical values for the pump power atthreshold are listed in Table 11.48 for CW OPOs withthe different resonators (TRO, DRO, SRO).

The pump power at threshold of singly resonantOPOs is significantly higher than 1 W, even in a high-finesse cavity with no significant output coupling ofthe resonant wave. Since most commercially availablesingle-frequency lasers provide maximum pump pow-ers of about 1 W, most CW OPOs demonstrated in thepast are doubly or triply resonant devices.

The threshold pump intensity can be calculated fromthe coupled wave equations in analogy to the parametricgain assuming all three waves (pump, signal and idler)are infinite plane waves, with uniform intensity acrossthe beam, neglecting focusing and double refraction.The gain is considered to be unidirectional and the pumpdepletion is always small at threshold.

At the steady-state threshold, the gain balancesthe resonator losses for each resonator round-trip. Thethreshold condition for an SRO is

Es(0) = (1−αs)Es(lc) . (11.175)

Where αs are the round-trip losses of the resonantsignal wave and lc is the crystal length. In the low-gainlimit and assuming small losses, the parametric gaincoefficient (11.167) can be written [11.1606]

Γ 2l2c = 2αs . (11.176)

Then we obtain the pump intensity at the SROthreshold

Ip,th = αsnpnsni

ωsωid2effl

2c

sinc−2(

∆klc

2

). (11.177)

Table 11.48 Typical pump power at threshold for CW TRO,DRO, SRO

OPO Pump threshold (mW) Ref.

TRO < 1 [11.1633]

DRO 100 [11.1642]

SRO 2600 [11.1643]

With Γ 2l2c = αιαs the pump intensity at the DRO

threshold is similarly derived as

Ip,th = αiαs

4

npnsni

ωsωid2effl

2c

sinc−2(

∆klc

2

). (11.178)

A low threshold is achieved when a long nonlinearcrystal with a high figure of merit is used inside a sig-nal resonator with highly reflecting mirrors. In generalthe pump threshold is lower for higher signal and idlerfrequencies. However, via the sinc function of the phasemismatch the pump threshold is exposed to an additionaland much stronger dependence on the signal and idlerwavelength. A minimum threshold is achieved for signaland idler wavelengths with ∆k = 0. As a consequencethe OPO always seeks to oscillate as close as possible tothe phase-matched wavelengths.

Usually CW OPOs are pumped with focusedbeams to enhance the pump power, therefore pump,idler and signal waves are expressed as Gaussianbeams [11.1644]. In this case the intensity shows aninhomogeneous distribution along the propagation axesz and the intensity and phases are not constant withineach of the xy-planes. Only for loose focussing canthese inhomogeneities be neglected and is equation(11.177) still a good estimation of the pump intensityat threshold. The pump power at threshold for tight fo-cussing has been derived in an detailed mathematicaltreatment [11.1614, 1645], which is beyond this scope.For optimized confocal focussing [11.1598], the pumppower at threshold is given by

Pp,th = 4πc2nsniαs

µ0ωsωiωpd2efflc

. (11.179)

As opposed to the l−2c dependence of the threshold in-

tensity for plane waves (11.177) for the pump powerof Gaussian beams decreases linearly with the crystallength.

Once above threshold a part of the pump power Pp isconverted into the appropriate signal and idler waves aslong as the effective gain is larger than one. An impor-tant parameter is the pump depletion, which occurs untilthe gain is saturated and the steady state is reached.The analytical description of the internal conversionefficiency ηint can be obtained under steady-state condi-tions assuming small cavity losses and optimum phasematching. The conversion efficiency

ηint = 1− Pout,p

Pin,p(11.180)

depends on the ratio of the incoming and the transmit-ted pump power and is a function of the pump ratio

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N0(N0 = Pp/Pp,th) [11.1645]. The maximum conver-sion efficiency for plane waves comes to 100% fora pump ratio of N0 = (π/2)2. The extension of thismodel to Gaussian intensity distributions with planewavefronts yields maximal conversion efficiency of 71%supposing a pump ratio of 6.5. The conversion efficiencyis reduced by the inhomogeneous intensity distribu-tion of the pump wave as well as distortions of thewavefronts.

Due to parasitical losses in the cavity (Fresnel lossesat the surfaces, diffraction and absorption), the measuredoutput power is lower than predicted by the internalefficiency. Therefore, it is useful to determine an external(utilizable) conversion efficiency, which in steady-stateconditions is

ηext,s = T

T + L

ωs

ωpη , (11.181)

assuming small transmission T of the output mirror andsmall losses L for the resonant signal wave.

In order to operate the CW OPO continuouslywith low fluctuations in output power and frequency,the cavity length needs to be stabilized. The mainsources of cavity length fluctuations are thermally in-duced fluctuations of the refractive index, mostly in thecrystal, and acoustically and thermally induced pertur-bations of the position of the cavity mirrors. Fluctuationscaused by these sources are usually significant justfor low-frequency ranges, i. e., ν < 10 kHz, which canbe compensated, for example, by an electronic sta-bilization device. (For higher-frequency components,i. e., ν > 10 kHz, the noise of the pump laser is sig-nificant instead of the cavity length fluctuations). Foran OPO with pump enhancement resonance, a changeof the cavity length induces a mismatch between thecavity resonance frequency and the pump laser fre-quency, resulting in an increased OPO threshold anddecreased output power. If the OPO is resonant for thesignal or the idler wave or for both simultaneously, thechange of the cavity length leads to fluctuations of thesignal and idler frequencies and to mode hops and clus-ter hops. One possible method to stabilize the cavitylength is to increase the passive stability in a monolithicsetup [11.1646]. However, pump-enhanced OPOs anddoubly resonant OPOs usually require active control ofthe cavity length [11.1647]. To stabilize a cavity reso-nance to a laser frequency several methods have beensuccessfully demonstrated [11.1648–1650]. Long-termstabilization of a doubly resonant parametric oscillatorwhere the frequency instability is below 1 kHz has beendemonstrated [11.1651]

One of the most crucial components of CW OPOsis the nonlinear material. For successful CW OPO op-eration it is of particular importance that the crystalsnonlinearity is high and noncritical phase match-ing is possible. Crystals used so far are noncriticalphase-matched LiNbO3, KTP, LBO, Ba2NaNb5O15and KNbO3 whereas especially KTP and its arsenateisomorphs KTiOAsO4 (KTA) and RbTiOAsO4 (RTA)turned out to be attractive materials for CW OPOs.Wavelength tuning of NCPM OPOs requires eithertemperature tuning of the crystal or wavelength tun-ing of the pump laser. Because of the low-temperaturetuning capability of the KTP crystal family wave-length tuning in such OPOs depends on tunable pumpsources. Many investigations have been performed withDROs and pump-enhanced OPOs to achieve continu-ous tunability. A significant increase of the continuoustuning range was realized with the dual-cavity de-sign [11.1634,1636,1652]. Low-threshold operation hasbeen demonstrated with a TRO and pump-enhancedSRO KTP OPO pumped with a single-stripe GaAlAsdiode laser [11.1652, 1653]. High idler powers of840 mW in the wavelength range of 2.4–2.9 µm havebeen achieved with KTP and KTA OPOs designedas an intracavity SRO configuration and pumped byTi:sapphire lasers [11.1654]. An important step was therealization of a singly resonant CW OPO [11.1629].

In recent years the availability of novel quasi-phase-matched nonlinear material and high-power solid-statelasers enabled significant advances in the developmentof CW OPOs. The high optical nonlinearity of periodi-cally poled LiNbO3 (PPLN) and long interaction lengthsup to 50 mm enables the operation of CW SROs pumpedby CW solid-state lasers. Pumping such a 50 mm-longPPLN crystal with the 13.5 W output of a high-powerdiode-pumped Nd:YAG laser, infrared radiation tunablein the range of 3.25–3.95 µm with an output power of3.6 W has been reported [11.1655]. The measured pumpdepletion was as high as 93%. Wavelength tuning wasachieved through temperature tuning or by using differ-ent poling periods implemented on the crystal. A steptoward compact efficient and powerful CW OPOs wasperformed with the realization of a diode-laser-pumpedsingly resonant PPLN OPO [11.1656]. The SRO con-sists of a 38 mm-long PPLN in a four-mirror cavity(Fig. 11.195). Pumping with 25 W of 925 nm laser radi-ation from an AlGaAs master-oscillator power-amplifier(MOPA) system, 480 mW of single-frequency idler ra-diation was generated. The tuning range of the idlercovers 2.03–2.29 µm. In addition several investigationshave been performed with pump-enhanced SROs in

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794 Part C Coherent and Incoherent Light Sources

>0

*

!

BC

Fig. 11.195 Experimental setup of the diode laser pumpedsingly resonant OPO (SRO). The pump beam from theMOPA system propagates through a 60 dB isolator (iso)and focuses into the PPLN crystal. All mirrors of the ringcavity are highly reflecting for the signal wave and highlytransmitting for the pump and idler wave. The idler outputbehind M2 is filtered from the residual pump and signalradiation by dielectric filters [11.1656]

PPLN. Tunable radiation was generated in the rangeof 2.29–2.96 µm with an output power of 140 mWpumped by 800 mW delivered from a miniature diode-pumped single-frequency Nd:YAG laser [11.1657]. Anextension of the wavelength into the mid-IR between4.07–5.26 µm was demonstrated by pumping a PPLNPESRO by CW single-frequency Ti:sapphire radia-tion [11.1658]. Another pumping design was performedusing a high-power fiber laser. The Yb-doped fiber lasergenerated in a 40 mm-long PPLN crystal mid-IR idlerpowers in the range 2.98–3.7 µm. Pumping with 8.3 Wlaser power the OPO generated 1.9 W [11.1659].

Besides the successful demonstration of powerfulPPLN OPOs one severe disadvantage of PPLN is theappearance of photorefractive effects, which may dam-age the crystal. Some effort is made to optimize thegrowing technique in such a way that these defects canbe avoided.

Other QPM materials have also been developed,such as periodically poled KTP (PPKTP) [11.1662–1664], RTA (PPRTA) [11.1665] and Lithium tantalate(PPLT) [11.1660]. Due to the high purity of these ma-terials photorefractive defects do not occur even at highpump levels.

Wide wavelength tunability has been reported in therange 1.5–4 µm in PPLN [11.1666] and 1.55–2.3 µmin PPLT[11.1660] pumping the OPOs near degeneracy,which provides large parametric gain bandwidths fora given poling period, crystal temperature and pumpwavelength (Fig. 11.196). In PPLT the wavelength tun-ing is achieved by changing the crystal temperaturewithin an interval of only 10 K. Operating a CW OPOwith subsequent frequency-doubling wavelength tun-ability from the visible to the mid-IR range between550–2830 nm has recently been demonstrated. A com-

*&%

.# S.

5 6

&

;

(

*; *( -

$A

*%

B

4

Fig. 11.196 Temperature tuning curve of a near-degeneratePPLT OPO calculated for a pump wave of 925 nm anda grating period of 27.3 µm [11.1660]. Shaded areas: para-metric gain bandwidth of the 35 mm long PPLT crystal atdegeneracy is as large as 460 nm. The Sellmeier coefficientscan be found in [11.1661]

parison between the performance of PPKTP and PPLNwas performed within these investigations [11.1664].

These cited examples clearly indicate the dynamic inthis field. Further research in new ferroelectrics generat-ing radiation beyond 5 µm and the improvement of theexisting materials concerning dimensions, homogeneityand structure design will lead to high-efficiency compactintegrated optical systems.

Nanosecond Optical Parametric OscillatorsOptical parametric oscillators pumped with nanosecondpulses are the most well-established OPO systems gen-erating powerful tunable radiation in the spectral rangefrom the visible to the near infrared. Since the firstdemonstrated optical parametric oscillator [11.1600],which was a nanosecond pulsed OPO, reliable nanosec-ond OPOs were not established before the late 1980s.The appearance of new nonlinear materials with largenonlinearity and high damage thresholds (larger than1 J/cm2) such as BBO, LBO, and the improvement ofthe optical quality of KTP or KNbO3 was the start ofan intense OPO evolution. These materials together withthe optimized spatial beam quality and high peak powersof modern nanosecond Q-switched lasers made it pos-sible to reach the threshold of the single-resonant OPOconfiguration.

The basic principle of optical parametric oscillationfor nanosecond OPOs is the same as for CW oscilla-

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tors; however, due to the finite length of the pump pulse(typical pulse lengths are 2–30 ns) the OPO does notreach the steady-state regime. In nanosecond OPOs, thecavity length is about a few centimeters, which corre-sponds to less than 100 round-trips for the pump pulse.The steady-state threshold is no longer valid becauseof the transient character of pulsed OPOs. The dy-namic behavior was considered in a model developedby Brosnan and Byer [11.1667] using a time-dependentgain analysis. They calculated the buildup to thresholdfor a signal resonant optical parametric oscillator byassuming a Gaussian temporal profile for the incidentpump pulse intensity and a Gaussian spatial distribu-tion for the pump and the signal beams. Additionallytheir model assumes single pass of the pump waveand considers the effects of mode overlap and spatialwalk-off.

They introduced the time-dependent parametric gaincoefficient Γt

Γt = √gsΓ e−( t

τ )2

with gs = ω2p

ω2p +ω2

s,

(11.182)

where τ is the 1/e2 intensity half-width of the Gaussianpump pulse. The spatial coupling efficient gs describethe mode overlap between the resonant signal wave andthe pump wave. The influence of the Poynting vectorwalk-off defines the effective parametric gain length Λ

Λ= lωerf

(√π

2

lclω

)(11.183)

with the walk-off length, which depends on the birefrin-gence angle ρ,

lω =√π

2

ω

ρ

√√√√ ω2p +ω2

s

ω2p +ω2

s/2. (11.184)

The threshold energy fluence for the single-resonantOPO is derived from the coupled wave equations, as-suming low pump depletion, as

Jth = 2.25

κgsΛ2τ

×

(lr

2τcln

Ps,th

Ps,0+2α2lc + ln

1√R

+ ln 2

)2

(11.185)

with κ = 2ωsωid2eff

npnsniε0c3 , where R is the reflectivity of themirrors, α2 are the internal losses of the resonant waveinside the crystal, lr is the optical cavity length and lc is

the length of the crystal. The threshold identifies the de-tectable level of the signal energy, which corresponds toan energy of typical 100 µJ, giving a threshold power tonoise ratio of ln(Pth/Ps) = 33. The time required fromthe parametric gain of the signal field starting from thenoise level P0 to reach the oscillation threshold Pth iscalled the rise time of the parametric oscillator. Therise time is an important parameter in the design andoperation of a parametric oscillator and can be cal-culated from the time-dependent coupled amplitudesequations. For efficient conversion, the rise time hasto be as short as possible. Pearson et al. [11.1668]carried out a detailed investigation of the rise time asa function of various experimental parameters, such aspump pulse length, cavity length, cavity losses and thepump ratio N0, which is the factor between the pumppulse energy and the pump pulse energy at threshold,N0 = Pp/Pp,th. Important experimental parameters tominimize the rise time are high pump energy densitiesand a short cavity length. Beyond the influence of therise time the threshold energy density can be reducedby using long pump pulse durations τ , optimizing thereflectivity R and avoiding intracavity losses α. Addi-tionally the nonlinearity of the crystal should be largeand the length of the crystal should be as long as pos-sible with regard to the limitation through the Poyntingvector walk-off.

The spectral bandwidth of a singly resonant opti-cal parametric oscillator depends primarily on the gainbandwidth, which is determined by the dispersion, thebirefringence and the length of the nonlinear crystal.In addition, the bandwidth is determined by the spec-tral properties of the resonator, the correlation betweenthe resonator modes and the characteristics of the pumplaser, the wavelength, spectral bandwidth, intensity anddivergence. Because of the short pump pulse durationfor nanosecond pumped OPOs there is only a weakspectral-mode condensation and the nanosecond OPOdoes not reach the stationary state, unlike the CW OPO.Usually the nanosecond OPO bandwidth is estimatedwith the gain bandwidth, as there is no exact analyticaldescription of the spectral bandwidth of pulsed OPOs.Neglecting any saturation Brosnan and Byer [11.1667]deduced an expression for the spectral condensationdepending on the number p of cavity round-trips

∆ν(p) = 1√p∆ν . (11.186)

A more exact prediction concerning the theoreticalbandwidth can be calculated using numerically simu-lations [11.1669, 1670].

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Nanosecond pulsed optical parametric oscillators arepowerful devices of widely tunable coherent light. Withthe nonlinear crystals BBO and LBO such OPOs pro-vide coherent radiation in the entire spectral range fromthe UV (300 nm) to the near IR (2.5 µm). The advan-tageous properties of these crystals are high opticaldamage threshold, high nonlinearities, a wide trans-parency range and a large birefringence. Since the firstsuccessful operation of a BBO OPO reported by Fanet al. in 1989 [11.1671] significant progress in OPO tech-nology has been achieved. In the investigations reportedso far, BBO OPOs have been pumped with the second,third, and fourth harmonics of a Q-switched Nd:YAGlaser at 532 nm [11.1671,1672], 355 nm [11.1672–1677]and 266 nm [11.1672, 1678], and with the fundamentalof an XeCl excimer laser at 308 nm [11.1679,1680]. Thehighest output energies and efficiencies have been ob-tained with the third harmonic of Q-switched Nd:YAGlaser. The generated output energies are in the range100–200 mJ/pulse and external conversion efficienciesof up to 61% have been reported [11.1672, 1681].

Similar advances have been demonstrated forthe LBO OPO that covers almost the same spec-tral range [11.1669, 1682–1688]. The power densitiesat threshold for the 355 nm-pumped LBO OPO aretypically higher by a factor of two compared tothose of equivalent BBO OPOs due to the lowereffective nonlinear coefficient. However, LBO pos-sesses a smaller spatial walk-off angle than BBO[ρ(LBO) ≈ 0.25ρ(BBO)] allowing the use of longerLBO crystals to compensate the lower nonlinearity. Theexperimental comparisons of LBO and BBO [11.1669,1686] demonstrate the advantageous properties of LBOas there are large angular acceptance, low spatial walk-off (which can even be zero for temperature-tunednoncritical phase matching). These properties are valu-able if the OPO is to be operated with low-pump-energylaser sources (< 3 mJ) [11.1689], whereas BBO withrestricted angular acceptance and larger nonlinearityis more suitable for high-pump-energy laser sources(> 100 mJ).

In the past many investigations were performed pri-marily for the type I critically phase matched BBOOPO pumped by the third harmonic of a Nd:YAG laser.This well-established standard OPO consists typically ofa 12–15 mm-long BBO crystal placed in a linear signalresonant Fabry–Pérot resonator. Changing the phase-matching angle in the range of 10 the OPO coversa tuning range of 400–710 nm for the signal wave and710–3100 nm for the idler wave. Experimentally theidler tuning is restricted by an increase of absorption in

the crystal for wavelengths longer than 2600 nm. Thepump energy density at threshold is about 200 mJ/cm2.Pumped with 2–10 ns-long pulses such OPO devices op-erate routinely with conversion efficiencies in the rangeof 20–50% whereby the optical damage threshold of theOPO components limits the pump ratio. The spectralbandwidth of the type I phase-matched 355 nm-pumpedBBO OPO varies from 0.2 nm in the blue spectralrange (λ= 400 nm) to more than 10 nm at degeneracy(λ= 710 nm). The spectral bandwidth of the idler waveequals (expressed in wavenumbers or GHz) that of thecorresponding signal frequency.

In the past, many efforts have been made to re-duce the spectral bandwidth of OPOs. Significant linenarrowing, even to single-mode operation, can beachieved with the insertion of frequency selective el-ements (such as gratings or etalons) into the OPOresonator or with external injection seeding of the OPO.Although these methods can provide single-mode oper-ation, the disadvantages are obvious. First, the additionof frequency-selective elements increases the resonatorlength, the cavity losses and thus the oscillation thresh-old. Therefore, higher pump densities are required anda compromise is needed between the necessary pumppower and the damage threshold of the optical compo-nents. Second, continuous wavelength tuning may beaffected by the complexity of such devices. The spec-tral line narrowing via injection seeding avoids thesedisadvantages; the threshold is even reduced, becauseof a shorter rise time [11.1690]. Nevertheless, success-ful injection seeding in pulsed OPOs depends on manyfundamental parameters of the OPO operation. Besidesproper collinear alignment and divergence adaptationbetween the seed radiation and the seeded OPO wave,the frequency matching between the cavity mode andthe seed frequency is essential. Despite these difficultiessuccessful line narrowing was demonstrated with bothmethods [11.1675, 1681, 1691–1693]. A theoretical de-scription of the spectral characteristic of free-runningor injection-seeded OPO systems are given in severalnumerical models [11.1669, 1670, 1694, 1695].

The investigation and the control of the spatial beamquality of nanosecond OPOs is still a challenge. As a re-sult of large pump-beam diameters, high nonlinear gainin the crystal and a large Fresnel number of the opticalresonator the beam quality factor M2 of the generatedradiation are often high > 2–10 and thus far from thediffraction limit (M2 = 1). Most of the numerical modelssimulating the spatial behavior of nanoseconds OPO aretime integrated [11.1696–1699]. Recently time-resolvedexperimental and numerical investigation of the spec-

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tral and spatial dynamics of nanosecond OPO has beenreported [11.1700, 1701]. The result of the different in-vestigations is, that the OPO starts on the optical axiswith an almost Gaussian beam profile. However, duringthe buildup of the pulse the fields undergo an inhomoge-neous gain caused by the birefringence walk-off in theOPO crystal, pump depletion and back-conversion andas consequence the M2 value increases with the size ofthe pump energy, pump depletion and back-conversion.

Some investigations of the spectral and tempo-ral dynamics of the OPO has been performed usinga BBO OPO, which consists of a 2–3 mm-long crys-tal in a 3.5 mm-long cavity [11.1670, 1701]. Due to thewide mode spacing of about 1 cm−1 each single modecan be considered. Figure 11.197 shows the statisti-cal mode fluctuation of 10 successive pulses. Thus, theshort-cavity OPO proofs the macroscopic manifestationof the zero-point fluctuation of the vacuum.

Based on the numerical and experimental resultssome improvement on the spectral and spatial perfor-mance of nanosecond OPOs were developed based onnovel pumping and resonator configurations [11.1621,1702–1708].

For the generation of wavelengths up to 5 µm the ap-propriate crystals are KTP, KTA and KNbO3 [11.1709–1712]. Further extension to wavelengths beyond5 µm can be achieved in ZnGeP2, CdSe andAgGaSe2 [11.1713–1717]. Avoiding two-photon ab-sorption in the materials these crystals have to beoperated with pump wavelengths longer than 1.5 µm.Meanwhile nanosecond OPOs are demonstrated witha tuning range up to 14 µm. They provide conversion ef-

45 6-*($; -- --$

Fig. 11.197 Mode spectra of the signal wave (550 nm)recorded for 10 successive OPO pulses. The OPO is op-erated at 1.44 times above threshold. The mode spacing is28.8 GHz (0.96 cm−1) [11.1670]

ficiencies up to 40%, generating output powers of about5 W or pulse energies of 5 mJ [11.1716, 1717]. The dif-ficulty of finding the appropriate pump lasers for thesematerials has been overcome with cascaded OPO sys-tems published for AgGaSe2 and CdSe [11.1717–1720].

A new generation of nanosecond OPOs appearedwith the availability of periodically poled QPM ma-terials as there are PPLN, PPKTP, PPKTA andPPRTA. These crystals have large effective nonlinear-ities (8–16 pm/V), up to eight times higher than ina corresponding BBO crystal, and possess long in-teraction lengths. Typically the aperture of the QPMcrystals are 1 mm high. Therefore, these materialsare well suited for the generation of mid-IR radi-ation pumped with high-repetition low-power-pumppulses [11.1721–1723]. With the development of widelarge-aperture PPRTA crystals, QPM becomes suitablealso for high-power applications [11.1724, 1725]. Thelarge nonlinearity of PPLN enables the operation of effi-cient single-pass optical parametric oscillators [11.1721,1726–1728]. More recently tunable single-frequencygeneration has been demonstrated with injection-seededPPLN OPOs and PPLN OPGs [11.1729–1731].

Synchronously Pumped Optical ParametricOscillators

Optical parametric oscillators synchronously pumpedby mode-locked solid-state lasers are powerful devicesfor the generation of tunable ultrashort laser pulses. Thefirst CW synchronously pumped picosecond OPO wasdemonstrated by Piskarskas et al. in 1988 [11.1732] andthe first synchronously pumped femtosecond OPO byEdelstein et al. in 1989 [11.1733].

Many reports of such OPOs generating pulses inthe picosecond or femtosecond regime have been pub-lished in recent years [11.1734–1736]. Ultrafast lasersystems are attractive pump sources for optical paramet-ric oscillators. They combine high peak pulse intensitiesproviding sufficient nonlinear gain with a moderate en-ergy fluence preventing optical damage of the material.However, remembering the instantaneous character ofnonlinear polarization the optical gain is obtained exclu-sively during the time interval of the pump pulses length.In contrast to nanosecond OPOs, the time interval of ul-trashort pulses (τ < 100 ps) is too short to allow a finitenumber of cavity round-trips. Therefore, the basic oper-ation principle of ultrafast optical parametric oscillatorsis synchronous pumping to achieve macroscopic ampli-fication of the parametric waves from quantum noise.In this pumping scheme the length of the optical para-metric oscillator is matched to that of the pump laser

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resonator, so that the round-trip transit time in the OPOresonator corresponds to the repetition time of the pumppulses. The consecutive coincidence of parametric andpump wave in the nonlinear crystal amplifies the gainuntil it overcomes the threshold.

In general, two types of synchronously pumpedOPOs are to be distinguished: the pulsed (or quasi-CW) synchronously and the continuous synchronouslypumped OPOs. CW synchronously pumped OPOs arepumped with a continuous train of ultrashort pulses asthey are emitted by CW mode-locked neodymium lasersor Kerr lens mode-locked Ti:sapphire lasers, while thepump sources of pulsed synchronously pumped OPOsare mode locked Q-switched Nd lasers generating a trainof ultrashort pulses contained within a nanosecond ormicrosecond envelope.

The physical description of CW synchronouslypumped OPOs is equivalent to the steady-state for-malism for CW OPOs considering the peak pumppulse intensity determining the nonlinear gain. Pulsedsynchronously pumped OPOs show the transient be-havior of nanosecond OPOs so their analysis has totake into account the rise-time effects caused by thefinite time duration of the nanosecond pulsed enve-lope [11.1667]. An analysis of synchronously pumpedOPOs can be found for plane-wave conditions [11.1737]and for Gaussian beams in CW synchronously pumpedOPOs [11.1738, 1739].

Most of the early synchronously pumped OPOs werepulsed OPOs [11.1734] because of the significant higherpeak pump pulse powers compared to CW mode-lockedpump lasers at that time. Picosecond optical parametricoscillation even in the SRO configuration was gen-erated using the nonlinear materials KTP [11.1740],BBO [11.1622] and LBO [11.1741]. However due tothe pulsed pump characteristic the output of such OPOsdoes not consist of a really repetitive pulse train, andthe amplitude, intensity and the pulse duration of theoutput pulse can vary under the pulse envelope. In ap-plications where no high peak powers are needed CWsynchronously pumped OPO are advantageous becauseof their continuous train of output pulses. Meanwhilewith the availability of efficient CW mode-locked lasersand more-reliable nonlinear materials the research in thisfield is focused on CW synchronously pumped OPOs.

In practice, synchronously pumping is a reason-able pumping mechanism only for high repetition rates(> 50 MHz) otherwise the resonator length becomes toolong and unpractical. For example, typical repetitionrates of Kerr lens mode-locked Ti:sapphire lasers areabout 80 MHz, which corresponds to an OPO resonator

.

*

Fig. 11.198 Experimental setup for a linear standing waveresonator consisting of two plane mirrors M1 and M4 andtwo spherical mirrors M2 and M3. The pump laser beam isfocused into the crystal C between the spherical mirrors M2and M3. The total resonator length depends on the distancebetween two successively pump pulses

length of 1.85 m. The tolerance of the resonator lengthdetuning can be calculated from ∆lres = ∆τ(c/2), whichis half the geometrical pump pulse length. The theo-retical detuning calculated for 1 ps is about 150 µm,however experimentally even shorter detuning toler-ances of 30 µm are affordable.

Commonly used resonator designs for syn-chronously pumped OPOs are ring or standing-waveresonators, where the latter is favorable for compactnessreasons (its length is half that of the ring). A typi-cal standing-wave resonator for synchronously pumpedOPOs is shown in Fig. 11.198 consisting of two planeand two spherical mirrors.

Picosecond optical parametric oscillators. Picosecondoptical parametric oscillators synchronously pumped byCW mode-locked solid-state lasers are powerful devicesfor the generation of tunable ultrashort laser pulses es-pecially in the IR spectral region. For many applicationsin high-resolution spectroscopy a high spectral inten-sity and a narrow bandwidth is required. As the light ofsuch OPOs is almost Fourier limited (15 ps = 1 cm−1)picosecond pulses (with t> 10 ps) have a sufficient se-lectivity for spectroscopy, in contrast to pulses in thefemtosecond regime. Similar to the nanosecond pulsedOPOs the high peak powers of modern CW mode-lockedlasers easily overcome the threshold of SRO resonators.As a consequence, most synchronously pumped CWOPOs operate in the SRO configuration.

In the past, there have been various ways to generatetunable picosecond IR radiation by CW synchronouslypumped OPOs. In most synchronously pumped pi-cosecond OPOs, the use of noncritical phase matchingprevents a reduction of interaction length due to walk-off

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between the focused beams inside the nonlinear crystal.Wavelength tuning of noncritical OPOs requires a varia-tion of the pump wavelength or the crystal temperature.

A temperature-tuned picosecond LBO OPO hasbeen reported [11.1742] generating continuously tun-able radiation between 650 nm and 2.5 µm with a singlecrystal. The average output power achieved by this sys-tem was below 100 mW. By using tunable pump lasersystems like the Kerr-lens mode-locked Ti:sapphirelaser pump wavelength tuning offers a convenient tuningmechanism.

A noncritical-phase-matched picosecond KTP OPO[11.1743] generates tunable picosecond pulses inthe spectral ranges of 1–1.2 µm for the signal and2.3–2.9 µm for the idler wave via wavelength tuning ofthe pump between 720–853 nm. Average output powersof up to 700 mW in 1.2 ps pulses has been achieved witha pump power of 1.6 W at a 82 MHz repetition rate.

The combination of wavelength tuning of theTi:sapphire laser and temperature tuning in LBOhas been performed under the type I noncritical-phase-matching condition, providing continuously tun-able picosecond pulses in the spectral range of1–2.4 µm [11.1744]. The pump power was 1.2 W in1–2 ps pulses at a 81 MHz repetition rate and generatedan average OPO output power of about 325 mW. Usingthe arsenate isomorphs of KTP, such as KTA and RTA,the wavelength range of the OPO can be extended up to3.6 µm [11.1745]. Due to their wider transparency rangeand the lack of grey tracking inside the crystal thesematerials are advantageous for IR picosecond OPOs.

Further extension of the tuning range of picosecondOPOs into the mid-IR was demonstrated with a critical-phase-matched AgGaS2 OPO pumped by a Q-switchedmode-locked Nd:YAG laser [11.1746]. This OPO hadan idler tuning range of 3.5–4.5 µm and a maximumoutput power of 2 W.

Picosecond pulse generation in the visible spectralrange has been demonstrated in an LBO OPO pumpedwith the third harmonic of a CW mode-locked Nd:YLFlaser [11.1747]. The generated output power was about275 mW in the spectral range of 453–472 nm in 15 ps-long pulses at a repetition rate of 75 MHz.

In general, it should be pointed out that the de-vice Ti:sapphire-laser-pumped KTP or KTA OPO isa versatile tool for the generation of tunable picosec-ond pulses in the entire spectral region from the UVto the near-IR. The frequency-conversion schemes in-volved, such as second, third, and fourth harmonicsof the Ti:sapphire laser, optical parametric oscillationin KTP and frequency doubling of the OPO’s signal

wave, are summarized in experimental investigations forpicosecond and femtosecond radiation [11.1748, 1749].

High-power operation of a singly resonant pi-cosecond OPO has been reported pumping a critical-phase-matched KTP OPO with the fundamental of anmode-locked Nd:YLF laser at 1.053 µm [11.1750]. Thisdevice generates 12 ps-long signal pulses at 1.55 µm for40 ps-long pump pulses at a repetition rate of 76 MHz.A total average output power of 2.8 W was achievedwith 800 mW in the idler wave at 3.28 µm.

The new generation of mode-locked lasers arediode-pumped CW mode-locked Nd:YVO4 oscillator–amplifier laser systems. Such an all-solid-state lasersystem operating at 1.064 µm with 7 ps pulse lengths atrepetition rates of about 83 MHz delivers average outputpowers of 29 W. Pumping a noncritical-phase-matchedKTA OPO the OPO output power reaches the multiwattregion providing a signal wavelength of 1.53 µm withan output power of 14.6 W and an idler output power ashigh as 6.4 W at 3.47 µm in the mid-IR [11.1751]. Thetotal average output power of 21 W corresponds to anexternal efficiency of 70%. However, both systems op-erate at a specific signal and idler wavelength withoutany tunability.

With the availability of QPM nonlinear materials likePPLN, PPKTP, PPRTA new perspectives have arisenin the research of efficient tunable picosecond OPOs.With the advantageous properties of these materials astheir large nonlinearities, transparency in the mid IR andthe capability to choose an optimized phase-matchingcondition for the interacting waves, for example non-critical phase matching, they become ideal candidatesto realize singly resonant synchronously pumped OPOswith low thresholds, large tunability and efficient out-put powers. Compact all-solid-state devices have beenreported based on PPLN and PPRTA with a pumppower threshold as low as 10 mW [11.1752, 1753]. Thegenerated radiation covers the wavelength range from3.35–5 µm with total output powers of up to 400 mWin 1–5 ps pulses. The power of the idler wave wasup to 100 mW. The spectral range of the PPLN OPOhas been extended even beyond 5 µm [11.1754] andidler output powers of 0.5 mW are achieved. The op-eration of a PPLN OPO at GHz repetition rates wasdemonstrated by pumping the PPLN OPO with anall-solid-state neodymium laser at 10 GHz [11.1755].Rapid wavelength access was achieved with the useof a mode-locked diode laser master-oscillator power-amplifier (MOPA) system providing 7.8 ps pulses at2.5 GHz repetition rate. The InGaAs oscillator–amplifiersystem delivered an output power of 900 mW, which was

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converted to 78 mW idler power tunable in the spectralrange of 2.2–2.8 µm [11.1756].

A high-power picosecond PPLN-OPO was per-formed by synchronously pumping a PPLN crystal withthe fundamental wave of a CW mode-locked Nd:YLFlaser. The tuning range of the PPLN OPO is for the sig-nal radiation 1.765–2.06 µm and 2.155–2.61 µm for theidler radiation, respectively. High average output powersfor the signal and idler radiation of 2.55 W and 2.4 Whave been achieved in 45 ps-long pulses at a 76 MHzrepetition rate for an input pump power of 7.4 W, whichcorresponds to a total external conversion of about 67%.The measured pump depletion was 71% [11.1757].

Much emphasis is placed on further extension ofthe tuning range of picosecond systems into the IR.For this purpose OPO/OPA and OPG/OPA devices areof great interest [11.1758, 1759] as their output pow-ers and tuning ranges are sufficient to be used fordifference-frequency mixing into the mid-IR. The differ-ence frequency mixing in GaSe or CdSe [11.1760–1762]of signal and idler frequencies generated from an PPLNOPO seeded KTP-OPA provides picosecond pulses tun-able at 3–24 µm.

Femtosecond optical parametric oscillators. Tunablefemtosecond OPOs are attractive light sources fortime-resolved spectroscopy of chemical and biologicalreactions. Femtosecond pulses are characterized by theirhigh peak powers, which easily overcome the thresholdof a singly resonant OPO configuration. The short pulselength of about 100 fs is combined with a huge spectralbandwidth of several nanometers (≈ 10 nm) and disper-sion effects, such as group velocity dispersion, groupvelocity mismatch (i. e., temporal walk-off) and spec-tral acceptance bandwidths can no longer be neglected.The group-velocity dispersion causes a pulse broaden-ing whereas the temporal walk-off lowers the nonlineargain and/or changes the time characteristic of outputpulse. For efficient frequency conversion, the temporalwalk-off has to be minimized by an appropriate choiceof the crystal length. Typical crystal lengths of fem-tosecond OPOs are only 1–2 mm, and therefore highnonlinearities are desirable for low operation thresholds.The group velocity dispersion can be minimized via dis-persion compensation in the OPO cavity. Additionallyto these dispersion effects, other higher-order nonlin-ear effects come with the high peak pump powers, suchas self-phase modulation and cross-phase modulationwhich result in a chirped OPO output pulse [11.1613].

The start of an extensive development of fem-tosecond OPOs was initiated with the commercial

availability of the Kerr-lens mode-locked Ti:sapphirelaser. The first material used in synchronously pumpedfemtosecond OPOs was KTP [11.1763–1765] innon-collinear critical-phase-matching conditions andcollinear noncritical-phase-matching condition. Due tothe higher transmission in the mid-IRA OPOs withKTA [11.1766, 1767], RTA [11.1768], CsTiOAsO4(CTA) [11.1769, 1770], and KNbO3 [11.1771] gener-ate a wider wavelength range of 1–5 µm with pulsedurations of less than 100 fs. Typical maximum outputpowers of such devices are 100–200 mW for the signaland 50–100 mW for the idler wave.

The periodically poled nonlinear materials PPLN,PPRTA, and PPKTP are of particular importance forfemtosecond OPOs because noncritical phase match-ing becomes possible even for the extended wavelengthtuning. This feature results in compact, low-threshold,efficient OPO devices. The wavelength tuning of suchfemtosecond QPM OPOs is performed via grating, pumpwavelength or cavity tuning and covers the spectral rangefrom 1.7–5.4 µm in PPLN [11.1772], 1.06–1.22 µm(λs) and 2.67–4.5 µm (λi) in PPRTA [11.1773], and1–1.2 µm in PPKTP [11.1774]. Comparing these QPMmaterials, PPLN and PPRTA have an advantage becauseof their large transmission range, where PPLN possessesthe higher nonlinearity. However PPLN suffers fromprocess-induced photorefractive defects, which may besuppressed by heating the crystal to temperatures higherthan 100 C. Recently MgO-doped PPLN-OPO was in-vestigated [11.1775] allowing reliable OPO operation atroom temperature.

Tunable radiation in the range of 4–8 µm have beenachieved using a cascaded AgGaSe2 OPO, which hasbeen pumped with the idler of a CTA OPO. This systemgenerates average powers of 35 mW in 300–600 fs-longpulses at a repetition rate of 82 MHz.

Moreover, optical parametric generators and ampli-fiers are powerful tools for the generation of ultrashortpulses into further infrared regions [11.1776]. The gen-eration of femtosecond pulses in the spectral range3–12 µm with almost bandwidth limited pulses of100–200 fs have been reviewed for different parametricprocesses and a number of relevant materials [11.1777].

Many new perspectives for the generation of intenseultrashort pulses are given by parametric amplificationof chirped laser pulses. The basic principle of opti-cal parametric chirped pulse amplification (OPCPA) isthat stretched linearly chirped pump pulses are ampli-fied in a non-collinear phase-matched OPA. After thesingle-pass amplification, the pulses are compressed,providing high energy in short pulse lengths of 10 fs

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Lasers and Coherent Light Sources 11.10 Generation of Coherent Mid-Infrared Radiation by Difference-Frequency Mixing 801

or less. The non-collinear phase-matching scheme isimportant because of the huge spectral bandwidths offemtosecond pulses. For ideal OPCPA the pump pulseduration corresponds to the length of the chirped pulse.The spatial beam profile should be rectangular (flat-top),and the gain should be constant temporally and spatially.Dubietis et al. [11.1776] published the first proposaland demonstration of OPCPA. The first experimentaldemonstrations of OPCPA were performed in degener-ate LBO, KDP and BBO [11.1778–1782]. Amplificationof 1010 have been achieved in single-pass amplificationat degeneracy.

The use of non-collinear phase matching enablesthe efficient parametrical amplification of nondegeneratewavelengths despite large spectral bandwidth. Recentlydifferent sources of supercontinuum have been ampli-fied in BBO and other crystals via frequency-doubledTi:sapphire lasers with 1 kHz repetition rate [11.1783,1784]. Ultrashort pulses below 6 fs have been gener-ated, however the achieved pulse energies was lowerthan 5 µJ.

The generation of higher energies is possible by em-ploying visible picosecond pump pulses and temporallystretched seed pulses. At a center wavelength of 800 nmand a pump wavelength of about 530 nm in BBO andLBO gain bandwidths broader than 2000 cm−1 are ob-tained, which is sufficient to amplify Fourier-limitedpulses with 5 fs duration [11.1778]. Due to the largedamage thresholds of both materials, high pump inten-sities can be applied. Recently OPCPA was performedat 800 nm and with a Nd:YAG laser at 532 nm, gener-ating a gain of 4 × 106 in two BBO crystals [11.1785].The pulse energy was 2 mJ at a 10 Hz repetition rate.

QPM materials also turn out to be advantageous inOPCPA. Broad-bandwidth amplification is not restrictedany longer to collinear phase matching at degeneracy or

to non-collinear phase matching but can be obtainedat practically any center wavelength in the crystalstransparency window. Additionally QPM OPA can beincreased by using engineered grating profiles, such aschirped gratings [11.1625], fanned gratings [11.1626],and aperiodic gratings [11.1786].

Nondegenerate OPCPA has been demonstrated inPPKTP at a wavelength of 1573 nm pumped bya Nd:YAG laser [11.1787] and close to degeneracyat a wavelength of 1053 nm pumped by a Nd:glasslaser [11.1788].

This chapter describes the basic properties and cur-rent performance of CW, nanosecond and ultrashortoptical parametric oscillators. Today these parametricsystems are powerful tunable light sources providinga spectral range from the visible to the mid-IR in alltime regimes. The main progress in this field is based onthe development of more-sophisticated optical materialsand efficient, reliable laser systems. With the realizationof QPM materials there was an important breakthroughtowards compact widely tunable OPOs. Especially forCW OPOs, QPM materials are the key element forthe realization of widely tunable singly resonant OPOs.The high nonlinear gain of QPM materials enables thereduction of the optical parametric oscillation to opti-cal parametric generation, which leads to more-simpledevices. The combination of diode-pumped solid-statelaser systems and QPM crystals, which already leadsto direct diode-pumped OPOs, is the way to future,compact tunable laser systems that may be realized inintegrated designs.

Finally it should be mentioned that, besides theimportance of the generated OPO radiation for manyspectroscopic and technical applications, the OPO itselfis still an interesting physical system to be explored inthe field of quantum mechanics.

11.10 Generation of Coherent Mid-Infrared Radiationby Difference-Frequency Mixing

The fundamental infrared (IR) or mid-IR region of theelectromagnetic spectrum between, say 3–20 µm, is ofspecial interest for many applications, notably molecularspectroscopy [11.1789]. The reason is that most organicand inorganic molecules exhibit strong vibrational–rotational transitions in this wavelength region. This isillustrated in Fig. 11.199, where the absorption rangesof some important functional groups of molecules areplotted for wavelengths of 2–20 µm (top), whereasabsorption features of a few selected molecules are

depicted in the center. In view of applications in air mon-itoring the (relative) transmission through the terrestrialatmosphere is plotted at the bottom of Fig. 11.199.

This atmospheric absorption is obviously dominatedby the presence of water vapor and CO2. The atmo-spheric windows at 3–5 µm and 8–14 µm are clearlyvisible.

In addition to the ultraviolet–visible and the near-IRrange, the mid-IR region has attracted a lot of interestin recent times owing to the numerous applications of

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-G "'

- - - * *- -

.

N4NCN. N NC@ N.@*

@@

.

G 6Q

0:

% - * $- $-

+

=66

0:0:#)

0"6"#""0"6)6

,6= 6 =6

Fig. 11.199 Absorption ranges of important molecular functionalgroups (top) and of selected molecules (center). The size of the sym-bols is related to the absorption strength. Atmospheric transmissionis depicted at the bottom

trace-gas sensing in rather diverse fields including en-vironmental monitoring and climate research, industrialand process surveillance, workplace safety, agriculture(e.g., ethylene or ammonia detection, surveillance of fer-mentation), homeland security (e.g., chemical warfareor explosives detection) and medical diagnosis (e.g., hu-man breath monitoring). It has been demonstrated innumerous cases that laser spectroscopy offers the poten-tial for high sensitivity and selectivity, multicomponentcapability, and large dynamic range. These are all cru-cial properties for trace-gas sensing. An important aspectof laser-based sensors is that usually no sample prepa-ration, i. e., neither pretreatment nor pre-concentration,are required, in contrast to many competing techniques.Apart from an appropriate detection scheme, which en-ables sensitive absorption measurements, a mid-infrared

tunable laser source is a prerequisite for successfultrace-gas sensing and analysis. In addition to tunability,a narrow line width guarantees sufficient specificity ofdetection in multicomponent real-world gas mixtures.In the preferred mid-IR range, the choice of coherentsources is, however, rather limited. Table 11.49 liststunable lasers with wavelengths longer than 3 µm. Itessentially includes the conventional well-establishedline-tunable CO and CO2 gas lasers, continuously tun-able semiconductor lasers (lead-salt diode and quantumcascade lasers), color center lasers, novel crystallinesolid-state lasers and nonlinear optical devices [opticalparametric oscillators (OPOs) and difference-frequencygeneration (DFG) sources].

Recent progress both in solid-state lasers andin quantum cascade lasers (QCLs) appears promis-ing. Solid-state laser materials such as Ce2+:ZnSe orFe2+:ZnSe offer tuning ranges of 2.2–3.1 µm [11.1790]or 4–4.5 µm, respectively (though the latter only whenpulsed and with cryogenic cooling) [11.1791]. Newdevelopments in QCLs equipped with external cavi-ties yield a continuous tuning range of around 10%of the central wavelength and – at least partly –room-temperature continuous-wave operation [11.1792,1793]. Despite these attractive prospects the broad-est tuning range, best wavelength coverage, room-temperature operation and highest flexibility in terms ofwavelength selection is currently still achieved with non-linear optical devices (DFG and OPOs) [11.1794]. Herewe focus on DFG as the scheme that has been widelyused in recent times to access mid-IR wavelengths fortrace-gas sensing. In fact, these important applicationshave fostered further developments of DFG systems.

11.10.1 Difference-Frequency Generation(DFG)

Difference-frequency generation (DFG) representsa nonlinear optical effect that is related to the nonlinearsusceptibility of second order (χ(2)) of a material. Otherrelated effects are second-harmonic generation (SHG)and sum-frequency generation (SFG). Hence, DFG rep-resents a three-beam interaction process that is mostlyused to generate tunable coherent mid-infrared radiation,although it has also been employed for THz generation.

Birefringent, Quasi-Phase Matchingand Conversion Efficiency

In difference-frequency generation the light of two laserbeams is mixed in a nonlinear crystal and light witha frequency of the difference of the two incident fre-

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Table 11.49 Tunable continuous-wave mid-IR laser sources with wavelengths ≥ 3 µm. RT: room temperature, LN2:liquid nitrogen (77 K), TE: thermoelectric cooling, SHG: second-harmonic generation, QCL: quantum cascade laser,OPO: optical parametric oscillator, DFG: difference-frequency generation

Laser Wavelength (µm) Tuning characteristics Power Operation

CO 5–6 (2.7–4, overtone) only line tunable 50 mW to W LN2 cooling, also ≤ 0 C

CO2 9–11 (4.5–5.5, SHG) only line tunable W RT operation

Lead salt diode 4–30 ≈ cm−1 mode hop-free < 0.1 mW Cryogenic cooling

QCL 4–> 24, THz cm−1 to > 100 cm−1 per device mW LN2/TE cooling, also RT

Color center 1–3.3 ≈ 0.5 µm for single crystal 100 mW LN2 cooling

Solid-state 2.2–3.1 ≈ 0.5–1 µm ≤ 1 W RT operation

OPO 3–16 ∼ µm for specific setup ≤ 1 W RT operation

DFG 3–16 ∼ µm for specific setup µW to mW RT operation

quencies is generated. The generated frequency is givenby energy conservation:

ωp −ωs = ωi → ωi = ωp −ωs . (11.187)

The laser with the highest frequency ωp is called thepump laser, while the second laser is called the signallaser with a frequency ωs. The generated idler beam hasthe lowest frequency ωi. The phase-matching conditionis given by the conservation of momentum:

∆k = kp −ks −ki = 0 , (11.188)

where ∆k is the phase mismatch, and kp, ks, and ki arethe wavevectors of the pump, signal, and idler beam, re-spectively. In the case of collinear wave propagation thewavevectors can be replaced by |k| = nc/λ and (11.188)changes to:

np

λp− ns

λs− ni

λi= 0 , (11.189)

where n is the refractive index at the correspondingwavelength, λ is the wavelength with the subscripts p,

Table 11.50 Different phase matching types for DFG pro-cesses for positive and negative birefringent crystals. PM:phase matching, e: extraordinarily polarized, o: ordinarilypolarized, QPM: quasi-phase matching

PM Birefringence Pump Signal Idlertype beam beam beam

I Positive o e e

I Negative e o o

II Positive o e o

II Negative e o e

III Positive o o e

III Negative e e o

QPM e e e

s, and i corresponding to pump, signal and idler, re-spectively, and c is the speed of light in vacuum. Phasematching in a birefringent crystal can be achieved by

1. angle tuning2. temperature tuning3. wavelength tuning (changing the wavelength of the

pump and/or signal laser).

The most common way to achieve phase matching witha birefringent crystal is realized by angle tuning, i. e., byrotating the crystal until the phase-matching conditionis fulfilled.

For infinite plane waves the idler intensity is givenby [11.1795]:

Ii = 2ω2

i d2eff L2 Is Ip

c3ε0npnsnisinc2

(∆kL

2

)(11.190)

where deff is the effective nonlinear coefficient, L is thecrystal length, I is the intensity of the laser beam forpump, signal and idler beams, respectively, ε0 is the di-electric constant, and ∆k is the phase mismatch. Hencethe idler intensity scales with the product of the incidentintensities Is Ip and the square of the crystal length L2.This is valid as long as there is no pump depletion andnegligible walk-off between the beams. These effectslimit the useful crystal length and they are discussedin more detail in the section on nonlinear crystals. Theeffective nonlinear coefficient deff describes the non-linearity of the crystal seen by the incident light anddepends on the crystal structure, the direction of propa-gation, the polarization of the light, and the nonlinearcoefficients dij (Table 11.51) given by the tensor of thenonlinear susceptibility χ(2) = 2d.

To achieve phase matching it is taken advantage ofthe different refractive index for ordinarily and extraor-dinarily polarized light. Depending on the combination

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Table 11.51 Nonlinear coefficients dij measured at wavelength λm, transparency range and approximate damage thresholdfor ns pulses of selected nonlinear optical crystals used in DFG

Crystal Nonlinear coefficients (pm/V) [11.1796] λm Transparency Damage threshold

(µm) range (µm) pulsed (MW/cm2)

[11.1797] [11.1797, 1798]

AgGaS2 d14 = 57 d36 = 20 10.6 0.46–13 25

d36 = 23.6 1.064

AgGaSe2 d36 = 33 10.6 0.7–19 25

Ba2NaNb5O15 d31 = 12 d32 = 12 d33 = 16.5 1.064 0.37–5 4

[11.1797] [11.1797] [11.1797]

CdGeAs2 d36 = 235 10.6 2.4–18 20–40 [11.1799]

CdSe d15 = 18 d31 = −18 d33 = 36 10.6 0.57–25 60

[11.1800] [11.1800]

CsTiOAsO4 (CTA) d31 = 2.1 d32 = 3.4 d33 = 18.1 1.064 0.35–5.3 500 [11.1801]

GaAs d14 = 368.7 d36 = 83 10.6 1–17 [11.1799] 60 [11.1799]

[11.1800]

GaSe d22 = 54.4 10.6 0.62–20 30

HgGa2S4 d36 = 26 d31 = 6.7 1.064 0.55–13 60

[11.1797]

KNbO3 d15 = −17.1 d24 = −16.5 d31 = −18.3 1.064 0.4–> 4 180

d32 = −15.8 d33 = −27.4

KTiOAsO4 (KTA) d31 = 4.2 d32 = 2.8 d33 = 16.2 1.064 0.35–5.3 1200

d24 = 2.9

[11.1800]

KTiOPO4 (KTP) d15 = 1.91 d24 = 3.64 d31 = 2.54 1.064 0.35–4.5 150

d32 = 4.53 d33 = 16.9

LiB3O5 (LBO) d24 = 0.74 d31 = 0.8–1.3 d33 = 0 1.064 0.155–3.2 900

d15 = 1.03 d24 = −0.94 d31 = 1.09 1.079

d32 = −10 d33 = −0.94

LiIO3 d15 = −5.53 d31 = −7.11 d33 = −7.02 1.064 0.28–6 120

LiInS2 d31 = 9.9 d32 = 8.6 d33 = 15.8 10.6 0.35–12.5 [11.1802] 100 [11.1802]

LiNbO3 d31 = −5.95 d33 = −34.4 1.064 0.4–5.5 [11.1803] 300

d33 = −27

[11.1800]

d31 = −5.77 d33 = −33.4 1.150

d31 = 3.77 d33 = −31.8 1.318

d32 = −29.1 2.120

d15 = −5.95 –

[11.1803]

d22 = 3.07 –

[11.1803]

LiTaO3 d22 = 2 d31 = −1 d33 = −21 1.064 0.4–5 [11.1802] –

[11.1802] [11.1802] [11.1802]

RbTiOAsO4 (RTA) d31 = 3.8 d32 = 2.3 d33 = 15.8 1.064 0.35–5.3 [11.1804] 400 [11.1804]

RTiOPO4 (RTP) d31 = 4.1 d32 = 3.3 d33 = 17.1 1.064 0.35–4.3 [11.1804] 600 [11.1804]

ZnGeP2 d36 = 75 d14 = 69 d25 = 69 10.6 0.74–12 3

[11.1800] [11.1800]

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of polarization of the pump and signal beam and depend-ing on positive or negative birefringence of the crystal,the phase matching is called type I, type II or type III, asdefined in Table 11.50. For quasi-phase matching (seebelow) all three beams have the same polarization.

Equation (11.190) implies a possibility to generatelight at the idler frequency without phase matching. Iihas a maximum at ∆kL/2 = mπ/2, where m is a natu-ral number. This yields the coherence length lc = π/∆kafter which the newly generated idler light will inter-fere destructively with the light generated within theprevious coherence length. Thus, after twice the coher-ence length all generated light is destroyed. This can beavoided by changing the polarization of the material by180 after one or an odd multiple of the coherence lengthlc because this will change the sign of the nonlinear co-efficient, so the light will interfere constructively anda buildup of the generated light occurs, as illustrated inFig. 11.200. This means that idler power can be gener-ated without fulfilling the phase-matching condition byusing a periodically poled crystal. A quasi-phase match-ing (QPM) condition can be written by using the gratingperiod Λ= 2lc of the crystal:

np

λp− ns

λs− ni

λi− 1

Λ= 0 . (11.191)

For the same parameters as for the bulk material lessidler power will be generated for quasi-phase match-ing than for birefringent phase matching by a factor of(2/π)2. This factor is often included in deff by reduc-ing its value by a factor 2/π. Although the efficiency islower than in the case of birefringent phase matching,more power can often be generated this way becausethe highest nonlinear coefficient can be used [e.g.,for LiNbO3 d33 = −27 pm/V (Table 11.51) insteadof d22 = 3.07 pm/V, which is relevant for birefringenttype II phase matching]. Since all involved polarizationsare the same (extraordinarily polarized, Table 11.50),the walk-off angle is zero, therefore longer crystals canbe used, resulting in higher idler power. Furthermore,the selection of pump and signal wavelengths is ratherflexible. However, the production of periodically poledcrystals is difficult and not possible for all crystals. Un-til now only LiNbO3 (PPLN), RbTiOAsO4 (PPRTA),KTiOPO4 (PPKTP), and KTiOAsO4 (PPKTA) arecommercially available periodically poled. These areferroelectric crystals that are poled by applying a strongelectrical field. Another, non-ferroelectric material isGaAs. In the beginning thin plates of GaAs with alternatepolarization were stacked manually [11.1805]. Todayorientation patterned GaAs, where the periodic poling is

implemented during crystal growth [11.1806–1808], isbecoming available.

Equation (11.190)) is obtained under the assumptionof infinite plane waves and no absorption. For Gaussianbeams the relationship between idler power Pi, crys-tal length L , pump power Pp, signal power Ps, andabsorption coefficient α is given by [11.1809–1811]:

Pi = Pp Ps32π2d2

eff L

ε0cniλ2i (nsλp +npλs)

h(ξ, σ, µ, α, L) .

(11.192)

The focusing function h(ξ, σ, µ, α, L) for diffraction-limited Gaussian beams is given by:

h (ξ, σ, α, L)

= Re

⎛⎜⎝ e− αL

2

×

ξ∫−ξ

ξ∫−ξ

dτ ′ e−iσ(τ−τ ′)+ αL4ξ (τ+τ ′)

1+ ττ ′ − i 1+µ2

1−µ2 (τ− τ ′)

⎞⎟⎠ ,

(11.193)

where

ξ = L

b, (11.194)

µ= ks

kp= nsλp

npλs, (11.195)

σ = −πb

(np

λp− ns

λs− ni

λi− 1

Λ

). (11.196)

Here Λ is the grating period, and b is the confocal pa-rameter of both the pump and signal beam and is givenby the minimal beam waist w : b = kpw

2p = ksw

2s , σ de-

scribes the phase mismatch. The focusing function h isdiscussed further below (see also Figs. 11.204, 205, 206and 11.208). These equations are valid for both bulk andperiodically poled crystals, only deff changes by a factor2/π compared to the case of a bulk crystal. The focusingfunction describes two competing effects: the efficiencyis increased by focusing the beams because of higher in-tensities, but at the same time reduced because of lesscollinear wavevectors. A possibility to overcome thisproblem is to use waveguide periodically poled nonlin-ear crystals [11.1812]. Here the beams are confined ina waveguide in the crystal leading to collinear wavevec-tors and high intensities at the same time. With this

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806 Part C Coherent and Incoherent Light Sources

/

7"

C

""

) >! *"

0

>! *"

>!

.

Fig. 11.200 Idler power generation for phase-matchingcondition ∆k = 0 (A), non-phase-matching condition∆k = 0 (C) and quasi-phase-matching condition in the caseof a periodically poled crystal (B). lc denotes the coherencelength [11.1813]

technique continuous-wave (CW) idler powers in themilliwatt range can be achieved whereas with bulk or pe-riodically poled crystals the CW idler power is normallyin the microwatt range.

The limit ξ → 0 gives the result for the infinite planewave because b → ∞ for plane waves. In this case thefocusing function reduces to h ∼ ξ , resulting in an idlerpower proportional to L2, as in the case for the planewaves (11.190).

An often used nonlinear crystal for difference-frequency generation in the mid-infrared is LiNbO3because it can be made periodically poled (known asPPLN) and has a large nonlinear coefficient. Its effec-tive nonlinear coefficient is given by deff = 2/πd33Mij =−14.4 pm/V with d33 = −27 pm/V and Mij = 0.85 forLiNbO3 at 3.3 µm [11.1814]. Mij is the Miller factor,which describes the dispersion of the nonlinear coeffi-cient [11.1800, 1815, 1816]. The factor 2/π is neededwhen using quasi-phase matching.

To calculate the phase-matching condition the re-fractive index needs to be known for all wavelengths.It is given by the temperature-dependent Sellmeierequation, e.g., for extraordinary polarized light in

LiNbO3 [11.1817]:

n2e = 5.35583+4.629 × 10−7F

+ 0.100473+3.862 × 10−2F

λ2 − (0.20692−0.89 × 10−8F)2

+ 100+2.657 × 10−5F

λ2 −11.349272 −1.5334 × 10−2λ2 .

(11.197)

Here F = (T −24.5)(T +570.82) describes the temper-ature dependence when T is the temperature in C andλ is the wavelength in nm. This equation takes alsothe multiphonon absorption into account, which yieldsmore-accurate data for wavelengths between 4 µm and5 µm.

When choosing the grating period and the tempera-ture of the crystal, the thermal expansion of the crystalalso needs to be taken into account because it also in-fluences the grating period, although much less than thechange of the refractive index with temperature. Thethermal expansion is (at 25 C) aa = 15 × 10−6 /C andac = 7.5 × 10−6 /C [11.1818] where the indices a andc correspond to the crystal axis. The refractive index ofLiNbO3 strongly depends on its composition, the Sell-meier equation is for congruent composition, it will bedifferent for stoichiometric composition. Also the re-fractive index of MgO doped LiNbO3, as often used inexperiments, differs from that of ordinary LiNbO3.

Nonlinear CrystalsNumerous crystals show nonlinear optical effects, butonly a few are useful for difference-frequency gener-ation. The crystal material needs to be transparent atall wavelengths of pump, signal and idler beams, itshould have a high nonlinear coefficient and a highdamage threshold. In some cases, e.g., strong dispersionand weak birefringence, phase matching is not possible.These requirements limit the choice of nonlinear crys-tals. The optical properties of some nonlinear crystalsused in DFG systems are listed in Table 11.51. Fig-ure 11.201 shows the absolute nonlinear coefficient andthe transparency range of selected crystals for compari-son.

For an easier comparison and characterization ofcrystals and DFG setups reported in the literature thefollowing figure of merit (FoM) is useful:

FoM = Pi

Pp PsL, (11.198)

where Pi, Pp and Ps denote the power of the idler,pump and signal beams, respectively, and L is the crystal

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Lasers and Coherent Light Sources 11.10 Generation of Coherent Mid-Infrared Radiation by Difference-Frequency Mixing 807

$+"#Q

0 "99")<

4

.//

0>//

A>

A+0

004

A++0

A+

A+0

C

+0 04AC

@4*

04C

.4

4E

.0

Fig. 11.201 Transparency range of some selected nonlinearcrystals used for mid-IR DFG generation as a function ofthe absolute value of the nonlinear coefficient. For the peri-odically poled materials (PPLN, PPRTA, PPKTA, PPKTP),the nonlinear coefficient d33 is given [11.1813]

length. This way the comparison is independent of thelaser powers and crystal lengths chosen. In Table 11.52this FoM is listed for some representative setups in-volving various crystals and combinations of pump andsignal lasers. Some of the setups generate a CW idlerbeam by using two CW lasers for pump and signal; othersetups have a pulsed idler beam by using two pulsedlasers or one pulsed and one CW laser.

When choosing the pump and signal lasers severalissues have to be considered. The laser wavelengths needto be within the transparency range of the crystal, andthe wavelengths and the polarizations should be chosenso that phase matching is possible. Another importantaspect is the laser power. As the conversion efficiency israther low, high laser powers are preferred, but too-highpowers will damage the crystal, the surface of the crys-tal or the antireflection (AR) coating on the crystal. The

AR coating often has a lower damage threshold thanthe crystal itself. At high power of the pump laser othereffects such as optical parametric generation (OPG) oramplification (OPA) could become even stronger thanthe difference-frequency generation, resulting in an en-larged line width. If the signal laser has much morepower than the pump laser, pump depletion might bea problem. Therefore pump and signal laser and non-linear crystal have to be carefully matched. To estimatehow low the signal power needs to be to avoid pumpdepletion, the following formula can be used:

η= Ii

Ip= 8π2d2

eff L2 Is

ε0npnsnicλ2i

1 , (11.199)

where Ii, Ip, and Is denote the idler, pump, and sig-nal intensity, respectively, deff is the effective nonlinearcoefficient, L is the crystal length, λi is the idler wave-length, ε0 is the dielectric constant and n is the refractiveindex with the subscripts p, s, and i referring to pump,signal and idler, respectively. η is the conversion effi-ciency from pump beam to idler beam, where η= 1implies that 100% of the pump beam is converted tothe idler beam. This case gives the nonlinear interac-tion length Lnl, one of several characteristic lengths ofa DFG system

Lnl =√ε0npnsnicλ2

i

8π2d2eff Is

. (11.200)

Crystal lengths longer than Lnl will not increase the idlerpower because of pump depletion. The aperture lengthLan is the distance after which the beam is displaced by2w0 because of walk-off effects and is given by:

Lan = √πw0

ρn, (11.201)

wherew0 is the minimal beam waist and ρn is the walk-off angle. The diffraction length Ldiff is the length afterwhich the beam diameter has increased by a factor of√

2. Longer crystals will not increase the idler power.This effect can be calculated more precisely by usingthe focusing function in (11.192).

Ldiff = 4kw20 . (11.202)

For pulsed lasers the interaction length Lqs representsa further important issue

Lqs = √πτ

(1

vg1− 1

vg2

)−1

, (11.203)

where τ is the pulse duration, vg1 and vg2 are the groupvelocities of the pump and the signal beams, respec-

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Table 11.52 Figure of merit (FoM) for different setups and crystals. PP: periodically poled, OP: orientation patterned, ECDL:external-cavity diode laser

Crystal Pump laser Signal laser Idler Idler Figure ofwavelength power (µW) merit(µm) (µW/(W2cm))

LiNbO3 bulk ECDL 795–825 nm Nd:YAG 1064 nm 3.16–3.67 CW 0.030 0.4

[11.1819] 25–30 mW CW 1 Watt CW

PPLN [11.1820] ECDL 1030–1070 nm, Er-fiber laser 1545–1605 nm 2.9–3.5 CW 3500 200

with Yb-fiber amplifier 5 W CW

700 mW CW

PPLN [11.1821] Nd:YAG 1064 nm ECDL 1500–1600 nm 3.2–3.7 Pulsed 2000 4.4 × 105a

pulsed 6 ns rep. 4–8 kHz 5 mW CW Average 4.4 × 105b

5 kW peak power power

PPLN [11.1814] ECDL 808 nm Nd:YAG 1064 nm 3.3 CW 27 410

20 mW CW 660 mW CW

PPLN [11.1809] Diode laser master/slave Nd:YAG 1064 nm 4.15–4.35 CW 172 110

848–855 nm with Yb-fiber amplifier

78 mW CW 5 W CW

PPLN ECDL 850–870 nm Nd:YAG 1064 nm 4.3–4.7 CW 5–23 4–19

This setup 125 mW CW 2 W CW

(see Sect. 11.10.3)

Waveguide PPLN Diode laser 940 nm ECDL 1550 nm 2.30–2.44 CW 400 1.2 × 105

[11.1812] 17.5 mW CW 20 mW CW

AgGaS2 [11.1822] ECDL 679–683 nm Diode laser 786–791 nm 4.9–5.1 CW 0.1 31

40 mW CW 20 mW CW

AgGaSe2 [11.1823] Fabry-Perot diode laser ECDL 1504–1589 nm 7.1–7.3 CW 0.010 52

1290 nm 6 mW CW

8 mW CW

PPKTP [11.1824] Nd:YAG 1064 nm ECDL 1490–1568 nm 3.2–3.4 CW 0.170 22

222 mW CW with Er-fiber amplifier

34 mW CW

PPKTA [11.1825] Nd:YAG 1064 nm ECDL 1519 nm 3.45–3.75 CW 0.140 70

117.2 mW CW 17.4 mW CW

PPRTA [11.1826] Ti:Al2O3 laser 710–720 nm Ti:Al2O3 laser 874–915 nm 3.4–4.5 CW 10 250

100 mW CW 200 mW CW

OP GaAs [11.1808] DFB diode laser ECDL 1535–1570 nm 7.9 CW 0.038 6

1306–1314 nm with Er-fiber amplifier

1.5–3.3 mW CW 1 W CW

GaSe [11.1827] Nd:YAG 1064 nm OPA 1100–4800 nm 2.4–28 Pulsed 5 µJ 0.0041a

pulsed 20 ps pulsed 5 ps 2.1 × 106b

750 µJ rep. 10 Hz 35–50 µJ

LiInS2 [11.1828] Ti:Sapphire Ti:Sapphire 5.5–11.3 CW - 12.4

700–810 nm CW 800–900 nm CW

ZnGeP2 [11.1829] OPO signal 1760–1950 nm OPO idler 2710–2330 nm 5–12 Pulsed 25 µJ 0.2a

pulsed 7 ns, rep. 17 Hz pulsed 7 ns, rep. 17 Hz 1.6 × 106b

0.95 mJ 0.95 mJa FoM calculated with peak powersb FoM calculated with average powers

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Lasers and Coherent Light Sources 11.10 Generation of Coherent Mid-Infrared Radiation by Difference-Frequency Mixing 809

tively. Lqs describes the length after which the pulsesfrom the two beams are separated by τ . Another char-acteristic length for pulsed laser is the dispersion lengthLds, given by

Lds = τ2

gn, (11.204)

where gn is the group-velocity dispersion. This lengthdescribes the length after which the pulse duration hasdoubled. With these characteristic lengths first estima-tions for crystal lengths and pulse durations can bemade.

11.10.2 DFG Laser Sources

The literature on DFG laser sources, new nonlinearcrystal materials and applications has become exten-sive in recent years. By far the largest part of thesesystems have been, and still are, developed for spectro-scopic gas sensing and analysis application. A recentoverview of systems is given in reference [11.1813].In the following we describe a typical DFG setup thathas been implemented in our laboratory for measure-ments on isotopomers of trace gases. Precise isotoperatio measurements of trace gases represent an impor-tant contribution to the solution of key questions invarious areas, e.g., differentiation between natural andanthropogenic origin of specific compounds. Isotopiccompositions are of interest in such diverse fields asecological CO2 exchange, volcanic emission, medicaldiagnostics, extraterrestrial atmospheres, etc.

Detailed Discussion of a DFG Laser SourceIsotopic composition of trace gases such as CO2, CO orN2O are of special interest. These molecules with theirisotopomers exhibit strong absorption lines between 4.3to 4.7 µm (Fig. 11.202). Therefore a continuous-waveDFG source, which enables continuous tuning and nar-row line width (to differentiate between isotopomers),was implemented. In the following sections the theoret-ical calculations, the setup and the characterization ofthis system is described.

Calculations of optimal crystal length and beam pa-rameter. For the wavelengths of interest we choseLiNbO3 as nonlinear optical medium because it hasa transmission range from 0.4 µm to 5.0 µm, a largenonlinear coefficient, and it can be produced periodicallypoled.

However, LiNbO3 has an absorption band at 5 µmso the absorption between 4 µm and 5 µm cannot

-

G "'

6"')" "'

- *

.M';

C

.

$$*$%$;

M';

*M'

Fig. 11.202 Absorption lines of CO, N2O and CO2 asa function of wavenumber in the mid-IR region [11.1830]

be neglected. The absorption coefficient at 4.3 µm is0.25 cm−1, at 4.6 µm it is 0.55 cm−1 and at 4.7 µm0.75 cm−1 [11.1831]. There are other crystals that canbe used at this wavelength, e.g., AgGaS2 or AgGaSe2(Table 11.51), but they are not available periodicallypoled, so birefringent phase matching has to be used.This means the crystal length needs to be shorter be-cause of walk-off effects and it is not possible to usethe maximum nonlinear coefficient. Furthermore align-ment is more critical than with a periodically poledcrystal. Finally, wavelength tuning is also an importantissue. When working with such crystals angle tun-ing has to be used, which makes wavelength tuningover larger ranges more complicated. Often noncriticalphase matching is used, because of its larger accep-tance bandwidth, but for a certain idler wavelength thepair of signal and pump wavelength is fixed in thiscase. This limits the choice of lasers and normally twodiode lasers with lower power than other lasers have tobe used.

In contrast, when using quasi-phase matching, thephase matching is realized by choosing a grating pe-riod, so nearly every combination of lasers can beused (e.g., an external cavity diode laser (ECDL) thatcan be conveniently wavelength tuned, and a Nd:YAGlaser that delivers high laser power). Phase matchingis achieved by choosing the correct grating period Λ,which can be adjusted to other wavelengths by changingthe temperature of the crystal because of the tempera-ture dependence of the refractive index (11.197). Whenscanning the wavelength for spectroscopy, the crystaltemperature can be changed simultaneously to achieve

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$-

%

-

Fig. 11.203 A periodically poled crystal with several grat-ings for different wavelengths. λp,s,i: pump, signal, idlerwavelength, respectively; Λ: grating period

mid-IR wavelength ranges of several hundred nm withonly one grating period. Depending on wavelength thetemperature acceptance bandwidth can be quite large(Fig. 11.209b). Even larger wavelength ranges can beachieved by using a crystal with several gratings, asshown in Fig. 11.203. Typical grating periods for ourwavelength range are approximately 23 µm. As a resultit appears advantageous to use PPLN for the envis-aged 4.3–4.7 µm wavelength range rather than a bulkAgGaS2 or other crystal despite the non-negligible ab-sorption of PPLN.

According to (11.192) the idler power increases withthe crystal length but also the absorption increases,therefore there is an optimum crystal length for thiswavelength region. The maximum of the focusing func-tion h (11.193) is not at σ = 0 but, e.g., at σ = 1.3 fora pump wavelength of 863 nm, a signal wavelength of

-M'

* % ;

*

Fig. 11.204 Focusing function h(ξ) versus ξ calculated forλp = 868 nm, λs = 1064 nm, α = 0.75 cm−1, σ = 0, andL = 5 cm. The maximum is at ξ = 1.3 and for small ξ thefocusing function h is proportional to ξ . The symbols areexplained in (11.192, 193, 194, 195, 196)

1064 nm and a crystal length of 5 cm (Fig. 11.208), sothe maximum idler power is not obtained at perfect phasematching, but for a slightly different grating period.However, to simplify the calculations,σ was set to 0. Thefocusing function and hence the idler power has a clearmaximum at about ξ = 1.3 (Fig. 11.204). It depends onlyslightly on wavelength and absorption. Therefore fora chosen crystal length L there is an optimal confocalparameter b.

When the idler power is plotted as a function of crys-tal length L and of the confocal parameter b, it can beseen that a long crystal combined with a large confo-cal parameter yields higher idler power (Fig. 11.205).However, in a real experiment the confocal parameteris limited by the thickness of the crystal and crystalslonger than 6 cm are not easily commercially available.The problem with longer crystals is that crystal defectshave too much influence. The thickness of the crystalis limited by the production process of the periodicallypoled grating. This is done by applying an electric fieldto change the orientation in the crystal, which requiresvery strong field strengths. This limits the thickness to1 mm. Most crystals have a thickness of 0.5 mm, result-ing in a better grating quality than for a thickness of1 mm.

In Fig. 11.206 it can be seen how the idler powerchanges for different crystal lengths. The calculationswere made with σ = 0 and keeping ξ = L/b = 1.3at the maximum. The result is that at λp = 853 nm(λs = 1064 nm,λi = 4.3 µm) the optimal length is longer

""

2QG

-

-

-

(

* - % & ;

(

*

-%

&;

Fig. 11.205 Idler power Pi versus crystal length L and con-focal parameter b calculated at a pump wavelength of853 nm. The idler power increases with increasing crystallength and confocal parameter

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Lasers and Coherent Light Sources 11.10 Generation of Coherent Mid-Infrared Radiation by Difference-Frequency Mixing 811

.#6"

B5QG

-

-

* % ; .#6"

B5QG

* % ; .#6"

B5QG

* % ;

;-

;

*

%

-

*

;% ;%;

Fig. 11.206 Idler power as a function of crystal length for pump wavelengths λp of 853 nm, 863 nm, 868 nm. The signallaser is a Nd:YAG laser fixed at λs = 1064 nm. The calculations were made with σ = 0 and keeping ξ = L/b = 1.3 at themaximum. At 853 nm the ideal crystal length is longer than 10 cm, at 863 nm it is 5.9 cm, at 868 cm it is 4.0 cm

than 10 cm, at λp = 863 nm (λi = 4.6 µm) it is 5.9 cm,and at λp = 868 nm (λi = 4.7 µm) it is 4.0 cm. We chosea crystal length of 5 cm, implying b = 4 cm, whichgives a minimal beam waist of 0.13 mm. Such a Gaus-sian beam propagates within the crystal over the wholelength.

Based on (11.192), an idler power of 55–170 µW isexpected for a pump power of 150 mW, a signal powerof 2 W and a crystal length of 5 cm.

Setup and characterization. Our DFG system con-sists of an external cavity diode laser (EDCL) as pumplaser, a continuous-wave Nd:YAG laser as signal laserand a MgO-doped periodically poled LiNbO3 crystal(MgO:PPLN). The ECDL (Sacher TEC-120-850-150)has a power of 150 mW and a wavelength range of820–875 nm. To cover the idler wavelength range of4.3–4.7 µm, pump wavelengths of 852–868 nm areneeded. The Nd:YAG laser (Innolight Mephisto) hasa CW power of 2 W and a wavelength of 1064.5 nm.The MgO:PPLN crystal (HC-Photonics) is 5 cm longand 0.5 mm thick. It has several gratings with periodsof 21.45, 22.00, 22.50, 23.10, and 23.65 µm, each witha width of 1.2 mm. For the wavelength of interest inthis experiment, only the grating period of 23.1 µmis needed, it is quasi-phase matched for the differentwavelengths by changing the temperature from 30 C to130 C. The crystal is antireflection coated for the pump,signal and idler wavelengths.

The laser beams are focused by several lenses (in-cluding cylindrical lenses, not shown in Fig. 11.207) sothat the minimal beam radius is 0.13 mm within thecrystal. The λ/4 and λ/2 plates are used to match the

beam polarization for quasi-phase matching in the PPLNcrystal. A small part of the pump beam is directed toa wavemeter. The recorded pump wavelength then yieldsthe idler wavelength. After the crystal a germanium fil-ter is used to block the near-infrared light. The setup isdepicted in Fig. 11.207.

This DFG system was tested and characterizedby focusing the mid-IR beam onto a detector (VIGOPDI-2TE-5, TE-cooled) and recording the signal witha lock-in amplifier with a time constant of 100 ms.For modulating the laser power with a frequency of1.8 kHz, a chopper was placed after the Nd:YAG laser.To find the crystal temperature for phase matching,the temperature was increased in steps of 0.2 C or0.5 C while keeping the pump wavelength constant(Fig. 11.208).

The temperature for phase matching is 6.6–11.2%higher than theoretically calculated (Fig. 11.209a) and

=B

!

G

>/4

>

CND0

/4

)*

)

)

C

Fig. 11.207 Setup for difference-frequency generation. OI: opticalisolator, PBS: polarizing beam splitter, DBS: dichroic beam splitter,PPLN: periodically poled LiNbO3

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'

6"6

'- -

.# S.

B5QG

-

-;%

;%2-

*

%

;

;(- ( ;-

B5QG Fig. 11.208 (a) Idler power as a func-tion of phase mismatch σ (11.196),calculated for a pump wavelengthof 863 nm, a signal wavelength of1064 nm, and a crystal length of 5 cm.(b) Measured idler power as a functionof crystal temperature correspondingto phase mismatch, at a wavelength of863.5 nm and a crystal length of 5 cm.It can be seen that the measured andthe calculated curve have a similarshape

the temperature acceptance bandwidth increase forlonger wavelengths and is 2.5–3.7 C (Fig. 11.209b).The power of the CW mid-infrared beam amounts to23 µW at 4.3 µm and 5 µW at 4.76 µm (Fig. 11.209c).It decreases for increasing wavelengths because of thecrystal absorption near 5 µm. The generated idler poweris about four to ten times lower than calculated, mostprobably because of imperfections in the crystal, its grat-ing quality and non-Gaussian beam shape of the ECDLpump laser.

For measurements of the isotopic composition oftrace gases, the laser line width should be sufficientlynarrow to resolve adjacent molecular lines clearly, alsowhen recorded at reduced gas pressure. The line widthof our CW mid-IR source has been determined as ap-proximately 1 MHz and thus fulfills the requirements byfar.

Measurement of isotopic composition of trace gases.We measured the isotopic ratios of 13C/12C and18O/16O in CO2 and CO, and of 15N/14N in N2O. N2Ois of special interest because 14N15N16O and 15N14N16Ohave the same mass, so they cannot be distinguished by

;-G 6,.>

+ S.

;&

+6#

*

;-- ;% ;%-

%

;

G 6Q

0"""56S.

$%

*$G 6Q

B5QG

*$&*$ *$%*$-*$*

$%

$*

$

$

$;

-

-

*$&*$* *$- *$%

Fig. 11.209 (a) Quasi-phase-matching temperature as a function of pump wavelength. (b) Acceptance bandwidth asa function of the temperature. (c) Idler power as a function of the idler wavelength

conventional mass spectrometry but laser spectroscopyenables an easy identification. Our measurements weredone with the DFG system described above and directabsorption spectroscopy with an astigmatic Herriot cell(New Focus 5611) with an optical path length of 10 m.The setup is shown in Fig. 11.210.

One problem in measuring isotopomers is thatthe concentration of the main isotopomer is typicallya hundred times higher than that of the less-abundantisotopomer. There are two possibilities to overcome thisproblem, either to measure two lines of similar strengthresulting in a strong temperature sensitivity of the mea-surement, or by choosing lines with similar lower energylevels but with very different line strength. The astig-matic multipass Herriot cell offers the possibility to enterthe cell at a different angle than usual, so that the beamleaves the cell after only two passes (Fig. 11.210). Thismakes it possible to measure two lines of very differ-ent line strength by using two different path lengths (thebalanced-path-length detection scheme) [11.1832].

As an example, Fig. 11.211 shows the result ofa measurement on CO2 isotopomers in ambient air ata concentration of 350 ppm. The measurements were

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Fig. 11.210 Setup for transmission spectroscopy. In theastigmatic Herriot cell there are two paths, one of 10 m(solid line) and one of 40 cm (dashed line)

performed at a total pressure of 50 mbar and at roomtemperature. The experimental data are fitted withVoigt curves to calculate the concentrations of the iso-topomers.

The derived isotope ratios of 13C/12C for CO2are 1.3%±0.2% (only long path) and 1.4%±0.3%(balanced-path-length detection scheme), which is ingood agreement with the natural abundance of 1.1%.The isotope ratio 18O/16O of CO2 is deduced as0.44%±0.06% (only long path) and 0.47%±0.11%(balanced-path-length detection scheme), which againis in good agreement with the natural abundance of0.39%. A more-detailed discussion of these studies,which also includes measurements of CO and N2O iso-topomers, can be found in [11.1833]. A further study onN2O isotopomers, testing different detection schemesfor low concentrations (100 ppm) has been publishedin [11.1834].

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11.10.3 Outlook

Tunable coherent sources in the mid-infrared rangeplay an important role. This is manifested by the ever-growing number of publications. The development isfostered by numerous applications, primarily in gassensing. The requirements with respect to sensitive andselective monitoring devices with multicomponent capa-bility are manifold: access to a broad wavelength range,broad – preferentially continuous – wavelength tunabil-ity, narrow line width (i. e., much narrower than typicalmolecular absorption line widths), preferentially room-temperature (RT) or near-RT operation, and compactand robust setups for field applications. In this respect,DFG systems represent a very valuable choice. Sincetheir first realization in 1974 [11.1835], DFG-baseddevices have reached a mature level. Today, their wave-length range can be chosen between, e.g., 2 µm and19 µm only depending on the available pump and sig-nal sources and nonlinear crystals. The tuning of a DFGsource is straightforward and continuous, and the wave-length generated can be accurately determined via thenear-infrared input wavelengths. The line width is ba-sically given by the line width of the pump and signallaser. This enables narrow mid-IR line widths as re-quired for high-resolution spectroscopy. In contrast toalternative mid-IR sources, DFG systems are generallyroom-temperature devices except for the crystal, whichmay require heating in a small temperature-controlledoven. Line widths of continuous-wave DFG sourcesare in the MHz range, which makes them attractive

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for studies requiring high spectral resolution such asthe briefly discussed isotopomer measurements. DFGsources are rather low-power lasers with CW powersin the µW to mW range. With fiber amplifiers for thenear-IR pump and signal sources, higher powers can beachieved if necessary [11.1820]. Recent developmentsin waveguide technology in PPLN [11.1812] or KTPappear particularly attractive in view of higher conver-sion efficiencies. Last but not least, a DFG system canbe made compact in view of today’s diode lasers anddiode-pumped solid-state lasers used as pump and signalsources.

In view of the variety of applications a furtherimpetus for DFG systems can be expected. Cur-

rent developments in several areas appear promising:the availability of compact and powerful signal andpump lasers, fiber amplifiers, new nonlinear crystalmaterials including organic media to access furtherwavelength ranges, new materials with higher non-linear optical coefficients, birefringent bulk crystalswith better quality, larger crystal sizes and higherdamage thresholds, quasi-phase matching becomingavailable for more crystals than today, waveguidetechnology, etc. These developments will contributeto improve the performance of DFG devices, en-hance their distribution, lower their costs and hencefurther increase their role as attractive spectroscopictools.

11.11 Free-Electron Lasers

The free-electron laser (FEL) is a system consisting ofa relativistic electron beam and a radiation field inter-acting with each other while they propagate throughan undulator [11.1836, 1837]. The main componentsof the free-electron laser are electron accelerator, un-dulator, and optical resonator (optionally for an FELoscillator). The undulator is a periodic magnetic struc-ture with a planar, or helical magnetic field that causesperiodical transverse deflections of the electron-beamtrajectory [11.1838]. Optical resonators are fairly sim-ilar to those used in conventional lasers. FEL deviceshave been realized with nearly all types of accelera-tors: electrostatic accelerators, RF linear accelerators,induction accelerators, microtrons, storage rings, etc.The wavelength range covered by FELs spans from cen-timeters down to nanometers [11.1839] (Table 11.53).

Table 11.53 Parameter space of free-electron lasers as of2006

Radiation

Wavelength 13 nm–10 mm

Peak power up to 5 GW

Average power up to 10 kW

Pulse duration 10 fs to CW

Driving electron beam

Energy 200 keV–1 GeV

Peak current 1–3000 A

Undulator

Period 0.5–20 cm

Peak magnetic field 0.1–1 T

Undulator length 0.5–27 m

The scale of the FEL setup is mainly defined by the scaleof the driving accelerator. For FELs operating in the mil-limeter wavelength range this could be a room scale,while unique devices such as VUV and X-ray FELsuser facilities have a scale comparable with conven-tional third-generation synchrotron radiation facilities(Fig. 11.212).

11.11.1 Principle of Operation

The FEL is not actually a laser; it is most closely re-lated to vacuum-tube devices. As with vacuum-tubedevices, FEL devices can be divided in two classes:amplifiers and oscillators (Fig. 11.213). An FEL ampli-fier is a single-pass device, and there is no feedbackbetween the output and input. The FEL oscillator canbe considered as an FEL amplifier with feedback. Foran FEL oscillator in the optical wavelength range thefeedback is carried out by means of an optical resonator.FELs based on the oscillator principle are limited on theshort-wavelength side to ultraviolet wavelengths primar-ily because of mirror limitations. Free-electron lasing atwavelengths shorter than the ultraviolet can be achievedwith a single-pass high-gain FEL amplifier.

The field of the electromagnetic wave only has trans-verse components, so the energy exchange between theelectron and the electromagnetic wave is due to thetransverse component of the electron velocity. The lat-ter occurs due to periodical wiggling of the electron inthe undulator. The driving mechanism of free-electronlasers is the radiative instability of the electron beam,which develops due to the collective interaction of elec-

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trons with the electromagnetic field in the undulator.The basic principle of radiation-induced instability canbe described within the standard picture for the gener-ation of the synchrotron radiation. Electrons propagatealong a sinusoidal path and emit synchrotron radiationin a narrow cone in the forward direction. When anelectron beam traverses an undulator, it emits radiationat the resonance wavelength λ= (λw/2γ 2)(1+ K2/2).Here, λw is the undulator period, mc2γ is the electronbeam energy, K = eHwλw/2πmc is the dimensionlessundulator strength parameter, and Hw is the maximumon-axis magnetic field strength of the undulator. Al-though the electromagnetic wave is always faster thanthe electrons, a resonant condition occurs such that theradiation slips a distance λ relative to the electrons af-ter one undulator period. The fields produced by themoving charges in one part of the undulator react onmoving charges in another part of the undulator. Thus,we deal with some tail–head instability, which leads toa growing concentration of particles wherever a smallperturbation started to occur (Fig. 11.214). In the begin-ning – without microbunching – all N electrons can betreated as individually radiating charges producing thepower of the spontaneous emission ∝ N . With completemicrobunching, all electrons radiate almost in phase.This leads to a radiation power ∝ N2 and thus to an am-plification of many orders of magnitude with respect tothe spontaneous emission of the undulator.

The figure of merit for FEL performance is the ra-diation power gain during one pass of the undulator.The gain primarily depends on the value of the peakbeam current. Another quantity of importance for thedevelopment of the radiative instability is the electronbeam density in the six-dimensional phase space. Lowbeam quality (large energy spread and emittance) leadsto degradation of the FEL gain. Errors in the periodi-cal magnetic structure also degrade FEL performance.Presently accelerator techniques and undulator technol-ogy have reached such a level that it is possible to havepower gains of the order of 107 for the amplification ofthe radiation of a nanometer wavelength range.

11.11.2 Current Status and PerspectiveApplications of Free-Electron Lasers

Free-electron lasers hold several potential advantages:continuous tunability of the radiation wavelength, thepossibility to obtain high levels of average output power,and the possibility to obtain high conversion efficienciesof the net electrical power to radiation power. An im-portant feature of the FEL radiation is that it has a high

Fig. 11.212 Aerial view of the experimental hall for the FLASH userfacility in Hamburg (center) and the tunnel for the superconductingaccelerator and undulator (covered with grass). The hall in the up-per right corner houses the injector part of the linac. The total lengthof the FLASH facility is 300 m. The maximum energy of electronsis 1 GeV, and the minimum radiation wavelength is 6 nm. The undu-lator of the FLASH (photo in the upper left corner) is a permanentmagnet device (period 2.73 cm, gap 12 mm, peak field 0.47 T). Theundulator system is subdivided into six segments, each 4.5 m long

degree of transverse coherence. In other words, the FELradiation can always be focused on to a spot whose sizeis defined totally by diffraction effects. This feature re-veals a wide range of possibilities for FEL applicationsin the transportation of the radiation over long distancesand in obtaining high focused intensities.

However, FELs are relatively expensive devices,thus their applications are in the fields not coveredby conventional radiation sources, in the far-infraredwavelength range and the THz gap (the sub-millimeterwavelength range). Another field for FEL applicationsis the generation of short-wavelength radiation, from thevacuum ultraviolet down to X-ray range. Technologicaldevelopments are on the way, aiming at industrial appli-cations of powerful FELs, and the use of FELs for energytransfer in space is also under consideration. Organiza-

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Fig. 11.213 Free-electron laser configurations: oscillator(top), seeded amplifier (middle), and self-amplified sponta-neous emission (SASE) FEL (bottom)

tion of user operation at the FEL facilities is similar tothat used at synchrotron radiation facilities: the host or-ganization constructs and serves free-electron laser anduser beam lines, and users perform their experimentsaccording to the schedule of the facility.

Linac-Based FEL User FacilitiesThere are several operating linear accelerator (linac)-based FEL user facilities (in China, France, Germany,Japan, Korea, The Netherlands, Russia, and USA) work-ing for scientific applications [11.1840–1853]. Thewavelength range covered with these facilities spansfrom 200 nm to a few hundred micrometers. Typicalparameters of the FELs driven by RF linear acceler-ators are: pulse duration of a few picoseconds, peak

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power in the MW range, and micropulse repetition ratein the range 10 MHz to 3 GHz. FELs driven by nor-mal conducting RF accelerators operate in pulsed modewith a macropulse repetition rate of 10–100 Hz. Themacropulse duration is defined by the length of the RFpulse, typically 1–20 µs. Thus, the average radiationpower is in the watt range. FELs driven by superconduct-ing accelerators operate in continuous mode and producea high level of average output power, up to a few kW.Linac-based FEL facilities are recognized nowadays asa unique tool for scientific applications that require tun-able coherent radiation in the infrared wavelength range.The general tendency is that the wavelength region ofinterest for users at these facilities is moving into thefar-infrared and THz gap.

FEL User Facilities at Storage RingsSeveral storage rings are equipped with free-electronlasers [11.1854–1857]. Typical parameters of the ra-diation are: wavelength range of 190–700 nm, pulseduration of some tens of picoseconds, average powerof 10–300 mW, and peak power in the kW range. Op-eration in a locked-mode regime allows the peak powerto be increased by an order of magnitude. The generaltendency is that users’ interest in these facilities is grad-ually reducing due to the limited features compared withconventional lasers.

High-Average-Power FELsRecent progress in accelerator technology has pavedthe way for construction of high-average-power linearaccelerators. The quality of electron beams producedby such machines are sufficient to drive free-electronlasers. Application of the energy-recovery technique al-lows a high level of overall efficiency to be achieved.Pilot facility are already in operation in Japan, Rus-sia, and USA [11.1851–1853]. The demonstrated levelof the average radiation power is about 10 kW in the3–20 µm wavelength range [11.1852], and about 0.4 kW

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in the THz gap [11.1851]. Projects of next-generationFELs based on energy-recovery linacs aim at increas-ing the average power into the tens of kilowatt range,and also to build high-power FELs operating in the UVrange [11.1858–1863].

Potential industrial applications of high-average-power FELs involve material processing (for instance,the treatment of polymer surfaces), lithography, iso-tope separation, and chemical applications. Productionof pure isotopes on an industrial scale may have a bigimpact on future developments. For instance, isotope28Si is a radiation-resistant material of great interestfor space research and the nuclear energy industry. Thethermal conductivity of the pure isotope 28Si is 50%higher than that in the natural mixture, which seemsvery attractive for the semiconductor industry. The arealso many other isotopes of great practical interest suchas 13C (medical applications) and 15N (for the studyand current control of use of nitrogen fertilizers inagriculture and agrochemistry). The basic process ofisotope separation is a selective multiphoton dissocia-tion of molecules. The required resonance wavelengthrange is in the range 2–50 µm. The required energy ina pulse is no less than 0.1 mJ and monochromaticity of10−2 –10−4, depending on the type of reaction. Indus-trial production of isotopes requires FELs with averagepower exceeding 10 kW. Construction of such facilitiesis on the way, and the first results are expected in thenear future.

X-ray Free-Electron LasersAt the start of this century, we have seen a revolu-tion in synchrotron source intensities. This revolutionstemmed from the technique of free-0electron laserscombined with recent progress in accelerator technolo-gies, developed in connection with high-energy linearcolliders [11.1866]. X-ray FELS (XFELs) have madea new regime of intensities accessible, thus opening upa fundamentally new physical domain.

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Fig. 11.215 Left: Average energyin the radiation pulse versus un-dulator length for TTF FEL atDESY [11.1864]. Right: interac-tion of powerful VUV radiation withsolids [11.1865]. Ablation of a goldtarget after one pulse of the TTF FELat DESY. The radiation wavelengthis 98 nm, the pulse duration is 40 fs,and the peak power density is about100 TW/cm2

A new era of synchrotron radiation research hasbegun withthe first user experiments on a vacuumUV FEL based on self-amplified spontaneous emis-sion (SASE) [11.1865, 1867]. It is worth mentioningthat such an essentially quantum terminology [intro-duced after amplified spontaneous emission (ASE)]does not reflect the actual physics of the process. Theamplification process in the SASE FEL has its ori-gin in density fluctuations in the electron beam. Thelatter effect is completely classical. The results havebeen obtained at the TESLA test facility (TTF) atthe Deutsches Elektronen-Synchrotron DESY (Ham-burg, Germany), using radiation pulses at a wavelengthof 98 nm with a 40 fs pulse duration and a peakpower of 1.5 GW [11.1864, 1868] (Fig. 11.215). Com-pared to present-day synchrotron radiation sources itspeak brilliance is more than 100 million times higher(Fig. 11.216), the radiation has a high degree of trans-verse coherence and the pulse duration is reduced fromhundreds of picoseconds down to ten-femtosecond timedomain. While modern third-generation synchrotronlight sources are reaching their fundamental perfor-mance limit, recent success in the development of theVUV FEL at DESY has paved the way for the construc-tion of a novel type of light source that will combine mostof the positive aspects of both a laser and a synchrotron.

In an X-ray FEL the radiation is produced by theelectron beam during a single pass of the undula-tor [11.1869–1871]. The amplification process startsfrom shot noise in the electron beam. Any random fluc-tuations in the beam current correspond to an intensitymodulation of the beam current at all frequencies simul-taneously – including of course, the frequency to whichthe undulator is tuned. When the electron beam entersthe undulator, the presence of the beam modulation atfrequencies close to the resonance frequency initiatesthe process of radiation. The FEL collective instabilityin the electron beam produces an exponential growth(along the undulator) of the modulation of the electron

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Fig. 11.216 Peak brilliance of X-ray FELs versus third-generation storage rings light sources. Circles show theexperimental performance of the FLASH free-electron laserat DESY

density on the scale of undulator radiation wavelength(Figs. 11.214, 215). The fluctuations of current density inthe electron beam are uncorrelated, not only in time butalso in space. Thus, a large number of transverse radia-tion modes are excited when the electron beam enters theundulator. These radiation modes have different gains.Obviously, as undulator length progresses, the high-gainmodes will predominate increasingly and we can regardthe XFEL as a filter, in the sense that it filters from ar-bitrary radiation field those components correspondingto the high-gain modes. Hence, for a long enough un-dulator, the emission will emerge in nearly completetransverse coherence. An intensity gain in excess of 107

is obtained in the saturation regime. At this level, theshot noise of the electron beam is amplified up to com-plete microbunching, and all electrons radiate almost inphase, producing powerful, coherent radiation.

The amplification bandwidth of a high-gain FELamplifier is restricted by the resonance properties ofthe undulator, that is, by the number of undulator pe-riods Nw within one gain length (the distance over

which the power increases by the factor of e = 2.718).The spectrum of transversely coherent fraction ofthe radiation is concentrated within the narrow band,∆λ/λ (2πNw)−1. The typical amplification band-width of the XFEL is of the order of 0.1%. The electronbeam in an XFEL transfers enormous peak power. Forinstance, for typical XFEL parameters (electron energyof 17.5 GeV and peak current 5 kA) it is about 100 TW.The conversion efficiency of kinetic energy of electronsto the light is of the order of the amplification band-width, thus the peak power of X-ray radiation is in themulti-GW range (Table 11.54).

Experimental realization of XFELs has developedvery rapidly during the last decade. The first demon-stration of the SASE FEL mechanism took place in1997 in the infrared wavelength range [11.1874]. InSeptember 2000, a group at Argonne National Labora-tory (ANL) became the first to demonstrate saturation ina visible (390 nm) SASE FEL [11.1875]. In September2001, a group at DESY (Hamburg, Germany) demon-strated lasing to saturation at 98 nm [11.1864, 1868].In June 2006 saturation was achieved at 13 nm, theshortest wavelength ever generated by FELs. Theexperimental results have been achieved at FLASH(Free-Electron-LASer in Hamburg, Fig. 11.212). Regu-lar user operation of FLASH started in 2005 [11.1872].Currently FLASH produces GW-level, laser-like VUVradiation pulses with 10–50 fs duration in the wave-length range 13–45 nm. After the energy upgrade ofthe FLASH linac to 1 GeV planned in 2007, it will bepossible to generate wavelengths down to 6 nm.

Recently the German government, encouraged bythese results, approved funding of a hard-X-ray SASE

Table 11.54 Main parameters of present and future X-rayFELs [11.1872, 1873]

2006: 2013:FLASH European XFEL

Radiation

Wavelength 13–180 nm down to 0.1 nmPeak power up to 5 GW up to 150 GWAverage power 10 mW up to 500 WPulse duration 10–50 fs 0.2–100 fsDriving electron beam

Energy 0.3–0.7 GeV up to 20 GeVPeak current 1–3 kA up to 5 kAUndulator

Period 2.73 cm 3.6–8 cmPeak magnetic field 0.5 T 0.5–1.4 TUndulator length 27 m up to 200 m

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FEL user facility – the European X-ray free-electronlaser [11.1873]. The US department of energy (DOE)has given SLAC the go ahead for the engineering designof the linac coherent light source (LCLS) device to beconstructed at SLAC [11.1876]. These devices shouldproduce 100 fs X-ray pulses with over 10 GW of peakpower. New X-ray sources will be able to produce inten-sities of the order of 1018 W/cm2. The main differencebetween these projects is the linear accelerator: an ex-isting room-temperature linac for LCLS at SLAC, anda future superconducting linac for the European XFEL.The XFEL based on superconducting accelerator tech-nology will enable not only a jump in peak brillianceof ten orders of magnitude, but also an increase byfive orders of magnitude in the average brilliance. TheLCLS and European XFEL projects are scheduled tostart operation in 2009 and 2013, respectively.

11.11.3 Suggested further reading

Books1. T. C. Marshall: Free-Electron Lasers (Macmillan,

New York 1985)

2. C. A. Brau: Free-Electron Lasers (Academic,Boston 1990)

3. P. Luchini and H. Motz: Undulators and Free-Electron Lasers (Clarendon, Oxford 1990)

4. W. B. Colson, C. Pellegrini and A. Renieri (Eds.):Free Electron Lasers, Laser Handbook, Vol. 6 (NorthHolland, Amsterdam 1990)

5. G. Dattoli, A. Renieri, A. Torre: Lectures on the FreeElectron Laser Theory and Related Topics (WorldScientific, Singapore 1993)

6. H. P. Freund, T. M. Antonsen: Principles ofFree Electron Lasers (Chapman Hall, New York1996)

7. E. L. Saldin, E. A. Schneidmiller, M. V. Yurkov: ThePhysics of Free Electron Lasers (Springer, Berlin,Heidelberg 1999)

FEL Conference ProceedingsIn the period from 1985 to 2002 the FEL ConferenceProceedings have been published in Nuclear Instrumentsand Methods, Section A. Starting from 2004 they arecollected in electronic form on a WEB page dedicatedto accelerator physics http://www.JACoW.org.

11.12 X-ray and EUV Sources

The extension of lasing into the X-ray region has beeninvestigated by a wide variety of methods since thefirst demonstration of the visible laser in 1960. Al-though laser operation in the hard-X-ray region, whichcan be categorized as wavelengths shorter than 0.2 nm,has not been realized, much progress on the develop-ment of coherent light sources has been made in thesoft-X-ray (0.2–30 nm) and extreme-ultraviolet (EUV,30–100 nm) regions during these 20 years. There are twomajor methods to produce coherent short-wavelengthradiation in these spectral regions. One is the use oftransitions in highly charged ions in high-density plas-mas created by laser irradiation of various targets orelectric discharge. The other is the generation of veryhigh harmonics of intense laser pulses. Each methodhas its advantages. X-ray lasers based on high-densityplasmas can produce much higher energy per pulse andnarrower spectra. High harmonics can be generated withcompact, high-repetition rated lasers and produce a widerange of the spectrum from 100 nm to 3 nm. The use ofthese sources depends on the application and will becomplementary.

11.12.1 X-Ray Lasers

The development of an X-ray laser has been oneof the elusive dreams of laser physicists. Proposalsfor excitation schemes for X-ray lasers date back to1965, when the possibility of achieving soft-X-rayamplification by collisional recombination was first sug-gested by Gudzenko and Shelepin [11.1877]. This wasfollowed by proposals for photoionization pumpingof X-ray lasers in 1967 [11.1878] and of electron-impact excitation schemes [11.1879]. The latter wereinspired in part by the earlier success in the devel-opment of visible and ultraviolet ion lasers excitedby electron collisions. However, the drastic scaling ofthe pump power requirements with decreasing wave-length and the low reflectivity of optics at soft-X-raywavelengths, combined with the short lifetime of theexcited levels involved in the lasing process, madethe realization of soft-X-ray lasers a very challeng-ing task [11.1880, 1881]. Several experiments carriedout during the 1970s and early 1980s yielded the ob-servation of population inversion and gain [11.1882].

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Fig. 11.217 Schematic drawing of an exploding-foil-targetX-ray laser

Nevertheless, the experimental demonstration of largeamplification at soft-X-ray wavelengths was not realizeduntil 1984. Matthews et al. [11.1883] at the LawrenceLivermore National Laboratory reported soft-X-ray am-plification at wavelengths of 20.6 and 21.0 nm in Ne-likeSe by electron collisional excitation. An exploding-foil-target X-ray laser demonstrated by Matthews et al.and the observed spectra are shown in Figs. 11.217and 11.218, respectively. In parallel with this, Suck-ewer et al. [11.1884] at Princeton University reportedamplification at 18.2 nm in H-like C by recombinationexcitation. These pioneering works were soon followedby numerous successful soft-X-ray amplification exper-iments conducted using of the world’s most powerfullasers as pump sources [11.1885–1887]. Subsequent ex-periments achieved soft-X-ray laser operation in thesaturated-gain regime [11.1888], and brought severalpotential applications to fruition [11.1889,1890]. Theseapplications include microscopy, holography, diagnos-tics of dense plasmas, and the excitation of nonlinearphotoluminescence in crystals. In spite of their pio-neering works, these soft-X-ray lasers have not beenused as a laboratory tool because both schemes re-quired huge laser energies of several hundred joules

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and were operated at a low repetition rate. Since sat-urated amplification at a water window wavelength hasnot been accomplished even with the world largestlaser driver [11.1891], a new excitation scheme hasbeen sought for obtaining soft-X-ray lasers operatingat shorter wavelengths. Many efforts have been di-rected in order to reduce the driver laser energy. Thereduction of the pump laser energy is essential for therealization of tabletop X-ray lasers for potential appli-cations [11.1892].

Electron Collision ExcitationIons with specific numbers of electrons have a fullyoccupied outer-shell structure and survive over a widerange of plasma parameters. This is a significant ad-vantage for plasma X-ray lasers because it results ina high relative abundance of the lasing ions over a widerange of temperature and density. To date, amplifica-tion has been observed in those ions. For example, theelectronic structure of highly ionized selenium with 24electrons removed (Se24+) is similar to that of neutralneon (Ne-like), and the transitions are also similar tothose of neutral neon [11.1893]. Such a comparison isreferred to as isoelectronic scaling of energy levels andtransitions.

Figure 11.219 shows a simplified energy-level di-agram of a Ne-like ion scheme. The 3p laser upperstates are populated mainly by electron collision ex-citation from the 2p6 Ne-like ion ground state. Thepopulation inversions are maintained by the rapid ra-diative decay of the 3s lower states to the groundstate. The Ne-like scheme is well studied and is themost robust for electron collisional soft-X-ray lasers.However, it has the disadvantages that a large pumppower is required to produce population inversion ata given wavelength. The nickel-like scheme was firstproposed by Maxon et al. [11.1894] and proved tobe useful for shorter-wavelength amplification, below10 nm [11.1895]. Although the lasing scheme of the Ni-like ions is directly analogous to that of Ne-like ions,its higher quantum efficiency for a given ionization stateallows the required pumping energy to be significantlyreduced. Recently, gain-saturated amplification in theNi-like scheme has been obtained at wavelengths asshort as 7.3 nm [11.1896].

Transient Collision ExcitationThe transient collisional excitation (TCE) scheme firstproposed by Afanasiev and Shlyaptsev [11.1897] isa variation of the electron collisional excitation. TCEhas attractive properties such as:

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1. the gain coefficients are 1–2 orders of magnitudelarger than those obtained for the same transition inthe quasi-steady-state regime,

2. the required laser energy for amplification can be re-duced greatly, resulting in the realization of tabletopX-ray lasers.

In the TCE scheme, an intense picosecond laser pulseoverheats the preformed plasma at a rate faster than othercollisional processes. The preformed plasma, containingthe lasing ions such as Ne-like and Ni-like ions, is pro-duced by a preceeding low-intensity nanosecond laserpulse. Transient gains in excess of 100 cm−1 have beenpredicted theoretically. Similar to the quasi-steady-statescheme, the TEC was firstly demonstrated in an Ne-likeion [11.1898] and has been successfully applied to Ni-like ions [11.1899]. Saturated amplification with a gaincoefficient as high as 63 cm−1 has been reported fromNi-like Pd at 14.7 nm with a total pump energy as low as7 J [11.1900]. Figure 11.220 shows the evolution of the14.7 nm laser line along the plasma column. Recently,this scheme has been extended to lasing at 8.8 nm inNi-like La [11.1901].

Optical-Field Ionization X-Ray LasersThere are two kinds of plasma production schemefor X-ray laser media by use of high-power lasers.The optical-field ionization (OFI) scheme is completelydifferent from the conventional plasma productionscheme. The OFI is direct ionization by a strongoptical field, which modifies the Coulomb potentialof the atoms or ions to create electrons by tunnel-ing ionization [11.1902]. In contrast to conventionalX-ray laser schemes using electron collisional ioniza-tion, the OFI scheme requires an intense laser powerrather than the large laser energy. Recent advances inthe technology of ultrashort-pulse generation and am-plification have achieved laser powers high enoughto realize OFI X-ray lasers with a tabletop device.The requirement for low pump energy will also al-low soft-X-ray lasers to operate at high repetitionrates, which is critical for most of the promisingapplications.

By applying this scheme to low-atomic-number me-dia, a plasma consisting of fully stripped ions and freeelectrons is produced on a time scale much shorter thantheir collisional and radiative time. In other words, thetemperature of electrons produced by the OFI is notdetermined by the charge state of ions, but can be con-trolled separately by ionizing-laser parameters such asthe polarization and wavelength [11.1903]. By means

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of a quasistatic model, Burnett and Corkum [11.1904]showed that cold dense multiply ionized plasmas couldbe produced by OFI, which are suitable for recombina-tion X-ray lasers. Nagata et al. [11.1905] demonstratedamplification of the Lyman α(n = 2−1) transition at13.5 nm in H-like Li by the recombination pumpingscheme following the OFI. Using a sub-picosecond KrF

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822 Part C Coherent and Incoherent Light Sources

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excimer laser focused at 1017 W/cm−2 (Fig. 11.221),singly ionized lithium ions were further ionized to fullystripped states, resulting in population inversion with re-spect to the ground state of hydrogen ions. A small signalgain of 20 cm−1 and a gain-length product of 4 have beenobserved. Subsequently, Korobokin et al. [11.1906] useda plasma waveguide in a LiF microcapillary to improvethe propagation of the pump pulse and successfullyincrease the gain-length product to 5.5.

On the other hand, Corkum et al. also suggested theuse of circularly polarized laser pulses for collisionalexcitation following OFI [11.1907]. Lemoff et al. pro-posed the specific systems [11.1908] and demonstratedlasing at 41.8 nm in eightfold-ionized Xe [11.1909]. Inthis experiments, circularly polarized Ti:sapphire laserpulse with an energy of 70 mJ and a duration of 40 fswas focused into a Xe static gas cell to create Pd-likeXe. The intense pump pulse produce the hot electronswhich collisionally excite the Pd-like ions to the laserupper level. Recently, Sebbon et al. reported the sat-

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urated amplification of the 41.8 nm line in Pd-like Xewith 0.6 J, 30 fs Ti:sapphire laser pulses. 5×109 photonsper pulse were obtained at gain saturation [11.1910].They also reported the lasing at 32 nm in nickel-likekrypton [11.1911]. The experimental setup is shownin Fig. 11.222.

Discharge ExcitationDirect excitation of plasma X-ray lasers by an electricaldischarge has the potential advantage of high effi-ciency and compactness over a laser-produced-plasmadevice. The uniform high-density plasmas requiredfor X-ray amplification can barely by produced bya conventional electric discharge. Rocca et al. pro-posed [11.1912] and demonstrated [11.1913] a capillarydischarged soft-X-ray laser operating on the 46.9 nmtransition in Ne-like Ar in 1994. A fast electricaldischarge in a capillary was used to excite plasmacolumns up to 20 cm in length with a peak currentof 40 kA. Figure 11.223 shows the fast capillary dis-charge soft-X-ray laser setup. The variation of thespectra as a function of capillary length is shown inFig. 11.224. A recently developed compact dischargeoccupies a space of only 0.4 × 1 m2 in an optical ta-ble and produces an average output energy of 0.88 mJat a repetition rate of 4 Hz. In two-pinhole interferenceexperiments, a high degree of spatial coherence wasobserved by single-pass amplification in a 36 cm-longcapillary. This compact soft-X-ray laser has been usedin a variety areas including plasma physics, materialcharacterization, and the characterization of soft-X-rayoptics [11.1914].

11.12.2 High-Order Harmonics

SurveyHigh-order-harmonic generation (HHG) by the inter-action of femtosecond high-intensity laser pulses witha gaseous medium has been extensively studied since ithas promising potential for use as a coherent extremeultraviolet and a soft-X-ray (XUV) source as an alterna-tive to soft-X-ray lasers or synchrotron radiation sources.The harmonic spectrum has a very characteristic shape,that is, it falls off for the first few harmonics, then showsa plateau where all the harmonics have the same in-tensity strength, and finally ends with a sharp cut-off.A typical harmonic spectrum observed by the interac-tion of ultrashort high-intensity pulses with rare gasesis shown in Fig. 11.225. The temporal duration of theharmonics is considered to be shorter than that of thedriving laser pulse whose width is typically 100 fs or

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Lasers and Coherent Light Sources 11.12 X-ray and EUV Sources 823

less. These spectral and short temporal properties makethe high-order harmonics an unique coherent sourcein the XUV region and are opening new applicationfields.

Approach to Shorter-Wavelength GenerationSince the demonstration of HHG in rare gases usinga KrF excimer laser by McPherson et al. [11.1915] anda mode-locked Nd:YAG laser by Ferray et al. [11.1916]in 1987, much effort has been made to extend the har-monic wavelength to a shorter one with various pumpingsources. Macklin et al. [11.1917] generated the 109thharmonic (7.4 nm) in neon gas excited with a 806 nmTi:sapphire laser. L’Huillier and Balcou [11.1918] ob-served the 135th harmonic (7.8 nm) using a 1 ps Nd:glasslaser. In these experiments, ionization of the neutralrare-gas medium limits the achievable wavelength be-cause of the low effective interaction intensity dueto the occurrence of ionization during the interac-tion. Although ions having larger ionization energypotentially produce higher harmonics, simultaneouslyexisting free electrons cause a large phase mismatch be-tween the pump and high-harmonic waves, resulting inthe lowering of cut-off orders. This drawback, whicharises from the use of ions as a nonlinear medium,was compensated for to some extent with an ultravi-olet pump source [11.1919]. Nagata et al. [11.1920]and Preston et al. [11.1921] reported the generationof the 37th harmonic (6.7 nm) of a 248 nm KrF ex-cimer laser. In those studies with pumping pulseslonger than 100 fs, the shortest wavelength achievedwas limited to around 7 nm. More recently, however,progress of ultrashort-pulse laser technology has en-abled the use of extremely short pulses with highintensity, which allows the circumvention of the lim-itation due to ionization [11.1922, 1923]. That is,the effective interaction intensity before the occur-rence of ionization can be increase with the extremelyshort pulses. Consequently, harmonic wavelengths wellwithin the water window region was attained, usingan ultrashort Ti:sapphire laser operating with a 26 fspulse duration [11.1924]. With a sub-10 fs Ti:sapphirelaser, Spielmann et al. [11.1925] obtained coherentcontinuum emission shorter than 2.5 nm, which corre-sponds to a photon energy greater than 0.5 KeV. Onthe other hand, a few approaches from the mediumside were reported. Although ions [11.1926, 1927],molecules [11.1928, 1929] and clusters [11.1930, 1931]have been investigated as nonlinear medium in place ofrare gases, the output property of the harmonics washardly improved for practical use.

Theoretical ProgressTheoretical understanding of HHG must be based ontwo processes:

1. a single-atom response in the driving laser field, and2. a macroscopic response including propagation ef-

fects.

Both responses must be taken into account to makea comparison between the experimental and theoreticalresults.

The emission property of harmonic radiation froma single atom is determined by the induced atomic po-larization, or dipole acceleration, which is calculatedfrom the solution of the time-dependent Schrödingerequation (TDSE). Krause et al. [11.1932] showed thatthe photon energy Emax of the highest harmonic inthe plateau is give by Emax = Ip +3.17Up, where Ipis the ionization energy of the nonlinear medium andUp = 9.33 × 10−14 Iλ2 (eV) is the ponderomotive en-ergy of the electron subjected in the pump laser fieldI (W/cm2) at a wavelength of λ (µm). In a semi-classical picture [11.1933], the physical origin of thisexpression is explained in terms of tunneling ionizationof the atom, followed by acceleration of the ionizedelectron in the pump field and recombination withits parent ion. During the acceleration by the opti-cal field, the electron obtains the maximum energyof 3.17Up. When this electron recombines with theparent ion, the electron release this energy plus the ion-ization energy as a harmonic photon. Figure 11.226shows a semiclassical model of high-order-harmonicgeneration.

Since calculation based on the TDSE is quite timeconsuming, it is very difficult to combine the numer-ical results of TDSE with the propagation equations.The calculation load is considerably relieved by usingthe model of Lewenstein et al. [11.1934, 1935]. Thismodel is based on the strong-field approximation and isvalid in the region Up > Ip. L’Huillier et al. developeda propagation code coupled to the Lewenstein modeland successfully described the harmonics characteristicsobserved in various experiments [11.1936, 1937].

Phase MatchingIn order to increase conversion efficiency and improvethe spatial quality of high harmonics, phase matching isessential. It is, however, not easy to satisfy the phase-matching condition along the interaction length because,in contrast with low-order-harmonic generation in theperturbative regime, the dipole phase is dependent on

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824 Part C Coherent and Incoherent Light Sources

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the driving laser intensity [11.1938]. Furthermore, inaddition to the medium’s dispersion, nonlinear phenom-ena such as self-focusing and plasma defocusing of the

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pump pulse also make the experimental achievement ofphase matching quite troublesome.

Several techniques to control the phase-matchingconditions have been investigated. Phase matching canbe achieved by controlling the balance of

1. the Gouy phase shift and atomic dispersion[11.1939, 1940]

2. nonlinear phase shift and plasma dispersion[11.1941] or

3. waveguide dispersion of hollow fiber and atomicdispersion [11.1942, 1943]

A geometrical phase and a dipole phase that have oppo-site dispersions are compensated by adjusting the focal

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Lasers and Coherent Light Sources 11.12 X-ray and EUV Sources 825

position around a gas jet. This method is extended toa loosely focused geometry and successfully improvedconversion efficiency and beam quality [11.1944]. Theuse of a hollow fiber provides a few advantages forhigh-harmonic generation. Due to the flat phase frontof the driving (pumping) laser in the hollow fiber, onecan avoid undesirable phase modulation of the harmon-ics, which originates from the intensity-dependent phasechange around the focus [11.1938]. This facilitates theclear and easy identification of the phase-matching con-ditions by adjusting the medium density. The increaseof the intensity–interaction length product would alsolead to an improvement in conversion efficiency. Fur-thermore, the use of a hollow fiber results in lowerbeam divergence and better spatial coherence. Such im-provements are very important for practical applications.With this new technique, several groups have reportedharmonic generation. Tamaki et al. first demonstratedHHG of a Ti:sapphire laser pulse in an Ar-gas-filled hol-low fiber and showed hundredfold enhancement aroundthe 25th harmonic [11.1945], [11.1946]. Figure 11.227shows the observed harmonic spectra with and with-out the hollow fiber filled with 5 torr Ar gas. Rundquistet al. [11.1942,1947] also reported phase-matched gen-eration of the 29th harmonic and improvement of boththe generation efficiency and the beam quality in the hol-

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Fig. 11.227a,b Observed harmonic spectral distributionemerging (a) from a 3 cm hollow fiber and (b) in 0.5 cmfree space. Ar gas pressure is 5 torr

low fiber. However, the output energy from the hollowfiber was restricted to a few nanojoules because onlya few millijoules of laser pulse could be introduced dueto the limited aperture of the hollow fiber [11.1948].

Energy ScalingFor the development of a variety of applications of highharmonics (HH), one of the most important issues isenergy scaling. High-energy HH is expected to boostnew physics in the soft-X-ray region. Takahashi et al.report energy scaling of HH in Ar under the optimizedphase-matched condition [11.1944,1949]. Their scalingmethod demonstrated a linear increase of harmonic en-ergy with respect to the geometrical focusing area ofthe pump pulse, while keeping an almost perfect spatialprofile of the harmonic output. The maximum energyof the 27th harmonic attained was 0.33 nJ with a con-version efficiency of 1.5 × 10−5. The evolution of HHintensities in the spectral region from the 23rd- to 27th-order harmonics was also measured as a function ofmedium length. The result is shown in Fig. 11.228. Thesolid line shows theoretically fitted intensities for the23rd, 25th and 27th harmonics. The coherence lengthwas estimated to be ≈ 15 cm by fitting the theoreticalcurves. As was pointed out by Constant et al. [11.1943],the optimizing conditions for the medium, coherenceand absorption lengths are given by Lmed > 3Labs andLc > 5Labs, where Lmed, Lc, and Labs are the medium

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Fig. 11.228 Emitted photon number of the harmonics inargon as a function of the medium length. The solid line cor-responds to the calculated photon number for Lc ≈ 15 cmwith 1.8 torr argon in free space

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826 Part C Coherent and Incoherent Light Sources

'

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Fig. 11.229a,b Interferogram of a harmonic beam dif-fracted by double pinholes separated by 100 µm: (a) fringeimage, (b) intensity profile of the interferogram along a hor-izontal line passing through the center

length, the coherence length, and the absorption length,respectively. The 23rd- and 25th-order harmonics satis-fied the optimized condition for this relation. Therefore,those orders were saturated under the experimental con-ditions. On the other hand, the 27th-order harmonic didnot yet satisfy the above conditions, because of lowabsorption.

Spatial CoherenceSince HHG is based on nonlinear frequency conversion,the spatial and temporal coherences of high-order har-monics are expected inherently to succeed to those ofthe fundamental laser pulse. However, HHG conductedby use of a tight-focusing geometry in thin gas media didnot allow full phase matching to enhance the coherence.Typically multimode components are observed as broad-ened peaks in the spectrum or the pedestal of the spatialprofiles. Recent studies have revealed that macroscopic

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phase matching can realize highly efficient and spatiallywell-characterized (nearly Gaussian profile) high-orderharmonic light [11.1949,1950]. Several groups have re-ported the interferometric measurement of the spatialcoherence of high harmonics [11.1951–1954]. Whenthe macroscopic phase matching is achieved in thehollow fiber or loosely focused geometry, the mea-surements show that the harmonic beams have almostperfect spatial coherence. Figure 11.229 shows theinterferogram of the 27th-harmonic beam generatedwith an Ar-filled hollow fiber pumped with 20 fs,0.35 mJ Ti:sapphire laser pulses [11.1954]. This in-terferogram was obtained with two pinholes separatedby 100 µm, while the harmonic-beam diameter wasmeasured to be 130 µm. Figure 11.230 also shows the

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Lasers and Coherent Light Sources 11.13 Generation of Ultrahigh Light Intensities and Relativistic Laser–Matter Interaction 827

interferogram recorded by point-diffraction interferom-etry, which indicates that the harmonic beam can beconsidered as a spherical wave within a phase error ofλ/15 [11.1954].

ApplicationsIn addition to applications in atomic physics [11.1955],high harmonics have been used for solid-state spec-troscopy [11.1956] and plasma diagnostics [11.1957].High intensity of high harmonics is also expected togive rise to nonlinear phenomena in the XUV region. Asdescribed above, Takahashi et al. [11.1944, 1950] gen-erated a peak power of 130 MW at 62.3 nm in 0.6 torrXe and 10 MW at 30 nm in 2 torr Ar. When theseharmonic pulses are focused with multilayer mirrors,the focused intensity will reach 1014 W/cm2, which ishigh enough to cause nonlinear interactions. Theoret-ical predictions of nonlinear interaction between soft

X-rays and matter have been reported by a few groups,such as two-photon ionization of He+ [11.1958], Hedouble ionization for the autocorrelation of an extreme-ultraviolet (XUV) pulses [11.1959], the advantages ofhigh-intensity short-wavelength radiation for Coulombexplosion imaging [11.1960], and the ionization of clus-ter targets. These research field can be expected to opena new area in high-intensity physics.

Furthermore, coherent XUV and X-ray pulses arenot only useful owing to their short wavelengths, butare also important due to their potential to produceelectromagnetic radiation in the range of attoseconds.High-order harmonics have been expected to be thesource of attosecond pulses [11.1961, 1962]. Recently,Hentschel et al. [11.1963] demonstrated the genera-tion of isolated soft-X-ray attosecond pulses and theirtemporal characterization by a novel cross-correlationtechnique using intense few-cycle visible laser pulses.

11.13 Generation of Ultrahigh Light Intensitiesand Relativistic Laser–Matter Interaction

Modern laser technologies allow the amplification ofshort laser pulses to energies of some tens of kJ. Addi-tionally, ultrashort pulses containing only a few opticalcycles can be generated. By merging these techniquesnowadays focused laser beams can reach unprecedentedintensities in the range of 1021 W/cm2 and will reacheven higher values in the near future. At these inten-sities the electric and magnetic field strength is manyorders of magnitude higher than those that will ever bepossible in a static generation scheme. By applying thesefields to a target it becomes possible to gain access toa new interaction regime of light and matter: relativis-tic optics. This opens a new wide area in experimentalscience where classical optics meets plasma dynam-ics, relativistic quantum mechanics, and high-energyphysics.

11.13.1 Laser Systems for the Generationof Ultrahigh Intensities

Amplification of Ultrashort Pulsesto High Energies

For ultrashort pulses the energy density of light at thesurface and in the volume of all the optical elementsis limited by the onset of nonlinear effects and laserdamage due to the high peak power. Moreover, in a laseramplifier the energy extraction efficiency is a function ofthe ratio of the energy density and the saturation fluence

of the laser material. Hence, a short pulse cannot be am-plified efficiently. The chirped pulse amplification (CPA)

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Fig. 11.231 The CPA principle. An ultrashort pulse gener-ated in a mode-locked laser is stretched by adding a spectralphase that group delay different wavelengths of the pulse.After amplification retaining the pulse spectrum and chirpthe pulse is recompressed to its original length, resulting inultrahigh peak power

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828 Part C Coherent and Incoherent Light Sources

technique [11.1964] circumvents this difficulty. Theprinciple of CPA is depicted in Fig. 11.231. Ultrashortpulses contain a broad spectral bandwidth according totheir Fourier transformation (see Chap. 12). This factenable the possibility to add a phase shift to differentfrequencies or wavelengths of the laser pulse just afterits generation in a mode-locked laser cavity. The resultis a stretched pulse containing a chirp, i. e., the pulse du-ration is no longer bandwidth limited. The lengthenedpulse has accordingly a lower peak power. It can be am-plified much more efficiently and compressed to a veryshort pulse afterwards by adding a spectral phase withan opposite sign to that introduced by the stretcher.

After the invention of the CPA technique tremen-dous progress in the development of ultrashort pulselasers towards higher peak power has been witnessed.Nowadays, terawatt (TW) laser pulses can be producedusing tabletop laboratory-scale laser systems that op-erate at repetition rates of 10 Hz or higher. Whereas,even higher power up to the petawatt (PW) can begenerated by flashlamp-pumped lasers that have factory-building sizes, like the first PW laser, that was realizedin 1999 [11.1965]. The CPA technique is common to allof these devices.

Because pulse stretching and compression are re-lated to each other and are normally based on the sameprinciple, the stretcher and compressor are treated hereas a stretcher–compressor pair (SCP). Pulse stretchingnaturally occurs in the vicinity of dispersion. Hence,the propagation of pulses in dispersive media alwaysleads to pulse stretching. In order to keep the requiredspace and size of optical elements as small as possiblehighly dispersive optics, i. e., gratings will be used. Nev-ertheless other elements such as prisms, grating prisms(so called grisms), fibers and chirped fiber Bragg grat-ings as well as chirped mirrors can be applied. Herewe limit ourselves to the simplest case of flat reflec-tive phase gratings, because these offer the largest groupdelay.

The principle of pulse compression with identicalparallel gratings was first described by Treacy [11.1966].A pulse is diffracted by a grating, angularly split intoits different wavelengths and propagated to the secondparallel-aligned grating. The second grating removes theangular modulation, i. e., redirects all wavevectors ofthe different wavelets into the same direction. If all thewaves are treated as plane waves, the introduced lat-eral shift can be neglected. For a real, limited beamsize the same device of parallel gratings has to be usedonce more in order to combine the wavelets again spa-tially. As a side effect this doubles the stretching factor

of the arrangement. Section 12.1.3 contains a detaileddescription of these devices.

For ultrahigh-peak-power lasers a maximum stretch-ing/compression factor, i. e., the ratio of stretched pulselength to bandwidth limited pulse length is required inorder to maximize the fluence in the amplifier chain. Thestretching factor depends on the line density of the grat-ings, the center wavelength, and the bandwidth of thepulse. Very long stretching factors of 10 000 or more arerequired for most broadband rare-earth-doped laser ma-terials in order to allow fluences in the amplifier close toor above the gain-saturation fluence.

If the grating distance is chosen in this way, such thatthe unclipped spectral bandwidth is twice the spectralfull width at half maximum (FWHM) of the pulse, themaximum stretched pulse length τmax is only a functionof the grating size L:

τmax = 2L

ccos (α) , (11.205)

where cosα is a factor according to the diffraction angleα of the grating and has a maximum of unity in the un-realistic case of a diffracted beam parallel to the gratingsurface. The factor of 2 refers to the Littrow case, i. e.,the input and diffraction angle being equal. The latternormally maximizes the grating diffraction efficiency.For holographic metal-coated diffraction gratings withnear-rectangular groove shapes it has been shown thatan optimum diffraction efficiency of a metal grating atwavelength λ can be achieved for a grating constant ofabout

√2λ [11.1967], which results in a Littrow angle of

45. This leads to a minimum grating size for an SCP ofabout 24 cm if a femtosecond pulse has to be stretchedto 1 ns. Passing a stretcher or compressor setup againcan, in principle, increase the stretching factor, but in-troduces losses that are most often acceptable for thestretcher but not for the pulse compressor.

Very large grating distances and hence long stretchedpulses require a similar accuracy for the compensationof dispersion by a well-matched stretcher and compres-sor as in the case of ultra-broadband fs pulses, becausehigher-order dispersion terms increase together with thesecond-order term. Because most often aberrations ofthe telescope optics cannot be tolerated, aberration-freeall-reflective designs like the Öffner triplet are ap-plied [11.1968]. In special cases aberrations introducedby the stretcher may help to compensate for laser-material dispersion in the amplifier chain [11.1969].

The required grating sizes for the compressor ofa high-energy laser system is ruled by the generatedgroup delay in order recompress the pulse and the size

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Lasers and Coherent Light Sources 11.13 Generation of Ultrahigh Light Intensities and Relativistic Laser–Matter Interaction 829

Fig. 11.232 (a) Capability of laser materials for energy stor-age and generation of high energy pulses. The inversesaturation fluence is plotted against the fluorescence life-time. For a higher gain amplified spontaneous emission(ASE) become an issue whereas for lower gain damagelimits efficient energy extraction. Effective pumping will beuseful if the fluorescence lifetime exceeds a certain limit.The limits are marked with dashed lines. (b) Capability oflaser materials for energy storage and generation of highpeak power. The inverse saturation fluence multiplied byminimum possible pulse width plotted against fluorescencelifetime τ . Differently doped materials are marked withdifferent colors

required to fit the compressor to the beam diameter. Be-fore compression the laser beam has to be expanded toreduce the fluence in the beam to an amount well belowthe damage threshold of the compressor optics, partic-ularly the gratings. For a metal-coated grating damagethresholds may reach the 0.5 J/cm2 level [11.1967] forhalf-ps pulses, but will normally be less than 0.25 J/cm2.In the compressor setup these fluences result in intensi-ties that are well above the onset of nonlinear effects inair at normal pressures. In order to prevent the pulsefrom self-focusing and self-splitting into white-lightfilaments, pulse compression has to be performed ina vacuum vessel.

In the region of fs to ps pulses the damage thresh-old for metal coatings is nearly independent of pulseduration, in contrast to dielectrics where the main dam-age mechanism is linked to nonlinear absorption effects.For pulses longer than 100 fs dielectric gratings maysubstantially improve overall performance [11.1970].Damage thresholds two to four times that of gold-coatedgratings are reported. These dielectric phase gratings ontop of a multilayer mirror can, in principle, show a 100%diffraction efficiency to the minus-first order [11.1971].Therefore they are favored in high-energy high-peak-power laser systems.

The peak power of a laser system is limited mainlyby the possible sizes of the diffraction gratings. A work-around is to add identical smaller gratings to a mosaicgrating or tiled grating. The smaller grating tiles haveto be coherently phased together in order to make themwork like a single one. Using this technique to builda compressor opens the way to further power scalingof high-peak-intensity lasers. Compression of pulsesto 650 fs with a tiled grating replacing a meter-sizedgrating in a high-energy laser system has already beendemonstrated [11.1972], as well as pulse compression ofa chirped 2 ns pulse down to 150 fs [11.1973] by phasing

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Laser Materials for High Peak PowerA crucial issue conceiving a high-peak-power laser sys-tem is the choice of the gain material. Broad band widthand a high cross section for stimulated emission arerequired. Due to the limited peak power of the pumpsource a long fluorescence lifetime for energy storageis desired. At higher fluorescence lifetimes either theemission cross section, or the gain bandwidth, or bothdecrease. For efficient amplification the extracted laser

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830 Part C Coherent and Incoherent Light Sources

pulse energy density has to be close to the saturationfluence of the gain medium.

Figure 11.232 illustrates the possibility of generatinghigh peak power out of some established laser mater-ials. The inverse of the product of the saturation fluenceand the shortest pulse duration indicates the capabilityof a laser material for high amplification at maximumbandwidth. Assuming Gaussian-shaped gain spectra thecorrelation between the emission cross section, band-width and fluorescence lifetime of a laser material isgiven by

σem = c20

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∆ν√π

(11.206)

with c0 the velocity of light in vacuum, n the refractiveindex, h Planck’s constant, ν the center frequency, τ f thefluorescence lifetime, and ∆ν the bandwidth (FWHM).Applying the time–bandwidth product for Gaussian line-shape a criteria for the generation of high peak powerat a corresponding fluorescence lifetime depending onthe laser wavelength lambda and the refractive index nis obtained

τf

tp Fsat≤ 4.26 × 109 (λ[µm])3

n2

[cm2][J] . (11.207)

Here the saturation fluence Fsat is given byFsat = hν/σem.

In Fig. 11.232 suitable materials for amplificationto a high energy level are mapped. Furthermore, thereis an optimum region between high gain and low gain,where amplified stimulated emission and damage issues,respectively, are likely to occur.

Recently, it was observed that cryogenic lasermaterials may show enhanced performance. Eitherspectroscopic parameters and thermal behavior canbe improved by cooling to liquid-nitrogen tempera-tures [11.1974]. Ti:Sapphire for instance gain, higherefficiency and thermal conductivity. Quasi-three-levelsystems represented by the Yb-level system show higheremission cross sections and additionally reduced absorp-tion at the laser wavelength, because population of thelow laser level is suppressed.

Laser Amplifier Schemes for High Peak PowerExcept some examples of high-energy excimer lasersand upcoming free-electron lasers for the generation ofhigh peak power at very short wavelengths typical sci-entific lasers are based on solid-state lasing materialsas presented above, all of which are pumped opti-cally. Because a high flux of pump photons is neededto invert the energy-level population significantly only

three options for pumping exist: flashlamps, lasers, anddiode lasers. Diode lasers are treated separately becauseno enhanced additional pumping is required for them,providing a stabilized current is sufficient.

So far the flashlamp is the cheapest generator fora high pump photon flux. In the past flashlamp-pumpedsolid-state lasers have been scaled to 20 kJ pulse en-ergy in a single beam. Such systems are based onschemes where bundles of meter-size flashlamps pump-ing several Nd-doped glass discs arranged at Brewster’sangle in the laser beam path as sketched in Fig. 11.233.For fusion lasers like the National Ignition Facility inthe US (NIF) [11.1975], Laser Megajoule in France(LMJ) [11.1976] or Gekko in Japan [11.1977] severalsuch beam lines are bundled together to form megajoulelaser facilities. Such beam lines were the first with thecapability to generate petawatt laser pulses [11.1965].Typical for this kind of lasers are a double-pass mainamplifier and a single-pass booster amplifier section.

The drawback of flashlamp pumping is the low ef-ficiency of this scheme. Pump photons are not onlygenerated at the absorption wavelength of the lasermaterial but also at wavelengths not involved in thelasing process. This low efficiency results in sub-stantial heat left inside the laser materials, whichare themselves poor heat conductors. The maximumrepetition rate of this disc configuration scales withthe inverse square of the beam diameter. Amplifierswith beam diameters of some 10 cm can be oper-ated at repetition rates of some shots per day only.With these systems scientific investigations of plasmaeffects are extremely difficult and expensive. For Nd-doped glasses the gain bandwidth limits the shortestpulses to about 400 fs. To generate 100 TW to PWpeak power pulses, energies in the 100 J range areneeded, which require large amplifiers. Nevertheless,using these lasers for pumping an OPCPA (see below)opens the door for ultra-broadband amplification of few-

!6B

49

!6

!6

!6

Fig. 11.233 Typical flashlamp pumped slab amplifier con-figuration. The seed pulse is injected at the spatial filterbetween a double-pass section and a single-pass boosteramplifier

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Lasers and Coherent Light Sources 11.13 Generation of Ultrahigh Light Intensities and Relativistic Laser–Matter Interaction 831

cycle pulses to the 10 J level with reasonable repetitionrates [11.1978].

Using laser diodes as the pump source helps torelieve some of these difficulties because their emis-sion spectrum is much narrower and can be fitted tothe absorption line of the laser-active medium. Con-sequently less energy is wasted and less heat is leftinside the laser active medium. However, high-powerdiodes with energy-conversion efficiencies up to 75%and continuous-wave output in the 100 W range areonly available in the red and near-infrared spectrum.These diodes are based on double heterostructures ofternary and quaternary semiconductors incorporatingGaAs. This spectral range is preferred for pumping rare-earth-doped laser materials, where Nd and Yb are mostfavored. Broadband gain materials including transitionmetals such as Cr:LiSAF will attract more interest ifsuitable diodes providing high brightness are developed.Today diode-pumped infrared frequency-doubled lasersare used to pump Ti:sapphire, the laser material ableto amplify the widest spectrum and hence the shortestpulses.

The direct diode-pumped amplification of 150 fspulses to the joule and 10 J level has already beenshown by pumping an Yb-doped fluoride-phosphateglass at 940 nm [11.1979, 1980] with the edge-cooleddisc arrangement shown in Fig. 11.234a.

To accomplish experimentalists’ demands for higherrepetition rates advanced cooling technologies have tobe developed even for diode-pumped solid-state lasers(DPSSL). Figure 11.234 shows a variety of DPSSLpumping schemes with different cooling architectures.

In slab lasers, where the various pump schemes de-picted in Fig. 11.234b–d are employed, one dimensionis used to provide a small distance for heat removal.They differ in the way in which pump light is pro-vided. Expanding this idea to two dimensions resultsin the cladding pumped fiber amplifier (Fig. 11.234g),which is not or only radiatively cooled like the heatcapacity lasers (Fig. 11.234f), which are able to runas long as the temperature does not reach a certainlevel. These lasers provide bursts of pulses. Thin disclasers (Fig. 11.234e) allow a very high average powerand can produce very good beam profiles but are dif-ficult to scale to larger beam diameters and higherpulse energies. A compromise is the use of a thickerdisc at a moderate repetition rate in the same con-figuration. The thicker disc does not require multiplepump-beam passes for full absorption and providesa larger single-pass amplification than the thin-discamplifier.

95

@

Fig. 11.234a–h Amplifier schemes for diode pumping. Edge cool-ing: (a) discs – conductive cooling (b) transverse diode-pumpedstructured slab and (c) transverse diode-pumped slab. Water cool-ing: (d) zigzag slabs and (e) thin discs. No cooling: (f) heat-capacitylasers and (g) fiber amplifiers. Gas cooling: (h) thin-disc assemblies

By putting several thin discs together and using theamplifier slab interspace for gas cooling, as in the archi-tecture sketched in Fig. 11.234h, the gain and absorptionequals that of the thick disc but efficient cooling isenabled. A diode-pumped laser of this type was devel-oped at the Lawrence Livermore National Laboratoryin the US. It is called Mercury [11.1981] and allowsan output of 65 J, the highest ns pulse energy froma single DPSSL at a repetition rate of 10 Hz so farreported.

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The long propagation length in a fiber that con-fines the laser light field to the doped and amplifyingcore allows high amplification factors in a single passand optical-to-optical efficiencies close to the theoreticallimit given by the quantum efficiency. For TEM00-moderadiation the maximum pulse energy is limited to somemJ for CPA. Nevertheless, average powers from sev-eral 100 W to the kW region are possible. To reacha high extraction efficiency for the low-energy seedpulses provided by fs oscillators multipass amplifiersor regenerative amplifiers are adopted. Multipass ampli-fiers can be found with a variety of geometries. All pulseparameters, such as the polarization or propagation di-rection, can be used to separate pulses from successivepasses and finally for out-coupling.

The regenerative amplifier is a seeded oscillator withan active element, in most cases a fast switching Pock-els cell, that allows the extraction of the amplified pulsewhen the gain of the active medium is saturated. It ben-efits from a stability that stems from gain saturationand beam quality that is ensured by successive spatialfiltering in the laser cavity. The drawbacks of regenera-tive amplifiers are pre-pulses produced by leakage fromthe cavity in round-trips before extraction and long pathlengths in dispersive material that has to be compensatedby the SCP of a CPA system.

The last section of a high-power laser amplifier sys-tem is a booster amplifier with a low number of passes orone pass only. In order to extract the energy with high-efficiency, saturation fluence has to be reached across thefull beam diameter. The result is a so-called top-hat beamprofile with a uniform fluence in the center and a steepfluence rise at the edge. In gain media with a long propa-gation distance this may be achieved automatically aftersome distance and the length were uniform aperture fill-ing is not guaranteed hardly affects the overall efficiencyof the amplifier. For amplifiers with short propagation

Fig. 11.235 Scheme to achieve broadband optical paramet-ric amplification. In order to achieve broadband signalamplification in a parametric nonlinear optical process us-ing a narrow-band pump different idler wavelengths havedifferent propagating directions in a crystal to fulfill thephase-matching condition while all signal waves propagatecollinearly

distances of the pulse in the active medium the seed pulseitself has to be converted into a top-hat profile startingfrom a Gaussian. A simple method to implement this isby diffraction at a serrated aperture [11.1982].

Because a top-hat beam unlike the Gaussian changesits profile while propagating it has to be relay imaged tosuccessive optical elements of the laser system to avoidhot spots and laser damage. In high-power laser systemsrelay imaging is often combined with beam expandersand spatial filters [11.1983].

Broadband Optical Parametric Chirped PulseAmplification

Alternatively instead of conventional laser amplifiersa parametric amplification process (OPA) can be usedfor the generation of high-energy broadband pulses. Inthis second-order nonlinear interaction a pump photonis split into a signal and an idler photon. This processis the reverse of sum-frequency generation. Energy-conservation demands that the sum of the signal and idlerfrequencies equals the pump frequency. Additionally,fulfilling momentum conservation ensures coherenceof the involved waves while propagating through thenonlinear medium. The latter is known as phase match-ing. The nonlinear medium has to be a crystal in orderto show second-order nonlinearity and achieve phasematching. The phase-matching condition also deter-mines the wavelengths of the signal and idler waves.

For a certain direction of pump, signal, and idlerwaves only one combination of wavelengths can begenerated. In order to allow broadband amplificationof a signal input wave using a narrow-band pump thedifferent idler wavelengths should have different di-rections in the crystal. This principle is illustrated inFig. 11.235 and is called optical chirped pulse amplifi-cation (OPCPA) if the signal beam is a chirped stretchedpulse [11.1984–1987].

With the OPCPA scheme, energy can be transferredfrom a narrow-band nanosecond pulse to a stretchedbroadband signal pulse. The problem of amplificationof ultrashort pulses is thereby split into the task ofamplification of high-energy laser pulses and the taskof implementation of broadband OPA. Comparing theOPCPA with a conventional laser-pumped CPA severaladvantages can be realized. Firstly no energy is storedin the nonlinear crystal and, except parasitic absorption,no energy is lost in the crystal, which results in no heatbeing produced in the process. This fact allow easy high-repetition-rate scaling of the amplifiers without phasedistortions by thermal effects. Because no resonance toenergy levels of a dielectric medium is required, very

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large band widths at any center wave length can be gen-erated. It was shown that pulses shorter than 10 fs canbe amplified. Using crystals with high nonlinear opticalcoefficients like the most prominent crystal BBO and atsufficient intensities a single pass can provide an am-plification of the signal by many orders of magnitude.Regenerative amplifiers are not needed and the problemof pulse leakage from them disappears. Amplificationlasts only as long as the pump passes the crystal andtherefore pre-pulses and post-pulses as well as back-reflected pulses are not amplified any more and higherpulse contrasts are possible with the OPCPA technique.Lastly the amplified signal beam is an analog of theinput signal and unwanted phase distortions of the high-energy pump are carried away with the idler beam likethe different k vectors in Fig. 11.236.

In order to use all of these advantages a careful de-sign of OPCPA is required. For instance if an OPCPAstage is driven to its maximum amplification and theprocess starts to saturate, the impact of intensity varia-tions of the signal to the output is reduced but intensityvariations of the pump are transferred to the amplifiedbeam. Additionally, signal and idler waves are generatedstarting from noise, a problem similar to the amplifiedspontaneous emission (ASE) in a conventional laser am-plifier. To minimize this effect a pump pulse length nolonger than the seeding signal pulse and their perfectsynchronization is required, which is challenging for pspulses. Contrariwise ps pulses offer higher intensities

+

''''(''%

#

I""

09:

04,

Fig. 11.236 Typical pulse contrast characteristics. The mainfemtosecond laser pulse is typically preceded by ampli-fied stimulated emission (ASE), amplified pre-pulses anduncompressed chirp

at a fixed fluence, which reduces the required crystallength, with the advantage of increasing the amplifica-tion bandwidth without changing the crystal orientation.More requirements may arise at different wavelengthsinvolved, like absorption, or second-harmonic genera-tion from the signal or idler beams, amongst others.

Nevertheless, OPCPA is the technique that allowsthe implementation of an ultrashort pulse option to allcoherent high-energy light sources that emit pulses in thenanosecond and picosecond range and transfers them toan ultrahigh-peak-power laser system. A 200 TW 45 fslaser system was recently demonstrated [11.1988] basedon OPCPA in a large-aperture KDP crystal. These crys-tals are used for Pockels cells and frequency convertersin fusion laser systems and can be grown with meter-sizeapertures. This in principle allows multi-petawatt powerscaling.

High Demands on Pulse Preparationfor Advanced Experiments

The acceleration of electrons and protons by a laserrequires light intensities on the order of 1020 W/cm2,which is many orders of magnitude above the onset ofnonlinear effects and atom ionization, which results inCoulomb explosion even in low-density targets such asgases, triggered by lower intensity background radia-tion of the laser. Experiments are strongly affected bythis pre-pulse laser-target interaction [11.1989]. To pre-vent the target from being destroyed before the main partof the laser pulse arrives, laser output has to be cleanedfrom its leading pedestal. A typical laser pulse character-istic is shown in Figure 11.236. The long-term pre-pulsepedestal results from ASE of the laser active material,whose duration is closely linked to its fluorescence life-time. Additionally, regenerative amplifiers that are oftenused as the first devices in the amplifier chain, alwayshave a certain leakage of pulses from preceding round-trips. The pulses are again amplified in successive mul-tipass and booster amplifiers, generating high-energypre-pulses on target. Finally, non-bandwidth-limitedpulse recompression based on material dispersion, wave-length clipping, aberrations in stretcher and compressorsetups and uncompensated higher-order dispersionterms result in pre-pulses and pedestals as shown inSect. 12.1.2 for frequency-domain-filtered short pulses.

Numerous methods are used to remove this un-wanted pre-pulse laser light. Primarily, the totalamplification is split into stepwise amplification with ad-ditional possibilities for time-domain and spatial pulsefiltering. For this purpose fast Pockels cells [11.1990]and spatial filters, which are often combined with

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834 Part C Coherent and Incoherent Light Sources

beam expanders, are applied. Spatial filters and aper-tures prevent spontaneously emitted fluorescence lightfrom propagation to the subsequent amplifiers. Thesemeasures can often improve pulse contrast ratios con-siderably [11.1991].

Nonlinear optical effects that favor high inten-sities can be used as well. Examples of the lattercase are saturable absorbers and laser pulse switchedgates. The OPCPA inherently embodies such a gate ifa rectangular-shaped pump pulse is used. Therefore pre-pulse suppression in the OPCPA [11.1992] can be ashigh as its gain.

Saturable absorbers and self-phase-conjugating mir-rors are used for nanosecond laser systems. In the caseof a chirped pulse amplification system these techniquesrequire pulse compression before and stretching after fil-tering. This limits their application to low-energy pulsesor prohibits their use if very long stretching factors areneeded.

Nonlinear effects can also be used after final pulserecompression. One possibility is frequency conversionby second-harmonic generation. Another pre-pulse sup-pression method with high efficiency and low loss is theusage of a plasma mirror [11.1993]. Because the laserhas to be focused anyway a perfectly transmitting di-electric medium is placed near the focus of the beam.The leading edge of the pulse is transmitted until theintensity reaches the ionization threshold. At this stepa plasma is generated with increasing electron density.The plasma occurs on a time scale that does not allowit to expand. If the plasma frequency matches the laserfrequency this plasma acts as a perfect mirror and themain pulse is reflected to the target. With a single plasmamirror the contrast ratio for petawatt-class lasers can beimproved by factors of about 100 [11.1965].

Experiments with lasers at ultrahigh intensities re-quire the focusing of a high-peak-power laser ontoa very small spot. Phase distortions in the beamline can increase the minimum spot size in the fo-cal plane considerably. Adaptive mirrors are used tocorrect phase fronts [11.1994]. Closed-loop systemsallow near-diffraction-limited focusing of ultrashortpulses [11.1995,1996]. These techniques today allow in-tensities in the range of 1020 –1021 W/cm2 with terawattand petawatt laser systems.

11.13.2 Relativistic Opticsand Laser Particle Acceleration

The interaction of light and matter at intensities of1020 –1021 W/cm2 changes considerably when com-

pared to classical, even classical nonlinear optics. Atintensities of 1013 –1015 W/cm2 matter becomes ion-ized and the high intensity laser pulse interacts witha dynamically evolving plasma where the dominantmechanism for the interaction of light and matter is theinteraction with free electrons.

In classical optics the interaction of light and matteris described in the following way: the electric field ofan electromagnetic wave exerts a force on the bound orfree electrons, which then oscillate with the frequency ofthe wave. The dependence of the oscillating polarizationinduced in the material by the motion of the electronson the driving electric field then yields the linear andnonlinear optical constants of the material. Althoughthis picture is inherently classical it also provides thegeneral idea of a quantum-mechanical description oflight–matter interaction.

This classical description of light–matter interac-tion is based on two assumptions, which are valid forsufficiently low light intensities:

1. the force exerted on the electrons by the magneticfield of the electromagnetic wave may be neglected.

2. The speed of the oscillating motion of the electronis small compared to the speed of light.

The breakdown of these two approximations marks theonset of relativistic optics, which is based on a fullyrelativistic description of the interaction of an electro-magnetic wave with matter.

Relativistic Motion of an Electronin an Electromagnetic Wave

We first consider the equation of motion of a free electronin the field of an electromagnetic wave

d

dt

(γmr)= −e

(E+ r × B

), (11.208)

where t is the time, m the electron rest mass, r thevelocity of the electron, e the elementary charge, E theelectric, B the magnetic field of the electromagneticwave, and

γ =(

1− ∣∣r∣∣2 /c2)− 1

2,

where c is the speed of light in vacuum. For a planeelectromagnetic wave of frequency ω and wavenum-ber k = ω

c propagating in the z-direction we insertE = E0 cos (ωt − kz). Since ∇ × E = −∂B/∂t we ob-tain: |B| = |E| /c. Before solving (11.208) we introducethe normalized quantities: t =ωt, z = zω/c, β = r/c and

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a0 = eE0mωc . Then we rewrite (11.208) in components as-

suming linear polarization of the electric field in thex-direction:

d

dt(γβx)= a0 (1−βz) cos

(t − z),

d

dt

(γβy)= 0 ,

d

dt(γβz)= a0βx cos

(t − z). (11.209)

Although (11.209) may be solved for more-general ini-tial conditions we give here the solution for an electroninitially at rest in the origin of our coordinate system,i. e., r(0) = 0 and r(0) = 0, with the phase parameter

K)- - -

)

*

K)

)

*

$-

$-

$-

$-

$*$'$'$*

Fig. 11.237 (a) Trajectories of a free electron in a travel-ing electromagnetic wave for different laser field strengthsa0 = 1 and a0 = 2 calculated for the initial conditionsθin = 0 and βz0 = 0 in the laboratory frame. (b) The tra-jectories in the moving frame of the electron (indicated bythe primed coordinate) exhibit the characteristic figure-of-eight movement. Laser field strength and initial conditionscorresponding to (a) were used

Θ = t − z:

x = a0 (1− cosΘ) ,

y = 0 ,

z = a20

4

(Θ− 1

2sin 2Θ

), and

γ 2 = 1+ a20

2sin2Θ . (11.210)

The motion of the electron is displayed in Fig. 11.237in two different coordinate systems:

1. the laboratory frame and2. the average rest frame of the electron, which is

a coordinate system co-moving with the electron inthe z′-direction, where the coordinate z′ is given byz′ = z −a2

0/4Θ.

It is immediately apparent that the motion of the electronin the fully relativistic case is quite different from theclassical picture, where a solution of (11.208) is obtainedfor γ = 1 and B = 0.

The electron oscillates, as in the classical case, peri-odically in the x-direction, the direction of polarizationof the oscillating electric field. However, unlike in theclassical case the electron is also accelerated in thez-direction, the propagation direction of the electromag-netic wave. The magnitude of the electron motion isgoverned by the parameter a0, the dimensionless electricfield strength [11.1997]. In the classical picture γ = 1,B = 0 which of course becomes incorrect at high ve-locities; the condition a0 = 1 would correspond to anelectron that acquires a maximum oscillation velocity ofr = c. The condition a0 1, therefore corresponds tothe classical case of |r| c while a0 1 describes theextreme relativistic motion of the electron. In practicalunits a0 is given by:

a20 = Iλ2

1.37 × 1018 W cm−2 µm2(11.211)

where I is the light intensity and λ its wavelength. Foroptical wavelength where λ≈ 1 µm intensities aboveI ≈ 1018 W/cm2 are therefore called relativistic intensi-ties. In the extreme relativistic case a0 1 the distancethe electron travels in z-direction during one electricfield oscillation is much larger than its excursion in thex-direction, and a complete reversal of the conditionsfound in the weakly relativistic case a0 < 1 occurs. Theacceleration of electrons in the propagation direction ofthe electromagnetic wave appears to be in contradic-tion to the well-known fact derived from energy and

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836 Part C Coherent and Incoherent Light Sources

momentum conversation that free photons cannot accel-erate free electrons. In fact if a light pulse is consideredthe electron acquires kinetic energy in the rising part ofthe pulse and looses it again to the electromagnetic wavein the falling part of the pulse. After the pulse has left,the electron experiences a shift in the propagation di-rection of the pulse but has not acquired a net energyin agreement with the conservation laws for energy andmomentum. If, however, the field of the electromagneticwave is suddenly switched off while the electron movesin the propagation direction the electron cannot give itsenergy back to the field and maintains its energy ac-quired from the field. The switching of the field may beachieved, i. e., by shielding the field in a steep plasmagradient or by a guiding channel.

The second interesting feature of the solution of theequation of motion (11.209) is the anharmonic oscillat-ing motion of the electron in the x- and z-direction bestseen in the co-moving frame (Fig. 11.237b). This meansthat the electron radiates not only at the frequency ofthe diving electromagnetic wave but also emits otherfrequencies. While for a0 < 1 the spectrum containsessentially the even (polarized in the z-direction) andodd (polarized in the x-direction) harmonics, the spatialand spectral emission pattern becomes very complex forhigher intensities (a0 > 1). This phenomenon is callednonlinear Thomson scattering and has been observedexperimentally.

The Ponderomotive ForceThe total energy of the electrons driven by the electro-magnetic field of a laser is given by

E(r, t) = γ (r, t)mc2 . (11.212)

Due to the dependence of γ on the electron velocity|r| the energy will vary widely on the length scale of thelaser wavelength and the time scale given by the laserfrequency. If one is not interested in the fast, mostlyoscillatory motion of the electrons one may average overtime, yielding:

〈γ (r)〉 =√

1+ a20(r)

2. (11.213)

The slow spatial dependence indicated in a20(r) may,

for example, be due to the intensity variation over thefocus of the laser beam, which usually occurs over muchlarger length scales than a wavelength.

The spatial dependence of the time-averaged energygives rise to a force called the ponderomotive force

Fp = −∇ 〈E(r)〉 = −mc2∇ 〈γ (r)〉 . (11.214)

This force will act on particles that oscillate in spatial re-gions where the laser intensity is high and push them intoregions of lower laser intensity. It is instructive to con-sider the weakly relativistic limit of the ponderomotiveforce. With the definition of a0 we find for 〈γ 〉−1 1:

Fp = − e2

4mω2 ∇E20(r) . (11.215)

In the classical limit this constitutes the gradient of thetime-averaged kinetic energy of the electron oscillatingin the laser field [11.1998].

Another way to look at the ponderomotive force isto view regions of high laser intensity as regions ofhigh electromagnetic energy density W = ε0 E2(r, t)/2,which represents a pressure pushing the electrons fromregions of high pressure to those of low pressure.

The Optical Properties of a Relativistic LaserPlasma and Relativistic Channeling

When an intense laser pulse at relativistic intensitiespropagates through a gas consisting of atoms with lowatomic number, such as hydrogen or helium, the risingpart of the laser pulse fully ionizes the gas and the laserpulse interacts with a fully ionized plasma. For examplehelium is fully ionized already at an intensity of a fewtimes 1016 W/cm2 while relativistic intensities usuallyexceed 1018 W/cm2. The dielectric constant of a fullyionized plasma is given by

εr = 1− ω2p

ω2, (11.216)

where ω2p = e2ne

ε0γm and ne is the electron density.Collisions have been neglected here which is a rea-

sonable assumption for laser pulses of durations ofless than 1 ps and electron densities of ne ≤ 1022 cm−3.It should be noted that in a relativistic plasma theplasma frequency ωp depends on γ and therefore onthe laser intensity. The dielectric constant becomes neg-ative for electron densities where the plasma frequencyωp exceeds the laser frequency ω. In the nonrelativisticcase (γ = 1) an critical electron density ncrit is definedthrough the condition ω2

p = ω2 yielding [11.1998]

ncrit = ε0mω2

e2. (11.217)

For electron densities ne > ncrit the plasma is calleddense while it is underdense for ne < ncrit. In prac-tical units the critical electron density is given byncrit = 1.1 × 1021 (1 µm/λ)2 cm−3. With this definitionthe dielectric constant may be rewritten

εr = 1− ne

γncrit(11.218)

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Lasers and Coherent Light Sources 11.13 Generation of Ultrahigh Light Intensities and Relativistic Laser–Matter Interaction 837

M(G)"

;92#)$2

#

@=Z('"'

,""4""""

46599#

'

C77$&2@7

2

..>

Fig. 11.238 Experimental setup. The main pulse is focused to an intensity of a few 19 W/cm2 into a pulsed He-gas jet,where a relativistic channel due to relativistic self-focusing is forming. By splitting and frequency doubling part of themain pulse a probe pulse is generated. This probe pulse is used for imaging and observing the laser–plasma interactionon a time scale determined by the probe pulse duration of ≈ 100 fs. The electrons that are accelerated in the relativisticchannel are characterized by magnetic spectroscopy and nuclear reactions

Electromagnetic radiation propagates in a mediumwhere εr > 0 and is reflected from a medium with εr < 0.The condition for light propagation is therefore ne/γ <

ncrit.The index of refraction of the plasma is given by

n = √εr. In an underdense plasma n is real and the laser

pulse propagates. But n is a complicated function ofintensity for two reasons:

1. At the center of the propagating laser pulse the in-tensity is high. Therefore the ponderomotive forcepushes the electrons away from the center and theelectron density in the center decreases, increasingthe index of refraction.

2. The remaining electrons in the center gain higherenergies in the laser field than those sitting at theedges of the laser beam. This increases γ further andthus increases the index of refraction at the center ofthe laser beam.

This means that a relativistic laser pulse self-modulates the index of refraction in such a way thatself-focusing of the laser occurs. As in other self-focusing phenomena known from nonlinear opticsself-focusing is countered by diffraction and there-fore self-focusing depends on a critical power ratherthan on intensity. A laser pulse interacting witha fully ionized plasma will therefore undergo self-focusing if its total power P exceeds the critical power

Pcrit [11.1999].

P> Pcrit = 8πε0m2c5

e2

ncrit

ne= 17.4 GW

ncrit

ne.

(11.219)

The dynamic equilibrium of defocusing because ofdiffraction and self-focusing due to nonlinear effectsleads to a guiding effect for the high-intensity laserpulse in a plasma called relativistic channeling. This phe-nomenon has been observed experimentally. A typicalexperimental arrangement is shown in Fig. 11.238.

A Ti:sapphire laser (λ = 800 nm) with a totalpower of 8 TW is focused to an intensity of about2 × 1019 W/cm2 into a helium gas jet from a noz-zle with a very well-characterized gas density profile(Fig. 11.239).

Figure 11.239 shows the relativistic channel whichextends over a length of about 300 µm, corresponding toabout 15 Rayleigh lengths of the laser focusing optics.At the onset of the channel the electron density amountsto 5 × 1019 cm−3

0 . The critical density is 1.7 × 1021 cm−3

at λ= 800 nm. Therefore the total laser power of 8 TWexceeds the critical power of Pcrit ≈ 0.6 TW obtainedfrom (11.219) substantially and relativistic channelingis expected and observed. In addition a probe pulse atλ= 400 nm was used to measure the electron density inthe channel interferometrically (Fig. 11.238). The resultshown in Fig. 11.240 gives the electron density profileacross the channel at the onset of the channel. It is ap-

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838 Part C Coherent and Incoherent Light Sources

'

=Q'- -

,"#("'

*

%

;

&*Q

49"6

,9"6

C77

Fig. 11.239 The lateral plasma density profile above thenozzle has a Gaussian shape (dotted line). The density alongthe channel and its position is indicated by the bold line.The inset shows the corresponding observed channel of thesame spatial region. The extension of the emission indicatesa channel length of 274 µm, about 12 times the Rayleighlength of the laser. The relativistic channel starts near thesteepest density gradient and ends close to the maximumdensity (grad ne = 0)

parent that the electron density at the channel center isdepressed, due to the expulsion of electrons because ofthe ponderomotive force. This result also provides di-rect evidence for the guiding structure of a relativisticchannel.

Q

)"

$

$-

-

-

Q

- - -

Fig. 11.240 Plasma density at the start of the channel deter-mined by interferometry. The electron density in the wallsof the channel rose up to values of ne(wall) = 6 × 1019cm−3.The image in the upper right corner shows the interferogramfrom which the plasma density was obtained

Finally it should be noted that optics in the relativis-tic regime is always nonlinear. The index of refractionis always intensity dependent due to the ponderomo-tive force and the intensity dependence of the electronmass.

Electron AccelerationSince in the relativistic regime of laser–plasma inter-action charged particles may be accelerated to highenergies, it has been recognized for some time that laserplasmas are an ideal medium for high-field compactaccelerators. The plasma as an ionized medium maysustain much higher fields than it is possible to generatewith conventional accelerator technology, where mater-ial breakdown imposes a limit at less than 100 MV/m.Electric fields in the TV/m range may however begenerated in laser plasmas.

In many experiments over the past two decadesa variety of acceleration mechanisms have been iden-tified. A selection of these mechanisms will brieflybe described in the following with focus on schemeswhere only a single laser pulse is needed to acceler-ate initially resting electrons to relativistic energies. Allthese schemes have in common that a laser pulse is fo-cused into a gas jet, generating an underdense plasma(Fig. 11.238).

Laser wakefield acceleration. When a relativistic laserpulse impinges on a plasma the ponderomotive forceexpels electrons in both the transverse and longitudinaldirections. The electrons expelled from the laser pulsein the backward direction lead to a plasma wave calledthe laser wakefield (Fig. 11.241) [11.2000].

The plasma wave follows the driving laser pulse witha phase velocity determined by the laser pulse groupvelocity. The electric fields associated with the plasmawave are now longitudinal. An electron can ride on theplasma wave and be accelerated to relativistic energies inthe direction of laser propagation. This process is calledlaser wakefield acceleration (LWFA).

This process is most efficient when the laser pulselength cτ , where τ is the pulse duration, is shorter thanthe plasma wavelength λp = 2πc/ωp.

This condition is depicted in Fig. 11.241a. Whena large number of electrons acquire a velocity close tothe phase velocity of the plasma wave, wave-breakingoccurs. The fast electrons are surfing on the wake of theplasma wave.

Self-modulated laser wakefield acceleration. If thelaser pulse length is longer than the plasma wavelength

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cτ > λp the laser pulse undergoes a self-modulationinstability. The leading edge of the laser pulse drivesa plasma wave. The electron density modulation ofthe plasma wave in turn represents a periodic modu-lation of the refractive index. It acts on the long laserpulse such that the pulse is self-modulated and breaksup into a train of short pulses (Fig. 11.241b). Theseshorter pulses now match the conditions for LWFAand can resonantly drive a plasma wave. This self-modulated laser wakefield acceleration (SM-LWFA)is not as efficient as pure LWFA, but still high-energy even quasi-monoenergetic electron beams maybe generated.

Direct laser acceleration. Another acceleration pro-cess which is quite different in nature to wakefieldacceleration is direct laser acceleration (DLA) whichis closely related to the formation of a relativisticchannel. The ponderomotive force expels electronsfrom the laser beam axis and generates a radial qua-sistatic electric field. Electrons that are accelerated alongthe laser propagation generate an azimuthal magneticfield. The combination of these two fields results inan effective potential well for relativistic electrons.Electrons trapped in this well will oscillate at the fre-quency ωβ = ωp/

(2√γ), the betatron frequency. If

the trapped electron is moving fast enough along thelaser propagation, the laser oscillations may be inphase with the betatron oscillations in the frame of theelectron.

In this case, an efficient energy coupling is possible.The energy gained by the electron in this process directlyresults from the laser field and therefore the name directlaser acceleration is appropriate.

Bubble acceleration. A novel regime of laser wake-field acceleration, called bubble acceleration, wasproposed in 2002 on the basis of particle-in-cell(PIC) simulations [11.2002]. Short (τ < 7 fs) and in-tense (a0 > 1) laser pulses were predicted to producequasi-monoenergetic electrons with energies exceeding100 MeV.

The ponderomotive force also plays an importantrole for this acceleration mechanism. In a frame mov-ing with an intensive laser pulse propagating througha plasma the electrons are expelled from the center of thepulse in both the longitudinal and transverse directions.It turns out that behind the laser pulse an electron densitydepression, called a bubble, is generated and electronsthe stream around this bubble and enter it again from theback side. In this way a strong electric field in longitudi-

'

2

'

Fig. 11.241 (a) LWFA: a short laser pulse (cτ ≤ p) shown as a dashedline drives a plasma wave. (b) SM-LWFA: an initially long laserpulse (dashed line) breaks up into a train of shorter pulses thatmatch the condition of LWFA and resonantly drive a plasma wave(after [11.1999])

nal direction reaching TV/m is generated, which leadsto efficient electron acceleration.

Experiments. Numerous laser acceleration experimentshave been carried out, mostly using experimental setupslike the one shown in Fig. 11.238. Initially always quasi-exponential electron spectra were obtained with electrontemperatures that follow the scaling law

kTe ≈ mc2

⎛⎝√

1+ a20

2−1

⎞⎠ , (11.220)

which corresponds to the kinetic energy of the elec-tron in the laser field. These exponential spectra

,"#<

C 9" #T)) )UM

-M

M

$-M

M

$-M

;%*

Fig. 11.242 Monoenergetic electron spectrum generatedin an experimental setup similar to the one shown inFig. 11.232 [11.2001]

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are not only observed in underdense plasmas butalso in the overdense plasmas encountered in laser–solid interactions [11.2003]. For a typical intensity ofI = 1020 W/cm2 and λ= 1 µm, a0 = 8.5 we obtainkTe ≈ 2.6 MeV from (11.220).

Only recently, however, has it became possibleto generate monoenergetic electron spectra. As pre-dicted by numerical simulations the bubble regimemay be reached for high-power, gently focused laserpulses of short pulse duration. Although the require-ments of the pure bubble regime have not yet quitebeen reached experimentally the experiments show thatin the transition regime between LWFA and bubbleacceleration monoenergetic electron spectra may be gen-erated [11.2001, 2004, 2005]. A typical result is shownin Fig. 11.242.

Ion AccelerationRelativistic laser–matter interaction also leads to thegeneration of monoenergetic beam of ions of MeV ener-

/5=99

@""

CND0

""

+6956"% "

+>+>+,.?

Fig. 11.243 Laser acceleration of protons from the back side of a mi-crostructured target. A terawatt (TW) laser pulse is focused onto thefront side of the target foil, where it generates a blow-off plasmaand subsequently accelerates electrons. The electrons penetrate thefoil, ionize hydrogen and other atoms at the back surface and set upa Debye sheath. The inhomogeneous distribution of the hot electroncloud causes a transversely inhomogeneous accelerating field [targetnormal sheath acceleration (TNSA)]. Applying a small hydrogen-rich dot on the back surface enhances the proton yield in the centralpart of the accelerating field, where it is nearly homogenous. Theseprotons constitute the quasi-monoenergetic bunch

gies by a mechanism quite different from the mechanisminvolved in electron acceleration.

As discussed earlier the interaction of an intenselight field with matter yields the generation of a hotplasma and the subsequent acceleration of electronsup to relativistic energies. Protons and ions are ac-celerated by a well-controlled mechanism known astarget normal sheath acceleration (TNSA) followingthe initial electron acceleration (Fig. 11.243). Fast elec-trons are accelerated by an intense laser pulse (intensityI ≥ 1019 W/cm2) from the surface of a thin metal foil inthe forward direction. They penetrate the foil and ion-ize atoms along their paths. Within about a picosecond,those electrons leaving the target at the rear surface (thatis, the back surface with respect to the laser irradiation)build up a quasistatic electric field. The field acts nor-mally to the target surface, has cylindrical symmetry anddecreases in the transverse direction. Owing to the ultra-short duration of the electron bunch and its high charge,this field may reach values of several TV/m close tothe axis and thus the potential can attain several tens ofMeV. Protons and positively charged ions present on theback surface of the foil may be accelerated by this fielduntil they compensate the electron charge. In most cases,the origin of these parasitic protons has been identifiedto be a hydrocarbon contamination layer on the targetsurface.

As the duration of the acceleration is ultrashort andthe protons (as well as the ions) are at rest before accel-eration, comprising a very small phase-space volume,the transverse emittance of the proton beam reaches val-ues as low as a few 10−3 mm mrad for 10 MeV protons.However, laser-accelerated ion beams still show an en-ergy spectrum exhibiting a quasi-exponential shape witha distinct cut-off energy. This can be explained by theinhomogeneous distribution of electrons in the sheath,which causes an accelerating field that is inhomogeneousin the transverse direction. For a plane and unstructuredtarget, the transverse dimension of the electric field andhence the source size of the accelerated protons is muchlarger than the laser’s focal spot. Therefore, different par-asitic protons experience a range of potentials, resultingin a broad distribution of energies.

Following this understanding of the mechanism oflaser acceleration of protons, it was pointed out thatthe resulting proton energy spectrum has a strong cor-relation to the spatial distribution of the protons on thetarget surface. In order to generate high-quality protonbeams with monoenergetic features, a bilayered, mi-crostructured target, consisting of a thin high-Z metalfoil and a small proton-rich dot on the back surface

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Lasers and Coherent Light Sources 11.14 Frequency Stabilization of Lasers 841

was proposed. The transverse dimension of such dots issmaller than that of the acceleration sheath, and hencethe protons will only be subject to the central – thatis, homogeneous – part of the acceleration field. Inthis configuration, the protons all experience the sameelectric field and are accelerated in the same potential(Fig. 11.243). The resulting proton beam has a spectrumwith a strong monoenergetic peak [11.2006].

Recent experiments with a 10 TW, 600 mJ lasershowed that in this way quasi-monoenergetic protonbeams containing 108 protons with energies of a fewMeV and a relative width of ∆E/E ≈ 10% and a totalcharge of about 100 pC were generated.

In a different experiment quasi-monoenergetic car-bon ions of 4 MeV energy were generated from a verythin carbon layer on a metal target [11.2007].

Applications and Future DevelopmentsRelativistic optics leads to laser particle acceleratorsthat are considerably more compact than classical ac-celerators. Presently laser accelerators reach maximumenergies for monoenergetic electron beams approaching1 GeV and several MeV for monoenergetic ion beamswith a total charge of up to 1 nC per bunch. The scal-ing for both electron and ion acceleration with laser andtarget parameters appears to be sufficiently well under-stood that electron energies exceeding 1 GeV and ionenergies of a few hundred MeV in a single accelerationstage appear to be possible in the near future. There are

however at least two challenges laser accelerators thathave to overcome before widespread applications willbecome attractive.

In order to reach energies in the TeV rangemultistaging of both electron and ion acceleratorsappears necessary. Although energy, beam qualityand charge per bunch are already attractive forlow-energy applications of laser accelerators their av-erage power presently remains orders of magnitudebelow the average power of conventional acceler-ators. This is solely due to limitations in lasertechnology and may be overcome in the futurewhen high-average-power high-intensity lasers becomeavailable.

Laser accelerators have already been used fora number of demonstration experiments. Low-energynuclear reactions have been induced using high-intensitylasers [11.1999] and nuclear transmutation scenariosemploying lasers have been discussed [11.2008]. Laser-generated ion beams carry enough dose for radiationtherapy using ion beams, a very promising applicationbecause the average power of a laser accelerator is al-ready sufficient for this innovative method of cancertreatment.

With ever-increasing laser intensities other areasof physics such as gravitational physics, elementaryparticle physics or nonlinear quantum electrodynamics(QED) effects [11.1999] also come into the focus ofhigh-intensity lasers.

11.14 Frequency Stabilization of Lasers

Soon after the development of the first HeNe laser, it wasrealized that the radiation of a continuous-wave laserwould be ideally suited as a measurement tool, providedits frequency was stable and reproducible. Applicationsof frequency-stabilized lasers include high-resolutionlaser spectroscopy, quantum optics, optical frequencystandards, the determination of fundamental constants,and the detection of gravitational waves. In these var-ious cases, the requirements on frequency-stabilizedlasers are quite different. For example, optical frequencystandards and optical clocks need the smallest possi-ble uncertainty in the knowledge of the absolute laserfrequency whereas gravity-wave detectors need lasersof extremely low frequency noise while the absolutelaser frequency is less important. Hence, different sta-bilization methods have been developed to address thevarious tasks. The purpose of this chapter is to review

the general methods of laser frequency stabilization andto describe some representative examples of frequency-stabilized lasers. The radiations of a variety of stabilizedlasers are recommended as optical reference frequen-cies [11.2009].

Basically, the frequency of any laser is determinedwithin the bandwidth of its amplifying medium by theoptical length of the resonator. This optical length in turnis a function not only of its actual geometrical lengthbut also of the refractive index of the gain mediumitself, which may depend on several different param-eters. The width of the gain profile may vary betweena few 10−6 ν for gas lasers up to approximately 10% for,e.g., dye lasers or laser diodes. Different types of lasershave different noise characteristics. In most cases, thedominant frequency noise is of technical nature and isfar above the Schawlow–Townes limit [11.2010]. Since

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such technical frequency fluctuations are relatively slowthey can be controlled by a suitable electronic servo sys-tem even if the frequency excursions of the free-runninglaser are large. Different laser types and the specificrequirements on these lasers have led to a variety ofstabilization methods. Within the limited space of thisarticle, it is impossible to present a comprehensive anddetailed description of all these stabilization schemes.Hence, we concentrate on the description of the basicprinciples of laser frequency stabilization. For a deeperunderstanding, the reader is encouraged to study the ref-erences given in this article as well as textbooks on laserspectroscopy and optical clocks [11.2011–2014].

We start with a brief discussion of the terms char-acterizing the frequency behavior of a laser. Theseare noise, stability, line width, reproducibility, andthe uncertainty of the laser frequency. Section 11.14.2describes the basics of laser frequency stabilization. Ex-amples of stabilized lasers are presented in Sect. 11.14.3.Section 11.14.4 explains a universal method of opti-cal frequency measurements by means of mode-lockedfemtosecond lasers.

11.14.1 Characterizationof Noise, Stability, Line Width,Reproducibility, and Uncertaintyof the Laser Frequency

Noise, stability, line width, reproducibility, and the un-certainty of the frequency are important parameters ofany frequency-stabilized laser. In general, the laser fre-quency fluctuates about a mean value which itself maydrift and walk randomly. Such variations may be caused,e.g., by changes in the temperature, air pressure, vibra-tions, acoustics or by fluctuations within the active lasermedium itself.

The variations in the laser frequency can be investi-gated, e.g., by measuring the beat frequency νB = |ν1 −ν2| between two identical but independently stabilizedlasers 1 and 2 (Fig. 11.244). For this purpose, the beams

![ "#"

4" #7

>"9

/4

2

'

Fig. 11.244 Schematics of a beat-frequency measurement

of two lasers are coaxially combined by a beam splitterand focused onto a fast photodetector. The beat fre-quency νB shows up as an oscillation in the power of thecombined laser beams. The photodetector transformsit to an oscillation of the photocurrent, provided νB issmall enough that it can be processed by state-of-the-artelectronics.

Frequency fluctuations of a laser can be measuredin both the frequency and time domain. Measurementsin the frequency domain are often used at higher fluc-tuation (Fourier) frequencies whereas slower-frequencyfluctuations and drifts can be measured conveniently inthe time domain (see below).

Spectral Density of Frequency NoiseIn the frequency domain, fluctuations δν at the fre-quency f can be detected by a frequency discriminator,which converts frequency fluctuations to proportionalvoltage fluctuations. Since the noise components are notcorrelated, it is convenient to describe the fluctuationsδν( f ) by the mean square 〈δν( f )2〉 which occurs withina bandwidth B at the Fourier frequency f . This valueis equivalent to the spectral noise power. The powerspectral density is then defined as Sf = 〈δν2〉/B and theintegration of Sf over B results in the total power of thefrequency noise within the bandwidth B. For the relativepower spectral density Sy we get Sy = 〈(δν/ν)2〉/B.

To a good approximation, the frequency noise ofany oscillator can be modeled by a power series of theFourier frequency

Sy =2∑

α=−2

cα f α . (11.221)

Depending on the exponent a the five terms of (11.221)describe

• a random frequency walk for α= −2• flicker (1/ f ) frequency noise for α= −1• white frequency noise for α= 0• flicker (1/ f ) phase noise for α= 1• white phase noise for α= 2

Superimposed on the noise given in (11.221), we mayalso find peculiarities in the noise spectrum that are gen-erated by noise sources specific to the individual laser.Such noise may be generated, e.g., by environmentalsources such as the excitation of mechanical resonancesin the laser resonator, by instabilities of the mechani-cal setup, or by fluctuations in the refractive index ofthe laser medium. Furthermore, the laser frequency issensitive to optical feedback and careful isolation of the

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laser system from back-scattered light is important forsuccessful operation of stabilized lasers. Usually, thedifferent noise components show up at different Fourierfrequencies. Hence, the detection of these noise compo-nents helps us to identify the various noise sources andto reduce their influence. In addition, the knowledge ofthe frequency noise spectrum of a free-running laser isalso important for the design and optimization of thefrequency control system.

The noise spectra of various types of lasers maybe quite different. For example, most of the frequencynoise in gas lasers such as HeNe or CO2 lasers oc-curs at low Fourier frequencies. It is generated in partby mechanical instabilities of the laser setup. The cor-responding frequency excursions δν may amount toa few MHz at Fourier frequencies up to about 10 kHz.In addition, random-walk and flicker (1/ f ) frequencynoise shows up at these low frequencies. Neglectingfurther possible noise sources originating from plasmaresonances of the discharge, the noise spectrum athigher Fourier frequencies may be approximated bywhite frequency noise. In tunable CW dye lasers, onthe other hand, the frequency noise is much largerthan in gas lasers. It is generated mostly by variationsin the thickness and the refractive index of the dyejet, which is flowing at high speed through the smallactive volume of the laser. Hence, rather strong fre-quency fluctuations are observed at Fourier frequenciesaround 50 kHz. At higher Fourier frequencies the influ-ence of the dye jet decreases and hence the noise alsodecreases, eventually reaching the photon shot-noiselevel.

Allan Standard DeviationSlow variations, in particular drifts and random walk ofthe laser frequency can be measured conveniently in thetime domain by counting the mean frequency ν withina time interval τ . Such measurements are advantageousif changes in the frequency within time intervals betweenseconds and several hours are of interest. The relativefrequency instability in the time domain is characterizedby the Allan deviation [11.2015, 2016]

σy(τ) = 1

ν

[1

2(N −1)

N−1∑n=1

(νn+1 −νn)2

]1/2

,

(11.222)

where the root mean square of the difference betweentwo consecutive frequency measurements (νn+1 −νn) isused as a measure of the stability. The frequency values

νn+1 in (11.222)

νn+1 = 1

τ

(n+1)τ∫i=nτ

νi(t)dt (11.223)

represent the mean frequency averaged during the n-thinterval of duration τ . Since it is difficult to measurethe optical frequency directly, it can be down-convertedby beating the laser radiation against the radiation ofa second laser such that the beat frequency arrives in theradio-frequency range, where it can be counted directly.In this case, the measurement contains the instabilitiesof both lasers. If the two lasers are identical but indepen-dent, we can assume that both lasers contribute equallyto the standard deviation and consequently, the meas-ured value corresponds to

√2 times the Allan deviation

σy(τ) of a single laser.The dependence of σy(τ) on the averaging time τ

contains information about the frequency noise spectrumof the laser. If the relative power spectral density ofthe noise Sy is known, σy(τ) can be calculated by therelation [11.2016]

σ2y (τ) =

∞∫0

Sy( f )(sinπ f τ)4

(π f τ)2d f . (11.224)

For the different models of the noise process given in(11.221), we can describe the Allan variance σ2

y (τ) bythe relation

σ2y (τ) = dατ

β (11.225)

where β is determined by the relation [11.2016]

β =⎧⎨⎩

−α−1 for α ≤ 1

−2 for α > 1(11.226)

and dα is a constant. The dependence of σ2y (τ) on τ is

listed for the noise processes of (11.226) in Table 11.55.Of course, any frequency modulation will also show

up in the Allan standard deviation. In the case that the

Table 11.55 Dependence of the Allan deviation on τ for thedifferent noise processes discussed in (11.226)

Noise category α β σ2y (2,τ)

White frequency noise 0 −1 ∝ 1/τ

White phase noise 2 −2 ∝ 1/τ2

1/ f (frequency noise) −1 0 constant

1/ f (phase noise) 1 −2 ∝ 1/τ2

Random frequency noise −2 1 ∝ τ

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gate time τ is an integer multiple of the modulation pe-riod τm = 1/ fm, where fm is the modulation frequency,the influence of the modulation is eliminated. It will bemaximum for τm = (2n+1)/(2 fm). With increasing in-tegration times τ , the influence of the modulation washesout and can be neglected, eventually.

As a result of (11.227), we find that the noise pro-cess can be estimated from the slope of a log–log σy(τ)plot. For example, in a HeNe laser that is frequency sta-bilized to an absorption line of molecular iodine, σy(τ)decreases in a wide range of τ with the square root ofτ corresponding to a slope of −1/2, (Fig. 11.245) indi-cating white frequency noise within the correspondingtime interval τ . At longer integration times, σy(τ) flat-tens out due to flicker frequency noise and in some caseseventually increases again due to drifts and random fre-quency walks. For practical applications, the σy(τ) plotindicates the minimum integration time τ necessary toachieve a given statistical frequency uncertainty.

Line Width of Laser RadiationIn incoherent light sources such as spectral lamps, wherethe radiation consists of uncorrelated photons emittedwithin a certain narrow frequency band, the line width isdetermined by a superposition of the natural width of thetransition and Doppler broadening. It can be measured,for example, by a high-resolution scanning Fabry–Pérotinterferometer. In the case of laser radiation, the linebroadening is caused by frequency fluctuations of thecontinuously emitted coherent laser radiation. The fre-quency fluctuations generate noise sidebands that causebroadening of the laser spectrum. The line width is then

'

'

'

')

Fig. 11.245 Allan deviation σy(τ) versus integration time τof an iodine-stabilized HeNe laser

determined by the size and spectral extension of thesesidebands. In the case of pure harmonic phase modu-lation, the field E of the laser radiation is given by therelation

E(t) = E sin [Ω0t + δφ sin (ωt)]

= E

J0(δφ) sin(Ω0t)

+∞∑

n=1

Jn(δφ) exp[i(Ω0 +nω)t]

+∞∑

n=1

Jn(−δφ) exp[i(Ω0 −nω)t],

(11.227)

whereΩ0 = 2πν0 andω= 2π f represent the carrier andmodulation frequency, respectively; δφ = δν/ f is themodulation index with δν the amplitude of the frequencydeviation. The amplitude of the n-th sideband given bythe n-th order of the Bessel function Jn(δφ) decreasesstrongly with n for n > δφ. Basically, we can distinguishbetween the two limiting cases of

1. large frequency fluctuations δν at low Fourier fre-quencies f and

2. small, fast frequency fluctuations δ f in a broad band.

Case 1 is frequently observed with laser resonators,which pick up environmental acoustic or mechanicalnoise. The mean amplitudes 〈δν2〉1/2 of such fluctua-tions may be in the range between a few 10 kHz up toseveral megahertz, whereas they usually occur at Fourierfrequencies in a range below 1 kHz. Hence the modula-tion index δϕ 1. In this case, the line width is givenby [11.2017]

∆νFWHM =[8 ln (2)〈δν2〉

]1/2∼= 2.355〈δν2〉1/2 (11.228)

close to the frequency excursions between the peaks.If the servo gain of the frequency stabilization is

high enough to reduce the frequency fluctuations atlow Fourier frequencies, we may observe case 2. Un-der the assumption that we can neglect frequency driftsand frequency changes at very low Fourier frequencies,we arrive at frequency fluctuations (〈δν〉2)1/2 B andcorrespondingly to a small modulation index. In thiscase, only the amplitudes of the first-order sidebandsJn(δφ) and J−n(δφ) contribute significantly to the linewidth. If the spectral density Sf is constant, the lineprofile is Lorentzian and the line width is given by the

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Lasers and Coherent Light Sources 11.14 Frequency Stabilization of Lasers 845

relation [11.2017]

∆νFWHM = πSf . (11.229)

Frequency Reproducibility and Uncertaintyof Laser Radiation

Whenever the laser frequency is needed as a precise ref-erence, knowledge of the reproducibility and uncertaintyis essential. Since the frequency of the stabilized laserdepends on various operational and environmental pa-rameters it is necessary to analyze this dependence andto control these parameters carefully. The frequency re-producibility is a measure of the scatter in the frequencyvalues either of an ensemble of differently designedlaser systems of the same kind, developed for exam-ple in different laboratories. The reproducibility can thenbe investigated by frequency inter-comparisons betweensuch differently designed lasers. Alternatively, the termfrequency reproducibility is also used to describe thescatter in the frequency of a single laser caused by theuncertainty in the precise control and optimization ofthe operation parameters.

The frequency uncertainty is a measure of how ac-curately we can realize the reference frequency anddetermine its value. Basically, the total uncertaintycontains two contributions. The first represents theuncertainty to which the transition frequency of an un-perturbed atomic or molecular absorber at rest can berealized. This part contains all values in the uncertaintyof frequency shifts caused by environmental and/or op-erational conditions. Such shifts may be generated, e.g.,by a residual Doppler effect, by collisions, and by ex-ternal fields. The second term in the total uncertainty islinked to the determination of the frequency value. Suchfrequency measurements have to be referenced to theprimary standard of time and frequency, the Cs atomicclock. If the frequency measurement is phase-coherent,the measurement itself does not contribute to the totaluncertainty. Under this condition, the uncertainty of theoptical frequency is determined only by the uncertaintyof the standard itself and that of the Cs reference.

11.14.2 Basicsof Laser Frequency Stabilization

In any active frequency control system, the laser fre-quency is servo-controlled to a reference frequency. Thismay be provided by an eigenfrequency of a stable optical(Fabry–Pérot) resonator, by the gain profile of the laseritself, or by a suitable atomic or molecular transition.The first case has the advantage that it provides a good

signal-to-noise ratio, allowing a fast servo lock. How-ever, since the laser frequency is coupled to the length ofan artifact, it may change in time. Stabilization to suchartifacts is often applied as an intermediate step to re-duce the line width of the laser emission and to allow forhigh spectral resolution. In a second step, the frequencycan then be stabilized to an absolute frequency referenceprovided by a suitable atomic or molecular absorptionline.

The following sections give an introduction to ba-sic methods of laser frequency stabilization. We startwith the generation of the error signal and review fourprominent examples. The first two methods – the side-of-the-fringe stabilization and the phase-modulation(Pound–Drever–Hall) technique are used to stabilize thelaser frequency to a resonance of a stable optical cavity.Absolute frequency stabilization to an atomic referenceis then reviewed and the fundamentals of the electronicservo-control system are described.

Generation of the Error SignalBasically, the frequency stabilization of a laser convertsthe frequency deviation δν = νl −ν0 between the laserfrequency νl and the reference frequency ν0 to an errorsignal that is proportional to δν. This signal is amplifiedby a servo amplifier and used to control the laser fre-quency such that the error signal vanishes. Figure 11.246shows the basic scheme of a stabilization to an atomicreference frequency.

Side-of-the-fringe stabilization. In commercial sta-bilized lasers such as tunable dye lasers, it isa well-established practice to stabilize the frequency toa resonance of a stable optical cavity. The frequency

02" 2

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0

,

1

Fig. 11.246 Schematic of a laser frequency stabilization tothe center of an absorption line

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846 Part C Coherent and Incoherent Light Sources

discriminator is provided by the side of a transmissionfringe [11.2018]. The basic idea is that the sloping sideof the fringe converts the laser frequency fluctuations toamplitude fluctuations with high conversion efficiencyand a good signal-to-noise ratio. To prevent laser ampli-tude noise from entering the discriminator channel, twobalanced photodetectors are used (Fig. 11.247a). Onelooks at the frequency-selective transmission of the res-onator and the other views an attenuated portion of theinput beam.

The attenuation is adjusted such that the resultingdifference of the two photodetector signals crosses zeroat the half-point of the maximum transmission. In a fre-quency interval close to the zero crossing, the signaldepends almost linearly on the frequency deviation δν

and can be used as an error signal for stabilization. Thisside-of-the-fringe method is quite simple and very usefulfor many applications. However, if high spectral reso-lution and low drift rates are required, it turns out thatthe reference frequency is not defined precisely enough.For example, if the direction of the laser beam inci-dent to the resonator changes, the transmission will alsochange whereas the power in the other reference beamstays constant. Hence, the zero crossing of the error sig-nal changes, leading to a shift of the stabilized laserfrequency. A further difficulty arises from the fact that

9"

/4

>

<

0 /

>

Fig. 11.247a,b Frequency stabilization to the side ofa fringe. (a) Experimental setup, (b) either point A or point Bcan be selected as the reference

the speed of the servo control is ultimately limited bythe response time of the cavity [11.2019]. Furthermore,the locking range of the servo control is asymmetric.For example, if the laser frequency is locked to point A(Fig. 11.247b), the locking range towards lower frequen-cies corresponds almost to the total free spectral range ofthe cavity, whereas for higher frequencies it correspondsonly to approximately one full width at half-maximum.Hence, spontaneous and fast excursions towards higherfrequencies exceeding the half-width may force the laserto fall out of lock and relock at the next-higher interfer-ence order. To increase the locking range and to avoidsuch unwanted frequency jumps, commercial lasers usu-ally apply reference resonators of rather low finesseat the cost of a strongly reduced sensitivity. Neverthe-less, side-of-the-fringe stabilization has been used verysuccessfully in many stabilized lasers.

Phase-modulation technique. Many of the shortcom-ings of the side-of-the-fringe method are avoided if thelaser frequency is locked to the center of a symmetricresonance. This can be achieved in transmission or re-flection. In the latter case, the amplitude of the returnbeam is a superposition of the beam reflected directly atthe entrance mirror and a small part of the light storedinside, which is leaking out of the cavity. At resonance,the two beams interfere destructively and the intensityof the return beam has a minimum at the line center.It follows that an abrupt change of the laser frequencyshows up instantaneously in the interference signal be-tween these two beams and the transient behavior of theresonator does not limit the bandwidth of the servo con-trol [11.2020, 2021]. Hence, a servo bandwidth up toa few megahertz can be achieved even with ultrahigh-finesse cavities with line widths in the low kilohertzrange.

Fast stabilization techniques to line center have beendeveloped [11.2022–2024] utilizing either DC or radio-frequency (RF) methods. In the following, we describean RF method where the signal is detected in reflection.This method was introduced by Pound [11.2025] to sta-bilize the frequency of a microwave oscillator and laterapplied in the optical range by Drever (the PDH tech-nique) [11.2023]. To generate the error signal, the phaseof the laser radiation is modulated at a radio frequency.For an optimum signal size, the modulation frequencyshould be larger than the width of the cavity resonance.A high modulation frequency is also advantageous, sincethe influence of 1/ f noise decreases and can eventuallybe neglected, leading to a signal-to-noise level close tothe shot-noise limit.

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The basic experimental setup of the PDH stabiliza-tion scheme is shown in Fig. 11.248. A small fraction ofthe laser power (< 10%) is split off the main laser beamand used for the stabilization. It is phase modulated byan electrooptic modulator (EOM). The modulation fre-quency (typically ≈ 15 MHz) is several times larger thanthe width of the resonance. The modulation index δν/ fmis in the range 10–30%. After passing a polarizing beamsplitter (PBS) and a 45 Faraday polarization rotator(FR), the incident beam is mode-matched into the refer-ence resonator. The polarization of the returning beamis rotated by another 45. Hence, the incident and the re-turning beam are polarized perpendicular to each otherand separated by the PBS. The return beam is sent toa photodetector (PD).

If the laser frequency is tuned far from resonance,the carrier and the sidebands of the modulated beamare promptly reflected and no intensity modulation isobserved by the photodetector. If the carrier or eithersideband approaches the cavity resonance, its ampli-tude and phase changes depending on the detuning ofthe corresponding frequency component from the cav-ity resonance. The other frequency components whichare off-resonance are promptly reflected. Hence, in thereturning beam, the balance between the carrier and thesidebands is broken and power modulation is gener-ated. The corresponding amplitude modulation of thephotocurrent is phase-sensitively detected by the PD fol-lowed by a double-balanced mixer, which is driven at themodulation frequency. The corresponding demodulatedsignals – after passing a low-pass filter – are shownfor different modulation frequencies G versus the laser

9"

+8B

, /4

>

>/

#

/4

B

4 9

Fig. 11.248 Schematic of the PDH frequency stabilizationtechnique (see text) (BS: beam splitter, PBS: polariz-ing beam splitter, EOM: electrooptic modulator, RF:modulation-frequency generator, PD: photodetector, DBM:double balanced mixer)

' )6

4

- & (

''*'%

' )6

4

)

- & (

* %''% '*

* %

Fig. 11.249a,b Demodulated signals versus detuning ob-served with the PDH technique, (a) in phase (ϕ = 0)and (b) in quadrature (ϕ = π/2) with the modulation.G = (νm/νh) corresponds to the ratio between the modula-tion frequency νm and the half-width νh of the resonance

frequency in Fig. 11.249a in phase and Fig. 11.249b inquadrature with the modulation voltage at the EOM.

In the first case (Fig. 11.249a), we observe a disper-sive signal D(∆) with zero crossings at the resonance ofthe carrier and close to the resonances of the sidebands.It is given by the relation [11.2013].

D(∆)

∝ Ω2Γ∆(Γ 2 +Ω2 −∆2)

(∆2 +Γ 2)[(∆+Ω)2 +Γ 2][(∆−Ω)2 +Γ 2] .(11.230a)

The absorptive signal A(∆) (Fig. 11.249b) is given by

A(∆)

∝ ΩΓ∆(Γ 2 +Ω2 +∆2)

(∆2 +Γ 2)[(∆+Ω)2 +Γ 2][(∆−Ω)2 +Γ 2] .(11.230b)

In (11.230), Γ /2π = νh is the half-width at half-maximum of the cavity resonance, Ω/2π is the

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modulation frequency, and ∆/2π = (ν−ν0) is the fre-quency detuning.

The central zero crossing of the dispersion signalD(∆) (Fig. 11.249a) is used as the error signal for thefrequency control. The signal detected in quadrature(Fig. 11.249b) shows a resonance feature when eitherof the sidebands coincides with the cavity resonance.The different polarities of the two features are caused bythe phase difference π between the carrier and the up-per sideband and that between the carrier and the lowersideband. Independent of the detection phase, the de-modulated signal is always antisymmetric with respectto the central zero crossing. Consequently, a small phasedeviation from zero does not change the frequency of thecentral zero crossing. However, it will slightly changethe slope of the frequency discriminator and thereby thegain of the servo loop.

Since the returning beam results from an interferencebetween the frequency components promptly reflectedat the entrance mirror and part of the stored radiationleaking out of the cavity, any fast frequency changeshows up as a change in the interference signal instanta-neously even if the response time of the cavity is muchlonger. At low Fourier frequencies f , smaller than thehalf-width Γ /2π(2π f Γ ), the transient response ofthe cavity can be neglected and the demodulated sig-nal acts as a frequency discriminator. With increasingFourier frequency, the signal turns to a phase discrimi-nator for 2π f Γ . Hence, the amplitude A( f ) and thephase φ( f ) of the error signal, have a low-pass charac-teristic which rolls off with 1/ f for frequencies f abovethe cut-off frequency Γ /2π [11.2020]

A( f ) = A(0)1√

1+ (2π f/Γ )2,

ϕ( f ) = − arctan(2π f/Γ ) . (11.231)

If this integrating behavior for Γ > 2π f is taken intoaccount in the design of the servo amplifier, a servobandwidth much larger thanΓ/2π can be envisaged. Ba-sically, the servo bandwidth is limited to a few megahertzby delays in the servo system.

If the frequency of the laser is stabilized, the sizeof the residual error signal provides us with informationabout the quality of the servo electronics. However, thiserror signal is by no means a measure of the actuallyachieved frequency stability or laser line width. Thereare still frequency fluctuations that are not detected bythe electronic servo. In particular, at very high spectralresolution, small variations of the optical length in thecavity can no longer be neglected. The true line width of

the laser can be estimated only by methods independentof the actual servo loop, e.g., by beat-frequency com-parisons with a second laser or by the use of a secondindependent frequency discriminator.

To achieve a narrow line width, it is important toisolate the cavity from any distortions, e.g., from vibra-tions, acoustics, and temperature changes, caused by theenvironment. For this reason, the resonator is frequentlyvibration isolated and supported in a temperature-controlled vacuum chamber [11.2020, 2026]. A furthercause of possible frequency offsets is a spurious resid-ual amplitude modulation (RAM) of the intrinsic laserbeam. Such RAM may be generated, e.g., by an im-perfect phase modulator itself. Very often, RAM is alsointroduced by spurious interferences in the optical setupcaused, for example, by light backscattered at the sur-faces of optical elements. The servo cannot distinguishbetween the true error signal and the RAM. Hence, thetotal of the superimposed signals are servo controlled tozero, leading to a nonzero error signal and consequentlyto a frequency shift. Usually, this spurious RAM varieswith time and we experience frequency fluctuations thatare not identified by the error signal of the servo loop.At high spectral resolution, these fluctuations may be or-ders of magnitude larger than those estimated from thenoise in the error signal. Hence, it is important to iso-late the cavity from environmental disturbances and tominimize the RAM very carefully.

The narrowest line width achieved with the PDHmethod in a dye laser was as low as 0.5 Hz [11.2027]and even with diode lasers a line width in the range of ap-proximately 1 Hz could be achieved [11.2021]. The PDHstabilization technique is now used almost exclusively, ifa precision stabilization to a cavity resonance is requiredcombined with extremely high spectral resolution.

Stabilization to a Doppler-free atomic reference. Thestabilization methods discussed above describe methodsto stabilize the laser frequency to a reference providedby the length of an artifact. An absolute stabilizationrequires that the frequency is traced back to a natu-ral constant, e.g., to the center of a suitable atomicor molecular absorption line. Ideally, the frequencyof such transition should not depend on environmen-tal or operational parameters such as external fields,atomic collisions, laser power, etc. The atomic referenceline should be narrow and provide us with a sufficientsignal-to-noise ratio. The atoms may be contained in anabsorption cell, in an atomic beam or stored in a trap.At room temperature, the thermal velocity distributionof the atomic absorbers cause a Doppler broadening of

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the line which is in the range of δν/ν ∼= 10−6 and whichis usually much larger than the natural line width of thereference transition. An atom moving with a velocitycomponent vz in the direction of the laser beam seesa Doppler-shifted frequency

νD = νl

[1±νz/c+ν2/(2c2)

], (11.232)

where νl is the laser frequency and νz is the velocity com-ponent in the direction of the laser beam. In (11.232),the term νlνz/c represents the first-order Doppler effect.The positive sign is valid for atoms counter-propagatingto the laser beam whereas the minus sign belongs to theco-propagating ones. The third term νlν

2/(2c2) corre-sponds to the (relativistic) second-order Doppler effect.Whereas the first-order Doppler effect can be stronglysuppressed by the application of nonlinear Doppler-free spectroscopic methods such as, e.g., saturatedabsorption [11.2028, 2029] or Doppler-free two-photonexcitation [11.2030, 2031] the second-order effect canbe reduced only by cooling the absorbers.

As a typical example of Doppler-free spectroscopy,we concentrate on the saturated absorption and givea brief description of this method. For simplicity, weneglect the influence of the second order Doppler effect.Let us consider an atomic ensemble which is excited bytwo counter-propagating laser beams of the same fre-quency νl. If νl does not coincide with the line centerν0 of the transition, the two beams will excite and sat-urate two groups of atoms with velocity components νzand −νz , of which the Doppler-shifted frequencies νDcoincide with the laser frequency νl. Consequently, twosaturation holes are burned into the velocity distributionof the ground state. If we tune the laser frequency closerto the center of the line, |νz| decreases and the holes ap-proach each other until they overlap for νz = 0 when thelaser frequency is tuned to line center. In this singularcase, the number of atoms contributing to the absorp-tion decreases and hence less laser power is absorbed,resulting in a narrow Doppler-free absorption dip whichis used as a reference for the stabilization. Hence, (first-order) Doppler broadening is suppressed and the widthof this dip is determined by the natural width of the tran-sition, by power and collision broadening of the line, andultimately by the limited interaction time of the atomsmoving across the laser beam. To first order, the pro-file of this dip can be approximated by a Lorentzian lineshape.

As a practical example, let us consider an absorbinggas contained in a cell at low pressure. It can be placedinside or outside the laser cavity. If the cell is placed

'' )6

'

'' ' )6

'

Fig. 11.250 Generation of the error signal by a square-wave fre-quency modulation

inside, a Doppler-free feature caused by saturated ab-sorption will show up as a small symmetric dip in thetuning curve of the laser at the center of the referencetransition. This feature has to be converted to an errorsignal which is antisymmetric with respect to the de-tuning ∆ν = νl −ν0. A simple concept to generate theerror signal is to switch the laser frequency betweentwo discrete values (Fig. 11.250a). The difference in thecorresponding two absorption signals I2 − I1 leads to anantisymmetric error signal with a zero crossing at the linecenter (Fig. 11.250b). For small modulation amplitudes,the error signal is proportional to the first derivative ofthe absorption signal.

Instead of a square-wave modulation, most lasersuse a harmonic frequency modulation. Consequently,the laser power will contain harmonics of the modu-lation frequency. The amplitudes of these harmonicsdepend on the modulation width and on the detuning.The amplitude of a particular harmonic signal in thelaser power can be detected by a photodetector followedby a phase-sensitive detector which is gated by the cor-responding harmonic of the modulation frequency. Fora symmetric absorption line, the amplitudes of the oddharmonics are antisymmetric and have a zero crossingat ∆ν = 0. Hence, these odd harmonics are suitable aserror signals for frequency stabilization. In many cases,third-harmonic detection is applied in order to suppressa residual slope [11.2032,2033]. If the modulation widthis small compared to the line width, the derived signalrepresents the corresponding derivative of the saturationsignal, e.g., the first derivative for the first harmonic andthe third derivative for the third harmonic.

If we approximate the Doppler-free saturation fea-ture by a Lorentzian line profile I(ν).

I(ν) = A

1+ [(ν−ν0) /ν1/2]2 (11.233)

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we can estimate the amplitudes of the harmonics versusthe detuning for various modulation widths [11.2034].In (11.233), A and ν1/2 are the height and the half-width(HWHM) of the saturated absorption feature, respec-tively. The modulated laser frequency ν(t) can be writtenas

ν(t) = ν0 +∆ν+ δνA cos(ωt) , (11.234)

where ∆ν is the detuning of the mean laser frequencyfrom the line center ν0 and δνA is the modulation am-plitude. If the modulation frequency ω/2π is smallcompared to the line width, the detected signal followsthe laser frequency ν(t) closely and we can replace ν in(11.233) by ν(t) of (11.234), leading to a time-dependentabsorption signal

I(t) = A

1+[∆ν+ δνA cos(ωt)] /ν1/22 .

If we relate ∆ν and δνA to the half-width (HWHM) ν1/2we get with the reduced detuning dD = ∆ν/ν1/2 and thereduced modulation amplitude dA = δνA/ν1/2

I(t) = A

1+ [dD +dA cos(ωt)]2 . (11.235)

The signal I(t) is periodic in time and can be expressedby a Fourier series

I(t) = A0

2+

∞∑m=1

Am cos(mωt) , (11.236)

where the Fourier coefficient Am represents the sig-nal amplitude of the m-th harmonic of the modulationfrequency ω/(2π). It can be calculated by the relation

Am = 2

π

π∫0

I(τ) cos(mτ)dτ .

For the coefficients A1 and A3 we get [11.2034]

A1 = 1

dA

[(sign dD) P− −dD P+

](11.237a)

and

A3 = 1

d3A

(sign dD)

[4(

1−3d2D

)+3d2

A

]P−

+[4(

3−d2D

)+3d2

A

]dD P+ −16dD

(11.237b)

with P± = 1ρ

√2(ρ±α) and ρ =

√α2 +4d2

D and α =1+d2

A −d2D.

>

'$

$

$

* % ;

$ * % ;>

'$-

'$

'$

'$

'$*

Fig. 11.251 Amplitudes of the first harmonic (A1)and the third harmonic (A3) versus the detuningdD = ∆ν/ν1/2. The modulation amplitude was cho-sen as dA = 0.2, 0.6, 1.0, . . . , 3.0 for A1, and dA =0.5, 1.0, 1.5, . . . , 6.0 for A3

A1 and A3 are shown versus the detuning dD forvarious modulation amplitudes dA in Fig. 11.251a andFig. 11.251b, respectively. Both curves have zero cross-ings at dD = 0. In the central part, their amplitudesdepend linearly on the detuning dD. Hence, both sig-nals A1 and A3 can be used as discriminator signals forfrequency control. Their slope depends on the modula-tion width dA. For small amplitudes dA 1, the slope ofthe first-harmonic signal depends linearly on dA and thatof the third harmonic increases with the third power ofdA. The largest slope and consequently the highest sen-sitivity can be obtained with the first harmonic A1(dD).In many cases, however, the saturation signal I(dD)is superimposed on a frequency-dependent (sloping)background. Consequently, a signal is added to the first-harmonic signal A1(dD) and the zero crossing is shifted,leading to a frequency offset of the stabilized laser fre-quency. This shift can be strongly reduced if the third ora higher odd harmonic is applied for the stabilization.

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Lasers and Coherent Light Sources 11.14 Frequency Stabilization of Lasers 851

A typical example of a laser utilizing the third-harmonicdetection technique is the iodine-stabilized HeNe laser(see Sect. 11.14.3).

Servo Amplifier and FilterThis subsection describes how the error signal is usedto control the laser frequency. If the servo control is op-erating, the initial frequency νi of the free-running laseris corrected by the servo loop to νS, which is close tothe reference frequency ν0 (Fig. 11.252). The differenceδν = νS −ν0 represents the residual frequency devia-tion from the line center ν0 of the stabilized laser. Thedeviation δν is converted to an amplitude U = Cδν bythe error signal, where the constant C is proportional tothe sensitivity of the frequency discriminator. The sig-nal Cδν is amplified in the following servo amplifierby a factor g( f ) and is then transferred to a frequencytransducer, which corrects the laser frequency νi by−CDg( f )δν. Here D is the sensitivity of the transducer,which transforms the control voltage to a correspondingfrequency shift of the laser. The term g( f ) characterizesthe frequency-dependent gain of the electronic servo am-plifier. The servo loop provides a negative feedback tothe laser frequency and the corrected value νS can bewritten as

νS = νi −CDg( f )δν (11.238)

which can be transformed to

δν

∆ν= 1

1+CDg( f ), (11.239)

where ∆ν = νi −ν0 is the initial frequency offset of thefree-running laser.

We see that the servo-control circuit reduces thefrequency deviation ∆ν of the free-running laser bya factor of 1/[1+CDg( f )], where CDg( f ) representsthe gain of the open servo loop. To achieve a negligi-bly small value of the residual frequency deviation δν,

4' 4'

4'

4

#

Fig. 11.252 Equivalent circuit of a laser frequency stabiliza-tion

the servo gain should be as high as possible. In par-ticular, the deviation of the mean frequency averagedover long times should be close to zero. This requiresthat g( f → 0) → ∞. In any servo-control system, phaseshifts and time delays of the servo loop put limitationson the maximum gain and the bandwidth of the servo-control. A simple technique to provide high servo gainat low frequencies and to reduce the influence of phaseshifts and time delays at high frequencies is to reducethe servo gain with increasing Fourier frequency f . Sucha characteristic can be obtained by an integrating behav-ior of the servo gain, i. e., if CDg( f ) decreases with 1/ ffor increasing frequencies. In this case, the total transferfunction of the control system can be characterized bya unity-gain frequency fC , the frequency at which thegain of the open servo loop CDg( fC) = 1. At low fluc-tuation frequencies f fC , the frequency deviations∆ν( f ) will be reduced by a factor of f/ fC and an in-crease of the servo gain corresponds to a proportionalincrease of the unity-gain frequency fC . Consequently,in an integrating servo loop a high servo gain requiresa large servo bandwidth fC . In cases where the servobandwidth fC is limited and where large frequencyfluctuations at low frequencies have to be suppressed,an additional integration can be introduced, leadingto a double integrating behavior and consequently toa higher servo gain at low frequencies. The stability ofthe servo loop requires that this double integration is

![ "#

4

%

*

@7 :@7 @7

'*/)"

'/)"

. #9>@

9 9

#I#

9[ "#

Fig. 11.253 Typical servo gain versus Fourier frequency ofa PDH frequency stabilization to a cavity resonance (seetext)

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852 Part C Coherent and Incoherent Light Sources

effective only at frequencies well below the unity-gainfrequency fC . In practice, the maximum crossover fre-quency fdi between single and double integration shouldbe smaller than fC/4. Figure 11.253 shows a typical de-pendence of servo gain on the fluctuation frequency ofa frequency stabilization to a narrow cavity resonanceusing the PHD technique.

11.14.3 Examplesof Frequency-Stabilized Lasers

This section describes a few examples of stabilizedlasers. The radiations of some of them are recom-mended as reference frequencies for the realization ofthe length unit meter and as references for scientific ap-plications [11.2009]. Frequency-stabilized lasers havebeen developed covering a wide range of wavelengthsfrom the near UV to the far IR. Of these examples, HeNelasers are of great practical importance since they are of-ten employed in many applications, e.g., as wavelengthstandards in interferometers for dimensional metrology.Most of these lasers utilize the frequency of their gainprofile itself as the reference for the stabilization. In thesecases, the stabilized frequency depends on the dischargeparameters, the gas pressure, and gas mixture inside thelaser tube and the relative reproducibility is limited tosome 10−7. Hence, if the absolute value of the laser fre-quency is important, it should be compared from timeto time with a superior standard, e.g., a HeNe Laser ofwhich the frequency is stabilized to an absorption lineof molecular iodine.

With increasing requirements on reproducibility andaccuracy, the laser frequency needs to be stabilized tosuitable references such as the narrow absorption linesof atoms, molecules, or ions and line broadening causedby the movement of the absorbers needs to be reduced byDoppler-free techniques such as saturated absorption or

..

,>

.

.

,

E+

,

>

Fig. 11.254a,b Typical setups of extended-cavity diode lasers(ECDLs). (a) Littrow configuration, (b) Littman configuration (seetext)

two-photon excitation. Ultimately any frequency shiftscaused by a residual Doppler effect can be strongly re-duced if the laser radiation interacts with cold atomicabsorbers only.

This section is organized as follows. In the first part,we explain as a typical example the stabilization of a tun-able diode laser to the resonance of a stable opticalcavity. We then continue with HeNe lasers stabilizedto their gain profile. The next section represents laser ra-diations that are stabilized to suitable absorption linesof atoms, molecules, or ions. In view of its practicalimportance, we focus on HeNe lasers, which can be sta-bilized to the absorption lines of molecular iodine. Theabsorbing particles of these lasers are contained in anabsorption cell. In the next step, laser stabilization sys-tems based on atomic or molecular beams are discussed.The last part of this section presents a brief descriptionof lasers stabilized to cold ions and atoms.

Frequency Stabilization of Diode LasersTunable lasers are important tools in the field of preci-sion laser spectroscopy. Whereas in the past the visiblespectrum was covered mostly by dye lasers, diode lasersare increasingly taking over due to their small size, highefficiency, high power, and reliability. They are avail-able in a wide range of the visible and infrared spectrum.However, if high spectral resolution is required, the linewidth of most solitary laser diodes, which ranges from10 MHz to 300 MHz, is orders of magnitude too large.Several methods have been developed to improve thespectral purity of these lasers. Most of them apply opticalfeedback [11.2035, 2036]. This section discusses a typ-ical example, where the length of the laser resonatoris increased by an external reflector [11.2037, 2038].Compared to solitary laser diodes, the line width of suchextended-cavity diode lasers (ECDL) can be reduced tosome 100 kHz. Figure 11.254 shows two typical setupsof ECDLs in the Littrow and Littman configurations. Inboth configurations, the beam leaving the antireflection-coated facet of the laser diode (LD) is collimated anddirected onto a reflection grating. In the Littrow con-figuration (Fig. 11.254a), the laser grating acts as theend mirror of the extended laser resonator where thebeam is back-reflected into the laser diode in the firstdiffraction order. The zeroth-order beam is coupled out.Coarse frequency tuning can be performed by rotatingthe grating whereas a piezoelectric transducer (PZT)and/or an electrooptic modulator (EOM) can be used tofine-tune and control the laser frequency by changingthe optical length of the cavity. Frequency tuning canalso be achieved by changing the injection current of the

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laser diode; this method, however, will also change thelaser power. The Littrow configuration can be designedto be very compact. Unfortunately, the direction of itsoutput beam changes with any rotation of the grating.This shortcoming is avoided in the Littman configura-tion [11.2039] where the ECDL contains an additionalmirror (Fig. 11.254b). In this configuration the beamdeflected in the first diffraction order is directed ontothis extra mirror and back-reflected from there into thelaser diode. In the Littman configuration, the laser beampasses the grating twice and hence the wavelength selec-tivity is increased compared to the Littrow configuration.The output beam leaves the grating in the zeroth order. Inthis configuration, coarse frequency setting is achievedby rotating the end mirror, with the consequence thatthe direction of the output beam does not change. Fine-tuning of the frequency can be performed in the sameway as with the Littrow configuration by means of a PZTor an EOM.

The frequency noise of ECDLs occurs at Fourier fre-quencies low enough that a further line width reductionbelow 100 kHz can be achieved by active frequency sta-bilization to the resonance of an optical cavity [11.2040],as described in Sect. 11.14.2. Precise frequency tuningcan be performed, e.g., by changing the length of theoptical reference cavity. This method, however, stronglyperturbs the stability of the cavity. To achieve the high-est possible resolution, it is better to introduce a variablefrequency difference between the laser output beam andthe cavity resonance. Since the frequency-shifted beamis locked to the cavity resonance, which is not affected,the stabilization circuit will shift the laser frequency,with the consequence that it can be tuned very preciselyrelative to the stable cavity resonance. Such a shift canbe achieved, e.g., by an acoustooptic modulator or anelectrooptic modulator. Using the PDH technique anda carefully designed and isolated reference cavity ofultrahigh finesse, a line width as narrow as 1 Hz wasrecently observed, demonstrating the great potential ofdiode lasers for applications in precision spectroscopyand optical frequency standards [11.2021].

Frequency Stabilization of HeNe LasersHeNe lasers can operate at several different wave-lengths within the spectral range between the green(λ= 543 nm) and the infrared (λ= 3.39 µm). Regard-ing the various wavelengths, the emission at λ= 633 nmis probably most important. HeNe lasers are advan-tageous since their operation is simple, the intrinsicfrequency noise is small, and the output power in therange between 100 µW and some 3 mW is sufficient

for many applications in metrology. Provided the neongas inside the laser tube consists of a single isotope,the width of the Doppler-broadened gain profile is inthe range of approximately 1.5 GHz and single- or two-frequency operation can easily be achieved by choosinga sufficiently short length of the laser resonator. Ow-ing to these favored properties and their simplicity,frequency-stabilized HeNe lasers were developed veryearly [11.2041]. The first stabilization systems – whichare still used in many laboratories – apply the gain profileitself as the frequency reference.

Lamb-dip stabilization. The Lamb-dip-stabilizedlaser was one of the first frequency-stabilizedlasers [11.2042]. It makes use of the fact that the out-put power passes through a local minimum – the Lambdip – when the laser frequency is tuned across the centerof the gain curve (Fig. 11.255a). Similar to the Doppler-free minimum of saturated absorption, the Lamb dip iscaused by an increased saturation of the gain at the linecenter when both the forward- and backward-runningwaves inside the resonator interact with the same veloc-ity group (νz = 0) of atoms. Hence the number of atomscontributing to the laser action is decreased, leading toa reduced laser power at line center. To generate the errorsignal (Fig. 11.255b), the laser frequency is modulatedand the first harmonic in the output power is detected.

Two-mode stabilization. A simple stabilization sys-tem can easily be set up, if the HeNe laser is operatingat two adjacent longitudinal modes (Fig. 11.256a) and if

9[ "#

,

5

Fig. 11.255a,bTuning curve(a) and errorsignal (b) ofa Lamb-dip-stabilized HeNelaser

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9[ "#

5

9[ "#

,

0 / 0

Fig. 11.256a,b Tuning curve (a) and error signal (b) of a two-modestabilized HeNe laser

these modes are polarized perpendicular to each other.These conditions are provided, e.g., in commercial HeNelaser tubes of 30 cm length with internal mirrors. Per-pendicular polarization is observed since it minimizesthe gain competition of two modes. Furthermore, thetwo planes of polarization are fixed within the tube.These properties allow us to separate the two modesby a polarization divider, e.g., a Wollaston prism, andto monitor the power in each mode by a separate pho-todetector. The difference between the photocurrents ofthe two detectors I1 − I2 depends on the frequencies ofthe two modes (Fig. 11.256b). It crosses zero when theyare distributed symmetrically with respect to the centerof the gain curve and the powers in the two modes areidentical. Hence, the difference in the photocurrents isused as the error signal for frequency stabilization andno frequency modulation is necessary [11.2043]. Sev-eral means can be applied to transfer the control signalto the length of laser tube, such as heating the tube viathe discharge current or a separate external heater. Withthis simple laser stabilization technique, a relative repro-ducibility of better than 10−7 has been demonstrated.During a time interval of five days relative frequencyvariations as low as 10−9 have been demonstrated underfavored laboratory conditions [11.2044].

Zeeman stabilization. Another method of frequencystabilization widely applied in commercial HeNe lasersand laser interferometers utilizes the Zeeman effectin a short (≈ 10 cm) single-mode laser. If we applya longitudinal magnetic field, the Zeeman effect causesa splitting of the laser mode into two submodes ν+ andν− of opposite circular polarization and slightly differ-ent frequencies. Depending on the size of the magneticfield, the frequency difference ν+ −ν− is in the rangebetween 300 kHz and 2 MHz. The two submodes can beseparated by a quarter-wave plate followed by a polar-

izing beam splitter. Similar to two-mode stabilization,the difference of the output powers at ν+ and ν− can beutilized as the error signal [11.2045].

Alternatively, the error signal can also be generatedby utilizing the frequency difference δν = ν+ −ν−. Ithas been shown that δν changes with the mean laser fre-quency and that it has a minimum at the line center. Thisdependence is caused by nonlinear mode-pulling effectsin the gain medium. The error signal to find this min-imum is generated by modulating the laser frequencyand monitoring the beat frequency between the two Zee-man modes by means of a reversible counter [11.2046].To derive the error signal, the direction of counting isreversed at every half period of the modulation. The ac-cumulated counter content, which corresponds to theintegrated error signal, is then fed into a digital-to-analog converter after each period. Its output is usedto control the frequency of the laser so that the meanlaser frequency coincides with the minimum of the beatfrequency. With such lasers a relative frequency repro-ducibility of 10−8 can be achieved over a period of fivemonths under laboratory conditions [11.2046].

Absolute Frequency Stabilization UtilizingSeparate Absorption Cells

As indicated above, the frequency of HeNe lasers stabi-lized to their gain profile may change with time. Suchfrequency shifts can be reduced by several orders ofmagnitude if the frequency reference is provided bya suitable narrow absorption line of which the transi-tion frequency is largely independent on the parametersof operation. There are now several different laser sys-tems available in a wide range of the optical spectrumwhich are stabilized to such frequency references. Someof these have been recommended by the Comité Inter-national des Poids et Mesures (CIPM) [11.2009]. Theirradiations are listed together with the values of their fre-quencies, wavelengths, and their relative uncertainties inTable 11.56. Six of the recommended reference frequen-cies are stabilized to hyperfine-structure components ofmolecular iodine, which has a rich spectrum of narrowabsorption lines in the visible range. They belong to thetransition between the electronic B level and the groundstate (X level). Since the iodine molecule is heavy, its ve-locity – and hence the Doppler broadening – is rather lowat room temperature. Furthermore, the vapor pressure –causing a pressure-induced shift and broadening – canbe controlled conveniently via the temperature of a cool-ing finger attached to the iodine absorption cell. Four ofthe iodine references are used to stabilize HeNe lasersat λ≈ 543 nm, 612 nm, 633 nm, and 640 nm. From this

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Lasers and Coherent Light Sources 11.14 Frequency Stabilization of Lasers 855

Table 11.56 Recommended reference wavelengths/frequencies generated by stabilized lasers [11.2009]

Atom / Transition Wavelength (fm) Frequency Rel. std.Molecule uncertainty (1 σ)

1 115In+ 5s2 1S0–5s5p 3P0 236 540 853.54976 1 267 402 452 899.92 kHz 3.6 × 10−13

2 1H 1S–2S (Two-photon transition) 243 134 624.62603 1 233 030 706 593.61 kHz 2.0 × 10−13

3 199Hg+ 5d106s 2S1/2 (F = 0)–5d96s2 2D5/2 (F = 2) 281 568 867.591969 1 064 721 609 899.143 Hz 1.9 × 10−14

∆mF = 0

4 171Yb+ 6s 2S1/2 (F = 0)–5d 2D3/2 (F = 2) 435 517 610.73969 688 358 979 309 312 Hz 2.9 × 10−14

5 171Yb+ 2S1/2 (F = 0, mF = 0)–2F7/2 (F = 3, mF = 0) 466 878 090.061 642 121 496 772.6 kHz 4.0 × 10−12

6 127I2 R(56) 32-0, component, a10 532 245 036.104 563 260 223 514 kHz 8.9 × 10−12

7 127I2 R(127) 11-5 component a16 (or f) 632 991 212.58 473 612 353 604 kHz 2.1 × 10−11

8 40Ca 1S0–3P1; ∆m J = 0 657 459 439.29167 455 986 240 494 150 Hz 1.1 × 10−13

9 88Sr+ 5 2S1/2–42D5/2 674 025 590.8631 444 779 044 095.5 kHz 7.9 × 10−13

10 85Rb 5S1/2(Fg = 3)–5D5/2(Fe = 5) two-photon 778 105 421.23 385 285 142 375 kHz 1.3 × 10−11

11 13C2H2 P(16)(ν1 +ν3) λ= 1 542 383 712 194 369 569.4 MHz 5.2 × 10−10

12 CH4 P(7) ν3, component of the F(2)2 3 392 231 397.327 88 376 181 600.18 kHz 3 × 10−12

central hyperfine structure

13 12C16O2 R(10) (0001)–(1000) line of the CO2-laser 10 318 436 884.460 29 054 057 446 579 Hz 1.4−13

group, the one operating at 633 nm is of particular im-portance since it is used in many standard laboratoriesas a reference for interferometric length measurements,laser wavelength calibrations, and applications in pre-cision laser spectroscopy. It can also act as a referenceto calibrate HeNe lasers that are stabilized to their gainprofile by beat-frequency comparisons. Hence, the I2-stabilized HeNe laser at λ= 633 nm can be regarded asthe workhorse in many laboratories.

The iodine molecules are contained at low vaporpressure in a cell. Depending on the laser system, thelength of the cell may range from a few centimeters upto several meters. In either case, the content of impuri-ties has to be low in order to prevent any frequency shiftscaused by collisions with impurities. For example, in thecase of iodine-stabilized lasers, the quality of the iodinefilling limits the uncertainty ultimately achievable. Nev-ertheless, as a result of its rich spectrum, I2 is often usedas the frequency reference for different types of lasersoperating within the visible spectrum. Two types of I2-stabilized lasers, the HeNe laser (λ= 633 nm) and thefrequency-doubled Nd:YAG laser (λ= 532 nm) will bediscussed in the following.

Iodine-stabilized HeNe laser (λ = 633 nm). Fig-ure 11.257 schematically shows a typical setup of aniodine-stabilized HeNe laser. The laser head consistsof a HeNe discharge tube and an iodine absorptioncell mounted inside the laser resonator. The lengths ofthe discharge tube and the absorption cell are typically

20 cm and 10 cm, respectively, corresponding to a min-imum length of the resonator of approximately 35 cmand a free spectral range of ≈ 430 MHz. Despite of therather long resonator the laser emits only one single fre-quency, owing to the loss in the absorption cell. In orderto reduce the influence of acoustics and vibrations, theresonator should be made as rugged as possible. Thespacers should consist of a material of low thermal ex-pansion. The radii of curvature of the laser mirrors areusually in the range r = 0.6–4 m with a radius of r = 1 mmost commonly used. Both mirrors have a reflectivityof approximately 98%, allowing single-frequency oper-ation over most of the free spectral range. Output powersof up to 300 µW have been achieved, sufficient for mostapplications in laser spectroscopy and interferometry.

6"9 @C

9[ "#

#

#

#9

4>

E+ E+B"

E+

B

Fig. 11.257 Experimental scheme of an iodine-stabilized HeNe laser

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856 Part C Coherent and Incoherent Light Sources

The two mirrors are mounted on PZT actuators, allow-ing one to change the resonator length and thereby thelaser frequency by applying a voltage to the PZTs.

Figure 11.258a shows four absorption features in thetuning curve of the laser. They belong to the hyperfine-structure (HFS) components d, e, f, and g of the transitionR(127), v′ = 5, v′′ = 11 of 127I2. The separation betweenthe components d, e, f, g is approximately 13 MHz andthe full width at half-maximum of each line is in therange of 5 MHz, corresponding to a quality factor ofQ = ν/δν ≈ 108. The height is approximately 0.1% ofthe laser power. To generate the error signal for the stabi-lization, the length and thereby the frequency of the laseris modulated harmonically by applying a sine-wave volt-age to the left PZT (Fig. 11.257). For the stabilization,third-harmonic detection is used. The third-harmonicsignal in the output power is filtered out, demodulatedby a phase-sensitive detector (PSD) and passed througha low-pass filter. It then serves as error signal for thestabilization. A typical third-harmonic spectrum of mo-lecular iodine containing seven HFS components withinthe tuning range of the laser is shown in Fig. 11.258b.Each absorption feature is suitable as a reference for thestabilization.

With the servo loop operating, the length of the laseris stabilized such that the third-harmonic signal of thecorresponding HFS component vanishes at its centralzero crossing.

The stability and the reproducibility of theI2-stabilized HeNe laser can be investigated by beat-frequency measurements of two independent lasersystems. Figure 11.245 shows the measured Allan stan-dard deviation σ(τ) of an iodine-stabilized HeNe laserversus the integrating time τ . In the range betweenτ = 10 ms and τ = 100 s, σ(τ) decreases approximatelywith the square root of τ , corresponding to whitefrequency noise. The minimum instability of about2 × 10−13 is observed at an integrating time τ ≈ 1000 s.

9[ "#

5 >"

9[ "#

Z 6 9 @7

Fig. 11.258a,b Saturation dips (a) and error signals (b) of an iodine-stabilized HeNe laser (λ= 633 nm)

The reproducibility of the iodine-stabilized HeNelaser was investigated by measuring the dependence ofthe laser frequency on the various operation parame-ters and by frequency inter-comparisons between lasersof different institutes. The laser frequency depends onthe modulation width, the iodine vapor pressure, andslightly on the laser power. The coefficients are approx-imately −6 kHz/Pa and −10 kHz/MHz for the pressureand modulation dependence, respectively. Internationalfrequency comparisons between iodine-stabilized lasershave shown that the stabilized frequencies of the ma-jority of iodine-stabilized HeNe lasers coincide withina range of approximately 12 kHz, corresponding to2.5 × 10−11ν. For specified operation parameters (tem-perature of the cell: (25±5) C, temperature of thecold point: (15±0.2) C, width of frequency mod-ulation between the peaks: (6±0.3) MHz, one-wayintracavity beam power: (10±5) mW) a relative stan-dard uncertainty of the recommended frequency valueof 2.5 × 10−11 can be achieved [11.2009].

Iodine-stabilized frequency-doubled Nd:YAG lasers(λ= 532 nm). Frequency-doubled YAG lasers pumpedby diode lasers are of particular interest in applicationsas optical frequency standards for the following rea-sons. Small lasers providing us with high output powersof 100 mW at 532 nm are commercially available. Theintrinsic frequency noise of YAG lasers can be very low.Part of the emission range coincides with strong ab-sorption lines of molecular iodine (Fig. 11.259a) thatare suitable as reference frequencies for the stabiliza-tion. Figure 11.259b shows the Doppler-free absorptionspectrum observed within the continuous tuning rangeof 5 GHz of a commercial laser. The observed absorptionlines represent two sets of hyperfine components belong-ing to the 32-0, R(57) and the 32-0, P(54) ro-vibrationallines labeled #1104 and #1105, respectively [11.2047].Any of these transitions is suitable as a frequencyreference. Since the frequency difference to the recom-mended a10 component of the 32-0, R(56) transition isknown to about 2 kHz an accurate optical frequency canbe attributed to each HFS component.

Various methods have been used in different lab-oratories to generate the error signal. The methods ofmodulation transfer spectroscopy [11.2048, 2049] andphase-modulation spectroscopy [11.2050,2051] are verypowerful to achieve discriminant signal with a highsignal-to-noise ratio.

International frequency comparisons show that thefrequency reproducibility of these lasers is in the rangeof 5 kHz [11.2052,2053], mostly limited by potential im-

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Lasers and Coherent Light Sources 11.14 Frequency Stabilization of Lasers 857

@7

%/&

$-

$-

$-

' ' '

-&=]*

-*=]-

+<

%/&

$

$*

$%

$;

'

]* ]-

' '

Fig. 11.259a,b Absorption lines of molecular iodine withinthe emission profile of a commercial frequency-doubledNd:YAG laser. (a) Doppler-limited, (b) with Doppler-freeexcitation, the hyperfine structure is resolved

purities in the absorption cell. Regarding its high power,its compactness, and its high frequency reproducibility,the iodine-stabilized frequency-doubled Nd:YAG laserrepresents an optical frequency standard with importantapplications in precision length metrology, interferom-etry, and spectroscopy.

Laser stabilized to a two-photon transition inrubidium. The radiation of a laser stabilized tothe two-photon transition 5S1/2–5D5/2 in rubidium(Fig. 11.260) [11.2054] represents a reference in thenear infrared at a wavelength of 778 nm. In this spec-tral range, easy-to-handle laser diodes of low frequencynoise are available, allowing the development of a trans-portable optical frequency standard of high precision.Furthermore, the laser may provide a precise fre-quency reference for optical communication systemssince twice its wavelength coincides with the transmis-

&&;

&&;

-*;

*

&(- &;

> >

->)2-)

%)2)

-)-)

-4)

Fig. 11.260 Energy levels of Rb relevant for a two-photonstabilized diode laser

sion band at 1.55 µm. Its optical frequency has beendetermined [11.2055].

In a simple setup [11.2054], the collimated beam ofan extended-cavity diode laser passes through an absorp-tion cell filled with rubidium vapor. The cell is sealedwith Brewster windows at its ends and filled with nat-ural rubidium (73% 85Rb and 27% 87Rb). The beamis retro-reflected by a mirror or a cat’s eye in orderto allow Doppler-free two-photon excitation. Opticalfeedback into the diode laser is avoided by Faraday iso-lators. When the laser frequency is scanned through thetwo-photon resonance, the excitation of the upper level(52D5/2) is observed via the blue fluorescence (420 nm)of the 6P–5S transition in the cascade of the spontaneousdecay 5D → 6P → 5S. With this setup, a frequency sta-bility of σ(τ) = 3 × 10−13τ−1/2 has been achieved forintegrating times τ up to 2000 s.

It is known that the stabilized frequencies of stan-dards based on two-photon transitions suffer from lightshifts. Its magnitude depends linearly on the intensity.Therefore, it is important to control the laser power andto prepare a well-defined laser beam geometry for theexcitation. This requirement can be fulfilled if the ab-sorption cell is mounted inside a nondegenerate opticalresonator. The use of a resonator also allows a powerbuildup of the radiation and therefore an increase ofthe two-photon excitation. Furthermore, it provides ex-act retro-reflection of the laser beam, which is necessaryto suppress residual first-order Doppler shifts. In theexperiments [11.2054], the laser frequency was pre-stabilized in a first step to a resonance of this cavity.It could then be tuned through the two-photon res-onance by changing the length of the resonator viaa PZT actuator to which one of the mirrors is mounted.Compared to the results of the simple setup describedbefore, the observed frequency stability was approxi-mately the same. However the light shift could be bettercontrolled and consequently the frequency of the tran-

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858 Part C Coherent and Incoherent Light Sources

sition can be extrapolated more precisely to zero laserpower.

Laser-cooled ensembles of Rb atoms have also beenused for frequency stabilization to reduce the influenceof the second-order Doppler effect [11.2056]. Furtherevaluations will show if a standard based on cold Rbatoms leads to a significant reduction of the relative fre-quency uncertainty. Nevertheless, in its present state, theRb-stabilized laser represents an interesting precisionfrequency reference.

Stabilized Lasers Basedon Atomic or Molecular Beams

Atomic and molecular absorbers are often prepared inbeams rather than in absorption cells for various rea-sons. First, some gases such as hydrogen are not stableas atomic species and need to be prepared just shortlybefore excitation. Vapors of metals would soon coat thecell windows and strongly decrease their transparency.Second, the use of a collimated beam allows us to uti-lize an excitation geometry in which the laser beamcrosses the atomic beam at a right angle. In the caseof single-photon transitions, this may reduce the first-order Doppler effect by several orders of magnitude.Furthermore, for long-lived atomic states, the excitationand detection zones can be separated, leading to an in-creased signal-to-noise ratio since scattered light fromthe excitation beam can be better suppressed.

The mean velocity of molecules or atoms at roomtemperature ranges from approximately 100 m/s to morethan 2000 m/s. At this velocity, the interaction time ofthe particles crossing the laser beam at a right angle, istypically less than 10−5 s for beam diameters of a fewmillimeters. Hence, the corresponding line broadening ismore than 0.1 MHz. This broadening can be reduced byapplying Ramsey’s method of separated field excitationfirst introduced in the microwave range [11.2057]. Forsingle-photon transitions in the optical regime, wherethe wavelength of the radiation is typically much smallerthan the diameter of the atomic beam, additional stepsneed to be taken to use separated field excitation. Thiscan be achieved either by blocking well-defined tra-jectories of the atomic beam [11.2058] or by usingthree [11.2059] or more [11.2060] spatially separatedexcitation zones.

An alternative method is a longitudinal excitationwhere the atoms fly parallel or anti-parallel to the laserbeam. This method is suitable in the case of two-photontransitions. For example, one of the recommended radia-tions (Table 11.56) corresponds to the 1S–2S two-photontransition of atomic hydrogen. In the experiment, a beam

of cold atomic hydrogen is formed by guiding hydrogenmolecules through a gas discharge and reflecting theatomic particles at a plate which is cooled by liquid he-lium. The reflector directs the hydrogen atoms on theoptical axis of an optical resonator which is tuned closeto the 1S–2S two-photon transition. The longitudinalexcitation inside a resonator allows the strong powerbuildup necessary for a two-photon excitation, precisemode-matching of the forward- and backward-runningwaves inside the resonator, and an increased interactiontime of the atoms with the laser beam, leading to in-creased spectral resolution. The relative uncertainty ofthe transition frequency recommended by the CIPM isas low as 2.0 × 10−13 [11.2009].

In the case of single photon transitions, the transversexcitation by separated laser fields (Fig. 11.262) is fre-quently applied to achieve high spectral resolutionand a good signal-to-noise ratio. It was shown byBordé that this method leads to an atom interferom-eter (Ramsey–Bordé interferometer) where the laserbeams act as coherent beam splitters for the atomicbeam [11.2061–2063]. The prominent advantage of opti-cal Ramsey excitation with separated fields results fromthe fact that the transit-time broadening and the resolu-tion can be adjusted independently. The former can beincreased by choosing short interaction times in eachzone, thereby allowing a large fraction of the atomsto contribute to the signal. The resolution, however, ismainly determined by the time of flight between theinteraction zones. In the following, we give a brief de-scription of the Ramsey–Bordé interferometer. Utilizingthe internal energy structure of the atoms, absorption andinduced emission, processes can be used to split or de-flect atomic beams. If an atom absorbs a photon froma traveling wave, it also absorbs the photon momentumk (Fig. 11.261a). Hence, the excited atom suffers a pho-ton recoil. If the atomic wave was in the excited state

?

?

?

?

? 2 @

? 2

? 2

@

? 2 @

? 2

@? 2 @

Fig. 11.261 Laser beams as beam splitters for an atomicwave

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> >

Fig. 11.262 Ramsey–Bordé matter–wave interferometer(the figure shows only those partial beams that lead to aninterference structure)

before interaction with the laser radiation, the inverseprocess (induced emission) will occur (Fig. 11.261b).The probability ρ of observing such excitation or emis-sion process depends on the intensity, the frequency ofthe light field, and the duration of the interaction. In gen-eral, the atom will be left in a coherent superposition ofthe ground and the excited state (Fig. 11.261). Since theatoms have different momenta in both states, it is moreappropriate to use the picture of atomic wave packets.The interaction can act, e.g., as a 50% beam splitter(ρ = 0.5, π/2 pulse) or as a mirror (ρ = 1, π pulse). Inany case, the phase of the light beam is transferred to thedeflected partial atomic wave. If the light field is slightlydetuned from the atomic resonance, the correspondingenergy difference δω is also transferred to the deflectedpartial wave as kinetic energy and consequently thede Broglie wavelength is changed.

The combination of such beam splitters may leadto atom interferometers that are similar to an opticalMach–Zehnder interferometer (Fig. 11.262). During thefirst and each subsequent interaction, the matter waveis coherently split into partial waves with the internalstates |g〉 and |e〉, which are represented by the solid andthe dashed lines in Figs. 11.261 and 11.262. If an ab-sorption or emission is induced, the momentum of thecorresponding wave packet is changed by the photonmomentum k. At the final interaction, the loop of thetrajectories closes and the laser field superimposes thepartial waves, resulting in an atom interference. Sincethe atomic wave packets leave the two outputs of each in-terferometer in different internal states, the interferencestructure can be observed by monitoring the populationeither in the excited or the ground state.

The inter-combination transitions 1S0–3P1 of thealkaline-earth atoms are well known to represent ex-cellent references for optical frequency standards (see,for example, [11.2065] and references therein). Theyexhibit narrow natural line widths of about 0.04 kHz(Mg), 0.3 kHz (Ca), and 6 kHz (Sr). In all three cases

*.

.

9"

4

$*A@7%-&

@7*

Fig. 11.263 Partial energy-level diagram of 40Ca

the frequencies of the ∆mJ = 0 transitions are almostinsensitive to electric and magnetic fields.

The inter-combination lines have been investigatedin effusive beams of magnesium [11.2066], stron-tium [11.2067, 2068], and barium atoms [11.2069].Most work for optical frequency standard applica-tions has probably been performed with Ca atomicbeams [11.2070,2071]. This transition was investigatedby various groups [11.2064, 2072–2074]. The radiationof a Ca-stabilized laser is also recommended as a refer-ence frequency by the CIPM [11.2009]. As a typicalexample let us discuss the layout of a transportableCa-atomic-beam standard (Fig. 11.264) [11.2064].

A few milliwatts of the available output power of anECDL – pre-stabilized to a reference frequency of an op-tical cavity – is sent through a polarization-preservingsingle-mode fiber to a beam splitter/mirror configura-tion and is split into two beams 1 and 2 of equalpower. From each one of the beams (1 or 2) crossing

."

6

/

6

>

>

%-&$*%

Fig. 11.264 Scheme of a transportable diode-laser fre-quency stabilized to the 1S0–3P1 inter-combinationtransition by means of separated field excitation of a 40Cabeam [11.2064]

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the atomic beam perpendicularly an excitation geome-try with two pairs of counter-propagating laser beamsis obtained by the help of two cat’s-eye retro-reflectors.The co-propagating beams are separated by the distanceD = 10 mm and the distance between the two innermostcounter-propagating beams is d = 13 mm. The opticallayout with the fused silica block is designed in sucha way that the part 2 leads to the same excitation geom-etry as part 1, but its direction with respect to the atomicbeam is reversed (Fig. 11.264). During the experiment,the laser beams are propagating either in direction 1 or 2and the beams propagating in the opposite direction areblocked.

The excited atoms are detected by measuring the flu-orescence intensity from the decay of the excited atomsby a photomultiplier. The detected fluorescence intensityof the ∆mj = 0 transition versus laser frequency detun-ing ∆ω/2π = (ω−ωg)/2π, representing the frequencydifference between the laser frequency ω/2π and thefrequency ω0/2π of the Ca inter-combination line, canbe described by the expression [11.2060].

I(∆0)

=∞∫

0

A(P, ν,∆0) f (ν)

×

cos

[2T

(∆0 + δrec + ω0ν

2

2c2

)+ΦL

]

+ cos

[2T

(∆0 − δrec + ω0ν

2

2c2

)+ΦL

]dν

+ B(P,∆0) . (11.240)

In (11.240), A(P, ν,∆0) describes the contribution ofa particular atom with velocity ν to the signal, andB(P, ν,∆0) describes the amplitude of the backgroundof the Doppler-broadened line including the saturationdip, both depending on the laser power P and weakly onthe detuning δν=∆0/2π of the laser frequency. The fac-tor f (ν) represents the velocity distribution, where wehave neglected the influence of the velocity componentsperpendicular to the atomic beam.

ΦL =Φ2 −Φ1 +Φ4 −Φ3 (11.241)

is a residual phase between the four exciting laser beamshaving the phases Φi in the interaction zone, and T =D/ν is the flight time of the atoms between two co-propagating beams. The phase ΦL – which shifts thefrequency of the interference structure – can be detectedby reversing the directions of the laser beams (from 1 to 2

or vice versa, Fig. 11.264) and compensated by rotatingthe phase plate [11.2064].

The phase of each cosine function in (11.240) con-tains three contributions. BesidesΦL there are the terms∆0 resulting from the detuning, δrec = k2/(2mCac2)from the photon recoil where k is the wavevector ofthe laser field and mCa the mass of a Ca atom, andω0ν

2/(2c2) from the second-order Doppler effect.The signal for each velocity group ν consists of two

cosine functions with a period 1/(2T ) that is determinedby the distance D and the atomic velocity. The twocomponents are separated by the recoil splitting 2δrec =2π23.1 kHz. For an optimum superposition of the twocosines, the period should be an integer fraction of therecoil splitting. The FWHM line width of the signal isapproximately given by 1/(4T ).

From the measured fluorescence signal versus de-tuning δν = ν−νo (Fig. 11.265) one clearly recognizesin the central part of the saturation dip the two cen-tral minima of the two cosine terms separated by therecoil splitting. With increasing detuning the cosinestructure is washed out, since all velocity groups ν ofthe atomic beam contribute with a slightly different pe-riod. The inset of Fig. 11.265 shows the total detectedsignal where the Doppler broadening is determined bythe degree of collimation and the velocity distributionof the atomic beam, leading to a FWHM of the struc-ture of 7.5 MHz. With an evaluated relative uncertaintyof 1.3 × 10−12 combined with a relative stability ofσ(τ) = 9 × 10−13 at τ = 1 s, the transportable Ca fre-quency standard is one order of magnitude superior to

';':@7

&$-@7

! ""

;'* *

$:@7

Fig. 11.265 Atom interference structures observed in a ther-mal beam of Ca atoms [11.2064]

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Lasers and Coherent Light Sources 11.14 Frequency Stabilization of Lasers 861

the widely used transportable iodine-stabilized HeNelaser at λ= 633 nm.

Optical Frequency Standards Basedon Laser-Cooled Absorbers

In the case of frequency-stabilized lasers utilizingthermal absorbers, the uncertainty in the center ofthe reference line is ultimately limited by frequencyshifts caused by residual phase errors in the opti-cal excitation and by the second-order Doppler shiftδν/ν = −ν2/(2c2). Both frequency shifts scale with thevelocity of the absorbers and can be reduced by ordersof magnitude if the control signal is only generated byslow absorbers. In addition, the increased interactiontime of the atoms with the laser beam leads to reducedbroadening of the reference line. Hence, the most ac-curate optical frequency standards are based on coldabsorbers. Two typical examples are described in thefollowing subsections. The first describes a standard uti-lizing a single trapped Yb+ ion whereas the second partpresents a frequency standard based on an ensemble ofcold Ca atoms.

Optical frequency standards based on cold, storedions. The ideal reference for an optical frequency stan-dard would consist of an ensemble of identical butindependent atoms at rest in free space. This conditioncan be approximated in part by a single ion confinedin a small volume inside an axially symmetric configu-ration of electrodes shown schematically in Fig. 11.266(Paul trap) [11.2075]. Trapping is performed by apply-ing a suitable RF voltage to the electrodes. Such trapsallow us to store and cool single ions in the field-freeregion of the trap center.

Ions suitable for precision frequency standards pro-vide a closed transition for cooling and a reference(clock) transition. Cooling of the ion is performed byirradiating it with laser radiation which is tuned slightly

,"

"

,"

Fig. 11.266 Electrode configuration of an RF ion trap (Paultrap) [11.2075]

)

>T)U)>T-)U-)

>-)

>)

4)

!&)

%(

(-%;

*

*- *%&

Fig. 11.267 Simplified energy-level diagram of 171Yb

below resonance and the cooling occurs by repeated ab-sorption and emission [11.2077, 2078]. When the clocktransition is probed, the cooling radiation has to beturned off in order to suppress light shifts and strongline broadening. The RF trapping field, however, maystay unchanged, allowing ideally unlimited interroga-tion times. If several ions are confined in a trap, theirrepelling forces will lead to an ion cloud, extending intothe nonvanishing RF field and kinetic energy from theoscillating trapping field will be picked up. Such heat-ing can be avoided if only a single ion is trapped inthe center of the trap, where the trapping field vanishes.Since only one ion can contribute to the stabilization, itis important to detect the excitation of the reference tran-sition with high efficiency. An efficiency close to 100%

'

>

,8"#$;

$*

'% %

@7

Fig. 11.268 Absorption signal of a single trapped andcooled 171Yb+ ion [11.2076]

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862 Part C Coherent and Incoherent Light Sources

.)

9" >"

-

Fig. 11.269 Time sequence of atom cooling and trappingand probing of the reference transition

can be accomplished by the method of electron shelv-ing [11.2079], which utilizes the fact that the strongfluorescence on the cooling transition is interrupted ifthe ion is excited to the upper state of the referencetransition.

Several different ions are being investigated for ap-plications in frequency-stabilized lasers. Very promisingresults have been observed with standards based on theions of mercury [11.2080], strontium [11.2081, 2082],indium [11.2083], and ytterbium [11.2076, 2084]. Asa typical example, we consider an optical standardbased on a single 171Yb ion (Fig. 11.267). There arethree different clock transitions that can be accessedby the frequency-doubled radiation of diode lasers. The435 nm clock transition has recently been observedwith a line width as narrow as 30 Hz (Fig. 11.268).In the meantime, this linewidth could be further re-duced to 10 Hz [11.2085]. Utilizing this reference linea relative uncertainty of 9 × 10−15 has recently beenestimated [11.2086], demonstrating the great potential

'@7

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4

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Fig. 11.270 Interference structure of cold, free Ca atoms generatedby time-separated excitation

of optical frequency standards based on cold storedions.

Frequency standards based on cold neutral atoms.Neutral atoms can be cooled and trapped in a magne-tooptical trap (MOT) [11.2087] if they have a suitablecooling transition. In contrast to ions, a large number ofatoms in the range of 107 can be utilized to contribute tothe stabilization, leading to an increased signal-to-noiseratio. However, to avoid unwanted frequency shifts, alltrapping fields have to switched off when the referencetransition is probed. Hence, the interaction time of thefree atoms released from the trap is limited by theiracceleration in the gravitational field.

Candidates for cold-atom optical frequency stan-dards are silver [11.2088] and the alkali-earth atomsmagnesium [11.2089,2090], strontium [11.2091,2092],and calcium [11.2065, 2070, 2088]. As a typical exam-ple, we describe a frequency-stabilized laser based oncold neutral calcium atoms.

Besides of the narrow reference transition 1S0–3P1at λ= 657 nm (Fig. 11.263), 40Ca has a strong cool-ing transition 1S0–1P1 (λ= 423 nm) that can be used tocool and trap Ca atoms in a MOT [11.2093]. On thistransition, the Ca atoms can be cooled to approximately3 mK. Further cooling to several mikrokelvin can then beachieved in a second step by means of the narrow inter-combination transition 1S0–3P1 utilizing the method ofquench cooling [11.2094].

In order to achieve high spectral resolution combinedwith a good signal-to-noise ratio (SNR), the method ofseparated field excitation in the time domain can be ap-plied, similar to the spatially separated field excitation ofan atomic beam. Short pulses of 1 µs duration are usedto excite a significant part of the cold ensemble of atoms.The necessary high spectral resolution is then achievedby a sufficiently large time separation T between twoconsecutive pulses. If the lengths of the pulses are smallcompared to their separation, the width of the inter-ference fringes δν = 1/(4T ) is inversely proportionalto T .

Cooling and trapping of the atoms and prob-ing of the clock transition are performed sequentially(Fig. 11.269). After the atoms have been trapped andcooled for about 15 ms in the first step, the trappingfields (laser beams and magnetic quadrupole field) areswitched off, a small homogeneous magnetic quantiza-tion field (Helmholtz field) is turned on and the clocktransition is probed by two pulsed pairs of counter-propagating laser beams (Fig. 11.269). During this time,the cloud of cold, free Ca atoms expands according to

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Lasers and Coherent Light Sources 11.14 Frequency Stabilization of Lasers 863

the temperature of about 3 mK at a root-mean-square ve-locity of vrms ≈ 80 cm/s. In the third step, the excitationto the 3P1 state is detected by observing the fluorescenceof its spontaneous decay to the 1S0 ground state.

If the laser frequency is tuned across the atomicresonance, the fluorescence intensity contains a contri-bution that oscillates with the cosine of the laser detuning(Fig. 11.270). Similar to spatially separated field ex-citation, this oscillating behavior can be explained byan atom interference generated by the excitation withtime-separated fields [11.2093].

The error signal for the stabilization is generatedfrom the interference signal by modulating the laserfrequency and simultaneously measuring the fluores-cence intensity. In the most straightforward approach,the frequency is square-wave modulated between twodiscrete values with the mean frequency tuned close tothe center of the central fringe. The difference in thecorresponding fluorescence intensities is used as the er-ror signal. This method corresponds to a first-harmonicdetection (1f method) of a servo-control system usinganalog electronics and harmonic modulation. The max-imum slope is obtained for a total modulation widthof δνmod = 1/4T , i. e., if the frequency alternates be-tween the points of maximum slope of the interferencesignal. After detection, the error signal is used to stepthe frequency of the laser spectrometer which corre-sponds to a digital integrating servo control. The lineardrift of the eigenfrequency of the reference resonatorcan be determined by the servo control and compen-sated for by adding a corresponding feedforward signalto the signal controlling the laser frequency. Using Caatoms, cooled on the 1S0–1P1 transition to approxi-mately 3 mK, a relative uncertainty of δν/ν ∼= 2 × 10−14

has been determined [11.2095].

11.14.4 Measurementof Optical Frequencies

In many cases, the use of frequency-stabilized lasersrequires precise knowledge of their frequencies. Theyhave to be determined in relation to the primary stan-dard of time and frequency, the Cs atomic clock. Earlierconcepts of optical frequency measurements used a har-monic frequency chain of several lasers [11.2096]. Forexample, measurements of the Ca inter-combinationline with such harmonic chain led to a fractional un-certainty of 2.5 × 10−13 [11.2097]. In the meantimealternative powerful methods have been developed todetermine optical frequencies. An elegant and success-ful method is based on a mode-locked femtosecond

![ "#

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+

Fig. 11.271a,b Signal of a mode-locked fs laser, shown(a) in the time domain and (b) in the frequency domain

laser [11.2098–2101]. Such lasers emit a continuoustrain of very short pulses covering a wide range of theoptical spectrum of some 100 THz. This range can befurther increased to more than one octave utilizing thephase modulation induced by the pulsed laser beam inan optical fiber of low dispersion [11.2102].

In the frequency domain, the continuous train of fem-tosecond pulses (Fig. 11.271a) corresponds to a combof frequencies of which the inverse of the pulse sep-aration is equal to the pulse repetition frequency frep(Fig. 11.271b). The value of an arbitrary comb frequencyν(m) is determined by

ν(m) = ν(0)+m frep , (11.242)

where m is an integer representing the respective or-der number of the comb frequency ν(m) and frep isthe repetition frequency (Fig. 11.271b). Since the groupvelocity and the phase velocity of the short light pulsescirculating in the laser cavity are slightly different an ex-trapolation to m = 0 leads to the frequency ν(0) which– in general – does not coincide with the zero frequency(Fig. 11.271b). According to (11.242), the determinationof an arbitrary frequency ν(m) requires the knowledgeof frep, the integer m, and ν(0). The pulse repetition fre-quency frep can be measured phase-coherently againstthe primary standard of time and frequency. If frep islarger than approximately 100 MHz it can easily be iden-tified by an interferometric wavelength measurement ofmoderate relative uncertainty (≤ 10−7). In order to mea-sure ν(0), we take advantage of the fact that the combspans over a frequency range of more than one octave.This allows us to frequency double the low frequency

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part of the comb spectrum 2ν(0)+ 2m frep and beatit against the corresponding high-frequency part of thecomb spectrum ν(0)+n frep. This self-referencing of thefrequency comb leads to a beat

δν = 2ν(0)+2n frep − [ν(0)+mνrep]

= ν(0)+ (2n −m) frep (11.243)

that contains ν(0) plus integer multiples of the pulserepetition frequency frep. Since m, n and frep areknown, ν(0) can now be measured with very low un-certainty [11.2099, 2100]. Hence, any comb frequencyν(m) can now be connected phase-coherently to thefrequency of the primary standard of time and fre-quency, the Cs atomic clock. Therefore, we can concludethat the frequency of any other laser emitting withinthe spectral range of the frequency comb can also bedetermined phase-coherently by measuring the beat be-tween the laser frequency and a corresponding combfrequency ν(m). Femtosecond lasers have been appliedfor the first time to measure the frequency measurementsof 1S–2S two-photon transition in hydrogen, leadingto the recommended value of the frequency given inTable 11.56 [11.2098].

In the meantime femtosecond lasers are widelyused for optical frequency measurements in manylaboratories worldwide. The first experiments appliedtitanium-sapphire femtosecond lasers as frequencycomb generators. They are now going to be replacedby femtosecond fibre lasers in view of their high re-liability,ease of operation, and accuracy [11.2103]. Ithas been shown that frequency comb generators basedupon fs lasers are well suited for optical frequencymeasurements of the lowest possible uncertainty. Fur-thermore, it was demonstrated that frequency ratioscould be determined with fs lasers with relative uncer-tainties as low as 10−18 [11.2104]. Basically, in theirpresent state, frequency comb generators based on fslasers allow the measurement of optical frequencieswith uncertainties determined only by the uncertaintyof the primary standard of time and frequency andthe reproducibility of the stabilized laser whereas the

uncertainty of the comb generator itself can still beneglected.

11.14.5 Conclusion and Outlook

The aim of this chapter was to review the foundationsand methods of laser frequency stabilization includingmodern methods of optical frequency measurements.Regarding the wide field of lasers, laser spectroscopyand frequency stabilization and the fast development inthese fields, this chapter could not be comprehensivewithin the limited space available. However, we hopethat it presents a reasonable overview and gives someintroduction to those scientists who are entering the fas-cinating field of laser frequency stabilization and laserspectroscopy.

The examples of stabilized lasers in Sect. 11.14.3are discussed to illustrate the variety of methods ap-plied in this field. The development of stabilized lasersis still progressing in various directions. One is the de-velopment of efficient, reliable and small lasers, e.g.,diode lasers, diode-pumped solid-state lasers, or opticalparametric oscillators (OPOs), which may be impor-tant to achieve efficient, reliable long-term operation.Another branch is the design of stable optical refer-ence resonators to further decrease the laser line widthclose to the quantum limit. An important task is to iso-late them from seismic influences and environmentaldisturbances. Here, alternative novel methods are be-ing developed to compensate and reduce the seismicinfluence. A further reduction of the frequency uncer-tainty can be envisaged from the refinement of lasercooling. The introduction of optical frequency combscreated by mode-locked fs lasers has provided us witha universal tool to determine optical frequencies and op-tical frequency ratios. This field will certainly be furtherexplored by the development of frequency combs oper-ating continuously over long periods and by increasingthe total width of their emission ranges. It is expectedthat the combination of all these efforts will lead to op-tical frequency standards and clocks of unprecedentedlow uncertainty.

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