Testing the foundations of physicswith lasers

Candidate 119499

30th April 2015University of Sussex

Submitted for the degree of Master of Science in Physics

SupervisorDr. Matthias Keller

Date of the graduation16th July 2015

Contents

List of Figures iii

List of Tables vi

Preface ix

Abstract xi

1. Introduction 1

2. Laser Theory 32.1. Gain Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.1. Titanium:Sapphire . . . . . . . . . . . . . . . . . . . . . . . . 42.1.2. Gaussian Beams . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2. Cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2.1. Ring Laser Cavities . . . . . . . . . . . . . . . . . . . . . . . . 62.2.2. Cavity Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2.3. Mode Matching . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.4. Mode Competition . . . . . . . . . . . . . . . . . . . . . . . . 102.2.5. Pulsed Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3. ABCD Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4. Beam Waist Calculations . . . . . . . . . . . . . . . . . . . . . . . . . 14

3. Laser Setup 173.1. Diode seed Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.1.1. Polarization maintaining fibres . . . . . . . . . . . . . . . . . . 193.1.2. Faraday Isolator . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2. Titanium:Sapphire Laser . . . . . . . . . . . . . . . . . . . . . . . . . 213.2.1. Cavity Calculations . . . . . . . . . . . . . . . . . . . . . . . . 223.2.2. Mode Matching . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.3. Laser Lock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.3.1. Hansch-Couillaud locking scheme . . . . . . . . . . . . . . . . 273.3.2. Locking Electronics . . . . . . . . . . . . . . . . . . . . . . . . 27

3.4. Pump laser setup and Damage Thresholds . . . . . . . . . . . . . . . 373.5. Overview of Laser System . . . . . . . . . . . . . . . . . . . . . . . . 39

4. Characterisation 43

i

Contents Contents

5. Summary and Outlook 51

Acknowledgments 53

Bibliography 55

A. Appendix 59A.1. Population in a 2 level system . . . . . . . . . . . . . . . . . . . . . . 59A.2. Population in a 3 level system . . . . . . . . . . . . . . . . . . . . . . 59A.3. Round trip matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 61A.4. Custom prism holder . . . . . . . . . . . . . . . . . . . . . . . . . . . 62A.5. Optical Components Details . . . . . . . . . . . . . . . . . . . . . . . 63

ii

List of Figures

1.1. Energy levels of molecular nitrogen N2. Exciting N2 from X1Σ+g to

a1Πg (green) requires a higher order two photon absorption, eachat a wavelength of 254.6 nm, corresponding to 4.869 eV (blue). Theionization (purple) from a1Πg to X2Σ+

g takes N2 to N+2 (red), requiring

a photon wavelength of 212.2 nm, corresponding to 5.843 eV. [28] . . 2

2.1. 4 level system energy diagram, showing stimulated emission. Thepump (green) excites population from auxiliary level |1> to |2>.They then relax, and non-radiatively (pink) to the excited state |e>,creating a population inversion. Incoming photons (red) stimulateemission B, while using up the population in |e>, which falls to theground level |g>, which quickly depleted via another non-radiativerelaxation (pink) towards the auxilliary level |1>. The population in|e> also spontaneously emits photons Aeg, if no photon has stimulatedthe emission previously. Created using Inkscape 0.91. . . . . . . . . 4

2.2. Energy levels of Titanium:Sapphire, with respect to the displacementof the Ti3+ in its lattice. Absorption can take effect at 400 - 630 nm,while emission takes place at 660 - 1180 nm. Relaxations from B toC and D to A are non-radiative. It is a quasi 4-level system and canbe simplified to the standard 4-level system Fig. 2.1, with very broadtransition lines. [40] . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3. Rayleigh length Lz, beam waist ω0, total angular divergence θ, andbeam width ω(z) of a Gaussian beam. After [1]. . . . . . . . . . . . . 7

2.4. Example of a ring cavity. The left mirror is partially reflective, theright one highly reflective. The triangle represents a prism, with again medium represented above it. Created using Inkscape 0.91. . . 7

2.5. Resonator transmission over a wavelength range, and a cavity finesseof 14 (red) versus a cavity finesse of 30 (blue), with the assumptionof perfect mode matching. [31] . . . . . . . . . . . . . . . . . . . . . . 9

2.6. Hermite Gaussian (left) and Laguerre Gaussian mode intensities (right)[19] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.7. Spectral hole burning of an inhomogeneously broadened gain mate-rial. Multiple modes are lasing simultaneously. For a homogeneouslybroadened gain material, only one mode would be lasing. (p. 398 [37]) 11

2.8. Optical ray propagation through an system of optical elements thatcan be modeled with an ABCD matrix, after [2] . . . . . . . . . . . . 13

iii

List of Figures List of Figures

3.1. External cavity diode laser (ECDL) principle (left) and the Littrowconfiguration (right). In this case, the output coupler is a diffractiongrating. [29] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2. An overview of the existing seed setup. An extra lense between thediode laser and the fibre is used for beam shaping. . . . . . . . . . . . 18

3.3. Faraday isolator principle. The incoming light from the left is firstlinearly polarized to 0 by the input polarizer, then rotated clockwisein the direction of propagation to 45, and then transmitted througha output polarizer set to 45. However, for the reverse direction, therotation from the 45 brings the beam to a 90 polarization angle,and is not transmitted through the input polarizer. [4] . . . . . . . . 20

3.4. An overview of the seed setup in conjunction with the cavity for the848.8 nm Ti:Sa laser. The setup is the same for the 763.8 nm laser. . 21

3.5. Ti:Sa cavity setup. Made in SolidWorks 2010 (top) and Inkscape0.91 (bottom). The SolidWorks design is to scale, while the Inkscapedepiction is for illustration only. The physical distance that the beamtakes through the prism is 7.4 mm. . . . . . . . . . . . . . . . . . . . 22

3.6. Modelled resonator. PMirror is the partially reflective (R=85%) mir-ror that is used for the input. The prism is modelled as three separateparts with the same refracetive index, two media at brewster angles,and a medium in between. Contrary to the depiction, the secondhalf of the prism is angled the other way and modeled as such. Thehighly reflective, curved mirror, OMirror, is modeled as a lense, asthey are mathematically equivalent. The Ti:Sa crystal is modelled asa Brewster angled medium with a refraction index of n = 1.76. . . . 23

3.7. Tangential and sagittal optical planes. A beam (red) is reflected offan optical element (grey). For the saggital plane, the angle betweenthe beam and optical element is not influencing the beam parameters.[7] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.8. Hansch-Couillaud lock. The transmitted intensity through a cav-ity (top) shows the peaks of destructive interference, that mark theoptimal cavity length. The error signal resulting from the Hansch-Couillaud locking scheme (bottom) is similar to the first derivative ofthe transmitted intensity (middle). [39] . . . . . . . . . . . . . . . . . 28

3.9. Circuit diagram of photo diode circuit. The numbers 2, 3, 4, and 7denote the input pins of the LF 356N operational amplifier. The restof its 8 pins are unconnected. Created with EAGLE 7.1.0. . . . . . . 29

iv

List of Figures

3.10. Simplified schematic of a sample and hold circuit (bottom) versuscircuit diagram of the sample and hold circuit used in this project(top). The simplified schematic features an analogue input (AI) andoutput (AO), as well as two operational amplifiers. In the samplingstage, the switch is closed, and the AI is connected to the AO, andis charging the holding capacitor. When the switch is opened by theS&H control, the voltage across the capacitor, and thus at the AOis the same as in the sampling stage. Created with EAGLE 7.1.0(right), and taken from [9] (left). . . . . . . . . . . . . . . . . . . . . 30

3.11. An overview of the locking electronics. The polarization of the lightreflected off and transmitted out of the cavity is used to create anerror signal in the photodiode circuit. The sample and hold (S&H)amplifier circuit prevents false reading from the error signal due tosaturation. The proportional and integrating (PI) circuit manipulatesthe error signal for a more stable lock. The high voltage amplifierserves as a master gain, sending a feedback signal to the cavity piezo,which changes the cavity length, influencing the polarization of thereflected light, closing the feedback loop. . . . . . . . . . . . . . . . . 32

3.12. Sample and hold circuit test. The error signal (red) is measuredbefore (top) and after (bottom) the circuit. The pump sends out anelectronic signal when it is firing (purple). . . . . . . . . . . . . . . . 33

3.13. Photo diode (top), and sample and hold amplifier (bottom) circuits . 343.14. Oscillation of the locking system. The cavity reflection intensity

(blue), and the error signal (red) is shown over time. The bottomgraph shows the locked system, with a very high master gain, result-ing in an oscillation around the optimal cavity length. As the cavityreflection intensity oscillates between two different peaks, there is anoffset in the error signal. This can be countered via the input offsetadjuster of the PI circuit. In the middle graph, the master gain isgreatly reduced, but oscillation is still occuring. The top graph showsthe master gain at a value just below where the oscillations start -this results in a cavity with the most stable lock. . . . . . . . . . . . 36

3.15. Overview of laser system with just one of the lasers shown. The widthof the green pump beam demonstrates the intensity of the beam, notits physical size. However, it is not to scale, for example the beamcan be attenuated and distributed by varying amounts. A completeoverview that includes the second Ti:Sa laser is shown in Fig. 3.16.Created using Inkscape 0.91 and a component library [21] . . . . . . 38

3.16. Overview of the complete laser system. The width of the green pumpbeam demonstrates the intensity of the beam, not its physical size.However, it is not to scale, for example the beam can be attenuatedand distributed by varying amounts. Created using Inkscape 0.91and a component library [21] . . . . . . . . . . . . . . . . . . . . . . 41

v

List of Figures List of Figures

4.1. Pulse energy of the Nd:YAG laser as a function of the Q-switch delay.Q-switch delays of 100-125 µs are of particular interest, as their pulseenergies are similar to the energies allowed by the damage thresholds,see sec. 3.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.2. Cavity length scan, measuring reflection intensity (blue) and calcu-lated length change (red) . . . . . . . . . . . . . . . . . . . . . . . . 45

4.3. Cavity length scan, measuring reflection intensity (blue) and resultingerror signal (red) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.4. Reflected intensity of a cavity. The pump pulse is registered 65 nsbefore the Ti:Sa lasing peak.[22] . . . . . . . . . . . . . . . . . . . . 47

4.5. TTL signal from the pump (blue), and reflected intensity off the cav-ity (red). The intensity is measured with an AC coupled photodiode,therefore the background reflectance is not measured. A fitted expo-nential decay (green) has a decay time constant of 4.2 µs. . . . . . . 48

4.6. Beam focusing before (left) and after beam splitting (right). Thecurrently used method (left) has the advantage of just using one pairof lenses, while the suggested method leads to an increased pulseenergy, as the beam is slightly wider at the cavity mirrors, whichlimit the maximum pulse energy via their low damage threshold. . . 48

4.7. Pump distribution via polarizing beam splitter plates (left) and cube(right). The plates add extra reflections of substantial energy parallelto the main beam, as well as a 56 angle of incidence, which is cor-rected via a second polarizing beam splitter plate, and an additionalmirror, while the cube has a lower damage threshold. . . . . . . . . . 49

A.1. Custom prism holder, designed in Solidworks 2010. The dimensionsare in millimeters, and M20x1 denotes a metric screw thread of 20mm diameter, and a pitch of 1 mm per turn. . . . . . . . . . . . . . 62

vi

List of Tables

2.1. ABCD matrices (p. 585 [36]) . . . . . . . . . . . . . . . . . . . . . . 15

3.1. Overview of the properties of the optical components of the cavitymodel, with lengths L, radii of curvature r, angles of incident α, andrefractive indices n. In the prism parts, the first figure is for thewavelength λ = 763 nm, and the second for λ = 848 nm. . . . . . . . 25

3.2. Conversion of LIDT to a wavelength of λ1 = 532 nm, and a pulseduration of t1 = 6 ns. The LIDT0 are the unconverted values takenfrom their specifications, which can be found via their part numbersin the appendix sec. A.5, except for the Ti:Sa crystal which is takenfrom [14]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.3. Overview of maximum pulse energies Emax dependent on the beamdiameter. For this calculation, a Gaussian beam shape is assumed.The laser line mirrors serve both before and after the beam shaping.The power denotes the fraction of the pump beam that is arrivingat the components. It is variable, as the pump is divided among thetwo Ti:Sa lasers by an arbitrary amount, and subsequently dividedapproximately evenly among the two beams entering the Ti:Sa fromopposite sides. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

A.1. Optics involved in pumping . . . . . . . . . . . . . . . . . . . . . . . 63A.2. Seed and Ti:Sa laser setup parts . . . . . . . . . . . . . . . . . . . . . 63

vii

Preface

The introduction of the 2, 3, and 4 level lasing systems, sec. A.1, and sec.A.2, as wellas sec. 2.1 are based upon the lecture Lasers by Dr Marco Peccianti held in springof 2014 at the University of Sussex. All experimental results, and their analysis aremy own work, except where specifically stated. My supervisor has chosen most ofthe optical components and the main design similar to [22, 41]. He has reviewed,and made changes to the electronics, and the prism holder. This project is usingan existing setup of two diode lasers, which are improved via optimization in itstemperature control, choice of fibres, and input coupling. The pump laser wasalways operated with the help of a PhD student of the ITCM research group, inaccordance with university regulations. The figures and tables are the result of myown work, except where specifically stated.

ix

Abstract

In this project, two seeded, pulsed Titanium:Sapphire lasers are set up. Their wave-lengths are 848.8 nm, and 763.8 nm, which are frequency converted towards wave-lengths of 212.2 nm, and 254.6 nm, corresponding to a 2+1’ Resonance enhancedmultiphoton ionisation process of molecular nitrogen N2 towards the ionised stateX2Σ+

g . The ionized nitrogen N+2 will then be used to do high resolution spectroscopy

in another experiment to place an upper bound on the possible change of the proton-to-electron mass ratio µ. To efficiently, and reliably, achieve the required state, alow linewidth of the lasers is required. Custom locking electronics are built to imple-ment a Hansch-Couillaud locking scheme, which is stable for potentially hours. TheTi:Sa laser cavities are characterized to have a cavity finesse of 12±2, and 32±5. Acoupling efficiency of the seed into the cavity of up to 69% is achieved.The laser induced damage threshold of the optics are placing a limit of 6 mJ per Ti:Salaser on the potential pulse energy. Infrared lasing is not achieved in this project,despite an extended search for the cause. One possible cause is a pump pulse energybelow the lasing threshold of the Ti:Sa lasers. However, several improvements, aresuggested, which could increase the damage threshold and maximum pump pulseenergy of this setup.

xi

1. Introduction

Particle physics as a discipline is focused around and based upon 26 fundamentalconstants [15]. While all other constants are derived mathematically from these ini-tial 26, they themselves cannot be derived but can only be measured experimentally.

Among these fundamental constants are the mass ratios of the fundamental particles.While the term "constant" would imply that there is no change over time, some theo-ries predict alterations to the fundamental constants over time. One example coversthe fine-structure constant α [16]. One group has even repeatedly reported that theyhave measured such changes using the astronomical many multiplet (MM) method[24, 23, 25]. However, newer measurements by different groups [38] have found novariation using the MM method and offer explanations why earlier measurementsdisagree. They conclude by stating that this method cannot at present be used todetect variations in α. This is fundamentally due to the lack of direct control overthe experimental parameters, which leads to increased systematic errors in the mea-sured wavelengths. Another such fundamental constant is the proton-to-electronmass ratio, µ = mp/me. A different astronomical method has put a constraint on thechange in the proton-to-electron mass ratio ∆µ/µ = (0.0±1.0) ·10−7 over a timescaleof 7 billion years [26].

In a laboratory experiment, the experimental parameters can be directly controlled,leading to a higher precision of the measurements. However, in contrast to theastronomical approach, it is not possible to look back in time, leading to a muchhigher precision requirement in order to detect potential changes. This can beachieved via high resolution spectroscopy of atomic ions.

According to [10], the constant has a value of 1836.152 672 45(75). Peik et. al havederived upper limits on changes of the fine-structure constant d

dtlnα = (−0.26 ±

0.39) · 10−15/yr and the proton-to-electron mass ratio ddt

lnµ = (−1.2 ± 2.2) · 10−15/yr[18].

For the derivation of a more restrictive upper limit on a possible change of the proton-to-electron mass ratio, molecular ions measured in a lab setting are a promisingcandidate. Molecular nitrogen in particular, N2, with its miniscule systematic energylevel shifts, has a potential to improve upon the previous upper limits (p. 121[27]). The ITCM group at the University of Sussex is focusing its research onthe molecular ion N+

2 , which is created via photo ionisation. In order to do highresolution spectroscopy on N+

2 , it needs to be deterministically prepared in a pre-selected state. This is done via a two step ionisation process, taking neutral N2 to

1

Chapter 1 Introduction

the ionised N+2 . The energy levels of N2 can be seen in Fig. 1.1. In order to excite it

from its electronic ground state X1Σ+g to an intermediate quantum state a1Πg, two

photons of a wavelength of 254.6 nm are used. Afterwards, a photon at a wavelengthof 212.2 nm takes the molecule to its ionised state X2Σ+

g . This process is referred toas Resonance enhanced multiphoton ionisation (REMPI), specifically 2+1’ REMPI.The aim of this project is to produce a setup of two pulsed Titanium:Sapphire lasersthat produce high peak intensities and reach the required wavelengths via frequencyconversion.

Figure 1.1.: Energy levels of molecular nitrogen N2. Exciting N2 from X1Σ+g to

a1Πg (green) requires a higher order two photon absorption, each at a wavelengthof 254.6 nm, corresponding to 4.869 eV (blue). The ionization (purple) froma1Πg to X2Σ+

g takes N2 to N+2 (red), requiring a photon wavelength of 212.2 nm,

corresponding to 5.843 eV. [28]

2

2. Laser Theory

2.1. Gain Medium

A laser is based on the stimulated emission of photons in contrast to spontaneousemission. Incoming photons are essentially amplified by this process. The newphotons can, in turn, stimulate the emission of additional photons, creating anavalanche effect. The word laser stands for light amplification by stimulated emissionof radiation, and usually involves a gain medium inside a cavity. There are manygain media to choose from, and they can be described via the energy levels of thosesystems. For stimulated emission to be possible, the population of occupied stateshas to be bigger on an energy level energetically above another with an allowedradiative transition. The lower energy level is called the ground level |g>, with theupper energy level being referred to as the excited level |e>. For reasons explainedin the next paragraphs, a common choice for a gain medium is the 4 level system,depicted in Fig. 2.1. Having a higher population in |e> than in |g> is a necessarycondition for lasing, and is called a population inversion.

For a gain medium with just two relevant energy levels, the ground level |g>, andexcited level |e>, it is not possible to achieve a population inversion. As the pumpingprocesses are reversible, pumping a two level system at best leads to an equal pop-ulation of the ground level |g> and excited level |e>. A derivation can be found inthe appendix, sec. A.1. However, by adding an auxiliary level |a> above the excited|e> level, a population inversion is achievable. The auxiliary level |a> then relaxesin a non-radiative process to the excited level |e>. By pumping to the auxiliarylevel |a>, the excited level |e> is no longer affected by the pump in the way it wouldbe in a two level system, and the population in |e> can get much higher than thepopulation in |g>, depending on the amount of pumping. This is derived in sec. A.2.A 4-level system has both an auxiliary level above, |2>, the excited and below, |1>,the ground level, as depicted in Fig. 2.1. The resulting population inversion n is

n = WN

Bρ+W + Aeg(2.1)

with the pumping rateW , the total population N , the Einstein coefficient for spon-taneous emission Aeg, stimulated emission B , and the spectral energy distribution

3

Chapter 2 Laser Theory

ρ. In Fig. 2.1, the rates for non-radiative relaxation from the upper auxiliary energylevel |2>, A2e, as well as to the lower auxiliary energy level |1>, Ag1, are also used.

Compared to a 3-level system with the auxiliary level above the excited level |e>,this system has the advantage of lacking of a pumping threshold to create the popu-lation inversion. This makes 4-level systems ideal for most applications. Ultimately,the choice is limited by available gain media, as well as influenced by other con-cerns, like its mechanical, chemical, and optical properties. In this project, a Tita-nium:Sapphire will be used as a gain medium and introduced in sec. 2.1.1. Thereare additional requirements to create a laser beyond having a population inversion:vitally, a cavity is required. This will be introduced in sec. 2.2.

A2e

Ag1

AegW |e>

|g>

|1>

|2>

B

Figure 2.1.: 4 level system energy diagram, showing stimulated emission. Thepump (green) excites population from auxiliary level |1> to |2>. They then relax,and non-radiatively (pink) to the excited state |e>, creating a population inver-sion. Incoming photons (red) stimulate emission B, while using up the populationin |e>, which falls to the ground level |g>, which quickly depleted via anothernon-radiative relaxation (pink) towards the auxilliary level |1>. The populationin |e> also spontaneously emits photons Aeg, if no photon has stimulated theemission previously. Created using Inkscape 0.91.

2.1.1. Titanium:Sapphire

Lasers like the widely used Nd:YAG laser have a comparatively small gain bandwidthof only a few nanometers. This would make finding a gain material for a specificapplication wavelength difficult. Dye lasers, on the other hand, are available formost wavelengths usually required, due to the abundance of different dyes. Theirdownside is a large linewidth and a difficult handling of the dyes [17].

4

2.1 Gain Medium

Aluminum Oxide A2O3 (Sapphire) doped with Titanium ions Ti3+ is a gain mediumthat has many useful properties. It is a quasi 4 level system, see Fig. 2.2.

Figure 2.2.: Energy levels of Titanium:Sapphire, with respect to the displacementof the Ti3+ in its lattice. Absorption can take effect at 400 - 630 nm, whileemission takes place at 660 - 1180 nm. Relaxations from B to C and D to A arenon-radiative. It is a quasi 4-level system and can be simplified to the standard4-level system Fig. 2.1, with very broad transition lines. [40]

They can be used over a wide wavelength range of 660 - 1180 nm, while beingpumped at 400 - 630 nm. The reason for these large wavelength ranges lies in thecoupling of the Ti3+ to the vibrational energy levels of the sapphire lattice (p. 451[40]). The energy of an incoming pump photon is divided between the 3d electronof Ti3+, and phonons. This allows for a wider range of acceptable energies.A common choice for pulsed pumping is a frequency doubled Nd:YAG laser at 532nm (p. 565 [37]). The upper energy level of Titanium:Sapphire has a fluorescencelifetime of 3.2 µs. As there is a common pool of states, it is homogeneously broadenedby it to a linewidth of 150 THz (p. 167f [37]). Sapphire has a very high thermalconductivity, reducing thermal problems for high laser powers [33].Sapphire changes its refractive index from n = 1.7718 for 532 nm, to n = 1.7612for 763.8 nm, or n = 1.7589 for 848.8 nm [34]. The titanium doping of the sap-phire should only change its dispersive properties slightly. The difference between

5

Chapter 2 Laser Theory

homogeneous and inhomogeneous broadening will be explained in sec. 2.2.4.

2.1.2. Gaussian Beams

Many optical beams can be approximated using Gaussian functions, and are thusreferred to as Gaussian beams. Laser light generally has two properties that deter-mines the shape of a Gaussian beam: the size of the beam waist ω0, and its positionalong the propagation axis z. Together with the wavelength λ, and the overall in-tensity I, the laser beam is fully defined. Gaussian beams are described by severaladditional parameters shown in Fig. 2.3.The Rayleigh length LZ in Fig. 2.3 is related to the wavelength λ and the beamwaist ω0 via

LZ = πw2o

λ(2.2)

The total angular divergence θ in Fig. 2.3 for z LZ is given by

θ ≈ λ

πwo(2.3)

Left to diverge freely, the beam size of a Gaussian beam is calculated via

w2 (z) = w20

(1 +

(z

Lz

)2)

(2.4)

2.2. Cavities

2.2.1. Ring Laser Cavities

A laser works by sending light through a gain medium. In order to achieve higheramplification, the light can be redirected to pass through the gain medium not oncebut many times. This can be achieved via a cavity.Laser cavities can generally be divided into two categories: firstly, a cavity in whichthe beam is reflected into the opposite direction by the final mirror, resulting in astanding wave, and secondly, a ring laser cavity, in which the beam continuouslytravels in the same direction.

6

2.2 Cavities

LZ

Figure 2.3.: Rayleigh length Lz, beam waist ω0, total angular divergence θ, andbeam width ω(z) of a Gaussian beam. After [1].

In a standing wave cavity, the field intensity has periodic nodes of zero intensity.Inside a laser gain medium, these nodes mark points where the light is not beingamplified, therefore the gain medium is being used less efficiently. In order to avoidthis, a ring laser cavity is used. Here, the light forms a traveling wave and the entirelength of the gain medium along the optical axis is used.

Figure 2.4.: Example of a ring cavity. The left mirror is partially reflective, theright one highly reflective. The triangle represents a prism, with a gain mediumrepresented above it. Created using Inkscape 0.91.

An example of a ring cavity can be seen in Fig. 2.4. For lower intensities, the gainmedium amplifies the light by a constant factor. However, for higher intensities,the population inversion can get depleted and the gain medium saturates. In acontinuous wave laser, the intensity of the light increases until the gain is at the samelevel as the losses. Losses take the form of light scattered, absorbed, or otherwiserendered unusable for the intended purpose, as well as light transmitted out of the

7

Chapter 2 Laser Theory

cavity in the intended way.Some cavities use a second laser, called a seed laser. When the main laser is turnedon, there are potentially many longitudinal modes of slightly different wavelengthsthat could be amplified. A seed laser is used to give one of these modes a startingadvantage in the form of some intensity already in that mode. Via a process calledmode competition, the other modes are suppressed, and single longitudinal modelasing can be achieved. Mode competition will be explained in sec. 2.2.4. However,in such a case, the partially reflective mirror acts as a barrier for the seed light to getinto the cavity. In order to get as much light into the cavity as possible, destructiveinterference is used. With a continuous wave seed laser, some light gets through thepartially reflective mirror and builds up inside the cavity. The light that is reflectedoff the partially reflective mirror then interferes with the light that is transmittedout of the cavity. If the phase difference of oscillation between those two beamsis a multiple of π, then they interfere destructively, and a lot less light is reflected.This way, the required amount of seed power can be reduced, and/or the actual seedenergy inside the cavity increased.In order to get the phase difference to be a multiple of π, the cavity length can bechanged. Alternatively, the wavelength can be changed, see Fig. 2.5. In that figure,the transmission instead of the reflectance is observed. For certain wavelengths, thetransmission peaks. In the case of a ring cavity, as seen in Fig. 2.4, the reflectanceof the cavity has dips instead of peaks in an otherwise high background reflectance.Measurements of this nature can be seen in Fig. 4.2.A measure of the quality of a cavity is the finesse. The distance in wavelength, orfrequency, between two successive extrema is called the free spectral range (FSR).The full width of such dips, or peaks, at half the intensity of its extrema is calledthe full width at half-maximum (FWHM). Dividing the FWHM by the FSR yieldsthe cavity finesse F .

F := FWHMFSR (2.5)

The inverse of the cavity finesse F approximately gives the losses per round trip ofthe cavity L ≈ 1/F [31]. These losses include every instance in which photons arelost to the cavity, including the output, absorption, and scattering of the opticalelements.

2.2.2. Cavity Modes

As light resonates inside a cavity, losses are inevitable. One such loss mechanism isdue to the finite sizes of the involved optics in contrast to an, in principle, infiniteGaussian beam profile. Therefore the edges of the beam are cut off and lost in each

8

2.2 Cavities

Figure 2.5.: Resonator transmission over a wavelength range, and a cavity finesseof 14 (red) versus a cavity finesse of 30 (blue), with the assumption of perfectmode matching. [31]

of the reflections. As the light bounces back and forth inside a cavity, steady statescan be found. These are referred to as modes. They were first described by Fox andLi in 1960 [20]. The cavity in this project has a Brewster angle window that givesthe mode a x-y symmetry, as opposed to a circular symmetry that would result fromjust a perfectly aligned simple empty two mirror cavity (p. 389f [37]). A theoreticalderivation of these modes can be found on (p. 644f [36]). They are sorted via twoindices, l and m, and called Transverse ElectroMagnetic modes, TEMlm for short.The lowest order, TEM00 has a Gaussian shape. A depiction of some of the higherorder modes can be seen in Fig. 2.6.

Figure 2.6.: Hermite Gaussian (left) and Laguerre Gaussian mode intensities(right) [19]

For many applications, including this project, the highest performing TEM is TEM00,as it is the smallest of all TEMlm. This makes it easier to focus, and results in thesmallest losses. In addition, it can also concentrate the most power in a small area,a characteristic particularly useful for frequency conversion. It can also be beneficial

9

Chapter 2 Laser Theory

e.g. for reducing the size required for a optical component, which might also makeit cheaper. Due to its range of useful properties, this project will focus on TEM00.

2.2.3. Mode Matching

The TEMlm modes, introduced in sec. 2.2.2, are the result of many reflections insidethe cavity. If the cavity is seeded the beam shape governs which of the TEM modeswill be coupled to. The incoming beam should overlap as much as possible with thetarget TEM mode, usually TEM00. This can be expressed via

η = |´E?

1E2dA|2´|E1|2dA

´|E2|2dA

, (2.6)

see [32]. This overlap integral calculates the mode matching quality η out of thecomplex electric fields E1 and E2. It is proportional to the amplitude of the modethat will be excited in the cavity due to incoming light. Therefore, the optimal valueof the mode matching quality η = 1 is reached when the beam waist of the incomingGaussian beam and of a TEM00 overlaps both in size and position - the shape ofthe beam is then fully defined, given a matching wavelength.

2.2.4. Mode Competition

With each oscillation, the light takes away energy from the gain material. If thegain material is a source of homogeneous instead of inhomogeneous broadening, theenergy is taken away from a common pool of population of the upper level (p. 397[37]). In a cavity with such a gain material, two beams of different frequencies,which are called longitudinal modes, are both amplified, but both the initial relativeintensities, and the frequency dependance of the gain material cause the light tobe amplified differently. Consequently, one of the modes will saturate, using upthe entire population inversion that is created by the pump, leaving little for othermodes, which will then exponentially decline in strength due to losses. This processis called mode competition.In contrast, if the gain material is inhomogeneously broadening, the state populationis divided into many individual pools. Different longitudinal modes can therefore usethe different sources, without competing for the same source of energy. Therefore,multiple modes can exist inside a cavity with such a gain material, see Fig. 2.7, ifnot prohibited by other means, such as a Fabry-Perot-Etalon.

2.2.5. Pulsed Laser

After an initial build up time, the pump of a continuous wave laser supplies exactlythe difference between the energy that is carried away by the amplified light, minus

10

2.2 Cavities

Figure 2.7.: Spectral hole burning of an inhomogeneously broadened gain material.Multiple modes are lasing simultaneously. For a homogeneously broadened gainmaterial, only one mode would be lasing. (p. 398 [37])

losses. In contrast, a pulsed laser typically delivers peak intensities that are muchhigher than that of continuous wave lasers, but only at bursts of energy. This canbe very useful for increasing the efficiencies of certain non-linear optical processessuch as second harmonic generation, as well as the two photon absorption processfor the transition to the intermediate quantum state a1Πg of molecular nitrogen N2,since their efficiency is proportional to the field intensity squared I2. Therefore, theTi:Sa lasers in this project will be operated in a pulsed fashion.A drawback, however, are the laser induced damage thresholds (LIDT) of the opticsinvolved. While for continuous wave lasers, as well as some pulsed lasers with a pulseduration of > 100 ns, the damage mechanism is through thermal effects, as opposedto a breakdown of the dielectrics for pulse durations of 1 − 100 ns [3]. The LIDTof the optics in this project is discussed in sec. 3.4, and is limiting the maximumenergy of the Ti:Sa lasersThe Ti:Sa crystal needs to be supplied with energy to develop an optical gain. Thisis done in the form of another laser, called the pump laser. In order to get thesehigh bursts of energies, the Ti:Sa laser is operated via gain-switching. The cavity

11

Chapter 2 Laser Theory

is continuously being supplied with the seed wavelength, while the pump is onlyswitched on for very short, very intense pumping. A reason for not having the seedwavelength pulsed to deliver the same short bursts, is the scheme that is being usedto lock the cavity to the seed laser. Some of the seed light reflected off the cavityis used for this purpose, which will be explained in sec. 3.3. The pump pulse is tooshort for the longitudinal mode selected by the seed laser resonating inside the cavityto achieve peak intensity at the same time as the pump pulse does. This results inthe seed light seeing an extremely high gain, yielding a very fast amplification ofthat light, at the expense of a rapid decline of the upper population in the Ti:Sacrystal, ending the amplification (p. 450[37]). Overall, the method employed in thisproject delivers a short high intensity pulse at the seed wavelength.

2.3. ABCD Matrices

In order to get insight into the propagation of light, a paraxial optical ray approx-imation can be used. It is a useful tool to describe the properties of a ray of lightpropagating through both free space and various optical elements, and can even beused to derive whether a resonator is fundamentally stable or not.It builds upon the description of rays via two properties, the distance from theoptical axis r, and an angle α to it. Assuming very small angles, which are typicalin laser applications, the angle α, in radians, can be approximated with the paraxialapproximation,

sin α ≈ α; cos α ≈ 1; tan α ≈ α (2.7)

For a beam propagation over the length L through free space, the distances to theoptical axis before and after propagation, r1 and r2, the paraxial approximationbecomes apparent. Through geometric considerations, the additional distance afterpropagation, ∆r = r2 − r1 is related to the distance traveled via

tan α = ∆rL

(2.8)

Using the paraxial approximation, this becomes

α ≈ ∆rL

(2.9)

Overall, the propagation can then be described via

r2 = r1 + Lα1 and α2 = α1 (2.10)

12

2.3 ABCD Matrices

Figure 2.8.: Optical ray propagation through an system of optical elements thatcan be modeled with an ABCD matrix, after [2]

with the z-axis being aligned with the optical axis, and α being the angle of prop-agation, as demonstrated in Fig. 2.8. Note that this angle does not change in thiscase, α2 = α1. A compact way of combining the properties of the angle of propa-

gation α, and the length traveled r is using 2D vectors,(rα

). In this notation, the

propagation through free space in equation 2.10 can be written using a 2x2 matrix,(1 L0 1

). Overall, the propagation is then summarized via

(r2α2

)=(

1 L0 1

)(r1α1

)(2.11)

For more complex situations, the 2x2 matrix is different. The appropriate matricesfor all the optical elements used in this project are listed in Tab. 2.1. These 2x2matrices are called ABCD matrices. Generally, for a single optical element, the

13

Chapter 2 Laser Theory

equation is

(r2α2

)=(A BC D

)(r1α1

)(2.12)

in which r1, r2, α1, and α2 are the distances and angles from the centre of the opticalaxis, before and after passing through the optical element, respectively. The effectof an ABCD matrix on a Gaussian beam will be explained in sec. 2.4. For multipleoptical elements, the corresponding matrices M1,M2, . . ., are multiplied, with thefirst optical element M1 multiplied first:

(r2α2

)= Mn . . .M2M1

(r1α1

)(2.13)

In addition to simply giving the beam properties of the outgoing beam, the resultingsystem matrix Mtotal = Mn . . .M2M1 can be used in the context of a resonator.There, the overall stability of the cavity can be calculated via the trace of thematrix Mtotal :

g =: tr (Mtotal)2 = A+D

2 (2.14)

For a large number of propagations through the system, g2 > 1 signifies that thebeam can wander arbitrarily far from the optical axis, but for g2 ≤ 1, it is confinedand therefore stable. For physical cavities, this stability parameter should be g2 < 1,in order to actually focus the beam.

2.4. Beam Waist Calculations

A very important problem in building a laser is manipulating the position and sizeof the beam waist. This can be done with the help of two lenses with focal lengthsf1 and f2. Assuming an initial beam waist w0,1 at position z1, lenses can be used tobring the beam to a waist of ω0,2 at position z2, which is summarized by using thecomplex formalism q1 and q2.

1q (z) = 1

R (z) −iλ

πw (z)2 (2.15)

14

2.4 Beam Waist Calculations

Matrix Optical Element Abbreviations(1 00 1

)Reflection from a flat mirror(

1 L

0 1

)Propagation in a constant n L = length travelled

n = refractive index 1 0− 1f

1

Thin lens f = focal length(1 0− 2R

1

)Reflection from a curved mirror R = radius of curvature1 0

0 n1n2

Refraction at a flat interface n1 = initial refractive indexn2 = final refractive index

Table 2.1.: ABCD matrices (p. 585 [36])

The position of the two lenses are zl,1 and zl,2. Note that the position waist ofthe incoming beam is potentially behind the lenses. This problem has four degreesof freedom, f1, f2, zl,1, and zl,2, but only two restraining variables, ω0,2 and z2.Therefore, two of those four degrees can be fixed. The focal lengths of the lensesf1and f2 are chosen, because these are usually only available in discrete values. Inorder to find the positions of the lenses zl,1 and zl,2, the ABCD matrix approach isused. The system can be thought of as similar to Fig. 2.8:

Mtotal = Ml3Mf2Ml2Mf1Ml1 (2.16)

(A BC D

)=(

1 l30 1

)(1 0− 1f2

1

)(1 l20 1

)(1 0− 1f1

1

)(1 l10 1

)(2.17)

This can then be solved using a software based approach. In this project, a freewarecalled ReZonator [7] is used. It is based on ABCD matrices and Gaussian beams,and specialized for optical systems, as well as being easy to use.The effect of an optical system, represented by an ABCD matrix, on a Gaussianbeam is

q2 = Aq1 +B

Cq1 +D(2.18)

For a lense that has a distance to a beam waist ω1 that is equal to its focal length

15

Chapter 2 Laser Theory

f , it produces another waist of potentially different waist size ω2. This new waistω2 is spaced one focal length f apart. The waist sizes follow the formula

w21 · w2

2 = λ2

π2f2 (2.19)

In conclusion, ABCD matrices and Gaussian beams are a very useful tool for calcu-lating the properties of beams propagating through, or oscillating in optical systems.In conjunction with a computer algebra system, they can be used for calculating evenmore complex optical systems, including the one featured in this project.

16

3. Laser Setup

Two publications form the basis for the setup of the Ti:Sa lasers: [41] and [22].The setup is centered around the two Ti:Sa lasers, and two diode seed lasers. Thecombined design is explained in this chapter. The two Ti:Sa lasers in this projectare slaves to two diode lasers at the wavelengths λ1 = 763.8 nm and λ2 = 848.8 nm,and feature a very similar setup.

3.1. Diode seed Lasers

Figure 3.1.: External cavity diode laser (ECDL) principle (left) and the Littrowconfiguration (right). In this case, the output coupler is a diffraction grating. [29]

The seed lasers are external cavity diode lasers (ECDL). One side of the diodesis coated to be highly reflective, the other side is coated to be anti-reflective, ascan be seen in Fig. 3.1. A lense then collimates the beam, i.e. it shapes the beamto minimal spread, limited by diffraction, which can be calculated using Gaussianbeams sec. 2.1.2. The beam then encounters a diffraction grating, which forms theend of a linear cavity. The first-order diffracted beam is used for optical feedback inthe diode - this step selects the wavelength that can resonate inside the diode laser.Overall, this diode laser setup is called a a Littrow configuration, see Fig. 3.1 [29].The wavelength can be tuned by a few nm via changing the angle of the diffractiongrating via a piezo, limited by the voltage specification of the piezo. They supplyseveral mW of lasing power, dependent on the applied current. Typically, the 848.8nm seed diode lases with ≈ 5 mW, while the 763.8 nm diode lases with ≈ 4 mW.In order to avoid any unwanted optical feedback from beyond the diffraction grating,the ECDL output is directed through two Faraday isolators, that allow only onepropagation direction of light, while severely attenuating the opposing direction.

17

Chapter 3 Laser Setup

A more detailed explanation of Faraday isolators can be found in sec. 3.1.2. Thisreduces the unwanted feedback by about 30 dB for each Faraday isolator. The seedlaser setup can be seen in Fig. 3.2.

2

Seed Setup

2848.8 nm Laser

FaradayIsolator

f f

PI Lock Wavemeter

Fibre

Figure 3.2.: An overview of the existing seed setup. An extra lense between thediode laser and the fibre is used for beam shaping.

The wavelength of the seed lasers dictates the wavelength of the Ti:Sa lasers. Inturn, their wavelengths will be frequency converted to be resonant to the opticaltransitions of molecular nitrogen N2, as introduced in chapter 1. The final frequencyconverted wavelengths can then be changed via the seed lasers to be resonant to the2+1’ REMPI. In order to measure their wavelengths, only a small portion, ≈ 4%is needed, for which a wedge plate is used. In a wedge plate, the back surface hasan angle towards the front surface. Therefore, the beams reflected from the frontand back surfaces are not parallel, which makes optical alignment easier, as the twobeams do not interfere with each other. The reflected portion of light is then coupledinto a fibre, which leads to a wavemeter, where the wavelength is measured. Thewavelength information is then used by an existing LabVIEW program, based on aproportional and integral (PI) circuit, as seen in Fig. 3.2.

A PI controller takes an error signal, in this case the mismatch between the wave-length information supplied by the wavemeter, subtracted from the wavelength se-lected in the LabVIEW. The proportional part of the PI circuit multiplies the currenterror by a constant, called the proportional gain. The integral part constantly in-tegrates the error, and multiplies it by the integral gain. The two parts are thensummed up, and fed back into the system, in this case via a digital-to-analogue con-verter, then to a three channel SVR 150/3 high voltage power supply produced byPiezomechanik GmbH. The high voltage amplifier is connected to the piezo, turningthe diffraction grating of the ECDL, thus changing the wavelength. A derivative partwould take the current rate of change and amplify that by the derivative gain. Sinceit carries difficulty in tuning, and the potential to destabilize the system, leading toa larger error, a derivative part is not included (p. 560f [13]). The proportional andintegral part of the PI controller are both chosen to be included to achieve maximumstability of the overall system. The magnitues of the proportional and integral gainsare chosen experimentally to achieve a high stability of the wavelength.

18

3.1 Diode seed Lasers

The wavelength of the diode seed lasers are also influenced by their temperature,and the applied current. Both of these have to be stabilized to avoid any changes inwavelength. For the diode laser with a wavelength of 848.8 nm, this is achieved viaa Toptica DTC 110 temperature control module, and a Toptica DCC 110 currentcontrol module. For the other diode with a wavelength of 763.8 nm, the modules area Toptica DCC 100, and a Toptica DTC 100. The temperature control modules arebased on a PID circuit. After the current is turned on, the temperature increases atfirst. The temperature controllers then actively cool the diode via a Peltier cooler,based on the eponymous Peltier effect. The differential gain increases the speed atwhich the temperature changes, at the cost of lower stability of the temperature.It is therefore set to a very low level to achieve a temperature stabilization withinminutes after turning on the current.

3.1.1. Polarization maintaining fibres

By using fibres between the seed laser, and the Ti:Sa laser setups, both of them canbe moved independently from one another, increasing the flexibility of the wholesetup. Additionally, different lasers could be used to seed the Ti:Sa lasers, or thecurrent seed lasers could be used for a different experiment. In both cases, this canbe achieved simply by changing the fibre connections, instead of having to changethe setup.The polarization of the seed laser light is used in the Ti:Sa laser setup, and has tobe linear. Therefore, polarization maintaining fibres are used to transport the diodeseed laser light to the Ti:Sa setup. For the 848 nm light, a PMJ-3A3A-850-5/125-3-5-1 fibre from OZ Optics is used, with an operating wavelength range of 810 to980 nm [6]. It has a fiber core diameter of 5 µm, with a cladding size of 125 µm, aminimum extinction ratio of 20 dB, and an insertion loss of less than 0.75 dB. For the763 nm seed laser, a P3-630PM-FC-10 from Thorlabs is chosen, capable of operatingfrom 620 nm to 850 nm, a mode field diameter of 4.2 µm and a maximum insertionloss of 1.2 dB [5]. The input coupling is reoptimized for maximum transmission.For coupling into the 848.8 nm fibre, a A375TM-B 7.5 mm collimator lense fromThorlabs is used.A coupling efficiency into the fibre of 30 %, i.e. the power ratio between before,and after the fibre, is achieved for both seed lasers. This is less than either of thespecifications of the fibres predict. Potentially different fibre collimator lenses couldimprove on this value, however, successful operation of the Ti:Sa lasers has beenachieved with as little as 40 µW (p. 584 [41]). Therefore, the coupling efficiency issufficient for this setup.Via the λ/2-waveplates, the polarization is adjusted to match the slow axis of thepolarization maintaining fibres. Without the λ/2-waveplates, the polarization of theseed laser light would be changed according to how much stress is applied to thefibre, leading to a more unstable polarization. After optimizing the polarization

19

Chapter 3 Laser Setup

angle, the extinction ratio of the transmitted preferred to the parasitic polarizationis measured to be 38.29 dB for the 763.8 nm setup and 33.51 dB for the 848.8 nmsetup.

3.1.2. Faraday Isolator

Faraday isolators are optical isolators, allowing only one propagation direction oflight, severely attenuating the opposing direction [30]. The kind featured in thisproject uses two polarizing beam splitter cubes (PBS), and a Faraday rotator. TheFaraday rotator is based upon the eponymous Faraday effect, and is proportional tothe applied magnetic field, length of the path through the material and a function ofthe wavelength. By varying the magnetic field or the orientation of the polarizers,the Faraday isolator can be tuned to a range of wavelengths, usually several hundrednanometers [30]. The two PBSs are rotated by 45 with respect to each other. Lightcoming in through the input is first linearly polarized, then rotated by 45 by theFaraday rotator to align with the output PBS. Most of the light is then transmitted.However, in the opposing direction, incoming light is rotated by 45, away from thetransmissive angle of the PBS. This is summarized in Fig. 3.3. The parts of thebeams that are not at the required angle of transmission are reflected away from theoptical isolator at a right angle to the optical axis. They could be accessed, howeverin this project, they are simply absorbed by a plastic plate.

Figure 3.3.: Faraday isolator principle. The incoming light from the left is firstlinearly polarized to 0 by the input polarizer, then rotated clockwise in the di-rection of propagation to 45, and then transmitted through a output polarizerset to 45. However, for the reverse direction, the rotation from the 45 bringsthe beam to a 90 polarization angle, and is not transmitted through the inputpolarizer. [4]

20

3.2 Titanium:Sapphire Laser

3.2. Titanium:Sapphire Laser

After the fibres, a lense works in conjunction with the fibre collimator lenses to formtelescopes for mode matching into the Ti:Sa laser, as seen in Fig. 3.4. The telescopeis chosen to be separated by the Faraday isolator, because of its optical length ofl = 20 cm. Alternatives are to place the second lense before the Faraday isolatorwhich would increase the physical size of the setup and increase the length to themode to be matched, or to introduce another lense behind the Faraday isolator tocomplete a telescope there. A λ/2-waveplate then adjusts the linearly polarized lightto the preferred direction of the resonator. Since the Faraday isolator has a higherbeam height than the cavity, two mirrors are used both for vertical beam shuttlingand for input coupling through the 85% reflective output coupling mirror.

848.8 nm Cavity

2

Seed Setup

2848.8 nm Laser

FaradayIsolator

f f

PI Lock Wavemeter

R=85%

PrismPiezo

R=99.5%Ti:Sa

Figure 3.4.: An overview of the seed setup in conjunction with the cavity for the848.8 nm Ti:Sa laser. The setup is the same for the 763.8 nm laser.

The resonator cavity itself consists of an 85% reflective output coupling mirror, aprism, a highly reflective R = 99.5% mirror, and a Brewster-cut Titanium:Sapphirecrystal with dimensions 5 · 5 · 20 mm3, as seen in Fig. 3.4. The Brewster angledsurfaces introduce a polarization dependent cavity finesse. This results in a pre-ferred polarization direction, which is used in sec. 3.3 to lock the cavity length tobe resonant with the seed laser light. The prism introduces a dispersive optical ele-ment, reducing the wavelengths that can resonate, as well as introducing additionalBrewster angled surfaces. It is made of the optical glass SF 11, has an apex angle of59, a height of 10 mm, and a side length of 15 mm. The partially reflective mirrorhas a reflectivity that is chosen to be similar to (p. 584 [41]), to get a comparablecavity finesse. Both cavity mirrors have a radius of curvature of r = 10 m, resultingin a stable cavity as well as an appropriate waist size of the light resonating in thecavity, as calculated in sec. 3.2.1. They are made of UV grade fused silica, are highlytransmissive, transmittance T > 90%, for 532 nm light. This transmittance is usedto supply the Ti:Sa with energy to create the population inversion, see sec. 3.4.Details on the calculation of the stability of the cavity can be found in sec. 3.2.1.The cavity has an optical round trip length of 30 cm. The physical size of the cavity

21

Chapter 3 Laser Setup

can be seen in Fig. 3.5. The prism is held in place with a custom built connectionto the piezo element, in turn held by a mirror holder. The schematics of the prismholder can be found in the appendix Fig.A.1.

60.4 mm80.4 m

m

60.9 mm

50 mm 31.0°

31.0°30.9°

Figure 3.5.: Ti:Sa cavity setup. Made in SolidWorks 2010 (top) and Inkscape 0.91(bottom). The SolidWorks design is to scale, while the Inkscape depiction is forillustration only. The physical distance that the beam takes through the prism is7.4 mm.

3.2.1. Cavity Calculations

The properties of the two cavities are calculated via ReZonator, which is an easyto use freeware for cavity calculations. It assumes a Gaussian beam and usesABCD matrix propagation, which were introduced in sec. 2.3, to calculate beamsizes, shapes, the free spectral range, and more. For a full introduction to theprogram, including tutorials, and detailed features, see [7].

22

3.2 Titanium:Sapphire Laser

The cavity is modeled as seen in Fig. 3.6, and can be compared to the actual setup,Fig. 3.5. The lengths L1 through L4 are chosen to be identical to the geometry of thecavity. The Ti:Sa crystal is modeled as a Brewster cut crystal, with a refraction indexn = 1.76, whose dispersion among the two wavelengths of the lasers λ = 763 nmand λ = 848 nm is neglected in this calculation, see [11]. As a result of the physicalrealization, the beam hits the components, including the mirrors, at an angle, whichis also modeled. In order to model the prism, it is divided into three sections,Brewster angled plane-parallel crystals on the outsides, filled with a medium on theinside. All the three sections have a refractive index of SF 11, n (763nm) = 1.7669,and n (848nm) = 1.7620, and a Brewster angle of θB (763nm) = 60.492, andθB (848nm) = 60.424, see [35]. The model shown in Fig. 3.6 just shows a singlepass through the resonator, however multiple passes are calculated via the ABCDmatrix of the system.

Figure 3.6.: Modelled resonator. PMirror is the partially reflective (R=85%) mir-ror that is used for the input. The prism is modelled as three separate partswith the same refracetive index, two media at brewster angles, and a medium inbetween. Contrary to the depiction, the second half of the prism is angled theother way and modeled as such. The highly reflective, curved mirror, OMirror, ismodeled as a lense, as they are mathematically equivalent. The Ti:Sa crystal ismodelled as a Brewster angled medium with a refraction index of n = 1.76.

The cavity has an overall optical length of 300 mm, and a free spectral range of1.0 GHz. The round-trip matrices, starting with the input mirror, are listed inthe appendix sec. A.3. ReZonator models beams as propagating through planarresonators. It takes the tangential and sagittal planes into account - in tangentialplane, the beams are reflected or refracted, while the sagittal plane is normal to thetangential plane, see Fig. 3.7.The stability parameter g is calculated via

g = (A+D)2 , (3.1)

with A and D denoting the matrix elements. It is stable if |g| < 1, as seen in sec. 2.3.As the only difference in the matrices of the two wavelengths is in matrix elementsB and C, the stability parameter is wavelength independent in this case:

gT = 0.94616 (3.2)

23

Chapter 3 Laser Setup

Figure 3.7.: Tangential and sagittal optical planes. A beam (red) is reflected offan optical element (grey). For the saggital plane, the angle between the beamand optical element is not influencing the beam parameters. [7]

gS = 0.94837 (3.3)

As both stability parameters are below 1, and above -1, the resonator is stable. Thewaists ω0 of the beams are

ω0,763,T = 438µm (3.4)

ω0,763,S = 450µm (3.5)

ω0,848,T = 462µm (3.6)

ω0,848,S = 474µm (3.7)

These waists inform the decision of the beam size to which the pump beam oughtto be focused. The pump produces a beam of 7 mm in diamter, which is focused

24

3.2 Titanium:Sapphire Laser

to a diameter of (1.2 ± 0.2) mm. This is achieved via a telescope of two lenses.The telescope lenses are UV grade Fused Silica (UVFS) lenses with a focal length off = −100 mm and f = 300 mm. By having a bigger pump beam, it becomes easierto overlap the pump beam with the infrared beam inside the Ti:Sa crystal.

The overlapping itself is achieved via with a two step iterative method. It is notsufficient to overlap the beams on their way to the Ti:Sa crystal, as dispersionchanges the path through the Ti:Sa crystal sufficiently for them to stop the beamsfrom overlapping after a few millimeters inside the Ti:sa crystal. Therefore, thepump beam is first overlapped with the infrared beam at the entrance to the Ti:Sacrystal, by adjusting the second closest pump mirror. Then, the two beams areoverlapped at the exit of the Ti:Sa crystal, using the closest pump mirror. Byrepeating this process multiple times, the beams are overlapped inside the Ti:Sacrystal.

Component L[mm] n α [] r [mm]Input Mirror 15.4955 10000Length 4 60.415 1

Prism Part 1 1 1.7669/1.7620 60.492/60.424

Prism Part 2 5.386 1.7669/1.7620Prism Part 3 1 1.7669/1.7620 -60.492/ -60.424

Length 3 80.415 1Highly Reflective Mirror 15.4955 10000

Length 2 60.92 1Ti:Sa Gain Medium 20 1.76

Length 1 50 1Table 3.1.: Overview of the properties of the optical components of the cavitymodel, with lengths L, radii of curvature r, angles of incident α, and refractiveindices n. In the prism parts, the first figure is for the wavelength λ = 763 nm,and the second for λ = 848 nm.

3.2.2. Mode Matching

In principle, it is possible to shape a beam arbitrarily using just two lenses. However,in practice, the physical sizes of the components involved have to be taken intoconsideration, among other factors such as the total size of the setup, and otheroptical components. In the laser setups in this project, the additional restrictionsare a λ/2-waveplate, an optical isolator with a physical length of 12.4 cm, with anoptical beam length of about 20 cm, and a pair of mirrors used for vertical beamshuttling and input coupling. These three components need to be placed betweenthe fibre and the cavity input mirror, as seen in Fig. 3.4.

25

Chapter 3 Laser Setup

The λ/2-waveplate can be placed anywhere between the input mirror and the fibre.Abiding by these restrictions, mode matching is achieved via the fibre collimatorlense in conjunction with a second lense.The second lense is placed in its position and the beam size measured at the positionof the cavity. In order to measure the waist of the beam, the distance of the fibrecollimator lense to the fibre end is adjusted until a minimum of the beam size isfound, i.e. an adjustment in either direction results in a higher measured beam size.To adjust the size of the waist, four parameters are used: the position of the secondlense, its focal length, the focal length of the fibre collimator lense, and the positionof the cavity. As the focal lengths are only available in discrete values, though alarge range of them, they are used for coarse adjustment of the waist size. On theother hand, the positions of the second lense and that of the cavity usually havea lower range of adjustment. However, they can be adjusted continuously, and arehence used for fine tuning the coarse waist size set by the focal lengths.For the 848 nm laser, a fibre collimator lense with a focal length of f = 4.5 mm,alongside a second lense with a focal length of f = −100 mm. The 763 nm laser usesa fibre collimator lense of f = 4.5 mm and a second lense of f = −125 mm. Theycome with an anti reflective coating for a wavelength range of 650-1050 nm. Usingthese components, mode matching to the calculated cavity waists of sec. 2.2.3, rang-ing from 438 to 474 µm is achieved with less than 5% waist size mismatch and lessthan 10 cm waist position difference, measured with a Thorlabs CCD camera, whichis connected via USB to a laptop running an existing custom LabView program.In order to gauge the coupling efficiency of the seed laser into the cavity, the cavityreflection is measured. This is done via an existing photodiode circuit. Via the piezovoltage, the cavity length is scanned across several free spectral ranges. Reflectiondips are then observed and used to optimize the cavity input coupling. The ratio ofthe power reflected power off resonance to on resonance is then measured. This istaken to be an indicator of how much of the seed power is actually going into thecavity.In order to optimize the cavity reflection dip, the waist sizes are subsequently ad-justed via the collimator lense distance. This procedure is intended to adjust theinput coupling via the two mirrors in front of the cavity, optimizing the cavity re-flection dip. Afterwards, the collimator lense mount is turned in its socket by a fewdegrees, resulting in a slightly different waist size and position. With the collimatorlense mounts used, this results in a slightly different angle of the lense, divertingthe beam, potentially losing the input coupling. In order to quickly regain this cou-pling, the collimator lense mirror mount can be adjusted, to counteract the effectsof turning the collimator lense mount. However, the input mirrors must still beadjusted to gain the optimal cavity reflection dip. In case the input coupling is lostcompletely, a CCD camera is used to look at the cavity transmission through thehighly reflective second cavity mirror. Then, the beams of the first and subsequentcavity round trip have to be overlapped.

26

3.3 Laser Lock

3.3. Laser Lock

3.3.1. Hansch-Couillaud locking scheme

Mode competition is used for getting a low linewidth of the Titanium:Sapphirelasers. The continuous wave infrared laser is coupled into the cavity, and when thepump is switched on, the crystal begins to amplify all the modes. However, theinfrared laser is giving the mode it is coupled to an advantage, acting as a seed,and mode competition results in a very low linewidth, that can even be close tothe Fourier limit [22]. An alternative way to get the linewidth down would be tointroduce additional optics inside the cavity. However, this would result in morelosses, which is why the seed lasers are chosen. To perform its function of selectingone longitudinal, and one TEM mode, the seed lasers light needs to arrive inside thecavity.

The linear polarization direction of the light coming from the seed laser is adjustedvia the λ/2-waveplate to nearly match that of the preferred cavity polarization di-rection. Therefore, a small amount of the perpendicular polarization direction stillarrives at the output coupling mirror. However, the cavity finesse for this polariza-tion direction is much lower, and most of it is reflected by the mirror. It thereforemixes with the output from the light that is resonating inside the cavity. Any phasechanges between the two perpendicularly polarized beams results in an ellipticallypolarized beam. A small part of this beam is then redirected onto a λ/4-waveplatewhose fast axis is rotated to form a 45- angle with the polarization axis of a subse-quent beam splitter, as seen in Fig. 3.11. Its output is detected via two photodiodes,which will be explained in sec. 3.3.2. The signals of the photodiodes are subtracted,and sent onto a PI circuit. That signal is then fed back via a high voltage source toa piezo element at the prism inside the cavity. This is known as a Hansch-Couillaudlock [39], and can be seen in Fig. 3.8. This electronic signal is called an error signal.The sign of the error signal indicates if the cavity length is too short or too long.The sign is found experimentally, as there are multiple steps in the locking schemewhere the sign flips, for example by turning the λ/4-waveplate in a different directionto form a negative 45 angle instead of a positive one. The error signal can beoptionally inverted by using the PI circuit, in case the cavity length is adjusted inthe wrong direction. If the cavity length is too short, or too long, the transmittedintensity drops, and the error signal increases. The error signal is then used tocorrect the cavity length back towards the optimal value.

3.3.2. Locking Electronics

In order to achieve a functioning Hansch-Couillaud lock, a photo diode circuit as wellas a sample and hold amplifier have to be designed and produced. The goal is to usethe polarization properties of light reflected off the cavity in order to get information

27

Chapter 3 Laser Setup

Figure 3.8.: Hansch-Couillaud lock. The transmitted intensity through a cavity(top) shows the peaks of destructive interference, that mark the optimal cavitylength. The error signal resulting from the Hansch-Couillaud locking scheme(bottom) is similar to the first derivative of the transmitted intensity (middle).[39]

on whether the cavity is too short, or too long in comparison to the optimal length.The polarity of the error signal encodes the direction that the cavity length has tobe adjusted, while the amplitude is encoding by how much. The locking electronicsinvolved are explained in detail in this section. The designs are done with the freePCB software EAGLE version 7.1.0. An overview of the locking electronics can befound in Fig. 3.11.The photo diode circuit consists of two photo diodes. The circuit can be seen inFig. 3.9. The signal of the photodiodes is subtracted, and the result amplified byan LF 356N operational amplifier. The light incident to the photo diodes is splitby a polarizing beam splitter, so the photodiodes receive perpendicularly linearlypolarized light. For the 848 nm laser, SFH213 photodiodes are used, and SFH203 forthe 763 nm laser. The potentiometer R1 that can be seen in Fig. 3.9, has a maximumresistance of 220 kΩ, which selects the amplification of the operational amplifier. A±15 V connector supplies both the photodiodes, and the operational amplifier withpower, as well as with a grounding connection. C1 and C2 are capacitors of 1 µF,while C3 and C4 have 100 nF. C1 and C3, filter the positive power supply, while C2and C4 filter the negative power supply. This two stage approach lowers potentialpower supply noise in comparison to using just one stage.The Ti:Sa crystal is supplied with energy via a Nd:YAG pump that is pulsed at 10

28

3.3 Laser Lock

Figure 3.9.: Circuit diagram of photo diode circuit. The numbers 2, 3, 4, and 7denote the input pins of the LF 356N operational amplifier. The rest of its 8 pinsare unconnected. Created with EAGLE 7.1.0.

Hz, with a pulse duration of ≈5 ns, and a pulse energy of several mJ. It is explainedin sec. 3.4. This is important for the locking electronics, as a small portion of thepump pulse is scattered into the photodiode circuit, which is enough for it to go intosaturation, creating a false error signal. This is avoided by using a sample and hold(S&H) amplifier circuit. The operational principle is shown, and explained, at thebottom of Fig. 3.10.

The signal from the photodiode circuit is sent to the S&H amplifier circuit, thecircuit diagram can be seen in Fig. 3.10. It features an Intersil HA-5320 (SOIC)precision sample and hold amplifier, which uses a transistor-transistor logic (TTL)signal to trigger the holding stage. The TTL signal is a rectangular pulse, with avoltage height of V = 5 V for the holding stage and 0 V for the sampling. When theTi:Sa is being pumped, the resulting laser pulse saturates the photo diode circuit,which could unlock the system, if its signal is forwarded to the next stage. However,the pump laser does give out a rectangular signal at the beginning of its firingsequence, which is used to trigger the holding stage, preventing the system becomingunlocked. The rectangular pulse can be seen in Fig. 3.12, and is 4 V when the laseris not firing. In contrast, the TTL pulse that the S&H circuit is using should be 0V for the sampling stage, i.e. when the pump laser is not firing, and 5 V when thelaser is firing Therefore, a function generator (Stanford Research Systems, modelDS345) is used to convert the electronic signal from the pump laser to the shapethat the sample and hold amplifier circuit can use.

The 16 pins of the S&H amplifier can be seen in Fig. 3.10. Pin 1, the inverting

29

Chapter 3 Laser Setup

Figure 3.10.: Simplified schematic of a sample and hold circuit (bottom) versuscircuit diagram of the sample and hold circuit used in this project (top). Thesimplified schematic features an analogue input (AI) and output (AO), as well astwo operational amplifiers. In the sampling stage, the switch is closed, and the AIis connected to the AO, and is charging the holding capacitor. When the switchis opened by the S&H control, the voltage across the capacitor, and thus at theAO is the same as in the sampling stage. Created with EAGLE 7.1.0 (right), andtaken from [9] (left).

30

3.3 Laser Lock

input, is directly connected to the output, thus fixing the amplification to a gain of1. Pin 2, the non-inverting input, is connected to the error signal supplied by thephotodiode circuit. Across pins 3 and 4, a 10 kΩ potentiometer can be used to adjustany output offset. Pins 5 and 11 form the power supply of the S&H amplifier. Pins6 and 15 are ground connections that the S&H amplifier uses internally. Pin 7 isthe output, that is either at the same voltage as the positive input (sampling stage),or as it was at the beginning of holding stage. Pins 8, 9, 12, and 14 are grounded,but not connected internally in the S&H amplifier. A TTL signal received at pin 16triggers the holding stage. Pin 13 is connected to the holding capacitor, and pin 10 isconnected to a noise reducing capacitor of 100 nF, in accordance with the manual ofthe S&H amplifier (p. 5 [12]). The power lines are filtered, similar to the photodiodecircuit, via 100 nF and 1 µF to ground to protect against power line and backgroundnoise. The connections of the error signal from the photodiode, the TTL signal tothe sample or hold (S/H) control, and the output signal are transferred via BayonetNeill–Concelman (BNC) connectors, and shielded via grounding against backgroundnoise.

The capacitance of the holding capacitor influences the performance of the S&Hamplifier. A higher capacitance increases the acquisition time, while decreasing theoffset error between the sample, and holding stages, as well as decreasing the voltagedroop during the hold mode (p. 7 [12]). A 1 nF holding capacitor is used to get anoffset error of 0.5 mV, a voltage droop of < 1.0 mV/100 ms, while having an acquisitiontime of < 5µs, which is sufficient for the timescales involved.

Both sides of the photo diode, and the sample and hold amplifier circuits are printedby an office laser printer on a transparency. They then cover both sides of a photosensitive printed circuit board (PCB) produced by Kelan, part number 141-301,while being exposed to a UV light source. Subsequently, the PCB is developedto remove the photosensitive layer that was not exposed due to the black circuitdiagram on the transparency. The photoresist used is a positive tone resist. Itis then etched, removing the copper layer not covered by the photosensitive layer.Afterwards, it is washed with Acetone, after which the holes are drilled, and thesurface mounted device (SMD) components soldered on. Metal boxes are used tohouse the PCBs, and are prepared via holes for BNC connectors, ±15 V powersupply connectors, and mounting holes. These connectors are then linked to thePCB via cables which are soldered on both sides. For use in both lasers, two photodiodes, and two sample and hold amplifier circuits are built. These are shown inFig. 3.13.

After the sample and hold amplifier, the signal is fed into an existing two channelproportional and integrating (PI) circuit. This stage also allows for offset correctionsfor both an input and output offset, as well as adjustments to the gains of theproportional circuit, the integrating circuit, and a master gain.

The PI circuit sends its signal to a three channel SVR 150/3 high voltage powersupply produced by Piezomechanik GmbH, with a maximum current of 30 mA.

31

Chapter 3 Laser Setup

848.8 nm Cavity

4

S&H

LockingElectronics

Figure 3.11.: An overview of the locking electronics. The polarization of the lightreflected off and transmitted out of the cavity is used to create an error signal inthe photodiode circuit. The sample and hold (S&H) amplifier circuit prevents falsereading from the error signal due to saturation. The proportional and integrating(PI) circuit manipulates the error signal for a more stable lock. The high voltageamplifier serves as a master gain, sending a feedback signal to the cavity piezo,which changes the cavity length, influencing the polarization of the reflected light,closing the feedback loop.

There, another master gain can be set for up to a factor of 30. The amplified signalis then sent to a piezo which moves a prism that changes the cavity length, thusaffecting the cavity resonance and the produced error signal, closing the feedbackloop. For every 1 µm the piezo moves, the effective round trip length changes by0.84 µm, derived from geometrical considerations and affected by its precise angleand refractive index. To be as precise as λ/40, which is a guide for the requiredprecision to hold the cavity on lock, the piezo has to move less than 23 nm. In orderto have a swing of more than 4 free spectral ranges of the cavity, it needs to movemore than 3.7 µm.

A mirror-shifter STr-25, also from Piezomechanik GmbH is chosen to fulfill thispurpose. The high voltage power supply can deliver a voltage of -30 to 150 V, whichspans the whole range that the piezo can move. The piezo has a capacitance ofC = 2.5µF, while the power supply is specified to be able to adjust the voltage by∆U = 10 V at a frequency of f = 60 Hz. However, much smaller adjustments ofonly 0.1 V are required to keep the laser locked. Therefore, it is unclear at present ifthis is the factor limiting the bandwidth of the whole feedback system at around 800Hz, or if this limitation originates elsewhere. A high voltage supply with a highercurrent, such as an LE 150/100 EBW from Pizeomechanik GmbH, featuring a peakcurrent of 1200 mA, can perform voltage adjustments of ∆U = 10 V at a frequencybeyond 10 kHz. That would likely put it beyond the resonance frequency of 15 kHzof the piezo at the small voltage adjustments required. This would result in the

32

3.3 Laser Lock

-4

-2

0

2

4

6

8

10

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0 5 10 15 20 25

Pu

mp

TTL

Sig

nal

[V

]

Erro

r Si

gnal

[V

]

Scan time [μs]

-4

-2

0

2

4

6

8

10

-0.25

-0.2

-0.15

-0.1

-0.05

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0.1

0 5 10 15 20 25

Pu

mp

TTL

Sig

nal

[V

]

Erro

r Si

gnal

[V

]

Scan time [μs]

Figure 3.12.: Sample and hold circuit test. The error signal (red) is measuredbefore (top) and after (bottom) the circuit. The pump sends out an electronicsignal when it is firing (purple).

33

Chapter 3 Laser Setup

Figure 3.13.: Photo diode (top), and sample and hold amplifier (bottom) circuits

bandwidth of the system being limited by the piezo.

As changes affecting the length of the cavity have to be actively countered via thelocking system, see sec. 3.3, it is helpful to have as high a stability as as possible. Thesetup is created on a breadboard with rubber feet on top of an air cushioned opticaltable from Thorlabs. In order to maximize the mechanical stability of the cavity, itsheight above the optical table has to be minimized. The limiting factor in the cavityis the holder of the prism, as it has a distance of 25.4 mm to its bottom, without apedestal. The part numbers, manufacturer, and other details of all the componentsinvolved the the optical setup, as well as the mirror holders and mechanical spacers,are included in the appendix sec. A.5.

In order for the laser to be locked, the cavity reflection dip first has to be optimized.After that has been done, the error signal has to be calibrated. This is first done viathe wedge plate mirror mount, which is adjusted until a strong signal is received onboth of the photodiodes. A good way of achieving that is to scan the cavity length

34

3.3 Laser Lock

across several free spectral ranges again. The received signal from both the cavityreflection and the error signal is compared - the result is expected to look similar toFig. 3.8. The angle of the λ/4-plate in front of the error signal circuit combined withadjustments to the wedge plate angles are used to achieve this.The PI circuit has both an input offset and output offset adjustment. The lengthwhere the error signal is zero, is the length where the feedback system locks to.The PI circuit input offset adjustments are used to choose that point precisely. Theoutput offset is used to counter any unwanted effects of the PI circuit itself.The output from the PI circuit is then connected to the high voltage amplifier,where the master gain is gradually increased until the measured cavity reflectionvoltage goes down. If the voltage is equivalent to the minimum voltage of thereflection dips, a lock is achieved. If the voltage is higher, the lock is achieved on apotential secondary reflection dip of a higher TEM mode, which can be counteredby manually turning the piezo voltage close to the reflection dip to be locked onto,and then turning up the master gain.Turning up the master gain further, the feedback system goes into oscillation, asshown in Fig. 3.14. The frequency of oscillation is a measure of the bandwidth of thefeedback system in total. It is limited by the part with the lowest bandwidth, andis at ≈ 800 Hz. The proportional and integral gain on the PI circuit are optimizedin tandem with the master gain for highest stability of the lock. Vibrations are veryvisible on the cavity reflection intensity at this stage. While sound vibrations werenot found to adversely affect the system beyond their duration, sudden vibrations ofthe table sometimes send the system lock a free spectral range to the next resonance,or completely out of lock. Shielding from external vibrations is therefore importantfor extended operation. The system is found to stay in lock for at least an hour,potentially many hours.

35

Chapter 3 Laser Setup

-0.125

-0.1

-0.075

-0.05

-0.025

0

0.025

0.1

0.125

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0.2

0.4

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1

1.2

1.8

2

0 5 10 15 20 25

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-0.025

0

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1

1.2

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-0.05

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Erro

r si

gnal

[V

]

Ref

lect

ion

Inte

nsi

ty o

n P

ho

tod

iod

e [V

]Scan time [ms]

Figure 3.14.: Oscillation of the locking system. The cavity reflection intensity(blue), and the error signal (red) is shown over time. The bottom graph showsthe locked system, with a very high master gain, resulting in an oscillation aroundthe optimal cavity length. As the cavity reflection intensity oscillates between twodifferent peaks, there is an offset in the error signal. This can be countered viathe input offset adjuster of the PI circuit. In the middle graph, the master gain isgreatly reduced, but oscillation is still occuring. The top graph shows the mastergain at a value just below where the oscillations start - this results in a cavitywith the most stable lock.

36

3.4 Pump laser setup and Damage Thresholds

3.4. Pump laser setup and Damage Thresholds

The two Ti:Sa crystals are pumped via a single frequency doubled Nd:YAG laser(Continuum, model Surelite SL II-10) at wavelength of 532 nm. It has a maximumpump energy of 300 mJ per pulse, a repetition rate of 10 Hz, and a pulse durationof 4-6 ns [8]. An overview of the pump setup, as well as the other parts of thelaser, can be seen in Fig. 3.15. Its pulse first passes through a telescope, and a λ/2-waveplate, then a polarizing beam splitter. The combination of the λ/2-waveplateand the polarizing beam splitter allows an arbitrary division of the pulse energyacross the two Ti:Sa crystals. After the beam is split, each partial beam encountersa non-polarizing beam splitter, splitting the original pump beam into four individualbeams that are directed to enter the two Ti:Sa crystals from both their sides. Thisis done to ensure an even distribution of the pump energy across the Ti:Sa crystals.In order to redirect the beam from the pump laser to the Ti:Sa crystals, a total of12 mirrors are used.The laser induced damage threshold (LIDT) of the various optical components mustbe examined to find the maximum power per pulse that the pump can safely deliver.The pulse durations and wavelengths of the testing conditions t0 and λ0 for thecomponents generally differ from the target conditions t1 and λ1. However, theadjusted maximum pulse energy E1 can be approximated by accounting for thewavelength and pulse lengths differences via the following formulae [3]:

E1 = E0

√λ1

λ0(3.8)

E1 = E0

√t1t0

(3.9)

However, this only works as an approximate guideline. The pulse lengths for whichequation 3.9 holds, works only between 1 - 100 ns, due to different damage mech-anisms taking effect at both higher, and lower timescales [3]. Combining the twoequations yields

E1 = E0

√λ1

λ0· t1t0

(3.10)

As the pump laser has a maximum energy of 300 mJ per pulse, at a repetition rateof 10 Hz, and a pulse duration of 4-6 ns, a safe pump pulse energy has to be found.For the following calculations, a pulse duration of 6 ns, and a beam diameter of 7

37

Chapter 3 Laser Setup

848.

8 nm

Cav

ity

Fre

quen

cyC

onve

rsio

n84

8.8

-> 2

12.2

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Pum

p D

istr

ibut

ion

Nd:

YA

G P

ump

Atte

nuat

ion

& F

ocus

ing

PB

S

Bea

mD

ump

PB

S

To 7

63.8

nm

lase

r

BS

TT

L P

ulse

4

2

See

d S

etup

284

8.8

nm

Lase

r

Far

aday

Isol

ator

ff

S&H

22

PI L

ock

Wav

emet

er

To E

xper

imen

t

Lock

ing

Ele

ctro

nics

Ho

ld

Figure 3.15.: Overview of laser system with just one of the lasers shown. The widthof the green pump beam demonstrates the intensity of the beam, not its physicalsize. However, it is not to scale, for example the beam can be attenuated anddistributed by varying amounts. A complete overview that includes the secondTi:Sa laser is shown in Fig. 3.16. Created using Inkscape 0.91 and a componentlibrary [21]

38

3.5 Overview of Laser System

mm before and 1 mm after focusing will be assumed. Furthermore, it has a flattop intensity profile, in contrast to a Gaussian one, which influences the maximumpulse energy by a factor of 2 [3]. The maximum energy that can theoretically bedelivered by the pump laser is much too high for the optics to handle after focusing,and their lowest damage threshold is derived as follows. The optical componentsthat the λ = 532 nm pump light passes through are listed in Tab. 3.2. Conversionsof the LIDT to the wavelength and timing used in this project is done via equation3.10. The converted LIDT1 can be seen in Tab. 3.3.

LIDT0 J/cm2 λ0 [nm] t0 [ns] LIDT1 J/cm2

Laser Line Mirrors 5 1064 8 3.1First telescope lense 4 532 10 3.1Second telescope lense 4 532 10 3.1

λ/2-waveplates 10 1064 10 5.5Polarizing beam splitter 6 1064 10 3.350:50 beam splitters 5 1064 8 3.1

Pump separation mirrors 1 532 10 0.77Cavity mirrors 1 532 10 0.77Ti:Sa crystals 270 532 5 296

Table 3.2.: Conversion of LIDT to a wavelength of λ1 = 532 nm, and a pulseduration of t1 = 6 ns. The LIDT0 are the unconverted values taken from theirspecifications, which can be found via their part numbers in the appendix sec. A.5,except for the Ti:Sa crystal which is taken from [14].

Accounting for the sizes of the beams, the LIDT1 is used to calculate the maximumpulse energy, see Tab. 3.3. The maximum pulse energy is bottle-necked by thecavity mirrors, through which only 3 mJ per mirror are allowed, staying at 50% ofthe LIDT. Using each laser separately, a pump energy of 6 mJ is therefore allowed.At simultaneous operation, the pump energy can be increased to 12 mJ, assumingthe energy is distributed evenly onto both lasers. However, the pulse energies atsimulatenous operation reaches 50% of the LIDT of multiple other optics as well,making component failure more likely when both lasers are used.

3.5. Overview of Laser System

Sapphire changes its refractive index from n = 1.7718 for 532 nm, to n = 1.7612for 763.8 nm, or n = 1.7589 for 848.8 nm [34]. The titanium doping of the sapphireshould only change its dispersive properties slightly. While the change in refractiveindices is not huge, its effect has to be considered when overlapping the infrared withthe pumping beam. If the two are just overlapped via the same angle of incidenceand position of entry, this dispersive effect is enough to cause the two beams to be

39

Chapter 3 Laser Setup

LIDT1 J/cm2 ω [mm] Emax [mJ] Power I/I0

Laser Line Mirrors 3.1 7 & 1 1200 & 24 1/4 to 1First telescope lense 3.1 7 1200 1Second telescope lense 3.1 1 24 1

λ/2-waveplates 5.5 1 43 1/2 to 1Polarizing beam splitter 3.3 1 26 150:50 beam splitters 3.1 1 24 1/2 to 1

Pump separation mirrors 0.77 1 6.1 1/2 to 1/4

Cavity mirrors 0.77 1 6.1 1/2 to 1/4

Ti:Sa crystals 296 1 1480 1/2 to 1/4

Table 3.3.: Overview of maximum pulse energies Emax dependent on the beamdiameter. For this calculation, a Gaussian beam shape is assumed. The laserline mirrors serve both before and after the beam shaping. The power denotesthe fraction of the pump beam that is arriving at the components. It is variable,as the pump is divided among the two Ti:Sa lasers by an arbitrary amount, andsubsequently divided approximately evenly among the two beams entering theTi:Sa from opposite sides.

too far apart for lasing. Therefore, a different method has to be used; aligning thebeam via both the points of entry and exit. This accounts for any dispersive effectsinside the Ti:Sa crystal via different angles of incidence and exit. Depending on theexact distances of the optics the infrared and the pumping beam have to pass, theslight difference in angles might become significant enough for one beam to passnext to a piece of optics, while the other one hits it centrally.An overview of the whole setup can be seen in Fig. 3.16. The bottom left partbelongs to the 848.8 nm setup, while the top right part is the twin setup for 763.8nm. The pumping setup can be seen in the top left and top middle part. Forclarity, the overview is divided into functional sections, which have individuallybeen introduced in earlier chapters. The optics involved in the frequency conversionare not set up yet, but are shown for completeness of the setup.The beam eminating from the Nd:YAG pump is linearly polarized, and can beattenuated with the use of a λ/2-waveplate, that changes the angle of the linearlypolarized light, see Fig. 3.16. This angle governs how much of the light is transmittedthrough a subsequent PBS, or reflected into a beam dump.

40

3.5 Overview of Laser System

848.

8 nm

Cav

ity

Fre

quen

cyC

onve

rsio

n84

8.8

-> 2

12.2

nm

Pum

p D

istr

ibut

ion

Nd:

YA

G P

ump

Atte

nuat

ion

& F

ocus

ing

PB

S

Bea

mD

ump

PB

S

BS

BS

TT

L P

ulse

4

2

See

d S

etup

284

8.8

nm

Lase

r

Far

aday

Isol

ator

ff

S&H

S&H

22

2

PI L

ock

Wav

emet

er

2

See

d S

etup

2

763.

8 nm

Las

erF

arad

ayIs

olat

orf

fP

I Loc

k

Wav

emet

er

To E

xper

imen

t

763.

8 nm

Cav

ity

Lock

ing

Ele

ctro

nics

Ho

ld

Fre

quen

cyC

onve

rsio

n76

3.8

-> 2

54.6

nm

Figure 3.16.: Overview of the complete laser system. The width of the green pumpbeam demonstrates the intensity of the beam, not its physical size. However, it isnot to scale, for example the beam can be attenuated and distributed by varyingamounts. Created using Inkscape 0.91 and a component library [21] 41

4. CharacterisationAs mentioned in sec. 3.4, the laser that is used to pump both Ti:Sa crystals simul-taneously is a frequency doubled Nd:YAG laser from Continuum, model Surelite SLII-10. It has a maximum energy of 300 mJ per pulse, at a repetition rate of 10Hz, and a pulse duration of 4-6 ns [8]. However, it is optimized to emit its thirdharmonic, 355 nm, because of being used in another experiment. Therefore, its max-imum energy at the second harmonic, 532 nm, is substantially reduced, however stillbeyond the damage threshold of the optics. The pulse energy can be changed viaa built-in Q-switch delay after the flashlamp inside the laser has fired. In order toensure that only a pulse energy below the damage threshold of the optics is used,the pulse energy as a function of the Q-switch delay needs to be calibrated.To this end, an ES220C pyroelectric detector from Thorlabs, connected via a PM100USBEnergy Meter Interface to a Thorlabs measurement suite is used. The pyroelectricdetector itself comes with a NIST calibration certificate, taking into account thewavelength dependent absorption of the ceramic detector coating. A Q-switch delayof 180 µs is the highest delay recommended for safe operation of the laser, while aQ-switch delay of less then 100 µs falls below the detection threshold of the pyro-electric detector. The measured calibration can be seen in Fig. 4.1. For each point,600 shots are recorded, and the standard deviation is taken to be the measurementerror.As calculated in sec. 3.4, the maximum pulse energy that the pump can safely deliver,without damaging the optics, is 6 mJ for operating a single Ti:Sa laser, and 12 mJfor simultaneous operation. The corresponding Q-switch delays are 112 µs, and 120µs.In order to measure the coupling efficiency of the seed laser into the cavity, the cavityreflection dip is measured. For 848.8 nm, a typical maximum coupling efficiency of 52% is reached, while for 763.8 nm, 42 % are typical, however peak coupling efficienciesof up to 69 % are also observed. The input coupling has to be reoptimized almostevery day, as the coupling efficiencies are found to drop over time. The cause for thisis not investigated, but could be due to a slow drift of the mirror mounts involved.The cavity finesse of the 848 nm laser Ti:Sa laser cavity is measured to be 20±5,which is in line with [41]. However, subsequent measurements of the 848 nm cavity,have shown a decrease in the cavity finesse to 12±2. The cause of this is underinvestigation, and might be due to particles of dust, or thin layers of dirt on theoptics inside the cavity. The cavity finesse of the 763 nm laser is placed at 32±5,surpassing the finesse measured in [41].

43

Chapter 4 Characterisation

0

10

20

30

40

50

60

70

100 110 120 130 140 150 160 170 180

Ener

gy p

er p

uls

e [m

J]

Q-switch delay [μs]

Figure 4.1.: Pulse energy of the Nd:YAG laser as a function of the Q-switch delay.Q-switch delays of 100-125 µs are of particular interest, as their pulse energies aresimilar to the energies allowed by the damage thresholds, see sec. 3.4.

Due to the design of the setup, the 50:50 beam splitters are not used at their designedpolarization, unpolarized light, but with polarized light. This introduces an unevendistribution of the pump beam. It could be countered by using a 70:30 beam splittersuch that the polarization counteracts the uneven distribution, coming reasonablyclose to a 50:50 beam splitter. Alternatively, additional λ/2-waveplates after thebeam splitter can be introduced. The polarization in front of the beam splitters canthen be adjusted to counteract the uneven distribution, while after the 50:50 beamsplitters it is adjusted for maximum transmission into the Brewster angle cut Ti:Sacrystal.

A guide for what timescales the laser pulses in this project can be expected toappear in is needed. For [22], the delay between the pump pulse peak and theIR peak (buildup time) was 65 ns, with the IR peak full width at half maximum(FWHM) being 35 ns, as shown in Fig. 4.4. [41] found the buildup time changingfrom 30 ns at around 7 mJ total IR output energy converging to 21 ns for higherpumping levels. They also found the pulse length to be changing from about 26 nsat 2 mJ to be converging to 11 ns. Therefore, the pulse of 848.8 nm, or 763.8 nm,is expected to arrive in the timeframe of 10-100 ns after the pump pulse.

The delay of the electronic signal from the pump falling to half its voltage to thepumping beam reaching its maximum intensity varies with the Q-switch delay. Fora Q-switch of 115 µs, the time is 155±5 ns, and for a Q-switch of 180 µs, it is 125±5ns. This could be used to differentiate the pump pulse from the IR pulse, if thepump pulse is completely filtered from arriving at the photodiode.

44

Characterisation

0

200

400

600

800

1000

1200

1400

1600

1800

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 5 10 15 20 25

Cal

cula

ted

ro

un

d t

rip

len

gth

ch

ange

[n

m]

Ref

lect

ion

Inte

nsi

ty o

n P

ho

tod

iod

e [V

]

Scan time [ms]

Figure 4.2.: Cavity length scan, measuring reflection intensity (blue) and calcu-lated length change (red)

The pump is fired into the locked cavities, however, no additional reflection intensityspike is found in the given timeframe, nor in other timeframes up to 25 µs. Instead,fluorescence is found, which should be a precursor to the actual lasing. A possiblecause for the lack of detectable infrared lasing may be too low a gain with respectto the cavity losses.The cavity finesse for the 848.8 nm laser cavity is significantly lower than expected.This could be due to a layer of dust or residue on the cavity mirrors, prism, or Ti:Sacrystal. A higher cavity finesse yields lower losses per round trip. That means thatthe gain medium can achieve the same performance with a lower gain. Alternatively,a higher cavity finesse results in a lower lasing threshold, as the same gain then canbe higher than the lowered cavity losses. Cleaning of these optics very much is nota trivial task. However, the cavity finesse for the 763.8 nm laser is even higher thanthat of [41], but no lasing could be achieved in either laser.The damage threshold of the optics limits the energy per pulse to 6 mJ per laser, ascalculated in sec. 3.4. However, the lasing thresholds for similar setups can be 6.2mJ (p. 3 [41]). If the lasing threshold is similar, or even higher than in that setup,then the damage threshold of the optics are too low for the lasing to be achieved.One possible way to increase the pulse energy is explained in this paragraph. TheLIDT is mainly a concern for optics that come after the beam has been focused.In order to minimize the number of optical elements that are exposed to pumplaser intensities close to their damage threshold, the beam can be focused after the

45

Chapter 4 Characterisation

-0.125

-0.075

-0.025

0.025

0.075

0.125

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 5 10 15 20 25

Erro

r Si

gnal

[V

]

Ref

lect

ion

Inte

nsi

ty o

n P

ho

tod

iod

e [V

]

Scan time [ms]

Figure 4.3.: Cavity length scan, measuring reflection intensity (blue) and resultingerror signal (red)

beam distribution. That way, the λ/2-waveplates, polarizing beam splitters, and50:50 beam splitters will be subjected to approximately 1/50th the intensity whencompared to the focused beam. This will increase the lifetime of those components,however at the cost of needing three additional lenses of each focal length, f = −100mm and f = 300 mm, to perform the beam focusing after the pump distribution.This can even be taken further, by placing the lenses as close to the cavity aspossible. In case of the highly reflective cavity mirrors, the f = −100 mm lenses canbe placed directly in front of them. Then, by adjusting the distance of the f = 300mm lenses, the beam is brought to focus behind the Ti:Sa crystal, resulting in awider beam spot on the highly reflective cavity mirrors, see Fig. 4.6. As the pulseenergy is limited by the damage threshold of the cavity mirrors, having even a smallincrease in spot size on these mirrors directly leads to a higher allowed pulse energy.In case of the partially reflective cavity mirrors, the lenses can only be placed afterthe beam separation, decreasing the effectiveness of this method. However, overall,this method could significantly increase the allowed pulse energy.

In the current version of the setup, the polarizing beam splitters work at an angleof 56, and a second one is used to get back to angles of 45 and 90. However, itis found that the polarizing beam splitter plates introduce additional reflections ofsubstantial energy. Subsequently, an iris is used to clip the additional reflections,reducing the available energy in the beam. By using a polarizing beam splitter cubeinstead of a plate, the design of the pump distribution can be simplified, as seen in

46

Characterisation

Figure 4.4.: Reflected intensity of a cavity. The pump pulse is registered 65 nsbefore the Ti:Sa lasing peak.[22]

Fig. 4.7. Only one PBSC is needed instead of two PBS and a pump mirror. Thiscomes at the cost of a lower LIDT, which is 2 J/cm2 instead of 6 J/cm2. However, byfocusing the beam after the pump distribution, the pulse energies are still limitedby the cavity mirrors, with a large safety margin on the PBSC.One way is to turn the λ/2-waveplate in front of the BS such that the beam isexactly 45 between s- and p- polarization. With two additional λ/2-waveplates perlaser after the BS, the polarization is then adjusted to match the Brewster angletransmission polarization on the Titanium:Sapphire crystals. However, this case isdifferent to just using unpolarized light with a random distribution of polarizationdirections, therefore the light might have additional unwanted polarization effects,such as a circularly polarized portion.Alternatively, a 50:50 BS cube can be used, for example a Thorlabs BS010. Thiswould, however, reduce the LIDT by a factor of 20, making this only a viable optionif the beam is focused after this element. In that case, the LIDT would result in amaximum pulse energy of 59 J, and thus a practical maximum of 30 J.It would also be possible to use a 70:30 or 30:70 BS plate, depending on the definitionof the polarization directions, and intentionally using the inherent asymmetry of theoptical element. This way, the effects of the polarization counteract the asymmetryof the BS plate, creating an approximately even distribution. For a Thorlabs BSS1070:30 BS, using 532 nm linearly polarized light, the split ratio is 51:46, which includeslosses. This is close enough to an optimal 50:50 beam splitter to fulfill its purposeof an even distribution. The LIDT is reduced to just 1 J/cm2, however, a highbeam energy can still be achieved by placing the telescopes after the beam splitters.Overall, this method has the advantage of being the cheapest one, with a sufficientperformance, while still having a large safety margin to the maximum pulse energy.

47

Chapter 4 Characterisation

-4

-2

0

2

4

6

8

10

-1

0

1

2

3

4

5

6

0 5 10 15 20 25

Pu

mp

TTL

Sig

nal

[V

]

Ref

lect

ion

Inte

nsi

ty o

n P

ho

tod

iod

e [m

V]

Scan time [μs]

Figure 4.5.: TTL signal from the pump (blue), and reflected intensity off the cavity(red). The intensity is measured with an AC coupled photodiode, therefore thebackground reflectance is not measured. A fitted exponential decay (green) has adecay time constant of 4.2 µs.

Nd:YAG Pump Attenuation & Focusing

PBS

BeamDump

2

848.8 nm Cavity

Nd:YAG Pump Focusing

Figure 4.6.: Beam focusing before (left) and after beam splitting (right). Thecurrently used method (left) has the advantage of just using one pair of lenses,while the suggested method leads to an increased pulse energy, as the beam isslightly wider at the cavity mirrors, which limit the maximum pulse energy viatheir low damage threshold.

48

Characterisation

Pump Distribution

PBS

BS

BS

22

Pump Distribution

PBS BS

BS

2

2

Figure 4.7.: Pump distribution via polarizing beam splitter plates (left) and cube(right). The plates add extra reflections of substantial energy parallel to the mainbeam, as well as a 56 angle of incidence, which is corrected via a second polarizingbeam splitter plate, and an additional mirror, while the cube has a lower damagethreshold.

49

5. Summary and Outlook

In this project, two seeded Titanium:Sapphire lasers have been set up. Their wave-lengths were 848.8 nm, and 763.8 nm, which were chosen to enable frequency con-version towards wavelengths of 212.2 nm, and 254.6 nm, corresponding to a 2+1’REMPI ionization process of molecular nitrogen N2. The molecular ion N+

2 will thenbe used to do high resolution spectroscopy, to get an upper bound on the change ofthe proton-to-electron mass ratio µ. To this end a specific state was needed, X2Σ+

g .To efficiently, and reliably, achieve the required state, a low linewidth of the laserswas required.

A theoretical background was given on laser cavities, Gaussian optics, Titanium:Sapphireas a gain medium, and the Faraday isolator. The laser setup was explained in detail.

The locking electronics were introduced, which consisted of the custom photodiodecircuit, producing the error signal, a custom sample and hold amplifier, protectingthe system from intensity fluctuation noise when the pump pulse arrives, a PI circuit,shaping the error signal for maximum locking stability, a high voltage amplifier, anda piezo element, which changed the cavity length by moving the attached prism.

The mode properties of the laser cavity were calculated, which were used for modematching, and damage threshold purposes. The maximum pump intensity allowedby the damage threshold of the optics was calculated. The existing diode seed laserswere introduced, along with their fibres. The procedure for and quality of the modematching was presented. The feedback system was introduced, a stable laser lockwas achieved, and oscillation looked at.

The maximum pulse energy is currently restricted by the cavity mirrors to 6 mJper laser. However, with both lasers running, 50% of the LIDT of several otheroptics involved in the pump distribution are also reached, and thus creating a riskfactor. By placing the beam focusing optics after the beam distribution, the pumpdistribution LIDTs are no longer a concern.

The pump distribution among the two lasers has the problem of creating additionalreflections of substantial energy. These can be countered by using a polarizingbeam splitter cube, whose reduced LIDT is countered via the previous suggestionof placing the beam focusing optics after the beam distribution.

The 50:50 beam splitters are distributing the beam very unevenly, at about 33:67.Three ways to get around this problem are suggested - using additional λ/2-waveplatesafter the BS, using a BS cube instead of a BS plate, or using a 70:30 BS plate.

51

Chapter 5 Summary and Outlook

Infrared lasing has not been achieved in this project, despite an extended search forthe cause. One possible cause is a pump pulse energy below the lasing threshold ofthe Ti:Sa lasers. However, several improvements, as detailed above, can be made,which could increase the damage threshold and maximum pump pulse energy of thissetup.

52

Acknowledgments

I would like to express my very great appreciation to Dr Matthias Keller for givingme the opportunity to work on this fascinating subject, and for his assistance, andguidance.I would like to offer my special thanks to Markus Vogt, Jack Morphew, StephenBegley, and Amy Gardner for many fruitful discussions and valuable guidance.Furthermore, I would like to thank Dr Hiroki Takahashi, William Groom, and EzraKassa for their help.To all members of the ITCM group, who have provided a wonderful working atmo-sphere, I extend my sincere thanks.Lastly, I would like to express my sincere and very special thanks to Anna Moosbauerfor providing absolutely invaluable feedback and support.

53

Bibliography

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[2] Gaussian optics and paraxial ray matrices. 2015.03.29. Available from: http://seghil.github.io/OpenCavity/before_starting.html.

[3] Laser Induced Damage Threshold Tutorial. 2015.03.08. Available from: https://www.thorlabs.com/tutorials.cfm?tabID=27503.

[4] Optical Isolator Tutorial. 2015.03.21. Available from: http://www.thorlabs.de/newgrouppage9.cfm?objectgroup_id=2996.

[5] Polarization-Maintaining FC/APC Fiber Optic Patch Cables. 2015.03.21.Available from: http://www.thorlabs.de/newgrouppage9.cfm?objectgroup_id=3345&pn=P3-630PM-FC-10.

[6] Polarization Maintaining Patchcords. 2015.03.21. Available from: http://shop.ozoptics.com/polarization-maintaining-fiber-patchcord?pagenumber=3.

[7] reZonator. 2015.02.25. Available from: http://www.rezonator.orion-project.org/.

[8] Surelite Series. 2015.03.14. Available from: http://www.continuumlasers.com/index.php?option=com_content&view=article&id=646&Itemid=574#liintroTab.

[9] A simplified diagram of a sample and hold circuit., June 2009. Available from:http://commons.wikimedia.org/wiki/File:Sample-hold-circuit.svg.

[10] Fundamental Physical Constants, Committee on Data for Science and Technol-ogy, 2010. 2015.04.02. Available from: http://physics.nist.gov/cgi-bin/cuu/Value?mpsme.

[11] Ti:Sapphire Crystal, 2010. 2015.03.03. Available from: http://www.redoptronics.com/Ti-Sapphire-crystal.html.

[12] 1 Microsecond Precision Sample and Hold Amplifier HA-5320 Datasheet,April 2013. Available from: www.intersil.com/content/dam/Intersil/documents/ha-5/ha-5320.pdf.

[13] Chong G.C.Y. Ang, K.H. and Y. Li. PID control system analysis, design, andtechnology. IEEE Trans Control Systems Tech, 13(4):559–576, 2005. Availablefrom: http://eprints.gla.ac.uk/3817/1/IEEE3.pdf.

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[15] John Baez. How Many Fundamental Constants Are There?, April 2011.2015.04.02. Available from: http://math.ucr.edu/home/baez/constants.html.

[16] Jacob D. Bekenstein. Fine-structure constant: Is it really a constant? PHYS-ICAL REVIEW D, 25(6):1527 – 1539, September 1982.

[17] Frank J. Duarte, editor. Dye Laser Principles: With Applications. AcademicPress, 1990.

[18] H. Schnatz Chr. Tamm S. Weyers R. Wynands E. Peik, B. Lipphardt. Labora-tory Limits on Temporal Variations of Fundamental Constants: An Update.[physics.atom-ph], 2006. Available from: http://arxiv.org/abs/physics/0611088v1.

[19] Sebastien Forget. Hermite-Gaussian Modes, September 2007. 2015.01.27.Available from: http://www.optique-ingenieur.org/en/courses/OPI_ang_M01_C03/co/Contenu_13.html.

[20] A. G. Fox and Tingye Li. Resonant Modes in a Maser Interferometer. BellLabs Technical Journal, 1960.

[21] Alexander Franzen. ComponentLibrary (a vector graphics library for illus-trations of optics experiments). 2015.03.29. Available from: http://www.gwoptics.org/ComponentLibrary/.

[22] Thomas Walther Matthew P. Larsen Edward S. Fry. Generation of Fourier-transform-limited 35-ns pulses with a ramp-hold-fire seeding technique in aTi:sapphire laser. Applied Optics, 40(18):3046 – 3050, June 2001.

[23] M. T. Murphy V. V. Flambaum R. F. Carswell J. K. Webb, J. A. King andM. B. Bainbridge. Indications of a Spatial Variation of the Fine StructureConstant. Phys. Rev. Lett., 107(19):191101, October 2011. Available from:http://dx.doi.org/10.1103/PhysRevLett.107.191101.

[24] V. V. Flambaum V. A. Dzuba J. D. Barrow C. W. Churchill J. X. ProchaskaJ. K. Webb, M. T. Murphy and A. M. Wolfe. Further Evidence for Cosmo-logical Evolution of the Fine Structure Constant. Phys. Rev. Lett., 87(9):91301,August 2001.

[25] Christopher W. Churchill Michael J. Drinkwater John K. Webb, VictorV. Flambaum and John D. Barrow. Search for Time Variation of the FineStructure Constant. Phys. Rev. Lett., 82(5):884, February 1999. Available from:http://dx.doi.org/10.1103/PhysRevLett.82.884.

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[28] Roland Thissen Veronique Vuitton Christian Alcaraz Pascal Pernot NadiaBalucani Piergiorgio Casavecchia Andre Canosa Sebastien Le Picard Jean-Christophe Loison Zdenek Herman Jan Zabka Daniela Ascenzi Paolo TosiPietro Franceschi Stephen D. Price Odile Dutuit, Nathalie Carrasco andPanayotis Lavvas. CRITICAL REVIEW OF N, N+, N2+, N++, And N2++MAIN PRODUCTION PROCESSES AND REACTIONS OF RELEVANCETO TITAN’S ATMOSPHERE. The Astrophysical Journal Supplement Series,204(2):20, January 2013. Available from: doi:10.1088/0067-0049/204/2/20.

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[31] Ruediger Paschotta. Finesse - Encyclopedia of Laser Physics and Technolog.2015.04.20. Available from: http://www.rp-photonics.com/finesse.html.

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58

A. Appendix

A.1. Population in a 2 level system

A laser mainly utilizes two energy levels of a gain material - incoming photonsstimulate an emission of additional photons from electrons that jump to the lowerenergy level. A necessary condition for building a laser is having a higher populationin the upper than in the lower energy level (p. 244 [42]). This is known as apopulation inversion, as generally the electron population occupies the lowest energylevels. In order to achieve a population inversion, one might think it is enough tosimply pump such a two-level system. The equation that governs the rate of changeof the ground population dNg

dtis

dNg

dt= Wn+BNeρ−BNgρ+ ANe (A.1)

with the pumping rate W , the Einstein coefficients for the spontaneous A, andstimulated B emission, the excited population Ng, and the population inversionn = Ne − Ng. Solving for the population inversion, while assuming the stationarycase dNg

dt= 0, yields

n = − ANe

W +Bρ, (A.2)

which is always negative. Therefore for even an arbitrary amount of external pump-ing, a population inversion is not possible, due to the pump itself stimulating emis-sion at a faster rate than the upper population can be pumped.

A.2. Population in a 3 level system

As seen in sec. A.1, a population inversion in a 2 level system is not achievable.However, by adding an additional level, in this case being energetically above the

59

Chapter A Appendix

excited state, a population inversion is achievable. The populations of the energylevels can be calculated via (p. 246 [42] & p. 507 [36])

dNi

dt= WNg −WNi − AigNi − SNi (A.3)

dNe

dt= SNi +BρNg −BρNe − ANe (A.4)

dNg

dt= WNi −WNg + ANe +BρNe −BρNg + AigNi (A.5)

Assuming a stationary condition, the derivatives of the populations dNi

dt, dNe

dt, and

dNg

dtare all 0. Furthermore, we assume that s W,Aig, A. Therefore terms in

equations A.3 and A.5 can be neglected:

0 = WNg − SNi (A.6)

0 = −WNg + ANe +BρNe −BρNg (A.7)

Using equation A.6 in equation A.4, one term can be interchanged:

0 = WNg +BρNg −BρNe − ANe (A.8)

Defining n = Ne − Ng as the population inversion, and subtracting equation A.8from A.7, one gets

−Bρn+W(N − n

2

)− A

(N + n

2

)= 0 (A.9)

And solving for the population inversion n finally yields

n = W − A2Bρ+W + A

(A.10)

Therefore, a population inversion n only occurs if the pumping rate W is greaterthan the stimulated emission rate A.

60

A.3 Round trip matrices

A.3. Round trip matrices

MT,763 =(

0.97622 257.68114−0.00041 0.9161

)(A.11)

MS,763 =(

0.97643 266.1589−0.00038 0.9203

)(A.12)

MT,848 =(

0.97622 257.6841−0.00041 0.9161

)(A.13)

MS,848 =(

0.97643 266.16198−0.00038 0.9203

)(A.14)

61

Chapter A Appendix

A.4. Custom prism holder

34

17 246

12

20

24

3.266

M20x1

145

5

24

126

Figure A.1.: Custom prism holder, designed in Solidworks 2010. The dimensionsare in millimeters, and M20x1 denotes a metric screw thread of 20 mm diameter,and a pitch of 1 mm per turn.

62

A.5 Optical Components Details

A.5. Optical Components Details

Name Company Partnumber

Laser Line Mirrors Eksma Optics 032-0530

First telescope lense Eksma Optics 112-1211E+3025-i0

Second telescope lense Eksma Optics 110-1223E+3025-i0

λ/2-waveplates Eksma Optics 461-4230

Polarizing beam splitter Eksma Optics 420-1254E

50:50 beam splitters Eksma Optics 042-7250A

Pump separation mirror Eksma Optics custom

Input mirror Eksma Optics custom

Table A.1.: Optics involved in pumping

Name Used for Company Partnumber Property

Prism Cavity Eksma Optics 320-8110 Apex angle = 59º

λ/2- plate Aligning polarization to cavity B. Halle Custom Quartz Zero Order 850 nm

λ/2- plate Aligning polarization to cavity B. Halle Quartz Zero Order 780 nm

Prism Mount Ti:Sa Mounting Thorlabs KM100PM/M ø 25.4 mm

Precision Mirror Mount Cavity mirrors Thorlabs KS05K/M ø 12.7 mm

Precision Mirror Mount Many beamline elements Thorlabs KS1 ø 25.4 mm

Pedestal mechanical structure Thorlabs RS075P/M l = 19 mm

Pedestal mechanical structure Thorlabs RS05P/M l = 12.5 mm

Mechanical Spacer mechanical structure Thorlabs RS3M l = 3 mm

Aplanar Fibre 848 nm Thorlabs P3-630PM-FC-10 Polarization maintaining

Aplanar Fibre 763 nm OZ Optics PMJ-3A3A-850-5/125-3-5-1 Polarization maintaining

Optical Isolator Seed setup LINOS FI-500/110-5SI Isolation > 30 dB

Piezoelement Cavity Piezomechanik GmbH Mirror-shifter STr-25 15 kHz bandwidth

Table A.2.: Seed and Ti:Sa laser setup parts

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