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Intense few-cycle laser fields: Frontiers of nonlinear optics

Thomas Brabec and Ferenc Krausz*

Institut fur Photonik, Technische Universitat Wien, Gusshausstrasse 27/387,

A-1040 Wien, Austria

The rise time of intense radiation determines the maximum field strength atoms can be exposed to

before their polarizability dramatically drops due to the detachment of an outer electron. Recent

progress in ultrafast optics has allowed the generation of ultraintense light pulses comprising merely

a few field oscillation cycles. The arising intensity gradient allows electrons to survive in their boundatomic state up to external field strengths many times higher than the binding Coulomb field and gives

rise to ionization rates comparable to the light frequency, resulting in a significant extension of thefrontiers of nonlinear optics and (nonrelativistic) high-field physics. Implications include the

generation of coherent harmonic radiation up to kiloelectronvolt photon energies and control of theatomic dipole moment on a subfemtosecond (1 fs1015 s) time scale. This review presents thelandmarks of the 30-odd-year evolution of ultrashort-pulse laser physics and technology culminating

in the generation of intense few-cycle light pulses and discusses the impact of these pulses on high-fieldphysics. Particular emphasis is placed on high-order harmonic emission and single subfemtosecond

extreme ultraviolet/x-ray pulse generation. These as well as other strong-field processes are governed

directly by the electric-field evolution, and hence their full control requires access to the (absolute)phase of the light carrier. We shall discuss routes to its determination and control, which will, for the

first time, allow access to the electromagnetic fields in light waves and control of high-field interactions

with never-before-achieved precision.

CONTENTS

I. Introduction 545

II. Evolution of Ultrashort-Light-Pulse Generation 547

A. Basic concepts and early implementations 547

B. Continuous-wave passively mode-locked solid-

state lasers 548

C. Kerr-lens and solitary mode locking: Routine

generation of femtosecond pulses 549

D. Chirped pulse amplification: Boosting the peak

power to unprecedented levels 551

E. Chirped multilayer mirrors: Paving the way

towards the single-cycle regime 552

III. Generation of Intense Light Pulses in the Few-

Cycle Regime 554

A. Principles of optical pulse compression 554

B. Self-phase-modulation in free space and guided-

wave propagation 555

C. Few-cycle pulse generation: Current state

of the art 558

D. Approaching the light oscillation period: Does

the absolute phase of light matter? 560

IV. Nonlinear Response of Atoms to Strong Laser

Fields 563

A. Perturbative nonlinear optics 564

B. The strong-field regime 565

C. The role of the pulse duration in strong-field

physics 567V. Propagation of Intense Light Pulses 567

A. Perturbative nonlinear optics 568

B. The strong-field regime 569

VI. Optical-Field Ionization of Atoms 570

VII. High-Order Harmonic Generation 571

A. Microscopic analysis: The single-atom dipole

moment 572

B. Macroscopic analysis: Propagation effects 574

C. Absorption-limited and dephasing-limited

soft-x-ray harmonic generation 575

D. Phase matching of soft-x-ray harmonics 577

E. Attosecond x-ray pulse generation 580

VIII. Phase Sensitivity of Strong-Field Phenomena 581

A. Is few-cycle pulse evolution phase sensitive? 581

B. Phase sensitivity of optical field ionization 582

C. Phase effects in high harmonic generation 582

IX. Outlook 583

Acknowledgments 584

Appendix A: Conversion Between Atomic and SI Units 584

Appendix B: Derivation of the First-Order Propagation Equation 584

Appendix C: Influence of The Lorentz Force on High Harmonic

Generation 585

References 585

I. INTRODUCTION

Lasers generating ultrashort light pulses have come ofage. Their applications range from testing ultrahigh-speed semiconductor devices to precision processing ofmaterials, from triggering and tracing chemical reactionsto sophisticated surgical applications in opthalmologyand neurosurgery. In the physical sciences, ultrashortlight pulses enable researchers to follow ultrafast relax-ation processes in the microcosm on never-before-

accessed time scales and study light-matter interactionsat unprecedented intensity levels.

Recent technological advances in ultrafast optics havepermitted the generation of light wave packets compris-ing only a few oscillation cycles of the electric and mag-netic fields. The spatial extension of these wave packetsalong the direction of their propagation is limited to afew times the wavelength of the radiation (0.5 1 min the visible and near-infrared spectral range). Thepulses are delivered in a diffraction-limited beam andhence focusable to a spot size comparable to the wave-length. As a consequence, radiation can be temporarily

*Electronic addresses: [email protected]@tuwien.ac.at

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confined to a few cubic micrometers at the focus of aparabolic mirror, forming a light bullet, as shown inFig. 1. Due to this extreme temporal and spatial confine-ment, moderate pulse energies of the order of one mi-crojoule can result in peak intensities higher than1015 W/cm2. The amplitude of the electric field at theseintensity levels approaches 109 V/cm. These fieldstrengths exceed that of the static Coulomb field experi-enced by outer-shell electrons in atoms. As a conse-quence, the laser field is strong enough to suppress thebinding Coulomb potential in atoms and triggers optical-field ionization (Keldysh, 1965).

This process temporarily enhances the nonlinearatomic polarization because the electron is far removedfrom the nucleus. Once the electron is set free, its re-sponse to the field becomes linear, terminating the non-linear light-matter interaction (aside from a small re-sidual nonlinearity originating from the ionpolarizability). Significant new nonlinearities emergeonly at substantially higher intensity levels by furtherionization and/or when the free electrons become rela-

tivistic, both of which are beyond the scope of thepresent work. The nonlinear atomic polarization re-sponse culminates as the optical-field ionization ratepeaks, shortly before the first weakly bound electron isdetached. The intensity level at which this occursstrongly depends on the temporal evolution of the inci-dent radiation.

Long laser pulses comprising many field oscillationcycles deplete the atomic ground state, i.e., detach anouter atomic electron with a probability approaching100% at moderate intensity levels. In few-cycle laserfields high intensities can be switched on within a few

optical periods lasting merely a few femtoseconds(1 fs1015 s). Hence detachment of the first electron iscompleted at substantially higher field strengths and theoptical-field ionization rate becomes comparable to thelaser field oscillation frequency. As a result, the elec-trons gain unprecedented kinetic energies (up to andbeyond the keV level) during the first field oscillationcycle following their detachment, and a substantial frac-

tion of the atoms is ionized during one laser oscillationperiod To . The implications are numerous and farreaching. Generally speaking, the nonlinear response ofthe ionizing atomic medium is extended to unprec-edented irradiation intensity levels and the induced in-stantaneous nonlinear current densities reach unprec-edented values. Previously inaccessible regimes ofnonlinear optics and high-field physics are being en-tered.

Somewhat more specifically, circularly polarized in-tense few-cycle fields can inject an ultraintense jet ofhigh-energy electrons with a rise time comparable to Tointo a plasma. This may open the way to the develop-ment of electron-pumped inner-shell x-ray lasers in thekeV range (Kim et al., 1999). The high temporal gradi-ent of the pulse front allows neutral atoms to survive tointensities of the order of 1016 W/cm2. The oscillationspectrum of the dipole moment of these atoms may ex-tend to frequencies some thousand times higher thanthat of the linearly polarized few-cycle driving laserfield, giving rise to the emission of (spatially) coherentharmonic radiation up to photon energies of 0.5 keV(Schnurer et al., 1998). The extremely high optical-fieldionization peaks impose appreciable modulation on thedriving few-cycle pulse within To during propagation.The resulting nonadiabatic self-modulation is predictedto permit phase-matched high-harmonic emission at ki-loelectronvolt photon energies (Tempea et al., 1999b).Few-cycle-driven harmonic x-ray radiation may be tem-porally confined to a single burst of subfemtosecond du-ration, which may result in unprecedented peak intensi-ties in the x-ray regime and a never-before-achievedtime resolution in physics.

Temporal confinement of irradiation of atomic sys-tems to a few cycles allows atom-field interactions in-duced by visible or near-infrared radiation to take placefor the first time in the strong-field regime (characterizedby the onset of optical-field ionization) without any no-table preionization in the multiphoton regime. This im-plies that the nonlinear atomic polarization Pnl at in-

stant t can be analytically expressed as a function of thedriving electric field E(t) with t t, which greatly fa-cilitates theoretical analysis of the interaction of intenseradiation with an extended atomic medium (propagationeffects). Even more importantly, the explicit depen-dence of Pnl on E indicates that the evolution of theatomic dipole moment (or the electron wave function) isgoverned directly by the electric field, E( t)Ea(t)cos(t0); hence it depends not only on theintensity envelope Ea

2(t), but also on the carrier phase of a few-cycle laser pulse. Strong-field processes drivenby few-cycle wave packets hold promise for gaining ac-

FIG. 1. Focusing of few-cycle ultrashort light pulses deliveredin a collimated laser beam by a parabolic mirror, producing alight bullet with transverse and longitudinal dimensions of

the order of a few microns. This extreme spatial and temporalconfinement of light creates optical-field strengths sufficient tolower the Coulomb barrier of atoms and to tunnel-ionize anouter electron at moderate pulse energy levels.

546 T. Brabec and F. Krausz: Intense few-cycle laser fields

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cess to this parameter and thereby directly to the electricfield in a light wave for the first time (Krausz et al.,1998). Phase control will be one of the prerequisites forgenerating subfemtosecond extreme ultraviolet (xuv;10100 eV) /x-ray harmonic radiation in a reproduciblemanner. Generally, phase-controlled few-cycle lightpulses will permit control of both the trajectories offreed electrons in strong-field interactions and bound-electron dynamics in atoms and molecules on a timescale of the light oscillation period.

This paper is devoted to the basic theoretical and ex-perimental concepts underlying the generation of in-tense few-cycle laser pulses and high-field femtosecondlight-atom interactions. Sections II and III review theunderlying physics and evolution of ultrafast laser opticsculminating in few-cycle pulse generation. Section IVaddresses basic aspects of the nonlinear response ofmatter to intense radiation relevant to the generation ofultrashort light pulses (the perturbative regime) andtheir application to triggering optical-field ionizationand concomitant phenomena (the strong-field regime).Section V introduces basic propagation equations for

the above regimes. Section VI analyzes optical field ion-ization over a wide parameter range including the tun-neling regime as well as the above-barrier regime. Sec-tion VII is devoted to few-cycle-driven high-orderharmonic radiation, which can lead to the emission ofsingle-attosecond xuv and x-ray pulses. Contrastingthese phenomena with related processes driven by mul-ticycle pulses beautifully shows the advantages of few-cycle drivers in the strong-field regime. In Sec. VIII itwill be shown that the sensitivity of strong-field pro-cesses to the carrier phase of few-cycle light wavepackets may open up access to the absolute phase oflight for the first time. Finally, in Sec. IX we present

some intriguing future prospects. In this paper we shalluse units of the mksa system except for some cases,in which atomic units significantly simplify the treat-ment. Conversion between the two systems is given inAppendix A.

II. EVOLUTION OF ULTRASHORT-LIGHT-PULSE

GENERATION

A. Basic concepts and early implementations

The development of ultrafast optics was triggered bythe invention of laser mode locking, one of the most

striking interference phenomena in nature (DiDo-menico, 1964; Hargrove et al., 1964; Yariv, 1965). Simul-taneous oscillation of a vast number of highly coherent,phase-locked longitudinal modes in a laser yields a re-sultant field equal to zero most of the time except forvery short intervals. The entire energy of the radiationfield is concentrated within these short periods as a re-sult of a short-lived constructive interference betweenthe oscillating waves. Because the frequency spacing be-tween adjacent phase-locked longitudinal modes isequal to 1/Tr, where Tr is the resonator round-trip time,this temporary field enhancement is repeated periodi-

cally with a period Tr at a fixed position on the resona-tor axis. As a consequence, laser mode locking leads tothe formation of a short light pulse circulating in theresonator. Each time the pulse hits a partially reflectingmirror, a small portion of its energy is coupled out of theoscillator, resulting in a train of ultrashort pulses at theoutput of the mode-locked laser, as illustrated in Fig. 2.Straightforward algebra relates the pulse duration and

peak power to the number of phase-locked modes assummarized in Fig. 2.

The large number of modes that can be locked usingstate-of-the-art mode-locking techniques give rise tolight wave packets with unprecedented characteristics.In the rest of this section we shall survey the landmarksin the evolution of ultrashort-pulse laser physics andtechnology from the invention of laser mode locking upto the present day. When doing so we shall be able tocite only a tiny fraction of the relevant literature becauseof space limitations. Nevertheless, the referenced publi-cations should provide a useful guide to the reader forfinding further readings. Our apologies go to the numer-

ous colleagues whose important contributions have notbeen credited. A comprehensive survey of the state ofthe art of primary (mode-locked laser) as well as second-ary (frequency-converted) sources of ultrashort pulses isoffered by conference proceedings, special issues oftechnical journals, and books.1 The physics of the propa-gation, manipulation, and interaction of ultrashort laserpulses with matter (at low and moderate intensities) isthoroughly presented in two recent textbooks (Akh-manov et al., 1992; Diels and Rudolph, 1996), whereasthe physics and technology of the (most important) lat-est generation of ultrafast sources, namely, femtosecondsolid-state lasers, have been reviewed by Krausz, Fer-mann et al. (1992), Keller (1994), Brabec et al. (1995),and French (1996).

The first generation of mode-locked lasers producingpulses of durations less than 100 ps used solid-state lasermaterials such as ruby, Nd:glass, or Nd:YAG as gainmedia. Mode locking was implemented either by activeloss or frequency modulation driven by an external elec-

1These include Heritage and Nuss, 1992; Martin et al., 1993;Barbara et al., 1994; Krausz and Wintner, 1994; Barbara et al.,1996; Zhang et al., 1996; Keller 1997; Barty, White et al., 1998;Elsasser et al., 1998.

FIG. 2. Radiation power as a function of time at the output ofa stationary mode-locked laser.

547T. Brabec and F. Krausz: Intense few-cycle laser fields



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tronic oscillator (Deutsch, 1965; DiDomenico et al. 1966;Osterink and Foster, 1968; Kuizenga and Siegman,1970a, 1970b), or by passive loss modulation with a fast-response saturable absorber (Mocker and Collins, 1965;DeMaria et al., 1966). These techniques were termed ac-tive and passive mode locking, respectively. The prin-ciple of operation of the latter, which provides muchmore efficient pulse shaping and hence shorter pulses, isillustrated in Fig. 3. Nonlinear autocorrelation tech-niques needed to be developed for the temporal charac-terization of these pulses (Maier et al., 1966; Armstrong,1967; Weber, 1967). The passively mode-locked,flashlamp-pumped Nd:glass laser became the majorworkhorse for investigations in the field of nonlinearoptics and opened up the entirely new field of time-resolved spectroscopy. The interested reader is referredto several excellent reviews devoted to these first-generation ultrafast sources and their characterization(DeMaria et al., 1969; Bradley and New, 1974; Green-how and Schmidt, 1974; Smith et al., 1974).

The picosecond response time of organic saturable ab-sorbers employed for passive mode locking has set alimit to the pulse duration. Continuous-wave operation

(Peterson et al., 1970) of an organic dye laser (Schaferet al., 1966; Sorokin and Lankard, 1966) and its modelocking by a saturable absorber (Ippen et al., 1972) in aspecifically designed cavity (Kogelnik et al., 1972) trig-gered the development of the second generation ofmode-locked lasers. In contrast with solid-state lasers,the response time of the absorber no longer constituteda limitation to the achievable pulse duration, owing tothe active role of gain saturation in pulse formation(New, 1972, 1974; Haus, 1975a). As a result, opticalpulses shorter than 1 ps could be produced for the firsttime (Ruddock and Bradley, 1976) and subsequent im-

provements in the cavity design allowed the breaking ofthe 100-fs barrier by utilizing a new concept, colliding-pulse mode locking (Fork et al., 1981, 1983). Intracavitydispersion control by means of low-loss Brewster-angledprism pairs (Fork et al., 1984; Martinez et al., 1984) wasthe next major breakthrough, which together with im-proved insight into the relevant physical processes andlimitations (Kuhlke et al., 1983; Stix and Ippen, 1983; DeSilvestri et al., 1984; Martinez, Fork, and Gordon, 1984;

Diels et al., 1985; Martinez et al., 1985) allowed repro-ducible generation of sub-100-fs pulses and the demon-stration of pulses as short as 27 fs directly from a laseroscillator (Valdmanis et al., 1985; Valdmanis and Fork,1986). This progress was achieved by using the organicdye Rhodamine 6G (Rh6G) as a gain medium emittingat around 620 nm. The dispersion-controlled colliding-pulse mode-locked Rh6G laser was the major workhorsefor femtosecond spectroscopy until the late 1980s. Nev-ertheless, a number of other cw dye lasers have alsobeen successfully mode locked to produce femtosecondpulses in the visible and near-infrared spectral range(French et al., 1987, 1989; French and Taylor, 1988). For

comprehensive reviews of ultrafast dye lasers see French(1985) and Shank (1988).

B. Continuous-wave passively mode-locked solid-state

lasers

In the 1980s, continued work on the development ofsolid-state laser materials gave rise to the emergence ofa number of new laser media. Various host crystals(YAG, sapphire, forsterite, LiSAF, etc.) doped withtransition-metal (titanium, chromium) ions now providelaser transitions with enormous bandwidths on the orderof 100 THz in the near-infrared wavelength range, as

shown in Fig. 4 (Moulton, 1982, 1986; Petricevic et al.,1989; Borodin et al., 1992; Smith et al., 1992). The devel-opment of novel techniques and devices suitable for pas-sive mode locking of lasers with long gain relaxation andtheir application in these broadband solid-state systemsled to the emergence of the third generation of ultrafastlaser sources. Continuous-wave passive mode locking ofbroadband solid-state laser oscillators was accomplishedby exploiting resonant (Keller, Knox, and t Hooft 1992)and nonresonant (Krausz, Fermann, et al., 1992) opticalnonlinearities. The most successful devices for imple-menting the former technique have been semiconductor

FIG. 3. Reliance of passive mode locking on a nonlinear ele-ment that exhibits an increased transmittivity for increased la-ser intensity. With such an element introduced in the oscilla-tor, an initial amplitude fluctuation arising in the free-runninglaser (due to some mode beating, for instance) can grow uponrepeated round trips in the cavity at the expense of the lower-intensity background. The resulting energy transfer continuesuntil all the intracavity radiation energy is confined within asingle pulse originating from the initial fluctuation. The steady-state pulse parameters are dictated by a balance between

pulse-shortening and pulse-broadening effects. Saturable ab-sorbers were the first (and for a long time the only) nonlinearoptical elements employed for passive mode locking.

FIG. 4. Fluorescence emission spectra of some transition-metal-doped broadband solid-state laser crystals.




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saturable absorber mirrors, referred to as SESAMs, andsaturable Bragg reflectors (Keller et al., 1990; Feldmannet al., 1991; Keller, Miller, et al., 1992; Rizvi et al., 1993;Tsuda et al., 1995; for reviews see Keller, Knox, and tHooft, 1992; Keller et al., 1996; Tsuda et al., 1996; Keller,1999). The simultaneous exploitation of a solitonlike in-terplay between negative intracavity group delay disper-sion (GDD) and self-phase modulation (SPM) inducedby the nonresonant Kerr effect allowed the generationof optical pulses significantly shorter than the pico- orsubpicosecond absorber recovery time (Kartner andKeller, 1995; Kartner et al., 1996).

Because of the inability of long-relaxation-time solid-state gain media to participate in ultrashort pulse forma-tion, fast-response artificial saturable absorbers basedon nonresonant optical nonlinearities were devised byseveral researchers. The first attempt dates back to theearly 1970s (Dahlstrom, 1972) and was followed by anumber of proposals based on third-order (3 ) type(Sala et al., 1977; Ouellette and Piche, 1986) and second-order (2 ) type (Stankov and Jethwa, 1988; Barr et al.,1991) nonlinearities [for a definition of(2 ) and (3 ), see

Eq. (19)]. The first breakthrough occurred when re-seachers recognized that coupled cavities containing aKerr nonlinearity [see Eq. (30)] introduced by a single-mode fused silica fiber that can be used for mode lock-ing. This concept was first employed in the soliton laser(Mollenauer and Stolen, 1984; Mitschke and Mol-lenauer, 1986, 1987) and subsequently was shown to in-troduce a fast saturable-absorber effect over a widewavelength range irrespective of fiber dispersion (Blowand Wood, 1988; Blow and Nelson, 1988; Kean et al.,1989; Mark et al., 1989). The technique was calledcoupled-cavity or additive-pulse mode locking (Ippenet al., 1989; Morin and Piche, 1990; Haus et al., 1991)

because coherent superposition of the intracavity pulsewith its self-phase-modulated replica introduced thesaturable-absorber effect. Additive-pulse mode lockingallowed cw self-starting passive mode locking (Ippenet al., 1990; Krausz et al., 1991) in a wide range of solid-state lasers. Implementing additive-pulse mode lockingin all-fiber lasers (Duling 1991; Hofer et al., 1991 1992;Richardson et al., 1991; Tamura et al., 1993; Fermannet al., 1994) led to the development of compact turnkeyfemtosecond sources; for recent reviews see Duling andDennis (1995) and Fermann (1995).

C. Kerr-lens and solitary mode locking: Routine

generation of femtosecond pulses

The discovery of self-mode-locking (Spence et al.,1991) in a titanium-doped sapphire (Ti:S) laser (Moul-ton, 1982, 1986) revolutionized ultrafast laser technol-ogy. Subsequent experimental (Keller et al., 1991;Spinelli et al., 1991) and theoretical (Piche, 1991; Salinet al., 1991; Brabec, Spielmann, Curley et al., 1992; Hauset al., 1992) work revealed that an intracavity aperturetranslates self-focusing (Zakharov and Shabat, 1972;Marburger, 1975) introduced by the Kerr nonlinearity ofthe laser crystal into an ultrafast saturable-absorber-like

self-amplitude-modulation (SAM), and the techniquehas been termed Kerr-lens mode locking (KLM). Theoperational principle of KLM is illustrated schematicallyin Fig. 5. The optical Kerr effect in the laser host crystalresults in a fast-response intensity-induced change,

n r,t n2I r,t , (1)

of the refractive index. Here n2 cm2/W is the nonlinear

index of refraction and I W/cm2 is the cycle-averaged

laser intensity. This effect transforms the radial intensityprofile of a laser beam in a lensing effect, which tends tomore strongly focus more intense temporal slices ofthe laser beam. An aperture of suitable size placed at asuitable position in the cavity can thus transmit a largerfraction of the laser beam at instants of higher intensity.The consequence is reduced loss for increased intensity,which constitutes a fast saturable-absorber effect (simi-lar to additive-pulse mode locking). This ultrafast SAMeffect can initiate and sustain the formation of an ul-trashort pulse in Ti:S and other solid-state lasers. Simul-taneously, n(r,t) directly modulates the phase, whichis termed self-phase-modulation (SPM).

The effect of SAM and SPM on the laser pulse circu-

lating in the resonator can be simply described by thechange they cause in the complex amplitude envelope

Ea(), where the time is measured in a frame of ref-erence retarded such that the pulse peaks at 0 at anytime [for definition see Eq. (14)] upon each round trip inthe cavity:

Ea kSA Mp Ea (2)

and

Ea ikSP Mp Ea , (3)

respectively. Here p() Ea()2 is the cycle-averaged

time-dependent radiation power carried by the laserbeam and kSA M,kSP M W

1 stand for the SAM andSPM coefficients, respectively. In KLM lasers, calcula-tion of the intensity-dependent beam radius in the frameof the quadratic approximation2 allows us to determinethe coefficient kSA M analytically. A more complete listof related literature is given in a recent paper of Kalosha

2See, for example, the calculations of (Piche, 1991; Salin et al.,1991; Brabec, Spielmann, Curley et al., 1992; Georgiev et al.,1992; Haus et al., 1992; Chilla and Martinez, 1993; Cerulloet al., 1994; Herrmann, 1994; Agnesi, 1995; Lin et al., 1995).

FIG. 5. Self-focusing-induced changes in the transmittivity ofan aperture: the ultrafast-response saturable-absorber effectresults in Kerr-lens mode locking.




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et al. (1998), which presents a comprehensive numericalanalysis of Kerr-lens mode locking. For the SPM coeffi-cient, straightforward algebra leads to

kSP MSP ML, SP M2n2

0w02 . (4)

Here 0 is the carrier wavelength of the laser pulse, L isthe length of the Kerr medium, which is assumed to be

shorter than the confocal parameter 2 z02w02

/0 , andw0 is the 1/e

2 beam radius.In a typical KLM Ti:S laser, the SPM coefficient is on

the order of kSP M106 W1, whereas SAM is weaker

by more than an order of magnitude: kSA M107 W1

(Krausz, Fermann et al., 1992). Typical intracavity pulsepeak powers of the order of 105106 W in the subpico-second and femtosecond regimes hence introduce anamplitude modulation of merely a few percent by meansof the KLM artificial saturable absorber. This compara-tively weak modulation is incapable of overcoming thepulse broadening caused by dispersion of the laser me-dium in the femtosecond regime, which stops pulse

shortening far from the limit set by the gain bandwidthin the Ti:S laser.Accessing the regime well below 100 fs in the Ti:S

laser becomes feasible when we introduce negativegroup delay dispersion (GDD) and exploit a highly effi-cient pulse-shortening mechanism resulting from an in-terplay between Kerr-induced SPM (e.g., in the lasercrystal) and a net negative cavity GDD introduced, forexample, by a pair of prisms (Fork et al., 1984). Thisinterplay is referred to as solitary mode locking (Brabecet al., 1991) and will be discussed as the key techniquefor the generation of few-cycle light pulses in a broadercontext in the next section.

Because kSP M/kSA M

1, solitary pulse shaping domi-nates pulse shortening in the presence of a net negativeintracavity GDD and determines the steady-state pulseduration in the KLM Ti:S laser. In the frame of the weak

pulse-shaping approximation, Ea( t)/ Ea(t) 1, whichdefines the range of validity of the powerful analyticmaster-equation approach of Haus et al. (1991, 1992),and in the limit kSP M/kSA M1, the master equation

yields a steady-state pulse of the form Ea()E0sech(/0) with

02 D

kSP MWp(5)

and a pulse duration of p1.760 [full width at halfmaximum ofp() Ea()

2]. Here Wp is the intracavitypulse energy and D(0) fs2 denotes the net intracav-ity GDD [for a definition, see Eq. (8)]. Further, Eq. (5)is valid under the assumption that D is independent offrequency, i.e., that high-order dispersion is negligible.This latter approximation tends to fail increasingly fordecreasing pulse durations, which rely on smaller andsmaller values of D according to Eq. (5), and this even-tually sets a limit to pulse shortening. The lowest-orderperturbation to a constant GDD is a small linear depen-dence on frequency, D()DD 3(0), where D

D(0), 0 is the center frequency of the mode-lockedlaser, and D3 fs

3 is referred to as third-order disper-sion, because it is equal to the third derivative of thephase with respect to frequency, as will be revealed byEq. (8). Computer simulations yielded the explanationthat mode locking becomes increasingly perturbed andeventually unstable as p D3 /D , setting a limit to theminimum pulse width achievable with solitary mode

locking in the presence of third-order dispersion (Bra-bec, Spielmann, Curley, et al., 1992). Overcoming thislimitation would result in a mode-locked spectrum ex-tending into a wavelength regime, where the cavityGDD D() becomes positive, preventing stable solitarypulse formation (Spielmann et al., 1994).

In the absence of high-order dispersion and limita-tions due to the finite high-reflectivity bandwidth of mir-rors, the separate action of negative GDD and SPM ispredicted to limit pulse shortening well before attain-ment of the gain-bandwidth limit of Ti:S, which wouldpermit pulse durations approaching 1 fs at low cavitylosses. Discrete solitary pulse shaping is a consequence

of the Kerr-induced phase shift kSP Mp( t) approaching(or even exceeding) unity at the pulse peak for decreas-ing D , leading to pulse shortening below 10 fs. As aresult, the weak pulse-shaping approximation breaksdown and a correction term proportional to kSP MWp ,which is negligible for large values of D , appears onthe right-hand side of Eq. (5) (Brabec et al., 1991;Krausz et al., 1992). This correction term is dependenton the position in the resonator, predicting a periodicpulse evolution in the (sub-)10-fs regime.

These findings resulted in a rapid evolution of prism-dispersion-controlled KLM Ti:S lasers, driven by thesearch for prism materials characterized by the smallestratio of third-order dispersion to GDD. Fused silica wasfound to be the best choice and, as a result, fused-silica-prism-controlled KLM Ti:S lasers were the first laser os-cillators producing pulses in the 10-fs regime (Asakiet al., 1993; Curley et al., 1993; Proctor and Wise, 1993;Spielmann et al., 1994; Zhou et al., 1994). Just as in fem-tosecond dye lasers (De Silvestri et al., 1984; Salin et al.,1990), high-order dispersion of the prisms limited thepulse duration (Lemoff and Barty, 1993a; Spielmannet al., 1994). These technological limitations were to beovercome with the discovery of chirped multilayer di-electric mirrors (Szipocs et al., 1994, 1995), which will betreated below. With these devices, resonators providinghigh reflectivity and approximately constant negative

GDD over the entire gain band of Ti:sapphire(600 1000 nm) can be constructed.

Compact mirror-dispersion-controlled (MDC) KLMTi:S laser oscillators now routinely generate high-qualitysub-10-fs pulses (Stingl et al., 1995; Kasper and Witte,1996) with peak powers exceeding 1 MW (Xu et al.,1997, 1998; Beddard et al., 1999). Recently, hybrid dis-persion control using prisms and chirped mirrors yieldedpulses below 6 fs from Ti:S lasers (Gallmann et al., 1999;Morgner et al., 1999; Sutter et al., 1999). Combining theconcept of mirror dispersion control with extension ofthe laser cavity by a telescope (Cho et al., 1999; Libertun




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et al., 1999) resulted in the generation of stable sub-10-fspulses with peak powers exceeding 3 MW for the firsttime directly from a laser oscillator (Poppe, Lenzner,et al., 1999). These technological advances led to a rapidevolution of femtosecond Ti:S lasers since the first dem-onstration of passive mode locking in a Ti:sapphire laser(French, Williams, et al., 1989) and gave rise to an im-provement of the pulse duration and peak power bymore than two and four orders of magnitude, respec-tively, within the last 30 years, as depicted in Fig. 6. Dueto the absence of intracavity components other than thegain medium in KLM/MDC Ti:S lasers, the noise perfor-mance of femtosecond sources could also be improvedsubstantially (Poppe et al., 1998). The KLM Ti:S laser isnow commercially available with both prism and mirrordispersion control and has become the major workhorsefor ultrafast time-resolved spectroscopy and nonlinearoptics. More recently, it was also demonstrated as apromising tool for metrology applications in high-resolution spectroscopy (Udem et al., 1999a; 1999b).

D. Chirped pulse amplification: Boosting the peak power

to unprecedented levels

Many intriguing applications of ultrashort pulses call

for intensities requiring peak powers far exceeding thepower levels that can be directly obtained from cwmode-locked oscillators. To this end, the pulses deliv-ered by mode-locked lasers need to be amplified. Fem-tosecond pulse amplification was first demonstrated byusing dye cells or jets pumped by Q-switched Nd:YAGand copper-vapor lasers (Fork et al., 1982; Sizer et al.,1983; Koroshilov et al., 1984; Knox et al., 1985); for areview see Knox, 1988). The maximum pulse energy thatcan be practically achieved in dye amplifiers is restrictedto less than 1 mJ. This limitation relates to the factthat the maximum possible energy that can be extracted

per unit beam cross-sectional area without significanttemporal distortion of the amplified pulse is given by thesaturation fluence

Fsa tl /e , (6)

where l and e are the center frequency and the peakstimulated-emission cross section of the laser transition,respectively. The above limit is set by the low saturationfluence implied by the high emission cross section

eof

dye lasers and the maximum dye volume that can beuniformly pumped in practice. The fundamental rela-tionship

felk l

2

n2(7)

connecting the characteristics of a laser transition (l isthe gain bandwidth, n is the refractive index of the gainmedium, and k is a numerical factor of the order of unitythat depends on the emission line shape) reveals that thehigh e also implies a short fluorescence lifetime f (Ko-echner, 1996). The related short energy-storage time and

strong amplified spontaneous emission constitute furtherdrawbacks of dye amplifiers. Excimer amplifiers havesimilar characteristics but offer significantly larger uni-formly inverted apertures, allowing amplification up tohundreds of millijoules in the ultraviolet spectral rangeat pulse durations of the order of 100 fs (Glownia et al.,1987; Szatmari et al., 1987; Watanabe et al., 1988; Tayloret al., 1990; Mossavi et al., 1993; for a review see Szat-mari, 1994). Owing to their high peak power, of the or-der of 1 TW, excellent beam quality, and short wave-length, these pulses can be focused to a spot size below 1m, resulting in peak intensities in excess of 1018 W/cm2

(Szatmari, 1994).Novel solid-state gain media developed in the 1980s

held promise for producing even higher peak powersdue to their far higher energy fluences and broaderbandwidths (see Fig. 4). However, amplification of fem-tosecond pulses in these media gives rise to catastrophiceffects due to an accumulated intensity-dependent phaseshift induced by the optical Kerr effect long before thesaturation fluence can be reached (see, for example, Ko-echner, 1996). This limitation has been overcome by theingenious concept of chirped pulse amplification (Strick-land and Mourou, 1985; Maine et al., 1988), the principleof which is illustrated in Fig. 7. By the early 1990s,chirped pulse amplification implemented withdiffraction-grating-based pulse compressors (Treacy

et al., 1969) and stretchers (Martinez, 1987) had made itpossible to generate pulses in the 100-fs range with peakpowers of several terawatts from laboratory-scale sys-tems based on Ti:S and Cr:LiSAF (Sullivan et al., 1991;Beaud et al., 1993; Ditmire and Perry, 1993), and the useof large-aperture Nd:glass power amplifiers resulted in1- ps pulses having peak powers of tens of TW (Patter-son and Perry, 1991; Sauteret et al., 1991; Yamakawaet al., 1991; Rouyer et al., 1993; Blanchot et al., 1995).

Advances in Ti:S seed oscillators (Asaki et al., 1993;Curley et al., 1993; Stringl et al., 1995) and stretcher/compressor design (Lemoff and Barty, 1993b; Zhou

FIG. 6. Evolution of the passively mode-locked Rh6G dye la-ser and the passively mode-locked Ti:sapphire laser in terms ofpulse duration and peak power from the mid 1960s to date. Acomparison of the shortest pulse durations achieved with Kerr-lens mode locked, mirror-dispersion-controlled Ti:sapphire la-sers with the round-trip time Tr1015 ns reveals that morethan one million phase-locked longitudinal resonator modesare oscillating in these systems (see Fig. 2).




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et al., 1995; Cheriaux et al., 1996) led to the developmentof Ti:S chirped pulse amplification systems generating

multiterawatt pulses in the 20-fs regime at 10 Hz (Bartyet al., 1994; Zhou et al., 1995) and triggered a subsequentrapid evolution of these systems (Chambaret et al., 1996;Yamakawa et al., 1998a) to a peak power of 100 TW(Yamakawa et al., 1998b) and (expected) peak intensi-ties at focus in excess of 1020 W/cm2. The evolution oflaboratory chirped pulse amplification systems is de-picted in Fig. 8 (for recent reviews see Perry andMourou, 1994; Morou, 1997). This and similarlaboratory-scale chirped pulse amplification architec-tures based on Ti:sapphire or Yb:glass (Nees et al., 1998)are expected to be scalable to petawatt peak power lev-els, which could be demonstrated with a large laser sys-

tem at Lawrence Livermore National Laboratory, Liver-more (Perry et al., 1999). Peak powers in excess of1021 W/cm2 can now be attained from both large-scale(Perry et al., 1999 and lab-scale lasers (Patterson et al.,1999). The excellent thermal conductivity and favorablethermo-optic properties of Ti:sapphire and the availabil-ity of kHz-rate Q-switched solid-state pump sourcesopened the way to implemeting chirped pulse amplifica-tion at kHz repetition rates. Related research recentlyculminated in the demonstration of kHz sources of sub-20-fs pulses with peak powers of several hundred giga-

watts (Backus et al., 1997; Nibbering et al., 1997; Na-bekawa et al., 1998; for a review see Backus et al., 1998)and suggest the feasibility of kHz-repetition-rate tera-watt sources (Durfee et al., 1998) in the near future.

E. Chirped multilayer mirrors: Paving the way towards the

single-cycle regime

In the preceding subsections we have addressed howoptical nonlinearities can be efficiently exploited for thegeneration of femtosecond laser pulses in broadbandsolid-state oscillators and how they can be efficientlyavoided when their energy is boosted in solid-state am-plifiers. Apart from optical nonlinearities, dispersion be-comes increasingly important as ever shorter pulses aresought. In fact, generating bandwidth-limited pulses ofdecreasing duration calls for controlling the frequency-dependent group delay over increasing bandwidth withincreasing precision. Hence precision broadband disper-sion control is a prerequisite for approaching the few-cycle regime in ultrashort pulse generation.

Dispersion can be quantified by expanding the groupdelay, which is equal to the first derivative of the phaseretardation () of the optical system with respect tofrequency, about the center of the pulse spectrum 0 inthe form

Tg 0 0 0

1

2 0 0

2

1

6 0

3

. (8)

Here (0) gives the time it takes for the peak of thepulse to traverse the dispersive medium. The higher-

order terms in the expansion describe the frequency de-pendence of the group delay and hence are responsiblefor dispersive effects. (0) is the lowest-order (linear)group delay dispersion or second-order phase disper-sion, most frequently referred to as group delay disper-sion (GDD) in the literature and denoted by D, a con-vention that we also adopt in this work; (0)D3and (0)D4 are termed third-order and fourth-order dispersion, respectively. Critical values of the dis-persion coefficients, above which dispersion causes asubstantial change of the pulse, obey the simple scaling(n)p

n . For instance, a GDD of p2 results in a

pulse broadening by more than a factor of 2. This scaling

reveals a dramatic increase in susceptibility todispersion-induced broadening and distortion for de-creasing pulse durations.

Most laser and optical materials exhibit some positiveGDD, implying a group delay that increases with fre-quency and hence imposing a positive frequency sweepor chirp on a pulse passing through the medium. Pulsestens of fs in duration tend to broaden significantly evenupon passage through transparent optical media (such asquartz or sapphire) of merely a few mm in length. Thisbroadening even gets accelerated at high intensities,leading to self-phase-modulation due to the optical Kerr

FIG. 7. Principle of chirped pulse amplification (Stricklandand Mourou, 1985). A low-energy ultrashort seed pulse is tem-porally stretched before amplification and recompressed afteramplification to avoid high peak powers in the amplifier sys-tem.

FIG. 8. Evolution of ultrashort-pulse amplification in terms ofpeak power and achievable peak intensity.




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effect, which broadens the spectrum by imposing a posi-tive temporal chirp. Clearly, restoring the duration ofpulses passed through optical materials calls for negativeGDD, just as does compressing pulses spectrally broad-ened by self-phase-modulation.

Treacy (1969) was the first to demonstrate negativeGDD by passing light pulses through a pair of diffrac-tion gratings, successfully achieving pulse compressionwith such a system. The Brewster-angled prism pairs ofFork et al. (1984) were the first low-loss sources of nega-tive GDD, which have been extensively employed fordispersion control inside laser oscillators since their dis-covery. In both cases, negative GDD is accompanied bysignificant amounts of intrinsic high-order dispersion,which cannot be lowered or adjusted independently ofthe (useful) lowest-order dispersion (GDD), limiting thebandwidth over which adequate dispersion control canbe provided. This drawback was overcome, to lowestorder, by combining prism and grating pairs exhibitingthird-order dispersion of opposite sign, thus allowingpulse compression to 6 fs in the mid 1980s (Fork et al.,1987) and recently to less than 5 fs (Baltuska et al.,1997a). Nevertheless, this approach cannot be used forfew-cycle pulse generation either in oscillators (because

of the high losses of gratings) or in external compressorsat high power levels (because of unwanted nonlinearitiesin the prisms).

Recently, modulation of the multilayer period of di-electric laser mirrors was demonstrated to result in awavelength-dependent penetration depth of the incidentradiation (Szipocs et al., 1994). This implies a corre-sponding dependence of the group delay on frequency(see Fig. 9), which can be tailored to yieldwithin cer-tain limitsrequired amounts of GDD as well as higher-order dispersion over almost the entire high-reflectivityband of the mirror, which can be substantially broader

than that of a standard quarterwave stack. These deviceshave been referred to as chirped multilayer mirrors (Szi-

pocs et al., 1994). Recent advances in the design andmanufacturing of such structures have led to chirpedmirrors exhibiting high reflectivity and approximatelyconstant negative GDD over unprecedented bandwidths(Matuschek, 1998; Tempea, Krausz, et al., 1998).

Figure 10 depicts the GDD versus wavelength for sev-eral chirped mirrors designed for second- and higher-order dispersion control in Ti:sapphire-based sub-10-fssystems. These results demonstrate the power of thistechnique for controlling dispersion over unprecedentedbandwidths with unprecedented precision. The chirpedmirrors current capability of exhibiting high reflectance

FIG. 9. Schematic illustration of the origin of dispersion in achirped multilayer mirror. Quasimonochromatic wave packetscarried at different wavelengths penetrate to different depthsbefore being reflected, as a consequence of a modulation ofthe multilayer period across the layer stack. In the illustratedexample, the increasing multilayer period with increasing dis-tance from the mirror surface implies that radiation with in-creasing wavelength has to penetrate deeper before being re-flected as a result of constructive interference of the partialwaves reflected from the interfaces between the low- and high-index layers. The result is a group delay that increases withincreasing wavelength, i.e., decreasing frequency, giving rise to

negative group delay dispersion. FIG. 10. Group delay dispersion (GDD) vs wavelength of ul-trabroadband chirped mirrors aiming at introducing GDD aswell as higher-order dispersion control in sub-10-fsTi:sapphire-based systems: , measured data (restricted to therange 6001000 nm) obtained from white-light interferometry;dashed curves, nominal (averaged) or target GDD curves. Thequasiperiodic fluctuation of the measured dispersion is inher-ent to chirped mirrors consisting of discrete layers and can beefficiently (by more than 50%) suppressed by complementarymirrors with opposite fluctuations. (a) Chirped mirror de-signed for nominally constant negative GDD of40fs2. (b)Chirped mirror designed for compensating the dispersion in-troduced by fused silica up to (and including) fourth orderover the wavelength range 580930 nm; dashed line, disper-

sion introduced by fused silica (with reversed sign) over a1-mm propagation length: D36fs2, D327fs

3, and D411fs4 at 0800 nm. (c) Chirped mirror designed for com-pensating high-order dispersion up to (and including) fourth-order dispersion over the wavelength range 5601000 nm;dashed line, least-squares fit to the data yielding D53fs2,D39 fs

3, and D4400 fs4 at 0800 nm. The high reflectivity

(99%) range of these mirrors extends over more than 200THz, exceeding the bandwidth of standard quarterwave mir-rors by more than a factor of two.




10/47

over a bandwidth 200 THz and controlling dispersionover a spectral range as broad as 150 THz has allowedthe generation of sub-6-fs pulses directly from laser os-cillators (Gallmann et al., 1999; Morgner et al., 1999;Sutter et al., 1999) sub-5-fs pulses from optical paramet-ric amplifiers (Shirakawa et al., 1999), and pulse com-pression at high (subterawatt) power levels down to 4 fs(Cheng et al., 1998). The latter pulses are carried at a

wavelength of 00.8m, implying a laser oscillationcycle of To2.6 fs. Hence chirped mirrors are able toprovide adequate dispersion control for the generationof light pulses comprising merely one and a half fieldoscillation cycles within their intensity half maximumand hold promise for pushing the limits of ultrafast op-tics into the single-cycle regime in the near future.

III. GENERATION OF INTENSE LIGHT PULSES IN THE

FEW-CYCLE REGIME

In this section we briefly review the key physicalmechanisms and techniques that allow the generation of

powerful light pulses with durations approaching thelight oscillation period To and peak intensities penetrat-ing far into the strong-field regime (see Fig. 24 below).After presenting the status of few-cycle optical pulsegeneration, we address a parameter that has not re-ceived attention until recently: the absolute phase oflight wave packets becomes important in the interactionof intense few-cycle radiation with matter, and its con-trol in ultrabroadband mode-locked laser oscillators willalso benefit precision measurements of the frequency oflight as well as atomic transitions.

A. Principles of optical pulse compression

The nonlinear refractive index n2 , as defined in Eq.(1), was recognized as a source of new frequency com-ponents during propagation of an intense short lightpulse through a Kerr medium. The time-dependent non-linearly induced phase shift nl()(2/0)n2I()L, where I() is the intensity, theretarded time as defined in the previous section, and Lthe propagation length, manifests itself as a self-phase-modulation (SPM) and tends to broaden the spectrum ofan initially bandwidth-limited pulse (Shimizu, 1967;Fisher et al., 1969; Laubereau, 1969; Laubereau and vonder Linde, 1970; Alfano and Shapiro, 1970). In high-power laser systems, this effect is undesirable because

the dependence ofI on the transverse space coordinatesintroduces phase aberrations leading to a degradation ofthe laser beam. However, under specific experimentalconditions nl can be utilized for the generation ofultrashort pulses.

The feasibility of pulse compression arises from thesequential emergence of new redshifted and blueshiftedspectral components at different positions of the pulseenvelope. As a consequence, subsequent passage of thepulse through a delay line, introducing shorter group de-lay for the new spectral components riding on the trail-ing edge of the pulse, as compared to the delays suffered

by the components emerging on the front edge, trans-lates the pulse carrying nl into a temporally com-pressed pulse, as illustrated in Fig. 11. The frequencysweep (or chirp) dnl /d is linear to good accuracy inthe vicinity of the pulse center (0), where most ofthe energy is concentrated, particularly ifnl emergesin the presence of GDD of the same sign as nl (Tom-linson et al., 1984). As a consequence, optimum tempo-ral compression calls for a group delay Tg() exhibitinga near-linear dependence on frequency in the dispersivedelay line. The nonlinear index n2 is generally positivefar from resonances, hence a negative group delay dis-

persion (dTg /d)00 is required for pulse compres-sion.

This interplay between Kerr-induced self-phase-modulation and negative GDD forms the basis of allpulse compression schemes that have been demon-strated to date. At low (nanojoule) pulse energy levels,compression can be most efficiently implemented bypassing the pulse through a single-mode fiber and sub-sequently through a grating pair (Grischkowski and Bal-ant, 1981; Johnson et al., 1984), and somewhat later,through gratings and prisms for improved high-orderdispersion control, resulting in nanojoule-energy 6-fspulses at 620 nm (Fork et al., 1987). This long-standing

record has recently been improved by using a single-mode-fiber/chirped-mirror compressor seeded with 15-fspulses generated by a cavity-dumped KLM Ti:sapphirelaser (Ramaswamy et al., 1993; Pshenichnikov et al.,1994). This compact all-solid-state system is capable ofgenerating nanojoule pulses down to 4.5 fs in duration atMHz repetition rates (Baltuska et al., 1997a; 1997b;Pshenichnikov et al., 1998), constituting a powerful toolfor ultrafast spectroscopy with unprecedented temporalresolution.

Nanojoule-energy few-cycle pulses with durations be-tween 5 and 6 fs have more recently been directly avail-

FIG. 11. Schematic representation of optical pulse compres-sion. The graphs depict qualitatively the spectral intensity dis-tribution (frequency domain), and evolution of the electricfield E(t) and change in instantaneous frequency with respectto the carrier frequency ()inst()0 (time domain).The bandwidth-limited ()0 input pulse propagatesfrom the left to the right. The positive frequency sweep()(d/d)nl() imposed on the intense light pulse byself-phase-modulation induced by the optical Kerr effect uponpassage through a transparent optical medium can be removedby subsequently passing the pulse through a dispersive delayline with negative group delay dispersion, resulting in a tem-poral compression.




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able from KLM Ti:S oscillators employing chirpedmultilayer resonator mirrors (Gallmann et al., 1999;Morgner et al., 1999; Sutter et al., 1999). For restoringthese pulse durations after amplification to millijoule en-ergy levels, the implementation of self-phase-modulation in a gas-filled multimode hollow waveguide(Nisoli et al., 1996) and subsequent temporal compres-sion in chirped mirrors (Nisoli, De Silvestri, et al., 1997;Nisoli, Stagira, et al., 1997; Sartania et al., 1997) havebeen proposed and successfully demonstrated. The for-mation of few-cycle pulses in Ti:S oscillators and hollow-fiber-based compressors is the consequence of the samephysical mechanisms (SPM-GDD interplay) exploited indistinctly different parameter regimes. In the followingsubsections we provide a unified treatment of the forma-tion of few-cycle pulses in solitary (bulk) laser oscillatorsand in multimode-waveguide/chirped-mirror-based com-pressors (at low and high energy levels, respectively)and present the state of the art of few-cycle pulse gen-eration.

B. Self-phase-modulation in free space and guided-wavepropagation

Self-phase-modulation (SPM) of ultrashort laserpulses and related spectral broadening can be most sim-ply induced by focusing into a bulk nonlinear medium. Ifthe length L of the medium is short compared to theconfocal parameter of the beam, the wave front is ap-proximately planar and the radial intensity distributionapproximately constant in the nonlinear medium. Yetthe radial intensity variation produces a spatially varyingnl , giving rise to self-focusing (for n20) and small-scale instabilities (Campillo et al., 1969). In addition, theextent of spectral broadening depends on the radial co-ordinate, frustrating a spatially uniform pulse compres-sion. Creating a near-flat-top intensity profile in a thinnonlinear medium by beam truncation and suppressingsmall-scale instabilities by spatial filtering is one possibleway of circumventing these problems (Rolland and Cor-kum, 1988). The penalty to be paid in this approach isvery high losses.

Spatially uniform spectral broadening can be achievedmuch more efficiently by guiding the self-phase-modulated beam and distributing nl over an extendedpropagation length. To shed light on the crucial role ofwave guiding and of a slow accumulation of nl wedecompose the laser beam into transverse eigenmodes

of the propagation medium and perform coupled-modeanalysis to contrast nonlinear propagation in a multi-mode waveguide with that in free space. A unified the-oretical treatment of these phenomena becomes feasibleby replacing the transverse eigenmodes of the wave-guide with the Hermite-Gaussian solutions of theparaxial wave equation (Haus, 1984) in the coupled-mode analysis of nonlinear wave propagation.

In most practical cases, the fundamental propagationmode is excited at the entrance of the nonlinear me-dium. The nonlinear index n2 has two major implica-tions in this scenario: (i) a spatially uniform phase shift

nl() is imposed on the fundamental mode and (ii) afraction of its energy is coupled to higher-order modesduring propagation. Self-focusing and related beam de-terioriation are consequences of the latter effect. It wasrecently shown that the coupled-mode equations can besolved analytically in the limit of small depletion of thefundamental mode for nonlinear light-pulse propagationin a multimode (hollow) waveguide (Tempea and Bra-

bec, 1998b) as well as in free space (Milosevic et al.,1999). In what follows we summarize the major resultsof these analyses and draw important conclusions forimplementing few-cycle pulse generation.

First, we address free-space propagation. If the funda-mental TEM00 mode is focused into a nonlinear me-dium, the nonlinear polarization response of the me-dium couples energy from the incident fundamental intohigher-order TEMmn modes. The upper diagram of Fig.12 depicts qualitatively the fractional energy coupledfrom the TEM00 into the TEM01 mode during propaga-tion in a bulk nonlinear medium of length L positionedbetween zL/2 and zL/2, where z0 is the posi-

tion of the beam waist. A useful scale parameter for thenonlinear interaction is the nonlinear length, defined as

Lnl1

SP Mp0, (9)

where SP M is defined in Eq. (4) and p0 stands for thepeak power of the pulse. If Lnlz0 , where z0w0

2/0 denotes the Rayleigh range and w0 is the 1/e2

radius at the beam waist, only a small fraction of theenergy of the light pulse fed into the medium in thefundamental transverse mode is coupled to higher-ordermodes, and for a medium of length Lz0 a perturbativeapproach allows us to solve the coupled-mode equationsanalytically.

From this solution we can obtain the fractional power(z), coupled from the fundamental TEM00 mode tothe lowest-order higher transverse mode TEM01 , towhich coupling due to n2 is strongest, and the evolutionof the peak nonlinear phase shift nl(z ,r,) imposedon the propagating beam. In the limit of Lnlz0 , thefractional energy extracted temporarily from the TEM00mode is small and the energy coupled to higher-ordermodes is fed back to the fundamental mode behind thebeam waist in a lossless and dispersion-free medium. Asa consequence, the output pulse is delivered in a Gauss-ian beam and carries a spatially uniform phase shift (i.e.,

nl is independent of the radial coordinate r). (z)(solid line) and nl(z ,0) (dashed line) calculatedunder these conditions are plotted qualitatively in theupper diagram of Fig. 12. The energy flow between theTEM00 and TEMmn modes decreases with increasingmode order and changes its sign an increasing number oftimes for increasing mode order between the fundamen-tal and higher-order modes.

The maximum energy transferred to the mode TEM01at the beam waist and the peak nonlinear phase shiftcarried by the pulse exiting the medium are given by(Tempea and Brabec, 1998b)




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0 2

4

p0

pc ,fs

2

and nl,ou t 0 ,

(10)respectively, where

z0

Lnland pc ,fs

02

n2. (11)

The requirement of small depletion of the fundamentalmode can be formulated as 2/41, which can be rewrit-ten in terms of (p0 /pc ,fs )

21, where pc ,fs determines

the power at which self-focusing tends to introduce se-vere distortions to the laser beam propagating in freespace. This power identified by coupled-mode analysis isin reasonable agreement with the well-known critical

power for self-focusing in a bulk medium pcfs0

2/(8n2), where fs3.8 6.4 is a correction fac-tor acquired from numerical investigations (Sheik-Bahae et al., 1984).

The requirement of2/41, which is to be fulfilled forspatially uniform self-phase-modulation and weak beamdeterioration, implies that nl,ou t(0) can at maxi-mum be equal to . Such an SPM-induced peak non-linear phase shift, when carried by a near-bandwidth-limited pulse, can induce relative spectral broadening bya factor of 2 (Agrawal, 1995), which determines themaximum compression factor in spatially uniform tem-poral compression that is achievable after a single passthrough a bulk nonlinear medium. For generating alarger spatially uniform nonlinear phase shift, the pro-cess needs to be repeated several times by refocusing thebeam with a suitable array of lenses or mirrors. For largecompression factors requiring many passes, the resultantdiscrete guiding configuration tends to become im-practical.

Guiding of high-power laser pulses can be imple-mented much more conveniently and efficiently in a hol-low waveguide (Marcatili and Schmeltzer, 1964). Fillingthe waveguide with some gas can introduce a Kerr (orother) nonlinearity required for spectral broadening ofhigh-power laser pulses and their subsequent temporalcompression (Nisoli et al., 1996). In the limit of smalldepletion, the energy transferred from the fundamentallinearly polarized mode LP01 , which can be excited atthe waveguide entrance by a Gaussian input beam withnear-100% efficiency (Nisoli et al., 1998), into thehigher-order modes LPmn (m0 and n2, where form1 the mode-coupling constant is zero) is fully re-turned to the fundamental mode over a propagationlength comparable to z0 of the input beam (Tempea and

Brabec, 1998b).The lower part of Fig. 12 depicts qualitatively the evo-

lution of the energy coupled into the LP02 mode duringpropagation through the waveguide. The energy oscilla-tion period is twice the mutual coherence length be-tween LP01 and LP0n , Ln/ 0

(n)0

(1 ) . Under opti-mum input coupling conditions for exciting thefundamental mode, which is achieved for w0(2/3) a ,where a is the bore radius of the hollow waveguide andw0 is the beam waist of the incoming beam (Nisoli et al.,1998), the coherence length between the two lowest-order modes L2 can be approximately expressed as L21.1z0 . The propagation constant of the LP0n mode,

0(n) , defined in Eq. (27), decreases with increasingmode order and so does the maximum transferred en-ergy, which is proportional to Ln

2 .The bottom diagram of Fig. 12 depicts the periodic

energy exchange between the fundamental mode andthe LP02 mode, for which the oscillating energy ismaximum, as a function of the propagation length inthe waveguide. For low depletion of the fundamentalmode, the maximum fractional energy coupled to theLP02 mode and the evolution of the peak nonlinearphase shift, which is uniform across the beam, can bewritten as

FIG. 12. Schematic of pulse evolution in a Kerr nonlinearitywithout and with guiding. Upper graphs, free space propaga-tion in a Kerr nonlinear medium with a length L much longerthan the nonlinear length Lnl , which in turn is assumed to bemuch longer than the Rayleigh range z 0 of the beam (for defi-nitions see text) and the signal coupled from the fundamental

(TEM00) Gaussian mode into the next higher-order TEM01mode vs propagation distance: solid curve, the fractional en-ergy transfer (z) from the TEM00 into the TEM01 mode inthe limit of(0 )2/41. Under this condition coupling tohigher-order modes may be neglected. Dashed curve, the non-linear phase shift at the peak of the pulse, which is indepen-dent of the transverse coordinates as long as the condition ofweak coupling, (0 )1, is met. The dominant contribution tothe nonlinear phase shift nl,ou t(0) , which determinesKerr-induced spectral broadening, originates from the regionz0zz0 . Lower graphs, guided-wave propagation in a hol-low fiber filled with some noble gas to introduce a Kerr non-linearity. Here (z) represents the fraction of the energy ofthe pulse carried in the fundamental LP01 mode that is trans-

ferred into the next higher-order LP02 mode by means of n2 .In the limit of(L2)(/)21, only a small amount of en-

ergy is coupled from the fundamental into the next highermode, which oscillates between the two modes with a period-icity equal to twice the coherence length L2 . Under this con-dition a spatially uniform nonlinear phase shift can grow tolengths much longer than the coherence length. This is in con-trast with propagation in a bulk nonlinear medium and resultsfrom confinement of the beam, keeping the intensity high overpropagation lengths far beyond z0 .




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L 2

2

p0

pc ,wg

2

and

nl z , 0 z

L2

z

z0, (12)

respectively (see Fig. 12), where

pc ,wg 02

2n2(13)

is the critical power for self-focusing in the multimodehollow waveguide and pc ,wg1.5pc ,fs , i.e., self-focusingin waveguides and in free space sets in at comparablepower levels (Milosevic et al., 1999). For w0

23 a the

coupling coefficient is approximately defined by Eqs.(11), (9), and (4) with w0 replaced by a (Tempea andBrabec, 1998b). Most importantly, meeting the require-ment of weak depletion, (/)21, implies the feasibil-ity of accumulating a spatially uniform SPM-inducedphase shift over many times the Rayleigh range z0 of theincident Gaussian beam. As a matter of fact, Fig. 13

reveals that a spatially uniform phase shift of nlcan be imposed on the pulse over a propagation lengthof 2L2 without the onset of self-focusing, implying aspectral broadening of a factor of 10 over L20z0 .Enhancing SPM-induced spectral broadening by increas-ing the propagation length is limited only by the propa-gation loss suffered by the fundamental mode in theleaky waveguide and/or by distortion of the temporalpulse shape.

It is instructive to look at nl(z,r,) in the oppositelimit, LL2 , which is shown, under the same experi-mental conditions, in Fig. 14. The plot in Fig. 14 is the

result of superimposing an infinite number of high-ordermodes, many of which with LnL have comparable am-plitudes, on the fundamental LP01 mode. For propaga-tion lengths much shorter than the coherence length L2 ,the nonlinear phase shift mimics not only the temporalshape of the pulse but also the transverse intensity dis-tribution of the fundamental mode. It is this latter effectthat gives rise to self-focusing if the peak power be-comes so high that / approaches unity and depletionof the fundamental mode becomes appreciable. Analo-gously, strong self-focusing emerges in a bulk nonlinearmedium as p0pc ,fs . In order to avoid catastrophic ef-fects, L must be significantly shorter than z0 .

These are the experimental conditions relevant to pas-sively mode-locked femtosecond Ti:sapphire and othersolid-state laser oscillators. These systems are capable ofdelivering ultrashort pulses in the fundamental TEM00mode, which is in apparent contradiction to coupling anappreciable amount of energy from the fundamentalinto higher-order transverse modes by the radially vary-ing nonlinear phase shift nl(r,) induced by the non-linear index of the gain medium. This paradox can beresolved by considering the spatial filtering action of alaser resonator, which efficiently suppresses the emerg-

ing high-order modes and can lead to a TEM00 outputbeam even in the presence of strong nonlinearities in thecavity.

We may therefore conclude that the same nonlinearoptical process in strongly different parameter regimes,Lz0 and p0 comparable to pc on the one hand, andLz0 and p0pc on the other hand, is responsible, incombination with broadband GDD control, for the gen-eration of low-energy few-cycle light pulses in laser os-cillators and for the generation of few-cycle pulses atmuch higher power levels by guided-wave compressionfollowing amplification.

FIG. 13. Phase shift nl(zL,r,) imposed by the Kerr non-linearity of argon gas at a pressure of 4 bars (n241019 cm/W) in a fused-silica hollow waveguide having achannel diameter of 140 m and a length of Lnl29L229 cm upon a pulse of peak power p03.5 GW injected intothe fundamental LP01 mode at the entrance of the waveguide(Tempea and Brabec, 1998b). For these parameters, the pulsepeak power is lower than the critical power for self-focusing,

p c,wg8 GW. Under this condition, the power coupled intohigher modes remains small (0.25), allowing the accumu-lation of spatially uniform self-phase-modulation on the pulsepropagating primarily in the fundamental mode. This self-phase-modulation grows linearly with propagation distance un-til propagation losses become significant.

FIG. 14. Self-phase-modulation-induced phase nl(z,r,)imposed by the Kerr nonlinearity on a pulse in a hollow wave-guide as a function of the radial and the temporal coordinatesunder conditions described in the caption of Fig. 13 for apropagation length ofzL2/50 (Tempea and Brabec, 1998b).For zL 2 , the signals coupled into different higher-ordermodes are comparable; therefore all coupling channels mustbe taken into account, resulting in a spatially varying phaseshift that follows the transverse intensity profile of the funda-mental LP01 mode. For pulse peak powers p0 approaching

pc,wg this phenomenon causes self-focusing of the beam, de-

stroying propagation in the fundamental mode.




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C. Few-cycle pulse generation: Current state of the art

Optical pulse compression based on the interplay be-tween self-phase-modulation induced by the opticalKerr effect and negative group delay dispersion is thekey concept for light-pulse generation in the few-cycleregime. Currently it is most efficiently implemented in

mirror-dispersion-controlled (MDC) Kerr-lens mode-locked (KLM) Ti:sapphire laser oscillators and hollow-fiber chirped-mirror compressors.

In the former systems, schematically illustrated in Fig.15, SPM in the laser crystal and negative GDD in thechirped mirrors M1M4 act alternately many thousandtimes before the stationary pulse duration in the sub-10-fs regime is reached. The pulse is moderately chirpedand spectrally broadened upon passage through the Ti:Scrystal, which is converted in a temporal compression inthe broadband chirped resonator mirrors. In the steadystate this pulse, shortening process is stopped by the fi-nite bandwidth over which nearly constant negativeGDD can be introduced by the mirrors and/or balancedby the finite gain and resonator bandwidth.

Solitary pulse formation inside a laser oscillator needsto be assisted by a fast saturable-absorber-like self-amptitude modulation (SAM), which is introduced byKLM in the Ti:S oscillator, as described in the previoussection. The role of this SAM action is (i) to initiate andsupport the formation of a short pulse in the cavity and(ii) to stabilize the SPM/GDD-dominated shorteningprocess and eventually the stationary state against low-intensity noise. This noise arises inherently due to gainand positive feedback in the pulse wings and can be ef-ficiently filtered by SAM outside the short interval com-prising the mode-locked pulse (see Fig. 3).

When pulse compression external to a laser oscillatoris implemented, both SPM and GDD usually act onlyonce and in this sequence, as illustrated in Fig. 16.Hence spectral broadening in the Kerr medium is muchstronger than in the case of intracavity solitary pulseformation. The absence of quasiperiodic pulse evolutionand gain in extracavity pulse compression implies thatinstabilities and noise cannot grow. Hence filtering bySAM is not absolutely necessary. Summing up, bothSPM and GDD tend to impose a much stronger modifi-cation on the light pulse in extracavity compressors thandoes compression inside the oscillator. In addition, intra-

and extracavity spectral broadening in a KLM Ti:S laserand a multimode hollow waveguide take place in en-tirely different physical regimes, as described at the endof the previous subsection. Table I contrasts the param-eter regimes of KLM/MDC Ti:S oscillators andmultimode-hollow-waveguide/chirped-mirror compres-sors by listing typical values of the parameters relevantto few-cycle pulse formation.

KLM/MDC Ti:S laser oscillators routinely generatesub-10-fs pulses at typical pulse energy levels of a fewnanojoules and multi-MHz repetition rates. The simplestembodiment of this system is depicted in Fig. 15. It iscapable of producing pulses down to 7 fs in durationwith peak powers up to 1 MW (Xu et al., 1997; 1998;Beddard et al., 1999) and exhibits unparalleled noisecharacteristics in the sub-50-fs regime while beingpumped with a low-noise diode-pumped solid-state laser(Poppe et al., 1998). Incorporating a pair of low-dispersion prisms as a source of adjustable GDD in ad-dition to the chirped mirrors and employing speciallydesigned output couplers in KLM/MDC Ti:S lasers re-cently pushed the record below 6 fs in laser oscillators(Morgner et al., 1999; Sutter et al., 1999). Figure 17shows the intensity envelope and phase of sub-6-fs

pulses (Gallmann et al., 1999) obtained from a SPIDERmeasurement (spectral phase interferometry for directelectric-field reconstruction; Walmsley and Wong, 1996).The pulses are generated by a self-starting KLM/MDCTi:S laser developed by U. Keller and co-workers at theEidgenossische Technische Hochschule (ETH) Zurichwith energies of 24 nJ at peak power levels of approxi-mately 0.5 MW (Sutter et al., 1999).

The pulse energy can be enhanced by extending thecavity with a telescope and reducing the repetition rate(Cho et al., 1999; Libertun et al., 1999). Implementingthis technique in a KLM/MDC Ti:S laser has yielded

FIG. 15. Schematic of a mirror-dispersion-controlled Ti:sap-phire laser made up of chirped mirrors (M1-M4), a broadbandoutput coupler (OC), and a thin, highly doped Ti:sapphirecrystal (Ti:S); for more details see, for example, Xu et al.(1997).

FIG. 16. Schematic experimental setup of a hollow-fiberchirped-mirror high-energy pulse compressor. The input pulseis coupled into a gas-filled hollow fiber, where its spectrum isbroadened by self-phase-modulation imposed by the opticalKerr effect in the gas. Subsequently, the chirp of the broad-ened pulse is removed upon reflection off broadband chirpedmirrors [see diagrams (a) and (b) in Fig. 10], which leads to atemporal compression of the laser pulse (courtesy of A.Poppe).




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sub-10-fs pulses with energies in excess of 30 nJ, giving

rise to peak powers of3 MW at a repetition rate of 25MHz (Poppe, Lenzner, et al., 1999). This value ap-proaches peak power levels which until recently couldonly be attained with cavity-dumped Ti:S lasers at high(1 MHz) repetition rates (Ramaswamy et al., 1993;Pshenichnikov et al., 1994). The lower repetition rate (ascompared to 80100 MHz typically) of extended-cavityand cavity-dumped Ti:S oscillators not only enhancesthe peak power to levels allowing further pulse compres-sion below 5 fs (Baltuska et al., 1997a, 1997b; Pshenich-nikov et al., 1998), but also reduces thermal load to thesample in ultrafast spectroscopy.

The highest peak intensities that can be expected

from oscillators in the few-cycle sub-10-fs regime ap-proach 1014 W/cm2 (Xu et al., 1998; Jasapara and Ru-dolph, 1999; Poppe, Lenzner, et al., 1999). Althoughstrong-field effects may arise at these intensity levels insome systems, the exploration (and possibly exploita-

tion) of a wide range of strong-field phenomena calls for

substantially higher pulse energies, which can only beachieved by external amplification. This process is inca-pable of preserving the duration of few-cycle seed pulseseven if laser media with the broadest amplification bandavailable to date (see Fig. 4) are used. Gain narrowing isinherently coupled to a high gain, limiting the relativebandwidth /0 of the amplified pulses typically to afew percent or less. Because wave packets comprising

just a few field oscillation cycles are characterized by/00.1, the generation of intense few-cycle lightpulses requires pulse compression after amplification.

This can be implemented by spectral broadening in amultimode gas-filled hollow fiber (Nisoli et al., 1996;Nisoli, De Silvestri, et al., 1997; Nisoli, Stagira, et al.,1997) followed by temporal compression upon reflectionoff chirped mirrors (Sartania et al., 1997; Tempea andBrabec, 1998a), as illustrated schematically in Fig. 16. Instrong contrast with pulse shortening in the oscillator,broadening of the laser-pulse spectrum can be effecteddramatically upon one single passage through the non-linear medium, owing to guided-wave propagation as ex-plained above and shown in Fig. 18. Seeded with pulsesof 2025 fs in duration and 1.51.7 mJ in energy, thesystem described in the caption of Fig. 16 produces 5-fspulses of energy of 0.70.8 mJ, implying a peak power ofsome 0.15 TW, at a repetition rate of 1 kHz. The tech-nique is expected to be scalable to peak powers ap-

proaching the terawatt level. The pulses are delivered ina diffraction-limited beam and have recently been fo-cused to intensities approaching 1018 W/cm2. Figure 19shows the temporal intensity envelope and the phase(Cheng et al., 1999) as evaluated from a frequency-resolved optical gating measurement (Kane and Tre-bino, 1993; Trebino et al., 1997) with 0 arbitrarily cho-sen to be zero. The clean front edge of the pulse is ofprime importance for strong-field experiments. The timeit takes for the intensity to rise from 10 to 90% of thepeak is less than 5 fs and hence less than two field oscil-lation periods.

TABLE I. Parameter ranges for few-cycle pulse generation in Kerr-lens-mode-locked/mirror-dispersion-controlled Ti:S oscillatorsat low energy levels and in hollow-waveguide/chirped-mirror compressors at high energy levels.

Systemsparameters

KLM/MDC Ti:sapphireoscillator

Multi-mode hollow-waveguide/chirped-mirror compressor

Nonlinear interaction length z0 z0p0 /pc 1 1

pc 2.5 MW 10 GW1 TW

nl(

0) 0.1

0.5 5

10Spectral broadening/pass 1.11.3 510

Net negative group delay dispersion D 1030fs2 1030fs2

Compression factor/pass 1.11.3 510

Self-amplitude modulation depth 13 %

Output pulse energy 330 nJ 0.011 mJ

Output pulse duration 58 fs 47 fs

Output peak power 0.33 MW 1 GW0.2 TW

Peak intensity in focus up to 1014 W/cm2 up to 1018 W/cm2

FIG. 17. Spectral intensity and phase (insert) and intensityenvelope of sub-6-fs pulses produced by a semiconductor-saturable-absorber-initiated Kerr-lens mode-locking mirror-dispersion-controlled Ti:S laser (Sutter et al., 1999). The datahave been evaluated from a SPIDER (spectral phase interfer-ometry for direct electric-field reconstruction) measurement(Gallmann et al., 1999).




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To test the limits of currently available chirped mirrortechnology, sub-6-fs pulses from the compressor havebeen gently focused in atmospheric air at reduced en-ergy levels and slightly broadened temporally (resultingin some positive chirp) such that the nonlinear index of

air broadened the spectrum by some 30% uniformlyacross the laser beam, as predicted by the coupled-modeanalysis reviewed in the previous subsection. Subse-quent reflection off a few ultrabroadband chirped mir-rors [see diagrams (a) and (b) in Fig. 10] compensatedfor the GDD and third-order dispersion of air and theSPM-induced chirp and resulted in the interferometricautocorrelation (Diels, Fontaine, et al., 1985) depictedby the solid line in Fig. 20. The dotted line is calculatedfrom the inverse Fourier transform of the measuredspectrum (corrected for the spectrograph and detectorresponse) under the assumption of no spectral phasemodulation, yielding a pulse duration of 4 fs at a peak

power of 30 GW (Cheng et al., 1998). These are theshortest electromagnetic pulses demonstrated to date,comprising some one and a half field oscillation cycleswithin the full width at half maximum of their intensityenvelope. These unique characteristics have far-reachingimpacts on strong-field physics, some of which will beaddressed in the remaining part of this paper.

D. Approaching the light oscillation period: Does theabsolute phase of light matter?

Generating ultrashort pulses with durations approach-ing the light oscillation period To2/00 /c bringsup several questions for experimentalist and theoristalike: Are the techniques used for characterization inthe multicycle regime still adequate and do they providecomplete information about the characteristics of few-cycle wave packets? Does the description of ultrashortpulses in terms of carrier and envelope remain valid asthe pulse duration p approaches To? These questionsare not quite independent of each other, given the factthat the determination ofp , defined as the full width athalf maximum of the intensity envelope, relies on aphysically meaningful definition of the intensity enve-

lope.First let us address the latter question, in order to

provide a convenient mathematical framework for ad-dressing the former. The complex amplitude (envelope)

Ea(t) of pulsed radiation permits the electric field E(t)to be expressed as

E t Ea t ei0ti0c.c. (14)

For multicycle pulses, Ea(t) can be derived from E(t)

by introducing the complex field E(t)

(2)1/2 0E()eitd, in which E()

(2)1/2 E( t)e itdt. This obeys E(t)E( t)c.c.,

and hence E

a(t) can be determined from E

(t)Ea(t)e

i0ti0, provided that 0 and 0 are known.The tilde represents a complex quantity throughout the

paper. In order to avoid fast oscillations in Ea( t), thereference frequency 0 must be roughly at the center ofthe spectrum of the wave packet. The precise choice of0 is not critical for describing pulse propagation on thebasis of Eq. (14). There are several possible definitionsof0 , one of which,

0 0

E 2d

0 E 2d

, (15)

FIG. 18. Typical spectrum before (dashed line) and after (solidline) propagation through a hollow fiber having a channel di-ameter of 250 m and a length of approximately 1 m. Thewaveguide is seeded with 25-fs, 1.5-mJ input pulses and filledwith neon at a pressure of1 bar.

FIG. 19. Millijoule-scale pulses exiting the hollow-fiberchirped-mirror compressor: solid curve, intensity envelope;dotted curve, phase, with 0 chosen arbitrarily to be equal tozero; inset, spectrum. The data were retrieved from afrequency-resolved optical gating measurement (Cheng et al.,1999). The pulse duration evaluated is p5.3 fs.

FIG. 20. Intense pulses originating from double-stage pulsecompression, as described in the text and reported by Chenget al. (1998): solid curve, measured interferometric autocorre-lation; dotted curve, calculation from the inverse Fourier trans-form of the measured spectrum in the absence of a frequency-dependent phase, yielding a FWHM pulse duration of p4 fs; inset, spectrum.




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stands out for its minimization of the intensity-weighted

phase variation of Ea(t) (Diels and Rudolph, 1996). In

spite of this important characteristic, there is some arbi-trariness in the above definition of 0 . Therefore it isimportant to stress that the major conclusions from thefollowing considerations are not affected by the specificchoice of a definition for 0 . Now with 0 defined and

0 chosen arbitrarily so that the imaginary part of Ea( t)is zero at the reference instant t0 (which is, for prac-tical reasons, often adjusted to coincide with the center

of gravity of E( t) 2), the route to determining Ea( t)from E( t) is unambiguously prescribed.

The above definition of the complex amplitude enve-lope of a transient wave form (wave packet) is appar-ently valid irrespective of the duration of the wave

packet. However, to legitimize the concept of carrierand envelope we must require that 0 and Ea( t) remaininvariantunder a change of0 , the physical significanceof which is confined to determining the relative positionof the carrier wave with respect to the envelope. A shift

of 0 by some yields the new wave form E( t)

Ea( t)ei0ti(0), i.e., translates the carrier with

respect to the envelope. In order that the definition of

Ea( t) and 0 be self-consistent, the modified wave

E( t)E(t)e i must yield 00 , which also implies

Ea( t)Ea( t). Intuitively, one expects the concepts ofcarrier and envelope to fail as the temporal extension of

the transient wave form becomes comparable to the os-cillation cycle of the wave.

In order to develop some feel for the parameter rangein which one can rely on the carrier-envelope concept,we have computed 0 for E(t)Eanal sin(t) and 0 for

its phase-translated counterpart E(t)E( t)e i forsome of the most important analytic pulseforms. /2 was chosen to maximize the expected differencebetween 0 and 0 for short pulses. Figure 21 depicts 00 /0 as a function of the pulse duration normal-ized to the oscillation period Tosc2/ . As antici-pated, the requirement of phase invariance of carrier

frequency and hence that of the envelope is met with ahigh accuracy for pulse durations that are long com-pared to Tosc . It is, however, somewhat surprising thatthis requirement is still well satisfied for extremely shortwave forms with durations comparable to the carrier os-cillation period Tosc and tends to be increasingly vio-lated only as the pulse duration p [full width at half

maximum (FWHM)] of the intensity envelope Ea( t)2)

becomes shorter than the carrier period. It can be seenby inspection that this finding applies irrespective of theprecise choice of either 0 or the temporal profile of(bell-shaped) wave packets.

In the multicycle regime the complex envelope Ea(t)can also be used to calculate the cycle-averaged radia-

tion intensity I( t

intense few-cycle laser fields_frontiers of nonlinear optics.pdf

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