406 J. Opt. Soc. Am. B/Vol. 13, No. 2 /February 1996 Ditmire et al.

Calculation and measurement of high-orderharmonic energy yields in helium

T. Ditmire, K. Kulander, J. K. Crane, H. Nguyen, M. D. Perry

Laser Program, Lawrence Livermore National Laboratory, L-443, P.O. Box 808, Livermore, California 94550

Received March 14, 1995; revised manuscript received June 1, 1995

We present calculations of the energy yields of high-order harmonic radiation produced by 526-nm laser lightfocused into a helium gas jet. The intensity-dependent dipole moments of helium calculated by numericalintegration of the Schrodinger equation have been used in a numerical solution of the wave equation todetermine the energy yields of harmonics in the 30–17-nm wavelength range. These calculations arecompared with measured absolute energy yields and are shown to be in good agreement with experiment. 1996 Optical Society of America

1. INTRODUCTIONHigh-order harmonic generation1,2 is one of the poten-tial sources of bright extreme ultraviolet (XUV) radia-tion for a host of future experiments in radiation-matterinteraction physics.3 When compared with other XUVsources, such as synchrotrons, free-electron lasers, andx-ray lasers, high-order harmonic-based sources exhibit anumber of advantages. These include short pulse dura-tion (30 fs to 100 ps), very high peak brightness (.1023

photons mm22 mrad22 s21), good spatial coherence, andbroad spectral tunability. The relatively small size andhigh repetition rate of the lasers required for generatingshort-wavelength harmonics make this source an attrac-tive alternative for experiments requiring coherent XUVradiation.

In the past two years the emphasis has shiftedfrom understanding the basic physics of high harmonicgeneration4,5 to the characterization of the harmonics forpotential use in applications. A number of groups haveinvestigated the spatial coherence of the harmonics.6 – 9

The spectral and the temporal characteristics have alsobeen investigated.10 – 12 We have recently reported mea-surements of the energy yields of the high-order harmon-ics under a variety of conditions.13 The use of high-orderharmonics in applications has grown recently as well.Balcou et al. have recently demonstrated the use of har-monics in the measurement of rare-gas photoionizationstudies with photon energies above 100 eV.14 Haightand Peale have utilized harmonics in an experimentthat takes advantage of the harmonics’ short-pulse char-acter by making time-resolved measurements of sur-face states in semiconductors by means of photoelectronspectroscopy.15,16 The many advantages offered by ahigh-order harmonic-based source ensure an increasingnumber of applications in the future.

The theoretical understanding of harmonic genera-tion has also greatly increased over the past few years.Single-atom calculations have been successful in pre-dicting the presence of the harmonic plateau and theposition of the short-wavelength cutoff of the harmonicplateau.17,18 The effects of propagation of the high-intensity pulse through the gas target have been studied

0740-3224/96/020406-06$06.00

as well, and the importance of phase matching has beenwell established.19 – 21 With this understanding comesthe potential for calculations of the total energy yieldsthat are attainable in the soft-x-ray region with high-order harmonic generation. In this paper we presentcalculations of the energy yields of harmonics producedin helium by a 526-nm 600-fs laser pulse. We haveconstructed a model similar to that of L’Huillier et al.20

and applied it to calculations of absolute harmonic energyyields in the high focused intensity regime sI . Isatd. Wehave incorporated calculated dipole moments for heliuminto a numerical solution of the wave equation for theharmonic field. These results are compared with mea-sured absolute energy yields of harmonics, providing agood test of the theoretical understanding of harmonicgeneration.

2. HARMONIC YIELD MODELDESCRIPTIONTo calculate the yield of the qth harmonic, we numericallyintegrate the wave equation for the harmonic field propa-gating through the gas target. The wave equation for afield oscillating at a frequency vq is

=2Aq 1nq

2vq2

c2Aq ­ 2

4pvq2

c2Pq , (1)

where Aq is the field strength of the oscillating qthharmonic, nq is the spatially and temporally varyingrefractive index of the media at vq, and Pq is thedipole moment induced by the laser field at a fre-quency vq. Equation (1) ignores the group-velocitydispersion of the harmonic pulse as well as the group-velocity walk-off of the harmonic pulse with the laserpulse. Both of these effects are negligible for pulses of100 fs or longer in a low-density (# 1019 atomsycm3) gasmedium.

We can introduce the slowly varying envelopes for theharmonic field and the polarization into Eq. (1). Theseare given by

1996 Optical Society of America

Ditmire et al. Vol. 13, No. 2 /February 1996 /J. Opt. Soc. Am. B 407

aqsx, td ­ Aqsx, tdexp

(2i

Z z

2`

fqk0sx, td 1 Dksx, tdgdz0

),

(2a)

pqsx, td ­ Pqsx, tdexp

"2i

Z z

2`

qk0sx, tddz0

#, (2b)

where k0 is the wave number of the fundamental laserfield and the phase mismatch of the harmonic with thelaser field is defined as Dk ; kq 2 qk0. Using the slowlyvarying envelope approximation in Eq. (1), we arrive atthe paraxial wave equation for the qth harmonic:

='2aq 1 2ikq

≠aq

≠z2 2kqDkaq 1 ikqNsabsaq

­ 24pvq

2

c2 pq . (3)

To derive Eq. (3) we have explicitly separated the realand the imaginary parts of the harmonic wave number,where kq is the real part of the harmonic wave number,sabs is the absorption cross section for harmonic photons,and N is the gas density. We assume that the phasemismatch is dominated by the presence of free electronsfrom ionization-induced plasma formation during the har-monic generation and ignore the small contribution of theneutral atoms to the phase mismatch.2 If the electrondensity ne is much lower than the critical density for boththe harmonic and the fundamental fields, this phase mis-match can be approximated as

Dke >e2nelq

c2me

0@1 2l0

2

lq2

1A . (4)

In deriving Eq. (3) we have ignored all the terms thatvary as ='kq; this is equivalent to saying that

≠nq

≠x­

≠nq

≠yø 0 (5)

(where nq is the refractive index of the qth harmonic).In doing this we have ignored any refraction of the har-monic field by a spatially varying refractive index arisingfrom the plasma formation. This is a good approxima-tion because short-wavelength radiation will be resistantto refraction by plasmas because of the large critical den-sity associated with soft-x-ray radiation.22 This refrac-tion, however, may have an effect on the spatial profile ofthe laser fundamental. These calculations, however, donot account for refraction of the laser by ionization. Al-though the refraction of the fundamental can have someeffects on the far-field profile of the harmonic,23 it hasless of an effect on the energy yields, as the majority ofthe harmonic radiation is produced in regions were thereis only a small amount of ionization [see Fig. 1(c)], andthe fundamental beam undergoes little spreading withinthe short length of the gas medium.24

We treat the focused fundamental laser beam as aGaussian spatial profile. Though our experimental re-sults were taken with a flattop profile focused into thegas medium, resulting in Lommel function profiles at thefocus,25 the use of a Gaussian is a reasonable approxima-tion when the confocal parameter is much longer than the

interaction region, a condition satisfied by all our experi-ments. Consequently, we take the laser intensity as

I sx, td ­ I01

1 1 4z2yk02w0

4exp

"2

2sx2 1 y2dw0

2s1 1 4z2yk02w0

4d

#3 exps24 ln 2t2ytFWHM

2d , (6)

where w0 is the 1ye2 radius of the laser at best focus. Thespatially and temporally varying harmonic polarization isthen given by

pqsx, td > 2Nsx, tdjdqfI sx, tdgjexp

"2iq

3 tan21s2zyk0w02d 2 iq

2k0sx2 1 y2dzk0

2w04 1 4z2

#, (7)

where jdqfI sx, tdgj is the intensity-dependent dipole mo-ment oscillating at a frequency q. For our calculationswe use values for jdqfI sx, tdgj that have been calculated byKrause et al. by numerical integration of the Schrodingerequation for helium irradiated by 526-nm laser light.18

We ignore the intensity dependence of the phase of thedipole, dqfI sx, tdg, which has been shown to be importantin affecting the far-field spatial profiles of harmonics inthe 400–100-A range.21 This is a good approximation forour calculations because we consider only weak focusing,in which the laser intensity varies little in the z directionover the length of the medium and therefore has very lit-tle effect on the phase matching.

Finally, we calculate the time-dependent density of theneutral atoms and the free electrons by simultaneouslysolving the rate equation

dN sx, tddt

­ 2WtifI sx, tdgN sx, td (8)

for the density of helium atoms. Wti is the tunnel ioniza-tion rate that we take to be given by the rate of Ammosovet al.26 We assume that there is no harmonic generationfrom the singly ionized helium.

The computer algorithm solves Eq. (3) for aq on a three-dimensional sx, y, zd grid for each time slice in the laserpulse. The grid is 128 3 128 3 100 points. Equation (3)is solved by the split operator method of Feit and Fleck.27

The incident laser pulse is assumed to be Gaussian with a600-fs full width at half-maximum, focused at the centerof the medium. We solve Eq. (3) for independent timeslices separated by 25 fs. The harmonic field is calcu-lated assuming an initial neutral atom medium that is800 mm long and uniform along the entire length. Theoutput harmonic energy Eq emerging from the medium isthen found from

Eq ­c

8p

Zjaqsx, tdj2dxdydt . (9)

All our calculations and measurements were conductedfor situations of weak focusing. This condition is fulfilledwhen the laser confocal parameter is much longer thanthe gas-jet interaction length. This is the ideal situa-tion for optimal harmonic conversion efficiency. It can beshown that the qth harmonic has an effective geometriccoherence length that is Lcoh ø pby2q, where b ­ k0w0

2 is

408 J. Opt. Soc. Am. B/Vol. 13, No. 2 /February 1996 Ditmire et al.

the focused laser confocal parameter. Thus when Lcoh islonger than the gas medium, geometric phase mismatchfrom the focused Gaussian beam is minimized and thephase mismatch is dominated by Dke in Eq. (3).2 Thoughthe geometric effect is accounted for in our solution ofEq. (3), the limit to harmonic conversion is found to stemfrom free-electron production by ionization. Our experi-ments and calculations were conducted for parameters inwhich Lcoh ranges from 1 to 10 mm, comparable with orlonger than our 0.8-mm gas-jet length.

An example of the harmonic field that was calculatedas it was emerging from the helium medium, which il-lustrates the importance of ionization in calculating theharmonic yield, is shown in Fig. 1. Here we show the in-tensity distribution of the 21st harmonic of 526-nm lightemerging from the gas medium calculated for two pointsin the laser pulse, 450 fs before the peak and right atthe peak for a focal spot diameter of 160 mm s1ye2d and afocused peak intensity of 3 3 1015 Wycm2. The gas den-sity was taken to be 4 3 1018 atomsycm3. Because thepeak intensity is approximately four times the ionizationsaturation intensity for the helium, the medium beginsto ionize at the peak of the focus at approximately 400 fsbefore the pulse peak. As the intensity increases laterin the pulse, the radial extent of the ionization increases.Consequently, the harmonics produced in the pulse be-fore the onset of ionization [Fig. 1(a)] exhibit a Gaussianprofile.

After ionization, however, the harmonics are producedin a small annulus limited on the inside by ionizationand falling off rapidly on the outside because of the highlynonlinear dependence of the polarization with laser inten-sity [Fig. 1(b)]. The size of this annulus expands later inthe pulse as the intensity at focus increases. Previously,L’Huillier et al.20 found that the harmonic field producedfrom a tightly focused laser ( i.e., when the confocal pa-rameter is comparable with the gas-jet interaction length)exhibited a ring structure. These were due to the geo-metric phase mismatch. The ring structure that we seein Fig. 1(b) is not due to this geometric phase interfer-ence, which is negligible because we are considering a casein which the laser focus is weak, but is instead due tothe free-electron production and neutral-atom depletionon axis from ionization. A slice through the harmonicprofile at the peak of the laser pulse is compared withthe radial dependence of the electron density at z ­ 0in Fig. 1(c). From this we can see that the majority ofthe harmonic radiation is produced in the annular spa-tial region just outside the region of complete neutral-atom depletion where there is some ionization and freeelectrons.

3. EXPERIMENT DESCRIPTIONThe experiments on harmonic-conversion yield were per-formed with a Nd:glass laser that produces 600-fs pulsesat 1053 nm with energies up to 8 J and pulses of itssecond harmonic at 526 nm with energies up to 4 J.28

The laser was focused into the plume of a pulsed, su-personic nozzle, gas jet. This jet produces localizedatomic densities from 1018 to 5 3 1018 atomsycm3 andexhibits a linear density dependence with gas-jet backingpressure, verified by backward stimulated-Raman scat-ter measurements.29 The interaction length through

the gas jet is 0.8 mm and is roughly flattopped in pro-file for our gas-jet backing pressures. Our measure-ments were conducted with an estimated atom density of

(a)

(b)

(c)Fig. 1. Calculated intensity of the 21st harmonic of 526-nmlight produced in helium emerging from the gas jet found at twopoints in the laser pulse: (a) 450 fs before the pulse peak, (b) atthe pulse peak for the focal spot diameter of 160 mm (1ye2) anda focused peak intensity of 3 3 1015 Eycm2. (c) A slice throughthe harmonic profile at the peak of the laser pulse compared withthe radial dependence of the electron density at z ­ 0.

Ditmire et al. Vol. 13, No. 2 /February 1996 /J. Opt. Soc. Am. B 409

4 3 1018 atomsycm3 6 30%. The shot-to-shot variationin atom density was less than ,1%. The harmonic radi-ation is sampled by an astigmatic-compensated, grazing-incidence, XUV spectrometer. A calibrated aluminumfoil filter prevents scattered laser light from reaching thedetector.

The harmonics are detected with an absolutely cali-brated x-ray CCD detector (Princeton Instruments).This detector uses a thermoelectrically cooled Tektronix1024B backilluminated CCD chip. The quantum effi-ciency of the chip (which is lowered in the spectral regionof interest by a thin layer of SiO2 on the surface of a chipapproximately 5 to 10 nm thick) was calibrated at theBrookhaven National Synchrotron Light Source.30 Thequantum efficiency ranges from 0.7 for wavelengths be-tween 20 and 10 nm and drops to below 0.2 at 30 nm.The response of this camera drops rapidly with wave-lengths longer than 31 nm. The correlation factor be-tween accumulated charge and detected counts on thecamera was measured when the CCD chip was exposedto single photon hits of Ka emission from tin at a photonenergy of 25 keV.

The throughput of the XUV spectrometer was mea-sured when signals of harmonics on the x-ray CCD pro-duced under identical conditions were compared with andwithout the spectrometer in the system. Data takenwith the spectrometer yielded the relative energy in eachharmonic, whereas data taken without the spectrometeryielded the total integrated harmonic yield per laser shot.Two calibrated aluminum filters (860-nm total thickness)were used to pass the harmonics and completely block allthe laser light for data taken without the spectrometer.Comparison of these shots with harmonics shorts takenwith the spectrometer at the same laser intensity yieldedthe spectrometer throughput of those harmonics between30 nm and the aluminum L-edge at 17 nm. This mea-surement was repeated for a range of laser intensities andfor three different laser focal configurations, fy25, fy50,and fy70 focusing.

4. COMPARISON OF YIELDCALCULATIONS WITH ENERGYMEASUREMENTSIn Fig. 2 we show energy yields of the 17th through the29th harmonics of 526-nm light produced in helium witha peak focused intensity of 3 3 1015 Wycm2. The har-monics of the 526-nm light were generated with a 160-mJflattop beam focused with an fy50 geometry to a spot sizeof 155 mm (610 mm; 1ye2 diameter). These yields repre-sent the average of five shots in a 610% peak intensity bincentered at 3 3 1015 Wycm2. The measured harmonicyield in the plateau varies between 6 and 8 nJyharmonic.The error bars result from uncertainties in the spectrome-ter throughput measurement and in the CCD chip quan-tum efficiency.

The calculated yield under these conditions is shown inFig. 2 for comparison. The calculations predict energyyields of between 15 and 20 nJ in the plateau, fallingto ,1 nJ for the 29th harmonic. The calculated yieldsfall to within a factor of 2–4 of the measured yields.This close agreement between calculation and measure-ment is quite remarkable, as the calculation is based on

a first-principle solution of the Schrodinger equation forthe single-atom response and the solution of the waveequation for the energy yield with no fitting parameters,confirming the validity of the assumptions made in thesingle-atom and propagation calculations. The locationof the cutoff is at the 27th harmonic, a wavelength of195 A (photon energy of 64 eV). This is also mirroredin the energy yield calculation.

The falloff in the harmonic energy above the 27th is inpart due to the calculated single-atom response of the he-lium because the 27th harmonic is near the single-atomcutoff for helium with a 526-nm drive.18 The dramaticdrop in energy yield is also due, in large part, to the in-creased importance of the free-electron phase mismatchfor the shorter-wavelength harmonics. This can be seenif we make some simple approximations to the harmonicyields and derive a simple scaling relation for the har-monic conversion. If we assume that the dipole momentof the qth harmonic varies with the incident laser fieldwith an effective nonlinear order p we can write

jdqj ­ xqa0p 1

s1 1 4z2yk02w0

4dp /2

3 exp

"2

psx2 1 y2dw0

2s1 1 4z2yk02w0

4d

#, (10)

where xq is the single-atom polarizability at the frequencyvq and a0 is the strength of the incident laser field.L’Huillier et al.2 showed that, in the weak focusing limit,when b .. l and the medium is uniform, the harmonicfield has the solution

aqs x d ­ 22ipkqNlxqa0

p

s1 1 2ipzykqw02d

3 exp

242psx2 1 y2d

w02s1 1 2ipzykqw0

2d

353

sinfsDk 1 2qyk0w02 2 2pykqw0

2dly2gsDk 1 2qyk0w0

2 2 2pykqw02dly2

. (11)

Because the free-electron phase mismatch is much largerthan the geometric phase mismatch in Eq. (11), we canignore the final two terms in the sinc function. Whenthis solution for the harmonic field is used in Eq. (9), the

Fig. 2. Calculated and measured energy yields of the 17ththrough the 29th harmonics of 526-nm light produced in he-lium with a peak focused intensity of 3 3 1015 Wycm2.

410 J. Opt. Soc. Am. B/Vol. 13, No. 2 /February 1996 Ditmire et al.

Fig. 3. Calculated and measured energy yield of the 21st har-monic of 526-nm light at a constant intensity of 3 3 1015 Wycm2

as a function of confocal parameter k0w02.

Fig. 4. Calculated and measured energy yield of the 21st har-monic of 526-nm light generated in helium as a function of peaklaser intensity for fy50 focusing.

Fig. 5. Calculated and measured energy yields of the 27th har-monic of 526-nm light generated in helium as a function of peaklaser intensity for fy50 focusing.

harmonic energy yield for a Gaussian-pulse shape with afull width at half-maximum pulse width of t becomes

Eq ­

24p5/2ckq2xq

2t

8p

ln 2p3/2

√8p

c

!p35N2l2w0

2I0q sin2sDkly2d

sDkly2d2,

(12)

where I0 is the peak laser intensity. If sl02ylq

2d .. 1,the free-electron phase mismatch Dke scales like ,1ylq.Equation (12) then implies that the yield will drop withshorter harmonic wavelengths because of the decrease ofsin2sDkly2dysDkly2d2.

On the basis of this analysis, in the weak focusingregime, Eq. (12) also suggests that the harmonic yieldwill increase linearly with the focal area pw0

2. Figure 3shows the energy yield of the 21st harmonic of 526-nmlight at a constant intensity of 3 3 1015 Wycm2 for threedifferent f-number configurations. The harmonics weregenerated with 75 mJ at fy25, 160 mJ at fy50, and260 mJ at fy70. The energy yield is plotted versus theconfocal parameter b ­ k0w0

2. The yield is roughly lin-ear with b, confirming the scaling predicted by Eq. (12).The calculated yield is also shown on the plot in Fig. 3for the same intensity with the confocal parameter rang-ing from 1 to 20 cm. Though the calculated yield issomewhat higher, it is close to the measured values, andits increase with increasing b follows the experimentaldata. The difference in the slope between the measuredand the calculated values may arise from errors in pre-dicting the single-atom dipole moment as well as the in-accuracies derived from modeling the actual focal sectionand pulse shape as Gaussian. As Eq. (12) shows, themagnitude of the slope will be sensitive to both of thesequantities. Nonetheless, the predicted yields are compa-rable with the measured values, and the linear scaling isreproduced. Also note that, although the energy yield in-creases with b, the conversion efficiency is constant withconfocal parameter for any given intensity. Because theharmonic energy yield increases linearly with pw0

2, acommensurate increase in laser energy is required formaintaining a given peak intensity as b increases.

We find that the best harmonic yields and conversionefficiency are achieved with intensities well above thesaturation intensity. Figure 4 shows the yields of the21st harmonic of 526-nm light generated in helium as afunction of peak laser intensity for fy50 focusing. Forthis focusing geometry we measured harmonic energiesnear 60 nJ at a peak intensity of 1.1 3 1016 Wycm2 witha drive energy of 590 mJ. This corresponds to a conver-sion efficiency of .1027 from the laser into the harmonicat 250 A. The calculated yield of the 21st harmonic isshown in Fig. 4 as a solid curve. The agreement betweenthe calculated energies and the measured energy is quiteremarkable. The calculated yield is within a factor of 3of the measured yield over nearly the entire measuredrange of peak intensities above the ionization saturation.

We also find good agreement with the shorter-wavelength harmonics in the cutoff of the harmonicsspectrum. The yields of the 27th harmonic (at 195 A)are shown in Fig. 5 for fy50 focusing. We measured har-monic energies of as much as 15 nJ at this wavelengthat an intensity of 1.1 3 1016 Wycm2. This correspondsto a conversion efficiency of 3 3 1028. The calculatedyield is shown on this plot as a solid curve, illustratingthe validity of our calculations for harmonics in the cutoffas well as the plateau.

5. CONCLUSIONIn conclusion, we have presented calculations and mea-surements of the energy yields of harmonics produced inhelium with a 526-nm 600-fs laser in the soft-x-ray spec-tral range of 31–17 nm. We have described a numeri-cal model that utilizes single-atom dipole moments calcu-lated in helium in a wave equation propagation model for

Ditmire et al. Vol. 13, No. 2 /February 1996 /J. Opt. Soc. Am. B 411

the harmonic field. We have shown that calculations ofthis type agree quite well with the measured harmonicenergy yields and that such a model is a useful tool inpredicting the harmonic yields that are attainable undera variety of conditions. Experimentally, we have mea-sured energy yields of as much as 60 nJ in the 31–23-nmspectral range corresponding to a conversion efficiencyof 1027. We have also measured harmonics with ener-gies in excess of 15 nJ for wavelengths below 20 nm. Inalmost all the cases for the harmonics studied experi-mentally, our calculated harmonic yields agree with themeasured energies to within better than a factor of 4.

ACKNOWLEDGMENTWe acknowledge many useful conversations with MikeFeit and Luiz DaSilva. This research was performedunder the auspices of U.S. Department of Energy contractW-7405-Eng-48.

REFERENCES1. A. McPherson, G. Gibson, H. Jara, U. Johann, T. S. Luk,

I. A. McIntyre, K. Boyer, and C. K. Rhodes, “Studies ofmultiphoton production of vacuum-ultraviolet radiation inthe rare gases,” J. Opt. Soc. Am. B 4, 595 (1987).

2. A. L’Huillier, L. Lompre, G. Mainfray, and C. Manus, “Highorder harmonic generation in rare gases,” in Atoms in In-tense Laser Fields, M. Gavrila, ed. (Academic Press, Boston,1992), p. 139.

3. “Free electron lasers and other advanced sources of light:scientific research opportunities,” National Research CouncilReport (National Academy, Washington, D.C., 1994).

4. A. L’Huillier and Ph. Balcou, “High-order harmonic genera-tion in rare gases with a 1-ps, 1053 nm laser,” Phys. Rev.Lett. 70, 774 (1993).

5. J. J. Macklin, J. D. Kmetec, and C. L. Gordon II, “High-orderharmonic generation using intense femtosecond pulses,”Phys. Rev. Lett. 70, 766 (1993).

6. P. Salieres, T. Ditmire, K. S. Budil, M. D. Perry, andA. L’Huillier, “Spatial profiles of high-order harmonicsgenerated by a femtosecond Cr:LiSAF laser,” J. Phys. B27, L217 (1994).

7. J. W. G. Tisch, R. A. Smith, J. E. Muffett, M. Ciarrocca,J. P. Marangos, and M. H. R. Hutchinson, “Angularly re-solved high-order harmonic generation in helium,” Phys.Rev. A 49, R28 (1994).

8. A. L’Huillier and Ph. Balcou, “Recent advances in strong-field harmonic generation,” Laser Phys. 3, 654 (1993).

9. J. Peatross and D. D. Meyerhofer, “Angular distribution ofhigh-order harmonics emitted from rare gases at low inten-sity,” Phys. Rev. A 51, R946 (1995).

10. C.-G. Wahlstrom, J. Larsson, A. Persson, T. Starczewski,S. Svanberg, P. Salieres, Ph. Balcou, and A. L’Huillier,“High-order harmonic generation in rare gases with an in-tense short-pulse laser,” Phys. Rev. A 48, 4709 (1993).

11. M. E. Faldon, M. H. R. Hutchinson, J. P. Marangos, J. E.Muffett, R. A. Smith, J. W. G. Tisch, and C.-G. Wahlstrom,“Studies of time-resolved harmonic generation in intenselaser fields in xenon,” J. Opt. Soc. Am. B 9, 2094 (1992).

12. T. Ditmire, K. S. Budil, J. K. Crane, H. Nguyen, M. D.Perry, P. Salieres, and A. L’Huiller, “Coherence properties

of high order harmonic radiation,” in Ultrafast PhenomenaIX, Vol. 60 of the Springer Series on Chemical Physics, P. F.Barbara, W. H. Knox, G. A. Mourou, and A. H. Zewail, eds.(Springer-Verlag, Berlin, 1994), pp. 228–232.

13. T. Ditmire, J. K. Crane, H. Nguyen, L. B. DaSilva, andM. D. Perry, “Energy-yield and conversion-efficiency mea-surements of high-order harmonic radiation,” Phys. Rev. A.51, R902 (1995).

14. Ph. Balcou, P. Salieres, K. S. Budil, T. Ditmire, and M. D.Perry, “High order harmonic generation in rare gases: anew source in photoionization spectroscopy,” Z. Phys. D. 34,107 (1995).

15. R. Haight and D. R. Peale, “Antibonding state on theGe(111): as surface: spectroscopy and dynamics,” Phys.Rev. Lett. 70, 3979 (1993).

16. R. Haight and D. R. Peale, “Tunable photoemission withharmonics of subpicosecond lasers,” Rev. Sci. Instrum. 65,1853 (1994).

17. P. B. Corkum, “Plasma perspective on strong-field multipho-ton ionization,” Phys. Rev. Lett. 71, 1994 (1993).

18. J. L. Krause, K. J. Schafer, and K. C. Kulander, “High-order harmonic generation from atoms and ions in the highintensity regime,” Phys. Rev. Lett. 68, 3535 (1992).

19. L. A. Lompre, A. L’Huillier, M. Ferray, P. Monot, G.Mainfray, and C. Manus, “High-order harmonic genera-tion in xenon: intensity and propagation effects,” J. Opt.Soc. Am. B 7, 754 (1990).

20. A. L’Huillier, Ph. Balcou, S. Candel, K. J. Schafer, andK. C. Kulander, “Calculations of high-order harmonicgeneration processes in xenon at 1064 nm,” Phys. Rev. A46, 2778 (1992).

21. P. Salieres, A. L’Huillier, and M. Lewenstein, “Coherencecontrol of high-order harmonics,” Phys. Rev. Lett. 74, 3776(1995).

22. R. Rankin, C. E. Capjack, N. H. Burnett, and P. B. Corkum,“Refraction effects associated with multiphoton ionizationand ultrashort-pulse laser propagation in plasma wave-guides,” Opt. Lett. 16, 835 (1991).

23. T. Ditmire, J. K. Crane, H. Nguyen, and M. D. Perry,“Plasma effects on high-order harmonic generation,” J. Non-linear Opt. Phys. Mater. 4, 737 (1995).

24. T. Auguste, P. Monot, L. A. Lompre, G. Mainfray, andC. Manus, “Defocusing effects of a picosecond terawatt laserpulse in an underdense plasma,” Opt. Commun. 89, 145(1992).

25. M. Born and E. Wolf, Principles of Optics: ElectromagneticTheory of Propagation, Interference, and Diffraction of Light(Pergamon, Oxford, 1989), p. 438.

26. M. V. Ammosov, N. B. Delone, and V. P. Krainov, “Tun-nel ionization of complex atoms and of atomic ions in an al-ternating electromagnetic field,” Sov. Phys. JETP 64, 1191(1986).

27. M. D. Feit and J. A. Fleck, “Solution of the Schrodingerequation by a spectral method,” J. Comp. Phys. 47, 412(1982).

28. F. G. Patterson, R. Gonzales, and M. D. Perry, “Compact10-TW, 800-fs Nd:glass laser,” Opt. Lett. 16, 1107 (1991).

29. M. D. Perry, C. Darrow, C. Coverdale, and J. K. Crane,“Measurement of the local electron density by means ofstimulating Ramen scattering in a laser-produced gas jetplasma,” Opt. Lett. 17, 523 (1992).

30. L. B. DaSilva, Laser Program, Lawrence Livermore NationalLaboratory, Livermore, Calif. 94550 (personal communica-tion, 1994).