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Calculation and manipulation of the chirp rates of high-order harmonics

Murakami, M; Mauritsson, Johan; L'Huillier, Anne; Schafer, KJ; Gaarde, Mette

Published in:Physical Review A (Atomic, Molecular and Optical Physics)

DOI:10.1103/PhysRevA.71.013410

2005

Link to publication

Citation for published version (APA):Murakami, M., Mauritsson, J., L'Huillier, A., Schafer, KJ., & Gaarde, M. (2005). Calculation and manipulation ofthe chirp rates of high-order harmonics. Physical Review A (Atomic, Molecular and Optical Physics), 71(1).https://doi.org/10.1103/PhysRevA.71.013410

Total number of authors:5

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Calculation and manipulation of the chirp rates of high-order harmonics

M. Murakami,1 J. Mauritsson,1 A. L’Huillier, 2 K. J. Schafer,1 and M. B. Gaarde11Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803-4001, USA

2Department of Physics, Lund Institute of Technology, P. O. Box 118, S-22100 Lund, SwedensReceived 17 September 2004; published 14 January 2005d

We calculate the linear chirp rates of high-order harmonics in argon, generated by intense, 810 nm laserpulses, and explore the dependence of the chirp rate on harmonic order, driving laser intensity, and pulseduration. By using a time-frequency representation of the harmonic fields we can identify several differentlinear chirp contributions to the plateau harmonics. Our results, which are based on numerical integration of thetime-dependent Schrödinger equation, are in good agreement with the adiabatic predictions of the strong fieldapproximation for the chirp rates. Extending the theoretical analysis in the recent paper by Mauritssonet al.fPhys. Rev. A70, 021801sRd s2004dg, we also manipulate the chirp rates of the harmonics by adding a chirpto the driving pulse. We show that the chirp rate for harmonicq is given by the sum of the intrinsic chirp rate,which is determined by the new duration and peak intensity of the chirped driving pulse, andq times theexternal chirp rate.

DOI: 10.1103/PhysRevA.71.013410 PACS numberssd: 32.80.Rm, 42.65.Ky

I. INTRODUCTION

High-order harmonics, which can be generated in the in-teraction between an intense ultrashort pulse and a gas ofatomsf1g, represent a unique and versatile source of extremeultraviolet sxuvd radiation. Applications of the harmonic ra-diation range from xuv pump-probe spectroscopyf2,3g andinterferometry f4g to the generation of attosecond pulses,both in the form of isolated attosecond burstsf5g and in theform of trains of attosecond pulsesf6–9g. A characteristicfeature of the harmonic pulses is that they are generated witha time-varying phase and therefore exhibit a time-dependentfrequencyf10,11g.

Much effort has been put into the manipulation and char-acterization of this time-dependent frequencyf10–13g. Itsmanipulation is important for the use of the harmonic radia-tion as an xuv source since, in addition to its effect on thespectrum of the individual harmonic, the time-dependent fre-quency and its variation with harmonic order strongly influ-ence the time structure of the attosecond pulse trains gener-ated by superposing a number of harmonicsf14g. Thecharacterization of the time-dependent frequency is also offundamental interest, since the time dependence of the har-monic phase to a large part is intrinsic to the generationprocess, originating in the electron dynamics of each atomdriven by the strong fieldf15g. The time-dependent fre-quency therefore yields important information about the har-monic generation process itself. Experimentally, other factorssuch as ionization-induced blueshifting and phase matchingalso affect the chirp ratesf12g, but these can be kept to aminimum by keeping the peak intensity below saturationf13g.

The time-dependent frequencies of the harmonics andtheir variation with harmonic order can be understood in theframework of the semiclassical model of harmonic genera-tion f16,17g, in which an electron is released into the con-tinuum via tunnel ionization, accelerated by the laser field,and on returning to the ion core and recombining can transferits kinetic energy to radiation. In this model, the energy and

phase of the emitted light are determined by the electron’skinetic energy and its time of return, respectively. The time-dependent phasefqstd is proportional to the laser intensityIstd, fqstd=−aq

i Istd, where the phase coefficientaqi is char-

acteristic of the space-time quantum pathi the electron hasfollowed f18g. The temporal variation discussed above refersto theslow variation of the pulse envelope. In this adiabaticlimit, the time-dependent frequencyvqstd=−dfqstd /dt is de-termined by the cycle to cycle variation of the intensity en-velope. The time-dependent frequency is approximately lin-ear close to the peak of the laser pulse, and is oftencharacterized by its linear chirp ratebq

dip:

bqdip ~ − aq

i Ipeak/t2. s1d

Here Ipeak andt are the peak intensity and the full width athalf maximumsFWHMd duration, respectively, of the driv-ing pulse. The proportionality constant depends on the pulseshapef19g. In the adiabatic picture, the chirp rate thus scaleslinearly with the pulse peak intensity, and quadratically withthe inverse of the pulse duration. The variation of the chirprate from harmonic to harmonic originates in the variation ofthe return time, and therebyaq

i , with orderf7,8,14g.In this paper we theoretically explore the time-frequency

sTFd behavior of high-order harmonics, generated in argonby 810 nm laser pulses with durations ranging from 13T1 to36T1, whereT1 is the laser period. Using a time-frequencyanalysis, we can resolve more than one linear chirp contri-bution for the harmonics in the plateau. We calculate thelinear chirp rate as a function of the harmonic order and findresults that are in good agreement with the predictions basedon the strong field approximationsSFAd f15,18g and withrecent measurementsf13g. In agreement with previous re-sults f20g, we find that the most important contributions tothe plateau harmonics come from the firstsshortestd and thethird and longer quantum paths. Finally, as was found experi-mentally inf13g, we show that the harmonic chirp ratebq can

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be manipulated by adding a linear chirp with rateb1 to thedriving laser pulse according to

bq = qb1 + bqdip, s2d

when one takes into account the change in the driving pulseduration and peak intensity caused by its chirping.

The paper is outlined as follows. In Sec. II we discuss ourtheoretical methods. Sections III and IV present the TF re-sults and the chirp manipulation results, respectively. SectionV summarizes our conclusions.

II. THEORETICAL METHOD

We solve the time-dependent Schrödinger equationsTDSEd by numerical integration within the single-active-electron approximation. We follow exactly the same proce-dure as outlined in detail inf17,21,22g: the initial state of theatom sthe ground stated is found as the solution to the field-free time-independent Schrödinger equation, and the time-dependent one-electron wave function is then calculated bydirect numerical integration of the TDSE in the combinedatomic and laser potentials. The atom is described by apseudopotential, the construction of which is based onHartree-Slater calculations and is discussed in detail inf23–25g. We use a Gaussian pulse shape with a FWHM du-ration of t and integrate fromt=−2.5t to t=2.5t, evaluatingthe time-dependent acceleration formastd of the dipole mo-ment in each time step. The dipole spectrumdsvd is propor-tional to the Fourier transform ofastd. We find the time pro-file Eqstd of harmonicq by multiplying dsvd with a windowfunction centered around theqth harmonic and inverse Fou-rier transforming to the time domain. The results presentedbelow have been obtained with a square window of width2v1, wherev1 is the driving frequency, but they do not de-pend on the shape of the window function.

We are interested in the TF behavior ofEqstd= uEqstduexpfiFqstdg+c.c., whereFqstd is now the full time-dependent phase of theqth harmonic electric field. The har-monics in the cutoff region exhibit a simple TF behaviorwhich can be characterized by the linear chirp ratebq of thetime-dependent frequencyvqstd=−dFqstd /dt. We find thechirp rate by fitting a straight line tovqstd over approxi-mately the FWHM duration of the harmonic pulsessince theharmonic pulses are far from Gaussian in shape, the durationover which the frequency is fitted varies somewhat from har-monic to harmonicd. The harmonics in the plateau regionhave contributions from several quantum paths each withtheir own TF characteristics. We therefore need to simulta-neously represent the temporal and spectral characteristics ofthese harmonics. We choose the following TF representationsTFRd:

Sqst,vd = UE dt8eivt8Eqst8dEIRst8 − tdU2

, s3d

where the probe pulseEIRst8− td has a center frequency ofv1

and a duration shorter than that ofEqstd. When the delaybetween the probe and the harmonic pulses is varied, thespectrogramSqst ,vd traces how the “instantaneous” spec-

trum of the harmonic field changes during the pulse. IfEqstdcan be characterized by a linear chirp, for instance, thenSqst ,vd is distributed along a straight line in thet-v planewhose slope yields the chirp rate when corrected for the fi-nite duration of the probe pulsef11g. In caseEqstd has mul-tiple spectral contributions with different chirp rates, its TFRwill split up into several linear structures, from which we canfind the chirp rates one by one. The spectrogram can there-fore in many cases resolve the different TF behaviors due tothe different quantum path contributionsf19g. We note thatour choice of TFR is very close to the experimental approachfor the TF characterization of harmonics through the genera-tion of sidebands, discussed in detail inf13g.

III. LINEAR CHIRP RATES OF PLATEAUAND CUTOFF HARMONICS

The results in this section have been obtained in argondriven by 810 nm laser pulses with a FWHM duration of20T1 s54 fsd and a peak intensity of 231014 W/cm2. Thecutoff energy predicted for this system is 36v1 s55 eVd f26g.To calculate the TFR, we have used a probe pulse with aFWHM duration of 5T1.

In Fig. 1, we plot the TFR of nine consecutive harmonicsspanning part of the plateau and the cutoff region to makethe following observations. The TFR of the harmonics be-yond the cutoff is distributed along one direction only, indi-

FIG. 1. sColor onlined The time-frequency representation ofnine consecutive harmonics in argon, driven by a laser pulse with apeak intensity of 231014 W/cm2, and a pulse duration of 20T1,starting with the harmonic closest to the cutoff energy and movingdownward in photon energy. The probe pulse duration used in Eq.s3d is 5T1. The TFR is shown in false colors/gray scale. The scale isdifferent from harmonic to harmonic due to the large difference instrength between the plateau and cutoff harmonics.

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cating that these fields have a unique chirp rate. This chirprate is independent of order. For the harmonics in the pla-teau, there are two significant directions for the TFR tospread, one associated with a small negative chirp, and theother with a much larger negative chirp. The slopes of bothof these contributions clearly vary from harmonic to har-monic. We have determined these chirp rates separately andplot them in Fig. 2.

The closed circles in Fig. 2 show the chirp rates versus theharmonic energy, found from the primary contributions tothe TFR. To calculate them, we find the frequencyv atwhich Sqst ,vd takes its maximum value for eacht, and thenperform a linear fit to this set of pointsvstd. For harmonicsbeyond the cutoff, we also plot the rates found by directdifferentiation of the time-dependent phase ofEqstd sshownwith trianglesd. These are in excellent agreement with therates extracted from the TFR. The open circles in Fig. 2 havebeen found from the secondary linear structures in the TFR.

We interpret our results within the framework of the semi-classical modelf14,15,18g. The TFR resolves the differentquantum path contributions to the time-dependent frequency,and gives an intuitive measure of the electron dynamics atthe single-atom level. Quantitatively, the chirp rates are ingood agreement with predictions of the SFAsdotted curvesdfor the two shortest quantum paths. In agreement with resultsof our earlier work, and in contrast to the predictions of theSFA, we also find thatsid the shortest quantum path is domi-nant for most of the harmonics in argon, andsii d it is ingeneral the trajectories with return times longer than one ir

cycle which yield the next largest contributionf20g. For theharmonics in the lower plateau region, the contributions fromthe second and the third quantum paths are not distinguish-able with our resolutionssee alsof20gd. In an experiment it ispossible to spatially separate the contribution from the short-est trajectory from other contributions since its spatial diver-gence is smaller by approximately an order of magnitudef8,27g. In addition, as also seen in Fig. 1, the spectral distri-bution of the contributions from the longer trajectories isvery wide, often making their experimental observation dif-ficult.

IV. MANIPULATION OF THE CHIRP RATES

Next, we manipulate the harmonic chirps by adding achirp b1 to the driving pulse. In the adiabatic limit, we thenexpect the harmonics to exhibit a chirp given by the sum ofthe intrinsic chirp andq times the fundamental chirp. Recentexperimental and theoretical resultsf13g have supported thisexpectation. In Fig. 3 we detail our theoretical findings inf13g. We concentrate on the harmonics in the cutoff regionfor which we can directly find the linear chirp rates from thetime-dependent phase, as described above.

The chirp is added as it would be in an experiment, bystretching the driving pulse to be positively or negativelychirped, in a way that preserves the pulse energy. This meansthat the peak intensity of the chirped pulse decreases as

Ic = I0t0

tcs4d

when the pulse is stretched from its original durationt0 to adurationtc. We use a duration of the unchirped driving pulseof t0=13T1 s35 fsd. In Fig. 3sad the driving pulse has beenpositively chirped to a FWHM duration oftc=36T1. Thisleads to a linear chirp rate ofb1= +0.50 meV/fs and a re-duced peak intensity ofIc=0.7231014 W/cm2. The linearchirp rate of the odd harmonics 19 through 27sthe cutoffenergy is 19.5v1d found from the time-dependent phase isshown with filled circles. The adiabatic prediction for thechirp rate of the cutoff harmonics as given by Eqs.s2d ands1d, using tc and Ic, is shown with open circles. The filledtriangles showqb1. The harmonics in Fig. 3sbd have beencalculated with a negatively chirped pulse with rateb1=−0.68 meV/fs, a stretched pulse duration oftc=23T1, andreduced peak intensityIc=1.1331014 W/cm2 swith a cutoffenergy of 24.8v1d. In both calculations, the adiabatic predic-tion is in very good agreementsto within 10%d with the fullcalculation. In an experiment, one would also expect themeasured chirp rates to be influenced by ionization andphase matching although these can be minimized by thechoice of parametersf13g.

It is worth noting that when the driving pulse is chirped inthis way, which is experimentally the most straightforwardapproach, the change in the harmonic chirp rate is twofold.In addition to the externally added chirp, the rate of the in-trinsic chirp of the cutoff harmonics decreases with the thirdpower of the pulse durationfsee Eqs.s1d and s4dg. In addi-tion, the cutoff energy is lowered due to the smaller peak

FIG. 2. sColor onlined Chirp rates of the high harmonic fields inargon, obtained from the strongest component of TFRssfilledcirclesd, or by direct differentiationstrianglesd. The SFA predictionsfor the chirp rates originating in the first two quantum paths areshown with dotted curves. As illustrated in Fig. 1, an additionalcontribution to the TFR of harmonics is visible in the upper plateauregion, from which we have found another set of chirp rates, shownwith open circles. These chirp rates originate in the phase behaviorof the third quantum path; seef20g.

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intensity. Controlling the chirp of particular harmonics isthus a balance between several factors.

In Fig. 4 we use this manipulation to eliminate the chirpof a cutoff and a plateau harmonic. Insad we show the TFRof the 35th harmonic calculated with the same parameters asin Fig. 1, a pulse duration of 20T1 and a peak intensity of231014 W/cm2. Its linear chirp rate is −27 meV/fsssee Fig.2d. To compensate its chirp, we simply add a positive linearchirp with rate b1=b35/35=0.77 meV/fs to the drivingpulse, keeping the same intensity and pulse duration. This

means that we add bandwidth to the driving pulse. Experi-mentally, this could be achieved by positive stretching of ashorter pulses12.6T1d to the 20T1 duration used here. Theresulting TF behavior, which is now flat, is shown insbd.Since the intrinsic chirp of the harmonic has been compen-sated, the bandwidth of the harmonic is smaller than insadeven though the bandwidth of the driving pulse is largerf10g.

The plateau harmonics have several contributions withdifferent chirps, which means that adding a single chirp tothe driving pulse does not compensate the entire TF depen-dence. Here we manipulate the chirp of the strongest contri-bution to the plateau harmonics, from the short quantumpath. The TFR of the 27th harmonic calculated as in Fig. 4sadis shown in 4scd. In 4sdd we have added a positive chirp ofb1=0.31 meV/fs to the driving pulse, again keeping thesame pulse duration and peak intensity. This essentiallyeliminates the chirp of the dominant contribution, but doeslittle srelativelyd to the larger chirp.

While eliminating the chirp of the 27th harmonic for partin Fig. 4sdd, the manipulation also affects its neighboringharmonics. In Fig. 5 we show the chirp rates of harmonics 23through 31 generated by the chirped driving pulse of Fig.4sdd. The chirps of all the harmonics have been significantlyreduced, except for the 31st harmonic. This can be under-stood by looking at the order dependence of the chirp rate forthese harmonics shown in Fig. 2. The chirp rate due to thefirst quantum path increases almost linearly in magnitudebetween harmonics 23 and 29. This is nearly compensated by

FIG. 3. sColor onlined The chirp rates of cutoff harmonics con-trolled by a driving pulse which has been chirped by stretchingfrom an initial duration of 13T1 ssee also textd. In sad the drivingpulse is positively chirped with a rate ofb1=0.50 meV/fs to aduration of 36T1. This reduces the peak intensity to 0.7231014 W/cm2. The driving pulse insbd has a negative linear chirprate of b1=−0.68 meV/fs, a pulse duration of 23T1, and a peakintensity of 1.1331014 W/cm2. Filled circles show the calculatedchirp rates; open circles show the adiabatic SFA prediction for thechirp rates. The external chirpqb1 is shown with triangles.

FIG. 4. sColor onlined sad and scd: The TFRs of the 35th and27th harmonic, respectively, driven by an unchirped pulse as in Fig.1 speak intensity 231014 W/cm2 and pulse duration 20T1d. In sbdand sdd we show the 35th and 27th harmonics, respectively, gener-ated by chirped pulses with the same peak intensity and pulse du-ration, with positive chirp ratesb1=0.77 andb1=0.31 meV/fs.

FIG. 5. sColor onlined TheTFRs of the odd harmonics 23through 31, generated by the posi-tively chirped driving pulse in Fig.4sdd. This chirp rate compensatesthe intrinsic chirp of the 27thharmonic.

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the increase in the external chirp rate with ordersq3b1d. The31st harmonic has both a larger intrinsic chirp rate from thefirst quantum path and a larger contribution from the longerquantum pathsssee Fig. 1d. The manipulated 31st harmonicshown in Fig. 5 certainly exhibits two different TF behaviors.

V. SUMMARY

We have analyzed the TF behavior of many high-orderharmonics in argon. Using a TF analysis we could identifyseveral contributions with different linear chirps to the har-monic time profiles. Our results, which are based on numeri-cal integration of the TDSE using a realistic pseudopotentialfor the one-electron argon atom, are in good agreement withSFA predictions for the harmonic chirp rates. By adding achirp to the driving pulse we manipulated the harmonic chirp

rates and showed that, at least in the adiabatic regime as inthis study, the harmonic chirp rates are the sum of their in-trinsic and external chirp rates. When the driving pulse ischirped by stretching, the increase in the pulse duration rap-idly decreases the intrinsic chirp. Controlling the chirp ofany particular harmonic is thus a balance between pulse du-ration, intensity, and external chirp rate.

ACKNOWLEDGMENTS

The authors thank P. Johnsson, R. López-Martens, and K.Varjú for stimulating discussions. M.M. and M.G. acknowl-edge the support of the Louisiana Board of Regents throughGrant No. LEQSFs2004-07d-RD-A-09. J.M. and K.S. ac-knowledge the support of the National Science Foundationthrough Grant No. PHY-9733890. A.L. acknowledges thesupport of the Swedish Science Council.

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