REVIEW ARTICLE Femtosecond pulse shaping using spatial ...fsoptics/articles/Femtosecond... · Femtosecond pulse shaping using spatial light modulators ... waveforms with effective

REVIEW OF SCIENTIFIC INSTRUMENTS VOLUME 71, NUMBER 5 MAY 2000

REVIEW ARTICLE

Femtosecond pulse shaping using spatial light modulatorsA. M. Weinera)

School of Electrical and Computer Engineering, Purdue University, West Lafayette, Indiana 47907-1285

~Received 17 August 1999; accepted for publication 20 January 2000!

We review the field of femtosecond pulse shaping, in which Fourier synthesis methods are used togenerate nearly arbitrarily shaped ultrafast optical wave forms according to user specification. Anemphasis is placed on programmable pulse shaping methods based on the use of spatial lightmodulators. After outlining the fundamental principles of pulse shaping, we then present a detaileddiscussion of pulse shaping using several different types of spatial light modulators. Finally, newresearch directions in pulse shaping, and applications of pulse shaping to optical communications,biomedical optical imaging, high power laser amplifiers, quantum control, and laser-electron beaminteractions are reviewed. ©2000 American Institute of Physics.@S0034-6748~00!02005-0#

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I. INTRODUCTION

Since the advent of the laser nearly 40 years ago, thhas been a sustained interest in the quest to generattrashort laser pulses in the picosecond (10212s) and femto-second (10215s) range. Reliable generation of pulses bel100 fs in duration occurred for the first time in 1981 with tinvention of the colliding pulse modelocked~CPM! ring dyelaser.1 Subsequent nonlinear pulse compression of pufrom the CPM laser led to a series of even shorter pulses2–6

culminating in pulses as short as 6 fs, a record which stfor over a decade. Six femtoseconds in the visible cosponds to only three optical cycles, and therefore such pdurations are approaching the fundamental single optcycle limit. Further rapid progress occurred following thdemonstration of femtosecond pulse generation from sostate laser media in the 1990 time frame.7 Femtosecondsolid-state lasers bring a number of important advantacompared to their liquid dye laser counterparts, includsubstantially improved output power and stability and nphysical mechanisms for pulse generation advantageouproduction of extremely short pulses. Femtosecond sostate laser technology has now advanced to the pointpulses below 6 fs can be generated directly fromlaser.8–14Equally important, the use of solid-state gain medhas also led to simple, turn-key femtosecond lasers,many researchers are now setting their sights on pracand low cost ultrafast laser systems suitable for real-woapplications~in addition to the scientific applications fowhich femtosecond lasers have been used extensively!. De-tailed information on the current status of femtosecond tenology and applications can be found in several recent jonal special issues,15–17 books,18–21 and in another recenreview article in this journal.22

The focus of this article is femtosecond pulse shaping

a!Electronic mail: [email protected]

1920034-6748/2000/71(5)/1929/32/$17.00

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topic complementary to femtosecond pulse generation. Othe past decade powerful optical waveform synthesis~orpulse shaping! methods have been developed which allogeneration of complicated ultrafast optical waveformscording to user specification. Pulse shaping systems halready demonstrated a strong impact as an experimentalproviding unprecedented control over ultrafast laser waforms for ultrafast spectroscopy, nonlinear fiber optics, ahigh-field physics. Coupled with the recent advances andsulting widespread availability of femtosecond lasers, as was advances in femtosecond pulse characterization tniques, femtosecond pulse shaping is poised to impact mdiverse and additional applications. In the terminologyelectronic instrumentation, femtosecond lasers constituteworld’s best pulse generators, while pulse shaping isshort pulse optical analog to electronic function generatowhich widely provide electronic square waves, triangwaves, and indeed arbitrary user specified waveforms.

A number of approaches for ultrafast pulse shaping hbeen advanced. Here we concentrate on the most succeand widely adopted method, in which waveform synthesisachieved by spatial masking of the spatially dispersed optfrequency spectrum. A key point is that because wavefosynthesis is achieved by parallel modulation in the frequedomain, waveforms with effective serial modulation banwidths as high as terahertz and above can be generatedout requiring any ultrafast modulators. We will be particlarly interested in pulse shaping using spatial ligmodulators~SLMs!, where the SLM allows reprogrammabwaveform generation under computer control. A reviewticle by Froehly describes a variety of pulse shaping teniques investigated prior to 1983 for picosecond pulses.23 Amore recent review by Weiner provides a broad accounfemtosecond pulse shaping as well as related pulse procing techniques, including both the pulse shaping techniwhich is the focus of the current article as well as othapproaches.27 Other useful reviews include Ref. 24, whic

9 © 2000 American Institute of Physics

IP license or copyright, see http://rsi.aip.org/rsi/copyright.jsp

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1930 Rev. Sci. Instrum., Vol. 71, No. 5, May 2000 A. M. Weiner

describes early results on femtosecond pulse shaping ufixed masks and related experiments on picosecond pshaping performed in the context of nonlinear pulse copression, and Refs. 25 and 26, which include citationsrecent results on pulse shaping as well as holographicnonlinear pulse processing.

This review article is organized as follows. In Sec. II wdiscuss the basics of femtosecond pulse shaping, includidescription of the apparatus, examples of pulse shapingsults using fixed spatial masks, important results fromtheory of pulse shaping, and instrument control, alignmeand pulse measurement issues. In Secs. III and IV we disprogrammable pulse shaping using liquid crystal SLMs aacoustic-optic modulators, respectively; these are thetypes of SLMS which are most widely applied for femtoseond pulse shaping. Section V summarizes the relative adtages and disadvantages of liquid crystal and acousto-oSLMs for pulse shaping. Section VI covers developmentsdeformable, movable, and micromechanical mirrors for pushaping applications. These special mirror based approafor programmable pulse shaping, while important, are notas completely developed as the liquid crystal or acouoptic approaches, and for this reason are not includedcomparison in Sec. V. In Sec. VII we discuss further diretions in femtosecond pulse shaping, including shaping ofcoherent light, integration, direct space-to-time pulse shing, and generalized pulse shapers for holographicnonlinear pulse processing. We conclude in Sec. VIII by sveying some of the applications of femtosecond pulse shing, including optical communications, dispersion compention, laser control of terahertz radiation, coherent controquantum mechanical processes, and laser-electron beateraction physics.

II. FEMTOSECOND PULSE SHAPING BASICS

A. Linear filtering

The femtosecond pulse shaping approach describethis article is based on the linear, time-invariant filter, a cocept well known in electrical engineering. Linear filteringcommonly used to process electrical signals ranging frlow frequencies~audio and below! to very high frequencies~microwave!. Here we apply to linear filtering to generaspecially shaped optical waveforms on the picosecondfemtosecond time scale. Of course, the hardware needeprogrammable linear filtering of femtosecond laser pullooks very different from the familiar resistors, capacitoand inductors used for linear filtering of conventional eletrical signals.

Linear filtering can be described in either the time dmain or the frequency domain, as depicted in Fig. 1.27 In thetime domain, the filter is characterized by an impulsesponse functionh(t). The output of the filtereout(t) in re-sponse to an input pulseein(t) is given by the convolution ofein(t) andh(t)

eout~ t !5ein~ t !* h~ t !5E dt8ein~ t8!h~ t2t8!, ~2.1!


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where* denotes convolution. If the input is a delta functiothe output is simplyh(t). Therefore, for a sufficiently shorinput pulse, the problem of generating a specific output pushape is equivalent to the task of fabricating a linear filwith the desired impulse response. Note that instead ofterm ‘‘impulse response function,’’ which is common ielectrical engineering,h(t) may also be called a Green function, which is a common terminology in some other fields

In the frequency domain, the filter is characterized byfrequency responseH(v). The output of the linear filterEout(v) is the product of the input signalEin(v) and thefrequency responseH(v)—i.e.,

Eout~v!5Ein~v!H~v!. ~2.2!

Here ein(t), eout(t), and h(t) and Ein(v), Eout(v), andH(v), respectively, are Fourier transform pairs—i.e.,

H~v!5E dt h~ t !e2 ivt ~2.3!

and

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2p E dv H~v!eivt. ~2.4!

For a delta function input pulse, the input spectrumEin(v) isequal to unity, and the output spectrum is equal to thequency response of the filter. Therefore, due to the Foutransform relations, generation of a desired output wavefocan be accomplished by implementing a filter with thequired frequency response. Our pulse shaping approacdescribed most naturally by means of this frequency dompoint of view.

B. Pulse shaping apparatus and pulse shapingexamples using fixed masks

Figure 2 shows the basic pulse shaping apparatus, wconsists of a pair of diffraction gratings and lenses, arranin a configuration known as a ‘‘zero dispersion pulse copressor,’’ and a pulse shaping mask.28 The individual fre-quency components contained within the incident~usuallybut not always bandwidth limited! ultrashort pulse are angularly dispersed by the first diffraction grating, and then fcused to small diffraction limited spots at the back focplane of the first lens, where the frequency components

FIG. 1. Pulse shaping by linear filtering.~a! Time-domain view.~b! Fre-quency domain view.


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1931Rev. Sci. Instrum., Vol. 71, No. 5, May 2000 Femtosecond pulse shaping

spatially separated along one dimension. Essentially thelens performs a Fourier transform which coverts the angdispersion from the grating to a spatial separation at the bfocal plane. Spatially patterned amplitude and phase ma~or a SLM! are placed in this plane in order to manipulate tspatially dispersed optical Fourier components. After a sond lens and grating recombine all the frequencies intsingle collimated beam, a shaped output pulse is obtaiwith the output pulse shape given by the Fourier transformthe patterned transferred by the masks onto the spectrum

In order for this technique to work as desired, onequires that in the absence of a pulse shaping mask, the oupulse should be identical to the input pulse. Therefore,grating and lens configuration must be truly free of dispsion. This can be guaranteed if the lenses are set up as amagnification telescope, with the gratings located at the oside focal planes of the telescope. In this case the firstperforms a spatial Fourier transform between the plane offirst grating and the masking plane, and the second lensforms a second Fourier transform from the masking planethe plane of the second grating. The total effect of theseconsecutive Fourier transforms is that the input pulse ischanged in traveling through the system if no pulse shapmask is present.

Note that this dispersion-free condition also dependsseveral approximations, e.g., that the lenses are thin andof aberrations, that chromatic dispersion in passing throthe lenses or other elements which may be inserted intopulse shaper is small, and that the gratings have a flat stral response. Distortion-free propagation through the ‘‘zdispersion compressor’’ has been observed in many expments with pulses down to roughly 50 fs—see for examRefs. 28 and 29. For much shorter pulses, especially in10–20 fs range, more care must be taken to satisfy thapproximations. For example, both the chromatic aberraof the lenses in the pulse shaper and the dispersion exenced in passing through the lenses can become impoeffects. However, by using spherical mirrors insteadlenses, these problems can be avoided and dispersionoperation has been obtained.30

The first use of the pulse shaping apparatus shownFig. 2 was reported by Froehly, who performed pulse shing experiments with input pulses 30 ps in duration.23 Re-lated experiments demonstrating shaping of pulses a fewcoseconds in duration by spatial masking within a fiber agrating pulse compressor were performed independentlyHeritage and Weiner;31–33 in those experiments the gratin

FIG. 2. Basic layout for Fourier transform femtosecond pulse shapin


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pair was used in a dispersive configuration without interlenses since grating dispersion was needed in order to cpress the input pulses which were chirped through nonlinpropagation in the fiber. The dispersion-free apparatusFig. 2 was subsequently adopted by Weineret al. for ma-nipulation of pulses on the 100 fs time scale, initially usifixed pulse shaping masks28 and later using programmablSLMs.34,35 With minor modifications, namely, replacing thpulse shaping lenses with spherical mirrors, pulse-shapoperation has been successfully demonstrated for inpulses on the 10–20 fs time scale.30,36–38 A fiber-pigtailedpulse shaper, with fiber-to-fiber insertion loss as low asdB, has also been reported for 1.55mm operation for opticalcommunications applications.39–41 The apparatus of Fig. 2~without the mask! can also be used to introduce dispersifor pulse stretching or compression by changing the gratilens spacing. This idea was introduced and analyzedMartinez42 and is now extensively used for high-power femtosecond chirped pulse amplifiers.22,43

Pulse shaping using programmable SLMs will be dcussed beginning in Sec. III. Here we present severalamples using fixed spatial masks, the masking technolemployed in early femtosecond pulse shaping experimeFixed masks can provide excellent pulse shaping qualityhave been employed in experimental applications of pushaping in nonlinear fiber optics, fiber communications, aultrafast spectroscopy. Disadvantages of fixed masks arethey do not easily provide continuous phase variations~bi-nary phase variations are fine! and that a new mask must bfabricated for each experiment.

Figure 328 shows intensity cross-correlation traceswaveforms generated by using an opaque mask withisolated slits, resulting in a pair of distinct and isolated sptral peaks. Note that the intensity cross-correlation tracesproximately provide a measurement of optical intensity vsus time—see Sec. II G for further explanation. The tfrequencies interfere in the time domain, producing a higfrequency optical tone burst. The 2.6 THz period of thetensity modulation is identical to the separation of thelected frequency components and corresponds to a perioonly 380 fs. We have also performed experiments usingadditional phase mask to impose ap phase shift between thtwo spectral components. The resulting waveform is shoin Fig. 3 as the dotted line. The two distinct frequencies sinterfere to produce a tone burst. However, thep phase shift

FIG. 3. Intensity cross-correlation traces of optical tone bursts resulfrom a pair of isolated optical frequency components. Solid: optical frequcies in phase. Dotted: optical frequencies phase shifted byp.


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is expected to lead to an interchange in the positions ofpeaks and nulls of the time domain, and this effect is cleaseen in the data. It is worth noting that this effect, namelyshift in the time-domain interference features reflecting sptral phase variations, demonstrated here in the contexpulse shaping, has also been developed into a powerful pcharacterization tool for measuring the spectral phasefiles of unknown ultrashort pulses.44 It is also worth notingthat the data in Fig. 3 represent a time-domain analog ofwell known Young’s double slit interference experimenThis is one manifestation of the close analogy which exbetween time-domain Fourier optics discussed here andwell known and active field of spatial domain Fourier optic

We also consider generation of ultrafast square puusing fixed masks.28 The spectrum of a square pulse of dration T is shaped as a sinc function, given by

E~ f !5E0Tsin~p f T!

p f T. ~2.5!

The corresponding mask is specified by

M ~x!5sin~px/x0!

px/x0, ~2.6!

whereM (x) is the masking function,x05(T] f /]x)21, and] f /]x is the spatial dispersion at the masking plane. In orto implement the desired filtering function, both a phase aan amplitude mask are needed. The phase mask is usimpart the required alternating sign to the filter. The tramission function of the amplitude mask varies continuouwith position. Furthermore, due to the combination of faand slow temporal features~,100 fs rise and fall timespulse duration in the picosecond range!, the amplitude maskmust be capable of producing a series of sidelobes ovlarge dynamic range. The required phase and amplitmasks were fabricated on fused silica substrates usingcrolithographic patterning techniques, and the two mawere placed back to back at the masking plane of the pshaper. Phase masks were fabricated by using reactiveetching to produce a relief pattern on the surface of the fusilica. Amplitude masks consisted of a series of fine opametal lines deposited onto the substrate with linewidthsspacings varied in order to obtain the desired transmissThis approach to forming a variable transmission amplitumask is related to diffractive optics structures utilizedspatial manipulation of laser beams via computer generholography. Figure 4 shows a semilog plot of a power sptrum produced in this way. The data correspond to a mcontaining 15 sidelobes on either side of the central peakdynamic range approaching 104:1, as well as excellensignal-to-noise ratio, are evident from the power spectruThe dotted line, which is an actual sinc function, is in goagreement with the data, although the zeroes in the datawashed out due to the finite spectral resolution of the msurement device Figure 5~a! shows an intensity crosscorrelation measurement of a 2 pssquare pulse produced busing masks truncated after five sidelobes on either sidthe main spectral peak. The rise and fall times of the squpulse are found to be on the order of 100 fs. The rip


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present on the square pulse arises because of the truncof the spectrum and is in good qualitative agreement withtheoretical intensity profile@Fig. 5~b!#. Square pulses withreduced ripple have also been obtained, by avoiding truntion of the spectrum and instead using a more gentle speapodization. An experimental example of such a ‘‘smootsquare pulse is plotted in Fig. 5~c!.45

FIG. 4. Semilog plot of power spectrum of an optical square pulse.

FIG. 5. Optical square pulses.~a! Measurement of a 2 ps optical squarepulse.~b! Corresponding theoretical intensity profile.~c! Measurement of asquare pulse with reduced ripple.


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At this point we also discuss pulse shaping using phaonly filters. Phase-only filters have the advantage, in sittions where they are adequate, of no inherent loss. Herediscuss two examples of useful pulse shaping via losslphase-only filtering using fixed phase masks. There aremany examples of phase-only filtering using SLMs; thewill be discussed later.

One interesting example is encoding of femtosecopulses by utilizing pseudorandom phase patterns to scram~encode! the spectral phases.28,29 An example is shown inFig. 6.29 The clear aperture of the mask is divided intoequal pixels, each of which corresponds to a phase shieither zero orp. Figure 6~a! shows a measurement of thintensity profile of the encoded waveform. Spectral encodspreads the incident femtosecond pulses into a complicpseudonoise burst within an;8 ps temporal envelope. Thpeak intensity is reduced to;8% compared to that of anuncoded pulse of the same optical bandwidth. For compson, the theoretical intensity profile, which is the squarethe Fourier transform of the spectral phase mask, is showFig. 6~b!. The agreement between theory and experimenexcellent. Similar coding and decoding has been demstrated with longer phase codes~up to 127 pixels!28 and alsousing programmable SLMs.41 An important feature is thabecause the phase-only filtering is lossless, by using a sepulse shaper with a conjugate phase mask, the spectral pmodulation can be undone, with the result that the pseonoise burst is decoded~restored! back to the original ul-trashort pulse duration. This forms the basis of a propoultrashort pulse code-division multiple-access~CDMA! com-munications concept, in which multiple users share a comon fiber optic channel on the basis of different minimainterfering code sequences assigned to different transmireceiver pairs.41,46

In some cases only the temporal intensity profile ofoutput pulse is of interest, and this greatly increases thegrees of freedom available for filter design. In particulphase-only filters can be designed to yield the desired t

FIG. 6. Ultrafast pseudonoise bursts generated by using a pseudoraspectral phase filter~shown as inset!. ~a! Intensity cross-correlation trace.~b!Corresponding theoretical intensity profile.


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poral intensity profile. An important example is the useperiodic phase-only spectral filters to produce high quapulse trains.47,48 As in Fig. 3, where spectral amplitude fitering is used for pulse train generation, the repetition ratethe pulse train is equal to the periodicity of the spectral filtHowever, unlike the spectral amplitude filtering case,envelope of the pulse train depends on the structure ofphase response within a single period of the phase filteturns out that by using pseudorandom phase sequencessharp autocorrelation peaks~similar to those used in CDMAand other forms of spread spectrum communication!49 as thebuilding blocks of the phase filter, one can generate putrains under a smooth envelope. The intensity crocorrelation measurement of a resulting experimental putrain with 4.0 THz repetition rate, generated using 75 fs inpulses and binary phase masks based on periodic repetiof the so-calledM ~or maximal length! sequences,49 is shownin Fig. 7~a!.47 The pulse train is clean, and the pulses are wseparated. Pulse trains with similar intensity profiles~notshown! have been produced by spectral amplitude filterinbut with substantially reduced energies. Note that the optphase is constant from pulse to pulse in trains producedamplitude filtering, unlike the phase filtering case, whereoptical phase varies. Pulse trains such as that in Fig.~a!

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FIG. 7. Pulse trains generated by phase-only filtering.~a! Pulse train undera smooth envelope.~b! and ~c! Pulse trains under a square envelope.


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have been utilized for experiments demonstrating selecamplification of coherent optical phonons in crystals,50,51 la-ser control over coherent charge oscillations in multiquantum well semiconductor structures,52,53 and enhance-ment of terahertz radiation emitted from photoconductantennas.54 Similar pulse trains have also been generateding input pulses below 20 fs, both with fixed masks30 andwith SLMs,36 and with repetition rates in the vicinity of 2THz.36

Pulse trains with different envelopes can be generatedvarying the details of the phase response within a singleriod of the periodic phase filter.47,48For example, flat-toppedpulse trains have been generated by using filters based oso-called Dammann gratings.55–57 Damman gratings arecomputer generated holograms that have previously bused to split an individual laser beam into an equally spacequal intensity array of beams in space. The structure fDammann grating consists of a periodic binary phase fution, where the period of the phase modulation is selecteyield the desired beam separation in the spatial output aand the phase structure within a single modulation periodesigned using numerical global optimization techniquesprovide the desired number of beams, with as little energypossible outside the target array area. In spatial optics,output beam array can be obtained by passing a single ibeam first through the Dammann grating and then througlens, which takes the spatial Fourier transform. Pulsequences in the time domain can be formed by placing simmasks at the Fourier plane of a pulse shaper. One examptime domain data, obtained by placing a binary phase mfabricated according to a Dammann grating design intfemtosecond pulse shaper, is shown in Fig. 7~b!. The wave-form consists of a relatively uniform sequence of eigpulses, with one central pulse missing. Waveforms withmissing central pulse restored have been obtained by ading the phase difference on the mask to be less thanp—seeFig. 7~c!. These time domain results, achieved by usingphase filter originally designed for spatial beam forming aplications, underscores again the close analogy betweendomain and space domain Fourier optics.

It is worth noting that design of Dammann phase grings for spatial array generation is usually accomplishthrough numerical optimization techniques. New phase-ofilters designed to generate other femtosecond wavefocan also be found using numerical optimization codes. Seral authors have employed simulated annealing algoritto design either binary48 or gray-level58–60phase-only filters,which were tested in pulse shaping experiments using eibinary phase masks or liquid crystal modulators, resptively. Binary ~0-p! phase filters produce waveforms wisymmetrical intensity profiles, while gray-level phase filte~typically with four or more phase levels! can be used forgenerating pulse trains and other waveforms with asymmric intensity profiles. We emphasize that phase-only filteris generally sufficient only when the target time-domawaveform is not completely specified, e.g., when the timdomain intensity is specified but the temporal phases arefree.


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C. Results from pulse shaping theory

It is important to have a quantitative description of tshaped output waveformeout(t). In terms of the linear filterformalism, Eqs.~2.1!–~2.4!, we wish to relate the linear fil-tering functionH(v) to the actual physical masking functiowith complex transmittanceM (x). To do so, we note thathe field immediately after the mask can be written

Em~x,v!;Ein~v!e2~x2av!2/w02M ~x!, ~2.7!

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Herea is the spatial dispersion with units cm~rad/s!21, w0 isthe radius of the focused beam at the masking plane~for anysingle frequency component!, win is the input beam radiusbefore the first grating,c is the speed of light,d is the gratingperiod,l is the wavelength,f is the lens focal length, andu in

andud are the input and diffracted angles from the first gring, respectively.

Note that Eq.~2.7! is in general a nonseparable functioof both space~x! and frequency~v!. This occurs because thspatial profiles of the focused spectral components canaltered by the mask—e.g., some spectral componentsimpinge on abrupt amplitude or phase steps on the mwhile others may not. This leads to different amountsdiffraction for different spectral components and results inoutput field which may be a coupled function of space atime. This space-time coupling has been analyzed by sevauthors.61–63

On the other hand, one is usually interested in generaa spatially uniform output beam with a single prescribtemporal profile. In order to obtain an output field which isfunction of frequency~or time! only, one must perform anappropriate spatial filtering operation. Thurstonet al.64 ana-lyze pulse shaping by expanding the masked field iHermite–Gaussian modes and assuming that all of the spmodes except for the fundamental Gaussian mode are enated by the spatial filtering. In real experiments the Gauian mode selection operation could be performed by focusinto a fiber~for communications applications! or by couplinginto a regenerative amplifier~for high power applications!.This can be also be performed approximately by spatialtering or simply by placing an iris after the pulse shapisetup. In any case, if one takes the filter functionH(v) to bethe coefficient of the lowest Hermite–Gaussian mode inexpansion of Em(x,v), one arrives at the followingexpression:27,64

H~v!5S 2

pw02D 1/2E dx M~x!e22~x2av!2/w0

2. ~2.9!

Equation~2.9! shows that the effective filter in the frequencdomain is the mask functionM (x) convolved with theinten-sity profile of the beam. The main effect of this convolutiois to limit the full width at half maximum~FWHM! spectralresolution dv of the pulse shaper todv>(ln 2)1/2w0 /a.


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Physical features on the mask smaller than;w0 are smearedout by the convolution, and this limits the finest featurwhich can be transferred onto the filtered spectrum. One csequence of this picture is that wavelength componentspinging on mask features which vary too fast for the avaable spectral resolution are in part diffracted out of the mbeam and eliminated by the spatial filter. This can leadphase-to-amplitude conversion in the pulse shapprocess.41,64,65Conversely, in the limitw0→0, the apparatusprovides perfect spectral resolution, and the effective filtejust a scaled version of the mask.

The effect of finite spectral resolution and phase-amplitude conversion is illustrated by Fig. 8~a!.41 The datashow the power spectrum measured for a pulse spectencoded using a pseudorandom binary phase code in thetext of a fiber optic CDMA communications experiment. Tencoding process impresses a pseudorandom sequencephases, each either zero orp, onto the spectrum. Evethough the code itself is phase-only, a series of dipspresent in the encoded spectrum. Each dip correspondsphase jump in the spectral code and results from powerfracted out of the main beam which is not refocused intooutput fiber. Figure 8~b! shows the theoretical spectrumsimulated on the basis of Eq.~2.9!, the known phase codeand the appropriate values fora and w0 . The measuredpower lost in the encoding process was also comparedthe simulation results as a function of code length.41 For boththe spectrum and the power lost, the agreement betweenperiment and simulation is excellent. These experiments,others like them, show that Eq.~2.9! provides an appropriatetheoretical description, including the effect of diffractiolosses due to mask features, provided that a suitable spfilter is employed following the pulse shaping apparatus.

Note that in our treatment earlier we assume thatoutput Gaussian mode which is selected is matched toinput mode. The case where the input and output mode sare not matched is analyzed in Ref. 66; in some casescan give improvement in spectral resolution compared toexpected from Eq.~2.9!.

The effect of finite spectral resolution can be understoin the time domain by noting that the output pulseeout(t)will be the convolution of the input pulseein(t) with theimpulse responseh(t). The impulse response in turn is obtained from the Fourier transform of Eq.~2.9! and can bewritten as follows:

h~ t !5m~ t !g~ t !, ~2.10a!

FIG. 8. Experimental~a! and theoretical~b! power spectrum resulting fromspectral encoding using a pseudorandom binary phase code.


sn--

-nog

is

-

llyon-

f 31

iso aif-e

th

ex-nd

tial

eheesis

at

d

where

m~ t !51

2p E dv M ~av!eivt ~2.10b!

and

g~ t !5exp~2w02t2/8a2!. ~2.10c!

Thus, the impulse response is the product of two factors.first factorm(t) is the Fourier transform of the mask~appro-priately scaled! and corresponds to the infinite-resolution impulse response. The second factorg(t) is an envelope func-tion which restricts the time window in which the tailoreoutput pulse can accurately reflect the response ofinfinite-resolution mask. The FWHM duration of this timwindow ~in terms of intensity! is given by

T54aAln 2

w05

2Aln 2winl

cd cosu in. ~2.11!

The time window is proportional to the number of gratinlines illuminated by the input beam multiplied by the perioof an optical cycle. A larger time window can only be otained by expanding the input beam diameter. The shorfeature in the output shaped pulse is of course governedthe available optical bandwidth.

These results impose limits on the complexity of shappulses, which can be understood in terms of schematicquency and time domain plots in Fig. 9. The shortest temral feature that can be realized,dt, is inversely related to thetotal bandwidthB(Bdt>0.44), and the maximum temporawindow T is inversely related to the finest achievable sptral featured f (d f T>0.44). HereT, Dt, B, andd f all referto FWHMs. We defineh as a measure of the potential complexity of the shaped pulse, as follows:

h5B/d f 5T/dt. ~2.12!

Thus, h describes the number of distinct spectral featuthat may be placed into the available bandwidth, or equilently, the number of independent temporal features that mbe synthesized into a waveform. This measure is relatethe maximum time-bandwidth product:BT>0.44h. An ex-pression for the complexityh is given in terms of the gratingand input beam parameters as follows:

FIG. 9. Schematic time- and frequency-domain plots illustrating limitsthe complexity of shaped pulses.


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.

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h5Dl

l

p

~ ln 2!1/2

win

d cos~u in!, ~2.13!

whereDl is the bandwidth~in units of wavelength!.We consider a numerical example. If we assume an

nm center wavelength, an 1800 line/mm grating, an incidangle of 50° ~near Littrow!, a 100 fs pulse, andwin

52 mm, the resultant time window and complexity areT526.4 ps andh5264, respectively. These numbers are tycal in that several experiments have demonstrated timedows on the order of tens of picoseconds and complexitieone hundred to a few hundred. Larger time windowspossible using larger input beams; higher complexitiespossible using either larger input beams or shorter puwidths or both. Pulse shaping time windows in the 100range67 have been reported, as have experiments with puas short as 13 fs.36 However, in most cases the demonstrapulse shaping complexity is still in the range of a few hudred or less. Note that Eqs.~2.11! and ~2.13! are the upperbounds on the time window and complexity as set by funmental optics. Achieving these limits depends on havinsufficient number of spatial features available in the pushaping mask or SLM; and for increasing complexities,requirements on the precision of the pulse shaping maskthe optical system become increasingly stringent.

The earlier discussion, which results from an appromate treatment of diffraction at the mask, has been founadequately describe a great number of experimental resWe reiterate, however, that these results are only valid wa suitable spatial filter is employed so that the pulse shapconstant across the spatial beam profile. Practically, it ishelpful to avoid using a masking function whose infinresolution impulse response function exceeds the availtime window.

Several papers have analyzed the space-time coupeffects arising due to diffraction from the pulse shapimask.61–63Wefers and Nelson62 have obtained a simple analytical expression which describes this space-time couplAssuming an input electric field proportional tof in(x)ein(t),and assuming no aperturing or spatial filtering by the optsystem~except at the mask!!, the shaped electric field aftethe final grating is given by

eout~x,t !; f in~x1vt !E dt8ein~ t8!m~ t2t8!, ~2.14a!

where

v5cd cosu in

l. ~2.14b!

The integral gives the shaped pulse which would be obtaiin the perfect spectral resolution~infinite time window!limit—i.e., the convolution of the input pulse with the scaleFourier transform of the mask. Thef in(x1vt) multiplier rep-resents a time-dependent spatial shift of the shaped wform. It is interesting to note that for a spatial shiftDx alongthe output beam, the corresponding time offset is giventhe number of grating grooves intercepted by the spatial s(Dx/d cosuin) multiplied by the optical period (l/c). Equa-tion ~2.14! does not explicitly include any time window


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However, a time window automatically arises if one includan aperture selecting a portion of the shaped beam. FGaussian input beam and a spatial filter which selectsmatched Gaussian spatial mode of the shaped beam, ontains the results in Eqs.~2.7!–~2.11!.

D. Control strategies for pulse shaping

Femtosecond pulse shapers can be operated either uopen loop control or in a feedback~adaptive control! con-figuration. These concepts are illustrated in Fig. 10. Note ta femtosecond amplifier system may optionally be includunder either control strategy, depending on the experimerequirements.

Let us first consider the open loop configuration, F10~a!. Here the desired output waveform is specified byuser, and reasonable knowledge of the input pulse isusually available. Therefore, the desired transfer functionknown, and one sets the pulse shaping SLM~in the case ofprogrammable pulse shaping! to provide this transfer func-tion. If there is additional linear distortion present betwethe input and output~e.g., phase aberration in a chirped pulfemtosecond amplifier!, the pulse shaper can be programso that its transfer function also includes precompensafor such distortion. This open loop strategy requires preccalibration of the SLM. However, precise characterizationthe input pulse is often unnecessary, provided that the inpulse is known to be shorter than shortest features desirethe shaped output waveform. Open loop control, whichbeen used since the original pulse shaping experimentthe only method applicable with fixed masks. It has beused to obtain high quality shaped pulses for many differexperimental applications and remains in common useprogrammable pulse shaping with SLMs.

The ability to program a pulse shaper under compucontrol has led recently to several interesting demonstratof adaptive pulse shaping@Fig. 10~b!#—or pulse shaping un-der feedback control.37,38,68–73In these experiments one usually starts with a random spectral pattern programmed ithe pulse shaper, which is updated iteratively accordingstochastic optimization algorithm based on the differencetween a desired and measured experimental output. In

FIG. 10. Control strategies for femtosecond pulse shaping.~a! Open loopcontrol. ~b! Feedback or adaptive control. Note that these experimentsbe performed either with or without the amplifier pictured in the figure.


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way femtosecond waveform synthesis or chirp compensacan be achieved without the need to explicitly programpulse shaper. Adaptive pulse shaping experiments haveperformed using liquid crystal modulator arrays,37,68–72

acousto-optic modulators,73,74 and deformable mirror phasmodulator arrays.38 Experimental demonstrations have icluded chirp compensation and shaping of low energy pufrom femtosecond oscillators,37,38,68,70correction of residualchirps remaining after the pulse stretching-compressioncess in high energy chirped pulse amplifiers,69,71 and quan-tum control experiments demonstrating adaptive pulse shcontrol of fluorescence yields73 and photodissasociatioproducts.72 One example of data from an adaptive pucompression experiment is shown in Fig. 11.37 In this work aTi:sapphire laser was deliberately misaligned in order to gerate highly chirped output pulses@;80 fs, Fig. 11~a!#, al-though the;150 nm wide laser bandwidth~see inset! couldpotentially support sub-10 fs pulses. The state of a liqcrystal phase modulator array in a pulse shaper was atively controlled using a simulated annealing algorithmorder to maximize the average second harmonic signal msured after an external doubling crystal~the second harmonicis expected to be highest for the shortest, most intense pu!.The interferometric autocorrelation trace of the pulse

FIG. 11. Results from adaptive pulse shaping.~a! Interferometric autocor-relation measurement of highly chirped pulses from a Ti: sapphire laser~b!Measurement of pulse compressed to nearly the bandwidth limit throadaptive pulse shaping. Inset: optical power spectrum.


neen

es

o-

pe

-

dp-

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tained after 1000 iterations is shown in Fig. 11~b! ~note thedifferent time scale!. The pulse is compressed to 14 fs, mucloser to the bandwidth limit, and the pulse quality is greaimproved. The time required to run this experiment wroughly 15 min~;1 s per iteration!.

In adaptive pulse shaping neither the input pulse~seeearlier example! nor the calibration of the SLM nor the output laser waveform need to be specified. What does neespecified is the experimental outcome to be optimizedthe adaptive learning algorithm to be employed. The adtive control strategy can be appropriate when the calibraof the SLM is difficult or the state of the SLM depends incomplicated way on the input variables. It can also be pticularly useful when the appropriate laser waveform is nknown. One example is in the field of coherent control75

where shaped laser waveforms can be a tool for controlquantum mechanical motions, with potential applicatioe.g., to laser controlled chemistry. In many real systemsHamiltonian is not known with sufficient accuracy to predthe appropriate laser waveform. In these cases the adaapproach can be used to search for the laser waveform wgives the best experimental result, such as optimizingyield of a particular photochemical product.76,77

E. Pulse shaping in amplified systems

We comment briefly on pulse shaping in amplified femtosecond systems. Amplifiers are needed whenever mlocked oscillators alone are insufficient to produce shapulses at the desired energy levels. Briefly, the pulse shmay be placed before the amplifier~as in Fig. 10!, after theamplifier, or in some cases it may be part of the amplifier.the late 1980s several experiments were reported in whhigh energy shaped pulses were generated by placing a pshaper containing fixed masks after a microjoule level aplifier system.45,50,78,79In recent experiments with millijoulelevel amplifier systems and programmable pulse shaperspulse shaper has been placed either before80 or after71,73 theamplifier. Placing the pulse shaper before the amplifierthe advantage that the danger of damage to the SLM is mmized. Furthermore, in the case of a saturated amplifier,output energy is not strongly dependent on changes ininput energy; therefore, losses during the pulse shapingcess before the amplifier need not result in substantilower energies after the amplifier. One should note, howethat in the case of very high power amplifiers, one may neto consider the influence of pulse shaping on nonlinearitiethe amplifier. In chirped pulse amplifiers, the pulsestretched very substantially in order to lower the peak intsity and, hence, avoid nonlinearities in the amplifier. Chiing or stretching of a shaped pulse may in some situatimodulate or otherwise influence the intensity seen byamplifier, which may lead to nonlinear distortions in the aplified output waveform.81 Note that since chirped pulse amplifiers contain grating based pulse stretchers and compsors, it is also possible to perform pulse shaping withinamplifier by placing a SLM at the appropriate point in thstretcher or compressor. This approach has been reporteRef. 82, where a liquid crystal phase modulator array w

h


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isbn

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used within the stretcher to obtain millijoule level shappulses. However, this approach requires that the stretcheconstructed to match the aperture of the SLM, which cdown on design flexibility which may be needed to achievery large stretching ratios.

F. Pulse shaper alignment

The optical components within Fourier plane pulshapers have many degrees of freedom, and careful ament of the components is critical. A recipe for pulse shaalignment, used in much of the author’s work with fixemasks or liquid crystal SLMs, is given below. We assuthe pulse shaper is assembled on a flat working surface,an optical table, and a short pulse source is used foralignment.

~1! Make sure that the input beam from the laser is climated and propagating in a plane parallel to the optitable surface.

~2! Align the first diffraction grating such that the planof the grating surface and the rulings of the grating lie inplane normal to the optical table. This can be achievedobserving the specular reflection~zero order diffraction! aswell as the desired diffraction order and making sure tboth propagate in a plane parallel to the table. With corralignment the direction in which the optical frequency coponents spread should also be parallel to the table surfa

~3! Place the first lens approximately one focal lengaway from the first grating, with the lens centered on anormal to the diffracted beam. The height of the lens shobe adjusted to ensure that the transmitted beam still progates parallel to the table surface. However, the exact ptioning of the lens is usually not critical.

~4! The mask or SLM can be placed at approximatelyback focal plane of the first lens, making sure that the apture of the SLM is centered on the spatially dispersedquency spectrum. At this point the SLM should be set toquiescent state~no amplitude or phase modulations!. Alter-nately, the mask or SLM can be left out at this point ainserted after the rest of alignment is complete.

~5! Position the second lens two focal lengths away frthe first lens. The lens should be translated closer to orther from the first lens until the transmitted beam is comated. The height of the second lens is adjusted as per~3!.

~6! Position the second grating one focal length fromsecond lens. Adjust the tilt of the grating and the directionthe grating rulings in the same way as per step~2!. There arethen two further critical adjustments:

~6a! The rotation of the grating about the vertical axmust be set to exactly match that of grating one. This canachieved by observing the output beam in the far field amaking sure that there is no remaining spatial dispersioi.e., all the frequency components should be optimally ovlapped. To help visualize residual spatial dispersion, oneplace a card in the masking plane and sweep it acrossspatially dispersed frequencies. If there is residual spadispersion at the output, one can see an image of thesweeping across the output beam; if there is no spatial


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persion, the output will be uniformly attenuated as the carmoved in the masking plane.

~6b! The distance between the second and first gramust be set for exactly four focal lengths in order to obtazero group velocity dispersion.~In the case of a chirped inpupulse, one may vary the grating separation from this contion in order to compensate the input frequency chirp.! Thegrating distance can be fine tuned by measuring the oupulse duration, for example, via intensity cross correlatusing an unshaped pulse directly from the laser as a reence. The grating is translated until the output pulse widthminimized. In order to estimate how precisely the gratiseparation must be set, it is helpful to use the followiformulas, which specify the grating dispersion as a functof grating separation:42,83,84

]2f~v!

]v2 5l3~L24 f !

2pc2d2 cos2 ud~2.15a!

and equivalently

]T~l!

]l5

l~L24 f !

cd2 cos2 ud. ~2.15b!

Here f(v) is the spectral phase andT(l) is thewavelength dependent group delay.L-4 f represents the detuning of the grating separation from the dispersion-freefseparation condition. The grating separation should be sethat T varies by much less than one pulse width acrossbandwidth of the pulseDl, or equivalently so thatf variesby much less thanp.

~7! Now the pulse shaping SLM can be activated@in-serted first if not done in step~4!#. Its aperture should becentered on the desired spectral region, and it shouldplaced at the back focal plane of the first lens where invidual frequency components are focused to their minimsize. Both adjustments are facilitated by monitoring the oput spectrum from the pulse shaper in order to observefeatures placed by the mask or SLM onto the spectrum.

G. Measurement of shaped pulses

The task of producing a shaped ultrashort pulseclosely connected to the task of measuring such shapulse. The field of ultrashort pulse shape measurementundergone dramatic progress during the last decade, inallel with progress in pulse shaping. Although it is not posible within the scope of this article to give a comprehensdiscussion of pulse measurement techniques, neverthelethe following we do give a brief, nonexhaustive overviewmethods which have been most frequently applied to chaterization of shaped pulses.

The technique which has been most often used for msuring shaped pulses~especially until recently! is intensitycross correlation, which yields a signal

XI~t!;E dt I1~ t2t!I 2~ t !. ~2.16!

This quantity can be obtained by measuring the averageond harmonic power generated through the interactiontwo pulses with intensity profilesI 1(t2t) and I 2(t) in a


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second harmonic generation crystal as a function of the rtive delayt. WhenI 1 is a short pulse directly out of a modelocked laser, andI 2 is a shaped pulse significantly longthan I 1 , then the cross-correlation signalXI(t) provides agood approximation to the shaped intensity profile. Howevany very rapid features inI 2 varying on the same time scaas I 1 will be broadened. Furthermore, no phase informatis available. Note, however, that if pulse shaping is succeful in accurately generating a complicated desired intenprofile, as verified by intensity cross correlation, then ocan usually infer that the desired phase profile is also geated at least reasonably well, even without direct measment. This proved true, e.g., in dark soliton propagationperiments, where the nonlinear propagation of the dsoliton pulse in an optical fiber was sensitive both toshaped pulse intensity and phase profile.78 Nevertheless, thelack of any direct phase information is a significant limittion of the intensity cross-correlation method.

Unshaped pulses in ultrafast optics have been measfor many years via intensity autocorrelation~see for exampleRef. 85!, which is obtained from Eq.~2.16! by settingI 1(t)5I 2(t). Although the exact pulse shape cannot be demined from intensity autocorrelation data, for simple pushapes the approximate pulse duration can still be obtaiHowever, for complicated shaped pulses, the intensity acorrelation is usually not useful since most pulse shapeformation is lost.

In addition to intensity cross correlation, field croscorrelation measurements can also be useful, especiallylow power signals. The electric field cross correlationgiven by

Xe~t!;E dt e1~ t2t!e2* ~ t !1c.c., ~2.17!

where ‘‘c.c.’’ means complex conjugate. This quantity cbe obtained by recording the interference fringes betwtwo field e1(t2t) ande2(t) as a function of delayt at theoutput of an interferometer. Ife1(t) is a short reference puls~e.g., right out of the laser! and e2(t) is a longer shapedpulse, then the interference envelope provides an apprmate measurement of the shaped electric field amplitue2(t)u. If the phases of the fringes are also carefully mesured, then one can also determine the time-dependent pof e2(t). In this way the shaped electric field is completecharacterized.

An important breakthrough in the field of ultrafast optiwas the relatively recent development of techniques for coplete amplitude and phase characterization of ultraspulses. The best known of these techniques is cafrequency-resolved optical gating, or FROG.86,87 In FROGone gates the pulse to be measured with an identical tidelayed version of itself. Various gating mechanisms, sas second harmonic generation, nonlinear polarization rtion, and transient gratings, have been employed. The pospectrum of the output pulse resulting from the nonlingating interaction is measured with a spectrometer as a ftion of time delay between the two pulses. In the case ofpolarization rotation gating mechanism, the measured qutity S(v,t) can be expressed as


a-

r,

ns-yer-e--ke

ed

r-ed.

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for

n

xi-e-ase

-rtd

e-ha-errc-en-

S~v,t!;U E ein~ t !uein~ t2t!u2e2 j vtdtU2

, ~2.18!

whereein(t) is the pulse to be characterized. Equation~2.18!is a kind of spectrogram, a type of time-frequency transfowhich is well known in speech analysis for example. Oimportant feature of this measurement approach is that tdependent frequency shifts~chirps! can often be visualizeddirectly and intuitively. A second~and most important! fea-ture is that both the complete intensity and phase profilethe ultrashort pulse can be recovered, in practice by usiterative computer optimization techniques.

One example of FROG data, using polarization rotatgating, is shown in Fig. 12.87,88 Here the input pulses werderived from a Ti:sapphire chirped pulse regenerative amfier, with a mask placed at the Fourier plane of the pustretcher in order to block the central frequency componeas a simple example of pulse shaping. The modulation althe delay axis in the experimental FROG trace@Fig. 12~a!# isindicative of a strong temporal modulation, which is in faobserved in the retrieved multiple-pulse intensity profi@Fig. 12~b!#. The modulation along the wavelength axisthe FROG trace arises due to the double peaked nature oshaped spectrum.

A second example of FROG data, applied to optimiztion of high power pulses from a chirped pulse amplifisystem though pulse shaping, is presented in Sec. VII~Fig. 31!.

There are two ways in which FROG can be appliedcharacterization of shaped pulses. In the first, FROG isplied to measure the shaped pulse itself, as in the examabove. A limitation of this approach is that for sufficientcomplicated waveforms, convergence of the computer rtine for recovering the temporal phase and intensity profifrom the FROG data may become less dependable. S

FIG. 12. ~a! Measured FROG trace of amplified, shaped signal,~b! Time-dependent intensity~solid! and phase~dashed! retrieved from FROG data.


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examples of waveforms from a pulse shaper that have bcharacterized by FROG are published in Refs. 88–90. Insecond case, FROG is used to completely characterize pfrom the modelocked laser, which then serve as referepulses for measurement of the more complicated shapulses, using noniterative techniques such as intensityelectric field cross correlation or spectral interferometry~seelater!.

Another technique which can be applied to charactertion of shaped pulses is based on spectrally and temporesolved upconversion. Experiments of this type splitwaveform to be characterized into two parts: one is usedreference, while the other is spectrally sliced, e.g., by usinsimple pulse shaper containing a narrow slit at the Fouplane or by using an interference filter, and thereby stretcin time to a duration much longer than the reference puThe upconversion cross-correlation signal between the reence and spectrally sliced pulses is generated using a seharmonic crystal and then analyzed. In the earliest approthe temporal position of the spectrally sliced pulse is msured as a function of the frequency of the transmitted sptrum, which is varied by translating the slit in the pulshaper in a series of correlation scans.91,92 This yields thefrequency dependent delayt(v), which is related to thespectral phasef(v) by

t~v!52]f~v!

]v. ~2.19!

In another approach, the upconversion signal was spectresolved using a spectrometer as a function of crocorrelation delay for a fixed position of the slit~i.e., a fixedspectral slice!,93 which gives similar information. Compareto FROG, one advantage of these techniques is that dutheir noniterative nature, data can be analyzed fast enofor real-time display of the pulse characterization data.disadvantage is that the bandwidth of the spectral slicer mbe consciously chosen to be narrower than the most rapvarying spectral features of the waveform under test~and thismay not be knowna priori! in order to achieve accuratresults.

These early experiments cited earlier were used for cacterization of unshaped pulses. Recently related methhave been applied to shaped pulses. In one case the meof Refs. 91 and 92 was generalized by including a multipslit mask in the Fourier plane.44 Different pairs of slits giverise to various interference features in the time-domaintensity cross-correlation trace, and the temporal phase orsition of these interference features provide informationf(v), which can be extracted automatically via Fouranalysis. An improvement compared to the earlier workthat only a single cross-correlation scan is required. Texamples of pulse measurement data obtained using thicalled ‘‘direct optical spectral phase measurement’’~DP-SOM! are shown in Fig. 13. In one case an ‘‘odd pulse’’28 isgenerated by imposing an abruptp phase shift into half ofthe spectrum, resulting in a pulse doublet which is partiaresolved in the conventional intensity cross-correlation tr@Fig. 13~a!#. In the second case, a glass plate was inseinto half of the spectrum in the pulse shaper, leading t


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frequency-shifted pulse doublet, with the two pulses clearesolved in the cross correlation@Fig. 13~c!#. The spectralphase data extracted from the DPSOM measurementshown in Figs. 13~b! and 13~d!. In the latter trace, the lineaspectral phase evident over half of the spectrum correspoas expected to a simple time delay of that half of the sptrum, as per Eq.~2.19!.

Another case of this type of measurement which hbeen used for shaped pulse characterization has been nSTRUT ~for ‘‘spectrally and temporally resolved upconvesion technique’’!. This method is related to that of Ref. 93but adapted for a single-shot measurement geometryreal-time data analysis for application to femtosecond amfier systems.94,95 An example of STRUT data is discussedSec. IV.

Finally, we mention spectral interferometry,96 in which ashort reference pulse with spectrumE1(v) is combined withthe pulse to be characterized~e.g., a shaped pulse! with spec-trum E2(v). The two pulses are given a relative time delt, and the resulting power spectrum is measured, which ctains an oscillatory term of the form

S~v!;E1~v!E2* ~v!e2 j vt1c.c. ~2.20!

The spectrum contains fringes with period approximatt21, which can be analyzed for the case of a shobandwidth-limited~or well characterized! reference pulse toyield the complex spectrumE2(v). Note that it is the fre-quency dependent deviation of the spectral fringe perfrom the average period that yields spectral phase infortion; the average fringe period itself depends only onrelative delay and not on details of the pulse shape. Furtmore, since this is an interferometry technique, what is acally measured is the relative spectral phase of the pulse utest with respect to the reference pulse. Spectral interferetry has been used by several groups for characterizatioshaped ultrashort pulses;38,97–99 this technique is relativelyeasy to apply since a short pulse directly from the laseusually available in pulse shaping experiments. In orderobtain the exact spectral phase of the pulse under test,

FIG. 13. ~a! Intensity cross-correlation measurement and~b! spectral phaseof femtosecond ‘‘odd’’ pulse.~c! Intensity cross-correlation measuremeand ~d! spectral phase of frequency shifted pulse doublet.


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must carefully characterize the reference pulse, e.g., by uFROG, as noted earlier.87 On the other hand, if the referencpulse is already known to be approximately bandwidth liited, then one can get the approximate spectral phase oshaped pulse without characterizing the reference pulse.accuracy of this approach of course depends on how gthe reference pulse is. However, in some situations ononly interested in characterizing the relative spectral phbetween shaped and reference pulses; the exact form oshaped pulse is not of interest, and therefore, the phase oreference pulse is not relevant. This situation applies forample when one uses spectral interferometry to measuredispersion of an optical component or subsystem placeone arm of an interferometer. This approach has been ufor characterization of a pulse shaper response to broadincoherent light illumination100 and for characterization odispersion-compensated fiber links suitable for neadispersion-free femtosecond pulse transmission.101

III. PROGRAMMABLE PULSE SHAPING USING LIQUIDCRYSTAL SLMs „LC SLMs …

We now proceed to discuss programmable pulse shausing LC SLMs as a programmable mask. Pulse shapusing other programmable mask technologies, suchacousto-optic modulators, will be discussed in subsequsections. Note that some of the discussion in this sectsuch as SLM construction, is specific to pulse shaping usLC arrays, whereas other aspects of the discussion,pulse shaping using phase-only filtering, cover conceptsare also applicable to pulse shaping using other SLM tenologies.

A. Pulse shaping using electronically addressedliquid crystal modulator arrays

Liquid crystal modulator~LCM! arrays have been primarily configured for either phase-only or phase-anamplitude operation in pulse shaping applications. Figuredepicts an apparatus for programmable pulse shaping usLC SLM in phase-only operation.34,35 Compared to pulseshaping setups using fixed masks, the main differencesthat the fixed masks are replaced by the LC SLM, and aof half-wave plates are used to rotate the polarization inder to match that required by the modulator array. Thearray allows continuously variable phase control of easeparate pixel~whereas fixed masks usually provide only bnary phase modulation! and allows programmable control othe pulse shape on a millisecond time scale.

FIG. 14. Programable pulse shaping apparatus based on a LCM arr


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Figure 15 shows a schematic of an electronically adressed, phase-only LC SLM.35 A thin layer of a nematicliquid crystal is sandwiched between two pieces of glaThe nematic liquid crystal consists of long, thin, rod-likmolecules, which in the absence of an electric fieldaligned with their long axes along they direction. When anelectric field is applied~in the z direction!, the liquid crystalmolecules tilt alongz, causing a refractive index change foy-polarized light. A maximum phase change of at least 2p isrequired for complete phase control. In order to applyrequired electric field, the inside surface of each pieceglass is coated with a thin, transparent, electrically conduing film of indium tin oxide. One piece is patterned intonumber of separate electrodes~or pixels! with the corre-sponding fan out for electrical connections. In the origin‘‘homemade’’ modulator array in Fig. 15, there are 128 pels with 40mm center-to-center pixel spacing, 2.5mm gapsbetween pixels, and a total aperture of 5.1 mm. Current comercially available LC SLMS also have 128 pixels, but wi100 mm center-to-center pixel spacing for a 12.8 mm apture. The modulator array is controlled by a special drcircuit which generates 128 separate, variable amplitudenals to achieve independent, gray-level phase control o128 modulator elements. An additional point concerningdrive circuitry is worth noting. Each drive signal actualconsists of a variable amplitude bipolar square wave, tycally at a few hundred hertz or above, rather than a variaamplitude direct current~dc! level. The use of an alternatincurrent~ac! drive signal is required to prevent electromigrtion effects in the liquid crystal.102 Otherwise the use of asquare wave as opposed to a dc level does not changeoperation of the modulator, since the rotation of the liqucrystal molecules depends only on the amplitude~not thesign! of the applied voltage. Input data for the drive circuitloaded into local memory from a personal computer, thfacilitating the generation of complex phase patterns.

Figure 16 shows an example of real-time pulse shapdata measured using an earlier 32 element liquid cryphase modulator array and 75 fs pulses at 0.62mm from acoliding-pulse-modelocked dye laser.34 In the experimenthalf of the pixels were connected to a constant amplitudrive signal, while the other half were switched at a 20rate between the different drive levels. In one state,phases are all the same, and the output is a single p

.

FIG. 15. Schematic diagram of an electronically addressed, phase-onlSLM.


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similar to the input pulse. In the other state, half of the pixexperience a relative phase shift ofp. With the particular setof connections chosen in this experiment, this results ipulse doublet. The pulse intensity profiles~top trace! shownin Fig. 16 were obtained by using a real-time croscorrelation setup, which was driven in synchronism with ttime-varying drive signal~bottom trace!. The data demon-strate complete switching from a single pulse to a pulse dblet within 25 ms. Under appropriate drive conditions, suLCM arrays are capable of switching in;10 ms. It is im-portant to point out, however, that switch-on and switch-times are usually not the same and depend on the biasage levels.

In order to use LC SLMs for gray-level phase controlcareful phase versus voltage calibration is required. Thisbe accomplished by using the array as an amplitude molator for a continuous-wave~cw! laser. The laser is linearlypolarized, with its polarization rotated 45° relative to talignment direction of the liquid crystal, and focused ontosingle pixel of the multielement modulator. The phase cbration is obtained by measuring the transmission througsubsequent crossed polarizer and using the relation

T~V!5sin2S fy~V!2fx

2 D , ~3.1!

where T(V) is the fractional transmission through thcrossed polarizer, andV is the applied voltage. Herefx cor-responds to light polarized along the short axis of the liqcrystal molecules and is independent of the applied voltafy(V) corresponds to the voltage-dependent phase plaonto the optical spectrum when the modulator is positionwithin the pulse shaper. In Ref. 35 several calibration curwere measured at different points along the array. The resshows good uniformity, as required for high quality pulshaping. The LC array can also be calibratedin situ withinthe pulse shaper by using pulse characterization technisuch as spectral interferometry to measure the spectral pchanges imposed on an ultrashort pulse.

Gray-level phase control can be used to accomplisnumber of interesting pulse shaping tasks. For example,ing the modulator array to impart a linear phase sweep othe spectrum corresponds in the time domain to pulse p

FIG. 16. Real-time pulse shaping data. Top trace: pulse intensity promeasured using a real-time cross-correlation setup, showing switchingtween two distinct waveforms with 25 ms. Bottom trace: Time-varyindrive signal to the LCM array.


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tion modulation. Pulse position modulation relies on the fthat if f (t) andF(v) are a Fourier transform pair, then thdelayed signalf (t-t) is the Fourier transform ofF(v)3exp(2ivt). Thus, a pulse can be retarded~or advanced! byimposing a linear phase sweep onto its spectrum. The deltis given by the relation

t52df/2pd f , ~3.2!

where df and d f are, respectively, the imposed phachange per pixel and the change in optical frequency frone pixel to the next. For the experiments in Ref. 35 with128 element modulator,d f was approximately 0.092 THzand thereforet5210.87 (df/2p) ps. The modulator wasset to provide the required phase sweep modulo 2p, so thatfor each pixel the phase is in the range 0–2p. Figure 17shows cross-correlation measurements of temporally shipulses achieved by means of pulse position modulation. Dare shown for a phase change per pixel~df! of 0 and6p/4.The output pulses occur at 0 and61.38 ps, in close agreement with Eq.~3.2!. These data demonstrate the abilityshift the pulse position by many pulse widths by using sptral phase manipulation. Note that the total phase sweethese experiments is 32p. Because pixellated LCM arraycan be programed to provide the desired phase funcmodulo 2p, large phase sweeps can be achieved using molators with maximum phase changes as small as 2p. How-ever, in any case of a smoothly varying target phase fution, as here, the phase change from one pixel to the nshould remain small enough that the staircase phase patwhich is achieved by the LCM with its discrete pixels, issufficiently good approximation to the desired phase fution. This point is discussed further in connection with E~3.4! later.

Gray-level phase control can also be used to achiprogrammable compression of chirped optical pulses.34,35Asone example, we discuss recent experiments in which a pshaper equipped with a LC SLM was used to complemfiber dispersion compensation techniques in propaga;500 fs pulses over a 2.5 km optical fiber link.40 The linkconsisted of 2.1 km of standard single-mode fiber~SMF! and0.4 km of dispersion compensating fiber~DCF!. Since SMFand DCF have dispersion with opposite signs at the operawavelength of 1.56mm, the fiber lengths can be adjustedcancel all of the lowest order dispersion~i.e., phase varying

se-

FIG. 17. Measurements of temporally shifted pulses achieved by plalinear phase ramps onto the spectrum. Data are shown for phase changpixel of 0 and6p/4, resulting in delays of 0 and61.38 ps, respectively.


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lar00

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quadratically with frequency!. In addition, since the derivatives of the dispersion with respect and frequency also hopposite signs, the dispersion slope~i.e., cubic spectral phasvariations! can also be partially compensated. In the expements in Ref. 40,;500 fs input pulses are first broaden;400 fold in propagating through the SMF, then recopressed by the DCF to within a factor of 2 of the input puduration. Intensity cross-correlation measurements of theput pulse and output pulse after 2.5 km fiber propagationshown in Figs. 18~a! and 18~b!, respectively. The remainingbroadening and distortion of the output pulse are characistic of a cubic spectral phase variation. At this point, a LSLM in a pulse shaper can be used to apply a spectral pfunction @Fig. 8~d!# which is equal and opposite to the esmated residual phase variation of the output pulse fromfiber link. The result, shown in Fig. 18~c! is a completelyrecompressed pulse with the original pulse duration andobservable distortion. Thus, in these experiments all-fitechniques are used for coarse dispersion compensawhile a programmable pulse shaper is used to fine tuneremaining dispersion. Similar experiments extendedshorter pulses~400 fs! and longer fiber spans~10 km! haverecently been reported.103 Pulse shapers with LCM arrayhave also been used for compensation of residual disperin chirped pulse amplifier systems69,82and for adaptive com-pression of chirped pulses~see Sec. II D!.

Gray-level phase control for pulse position modulatiand programmable pulse compression, first demonstrateding LCMs, has also been extended to pulse shaping sysusing moveable or deformable mirrors or acousto-opmodulators. This will be discussed in later sections.

LC SLMs have also been used for phase-only filterand shaping of pulses with temporal resolution approach10 fs. Figure 1936 shows one example, where;13 fs pulsesfrom a modelocked Ti:sapphire laser with a center walength of approximately 800 nm are shaped usingall-reflective-optics30 pulse shaper. The LCM imparts a cubspectral phase variation, leading to an asymmetric distor

FIG. 18. ~a! Input pulse to the 2.5 km fiber link.~b! and ~c! Output pulsesfrom the fiber link with ~b! constant phase or~c! cubic phase correctionapplied to the LCM.~d! The cubic phase correction function.


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similar to that seen in Fig. 18~b! and in good agreement withthe calculated pulse shape.36

For the most general pulse shaping operation, indepdent control of spectral phase and amplitude is desirGiven the appropriate input polarization relative to the liqucrystal axis, a single LCM array together with a polarizer cbe used as an amplitude modulator; however, this leadsto a phase modulation which depends on the amplitmodulation level. Therefore, for independent phase andplitude control the use of two LC arrays is required. Orignally this was accomplished by modifying the setup of F14 to contain two telescopic lens pairs~four lenses! betweenthe gratings, with separate LC SLMs between the first asecond and third and fourth lenses, respectively.58 A consid-erably simplified and improved setup, which is currently tpreferred geometry for independent phase-amplitude molation, was achieved by combining two LC arrays intosingle device, as depicted in Fig. 20.104 The two arrays areattached permanently with their pixels aligned and anguorientations fixed. Each array consists of 128 pixels on 1mm centers with 3mm gaps. The dual-LC–SLM unit isplaced between a pair of Polacor polarizers passing light

FIG. 19. Shaping of 13 fs pulses using a phase-only liquid crystal phmodulator array. In this example, the LCM produces a cubic spectral phvariation, leading to the asymmetric distortion seen in the intensity crocorrelation plot.

FIG. 20. Schematic diagram of pulse shaping using an electronicallygrammable dual LCM array offering independent phase and amplitudetrol.


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larized along the spectral dispersion direction~horizontal inFig. 20, denotedx!. In this orientation there is no need trotate the polarization between the gratings and LCM arrtherefore, no wave plates are required. With the propagadirection denotedz and the second transverse direction~ver-tical in Fig. 20! denotedy, the long axes of the liquid crystamolecules in the two SLMs are aligned at645° with respectto thex axis. When a voltage is applied to a pixel in onethe SLMs, the liquid crystal molecules in that pixel are rtated towardz, resulting in a phase modulation for the component of light parallel to the liquid crystal axis in that SLM

For x-polarized light incident on a particular pixel of thdual SLM array, the output field is given by

Eout5Ein

2 Hexp~ j Df~1!!~x1y!1exp~ j Df~2!!~x2y! ~3.3a!

5Ein expF j ~Df~1!1Df~2!!

2 GFx cosS Df~1!2Df~2!

2 D1 j y sinS Df~1!2Df~2!

2 D G J , ~3.3b!

whereEin is the amplitude of the incidentx-polarized lightand Df (1) and Df (2) are the voltage dependent birefringences of the first and the second LCM array, respectivUsing an output polarizer alongx, the output phase and atenuation can be set independently by controllingDf (1)

1Df (2) and Df (1)2Df (2), respectively. Note that eacLCM array in the dual SLM can be calibrated by measurits amplitude modulation response as a function of voltawith the other LCM array held at constant voltage.

Figure 21105 shows four examples of pulse shaping usithe dual LC SLM array. The input pulses were 70 fsduration at 800 nm, and all the traces are intensity crocorrelation measurements using unshaped reference p

FIG. 21. Intensity cross-correlation traces of shaped pulses using theLCM array. ~a! 800 fs square pulse.~b! Sequence of five, equal-amplitudpulses.~c! Three pulse sequence with different chirp rates per pulse.~d! Tenpulse sequence with pulse timing, amplitude, and phase control. In~c! and~d!, the target intensity profiles are also shown~dashed lines!. @With permis-sion, from the Ann. Rev. Phys. Chem.46 ~1995!, by Annual Reviews http://www.AnnualReviews.org#


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e,

s-ses

directly from the laser. The data represent~a! an 800 fssquare pulse,~b! a five pulse sequence of equal amplitupulses in which both the pulse timings and phases are spfied, ~c! a three pulse sequence with different chirp ratespulse, and~d! a sequence of ten 70 fs pulses with pultimings, phases, and amplitudes all specified. In Figs. 21~c!and 21~d!, the desired intensity profiles are also shown~asdashed lines!. Clearly, waveforms with a rather high degreof complexity can be generated with excellent fidelity.

Finally, we note that dual LC SLMs may be useful focontrolling the time-dependent polarization profile of utrashort pulses. This possibility, as well as a simple expmental demonstration, are discussed in Ref. 104. Note, hever, that the diffraction efficiency of the gratings, whicusually have a strong polarization sensitivity, make polarition pulse shaping more difficult.

It is worth commenting on the degree of extinction posible with a LCM for pulse shaping. In my own group, whave sometimes used single layer LC SLMs~usually used forphase-only applications! as an amplitude only modulatorwhich is achieved by rotating the input polarization by 4and using an appropriately oriented output polarizer. Indata of Ref. 106, an on-off ratio exceeding 30 dB is achievfor a commercial single layer LC SLM working in the 1.5mm band. We also have experience using phase-ampliLCMs at 1.55mm, and are able to achieve on-off ratiosthe range 25–30 dB.107 In both cases, an accurate calibratioof the LCM response is essential. In our experience weachieve the best accuracy when the calibration is performwith the LCM placed inside the pulse shaper at exactlyposition where it will be used.

The pixellation of electronically addressable phasephase-and-amplitude LC SLMs leads to one significlimitation—namely, the SLM can only produce a staircaapproximation even when a smooth spectral profile issired. The requirement that the actual spectral modulashould approximate a smooth function despite the fixed,nite size of the individual modulator elements, limits thtemporal range over which pulse shaping can be successachieved. Essentially this is a sampling limitation: the sptrum must vary sufficiently slowly that is adequatesampled by the fixed modulator elements. For the casepulse position modulation, for example, we requireudfu!p, wheredf is the phase change per pixel. From Eq.~3.2!,then, we findutu!1/2d f ~whered f is the optical frequencyspan corresponding to a single pixel!. The effect of pixela-tion for pulse shaping in general has been analyzed in R35. The result is

eout~ t !;Fein~ t !* (n

h~ t2nd f 21!Gsinc~pd f t !, ~3.4!

whereh(t) is the desired impulse response function asEq. ~2.1! and sinc(t)5sin(t)/t. This expression assumes neligibly small interpixel gaps and a focused spot size atmasking plane much less than a LCM pixel, so that the teporal window functiong(t) in Eq. ~2.10! is much wider thanthe sinc function in Eq.~3.4!. The result of pixelation is toproduce an output pulse which is the convolution of the inpulse not only with the desired impulse response funct

ual


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ig

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h(t), but also with a series of replica impulse response futions,h(t2nd f 21), occurring at timest5nd f 21. The entireresult is weighted by a temporal window functiosinc(pd f t), which has its first zeros att56d f 21. If h(t) isnonzero only during a time intervalutu!d f 21, then the ac-tual shaped pulse will closely approximate the desishaped pulse, except for a few low amplitude replicas arotimes t56nd f 21. On the other hand, ifh(t) takes on asignificant amplitude for times extending out toutu;d f 21,then a sort of temporal aliasing, in which the replica wavforms blend into the main waveform centered att50, willoccur.

It is worth mentioning that in actual experiments, tfocused spot size at the masking plane if desired can beapproximately equal to the pixel size. In this case the specfilter function from Eq.~2.9! is a smoothed version of thspatial profile corresponding to the pixelated SLM. The cresponding Gaussian time window function significantlyduces the replica pulses predicted by Eq.~3.4!. This allowsthe experimentalist to give up some spectral resolution~op-timized by minimizing the focused spot size! for the purposeof smoothing of pixelation effects and elimination of replipulses~by increasing the focused spot size!.

Due to the effects just discussed, LC SLMs with pixcounts higher than the 128 used to date in pulse shawould be desirable. We expect that this will be achievedfuture pulse shaping instruments in view of the followinconsiderations:

~1! LC devices with substantially higher pixel counts ain widespread use in display applications, although the scific characteristics of these LC display devices are not ually directly suitable for pulse shaping.

~2! LC SLMs with a much higher degree of sophistiction than currently used for pulse shaping have been deoped for beam forming and beam scanning in laser rada108

For example, dedicated beam steering LC SLMs with o40 000 pixels, configured to achieve programmable linphase sweeps~modulo 2p! via the use of up to 512 independent address lines, have been reported. These devices sbe directly applicable to pulse shaping applications, sibeam forming and scanning operate based on the princof spatial Fourier optics, which are closely analogous toprinciples of pulse shaping. A variety of sophisticated LCdevices are discussed in Ref. 108, which also gives an exlent overview of relevant liquid crystal principles and tecnology.

It is also worth noting that programmable pulse shapusing ferroelectric liquid crystals~rather than the nematiliquid crystal discussed earlier! has also beendemonstrated.109 Ferroelectric liquid crystals offer approxmately two orders of magnitude faster response time~;100ms! but are limited to binary phase modulation and generahave significant insertion loss.

B. Pulse shaping using optically addressed LC SLMs

Pulse shaping using an optically addressed LC SLalso known as a liquid crystal light valve, was recen


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reported.98 The motivation for this work is to avoid pixelation of the pulse shaping SLM.

A simplified schematic of the LC light valve is shown iFig. 22. The light valve consists of two continuous transpent electrodes and continuous layers of a twisted nemliquid crystal and of photoconductive Bi12SiO20 ~BSO!.When illuminated with light in the blue green, the condutivity of the BSO layer depends on the local illuminatiolevel. When a voltage is applied between the two electroda local change in the BSO conductivity results in a changethe voltage drop across the LC layer. This leads to rotationthe LC molecules and a phase change for light passthrough the layer. The light valve is addressed by usindisplay device, such a liquid crystal television which sptially modulates light from an arc lamp. Gray-level intensichanges lead to gray-level phase shifts from the LC livalve. Pixelation effects are avoided because:~1! the lightvalve itself is a continuous device; and~2! the spatial fre-quency response of the light valve~;10 line pairs per mm!is not sufficient to resolve the small 40mm pixels from theLC television. Therefore, the discontinuous pattern onpixelated display device is transformed into a continuospatial response by the LC light valve. The liquid cryslayer of the light valve is placed at the focal plane ofgrating-lens pulse shaper for spectral phase filtering.

Control of the light valve is more complicated than fthe electrically addressed SLMs discussed earlier, sinceinteraction between nearby pixels of the input display devithe limited spatial frequency response of the light valve, athe effect of illumination conditions make it difficult to compute the input image needed to achieve a specified phpattern. To achieve a target phase pattern, an approxim~best guess! input picture was applied, and then spectralterferometry was used to measure the achieved spephase profile. A feedback loop then increases or decrethe local gray-level illumination intensity, dependinwhether the measured phase change at the corresponwavelength is higher or lower than the target value, unconvergence is achieved. Reference 98 demonstratedmethod to induce arbitrary phase patterns, e.g., fourth ophase targets, with a phase control accuracy of;3%. Thisapproach was also used to demonstrate compressiochirped pulses from a modelocked Ti:sapphire laser osctor from 150 to 50 fs98 and spectral phase modulation at t

FIG. 22. Schematic diagram of an optically addressed liquid crystal livalve.


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input of a millijoule-class chirped pulse amplifier~CPA! sys-tem allowing improvement of amplified pulse quality as was generation of antisymmetric pulse doublets at the amfier output.110

In these experiments the LC light valve was able to pvide a maximum phase shift of 12 rad. Unlike the pixelatLC modulators, where one can have 2p phase jumps between adjacent pixels and, hence, program desired pfunctions modulo 2p, for the nonpixelated light valvesabrupt phase jumps are not possible. Therefore, thephase sweep is in fact limited to a maximum of 12 rad. Dto the limited spatial frequency response of the LC ligvalve, the spectral resolution of the pulse shaper was 3~with the total optical bandwidth set to;50 nm!, which issignificantly coarser than that obtained in experiments uspixelated LC SLMs. For these reasons, the opticallydressed approach is better suited for correction of smallsidual phase in CPA systems than it is for generationstrongly modulated spectral phase profiles. On the ohand, liquid crystal light valves with significantly improvespatial frequency response have been reported,111 whichcould potentially be applied to improve spectral resolutfor pulse shaping applications.

IV. PROGRAMMABLE PULSE SHAPING USINGACOUSTO-OPTIC MODULATORS

Programmable pulse shaping based on the use oacousto-optic modulator~AOM! as the SLM has been deveoped by Warren and co-workers.80,112–114The apparatus isdepicted in Fig. 23. The AOM crystal, typically TeO2, isdriven by a radio-frequency~rf! voltage signal, which is converted into a traveling acoustic wave by a piezoelectric traducer. The acoustic wave travels across the modulatorvelocity vac, leading to a refractive index grating through thphotoelastic effect. The grating periodL is given byvac/n,wheren is the rf drive frequency. The grating can be phaamplitude, or frequency modulated through the use of a crespondingly modulated rf drive waveform; ideally the sptial grating s(x) would be related to the input voltagev(t)through s(x);v(x/vac). In practice there are a number omechanisms which distort this ideal relationship, as dcussed later. When the spatially dispersed optical frequecomponents diffract off the grating, the optical spectrum

FIG. 23. Programmable pulse shaping apparatus based on the useAOM as the SLM.


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modified according to the grating spatial modulation funtion. This results in the desired Fourier transform pulse shing operation.

AOMs are a relatively mature technology which habeen commercially available for many years. In the folloing we briefly describe some of the typical operating charteristics of AOMs used for pulse shaping. Further detailgiven in Refs. 80 and 113.

~1! The time for the acoustic wave to move acrossmodulator aperturel a is to5 l a /vac. This aperture time de-termines how quickly a new spatial pattern can be loadinto the device. For TeO2 the acoustic velocity isvac

54.2 mm/ms. For the experiments reported in Refs. 112 a80, the modulators had 5 mm and 4.3 cm apertures, restively. The corresponding aperture times are 1.2 and 10ms,which means that the acoustic grating pattern can be updon a microsecond time scale.

~2! The acoustic grating pattern, which propagates acrthe modulator, is not fixed. Although the grating does appfrozen during readout by a single femtosecond or picosecpulse, the pattern can shift significantly during the time btween pulses for typical modelocked laser sources. Forample, for a modelocked laser with 100 MHz repetition rathe grating in a TeO2 AOM travels by 42 mm betweenpulses, which is larger than the acoustic wavelength for tycal drive frequencies of 100–200 MHz. Therefore, the AOtechnique cannot in general be used for pulse shapinghigh repetition rate laser sources, since the pulse shwould change from pulse to pulse. Note, however, that thinot a limitation for amplified ultrafast laser systems whethe pulse repetition rate is usually slower than the acouaperture time, since this allows the acoustic pattern torefreshed before and synchronized to each amplifier puFor this reason AOM pulse shaping is usually restrictedapplications involving femtosecond amplifier systems.

~3! The number of independent acoustic features whcan be placed within the full aperture of the AOM providan upper limit to the pulse shaping complexityh @see Eqs.~2.12!–~2.13!#, which is the same as the number of spectfeatures which can be placed onto the spectrum. The numof independent pixels of resolution available is proportionto Neff;Dnto5Dnla /vac, whereto is the acoustic aperturetime, Dn is the modulation bandwidth of the AOM, anDnto is the AOM time-bandwidth product. For the largaperture (l a54.3 cm) AOM used in Refs. 80 and 113, thbandwidth is given asDn590 MHz, resulting in Dnto

'900. Therefore, an appropriate AOM should be ableplace up to nearly a thousand independent features ontospectrum, provided it is placed within an optical system wsufficient resolution. However, making use of all this potetial resolution depends on setting up the pulse shaping aratus for very high spectral resolution. To date the highexperimental resolution obtained via AOM pulse shapingpears to be on the order 100–150, which is achieved inperiments where;75 fs input pulses are reshaped over aps window. This is comparable to what has been demstrated in pulse shaping using LC SLMs.

~4! From an implementation perspective, it is necessto be able to generate the complicated rf voltage wavefo

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needed to excite appropriately shaped acoustic wavefoThis can be accomplished using electronic arbitrary waform generator~AWG! technology, which can produce complicated voltage waveforms under computer control. Tmost economical approach uses baseband AWGs, with bwidths of perhaps of few tens of megahertz; the outputs frthe AWGs are converted to the acousto-optic centerquency~typically a few hundred megahertz! using a simplerf oscillator and rf mixers. In these schemes two basebAWGs may be needed, one for each quadrature of the finsignal. An alternative but more expensive approach usesingle high-bandwidth AWG to produce the desired modlated rf waveform directly. For further discussion, see R113.

Figure 2480 shows an example of a relatively compleexperimental waveform. In this case the femtosecond psource was a modelocked laser oscillator with;100 MHzrepetition rate, so a boxcar integrator running at 20 kHz wused in the measurement to select those pulses for whichacoustic waveform was properly phased. The waveform csists of an initial pulse followed by a series of delayed pulwith linear, quadratic, cubic, and quartic spectral phase,spectively. Excellent agreement between experimentaltheoretical cross-correlation traces was observed. A numof other complex waveforms have been reported. Theseclude pulses with cubic spectral phase up to 90p,113 pulseswith a combined 50p cubic and 40p quintic spectralphase,113 and amplified pulses with hyperbolic secant amptude and hyperbolic tangent frequency sweep.114 The latterwaveform has potential applications for optical adiabarapid passage experiments in atomic systems.

A detailed time-frequency characterization of tfrequency-swept hyperbolic secant pulse was performe114

using the STRUT95 technique described briefly in Sec. II GNote that for the amplified pulse experiments, the AOM malso be programed to precompensate for residual spephase arising in the CPA system as well as any specamplitude distortion that may arise in the acousto-optic pushaper system.80,114 Experimental and theoretical STRUtraces for the frequency-swept pulse, with a target analytform given by

FIG. 24. Dashed line: intensity cross-correlation trace of waveform geated using an AOM in a pulse shaper. The waveform consists of an inpulse followed by a series of delayed pulses with linear, quadratic, cuand quartic spectral phase, respectively. Solid line: theoretical trace.


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are shown in Fig. 25.114 Clearly both traces show a closresemblance. For a simple interpretation of the STRUT drecall that the measurement provides information onfrequency-dependent delayt~v!. The instantaneous frequency shift as a function of time can be visualized appromately by rotating the data by 90°, so that wavelengthpears to plotted as a function of delay. In this orientationdata appear to take on the shape of hyperbolic tangent, wis consistent with the predicted frequency shift

Dv~ t !52]f~ t !

]t5rm tanh~rt ! ~4.2!

arising from the temporal phasef(t) in Eq. ~4.1!. Furtherdata from these measurements, including the recovered stral phase information and its derivative which givest~v!can be seen in Ref. 114.

r-alc,

FIG. 25. ~Color! ~A! Experimental and~B! theoretical STRUT traces forpulse with a hyperbolic tangent frequency sweep.


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It is useful to classify AOMs as operating in either thRaman–Nath or Bragg regime. In the Raman–Nath regthe AOM may be described as a thin grating, whereas inBragg regime the AOM is described as a thick~volume!grating. Quantitatively the AOM may be classified usingparameterQ, defined by

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whereL is the interaction length~thickness! in the grating,lm is the optical wavelength in the material, andL is theacoustic wavelength. The dependence of AOM operaversusQ has been discussed in the context of pulse shapin Refs. 80 and 113. ForQ<0.1, the AOM is in the Raman–Nath regime, which is characterized by a diffraction efciency which scales quadrically with acoustic amplitude aby the absence of any pronounced angular selectivity~whichcan lead to multiple diffracted orders from the acousto-opgrating!. For Q>4p, the AOM operates in the Bragg regime. In this regime the AOM has a strong angular selecity, and only light incident near a certain angle~the Braggangle! can be efficiently diffracted. The Raman–Nath regimgives the best spatial~hence, spectral! resolution due to thethin interaction region; however, the diffraction efficiencyrelatively low and even in the case of very strong acouwaves is limited to a theoretical maximum of 30%~due todiffraction into multiple orders!. In the Bragg regime thediffraction efficiency can be very high~theoretically 100%!,and the required acoustic amplitudes are reduced. Howethe thick interaction region degrades spatial~spectral! reso-lution and reduces the rf as well as the optical bandwAccording to Refs. 80 and 113, a value ofQ>4p appears tobe a good compromise for pulse shaping applications.

There are several factors which impact the fidelity athe pulse shaping resolution available with an AOM, apredicting the detailed effect of these factors in any givsituation is rather complicated. A detailed discussion is givin Refs. 80 and 113; we summarize the key points brielater.

The pulse shaping resolution of an AOM is limited boby the minimum acoustic feature size~hence, the time-bandwidth product! and by the finite optical beam size. Thbeam size in turn is limited not only by the factors discussin Sec. II C, but additionally due to the finite interactiolength in an AOM operating in or near the Bragg regimThere are three principal issues, as follows:

~1! The finite interaction thickness of the AOM imposa minimum optical spot size such that the depth of focremains greater than or equal to the thickness. If the optspot is further decreased, the depth of focus is also reduwith the result that the average spot size within the intertion region actually increases! The minimum ratio of taverage optical spot size to the minimum acoustic feasize can be shown to depend only on theQ parameter and therefractive index. ForQ54p the minimum average opticaspot size is approximately equal to the minimum acoufeature size, which means that the achievable pulse sharesolution is somewhat reduced~by less than a factor of 2!compared to that implied by the time-bandwidth product.


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~2! The input beam must be incident on the AOM at tBragg angleub5l/2L5ln/2vac, werel andL are the op-tical and acoustic wavelengths,n is the acoustic frequencyandvac is the acoustic velocity. Since the incidence angletherefore detuned from normal incidence, some opticalquencies will be focused before the AOM, others afterAOM. This effect becomes more important when either tacoustic center frequency~hence, the Bragg angle! or theAOM aperture is increased. A reasonable criterion is thatthe optical frequencies should be focused within a Raylerangepw0

2/l, wherew0 is the Gaussian beam radius at tAOM. For a fixed aperture, there is a maximum acousfrequency which satisfies this criterion. Since the modulatbandwidth of an AOM is generally proportional to the cenfrequency, this also limits the bandwidth and, hence,maximum time-bandwidth product. For the 4.3 cm modutor used in Refs. 80 and 113, the maximum bandwidth is 3MHz, which is larger than the center frequency used inexperiments.

~3! The horizontal displacement of the diffracted beain the interaction region must be much less than the mmum acoustic feature size to avoid limiting spatial/specresolution. This condition is satisfied in the experimentsRefs. 80 and 113.

The pulse shaping fidelity of an AOM pulse shaperimpacted both by acoustic attenuation and acoustic nonearities. The intrinsic attenuation coefficient due to the mterial generally scales quadratically with acoustic frequenwhich results in a time and frequency dependent wavefodistortion~since position along the AOM aperture is propotional to time delay!. Compensation for such a coupled timand frequency dependent distortion is challenging. Onother hand, at relatively low drive frequencies, other atteation mechanisms such as acoustic walkoff related totransducer geometry may dominate. In such cases the attation can be nearly frequency independent and can bebrated simply; precompensation for the acoustic attenuais then much easier to accomplish.

Acoustic nonlinearities can have a serious affectpulse shaping fidelity. Acoustic nonlinearities are in genefrequency, intensity, and material dependent, and are parlarly strong in the TeO2 material used in most pulse shapinexperiments. Such nonlinearities can lead to intensity depdent acoustic attenuation as well as acoustic intermodulaproducts. In TeO2 significant nonlinearities occur for rf powers well below those needed to reach full diffraction efciency. Even at diffraction efficiencies of 10%–15%, theare significant nonlinear effects. Therefore, nonlinearittend to restrict AOMs used in pulse shapers to the low dfraction efficiency regime and create complicated distortiofor which precompensation is nontrivial.

One approach to improve pulse shaping fidelity withothe need for a complicated theory-based precompensaprocedure is to use a feedback loop in which the driveplied to the AOM is modified in response to the experimendata. This adaptive pulse shaping approach was describeSec. II D. Adaptive pulse shaping using an AOM as the SLhas been demonstrated in quantum control experimentsamplified laser systems.73,74 This feedback approach is esp


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cially important for AOM pulse shaping due to the wavefordistortion effects discussed earlier.

The diffractive nature of the mask in AOM pulse shaing also gives rise to new dispersive effects which arepresent in pulse shaping using transmissive or reflecmasks or modulators. One effect is evident by referringFig. 23: the path lengths for optical frequency componentthe top part of the figure are clearly longer than those atbottom part of the figure.112 Such a frequency dependent palength constitutes a dispersion, which to lowest order iquadratic spectral phase variation. This dispersion cancompensated by changing one of the lens-grating separaor by using an external prism pair84 compressor. Note, however, that either procedure will introduce some higher or~cubic! dispersion, which becomes more important aspulses get shorter. A second effect involves the variationthe Bragg angle and the diffraction angle with opticfrequency.113 This effect should also become more importafor shorter pulses and larger optical bandwidths. To dateshortest pulses for which AOM pulse shaping has beenported are on the order of 50 fs.73 The effects describedearlier could conceivably preclude the extension of AOpulse shaping to shorter pulses. However, the short plimits of AOM pulse shaping have not yet been tested.

Although the traveling acoustic wave usually restricAOM pulse shaping to low repetition rate experiments wamplified or pulse selected laser systems, one applicationwhich AOM pulse shaping can be performed in conjunctwith high repetition rate cw-modelocked lasers was recereported.115 In this application a linear phase sweep is imposed onto the spectrum, which delays the optical pulsproportion to the amplitude of the phase sweep. This iwas previously demonstrated using a LCM array.34,35 In theAOM case, the linear phase sweep corresponds to a simshift in the applied rf drive frequency, which producespulse position modulation or tunable optical delay proptional to the rf frequency shift. Since the acoustic wavefois not spatially modulated, its motion across the apertureno effect ~except for an overall optical phase shift whicvaries from pulse to pulse!. Experiments using 350 fs pulsein the 1.55mm lightwave communications band achievedtunable optical delay of 0.34 ps/MHz. Tuning was demostrated over a delay range of 30 ps, although the pulse insity decreased away from the center position. The delaying range was approximately 15 ps to maintain an intenvariation under 3 dB. The delay can be switched in onlyms, which is the acoustic aperture time of the AOM usedthese experiments.

Reference 115 also demonstrates an interesting delaypendent dispersion effect, which would appear to applypulse shaping generally~not just to AOM pulse shaping!.This effect arises when the diffraction gratings are not ppendicular to the optical axis of the pulse shaping systFor a tilted grating, the distance between the grating andpulse shaping lens depends on the transverse position athe grating. As discussed in Sec. II C, a time delay in pushaping also corresponds to a lateral shift in the outbeam62 @see Eq.~2.14!#. This lateral shift therefore corre


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V. LIQUID CRYSTAL ARRAYS AND AOMs: PULSESHAPING COMPARISON

In this section we briefly summarize Secs. III and IV bcomparing LC and AOM pulse shaping technology, whiare the two programmable pulse shaping approaches whave been most widely investigated. General purpose pshaping using deformable mirror~DM! technologies are described in Sec. VI; however, at present there is significanless experimental experience with these devices thanLC or AOM pulse shaping. Therefore, DM pulse shapingnot included in the comparison in this section. For the Ltechnology, we concentrate on the nematic liquid crystechnology which has been used most extensively. We onize the discussion according to several important permance indicators, listed below. It is important to emphasthat both LC and AOM pulse shaping approaches each htheir own specialized advantages, and determination ofpreferred choice for a specific experiment may involvenumber of trade offs depending on the most importantquirements for that experiment.

(a) Pulsewidth. The LC approach has been tested doto 13 fs, and no obvious factors limiting the extensionshorter pulses have been identified. AOM shaping has bstudied most extensively on the 100 fs time scale, with oexperiment reporting pulses as short as 50 fs. There areeral factors which are expected to make the AOM appromore difficult for much shorter pulses, although compention of these factors may be possible. Further experimeare needed to determine the short pulse limits for bothand AOM pulse shapers.

(b) Reprogramming time. For AOMs the reprogrammingtime is determined by the acoustic aperture time, which isthe order of microseconds. For nematic LC arrays, the repgramming time is determined by the liquid crystal materresponse, typically on the order of 10 ms.

(c) Pulse repetition rate. Once programmed, the LCMremain static, and therefore can handle pulse trains at vally any repetition rate. The traveling wave nature of tAOMs generally limit these devices to pulse repetition raslower than the acoustic aperture time. Therefore, AOMsapplicable to amplified or pulse selected systems but geally not to cw modelocked systems.

(d) Modulation format. Both LC and AOM approachesoffer the ability to perform independent, gray-level spectamplitude and phase control.

(e) Pulse shaping complexity. Pulse shaping complexityis usually defined as the number of independent featuwhich may be placed onto the spectrum, which is equivalto the number of independent features which may be synsized onto the output temporal waveform@see Eq.~2.14!#.From the perspective of modulator technology, in the Lcase the pulse shaping complexity is limited by the numof pixels. Current LC arrays used for pulse shaping are 1pixel devices, although devices with larger pixel counts habeen demonstrated for other applications. In the AOM cthe complexity may approach the time-bandwidth produc


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the modulator, which can be of order 1000 with commcially available devices. However, the pulse shaping coplexity is also limited by the optical system placed arouthe modulator. Data representative of the highest complewaveforms published in the literature for LC and AOM pulshaping can be seen in Figs. 21 and 24, respectively. For21~d!, the total time window and individual pulse durationare;4 ps and 70 fs, respectively, for a ratio of roughly 5For Fig. 24, the corresponding numbers are;9 ps and 90 fs,respectively, giving a ratio of roughly 100. Thus, at presthe highest published complexity is somewhat less thafactor of 2 higher for AOM vs LC pulse shapers. This comparison could change in the future either due to advanceLCM technologies or due to improvements in the opticsystems surrounding AOM pulse shapers.

(f ) Efficiency. LCMs are inherently high efficiency devices, with transmissions approaching 100%. AOMstypically limited to low diffraction efficiency by acousticnonlinearities, which can be important even for diffractiefficiencies on the order of 10–15%. In both cases thereadditional losses due to diffraction off the gratings, etc.

(g) Pulse shaping fidelity. Excellent pulse shaping fidelity has been achieved for both LC and AOM approachwithout the need for feedback control. For the LC case,pixelated nature of the modulator can lead to temporal slobes for spectral patterns which vary too rapidly from opixel to the next. These sidelobes can be suppressed ioptical system is constructed to have a spectral resoluthat results in a slight smoothing of the programed specpattern; however, this reduces the effective number of pixavailable for pulse shaping. For the AOM case, the pushaping fidelity can be impacted by acoustic attenuationacoustic nonlinearities; in general, both of these effects mbe difficult to precompensate. However, under approprconditions very high quality waveforms with complexities othe order of 100 have been demonstrated for AOM pushapers, as stated earlier. For the LC and AOM data shin Figs. 21~d! and 24, respectively, the data are in excelleagreement with the experimental target in both casesqualitative terms, the pulse shaping fidelity appears tocomparable for each of these data sets, at least in the estion of this author. At present there is no standard definitof pulse shaping fidelity that can be used to charactepulse shaping fidelity quantitatively.

(h) Technology considerations. The AOM approach usestandard AOM technology; this is a mature technology wdevices available from many vendors. For the LC approamodulator arrays have been available in appropriate formfor approximately five years from a small number of vend~for mainstream applications such as displays, liquid crytechnology is much more highly developed!. The AOM ap-proach uses rf electronics and a single serial transduwhile 256 separate wires and a bank of low frequency etronics are used to control the 128-element amplitudephase LC arrays.

(i) Ease of control. First we consider open loop contro~the pulse shaper is not in a feedback loop!. For the LCapproach, the phase and amplitude response must bebrated. However, the calibration is not too difficult and c


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be performed accurately enough to give very good pushaping results. For the AOM approach, if everything weideal, no calibration would be necessary, at least in thediffraction efficiency limit, where the output field is directlproportional to the acoustic complex amplitude~hence, to thevoltage waveform!. However, in practice a variety of factorssuch as transducer impedance matching, acoustic attenuand nonlinearities, among others, play an important rolemust be taken into account. Precompensating for thesefects can be nontrivial.

In the case of adaptive pulse shaping~feedback control!,the control requirements are probably comparable for eitpulse shaping technology, provided that target waveforare consistent with limits imposed by the pulse shaping tenology and pulse shaping physics.

VI. PULSE SHAPING USING MOVABLE ANDDEFORMABLE MIRRORS

A. Moving mirrors

Several of the experimental examples discussed invious sections involve simple forms of pure phase modution. In particular, phase shifts which were linear, quadraand cubic with frequency were utilized, resulting, respetively, in pulse position modulation, chirp compensation, aa complex pulse distortion. These simple types of pure sptral phase modulation can also be produced using spepurpose reflective optics, which are moved or deformedorder to assume the specific shape needed to obtain thesired phase modulation. An advantage of such specialpose movable/deformable mirrors is the ability to provideprespecified type of phase modulation with very good presion; a disadvantage is that the arbitrary programmabavailable using SLMs is sacrificed.

We consider first the use of a movable mirror, whimakes possible a rapid scanning optical delay line. The pshaping arrangement, shown in Fig. 26, consists of a graplaced in the focal plane of a lens and a planar mirror inopposite focal plane.116 The mirror can be dithered about thpivot point under the influence of a piezoactuator and retsprings~not shown!. When the tilt angle of the mirror is zerothe setup is simply a folded equivalent of a pulse sha

FIG. 26. Rapid scanning optical delay line based on dithering a mirror atfocal plane of a folded pulse shaper.


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without any spatial filters. When the mirror is tilted, a linespectral phase shift is obtained, and this results in advancretardation of the output optical pulse. The result is similathe pulse position modulation achieved when a LC SLMAOM is used to impart a linear phase shift onto the sptrum.

Heritageet al. have implemented a rapid scanning opcal delay line by rapidly dithering the mirror through a smangular deflection and have used this setup to obtain femsecond intensity correlation measurements at a 400rate.116 This rate is much fast than possible using typicmechanical delay lines, in which the scan rate~and thereforethe data acquisition rate! is limited to a few tens of hertz. Fothe setup described in Ref. 116, a61 ps temporal delay isachieved with an angular deflection of only61.931024 rad; this corresponds to a total travel of only 4mm atthe edge of the mirror. In contrast, a standard linear acturequires a displacement of 150mm to achieve a same 1 pdelay. This technique was later extended to provide 10scans at a 2 kHz rate.117

Heritageet al. have also demonstrated a modificationthis geometry which allows scanning of 90 fs pulses ove100 ps time window.67 The ratio of the time window to thepulse width gives a pulse shaping complexity~h! exceeding1000, which may be the largest pulse shaping complexitydemonstrated in Fourier transform pulse shaping. The tporal waveforms produced in these time scanning expments are of course quite simple; they are simply shifversions of the original pulse. By ‘‘complexity,’’ however,have in mind the definition given in Eq.~2.14!, namely thetotal time window divided by the shortest temporal featuor equivalently the spectral bandwidth divided by the nrowest spectral feature. By this measure the complexithigh, since the output pulse can arrive at more than 1nonoverlapping temporal delays, and since in the frequedomain the linear phase sweep spans on the order of 10p.The trick in Heritage’s experiments is to retroreflect the oput beam in Fig. 26 back through the reflective pulse shaThis doubles the time delay while eliminating the spatwalkoff of the output beam. Recall that the spatial walkoffproportional to time delay as per Eq.~2.14!. With only asingle pass through the pulse shaper, the time windowusually assumed to be limited to approximately that vawhere the spatial walkoff equals the input spot size~see Sec.II C!. With double passing, this restriction is eliminated; ttime window is then determined by the spatial aperture ofgrating and lens, even for a relatively small input beamsimilarly large time window could also be obtained insingle pass geometry by expanding the input beam to fillgrating; however, this is usually less convenient. Note tthis double-pass concept will work for pulse position modlation using any modulator technology~mirrors, LCMs,AOMs!; however, it does not work for most other pulsshaping tasks.

Another useful feature is the ability to independencontrol the scan rates for delay and phase.117 In contrast,when delay is varied simply by translating a mirror to chanthe physical path length, the phase varies by 2p for everyoptical cycle of delay. In the apparatus of Fig. 26, howev


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the phase remains fixed as the delay is scanned, providedthe mirror rotates about a pivot point at the center~or refer-ence! optical wavelength. By translating the scanning mirrso the center wavelength is offset from the pivot point, ocan select a nonzero phase scanning rate. This attributegether with the rapid scanning delay capability, have mathe moving mirror pulse shaping approach the methodchoice for delay scanning in optical coherence tomogra~OCT! for optical biopsies.118 The rapid scanning is requirebecause OCT forms an image by performing sequential radelay scans for a large number of independent pixels; thfore, real-time imaging requires the individual delay scaoccur very quickly. The ability to independently contrphase and delay scan rates allows one to select the intermetric output frequency produced by the OCT apparatusbe compatible with high performance detection and dataquisition electronics.

We briefly compare this delay scanning approach, usa moving mirror~MM ! in the pulse shaper, to the case ofLC or AOM in the pulse shaper. One key difference is ththe moving mirror approach is a repetitive scan techniqwhile the LC or AOM approaches have random access cability. The throughput of the MM approach is very hig~similar to LC, higher than AOM!. The speed of the MMapproach is faster than for LC, and probably comparableAOM ~for the AOM approach the delay can be reset in mcroseconds; however many delay settings are needed toup a full scan!. The demonstrated temporal window is muhigher using a MM compared to the other approaches.the other hand, the MM approach provides only a sinfunctionality, while LC and AOM approaches allow generpurpose waveform control.

B. Deformable mirrors „DM…

In the first report of a DM in pulse shaping, the mirrowas used to impart a cubic spectral phase modulation.119 Thesetup is similar to that in Fig. 26, except that the MMreplaced by a DM. A cubic deflection is obtained by fixinthe position of a thin planar mirror at two symmetricalplaced support points and then applying equal and oppoforces on the two ends of the mirror. The desired cubicflection is the solution of the mechanical equations goveing small angle flexure of a thin elastic plate. The cubdeflection in space results in a cubic spectral phase modtion with a magnitude which can be adjusted by varyingstrength of the applied forces. This third-order disperserbeen applied to improve the quality of pulses from a higpower femtosecond semiconductor laser via third-order chcompensation.120

Recently pulse shaping has been demonstrated usiDM phase modulator which is controlled using electrostarather than mechanical forces.38 The device consists of agold coated, silicon nitride membrane suspended over anray of electrostatic actuators. The current device has amm wide active area controlled by 13 columns of 2 mm wiactuators. The maximum deflection and the response timereported to be 4mm ~corresponds to 20p at 800 nm wave-length! and 1 ms, respectively. Due to the relatively sm


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number of actuators, and the existence of a minimum radof curvature which can be induced on the membrane,device is mainly useful for providing smoothly varyinphase variations. Experiments have been performed wimodelocked Ti:sapphire laser, where the DM pulse sharecompressed a 15 fs pulse, which had previously bstretched to 90 fs by dispersion in glass, back to appromately the bandwidth limit,38 and with an amplified Ti:sapphire system, where the output pulse width was improvfrom 20 to 15 fs.121 The experiments also showed that longduration stretched pulses~220 fs! were incompletely com-pressed due to the maximum deflection limits of the mirro38

Two strategies were used to control the DM. In the firstDM phase response was calibrated using FROG measments of pulses emerging from the pulse shaper, anddesired phase functions were applied by using the previodetermined calibrations. In the second the DM is usedadaptive pulse shaping mode~see Sec. II D!, where a sto-chastic learning algorithm is used to adjust the actuatortings in order to optimize an experimental observable~e.g.,second harmonic generation intensity!. In the latter case precise knowledge of the DM phase response is not requiredDM pulse shaper has also be used for preliminary demstration of a method122 for compensating self-phase modultion in femtosecond CPAs.123

VII. FURTHER DIRECTIONS IN PULSE SHAPING

A. Integrated pulse shaping devices

Integration of pulse shaping devices would be desirafor applications such as optical communications where sand robustness are important. Considerable effort has bdirected toward developing integrated components to hamultiple optical wavelengths for wavelength division mulplexed~WDM! optical communications networks. With suiable modification such WDM components could also be uin conjunction with femtosecond input pulses as integrapulse shaping devices. A very simple example of pulse shing using a device known as an integrated acousto-opticable filter was reported in Ref. 124. More advancedamples of pulse shaping were reported recently usmodified arrayed waveguide grating routers.125–127There isalso substantial effort in the WDM communications commnity to implement multichannel equalizers to correct for twavelength dependent gain of erbium doped fiber amplifiHighly sophisticated arrayed waveguide grating devices hrecently been demonstrated for that purpose.128 Althoughthese devices have not been tested with femtosecond puthe devices are beginning to closely resemble what wouldneeded for integrated pulse shaping applications.

As an alternative to integrated pulse shapers, some ehas also been directed at improving the packaging and reing the size of bulk optics pulse shapers. For example,Refs. 129 and 130. Several interesting examples of pshaping, such as generation of short pulse bursts or cowaveforms, have also been demonstrated using distribdevices such as fiber Bragg grating reflectors131,132and ape-riodically poled second harmonic generation crystals133


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B. ‘‘Pulse shaping’’ with incoherent light

Pulse shapers can also be used for phase filteringbroadband incoherent light. Initial experiments were pformed using visible light with a few nanometebandwidth;134,135 subsequent experiments were performusing a programmable pulse shaper and amplified spontous emission from an erbium-doped fiber amplifier with seeral tens of nanometers bandwidth around the 1.55mm com-munications band.136 Although the result of phase filtering oincoherent light is still incoherent light, nevertheless, tphase filtering operation does affect the electric field crocorrelation function between the light before and afterpulse shaper. This can be measured by recording the inference fringes from a Michelson interferometer with a pgrammable pulse shaper in one arm. With this approachas been shown that the pulse shaper can manipulatecorrelation function much in the same way that a pushaper can manipulate the output intensity profile for cohent input pulses. An example of a shaped electric field crocorrelation function using a pulse shaper and broadbandcoherent light at 1.55mm, together with the correspondintheoretical trace, is shown in Fig. 27.100 Motivations for per-forming this work include the possibility of using coherencoding of inexpensive incoherent light sources~instead ofexpensive femtosecond pulse sources! for certain classes ooptical CDMA systems134 and for addressing of frequencydomain optical memories.109,135,137

C. Pulse shaping using two-dimensional masks

Most applications of pulse shaping make use of a odimensional spatial mask to produce a single output becorresponding to the desired temporal waveform. Howevby using cylindrical lenses instead of spherical lenses in

FIG. 27. Electric field cross-correlation function obtained using a pushaper and broadband incoherent light at 1.55mm. ~a! Experimental and~b!theoretical correlation traces of unshaped light.~c! Experimental and~d!theoretical correlation traces of shaped light using a length 31 quadresidue spectral phase code.


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as well as conventional139 fixed two-dimensional masks, awell as in preliminary experiments using programmable twdimensional liquid crystal arrays.140 Using a different ap-proach based on volume holography, formation of twdimensional images, where each spot in the imacorresponds to a different temporal waveform, has also bdemonstrated.141,142

D. Direct space-to-time pulse shaping

The concept of a direct space-to-time~DST! pulseshaper geometry, in which the output waveform is a direcscaled version of the pulse shaping mask, has also beenonstrated. This is in contrast to the Fourier transform retionship between the mask and the temporal waveform inconventional pulse shaping approach we have discussemost of this article. Operation of a DST pulse shaper wfirst demonstrated for simple waveforms on a picosectime scale by Emplitet al.143,144 In our group we have recently demonstrated femtosecond operation of a DST pshaper and generation of complex optical data packets.145 Adetailed description of this pulse shaping geometry is giin Ref. 145. The main motivation for pursuing the DSshaper is for applications in high-speed information technogy, such as parallel electronic to serial optical conversand generation of ultrafast optical data packets. For thapplications nanosecond to subnanosecond reprogratimes are desired. This should be possible by replacingfixed pulse shaping mask in Ref. 145 with an optoelectromodulator array, such as a hybrid GaAs–Si complemenmetal–oxide–semiconductor smart pixel array146 or an arrayof asymmetric Fabry–Perot modulators.147 Compared to theFourier transform pulse shaper, the DST has two key advtages important for these particular applications, as follo

~a! It avoids the need to perform a Fourier transformdetermine the masking function for each new packet, whwould be very difficult at high update rates, and

~b! Data packet generation using Fourier transform shing typically requires that both spectral amplitude and phbe precisely controlled. Pulse sequence generation wiDST shaper requires only intensity modulation, whichcompatible with existing optoelectronic modulator arrtechnologies.

E. Holographic and nonlinear Fourier pulseprocessing and space-time conversions

Pulse shaping can be extended to accomplish morephisticated pulse processing operations by including hographic or nonlinear materials in place of a mask or SLwithin a Fourier transform pulse shaping apparatus. Thissults in holographic and nonlinear Fourier processing teniques which can be used for pure time-domain procesas well as time-space conversions.


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We first discuss pure time-domain processing usingextension of pulse shaping called spectral holographywhich the pulse-shaping mask is replaced by a holograpmaterial. Spectral holography was first propostheoretically148 in the Russian literature, and subsequentperiments demonstrating the principles of time-domain pcessing via spectral holography were performed by Weiet al.149,150In analogy with off-axis spatial holography, twbeams are incident: an unshaped femtosecond referpulse with a uniform spectrum, and a temporally shapednal waveform with information patterned onto the spectruThe spectral components making up the reference and sipulses are spatially dispersed and interfere at the Fouplane. The resulting fringe pattern is stored by using a hographic medium which records a spectral hologram ofincident pulses. Readout using a short unshaped test pwhich passes through the pulse shaping apparatus resureconstruction of real and time-reversed replicas of the ornal signal waveform, in analogy with the real and virtuimages which are reconstructed in space-domain holograReadout using a shaped test pulse results in output wforms which are correlations or convolutions of the test asignal pulses. These operations, especially correlation,be useful for matched filtering and compensation of dispsion or timing drifts. Time-domain processing using spectholography has been demonstrated using both thermoplplates149–151~essentially a permanent recording medium! anddynamic photorefractive quantum wells~PRQWs!152 as theholographic material.

Holographic space-to-time conversion can be achieby placing a holographic mask at the Fourier plane of a pushaper. This can be achieved using either a fixed compugenerated hologram138 or a dynamic optically addressehologram.97,153,154In the case of an optically addressed hlogram, the pulse shaping mask is a hologram corresponto spatial images carried by the two input beams, which mbe cw. We have recently performed space-to-time conversexperiments using PRQWs as the dynamic holograpmaterial.97 The temporal profile resulting when a spectradispersed femtosecond pulse is diffracted from the hologcan be either a direct or Fourier-transformed version ofinput spatial image, depending whether a direct or Foutransform hologram of the image is recorded. In the casenonlinear recording conditions, additional pulse processfunctions can be realized; for example, in the case of sarated holographic recording, the temporal output wavefocan correspond to an edge-enhanced version of the inputtial image.97,155Recently, space-to-time conversion was ademonstrated by performing four-wave mixing~rather thanholographic recording! at the Fourier plane of the pulsshaper, using a cascaded second order nonlinear processecond harmonic generation crystal for the four-wave mixinteraction.156

Holographic and nonlinear Fourier pulse processingalso be used for time-to-space conversion of femtosecpulses. One method for time-to-space mapping is basedrecording of a spectral hologram using time-domain sigand reference pulses, as earlier, and then reading out usmonochromatic, cw laser. By passing the diffracted cw o


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put beam through a Fourier transform lens, the original teporal information is converted into spatial information. Dnamic PRQWs,157 bulk holographic crystals,154 and theexcitonic resonance in ZnSe crystals158 have all been em-ployed for time-to-space conversion experiments. Howenone of these materials are suitable for use in communicaapplications which typically require Gb/s frame rates. Alated time-to-space conversion scheme relies on secondmonic generation~SHG! using a nonlinear optical crystawithin a pulse shaper.159,160 This scheme provides the faresponse associated with the SHG nonlinearity, the possity of operation at convenient wavelengths and temperatuand the potential for high efficiency~nearly 60% harmonicconversion has been demonstrated using a modelocked olator source and a careful choice of nonlinear crystal!.25,161

VIII. APPLICATIONS OF PULSE SHAPING

We conclude this review article by discussing somethe applications of femtosecond pulse shaping. Both expments using fixed pulse shapers and programmable pshapers are included. Although this survey of applicationnot intended to be exhaustive, nevertheless it should iltrate the breadth of areas in which pulse shaping technois making an impact.

A. Fiber optics and photonics

1. Nonlinear optics in fibers

In ultrafast nonlinear guided wave optics, pulse shaphas been used to obtain femtosecond square pulses fooptical switching experiments with a fiber nonlinear couplwhere square pulses gave improved switching charactericompared to normal~unshaped! pulses by avoiding averaging over the pulse intensity profile.45 Specially shaped andphase modulated femtosecond ‘‘dark pulses’’ were also ufor the first clear demonstration of fundamental dark solipropagation in optical fibers78 and for subsequent dark solton experiments performed on a picosecond time scale.162 Abrief report of an experiment where pulse shaping was uto generate phase modulated input pulses for experimdemonstrating periodic evolution and refocusing of phmodulated higher order bright solitons has also bepublished.163

2. Optical communication systems

Pulse shaping has been used extensively for studieultrashort pulse CDMA communications based on specphase encoding~see Fig. 6! and decoding, where differenusers share a common fiber-optic medium based on theof different spectral codes. The CDMA receiver recognizthe desired data in the presence of multi-access interferon the basis of the strong intensity contrast between correand incorrectly decoded ultrashort pulses~see Fig. 28!.41

Femtosecond encoding and decoding were first demonstrusing fixed phase masks in the late 1980s,29 and a theoreticaanalysis of femtosecond CDMA was reported shorthereafter.46 Taking advantage of advances in fiber optechnology in the early 1990s, at Purdue we have now cstructed a femtosecond CDMA testbed, including encod


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and decoding, dispersion compensated transmission ofcoded pulses over kilometer lengths of fiber, and nonlinoptical thresholding to distinguish between correctly andcorrectly decoded pulses. Detailed discussion of our femsecond CDMA experimental results is given in Ref. 41. Eperiments on ultrashort pulse CDMA have recently bereported by industrial laboratories as well.127,132Pulse shap-ing also plays an important role in other optical CDMschemes based on either spectral amplitude164–166or spectralphase134 encoding of broadband incoherent light. Similarspectral encoding of broadband incoherent light can be ufor addressing of frequency domain optical memories baon spectral hole burning materials or volumholograms.109,135,137Finally, pulse shaping techniques havbeen extended for applications involving filtering of multipwavelength optical signals in WDM communications. Eamples include construction of WDM cross-connect switchwith flat-topped frequency response167 and multichannelWDM gain equalizers.130

3. Novel short pulse sources

Generation of multiple streams of modelocked pulssynchronized at multiple different wavelengths has bedemonstrated by introducing a pulse shaper into an extecavity modelocked semiconductor diode laser.168 The experi-mental setup is sketched in Fig. 29. The grating, lens, mirand spatial filter combination constitute a folded pushaper, in which the patterned spatial filter selects whspectral components are reflected back into the gain devThus, the oscillating wavelengths are controlled via thelected passbands of the pulse shaper which acts as ancavity spectral filter, while synchronization occurs via no

FIG. 28. Cross-correlation data obtained using ultrashort pulses at 1.55mmand a pair of fiber-pigtailed pulse shapers.~a! Pulse which is not coded.~b!Pulse which is coded and then correctly decoded.~c! Pulse which is codedbut then incorrectly decoded. Note the strong intensity contrast betwcorrectly and incorrectly decoded pulses.


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linear interactions between different wavelengths withindiode amplifier chip. Representative wavelength tuncurves are shown in Fig. 30 for spectral filters designedselect four wavelengths. Figure 30~a! shows a four wave-length spectrum with a fixed;1 nm spectral separationwhere the center wavelength is tuned over;18 nm in aregion centered around 830 nm. Figure 30~b! shows anotherexample where the center wavelength is held constant, wthe channel spacing is tuned from 0.8 to 2.1 nm. Timdomain measurements show that the pulses on eachvidual wavelength channel are 12 ps in duration with aGHz repetition frequency. Frequency-resolved optical gatmeasurements of these multiwavelength pulse trains halso been reported, which have been interpreted as evidof optical phase correlations between the different oscillatwavelengths.89 Such sources may be important in novel hbrid WDM-time-division multiplexed~TDM! communica-tions and photonic processing schemes, and possibly caextended to generate spectrally tailored ultrafast pulsequences directly from modelocked lasers. In contrast to pshaping and spectral tailoring outside a laser, in this intravity pulse shaping work all the available energy is pserved but caused to appear at the selected wavelength

FIG. 29. Schematic layout of a multiwavelength modelocked diode lawith an intracavity pulse shaper. At 1997 IEEE.

FIG. 30. Wavelength tuning curves for modelocked diode laser with incavity pulse shaper.~a! Tuning over 18 nm with wavelength separation heconstant.~b! Control of the wavelength channel spacing. At 1997 IEEE.


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4. Dispersion compensation

Dispersion compensation for fiber communications wdiscussed in Sec. III A. Briefly, a programmable pulse shawas used as a spectral phase equalizer in conjunctiondispersion compensated fiber links, resulting in distortiofree transmission of 400 fs pulses over 10 km of fiber.40,103

This approach may be important for ultrashort pulse CDMand several hundred Gb/s TDM communications, bothwhich are sensitive to higher order phase dispersion.

5. Space-time conversions

Space-time conversions, outlined briefly in Secs. VIIand VII E, may be important for generation, demultiplexinand processing of very high rate optical data sequenSpace-time conversion and its applications are discussedther in Refs. 25, 145, and 169.

B. Biomedical applications

We have already mentioned~Sec. VI A! that a pulseshaping approach, based on a moving mirror at the Fouplane, has become the method of choice for rapid time descanning in OCT biomedical imaging.117,118Pulse shaping isalso being investigated for applications in multiphoton cofocal microscopy.170 Here the goal is to tailor the pulse shapto maximize the efficiency of the nonlinear process, whichimportant in order to generate a bright image while miniming the exposure of sensitive biological samples to laserdiation.

C. Phase control in femtosecond amplifiers

Precise and complete chirp compensation is one ofkey issues in optimizing pulse quality in femtosecond CPsystems.22,43 Spectral phase control is important both in otaining the minimum pulse width and in suppressing winand sidelobes in the compressed pulse, since pulses withtremely high on-off contrast are needed for most high intsity applications. Pulse shaping is increasingly being conered for such phase control. In one early example, thirdfourth order spectral phase were both compensated by ading a custom air-spaced doublet lens placed with a pushaper,171 resulting in dramatically decreased tempowings. More recently, a number of groups are using pgrammable pulse shapers for such spectral phase comption in femtosecond amplifiers.69,71,114,121Results have beenpublished for several different programmable pulse shaptechnologies, and experiments on amplified pulses as sho;15 fs have been reported.121

It should be noted that phase compensation of CPA stems is possible down to at least the sub-30- fs regime wout the use of programmable techniques such as pshaping,22 although this requires considerable care. Pushaping is being applied for two reasons. The first isachieve phase compensation, and hence, shorter, higherity pulses, at durations beyond the experimental state ofart. The second is simply to be able to use a CPA syswhich has been constructed without completely optimizthe spectral phase, and still be able to make it perform at n

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Data from an experiment where a pulse shaper witphase-only LCM is placed after a CPA are plotted in Figs.and 32.71 The pulses are centered at 800 nm with a bawidth of ;8 nm, and the pulse energy after the pulse shais 200 mJ at a 1 kHz repetition rate. The duration of thamplified pulses without correction by the pulse shaper w195 fs, roughly a factor of 2 above the bandwidth limit. Busing the pulse shaper for phase correction, which wasjusted via adaptive feedback under control of an evolutionalgorithm, the pulses were compressed to 103 fs, close tobandwidth limit. FROG traces of the pulses based on S

FIG. 31. ~a! FROG trace of original, phase distorted amplified pulses.~b!FROG trace of optimized pulse after phase correction by programmpulse shaper.

FIG. 32. Retrieved time-domain intensity~upper plot! and phase~lowerplot! profiles of amplified pulses for original~open circles! and phase-corrected~full circles! amplified pulses.


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gating87 with and without phase correction are shownFigs. 31~a! and 31~b!, respectively. For interpreting thestraces, the main point is that a bandwidth-limited pulseexpected to have a simple elliptical SHG–FROG trace, wmajor and minor axes pointing along the wavelength atime axes. A simple linear chirp leads to an increase inarea of this ellipse, without changing its orientation. Mocomplex structure is indicative of higher-order phase vations. Thus, the irregular uncorrected FROG trace in F31~a! shows signs of a complex spectral phase, whichpears to be almost completely eliminated in the phacorrected plot. This is also evident in the time domain intesity and phase profiles~Fig. 32! retrieved from the FROGdata. The intensity profile is significantly smoother ashorter after phase correction, and the temporal phase prafter the correction is almost flat in the region where theresignificant intensity.

D. Bandpass filtering

A very simple application uses a pulse shaper withadjustable slit at the Fourier plane to implement a variabandpass optical filter. This technique has been used byeral groups to generate tunable, bandwidth limited pulfrom a femtosecond white-light continuum, which have beapplied, for example, for time-resolved spectroscopysemiconductors172,173 and nanocrystals174 and high intensitystudies of atomic gases.175 In some experiments a pair opulse shapers is used to carve synchronized pulses frowhite-light continuum at different independently tunabwavelengths.173 Such bandpass filtering can also be usedconjunction with coherent modelocked lasers, leadingpulsewidth control in the time domain, which was usedexample to measure the pulsewidth dependence of two pton absorption in nonlinear waveguide photodetectors176 andfor studies of polariton scattering in semiconductmicrocavities.177

E. Control of quantum dynamics and nonlinear optics

Pulse shaping plays a major role in the field of coheror quantum control, in which the experimentalist seeksmanipulate quantum mechanical wave packets via the phand intensity profiles of the exciting laser radiation. Sinmany of the quantum mechanical motions of interest ocon an ultrafast time scale, femtosecond pulse shaping is oneeded to create the required laser profiles. For a reviewthe quantum control field, see Ref. 75.

In one of the earliest femtosecond coherent controlperiments, terahertz rate trains of femtosecond pulses wused to achieve selective amplification of optical phonovia multiple pulse impulsive stimulated Raman scatter~ISRS!.50,51,178 The input pulse trains were generatedphase-only filtering in a pulse shaper.47 These ISRS experi-ments were sensitive only to the optical intensity profi~hence, the pulse timing! but not to optical phase. Relateexperiments were later extended for phase-sensitive manlation of coherent charge oscillations in semiconductor mtiple quantum wells.52,53

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Another simple example involves multidimensionspectroscopic studies using two and three pulse sequewith specified phase relationships and interpulse delayexcite specified coherent states in an atomic vapor.105 Thisleads to control of the fluorescent yield as a function ofdelays and phases, which were varied using a dual Larray in the pulse shaper. The variations in the fluoresceyield may be understood as arising from different degreeoverlap of the optical power spectrum, which is relativehighly structured and which depends on the pulse timiand phases, with the atomic absorption spectrum. A revof the application of such pulse sequences to multidimsional spectroscopy and optical control is given in Ref. 1Note that similar experiments using two-pulse sequencesbe performed conveniently by using a stabilized interferoeter to generate the phase-locked pulse pairs.179,180 Phase-locked three pulse sequences are more complex to genvia interferometers; at this level of complexity, pulse shapis the best choice.

More generally, one often desires more complicatedtrafast waveforms to generate wave packets and wave ftions that could not be obtained otherwise. In the weak ficase, Wilson’s group for example calculated ‘‘globally opmum weak light fields’’ giving the best possible overlap wispecified molecular dynamics outcomes.181–183Cases studiedinclude a ‘‘molecular cannon,’’ in which an outgoing continuum wave packet inI 2 is focused in both space and mmentum, and a ‘‘reflectron,’’ resulting in focusing of an incoming bound wave packet inI 2 , among others. In thestrong field regime, theory suggests a number of excitpossibilities for coherent control through shaped pulsespulse sequences, including breaking strong bonds andnipulating reaction pathways and curve crossings,184–188 toname just a few.

Experiments are beginning to yield interesting resuPulse shaping has been used in a series of experimenRydberg atoms, for example, where custom Rydberg wpackets have been produced as well as measured.74 Controlof fluorescent yields in dye molecules73 and optimization ofbranching ratios of different photodisassociation reactchannels of organometallic molecules has also breported.72 Many of these experiments were performed inadaptive pulse shaping configuration~see Sec. II D!. Otherworks were performed using open-loop pulse shaping. Thinclude a series of studies on coherent control of spinpopulation dynamics in quantum wells189 and experimentson coherent control of resonant two photon absorptionatomic argon as a function of the optical phamodulation.190

Pulse shaping can also be used for control of nonlinoptical processes which are not inherently quantum mechcal. For example, in SHG, spectral amplitude maskingchirped pulses was used to achieve both good efficiencya narrow second harmonic bandwidth simultaneously,spite the broad bandwidth of the excitation pulses.191,192Thisfocusing of spectral energy was related to focusing in fspace by a Fresnal zone plate. Similar experiments wereperformed for two photon absorption191,192 In a second ex-ample related to terahertz~THz! radiation, shaped optica


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pulse sequences were used to excite photoconductive anas, resulting in the ability to manipulate the emitted THradiation waveform.193 This technique has also resultedenhancements of the emitted peak THz power spectral dsity due to avoidance of saturation mechanisms through mtiple pulse excitation.54,194

F. Laser-electron beam interactions

Several schemes have been proposed in which pshaping would favorably enhance laser-electron interactioOne application involves laser generation of large ampliturelativistic plasma waves~with application to ultrahigh-gradient electron accelerators!. It has been shown theoretcally that by using an appropriate femtosecond pulsequence, one could in principle drive plasma oscillationslarger amplitudes than possible with a single pulse ofsame total energy.195,196 By using a pulse sequence, oncould compensate for the change is plasma frequency ariwhen the plasma enters the large amplitude, nonlineargime, in addition to avoiding various laser-plasma instabties through the use of lower peak intensities. The usepulse shaping to seed other desirable plasma instabilitiesalso been proposed.197 Further possibilities, such as usinpulse shaping~a! to generate energetically and temporamodulated electron beams to enhance the generation of swavelength radiation in free electron lasers198 or ~b! tomodify the spectra produced under the influence of nonlinDoppler shifts in relativistic Compton backscattering,199

have also been discussed.

ACKNOWLEDGMENTS

This work was supported in part by the Army ResearOffice under DAAG55-98-1-0514 and by the National Sence Foundation under 9626967-ECS and 9722668-PHY

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