FemtosecondLaserSpectroscopy

Edited byPeter Hannaford

FemtosecondLaserSpectroscopy

Springer

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Contents

Contributing authors

Foreword

Preface

xi

xv

xix

1. Phase Controlled Femtosecond Lasers for Sensitive,Precise and Wide Bandwidth Nonlinear Spectroscopy 1Jun Ye1.2.3.

4.

5.

6.

Introduction to femtosecond optical frequency combPrecision atomic spectroscopy – structure and dynamicsMolecular Spectroscopy aided by femtosecond opticalfrequency comb

18

12hyperfine interactions, optical frequency standards and

clocksExtension of phase-coherent fs combs to the mid-IR spectralregionFemtosecond lasers and external optical cavitiesReferences

14

192126

2. Supercontinuum and High-Order Harmonics: “Extreme”Coherent Sources for Atomic Spectroscopy and Attophysics 29Marco Bellini1.2.

IntroductionHigh-resolution spectroscopy with ultrashort pulses

2930

vi Femtosecond Laser Spectroscopy

3. High-order harmonics3.13.23.33.43.5

Basic principlesPhase coherence in harmonic generationSome insight into the microscopic generation processCollinear, phase-coherent, harmonic pulsesRamsey spectroscopy with high-order harmonics

4. Supercontinuum4.14.2

4.34.4

Basic principlesPhase preservation in the supercontinuum generationprocessCollinear, phase-coherent, supercontinuum pulsesMultiple-beam interference from an array of super-continuum sources: a spatial comb

Phase preservation in chirped-pulse amplificationFrequency combs, absolute phase control, and attosecond pulsesConclusionsReferences

5.6.7.

3333333537394242

4344

4954555757

3. The Measurement of Ultrashort Light – Simple Devices,Complex Pulses

Xun Gu, Selcuk Akturk, Aparna Shreenath, Qiang Cao andRick Trebino1.2.3.

4.5.6.

IntroductionFROG and cross-correlation FROGDithered-crystal XFROG for measuring ultracomplexsupercontinuum pulsesOPA XFROG for measuring ultraweak fluorescenceExtremely simple FROG deviceConclusionsReferences

4. Femtosecond Combs for Precision MetrologyS.N. Bagayev, V.I. Denisov, V.M. Klementyev, I.I. Korel,S.A. Kuznetsov, V.S. Pivtsov and V.F. Zakharyash1.2.3.4.

5.

IntroductionThe use of femtosecond comb for creation of an optical clockSpectral broadening of femtosecond pulses in tapered fibresFrequency stability of femtosecond comb by passage offemtosecond pulses through a tapered fibreConclusionsReferences

61

6163

6468758586

87

878994

102106107

Femtosecond Laser Spectroscopy vii

5. Infrared Precision Spectroscopy using Femtosecond-Laser-Based Optical Frequency-Comb SynthesizersP. De Natale, P. Cancio and D. Mazzotti1.

2.3.

4.

5.

Evolution of metrological sources in the IR: fromsynthesized frequency chains to fs-optical frequency combsMolecular transitions for IR frequency metrologyIR coherent sources3.13.2

Present coherent sourcesFuture IR sources and materials

Extending visible/near-IR fs combs to the mid-IR4.14.2

Visible/near-IR combsBridging the gap with difference frequency generationand optical parametric oscillators

Conclusions and perspectives for IR combsReferences

6. Real-Time Spectroscopy of Molecular Vibrations withSub-5-Fs Visible PulsesTakayoshi Kobayashi1.2.

IntroductionExperimental2.12.22.3

SampleStationary absorption and Raman spectraSub-5-fs real-time pump-probe apparatus

3. Results and discussion3.13.23.33.43.53.63.7

3.8

Real-time spectraDynamics of the electronic statesTwo-dimensional real-time spectrumDynamics of excitonic statesAnalysis of coherent molecular vibrationAnalysis of phase and amplitude of oscillationExciton-vibration interaction3.7.13.7.2

3.7.3

3.7.4

Quantum beat between different n exciton statesWave-packet motion on ground-state potentialenergy surfaceWave-packet motion on excited-state potentialenergy surfaceDynamic intensity borrowing

Theoretical analysis of results3.8.13.8.2

Herzberg-Teller type wave-packet motionEvaluation of amount of modulated transition

109

110112115115118120120

122126127

133

134137137137139144144146148150150152153153

154

155156159159

viii Femtosecond Laser Spectroscopy

3.8.3dipole momentEvaluation of magnitude of the oscillatorstrength transfer

4. ConclusionsReferences

7. Vibrational Echo Correlation Spectroscopy: A New Probeof Hydrogen Bond Dynamics in Water and MethanolJohn B. Asbury, Tobias Steinel and M.D. Fayer1.2.3.

IntroductionExperimental ProceduresResults and Discussion3.13.23.33.4

Hydrogen bond population dynamics in MeODPhotoproduct band spectral diffusion in MeODStructural evolution in water, an overviewLocal structure dependent evolution in water

4. Concluding remarksReferences

8. Spectrally Resolved Two-Colour Femtosecond Photon EchoesLap Van Dao, Craig Lincoln, Martin Lowe and Peter Hannaford1.2.

IntroductionPhysical Principles2.12.22.3

Bloch equation descriptionNonlinear optical response theorySpectrally resolved photon echoes

3.4.

ExperimentalSpectrally resolved photon echoes4.14.24.3

One-colour two-pulse photon echoesOne-colour three-pulse photon echoesTwo-colour three-pulse photon echoes4.3.14.3.2

Detection ofDetection of

5. Molecular systems5.15.2

Dye moleculesSemiconductor materials5.2.15.2.2

Gallium nitrideSemiconductor quantum dots

5.3 Biological molecules6. Summary and future directions

References

161

163164164

167

168170174174179184190193195

197

197199199202205207208208209211211216217217219219220221222223

Femtosecond Laser Spectroscopy ix

9. Optimal Control of Atomic, Molecular and ElectronDynamics with Tailored Femtosecond Laser PulsesTobias Brixner, Thomas Pfeifer, Gustav GerberMatthias Wollenhaupt and Thomas Baumert1.2.

IntroductionOne-parameter control on prototypes: atoms and dimersin the gas phase2.1 Control in the perturbative limit

2.1.12.1.22.1.3

Excitation schemeControl via the Tannor-Kosloff-Rice schemeControl via simple shaped pulses

2.2 Control in strong laser fields2.2.12.2.2

Coherent coupling of molecular electronic statesCoherent coupling of atomic electronic states– control beyond population transfer and

spectral interferences3. Many-parameter control in the gas phase

3.13.23.33.43.5

Closed-loop femtosecond pulse shapingControl of product ratiosBond-selective photochemistryOrganic chemical conversionMultiple optimization goals

4. Many-parameter control in the liquid phase4.14.2

Control of metal-ligand charge-transfer excitationControl of photo-isomerization

5. Coherent control of electron motion5.15.2

Coherent transfer to free electronsSelective optimization of high-order harmonicgeneration

6. ConcusionsReferences

10. Coherent Control of Atomic Dynamics with Chirped andShaped PulsesBéatrice Chatel and Bertrand Girard1.2.3.

IntroductionChirped pulses and pulse shapersObservation and control of coherent transients in one-photontransitions3.13.23.3

IntroductionControl of transient dynamics with shaped pulsesChirped pulses in the weak field regime: observation

225

226

228229229232235238238

241243244247248249251252253254255255

258262263

267

271271273

267269

x Femtosecond Laser Spectroscopy

3.4

3.5

3.63.7

of coherent transientsControl of coherent transients with simple spectralshapesControl of coherent transients with simple temporalshapes: temporal Fresnel lensesStrong field: saturation of coherent transientsReconstruction of wave function and laser pulse fromcoherent transients

4. Control of two-photon transitions – quantum ladder climbing4.1

4.2

4.3

Weak-field two-photon transition with a non-resonantintermediate stateWeak-field two-photon transition with a resonantintermediate stateAdiabatic excitation of a quantum ladder

5. ConclusionReferences

273

277

279282

283285

287

290297299300

11. Ultrafast Processes of Highly Excited Wide-Gap DielectricThin FilmsM. Mero, J. Zeller and W. Rudolph1.2.

IntroductionModelling of processes following fs pulse excitation2.1 Microscopic model based on the Boltzmann equation

2.1.12.1.22.1.32.1.42.1.5

PhotoionizationElectron-electron interactionImpact ionizationElectron-phonon-photon interactionCarrier-decay into defects

2.2 Phenomenological model of dielectric breakdown3.4.

Dielectric breakdown behaviour of oxide thin filmsTime-resolved reflection and transmission studies4.14.24.3

ExperimentsRetrieval of the dielectric constantInterpretation of the experiments

5.6.

Comparison of experiment and theorySummaryReferences

Index

305

305307307309310311312314316318322322322324325327328

331

Contributing Authors

Number in parentheses indicates first page of author’s contribution.

S. AKTURK (61), School of Physics, Georgia Institute of Technology,Atlanta, Georgia 30332-0430, USA

J. B. ASBURY (167), Department of Chemistry, Stanford University,Stanford, CA 94305, USA

S.N. BAGAYEV (87), Institute of Laser Physics, Siberian Branch, RussianAcademy of Sciences, Pr. Lavrentieva, 13/3, 630090 Novosibirsk, Russia

T. BAUMERT (225), Institute of Physics, University of Kassel, Heinrich-Plett-Str. 40, 34132 Kassel, Germany

M. BELLINI (29), Istituto Nazionale di Ottica Applicata (INOA), LargoFermi 6, 50125, Florence, Italy

T. BRIXNER (225), Physics Department, University of Würzburg, AmHubland, 97074 Würzburg, Germany

P. CANCIO (109), Istituto Nazionale di Ottica Applicata (INOA), LargoFermi 6, 50125 Florence, Italy, and European Laboratory for NonlinearSpectroscopy (LENS), Via Carrara 1, 50019 Sesto Fiorentino FI, Italy

xii Femtosecond Laser Spectroscopy

Q. CAO (61), School of Physics, Georgia Institute of Technology, Atlanta,Georgia 30332-0430, USA

B. CHATEL (267), Laboratoire de Collisions, Agrégats et Réactivité (CNRSUMR 5589), IRSAMC, Université Paul Sabatier, 31062 Toulouse CEDEX,France

L.V. DAO (197), Centre for Atom Optics and Ultrafast Spectroscopy,Swinburne University of Technology, PO Box 218, Hawthorn, Victoria3122, Australia

P. DE NATALE (109), Istituto Nazionale di Ottica Applicata (INOA), LargoFermi 6, 50125 Florence, Italy, and European Laboratory for NonlinearSpectroscopy (LENS), Via Carrara 1, 50019 Sesto Fiorentino FI, Italy

V.I. DENISOV (87), Institute of Laser Physics, Siberian Branch, RussianAcademy of Sciences, Pr. Lavrentieva, 13/3, 630090 Novosibirsk, Russia

M.D. FAYER (167), Department of Chemistry, Stanford University, Stanford,CA 94305, USA

G. GERBER (225), Physics Department, University of Würzburg, AmHubland, 97074 Würzburg, Germany

B. GIRARD (267), Laboratoire de Collisions, Agrégats et Réactivité (CNRSUMR 5589), IRSAMC, Université Paul Sabatier, 31062 Toulouse CEDEX,France

X. GU (61), School of Physics, Georgia Institute of Technology, Atlanta,Georgia 30332-0430, USA

P. HANNAFORD (197), Centre for Atom Optics and Ultrafast Spectroscopy,Swinburne University of Technology, PO Box 218, Hawthorn, Victoria3122, Australia

V.M. KLEMENTYEV (87), Institute of Laser Physics, Siberian Branch,Russian Academy of Sciences, Pr. Lavrentieva, 13/3, 630090 Novosibirsk,Russia

T. KOBAYASHI (133), Department of Physics, University of Tokyo, Hongo7-3-1, Bunkyo, Tokyo 113-0033, Japan

Femtosecond Laser Spectroscopy xiii

I.I. KOREL (87), Institute of Laser Physics, Siberian Branch, RussianAcademy of Sciences, Pr. Lavrentieva, 13/3, 630090 Novosibirsk, Russia

S.A. KUZNETSOV (87), Institute of Laser Physics, Siberian Branch, RussianAcademy of Sciences, Pr. Lavrentieva, 13/3, 630090 Novosibirsk, Russia

C. LINCOLN (197), Centre for Atom Optics and Ultrafast Spectroscopy,Swinburne University of Technology, PO Box 218, Hawthorn, Victoria3122, Australia

M. LOWE (197), Centre for Atom Optics and Ultrafast Spectroscopy,Swinburne University of Technology, PO Box 218, Hawthorn, Victoria3122, Australia

D. MAZZOTTI (109), Istituto Nazionale di Ottica Applicata (INOA), LargoFermi 6, 50125 Florence, Italy, and European Laboratory for NonlinearSpectroscopy (LENS), Via Carrara 1, 50019 Sesto Fiorentino FI, Italy

M. MERO (305), Department of Physics and Astronomy, University of NewMexico, Albuquerque NM 87131, USA

T. PFEIFER (225), Physics Department, University of Würzburg, AmHubland, 97074 Würzburg, Germany

V.S. PIVTSOV (87), Institute of Laser Physics, Siberian Branch, RussianAcademy of Sciences, Pr. Lavrentieva, 13/3, 630090 Novosibirsk, Russia

W. RUDOLPH (305), Department of Physics and Astronomy, University ofNew Mexico, Albuquerque NM 87131, USA

A. SHREENATH (61), School of Physics, Georgia Institute of Technology,Atlanta, Georgia 30332-0430, USA

T. STEINEL (167), Department of Chemistry, Stanford University, Stanford,CA 94305, USA

R. TREBINO (61), School of Physics, Georgia Institute of Technology,Atlanta, Georgia 30332-0430, USA

M. WOLLENHAUPT (225), Institute of Physics, University of Kassel,Heinrich-Plett-Str. 40, 34132 Kassel, Germany

xiv Femtosecond Laser Spectroscopy

J. YE (1), JILA, National Institute of Standards and Technology andUniversity of Colorado, Boulder, Colorado 80309-0440, USA

V.F. ZAKHARYASH (87), Institute of Laser Physics, Siberian Branch, RussianAcademy of Sciences, Pr. Lavrentieva, 13/3, 630090 Novosibirsk, Russia

J. ZELLER (305), Department of Physics and Astronomy, University of NewMexico, Albuquerque NM 87131, USA

Foreword

The embryonic development of femtoscience stems from advances madein the generation of ultrashort laser pulses. Beginning with mode-locking ofglass lasers in the 1960s, the development of dye lasers brought the pulsewidth down from picoseconds to femtoseconds. The breakthrough in solidstate laser pulse generation provided the current reliable table-top lasersystems capable of average power of about 1 watt, and peak power densityof easily watts per square centimeter, with pulse widths in therange of four to eight femtoseconds. Pulses with peak power densityreaching watts per square centimeter have been achieved in laboratorysettings and, more recently, pulses of sub-femtosecond duration have beensuccessfully generated.

As concepts and methodologies have evolved over the past two decades,the realm of ultrafast science has become vast and exciting and has impactedmany areas of chemistry, biology and physics, and other fields such asmaterials science, electrical engineering, and optical communication. Inmolecular science the explosive growth of this research is for fundamentalreasons. In femtochemistry and femtobiology chemical bonds form andbreak on the femtosecond time scale, and on this scale of time we can freezethe transition states at configurations never before seen. Even for non-reactive physical changes one is observing the most elementary of molecularprocesses. On a time scale shorter than the vibrational and rotational periodsthe ensemble behaves coherently as a single-molecule trajectory.

But these developments would not have been possible without thecrystallization of some key underlying concepts that were in the beginningshrouded in fog. First was the issue of the “uncertainty principle”, whichhad to be decisively clarified. Second was the question of whether one could

xvi Femtosecond Laser Spectroscopy

sustain wave-packet motion at the atomic scale of distance. In other words,would the de Broglie wavelength of the atom become sufficiently short todefine classical motion – “classical atoms” – and without significantquantum spreading? This too had to be clearly demonstrated and monitoredin the course of change, not only for elementary processes in molecularsystems, but also during complex biological transformations. And, finally,some questions about the uniqueness and generality of the approach had tobe addressed. For example, why not deduce the information from high-resolution frequency-domain methods and then Fourier transform to obtainthe dynamics? It is surely now clear that transient species cannot be isolatedthis way, and that there is no substitute for direct “real time” observationsthat fully exploit the intrinsic coherence of atomic and molecular motions.

Theory has enjoyed a similar explosion in areas dealing with ab initioelectronic structures, molecular dynamics, and nonlinear spectroscopies.There has been progress in calculating potential energy surfaces of reactivesystems, especially in their ground state. On excited-state surfaces it is nowfeasible to map out regions of the surface where transition states and conicalintersections are important for the outcome of change. For dynamics, newmethods have been devised for direct viewing of the motion by formulatingthe time-dependent picture, rather than solving the time-independentSchrödinger equation and subsequently constructing a temporal picture.Analytical theory has been advanced, using time-ordered density matrices, toenable the design of multidimensional spectroscopy, the analogue of 2-D(and higher) NMR spectroscopy. That the coupling between theory andexperiment is profound is evident in many of the chapters in this volume.

Other areas of studies are highlighted in this volume. The making offemtosecond combs for precision metrology and spectroscopy, and theadvances in nonlinear and multidimensional optical techniques are twoexamples of such frontiers. The ability to count optical oscillations of morethan cycles per second can potentially provide all-optical atomic clockswith a new limit of precision. Similarly, the ability to generate sub-femtosecond pulses pushes the limit and resolution toward new studies ofelectron dynamics. Besides these advances in precision (optical cycles) andpulse duration (pulse width) there are those concerned with the phase.Beginning in 1980, the phase of an optical pulse has been experimentallyunder control and pulses of well-defined phases etc) have beengenerated and utilized in, among other applications, the control of emissionfrom molecules. But only recently could composite phases be prescribedwith a feedback algorithm to control the outcome of a reactive channel, asshown in this volume. Coherent control is a frontier field stimulatingresearch in both theory and experiment.

Femtosecond Laser Spectroscopy xvii

Edited by Peter Hannaford this volume is a welcomed edition to thefield as it brings together the latest in some areas of developments with animpressive mix of new methodologies and applications. The use offemtosecond combs for precision measurements is well covered andcoherent control is presented with demonstrations for atomic, molecular andelectronic processes. Nonlinear optical methods, including novel geometriesof photon and vibrational echoes, are described for the investigation ofmolecular systems, in particular dye molecules, hydrogen-bonded networks,semiconductor quantum dots, and biomolecules. Measurements of ultrashortpulses, time-resolved reflection and transmission methods, and real-timespectroscopy with sub-5-femtosecond visible pulses provide the means forexploring new regimes and resolutions.

This book in the series on Progress in Lasers gives an exposé of somecurrent and exciting research areas in the technology of pulse generation andin the applications of femtoscience.

Ahmed ZewailCalifornia Institute of TechnologyPasadena, CaliforniaMay 2004

Preface

When I was first approached to edit a volume on Femtosecond LaserSpectroscopy in 2000, I did not anticipate that the field was about to explode,with the announcement of a series of remarkable new developments andadvances. This volume describes these recent developments in elevenchapters written by leading international researchers in the field. It includessections on:

Femtosecond optical frequency combs, which are currentlyrevolutionising ultrahigh precision spectroscopy and optical frequencymetrology;Soft X-ray femtosecond laser sources, which promise to have importantapplications in biomedical imaging;Attosecond laser sources, which will provide the next generation ofsources to study ultrafast phenomena such as electron dynamics;Novel methods for measuring and characterizing ultrashort laser pulsesand ultrashort pulses of light;Coherent control of atomic, molecular and electron dynamics withtailored femtosecond laser pulses;Real-time Spectroscopy of molecular vibrations with sub-5-fs pulses; andMultidimensional femtosecond coherent spectroscopies for studyingmolecular and electron dynamics.

Indeed, it is gratifying to see that with the recent advent of attosecond lasersources the title of this volume may soon be rendered obsolete.

I would like to thank each of the contributors for their cooperation inpreparing this volume, and Ahmed Zewail for writing the Foreword. Iappreciate the amount of work that goes into writing chapters of this typewhen the authors are heavily burdened with other demands on their time. I

xx Femtosecond Laser Spectroscopy

feel honoured and privileged to have been associated with such an eminentgroup of researchers. I also thank my co-workers in the UltrafastSpectroscopy group at Swinburne University of Technology – Lap Van Dao,Martin Lowe, Craig Lincoln, Shannon Whitlock, Xiaoming Wen, Tra MyDo, Petrissa Eckle and David McDonald – for their help and encouragementduring the preparation of this volume and for critical reading of some of thechapters. I thank Tien Kieu, Grainne Duffy and David Lau for theirassistance with the preparation of the camera-ready chapters, and GustavGerber for kindly allowing the use of Figure 9-12 on the front cover of thisvolume. Finally, I thank the publishers of the following journals and booksfor permission to reproduce material in this volume: Applied Physics B,Applied Physics Letters, Journal of Chemical Physics, Journal of Physics B,Laser Spectroscopy Proceedings, Nature, Optics Express, Optics Letters,Optical Review, Review of Scientific Instruments, Physical Review Letters,Physical Review A, and SPIE Proceedings.

Peter HannafordSwinburne University of TechnologyMelbourne, June 2004

Chapter 1

PHASE CONTROLLED FEMTOSECONDLASERS FOR SENSITIVE, PRECISE, AND WIDEBANDWIDTH NONLINEAR SPECTROSCOPY

Jun YeJILA, National Institute of Standards and Technology and University of ColoradoBoulder, Colorado 80309-0440, USAYe@jila.colorado.edu

Abstract: Recent progress in precision control of pulse repetition rate and carrier-envelope phase of ultrafast lasers has established a strong connection betweenoptical frequency metrology and ultrafast science. A wide range ofapplications has ensued, including measurement of absolute opticalfrequencies, precision laser spectroscopy, optical atomic clocks, and opticalfrequency synthesis in the frequency-domain, along with pulse timingstabilization, coherent synthesis of optical pulses, and phase-sensitive extremenonlinear optics in the time-domain. In this chapter we discuss the impact ofthe femtosecond optical frequency comb to atomic and molecularspectroscopy. Measurements performed in the frequency-domain provide aglobal picture of atomic and molecular structure at high precision whileproviding radio-frequency clock signals derived from optical standards. Time-domain analysis and experiments give us new possibilities for nonlinearoptical spectroscopy and sensitive detections with real-time information.

Key words: Phase control, femtosecond lasers, optical comb, precision frequencymetrology, nonlinear spectroscopy.

1. INTRODUCTION TO FEMTOSECOND OPTICALFREQUENCY COMB

Precise phase control of femtosecond lasers has become increasinglyimportant as novel applications utilizing the femtosecond laser-based optical

2 Chapter 1

comb are developed that require greater levels of precision and higherdegrees of coherence and control [1]. Improved stability is beneficial forboth frequency-domain applications, where the relative phase or “chirp”between comb components is unimportant (e.g., optical frequencymetrology), and, perhaps more importantly, time-domain applications wherethe pulse shape and/or duration are vital, such as in nonlinear opticalinteractions [2]. For both types of applications, minimizing jitter in the pulsetrain and noise in the carrier-envelope phase is often critical to achieve thedesired level of precision and coherence. Phase-stabilized mode-lockedfemtosecond lasers have played a key role in recent advances in opticalfrequency measurement [3-5], carrier-envelope phase stabilization [2, 6, 7],all-optical atomic clocks [8, 9], optical frequency synthesizers [10], coherentpulse synthesis [11], and broadband, phase-coherent spectral generation [12].

Mode-locked lasers generate short optical pulses by establishing a fixedphase relationship between all of the lasing longitudinal modes. Tounderstand the connection between the time-domain and frequency-domaindescriptions of a mode-locked laser and the pulse train that it emits, a keyconcept is the carrier-envelope phase. This is based on the decomposition ofthe pulses into an envelope function, Ê(t), that is superimposed on a

pulse is written as The carrier-envelope phase, is thephase shift between the peak of the envelope and the closest peak of thecarrier wave. In any dispersive material, the difference between group andphase velocities will cause to evolve. This group-phase velocitymismatch inside a laser cavity produces a pulse-to-pulse phase shiftaccumulated over one round-trip as

When is constant, the spectrum of a femtosecond optical combcorresponds to identical pulses emitted by the mode-locked laser. For asingle pulse, the spectrum is the Fourier transform of its envelope functionand is centered at the optical frequency of its carrier. Generally, for anypulse shape, the frequency width of the spectrum will be inverselyproportional to the temporal width of the envelope. For a train of identicalpulses, separated by a fixed interval, the spectrum can easily be obtained bya Fourier series expansion, yielding a comb of regularly spaced frequencies,where the comb spacing is inversely proportional to time interval betweensuccessive pulses, i.e., the inverse of the repetition rate of the laser. TheFourier relationship between time and frequency resolution guarantees thatany spectrometer with sufficient spectral resolution to distinguish theindividual comb lines cannot have enough temporal resolution to separatesuccessive pulses. Therefore, the successive pulses interfere with each otherinside the spectrometer and the comb spectrum occurs because there arecertain discrete frequencies at which the interference is constructive. Using

continuous carrier wave with frequency so that the electric field of the

1. Phase Controlled Femtosecond Lasers for Sensitive... 3

the result from Fourier analysis that a shift in time corresponds to a linearphase change with frequency, we can readily see that the constructiveinterference occurs at where n is an integer.

When is evolving with time, such that from pulse to pulse (with atime separation of there is a phase shift of then in thespectral domain a rigid shift will occur for the frequencies at which thepulses add constructively. This shift is easily determined asThus the optical frequencies, of the comb lines can be written as

where n is a large integer of order that indexes the comb line, andis the comb offset due to pulse-to-pulse phase shift,

The relationship between time- and frequency-domain pictures issummarized in Fig. 1-1. The pulse-to-pulse change in the phase for the trainof pulses emitted by a mode-locked laser can be expressed in terms of theaverage phase and group velocities inside the cavity. Specifically,

where is the round-trip length of the laser cavityand is the “carrier” frequency.

Figure 1-1. Summary of the time-frequency correspondence for a pulse train with evolvingcarrier-envelope phase.

Armed with the understanding of the frequency spectrum of a mode-locked laser, we now turn to the question of measuring the absolutefrequencies of comb lines. For a frequency measurement to be absolute, itmust be referenced to the hyperfine transition of that defines thesecond. From the relations listed above we see that determining the absoluteoptical frequencies of the femtosecond comb requires two radio frequency(RF) measurements, that of and Measurement of is straightforward;

4 Chapter 1

we simply detect the pulse train’s repetition rate (from tens of MHz toseveral GHz) with a fast photodiode. On the other hand, measurement ofis more involved as the pulse-to-pulse carrier envelope phase shift requiresinterferometric measurement, whether it is carried out in the time-domain[13] or frequency-domain [14]. When the optical spectrum spans an octavein frequency, i.e., the highest frequencies are a factor of two larger than thelowest frequencies in the spectrum, measurement of is greatly simplified.If we use a second harmonic crystal to frequency double a comb line, withindex n, from the low frequency portion of the spectrum, it will haveapproximately the same frequency as the comb line on the high frequencyside of the spectrum with index 2n. Measuring the heterodyne beat betweenthese two families of optical comb lines yields a difference frequency by

which is just the offset frequency.Thus, an octave-spanning spectrum enables a direct measurement of [6].Note that an octave-spanning spectrum is not required; it just represents thesimplest approach. We designate this scheme, shown in Fig. 1-2(a), as “self-referencing” since it uses only the output of the mode-locked laser. Self-referencing is not the only means of determining the absolute opticalfrequencies given an octave-spanning spectrum. For example, the absoluteoptical frequency of a CW laser can be determined if its frequency lies closeto comb line n in the low frequency portion of the femtosecond combspectrum. Then the second harmonic of the CW laser will be positionedclose to the comb line 2n. Measurement of the heterodyne beat between theCW laser frequency, and the comb line n givesand between the second harmonic of the CW laser and comb line 2n gives

Mixing the beats with appropriate weightingfactors giveswhich represents the second detection scheme shown in Fig. 1-2(b) [9].

An octave-bandwidth optical comb is not straight-forward to produce. AFourier-transform limited pulse with a full width at half maximum (FWHM)bandwidth of an octave centered at 800 nm would only be a single opticalcycle in duration. Such short pulses have not been achieved. Fortunately,neither a transform-limited pulse nor a FWHM of an octave is actuallyneeded. The pulse width is unimportant as the measurement and controltechniques are purely frequency domain approaches. Experimentally, it hasbeen found that even if the power at the octave spanning points is 40 dBbelow the peak, it is still possible to observe strong f-to-2f heterodyne beats.Still, the necessary comb bandwidth is larger than that from a typical sub-10-fs mode-locked Ti:sapphire laser. One approach to produce this sufficientspectral bandwidth is based on self-phase modulation directly inside theTi:Sapphire crystal itself [15] or inside an additional glass plate locatedinside the laser cavity with secondary coincident time and space foci [16].

1. Phase Controlled Femtosecond Lasers for Sensitive... 5

These techniques require carefully designed mirrors and laser cavities.Additional spectral bandwidth can also be obtained by minor changes in thecavity configuration of a high repetition rate laser, although it has not yetyielded sufficient intensity at the octave points for the observation of f-to-2fbeats [17]. Another widely adopted approach is to generate the extra combbandwidth using microstructure fibers that have zero group velocitydispersion (GVD) within the emission spectrum of a Ti:sapphire laser [18].The large index contrast for waveguiding inside microstructure fibers hastwo consequences: first, the ability to generate a zero in the GVD at visibleor near-infrared wavelengths and, secondly, the possibility of using a muchsmaller core size. This allows broadband continuum generation with only

pulse energies.

Figure 1-2. Two equivalent schemes for measurement of using an octave-spanning opticalfrequency comb. In the self-referencing approach, shown in (a), frequency doubling andcomparison are accomplished with the comb itself. In the second approach, shown in (b), thefundamental frequency and its second harmonic of a CW optical standard are usedto determine These two basic schemes are employed for absolute optical frequencymeasurement and implementation of optical atomic clocks.

For the purpose of using a femtosecond optical comb for absolute opticalfrequency measurements, it is straight-forward to establish the frequencyvalues of all of the comb components. The comb’s frequency spacingcan be phase locked with high precision via detection of higher harmonics of

relative to an RF standard. The value of is determined and controlled

6 Chapter 1

using schemes shown in Fig. 1-2. Control of requires a servo transduceracting differentially on the intracavity group and phase delays. One commonmethod for adjusting is to swivel the end mirror in the arm of the lasercavity that contains the prism sequence [19]. An alternative method ofcontrolling is via modulation of the pump power, with likely contributionsfrom the nonlinear phase, spectral shifts, and the intensity dependence in thegroup velocity [20]. It is worth noting that a scheme implemented by Telleet al. [21] allows the frequency comb to be free running (without any activestabilization) while making highly precise measurement or connection for anoptical frequency interval located within the comb bandwidth.

The dramatic simplification of a complex optical frequency chain to thatof a single mode-locked laser has greatly facilitated optical frequencymeasurement. Another important aspect of this new technology is its highdegree of reliability and precision and lack of systematic errors. Forexample, recent tests have shown that the repetition rate of a mode lockedlaser equals the mode spacing of the corresponding comb to within themeasurement uncertainty of The uniformity of the comb mode spacinghas also been verified to a level below even after spectral broadeningin a fiber [3, 4]. Comparison between two separate fs comb systems, bothlinked to a common reference source (microwave or optical), allows one toexamine the intrinsic accuracy of a femtosecond-comb-based frequencymeasurement system, currently at a level of a few parts in with nomeasurable systematic effects [22]. Direct comparisons of absolute opticalfrequency measurements between the femtosecond comb technique and thetraditional harmonic frequency chain approach have also produced assuringmutual confirmations at the to level [5, 23].

As the measurement precision for optical frequencies continues toadvance, the stability limitation imposed by available RF standards used forfs comb stabilization becomes an important issue [23, 24]. Instead ofoperating a fs comb using RF references, it appears to be advantageous tooperate the comb by stabilizing it to an optical frequency standard. The fscomb in turn produces optically derived stable clock signals in the RFdomain, leading to a so-called “optical atomic clock” [8, 9, 25]. Recentexperimental demonstrations support the concept that, in the future, the moststable and accurate frequency standards will be based on optical transitions[26, 27]. Stepping down the stability level by one or two orders ofmagnitude, portable optical frequency standards that offer compact, simple,and less expensive system configurations have also shown competitiveperformance with (in)stability near at 1 to 10 s averaging time [28].

To realize an optical atomic clock, an optical comb needs to be stabilizedto a pre-selected optical frequency source at a precision level that exceedsthe optical standard itself. can be extracted in a straight-forward manner

1. Phase Controlled Femtosecond Lasers for Sensitive... 7

using either schemes shown in Fig. 1-2. is then stabilized with respect toeither or an auxiliary stable RF source. It is worth noting thatstabilization of at a few mHz is more than adequate, as it yields fractionalfrequency noise of for an optical carrier. A heterodyne beat betweenone of the comb components and the optical standard yields informationabout fluctuations in For the particular case shown in Fig. 1-2(b),

producing adirect link between the frequencies and After appropriate processing,this error signal is used to stabilize the phase of coherently to therebyproducing an output clock signal in the RF domain derived from

Besides the capability of deriving RF signals from an optical frequencystandard, a fs comb can, of course, also be used to transfer the stability ofoptical standards across vast frequency gaps to other optical spectral regions.Easy access to the resolution and stability offered by optical standards willgreatly facilitate the application of frequency metrology to precisespectroscopic investigations. For spectroscopy applications, we indeed oftendesire a single-frequency and reasonably powered optical carrier wave thatcan be tuned to any desired optical spectral feature of interest. Realization ofsuch an optical frequency synthesizer (analogous to its RF counterpart) willadd a tremendously useful tool for modern spectroscopy experiments.Ideally, one would realize a high precision optical frequency synthesizerbased on a stable fs comb linked to an optical frequency standard. One couldforesee an array of diode lasers, each covering a successive tuning range of~ 10 to 20 nanometers that would collectively cover most of the visiblespectrum. Each diode laser frequency would be controlled by the stabilizedoptical comb, and therefore be directly related to the absolute time/frequencystandard in a phase coherent fashion, while the setting of the opticalfrequency would be accomplished via computer control.

In our preliminary implementation of such an optical frequencysynthesizer [10], the fs comb system is referenced by an opticalfrequency standard at 532 nm. A CW diode laser, as well as a CWTi:sapphire laser, is used to tune through targeted spectral regions (forexample, Rb D1 and D2 lines at 795 and 780 nm for the diode laser andhyperfine transitions in the region of 490 - 520 nm) with desired frequencystep sizes, while maintaining absolute reference to the stabilized opticalcomb. A self-adaptive search algorithm first tunes the CW laser to aspecified wavelength region with the aid of a wavelength measurementdevice (100 MHz resolution). A heterodyne beat signal between the laser’sfrequency and that of a corresponding comb line is then detected andprocessed. For fine-tuning, an RF source provides a tunable frequency offsetfor the optical beat. Once the laser frequency tuning exceeds we reset theRF offset frequency back to the original value to start the process over again.

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The laser frequency can thus be tuned smoothly in an “inch-worm” manneralong the comb structure. We have demonstrated two fundamental aspects ofan optical frequency synthesizer; namely continuous, precise tuning of theoptical frequency as well as arbitrary frequency setting on demand. Theentire search process takes about a minute.

2. PRECISION ATOMIC SPECTROSCOPY –STRUCTURE AND DYNAMICS

The first example of spectroscopic application of a precisely stabilizedfemtosecond comb centers on investigation of a two-photon transition inlaser cooled Rb atoms. Phase coherence among the successive pulsesinteracting with a cold atomic sample brings immediately to mind theapproach of Ramsey interference for precision atomic spectroscopy.However, the difference here is that the bandwidth associated with thefemtosecond pulse is so broad that one is enabled to explore the structure ofa large number of atomic states all at once, along with the possibility ofstudying coherent accumulation in a multi-level system [29]. It is thuspossible to simultaneously explore the global structure and dynamics of anatomic system. The multi-pulse interference in the time domain gives aninteresting variation and generalization of the two-pulse based temporalcoherent control of the excited-state wave-packet.

From the frequency domain perspective, it is also straight-forward toappreciate the fact that the spectroscopic resolution and precision will not becompromised by the use of ultrafast pulses since they are associated with aphase-stabilized, wide-bandwidth femtosecond comb. Phase coherenceamong various transition pathways through different intermediate statesproduces multi-path quantum interference effects on the resonantly enhancedtwo-photon transition probability in the cold Rb atoms. The two-photontransition spectrum can be analyzed in terms of the pulse repetition rateand the carrier-envelope offset frequency [30]. Both can be stabilized tohigh precision. With a set of measurements taken at a few different, butpredetermined, combinations of and one can essentially derive allrelevant atomic energy level positions in absolute terms.

Doppler-free two-photon spectroscopy is carried out usually with twoequal-frequency cw laser beams propagating in opposite directions. The two-photon transition rate can be resonantly enhanced via the intermediate stateswith two different laser frequencies [31] or accelerated atomic beams [32],with a small residual Doppler effect. High-resolution two-photonspectroscopy using pulsed light has also been demonstrated [33], with therecent extension to cold atoms [34]. The unique feature of the present work

1. Phase Controlled Femtosecond Lasers for Sensitive... 9

is that the wide bandwidth optical comb allows all relevant intermediatestates to resonantly participate in the two-photon excitation process,permitting the phase coherence among different comb components to inducea stronger transition rate through quantum interference. Following the initialproposal and the subsequent theoretical investigations, we are exploringexperimentally this novel, high-resolution spectroscopy using a femtosecondlaser [35].

Figure 1-3. Top: Schematic of the relevant energy levels of the atom and the frequency-domain perspective of the atom-light interaction. Bottom: Time-domain picture showing a

sequence of mode-locked pulses, with the relevant interaction parameters in and

Figure 1-3 shows the relevant energy levels involved in the two-photon transition from the ground state to the excited state Thedipole-allowed intermediate states, and are located ~ 2 nm and17 nm below the virtual level, respectively. Also shown is a regularly spacedcomb of optical frequencies around 800 nm. The experimental bandwidth ofthe comb is ~ 50 nm, emitted from a 10 fs, 100 MHz repetition-rate mode-locked Ti:sapphire laser. Adjustment of and allows the combcomponents to line up with corresponding hyperfine states of andto resonantly enhance the two-photon transition. This dependence of themulti-path quantum interference on and leads to simultaneousstabilization of both quantities, and thereby the entire comb. The frequency-

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domain analysis is complemented by the time-domain multi-pulse Ramseyinterference picture, as illustrated in Fig. 1-3, where the relevant quantitiesfor interaction are and Both the frequency-domain and thetime-domain analyses produce the same result on the two-photon transitionspectra when one assumes a static distribution among the relevant atomicstates. However, to follow the time evolution of the system, it is necessary toexplore the interaction dynamics from one pulse to the next, taking intoaccount both the atomic coherence and the optical coherence. The generalLiouville equation for the density-matrix components of the atomic states,along with phenomenological decay terms, are used to derive a set of Blochequations describing the evolution of all relevant levels associated with theground, the excited, and the intermediate states.

Figure 1-4. Density-matrix calculation of the excited state population due to the two-photontransition induced in by the phase-coherent fs pulse train. The nominal values of and

are indicated. The spectrum given by the dashed line corresponds to the case with a staticpopulation while the spectrum given by solid line shows dynamic evolution after 4000 pulses.

Figure 1-4 illustrates the calculated population of the 5D states due to thetwo-photon transition, showing clear evidence of population transfer whenthe number of interacting pulses increases. Not surprisingly, the mostdominant transition pathway when a large number of pulses is involved is

which represents a so-calledclosed transition. The horizontal axis represents a scanning of from itsnominal value indicated in the figure. The actual optical frequency for thetwo-photon transition is near 385 THz, which represents a harmonic order of

of Therefore, a change in by ~ 26 Hz implies a repeat in

1. Phase Controlled Femtosecond Lasers for Sensitive... 11

the optical comb spectrum near the two-photon transition region, and hencea repeat in the two-photon spectra.

Figure 1-5 shows experimental two-photon spectra resonantly enhancedby the intermediate states. We clearly confirm the predicted effect ofpopulation transfer by the pulse sequence when we compare the two spectraobtained under the influence of 1,200 and 250,000 pulses, respectively.Basically, the only transition pathway survived at the limit of a large numberof pulses is It is interesting to notethat we have also observed the pure two-photon transition pathway (energiesof the two photons are degenerate) that is not resonantly enhanced by theintermediate states. The signal is indicated in Fig. 1-5 by a small peak(around 19 Hz) in the 2.5 ms evolution curve represented by diamonds,which repeats every 13 Hz in the scan of the value. This observation isconsistent with the fact that a pure two-photon transition would repeat itssignal every time the pulse spectrum is shifted by half of the repetitionfrequency. More recent work has pushed the spectroscopy resolution to thelimit of the natural linewidth of 660 kHz associated with the D-state lifetime,owing to the use of ultracold atoms and careful control of photon momentumtransfer. The work on this simple two-photon transition dynamics thusprovides a solid link between the time-domain picture of carrier-envelopephase and the frequency-domain picture of and One practicalconsequence of these results is that we can now control both degrees offreedom for the femtosecond comb directly by a transition in cold atoms.

Figure 1-5. Experimental observation of resonantly enhanced two-photon transition in coldatoms with a clear influence by the pulse sequence on the atomic state dynamics.

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3. MOLECULAR SPECTROSCOPY AIDED BYFEMTOSECOND OPTICAL FREQUENCY COMB

Before we study examples of molecular spectroscopy aided by thetechnology of the precision frequency comb, we would like to discuss brieflythe implications of the frequency-domain control of the femtosecond laser tothe time-domain experiments. Prior to the development of femtosecondcomb technology, mode-locked lasers were used almost exclusively fortime-domain experiments. Although the femtosecond comb technology hasprimarily impacted on the frequency-domain applications described earlier,it is having an impact on time-domain experiments and promises to bringabout just as dramatic advances in the time-domain as it has in opticalfrequency metrology and optical clocks. Indeed, it is fascinating to blur theboundary between traditional CW precision spectroscopy and ultrafastphenomena. The time-domain applications put stringent requirements on thecarrier-envelope phase coherence. Stabilization of the “absolute” carrier-envelope phase at a level of tens of milliradians has been demonstrated andthis phase coherence is maintained over an experimental period exceedingmany minutes [36], paving the groundwork for synthesizing electric fieldswith known amplitude and phase at optical frequencies. Working with twoindependent femtosecond lasers operating at different wavelength regions,we have synchronized the relative timing between the two pulse trains at thefemtosecond level [37], and also phase locked the two carrier frequencies,thus establishing phase coherence between the two lasers. By coherentlystitching optical bandwidths together, a “synthesized” pulse has beengenerated [11]. With the same pair of Ti:sapphire mode-locked lasers, wehave demonstrated widely tunable femtosecond pulse generation in the mid-and far-IR using difference-frequency-generation [38]. The flexibility of thisnew experimental approach is evidenced by the capability of rapid andprogrammable switching and modulation of the wavelength and amplitude ofthe generated IR pulses. A fully developed capability of producing phase-coherent visible and IR pulses over a broad spectral bandwidth, coupled witharbitrary control in amplitude and pulse shape, represents the ultimateinstrumentation for coherent control of molecular systems. A pulse train withgood carrier-envelope phase coherence is also very promising forexperiments that are sensitive to i.e., the “absolute” pulse phase [2].This can be manifested in “extreme” nonlinear optics experiments, orcoherent control.

The capability to precisely control pulse timing and the pulse-carrierphase allows one to manipulate pulses using novel techniques and achieveunprecedented levels of flexibility and precision, as will be demonstrated inthe work on time resolved spectroscopy of molecules. For example, the