Calculation and measurement of high-order harmonic energy yields in helium

Transcription

1 406 J. Opt. Soc. Am. B/Vol. 13, No. /February 1996 Ditmire et al. Calculation and measurement of high-order harmonic 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 Received March 14, 1995; revised manuscript received June 1, 1995 We present calculations of the energy yields of high-order harmonic radiation produced by 56-nm laser light focused into a helium gas jet. The intensity-dependent dipole moments of helium calculated by numerical integration of the Schrödinger equation have been used in a numerical solution of the wave equation to determine the energy yields of harmonics in the nm wavelength range. These calculations are compared with measured absolute energy yields and are shown to be in good agreement with experiment Optical Society of America 1. INTRODUCTION High-order harmonic generation 1, is one of the potential sources of bright extreme ultraviolet ( XUV) radiation for a host of future experiments in radiation-matter interaction physics. 3 When compared with other XUV sources, such as synchrotrons, free-electron lasers, and x-ray lasers, high-order harmonic-based sources exhibit a number of advantages. These include short pulse duration (30 fs to 100 ps), very high peak brightness (.10 3 photons mm mrad s 1 ), good spatial coherence, and broad spectral tunability. The relatively small size and high repetition rate of the lasers required for generating short-wavelength harmonics make this source an attractive alternative for experiments requiring coherent XUV radiation. In the past two years the emphasis has shifted from understanding the basic physics of high harmonic generation 4,5 to the characterization of the harmonics for potential use in applications. A number of groups have investigated the spatial coherence of the harmonics. 6 9 The spectral and the temporal characteristics have also 10 1 been investigated. We have recently reported measurements of the energy yields of the high-order harmonics under a variety of conditions. 13 The use of high-order harmonics in applications has grown recently as well. Balcou et al. have recently demonstrated the use of harmonics in the measurement of rare-gas photoionization studies with photon energies above 100 ev. 14 Haight and Peale have utilized harmonics in an experiment that takes advantage of the harmonics short-pulse character by making time-resolved measurements of surface states in semiconductors by means of photoelectron spectroscopy. 15,16 The many advantages offered by a high-order harmonic-based source ensure an increasing number of applications in the future. The theoretical understanding of harmonic generation has also greatly increased over the past few years. Single-atom calculations have been successful in predicting the presence of the harmonic plateau and the position of the short-wavelength cutoff of the harmonic plateau. 17,18 The effects of propagation of the highintensity pulse through the gas target have been studied as well, and the importance of phase matching has been 19 1 well established. With this understanding comes the potential for calculations of the total energy yields that are attainable in the soft-x-ray region with highorder harmonic generation. In this paper we present calculations of the energy yields of harmonics produced in helium by a 56-nm 600-fs laser pulse. We have constructed a model similar to that of L Huillier et al. 0 and applied it to calculations of absolute harmonic energy yields in the high focused intensity regime I. I sat. We have incorporated calculated dipole moments for helium into a numerical solution of the wave equation for the harmonic field. These results are compared with measured absolute energy yields of harmonics, providing a good test of the theoretical understanding of harmonic generation.. HARMONIC YIELD MODEL DESCRIPTION To calculate the yield of the qth harmonic, we numerically integrate the wave equation for the harmonic field propagating through the gas target. The wave equation for a field oscillating at a frequency v q is = A q 1 n q v q c A q 4pv q P c q, (1) where A q is the field strength of the oscillating qth harmonic, n q is the spatially and temporally varying refractive index of the media at v q, and P q is the dipole moment induced by the laser field at a frequency v q. Equation (1) ignores the group-velocity dispersion of the harmonic pulse as well as the groupvelocity walk-off of the harmonic pulse with the laser pulse. Both of these effects are negligible for pulses of 100 fs or longer in a low-density ( atoms cm 3 ) gas medium. We can introduce the slowly varying envelopes for the harmonic field and the polarization into Eq. (1). These are given by /96/ $ Optical Society of America

2 Ditmire et al. Vol. 13, No. /February 1996/J. Opt. Soc. Am. B 407 a q x, t A q x, t exp i p q x, t P q x, t exp i ( Z z ` Z z ` ) qk 0 x, t 1Dk x,t dz 0, (a) qk 0 x, t dz 0, (b) where k 0 is the wave number of the fundamental laser field and the phase mismatch of the harmonic with the laser field is defined as Dk k q qk 0. Using the slowly varying envelope approximation in Eq. (1), we arrive at the paraxial wave equation for the qth harmonic: = a q 1 ik q a q z k qdka q 1 ik q Ns abs a q 4pv q p c q. (3) To derive Eq. (3) we have explicitly separated the real and the imaginary parts of the harmonic wave number, where k q is the real part of the harmonic wave number, s abs is the absorption cross section for harmonic photons, and N is the gas density. We assume that the phase mismatch is dominated by the presence of free electrons from ionization-induced plasma formation during the harmonic generation and ignore the small contribution of the neutral atoms to the phase mismatch. If the electron density n e is much lower than the critical density for both the harmonic and the fundamental fields, this phase mismatch can be approximated as 0 1 Dk e e n e l q c m l 0 1 A. (4) l q In deriving Eq. (3) we have ignored all the terms that vary as = k q ; this is equivalent to saying that n q x n q 0 (5) y (where n q is the refractive index of the qth harmonic). In doing this we have ignored any refraction of the harmonic field by a spatially varying refractive index arising from the plasma formation. This is a good approximation because short-wavelength radiation will be resistant to refraction by plasmas because of the large critical density associated with soft-x-ray radiation. This refraction, however, may have an effect on the spatial profile of the laser fundamental. These calculations, however, do not account for refraction of the laser by ionization. Although the refraction of the fundamental can have some effects on the far-field profile of the harmonic, 3 it has less of an effect on the energy yields, as the majority of the harmonic radiation is produced in regions were there is only a small amount of ionization [see Fig. 1(c)], and the fundamental beam undergoes little spreading within the short length of the gas medium. 4 We treat the focused fundamental laser beam as a Gaussian spatial profile. Though our experimental results were taken with a flattop profile focused into the gas medium, resulting in Lommel function profiles at the focus, 5 the use of a Gaussian is a reasonable approximation when the confocal parameter is much longer than the interaction region, a condition satisfied by all our experiments. Consequently, we take the laser intensity as 1 I x, t I z k 0 w exp x 1 y 4 0 w z k 0 w 04 3 exp 4 ln t t FWHM, (6) where w 0 is the 1 e radius of the laser at best focus. The spatially and temporally varying harmonic polarization is then given by p q x, t N x,t jd q I x,t jexp iq 3 tan 1 z k 0 w 0 iq k 0 x 1 y z, (7) k 0 w z where jd q I x, t j is the intensity-dependent dipole moment oscillating at a frequency q. For our calculations we use values for jd q I x, t j that have been calculated by Krause et al. by numerical integration of the Schrödinger equation for helium irradiated by 56-nm laser light. 18 We ignore the intensity dependence of the phase of the dipole, d q I x, t, which has been shown to be important in affecting the far-field spatial profiles of harmonics in the Å range.1 This is a good approximation for our calculations because we consider only weak focusing, in which the laser intensity varies little in the z direction over the length of the medium and therefore has very little effect on the phase matching. Finally, we calculate the time-dependent density of the neutral atoms and the free electrons by simultaneously solving the rate equation dn x, t W ti I x, t N x, t (8) dt for the density of helium atoms. W ti is the tunnel ionization rate that we take to be given by the rate of Ammosov et al. 6 We assume that there is no harmonic generation from the singly ionized helium. The computer algorithm solves Eq. (3) for a q on a threedimensional x, y, z grid for each time slice in the laser pulse. The grid is points. Equation (3) is solved by the split operator method of Feit and Fleck. 7 The incident laser pulse is assumed to be Gaussian with a 600-fs full width at half-maximum, focused at the center of the medium. We solve Eq. (3) for independent time slices separated by 5 fs. The harmonic field is calculated assuming an initial neutral atom medium that is 800 mm long and uniform along the entire length. The output harmonic energy E q emerging from the medium is then found from E q c Z ja q x, t j dxdydt. (9) 8p All our calculations and measurements were conducted for situations of weak focusing. This condition is fulfilled when the laser confocal parameter is much longer than the gas-jet interaction length. This is the ideal situation for optimal harmonic conversion efficiency. It can be shown that the qth harmonic has an effective geometric coherence length that is L coh pb q, where b k 0 w 0 is

3 408 J. Opt. Soc. Am. B/Vol. 13, No. /February 1996 Ditmire et al. the focused laser confocal parameter. Thus when L coh is longer than the gas medium, geometric phase mismatch from the focused Gaussian beam is minimized and the phase mismatch is dominated by Dk e in Eq. (3). Though the geometric effect is accounted for in our solution of Eq. (3), the limit to harmonic conversion is found to stem from free-electron production by ionization. Our experiments and calculations were conducted for parameters in which L coh ranges from 1 to 10 mm, comparable with or longer than our 0.8-mm gas-jet length. An example of the harmonic field that was calculated as it was emerging from the helium medium, which illustrates the importance of ionization in calculating the harmonic yield, is shown in Fig. 1. Here we show the intensity distribution of the 1st harmonic of 56-nm light emerging from the gas medium calculated for two points in the laser pulse, 450 fs before the peak and right at the peak for a focal spot diameter of 160 mm 1 e and a focused peak intensity of W cm. The gas density was taken to be atoms cm 3. Because the peak intensity is approximately four times the ionization saturation intensity for the helium, the medium begins to ionize at the peak of the focus at approximately 400 fs before the pulse peak. As the intensity increases later in the pulse, the radial extent of the ionization increases. Consequently, the harmonics produced in the pulse before the onset of ionization [Fig. 1(a)] exhibit a Gaussian profile. After ionization, however, the harmonics are produced in a small annulus limited on the inside by ionization and falling off rapidly on the outside because of the highly nonlinear dependence of the polarization with laser intensity [Fig. 1(b)]. The size of this annulus expands later in the pulse as the intensity at focus increases. Previously, L Huillier et al. 0 found that the harmonic field produced from a tightly focused laser (i.e., when the confocal parameter is comparable with the gas-jet interaction length) exhibited a ring structure. These were due to the geometric phase mismatch. The ring structure that we see in Fig. 1(b) is not due to this geometric phase interference, which is negligible because we are considering a case in which the laser focus is weak, but is instead due to the free-electron production and neutral-atom depletion on axis from ionization. A slice through the harmonic profile at the peak of the laser pulse is compared with the radial dependence of the electron density at z 0 in Fig. 1(c). From this we can see that the majority of the harmonic radiation is produced in the annular spatial region just outside the region of complete neutralatom depletion where there is some ionization and free electrons. 3. EXPERIMENT DESCRIPTION The experiments on harmonic-conversion yield were performed with a Nd:glass laser that produces 600-fs pulses at 1053 nm with energies up to 8 J and pulses of its second harmonic at 56 nm with energies up to 4 J. 8 The laser was focused into the plume of a pulsed, supersonic nozzle, gas jet. This jet produces localized atomic densities from to atoms cm 3 and exhibits a linear density dependence with gas-jet backing pressure, verified by backward stimulated-raman scatter measurements. 9 The interaction length through the gas jet is 0.8 mm and is roughly flattopped in profile for our gas-jet backing pressures. Our measurements were conducted with an estimated atom density of (a) (b) (c) Fig. 1. Calculated intensity of the 1st harmonic of 56-nm light produced in helium emerging from the gas jet found at two points in the laser pulse: (a) 450 fs before the pulse peak, (b) at the pulse peak for the focal spot diameter of 160 mm (1 e ) and a focused peak intensity of E cm. (c) A slice through the harmonic profile at the peak of the laser pulse compared with the radial dependence of the electron density at z 0.

4 Ditmire et al. Vol. 13, No. /February 1996/J. Opt. Soc. Am. B atoms cm %. The shot-to-shot variation in atom density was less than 1%. The harmonic radiation is sampled by an astigmatic-compensated, grazingincidence, XUV spectrometer. A calibrated aluminum foil filter prevents scattered laser light from reaching the detector. The harmonics are detected with an absolutely calibrated x-ray CCD detector ( Princeton Instruments). This detector uses a thermoelectrically cooled Tektronix 104B backilluminated CCD chip. The quantum efficiency of the chip (which is lowered in the spectral region of interest by a thin layer of SiO on the surface of a chip approximately 5 to 10 nm thick) was calibrated at the Brookhaven National Synchrotron Light Source. 30 The quantum efficiency ranges from 0.7 for wavelengths between 0 and 10 nm and drops to below 0. at 30 nm. The response of this camera drops rapidly with wavelengths longer than 31 nm. The correlation factor between accumulated charge and detected counts on the camera was measured when the CCD chip was exposed to single photon hits of Ka emission from tin at a photon energy of 5 kev. The throughput of the XUV spectrometer was measured when signals of harmonics on the x-ray CCD produced under identical conditions were compared with and without the spectrometer in the system. Data taken with the spectrometer yielded the relative energy in each harmonic, whereas data taken without the spectrometer yielded 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 all the laser light for data taken without the spectrometer. Comparison of these shots with harmonics shorts taken with the spectrometer at the same laser intensity yielded the spectrometer throughput of those harmonics between 30 nm and the aluminum L-edge at 17 nm. This measurement was repeated for a range of laser intensities and for three different laser focal configurations, f 5, f 50, and f 70 focusing. 4. COMPARISON OF YIELD CALCULATIONS WITH ENERGY MEASUREMENTS In Fig. we show energy yields of the 17th through the 9th harmonics of 56-nm light produced in helium with a peak focused intensity of W cm. The harmonics of the 56-nm light were generated with a 160-mJ flattop beam focused with an f 50 geometry to a spot size of 155 mm (610 mm; 1 e diameter). These yields represent the average of five shots in a 610% peak intensity bin centered at W cm. The measured harmonic yield in the plateau varies between 6 and 8 nj harmonic. The error bars result from uncertainties in the spectrometer throughput measurement and in the CCD chip quantum efficiency. The calculated yield under these conditions is shown in Fig. for comparison. The calculations predict energy yields of between 15 and 0 nj in the plateau, falling to 1 nj for the 9th harmonic. The calculated yields fall to within a factor of 4 of the measured yields. This close agreement between calculation and measurement is quite remarkable, as the calculation is based on a first-principle solution of the Schrödinger equation for the single-atom response and the solution of the wave equation for the energy yield with no fitting parameters, confirming the validity of the assumptions made in the single-atom and propagation calculations. The location of the cutoff is at the 7th harmonic, a wavelength of 195 Å (photon energy of 64 ev). This is also mirrored in the energy yield calculation. The falloff in the harmonic energy above the 7th is in part due to the calculated single-atom response of the helium because the 7th harmonic is near the single-atom cutoff for helium with a 56-nm drive. 18 The dramatic drop in energy yield is also due, in large part, to the increased importance of the free-electron phase mismatch for the shorter-wavelength harmonics. This can be seen if we make some simple approximations to the harmonic yields and derive a simple scaling relation for the harmonic conversion. If we assume that the dipole moment of the qth harmonic varies with the incident laser field with an effective nonlinear order p we can write p 1 jd q j x q a z k 0 w 04 p/ p x 1 y 3 exp, (10) w z k 0 w 04 where x q is the single-atom polarizability at the frequency v q and a 0 is the strength of the incident laser field. L Huillier et al. showed that, in the weak focusing limit, when b.. l and the medium is uniform, the harmonic field has the solution p x q a 0 a q x ipk q Nl 1 1 ipz k q w exp4 p x 1 y 5 w ipz k q w 0 3 sin Dk 1 q k 0w 0 p k q w 0 l. (11) Dk 1 q k 0 w 0 p k q w 0 l Because the free-electron phase mismatch is much larger than the geometric phase mismatch in Eq. (11), we can ignore the final two terms in the sinc function. When this solution for the harmonic field is used in Eq. (9), the Fig.. Calculated and measured energy yields of the 17th through the 9th harmonics of 56-nm light produced in helium with a peak focused intensity of W cm.

5 410 J. Opt. Soc. Am. B/Vol. 13, No. /February 1996 Ditmire et al. Fig. 3. Calculated and measured energy yield of the 1st harmonic of 56-nm light at a constant intensity of W cm as a function of confocal parameter k 0 w 0. Fig. 4. Calculated and measured energy yield of the 1st harmonic of 56-nm light generated in helium as a function of peak laser intensity for f 50 focusing. Fig. 5. Calculated and measured energy yields of the 7th harmonic of 56-nm light generated in helium as a function of peak laser intensity for f 50 focusing. harmonic energy yield for a Gaussian-pulse shape with a full width at half-maximum pulse width of t becomes! 3 E q 4 p5/ ck q x p q t 8 p 8p 5 N l w q 0 I sin Dkl 0, ln p 3/ c Dkl (1) where I 0 is the peak laser intensity. If l 0 l q.. 1, the free-electron phase mismatch Dk e scales like 1 l q. Equation (1) then implies that the yield will drop with shorter harmonic wavelengths because of the decrease of sin Dkl Dkl. On the basis of this analysis, in the weak focusing regime, Eq. (1) also suggests that the harmonic yield will increase linearly with the focal area pw 0. Figure 3 shows the energy yield of the 1st harmonic of 56-nm light at a constant intensity of W cm for three different f-number configurations. The harmonics were generated with 75 mj at f 5, 160 mj at f 50, and 60 mj at f 70. The energy yield is plotted versus the confocal parameter b k 0 w 0. The yield is roughly linear with b, confirming the scaling predicted by Eq. (1). The calculated yield is also shown on the plot in Fig. 3 for the same intensity with the confocal parameter ranging from 1 to 0 cm. Though the calculated yield is somewhat higher, it is close to the measured values, and its increase with increasing b follows the experimental data. The difference in the slope between the measured and the calculated values may arise from errors in predicting the single-atom dipole moment as well as the inaccuracies derived from modeling the actual focal section and pulse shape as Gaussian. As Eq. (1) shows, the magnitude of the slope will be sensitive to both of these quantities. Nonetheless, the predicted yields are comparable with the measured values, and the linear scaling is reproduced. Also note that, although the energy yield increases with b, the conversion efficiency is constant with confocal parameter for any given intensity. Because the harmonic energy yield increases linearly with pw 0, a commensurate increase in laser energy is required for maintaining a given peak intensity as b increases. We find that the best harmonic yields and conversion efficiency are achieved with intensities well above the saturation intensity. Figure 4 shows the yields of the 1st harmonic of 56-nm light generated in helium as a function of peak laser intensity for f 50 focusing. For this focusing geometry we measured harmonic energies near 60 nj at a peak intensity of W cm with a drive energy of 590 mj. This corresponds to a conversion efficiency of.10 7 from the laser into the harmonic at 50 Å. The calculated yield of the 1st harmonic is shown in Fig. 4 as a solid curve. The agreement between the calculated energies and the measured energy is quite remarkable. The calculated yield is within a factor of 3 of the measured yield over nearly the entire measured range of peak intensities above the ionization saturation. We also find good agreement with the shorterwavelength harmonics in the cutoff of the harmonics spectrum. The yields of the 7th harmonic (at 195 Å) are shown in Fig. 5 for f 50 focusing. We measured harmonic energies of as much as 15 nj at this wavelength at an intensity of W cm. This corresponds to a conversion efficiency of The calculated yield is shown on this plot as a solid curve, illustrating the validity of our calculations for harmonics in the cutoff as well as the plateau. 5. CONCLUSION In conclusion, we have presented calculations and measurements of the energy yields of harmonics produced in helium with a 56-nm 600-fs laser in the soft-x-ray spectral range of nm. We have described a numerical model that utilizes single-atom dipole moments calculated in helium in a wave equation propagation model for

6 Ditmire et al. Vol. 13, No. /February 1996/J. Opt. Soc. Am. B 411 the harmonic field. We have shown that calculations of this type agree quite well with the measured harmonic energy yields and that such a model is a useful tool in predicting the harmonic yields that are attainable under a variety of conditions. Experimentally, we have measured energy yields of as much as 60 nj in the 31 3-nm spectral range corresponding to a conversion efficiency of We have also measured harmonics with energies in excess of 15 nj for wavelengths below 0 nm. In almost all the cases for the harmonics studied experimentally, our calculated harmonic yields agree with the measured energies to within better than a factor of 4. ACKNOWLEDGMENT We acknowledge many useful conversations with Mike Feit and Luiz DaSilva. This research was performed under the auspices of U.S. Department of Energy contract W-7405-Eng-48. REFERENCES 1. A. McPherson, G. Gibson, H. Jara, U. Johann, T. S. Luk, I. A. McIntyre, K. Boyer, and C. K. Rhodes, Studies of multiphoton production of vacuum-ultraviolet radiation in the rare gases, J. Opt. Soc. Am. B 4, 595 (1987).. A. L Huillier, L. Lompré, G. Mainfray, and C. Manus, High order harmonic generation in rare gases, in Atoms in Intense Laser Fields, M. Gavrila, ed. (Academic Press, Boston, 199), p Free electron lasers and other advanced sources of light: scientific research opportunities, National Research Council Report ( National Academy, Washington, D.C., 1994). 4. A. L Huillier and Ph. Balcou, High-order harmonic generation 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-order harmonic generation using intense femtosecond pulses, Phys. Rev. Lett. 70, 766 (1993). 6. P. Salières, T. Ditmire, K. S. Budil, M. D. Perry, and A. L Huillier, Spatial profiles of high-order harmonics generated by a femtosecond Cr:LiSAF laser, J. Phys. B 7, L17 (1994). 7. J. W. G. Tisch, R. A. Smith, J. E. Muffett, M. Ciarrocca, J. P. Marangos, and M. H. R. Hutchinson, Angularly resolved high-order harmonic generation in helium, Phys. Rev. A 49, R8 (1994). 8. A. L Huillier and Ph. Balcou, Recent advances in strongfield harmonic generation, Laser Phys. 3, 654 (1993). 9. J. Peatross and D. D. Meyerhofer, Angular distribution of high-order harmonics emitted from rare gases at low intensity, Phys. Rev. A 51, R946 (1995). 10. C.-G. Wahlström, J. Larsson, A. Persson, T. Starczewski, S. Svanberg, P. Salières, Ph. Balcou, and A. L Huillier, High-order harmonic generation in rare gases with an intense 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. Wahlström, Studies of time-resolved harmonic generation in intense laser fields in xenon, J. Opt. Soc. Am. B 9, 094 (199). 1. T. Ditmire, K. S. Budil, J. K. Crane, H. Nguyen, M. D. Perry, P. Salières, and A. L Huiller, Coherence properties of high order harmonic radiation, in Ultrafast Phenomena IX, 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 T. Ditmire, J. K. Crane, H. Nguyen, L. B. DaSilva, and M. D. Perry, Energy-yield and conversion-efficiency measurements of high-order harmonic radiation, Phys. Rev. A. 51, R90 (1995). 14. Ph. Balcou, P. Salières, K. S. Budil, T. Ditmire, and M. D. Perry, High order harmonic generation in rare gases: a new source in photoionization spectroscopy, Z. Phys. D. 34, 107 (1995). 15. R. Haight and D. R. Peale, Antibonding state on the Ge(111): as surface: spectroscopy and dynamics, Phys. Rev. Lett. 70, 3979 (1993). 16. R. Haight and D. R. Peale, Tunable photoemission with harmonics of subpicosecond lasers, Rev. Sci. Instrum. 65, 1853 (1994). 17. P. B. Corkum, Plasma perspective on strong-field multiphoton ionization, Phys. Rev. Lett. 71, 1994 (1993). 18. J. L. Krause, K. J. Schafer, and K. C. Kulander, Highorder harmonic generation from atoms and ions in the high intensity regime, Phys. Rev. Lett. 68, 3535 (199). 19. L. A. Lompré, A. L Huillier, M. Ferray, P. Monot, G. Mainfray, and C. Manus, High-order harmonic generation in xenon: intensity and propagation effects, J. Opt. Soc. Am. B 7, 754 (1990). 0. A. L Huillier, Ph. Balcou, S. Candel, K. J. Schafer, and K. C. Kulander, Calculations of high-order harmonic generation processes in xenon at 1064 nm, Phys. Rev. A 46, 778 (199). 1. P. Salières, A. L Huillier, and M. Lewenstein, Coherence control of high-order harmonics, Phys. Rev. Lett. 74, 3776 (1995).. R. Rankin, C. E. Capjack, N. H. Burnett, and P. B. Corkum, Refraction effects associated with multiphoton ionization and ultrashort-pulse laser propagation in plasma waveguides, Opt. Lett. 16, 835 (1991). 3. T. Ditmire, J. K. Crane, H. Nguyen, and M. D. Perry, Plasma effects on high-order harmonic generation, J. Nonlinear Opt. Phys. Mater. 4, 737 (1995). 4. T. Auguste, P. Monot, L. A. Lompre, G. Mainfray, and C. Manus, Defocusing effects of a picosecond terawatt laser pulse in an underdense plasma, Opt. Commun. 89, 145 (199). 5. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light ( Pergamon, Oxford, 1989), p M. V. Ammosov, N. B. Delone, and V. P. Krainov, Tunnel ionization of complex atoms and of atomic ions in an alternating electromagnetic field, Sov. Phys. JETP 64, 1191 (1986). 7. M. D. Feit and J. A. Fleck, Solution of the Schrödinger equation by a spectral method, J. Comp. Phys. 47, 41 (198). 8. F. G. Patterson, R. Gonzales, and M. D. Perry, Compact 10-TW, 800-fs Nd:glass laser, Opt. Lett. 16, 1107 (1991). 9. M. D. Perry, C. Darrow, C. Coverdale, and J. K. Crane, Measurement of the local electron density by means of stimulating Ramen scattering in a laser-produced gas jet plasma, Opt. Lett. 17, 53 (199). 30. L. B. DaSilva, Laser Program, Lawrence Livermore National Laboratory, Livermore, Calif ( personal communication, 1994).