generation and control of ultrafast pulse trains for quasi-phase-matching high-harmonic generation

1UcaoFlospbp

omcstmcdlcphtltaqo

Robinson et al. Vol. 27, No. 4 /April 2010 /J. Opt. Soc. Am. B 763

Generation and control of ultrafast pulse trainsfor quasi-phase-matching high-harmonic

generation

Tom Robinson,1,* Kevin O’Keeffe,1 Matt Zepf,2 Brendan Dromey,2 and Simon M. Hooker1

1Department of Physics, University of Oxford, Clarendon Laboratory, Parks Road,Oxford OX1 3PU, United Kingdom

2Department of Physics and Astronomy, Queens University, Belfast BT7 INN, United Kingdom*Corresponding author: [email protected]

Received December 22, 2009; revised February 8, 2010; accepted February 8, 2010;posted February 16, 2010 (Doc. ID 121933); published March 26, 2010

Two techniques are demonstrated to produce ultrashort pulse trains capable of quasi-phase-matching high-harmonic generation. The first technique makes use of an array of birefringent crystals and is shown to gen-erate high-contrast pulse trains with constant pulse spacing. The second technique employs a grating-pairstretcher, a multiple-order wave plate, and a linear polarizer. Trains of up to 100 pulses are demonstrated withthis technique, with almost constant inter-pulse separation. It is shown that arbitrary pulse separation can beachieved by introducing the appropriate dispersion. This principle is demonstrated by using an acousto-opticprogrammable dispersive filter to introduce third- and fourth-order dispersions leading to a linear and qua-dratic variation of the separation of pulses through the train. Chirped-pulse trains of this type may be used toquasi-phase-match high-harmonic generation in situations where the coherence length varies through themedium. © 2010 Optical Society of America

OCIS codes: 320.5540, 320.1590, 190.4160.

aswhvuaustHiaambftmmegsa

hspttb

. INTRODUCTIONltrafast optical pulses have allowed fundamental pro-

esses in materials and chemical systems to be measurednd controlled with femtosecond precision [1]. Sequencesf such pulses also have many practical applications.emtosecond pulse trains have been used to study mo-

ecular motion in solids by selectively driving vibrationsf a crystal lattice [2], in order to study phonon-assistedtructural changes or chemical reactions. More recently,ulse trains have been used to generate bursts of narrow-and, tunable terahertz-frequency radiation, by excitinghotoconductive dipole antennas [3,4].In this paper we discuss techniques to generate trains

f femtosecond-duration pulses suitable for quasi-phase-atching (QPM) high-harmonic generation (HHG) with

ounterpropagating light [5–9]. HHG is an attractiveource of coherent, ultrafast soft x-ray radiation with pho-on energies extending to above 1 keV [10]. However, aajor limitation of this source is the low conversion effi-

iency mainly due to the phase mismatch between theriving laser pulse and the generated harmonics. This isargely a result of the ionization inherent in the HHG pro-ess leading to considerable free-electron dispersion. Thehase mismatch causes the intensity of each generatedarmonic to oscillate with the propagation distance be-ween zero and a maximum value; the period of this oscil-ation is 2Lc, where the coherence length Lc=� /�k andhe wave vector mismatch �k=qk0−kq, where k0 and kqre the wave vectors of the fundamental radiation and theth harmonic, respectively. In these circumstances theutput harmonic intensity is at most that generated over

0740-3224/10/040763-10/$15.00 © 2

single coherence length. QPM mitigates this effect byuppressing the harmonic generation in those regionshere the generated harmonics are out of phase with thearmonic beam at that point. Several methods have pre-iously been demonstrated to achieve this including these of modulated waveguides [11], separated gas jets [12],nd mode beating in capillary waveguides [13]. Trains ofltrafast pulses may also be used to achieve QPM [5–9],ince the standing phase modulation induced by rela-ively weak counterpropagating light suppresses theHG in regions where the driving and counterpropagat-

ng pulses overlap [14]. This rapid phase modulation willlso be present in the generated harmonic radiation withmagnitude which is q times greater for the qth har-onic. For a sufficiently intense counterpropagating

eam there will then be negligible harmonic emissionrom the region of overlap [15]. By using a train of coun-erpropagating pulses, the emission can be suppressed inultiple zones leading to significant increases in har-onic intensity. This was previously achieved by Zhang

t al. [5], who used counterpropagating pulses that wereenerated by placing blocks of glass in the pulse compres-or, delaying different frequency components and formingseries of up to three pulses.To maximize the enhancement of a particular high-

armonic order through QPM, the pulses in the trainhould ideally have sufficient energy to completely sup-ress the HHG. The trains should also have high con-rast, so that unwanted suppression does not occur be-ween the pulses; comprise many pulses, so that QPM cane applied to multiple out-of-phase zones; and be adjust-

010 Optical Society of America

ag

tbepleltnatpsbtm

omwt

2TTcipptoftvrt

tstpctocotmdhiteptem

psaraaffgciupqtpeeT

Fcap

764 J. Opt. Soc. Am. B/Vol. 27, No. 4 /April 2010 Robinson et al.

ble, so that they can be matched to a medium with a lon-itudinally varying coherence length.

Several methods have been employed to generate pulserains. Sequential Michelson interferometers [16] haveeen used to produce up to 16 pulses. For n interferom-ters, a train of 2n pulses may be generated and the inter-ulse separation can be varied by adjusting the arm-engths. This technique is capable of producing high-nergy pulse trains reproducibly and efficiently (50% forinearly polarized pulses), but for large numbers of pulseshe scaling of the number of interferometers required isot practical due to space requirements and difficulties oflignment. An alternative method is to use pulse-shapingechniques in which phase and amplitude masks arelaced in the focal plane of a zero-dispersion pulsetretcher [17–19]. The generation of up to 20 pulses haseen demonstrated using this technique [17]; however,he energy per pulse is limited by potential damage to theask.All of these methods are limited either by the number

f pulses which can be practically generated, the maxi-um energy per pulse, or by the tunability of the pulseidths and separations. In this paper we describe in de-

ail two techniques which can overcome these limitations.

. THE BIREFRINGENT CRYSTAL ARRAYECHNIQUEhe first technique makes use of multiple birefringentrystals [20]. Consider a linearly polarized pulse of lightncident on a birefringent crystal of length L, with thelane of polarization at 45° to the ordinary axis. Thisulse can be resolved into two pulses with equal intensi-ies, one polarized parallel to the ordinary axis and thether parallel to the extraordinary axis. Due to the bire-ringence of the crystal, these two pulses will propagatehrough the crystal with different group velocities, vg

o and

ge. On leaving the crystal, the two pulses will be sepa-ated by a time �t=L�1/vg

e −1/vgo� resulting in two or-

hogonally polarized, temporally separated pulses.Similarly, the number of pulses will again be doubled if

hese pulses are passed through another crystal with theame birefringence, but with double the length and withhe optical axis oriented so that it is at 45° to the plane ofolarization of both incoming pulses. Each additionalrystal, of twice the length of the previous one, will doublehe length of the pulse train, so n crystals give 2n pulsesf equal intensity. Note that the material dispersion of therystals will also broaden the pulses, although this can beffset by pre-compensating the dispersion encountered byhe pulse. In practice, materials of different birefringenceay be used as long as their lengths are chosen so that �t

oubles for each successive crystal. The first and secondalf of the pulse train will always have orthogonal polar-

zations, although the polarizations of all the pulses in therain can be made parallel by using a linear polarizer ori-nted at 45° to the polarizations of the pulses. Hence ifulses with the same polarization state are required, thisechnique is 50% efficient. Figure 1(a) illustrates the gen-ration of a train of eight equally spaced pulses using thisethod.

The pulse train data presented here were produced byassing a laser pulse of FWHM duration 200 fs throughix calcite crystals of lengths 0.42, 0.83, 1.66, 3.33, 6.65,nd 13.3 mm. Further details on the experimental ar-angement can be found in [20]. Calcite is suitable for thispplication since its large birefringence minimizes themount of material required and hence reduces the ef-ects of higher-order dispersion. The Sellmeier equationor calcite yields no=1.658 and ne=1.486, for �=800 nm,iving a pulse delay of 575 fs per millimeter. The thinnestrystal leads to a pulse splitting of 250 fs so that for ourncident laser pulses, the split pulses overlap. This can besed to generate pulses longer than the individual inputulses. For QPM under conditions of constant Lc we re-uire a pulse train in which the pulse separation is equalo the pulse width. This is possible by forming “macro-ulses” from a number of overlapping single pulses. Forxample, Fig. 1(b) shows a train of macropulses in whichach macropulse is formed from four overlapping pulses.his was achieved by alternating the crystal alignment in

ig. 1. (Color online) Crystal arrangements and cross-orrelations for (a) a train of eight equally spaced pulses and (b)train of eight macropulses, each macropulse consisting of four

ulses. Calcite crystals were used with L=0.42 mm.

otap

wtlpcdsb

Ft�ti

pdtdi

AUEpcalrobsitospiew

mttgamttd

htmFspatcgioatca

sittitcp

taeoa

Fce

Ftdg


rder of increasing thickness, but removing the thirdhinnest crystal. This method produces pulses which haven approximately square temporal profile making themarticularly suited to QPM.It is seen that the peak pulse intensity is not constant

ithin these trains. This is largely due to misalignment ofhe optical axes of the crystals relative to the plane of po-arization of the pulse. For equal splitting, the plane ofolarization must be at exactly 45° to the optic axis of therystal. For an uncertainty in crystal orientation of � ra-ians over n crystals, the ratio of the intensities of thetrongest and weakest pulses, �, is given approximatelyy

� = � cos � + sin �

cos � − sin ��2n

� �1 + �

1 − ��2n

. �1�

or the data presented here, the uncertainty in orienta-ion was approximately 2°. Over six crystals, this gives�2, which approximately matches the observed varia-ion in pulse intensity for the case of eight pulses shownn Fig. 1(a).

The technique described in this section is flexible, com-act, and produces pulse trains which are highly repro-ucible. All that is required to generate a certain train iso orient the crystals appropriately, which is easily repro-ucible with calibrated rotation-mounts, without requir-ng repeated characterization of the train.

. Quasi-Phase-Matching of High-Harmonic Generationsing Pulse Trainsxperiments to investigate QPM of HHG using counter-ropagating pulse trains generated by the birefringentrystal array technique were performed. A chirped-pulsemplification Ti:sapphire laser system gave linearly po-arized pulses with an energy of 0.5 mJ and FWHM du-ation of 50 fs and a spectrum centered at 808 nm. Theutput was split into the driving and counterpropagatingeams with total pulse energies of 0.27 and 0.23 mJ, re-pectively. An array of birefringent crystals, as describedn the previous section, was used to form trains of one orwo macropulses of length 0.6 mm, each comprising eightverlapping pulses. Cross-correlation measurementshowing an individual macropulse and a train of two suchulses are shown in Fig. 2. The two beams were couplednto opposite ends of a glass capillary with an inner diam-ter of 102 �m and length of 20 mm, filled with argon gas,hich acted as the HHG medium. The generated har-

ig. 2. The pulse trains used for the QPM experiment. Cross-orrelations of (a) one and (b) two macropulses, each consisting ofight individual pulses, are shown.

onic light passed through a hole in the mirror couplinghe counterpropagating beam into the capillary and en-ered a flat-field spectrograph consisting of a gold-coatedrating with a nominal line spacing of 1200 lines/mm and

cooled soft x-ray CCD (Andor DO440-BN). The har-onic spectrum was recorded as the point of overlap of

he driver pulses and the pulse train was scannedhrough the capillary by changing the path length of theriver beam using a timing slide.Figure 3 shows the normalized intensity of the 27th

armonic, generated in 50 mbar argon, as a function ofhe point of overlap of the driving pulse and the singleacropulse (of energy 0.12 mJ on the target) shown inig. 2(a). Clear oscillations of the harmonic intensity areeen corresponding to the macropulse successively sup-ressing HHG in regions which contribute constructivelynd destructively to the harmonic output as it is scannedhrough the capillary [9]. The inset shows the variation inoherence length measured along the capillary. The stron-est oscillations are observed where the coherence lengths approximately 0.3 mm, i.e., equal to the effective lengthf the macropulse. The HHG spectrum exhibited a cutofft 23 nm, indicating a peak intensity and ionization frac-ion of 2�1014 W cm−2 and 0.15, respectively. This gives aalculated coherence length [5] for the 27th harmonic ofpproximately 0.27 mm, close to the measured value.Toward the end of the capillary, the coherence length is

een to increase markedly due to a reduction in the driv-ng pulse intensity and a drop-off in the gas density nearhe exit of the capillary. Note that we might expect a fac-or of 4 increase in the harmonic intensity when suppress-ng one out-of-phase zone. However, due to limitations onhe pulse energy from our laser system, the HHG was notompletely suppressed by the counterpropagating macro-ulse and the resulting enhancement is reduced.Figure 4 shows the intensity of the 27th harmonic, this

ime as a function of the collision point with a train of onend two macropulses, where each macropulse now has annergy of 0.06 mJ on the target. As expected, for the casef two macropulses, a greater enhancement is observednd the oscillation of harmonic intensity is seen to extend

ig. 3. Intensity of the 27th harmonic recorded as a function ofhe position of the point of overlap of a single macropulse and theriving pulse in the capillary. The inset shows the measured lon-itudinal variation of L .
c

fps

taiwFwsipitmgscs

3TIlpswanwp

talFoptOt

woibi

aGeg

wF

tfpiFbp

wqcg

w

Tpwott

dpfm

Farlc


urther into the capillary. Additional counterpropagatingulses were not used, since the laser pulse energy was in-ufficient.

We have shown that the pulse trains generated usinghe birefringent crystal array method can be used tochieve optical QPM of HHG. Since they have constantnter-pulse separations, they are best-suited to situationshere the coherence length does not vary. As shown inig. 3, in general Lc will not be constant, especially to-ard the end of the generation region where there is a

trong variation in the density of the medium and whereonization is reduced due to the depletion of the drivingulse energy. Indeed, most of the harmonic radiation leav-ng the capillary will be generated in the region close tohe exit due to the absorption of harmonic light by the gasedium. Therefore if optical QPM is to be extended to a

reater number of coherent zones, pulse trains of varyingpacing are required. A technique capable of generatinghirped-pulse trains of this kind is described in the nextection.

. THE SINGLE BIREFRINGENT CRYSTALECHNIQUE

n this technique, first proposed by Yano et al. [21] andater investigated by the present authors [22], a linearlyolarized laser pulse is passed through a grating-pairtretcher introducing a frequency-chirp. A multiple-orderave plate is placed after the stretcher, with its fast axist 45° to the plane of polarization of the laser pulse. Fi-ally, a linear polarizer is placed after the wave plate,ith its transmission axis parallel to the original plane ofolarization of the light pulse.In order for the light to be transmitted by the polarizer,

he wave plate must introduce a phase-shift of 2m� (m,n integer) between the field components polarized paral-el and perpendicular to the fast axis of the wave plate.or a wave plate of a given thickness, this condition willnly be met for certain frequencies, and since the laserulse is frequency-chirped, the radiation transmitted byhe polarizer will be modulated to form a series of pulses.ne consequence of this is a variation of the frequency of

he carrier wave of each pulse across the pulse train,

ig. 4. (Color online) Intensity of the 27th harmonic recorded asfunction of the position of the point of overlap of a single mac-

opulse and two macropulses with the driving pulse in the capil-ary. The energy per macropulse is 0.06 mJ on target in bothases.

hich is approximately equal to the bandwidth of theriginal pulse. This variation of carrier frequency may bemportant if the pulses are being used for spectroscopy,ut for the case of QPM, where the pulses are simply be-ng used to disrupt the HHG, it is unimportant.

In order to calculate the form of the pulse trains gener-ted with this technique, consider a linearly chirpedaussian pulse of instantaneous frequency �0+2b0t and

lectric field amplitude A. The field in the plane z=0 isiven by [23]

E�z = 0,t� = A exp�− a0t2�exp�− i��0t + b0t2�� 2�

=A exp�− 0t2�exp�− i�0t�, �3�

here 0�a0+ ib0 and a0p2=2 ln 2, in which p is the

WHM duration of the intensity of the laser pulse.We can calculate the effect of propagating such a pulse

hrough a dispersive medium by taking the Fourier trans-orm of the electric field, multiplying by the spectralhase encountered in the medium, and then taking thenverse Fourier transform to return to the time domain.or the case of a quadratically dispersive medium locatedetween the planes z=0 and z=�, the amplitude of theulse at z=� may be written as

E��,t� = A��

0exp�i��0� − �0t��exp�− ��t − �0��2�,

�4�

here ��−1=0−1−2i�0��, and where for an angular fre-

uency of �0 the wave vector, group velocity, and recipro-al group velocity dispersion (GVD) of the medium areiven by �0, 1 /�0�, and �0�, respectively.

Equation (4) may be written more conveniently as

E��,t� = f��,t�exp�− i�0t�exp�i��,t��, �5�

here

f��,t� = A��

0exp− R��t − �0��2�, �6�

��,t� = �0� − �t − �0��2, �7�

=b + 2�0��a2 + b2�

�1 + 2�0��b�2 + �2�0��a�2. �8�

he amplitude of the radiation passing through the linearolarizer is given by Epol�� , t�= �Ee�� , t�+Eo�� , t�� /2,here the amplitudes Ee and Eo are given by expressionsf the form of Eq. (5) and the subscripts “o” and “e” denotehe ordinary and extraordinary polarizations, respec-ively.

Provided the difference of the second- and higher-orderispersion terms for the two polarizations in the wavelate are not too large, the pulse envelopes fo�� , t� and

e�� , t� will be approximately equal, and consequently weay write

watwsf

si

T

wpiapct

wpAte3ttatF

attc2pblcpstF

ptpl

stm0popgdbag

paccscssif

ncu1s

sFtcppmpspdp

da3tits

wensawatv


Epol � 2f̄��,t�exp�i�̄��,t��cos��o��,t� − �e��,t�

2 , �9�

here f̄�� , t� and �̄�� , t� are the average pulse envelopend phases for the two polarizations. We see that the in-ensity of the chirped pulse leaving the combination of theave plate and polarizer is modulated by a cosine-

quared function. In other words, a train of pulses isormed.

Peaks of intensity occur when the argument of the co-inusoidal function is equal to m�, where m is an integer,.e., when

�0o� − �0e� + e�t − �0e��2 − o�t − �0o��2 = 2m�. �10�

he interval between adjacent pulses is thus given by

� =�

��0e� e − �0o� o�� + � o − e�t�11�

��

b�

1

�0e� − �0o�, �12�

here the approximation holds if the GVD in the wavelate is small, whereupon o� e�b. We see that if thenitial stretched pulse has a constant frequency-chirp,nd the GVD in the wave plate can be neglected, theulses in the train have a constant separation. If theseonditions are not met the spacing of pulses within therain will vary as discussed in Subsection 3.A.

The relative variation of the separation of the pulsesill be most severe when the duration of the stretchedulse is long and the thickness of the wave plate is small.s an example, for calcite we find for the ordinary and ex-

raordinary polarizations, respectively, at 800 nm: �0�qual to 5.58 and 4.98 ps mm−1; �0� equal to 76.2 and9.6 fs2 mm−1. For the longest stretched pulse used inhis work (33 ps) and the thinnest wave plate (0.826 mm),he measured mean pulse spacing of 14 ps corresponds tob-parameter of 0.45 ps−2. From Eq. (11) it is then found

hat the pulse spacing varies by less than 0.2% over theWHM duration of the pulse train.This technique was investigated experimentally, using10 Hz Ti:sapphire chirped pulse amplification laser sys-

em, delivering linearly polarized pulses with energies upo 125 mJ at a center wavelength of 808.8 nm. Prior to theompressor, the beam was split using a beam splitter of5% reflectivity. The reflected beam was used to form theulse trains studied in this work, while the transmittedeam was used as a probe. The probe beam could be de-ayed relative to the driver beam using a computer-ontrolled timing slide. The duration of the driver androbe pulses was measured after the compressor byingle-shot autocorrelation and minimized by adjustinghe separation of the compressor gratings to give aWHM duration of 70 fs.The probe beam passed through a rotatable half-wave

late and a linear polarizer oriented vertically to enablehe energy of the probe pulses to be adjusted. The driverulse was re-stretched using a pair of gratings with 1200ines/mm; by varying the separation of the gratings,

tretched pulse durations of between 5 and 33 ps were ob-ained. The stretched pulse then passed through a calciteultiple-order wave plate (with a thickness ranging from

.8 to 26.4 mm), oriented with its fast axis at 45° to thelane of polarization, and then through a linear polarizerriented parallel to the plane of polarization of the beamrior to the wave plate. Measurements of the pulse trainsenerated were performed by a cross-correlation of theriver and probe pulses achieved by overlapping the twoeams in a 1-mm-thick �-barium borate (BBO) crystalnd observing the intensity of the second-harmonic lightenerated as the timing slide was adjusted.

The results of the cross-correlations performed withulse trains generated from each of the stretched pulsesre shown in Fig. 5. In each case the first plot shows theross-correlation of the stretched pulse, i.e., without thealcite crystal or polarizer present in the beam. The sub-equent plots show the cross-correlations measured withalcite crystals of various thicknesses inserted into thetretched beam. The data-acquisition rate was adjustedo that there were approximately 30 data points per pulsen the train. The noise in the cross-correlations arisesrom fluctuations in the energies of the two beams.

It is seen that as the crystal thickness is increased, theumber of pulses obtained from the stretched pulse in-reases and the separation between pulses decreases. Fig-re 6 shows how the reciprocal of the pulse separation,/�, varies as the crystal thickness is increased andhows the linear dependence expected from Eq. (12).

Equation (9) indicates that the pulse train intensityhould be a cosine-squared function of time. Hence theWHM duration of the individual pulses is simply half

he peak-to-peak separation of pulses in the cross-orrelation. The average timing-slide separation betweeneaks in the pulse train obtained with a 10 ps stretchedulse and 6.61-mm-long crystal shown in Fig. 5(b) is 0.06m giving a FWHM pulse duration of the individual

eaks of 200 fs. Pulses of shorter duration could not be re-olved because of the relatively long duration (80 fs) of therobe pulse. Therefore the shortest pulses reported hereo not represent a lower-limit on the duration of theulses generated using this method.Any higher-order dispersion in the system will intro-

uce a non-linear chirp to the pulse. Figure 7 plots � asfunction of time for pulse trains produced by passing a

3 ps stretched pulse through wave plates of varyinghickness. It is clear that � varies slightly with positionn the pulse train; this variation originates from thehird- and higher-order dispersion from the grating-pairtretcher and the wave plate material.

These pulse trains are again best-suited to conditionshere the coherence length is constant through the gen-rating region. However we have already seen that this isot generally the case, as illustrated by Fig. 3, whichhows that for the 27th harmonic, Lc varies by a factor ofpproximately 1.5 over the region in which the harmonicas generated strongly. Experiments by Lytle et al. [9]lso showed similar variations in Lc. In order to fully op-imize the QPM process, pulse trains with programmablyariable separation are therefore required. This is demon-

su

ACdq

wa

T

w

wpba3

Fp hickne

Foa

Fapd


trated in the next section, where dispersion control issed to manipulate the pulse spacing.

. Chirped-Pulse Trainsonsider a pulse (prior to the wave plate and polarizer),escribed as a superposition of harmonic waves of fre-uency �, given by

E = a��exp�i��,t��, �13�

here a�� is the spectral amplitude, �� , t�=��−�t,nd �� is the spectral phase. We can expand �� in a

ig. 5. Normalized cross-correlation traces obtained with stretchs passed through multiple-order calcite wave plates of various t

ig. 6. (Color online) Plot of 1/� as a function of the thicknessf the calcite crystal for stretched pulses of durations 5, 10, 15,nd 33 ps showing the expected linear dependence of Eq. (12).

aylor series about a center frequency �0, giving

�� = �n=0

1

n!��n�� − �0�n, �14�

here ��n��n� /��n ��0.

If we assume that the group delay dispersions of theave plate and polarizer are small, the frequenciesassed by this combination are evenly spaced and giveny �p−�0=p��, where p=0, ±1, ±2, ±3. . .. As an ex-mple, for the linearly chirped pulse considered in Section, the instantaneous frequency is given by �=�0+2bt and

ses of FWHM durations of (a) 5 ps, (b) 10 ps, (c) 15 ps, and (d) 33sses.

ig. 7. (Color online) Variation in the pulse separation, �,cross the pulse train generated when a 30 ps stretched pulseasses through crystals of varying thicknesses. Linear fits to theata are also shown.

ed pul

hpTqi

T

Is

tac(apAlds

wpgpcmwttnpt

gti

siFfswzsd

EsuadIt

Qlltpw

Fc(m

Fp=a


ence from Eq. (12), the frequencies passed by the wavelate and polarizer are separated by ��=2� / l��o�−�e��.he time taken for the pth pulse corresponding to a fre-uency �p−�0=p�� to traverse a wave plate of length Ls

�p� = Ldk

d�=

d�

d�= �

n=1

1

�n − 1�!��n��p��n−1. �15�

he pulse spacing is then given by

��p� =d

dp= �

n=2

1

�n − 2�!��n�pn−2��n−1. �16�

t can be seen that dispersion of order n gives a pulsepacing within the train which varies as pn−2.

It is therefore apparent from Eq. (16) that it is possibleo vary � within the pulse train by controlling the third-nd higher-order dispersions. To demonstrate this prin-iple, an acousto-optic programmable dispersive filterAOPDF) [24] was used to manipulate the spectral phasend amplitude of the chirped pulses used to generateulse trains [25], as shown schematically in Fig. 8(b). TheOPDF (FASTLITE Dazzler HR-800) employed a 7-cm-

ong TeO2 crystal capable of introducing a programmableelay of up to 8 ps. The generated pulse trains were mea-ured by cross-correlation as before.

Figure 9 shows the cross-correlation signals obtainedith a 0.8-mm-thick calcite crystal and a third-order dis-ersion, ��3�= ±1�106 fs3, introduced by the AOPDF. Theratings were separated by 20 mm to give a stretchedulse of duration 1.4 ps. The generated pulse trains areomprised of six pulses, each with a duration of approxi-ately 550 fs. It can be seen that the pulse separationithin the train varies strongly and that the direction of

his variation depends on the sign of ��3�. Also plotted ishe pulse separation, �p,p−1 as a function of the pulseumber, p, showing that �p,p−1 varies linearly, as ex-ected and by approximately 160% over the duration ofhe pulse train.

Figure 10 shows cross-correlations of the pulse trainsenerated using a 1.6-mm-long calcite crystal as thehird-order dispersion introduced by the AOPDF was var-ed. The same grating separation was used giving a 1.4 ps

ig. 8. (Color online) Experimental setup used to generate andontrol trains of ultrashort pulses with (a) constant spacing andb) variable spacing using an AOPDF. G, grating stretcher; W,ultiple-order wave plate; P, polarizer.

tretched pulse. Linear fits of �p,p−1 as a function of p (asn Fig. 9) yielded �p,p−1� , the rate of change of � with p.igure 11 shows the measured �p,p−1� as a function of ��3�

or crystals of lengths 0.8 and 1.6 mm. Note that there istill a small variation in � across the pulse trains, evenhen the third-order dispersion applied by the AOPDF is

ero. This is due to third-order dispersion from otherources including the grating-pair stretcher and materialispersion as was seen in Fig. 7.Higher-order dispersion was also investigated. From

q. (16), it is apparent that fourth-order dispersionhould lead to a quadratic variation in pulse spacing. Fig-re 12 shows the cross-correlation of a pulse train gener-ted in a 0.8-mm-thick crystal, with only a fourth-orderispersion of ��4�= ±1�108 fs4 introduced by the AOPDF.t is seen that the variation in pulse spacing matches wello a quadratic fit.

In order to illustrate the use of these pulse trains forPM, Fig. 13 shows a simulation of the QPM of harmonic

ight in a medium in which the coherence length variesinearly with position (see inset), from L0 to 1.5L0, similaro the variation encountered experimentally. The upperlot shows the pulse trains used in the simulations; oneith a constant pulse spacing, appropriate for QPM with

ig. 9. (Color online) Cross-correlation signal recorded for a 1.4s stretched pulse passed through a 0.8 mm crystal and ��3�

±1�106 fs3 applied by the AOPDF. The lower plot shows �p,p−1s a function of p for the two sets of data along with linear fits.

apmdssioa

mat

sdpapbwvtlwreA

ip

F1vn

Fwc

F1=pa


constant coherence length of Lc=1.25L0; and a “chirped”ulse train, with a linear variation in pulse spacing toatch the variation in Lc in the medium. A linear depen-

ence of the suppression effect on the pulse train inten-ity is assumed, up to a value of 1, above which suppres-ion is complete. The lower plot shows the harmonicntensity as a function of the position x of the leading edgef the pulse train in the medium. It is seen that QPM withuniform pulse train is predicted to increase the har-

ig. 10. (Color online) Cross-correlation signals recorded for a.4 ps stretched pulse passed through a 1.6 mm crystal and witharious third-order dispersions applied by the AOPDF. The sig-als have been displaced vertically for clarity.

ig. 11. (Color online) The rate of change of pulse separationith pulse number, �p,p−1� , is plotted as a function of ��3� for cal-

ite crystals of thicknesses 0.8 and 1.6 mm.

onic signal by a factor of �70 from that obtained in thebsence of QPM. Use of a chirped-pulse train increaseshe harmonic signal by a further factor of �2.

Since arbitrary spectral and temporal shaping is pos-ible with an AOPDF, the pulse trains could be generatedirectly simply by applying the correct amplitude orhase filter with the AOPDF [18]. However, if the pulsesre required to have a high energy or a large number ofulses is required, the total energy of the pulse train muste correspondingly high. In such cases, the pulse shaperould have to be placed before the laser amplifier to pre-ent damage to the device. However, the strong modula-ion of the amplitude or phase of the stretched pulse couldead to the non-linear distortion of the amplified outputaveform and possibly damage to the amplifier. For these

easons it is expected to be difficult to generate high-nergy pulse trains using pulse shapers such as anOPDF alone.The single crystal technique presented here is unique

n that it allows the generation of high-power adaptiveulse trains. To achieve this, a laser amplifier could be

ig. 12. (Color online) Cross-correlation signal recorded for a.4 ps stretched pulse passed through a 0.8 mm crystal and ��4�

±1�108 fs4 applied by the AOPDF. The lower plot shows theulse spacing, �p,p−1, as a function of p for the two sets of datalong with quadratic fits.

ppsp

4ToaspmghTvQv

aaotttT

vsQtptl

arhsptm

ATSCE

R

1

1

FLtsst


laced between the AOPDF and the multiple-order wavelate; since the strong modulation of the amplitude and/orpectral phase of the pulse is only imposed by the wavelate and polarizer, the amplifier will be protected.

. CONCLUSIONwo methods have been demonstrated to produce trainsf ultrashort pulses. In the first technique, employing anrray of birefringent crystals and a polarizer, the pulsepacing can be varied by choosing the crystal lengths ap-ropriately. A train of two pulses generated with thisethod was used to quasi-phase-match high-harmonic

eneration in a capillary waveguide leading to an en-ancement of harmonic radiation at 29 nm by up to 50%.hese measurements showed that the coherence lengtharied along the capillary and hence that for optimizedPM it would be necessary to employ a pulse train ofarying pulse spacing.

Chirped-pulse trains of this type were generated usingsecond technique. In this, a multiple-order wave plate

nd linear polarizer were used to modulate the intensityf a chirped pulse, producing a train of pulses. The addi-ion of an AOPDF was shown to enable the creation ofrains with non-uniform inter-pulse separation allowinghe generation of high-energy adaptable pulse trains.his technique could therefore be used for QPM where L

ig. 13. (Color online) Simulation of QPM in a medium wherec varies linearly with position. The upper plot shows the pulse

rains used (an intensity of 1 corresponds to complete suppres-ion of the harmonic generation), while the lower plot shows theimulated harmonic intensity as these pulse trains are scannedhrough the medium.

c

aries with position through the generating medium. Ithould be noted that there will be some degradation of thePM effect due to the cosine-squared shape of these

rains leading to some suppression of the HHG in the in-hase regions. However, this effect will be outweighed byhe ability to tune the pulse train to achieve QPM over aarge number of zones.

Programmable pulse trains of this type could be used tochieve QPM over multiple harmonic orders or to achieveandom QPM [26], which in principle could allow en-ancement of harmonic flux over a broad bandwidth, re-ulting in the generation of high peak-power attosecondulses. The adaptability of these pulse trains also makeshem ideal for use with genetic algorithms in order toaximize the harmonic output.

CKNOWLEDGMENTShis work was funded by the Engineering and Physicalciences Research Council (EPSRC) (grant number EP/005449). T. Robinson acknowledges the support of anPSRC Ph.D. Plus award.

EFERENCES1. A. H. Zewail, “Femtochemistry: Atomic-scale dynamics of

the chemical bond,” J. Phys. Chem. A 104, 5660–5694(2000).

2. A. M. Weiner, D. E. Leaird, G. P. Wiederrecht, and K. A.Nelson, “Femtosecond pulse sequences used for optical ma-nipulation of molecular motion,” Science 247, 1317–1319(1990).

3. Y. Liu, S. Park, and A. M. Weiner, “Enhancement ofnarrow-band terahertz radiation from photoconducting an-tennas by optical pulse shaping,” Opt. Lett. 21, 1762–1764(1996).

4. A. S. Weling, B. B. Hu, N. M. Froberg, and D. H. Auston,“Generation of tunable narrow-band THz radiation fromlarge aperture photoconducting antennas,” Appl. Phys.Lett. 64, 137–139 (1994).

5. X. Zhang, A. L. Lytle, T. Popmintchev, X. Zhou, H. C.Kapteyn, M. M. Murnane, and O. Cohen, “Quasi-phase-matching and quantum-path control of high-harmonic gen-eration using counterpropagating light,” Nat. Phys. 3, 270–275 (2007).

6. A. L. Lytle, X. Zhang, R. L. Sandberg, O. Cohen, H. C.Kapteyn, and M. M. Murnane, “Quasi-phase matching andcharacterization of high-order harmonic generation in hol-low waveguides using counterpropagating light,” Opt. Ex-press 16, 6544–6566 (2008).

7. A. L. Lytle, X. Zhang, P. Arpin, O. Cohen, M. M. Murnane,and H. C. Kapteyn, “Quasi-phase matching of high-orderharmonic generation at high photon energies using counter-propagating pulses,” Opt. Lett. 33, 174–176 (2008).

8. O. Cohen, A. L. Lytle, X. Zhang, M. M. Murnane, and H. C.Kapteyn, “Optimizing quasi-phase matching of high har-monic generation using counterpropagating pulse trains,”Opt. Lett. 32, 2975–2977 (2007).

9. A. L. Lytle, X. Zhang, J. Peatross, M. M. Murnane, H. C.Kapteyn, and O. Cohen, “Probe of high-order harmonic gen-eration in a hollow waveguide geometry using counter-propagating light,” Phys. Rev. Lett. 98, 123904 (2007).

0. J. Seres, E. Seres, A. J. Verhoef, G. Tempea, C. Streli, P.Wobrauschek, V. Yakovlev, A. Scrinzi, C. Spielmann, and F.Krausz, “Laser technology: Source of coherent kiloelectron-volt x-rays,” Nature 433, 596 (2005).

1. E. A. Gibson, A. Paul, N. Wagner, R. Tobey, D. Gaudiosi, S.Backus, I. P. Christov, A. Aquila, E. M. Gullikson, D. T. At-twood, M. M. Murnane, and H. C. Kapteyn, “Coherent softx-ray generation in the water window with quasi-phase

1

1

1

1

1

1

1

1

2

2

2

22

2

2


matching,” Science 302, 95–98 (2003).2. J. Seres, V. S. Yakovlev, E. Seres, C. Streli, P. Wobrauschek,

C. Spielmann, and F. Krausz, “Coherent superposition oflaser-driven soft-x-ray harmonics from successive sources,”Nat. Phys. 3, 878–883 (2007).

3. B. Dromey, M. Zepf, M. Landreman, and S. M. Hooker,“Quasi-phasematching of harmonic generation via multi-mode beating in waveguides,” Opt. Express 15, 7894–7900(2007).

4. J. Peatross, S. Voronov, and I. Prokopovich, “Selective zon-ing of high harmonic emission using counter-propagatinglight,” Opt. Express 1, 114–125 (1997).

5. M. Landreman, K. O’Keeffe, T. Robinson, M. Zepf, B.Dromey, and S. M. Hooker, “Comparison of parallel andperpendicular polarized counterpropagating light for sup-pressing high harmonic generation,” J. Opt. Soc. Am. B 24,2421–2427 (2007).

6. C. W. Siders, J. L. Siders, A. J. Taylor, S. Park, and A. M.Weiner, “Efficient high-energy pulse-train generation usinga 2 n-pulse Michelson interferometer,” Appl. Opt. 37, 5302–5305 (1998).

7. A. M. Weiner, J. P. Heritage, and E. M. Kirschner, “High-resolution femtosecond pulse shaping,” J. Opt. Soc. Am. B 5,1563–1572 (1988).

8. A. M. Weiner, D. E. Leaird, J. S. Patel, and J. R. Wullert,“Programmable femtosecond pulse shaping by use of a mul-

tielement liquid-crystal phase modulator,” Opt. Lett. 15,326–328 (1990).

9. M. M. Wefers and K. A. Nelson, “Programmable phase andamplitude femtosecond pulse shaping,” Opt. Lett. 18, 2032–2034 (1993).

0. B. Dromey, M. Zepf, M. Landreman, K. O’Keeffe, T. Robin-son, and S. M. Hooker, “Generation of a train of ultrashortpulses from a compact birefringent crystal array,” Appl.Opt. 46, 5142–5146 (2007).

1. R. Yano and H. Gotoh, “Tunable terahertz electromagneticwave generation using birefringent crystal and gratingpair,” Jpn. J. Appl. Phys. 44, 8470–8473 (2005).

2. T. Robinson, K. O’Keeffe, M. Landreman, S. M. Hooker, M.Zepf, and B. Dromey, “Simple technique for generatingtrains of ultrashort pulses,” Opt. Lett. 32, 2203–2205(2007).

3. A. E. Siegman, Lasers (University Science Books, 1990).4. P. Tournois, “Acousto-optic programmable dispersive filter

for adaptive compensation of group delay time dispersion inlaser systems,” Opt. Commun. 140, 245–249 (1997).

5. K. O’Keeffe, T. Robinson, and S. M. Hooker, “Generationand control of chirped, ultrafast pulse trains,” J. Opt. 12,015201 (2010).

6. A. Bahabad, O. Cohen, M. M. Murnane, and H. C. Kapteyn,“Quasi-periodic and random quasi-phase matching of highharmonic generation,” Opt. Lett. 33, 1936–1938 (2008).

generation and control of ultrafast pulse trains for quasi-phase-matching high-harmonic generation

Documents