Phase distortions in sum- and difference-frequency mixing in crystals

A. V. Smith and M. S. Bowers Vol. 12, No. 1/January 1995/J. Opt. So. Am. B 49 Phase distortions in sum- and differene-frequeny mixing in rystals Arlee V. Smith Lasers, Optis, & Remote Sensing Department, Sandia National Laboratories, Albuquerque, New Mexio 87185 0601 Mark S. Bowers Aulight Corporation, Suite 100, 40 Lake Bellevue, Bellevue, Washington 98005 Reeived May 19, 1994; aepted July 17, 1994; revised manusript reeived August 25, 1994 We show that if two waves are inident on a quadratially nonlinear rystal, with the third wave generated entirely within the rystal, a phase-veloity mismath Dk fi 0 leads to intensity-dependent phase shifts of the generated wave only if there is walk-off, linear absorption, or signifiant diffration of at least one of the waves as well as signifiant energy exhange among the waves. The result is frequeny broadening and wave-front distortion of the generated wave. Although the indued phase distortions are usually quite small, they may be signifiant in appliations that require high spetral resolution or pointing auray. 1. INTRODUCTION In the limit of low mixing effiieny, three-wave mixing in a quadratially nonlinear medium has been desribed analytially. For lowest-order Gaussian beam profiles, Boyd and Kleinman 1 presented suh a treatment, inluding walk-off and diffration. In this weak mixing limit the phase profiles of the output waves are independent of the beam intensities. When energy exhange beomes signifiant we might expet the output phases to depend on the strength of the mixing, whih is determined in part by the input intensities. In this ase analytial solutions for the full problem with walk-off, diffration, and depletion are not available. However, reent experiments and analyses of seond-harmoni generation have shown 2 5 that, when energy exhange is signifiant, a phase-veloity mismath Dk fi 0 among the interating waves does indeed lead to intensitydependent phase shifts of the fundamental wave. In some respets these phase shifts mimi those indued by antensity-dependent refrative index, ausing selffousing or self-defousing. 3 This raises the question whether there is a similar intensity-dependent shift in the phase of the seond-harmoni wave that ould lead to distortions of its wave front and, for pulsed light, to frequeny hirps. Suh effets ould ause line-shape distortions in high-resolution spetra under some onditions. They ould also ause time-dependent steering of the seond-harmoni beam. Here we onsider the effets of phase-veloity mismath in three-wave sum- and differene-frequeny mixing in nonlinear rystals for pulse durations of a few nanoseonds. We onsider only ases of moderate nonlinear drive suh as might be enountered when one is striving for effiient frequeny onversion with good output beam quality. We show that, when diffration, walk-off, absorption, and group-veloity mismath are insignifiant, and only two waves are inident upon the rystal, the generated third wave will not aquire an intensity-dependent phase shift, apart from possible 180- deg phase reversals. All the phase distortiontrodued by phase-veloity mismath will show up in the two waves that had nonzero input intensity. If diffration, walk-off, or absorption beomes important, or if all three waves are inident upon the rystal, this is no longer true. The output phases of all three waves will vary with the input intensities. Combined with spatial and temporal intensity variations of the input beams, this intensity dependene produes frequeny shifts and wave-front distortions for all three waves. First we will onsider plane waves and the influene of linear absorption, input amplitudes, and input phases on the output phases. Then, to illustrate the phase distortions introdued by walk-off, we will present results from a numerial model of nonlinear mixing that inludes spatial and temporal beam profiles, birefringene, linear absorption, diffration, and phase-veloity mismath. The model also permits signifiant energy exhange among the three waves. 2. EXACT PLANE-WAVE SOLUTIONS The equations that desribe the nonlinear interation for plane waves in SI units are 6 d s dz i d eff d i dz id eff d p dz id eff v s n s p i exp idkz 2as s, p i exp idkz 2ai i, n p s i exp 2iDkz 2a p p, (1) where the eletri field E v at frequeny s given by E v 1 /2 v exp 2i vt 2 kz 1 v exp i vt 2 kz, the phase veloity mismath Dk is defined by (2) 0740-3224/95/010049-09$06.00 1995 Optial Soiety of Ameria

50 J. Opt. So. Am. B/Vol. 12, No. 1/January 1995 A. V. Smith and M. S. Bowers and Dk k p 2 k s 2 k i, (3) v s 1. (4) The subsripts p, s, and i refer to the pump, signal, and idler waves, respetively, as is ustomary. The oeffiients d eff and a are the effetive nonlinear mixing oeffiient and the linear absorption oeffiient, respetively. Following the method of solution given by Armstrong et al., 7 we write the fields as where the e s are real valued. beome n e n exp iq n, (5) de s dz 2e d eff pe i de i dz 2e d eff pe s de p dz 1e d eff se i dq s dz d eff dq i dz d eff dq p dz d eff where u is defined by v s n s n p The mixing equations v s n s sin u2a s e s, sin u2a i e i, n p sin u2a p e p, e p e i e s os u, e p e s e i os u, e s e i e p os u, (6) u q p 2q i 2q s 1Dkz. (7) If the linear absorptions are all zero, the solution for u given by Armstrong et al. 17 is G2 Dkn! 2 pe p 2d eff os u, (8) e s e i e p where G is antegration onstant. It an be shown that, if e i z 0 and a i a s 1a p, os u Dk 2d eff is the solution for u. a p a i 1a s, os u 2 Dk 2d eff e i e s e p (9) Similarly, if e p z 0 0 and n p e p e s e i (10) is the solution. Thus, if the idler starts with zero intensity, from Eqs. (6) the equation for the idler phase beomes dq i dz 1 Dk 2. (11) This means that the phase of the idler wave is shifted relative to that of a solo idler wave by 1Dkz 2 but is independent of the intensities of the three interating waves. We should point out that, although the idler phase is orretly given by Eq. (11), the amplitude e i an hange sign if the mixing is strong enough to deplete the idler wave totally. When this ours it results in an apparent abrupt 180 ± phase shift as the amplitude e i passes through zero. Beause Eqs. (6) are symmetri in signal and idler, our disussion for the idler applies equally to the signal. If the pump-wave intensity starts from zero, the equation for the pump phase beomes dq p dz Dk 2, (12) 2 and the pump phase is independent of the intensities of the three waves. If the linear losses do not satisfy a i a s 1a p in the ase where e i z 0 0, the phase of the idler wave will vary with intensity. Similarly, if the ondition a p a i 1a s is violated for e p z 0 0 the pump phase will be intensity dependent. In ontrast to the intensity independene of the output phase that we just demonstrated for a wave that starts with zero intensity, if Dk is nonzero and the wave starts with nonzero intensity its input phase annot be adjusted to make its output phase independent of the input intensities. We show this for the idler wave by ombining Eq. (8) with the expression for dq i dz in Eqs. (6) to get ( dq i dz d eff e e 2 i 0 e s 0 e p 0 os u 0 i ) 1 Dkn p 2d eff e p 2 0 2e p 2 Applying the Manley Rowe relation. (13) e i 2 2e i 2 0 n p e p 2 0 2e p 2 (14) yields, for Eq. (13), dq i dz d eff e e 2 s 0 e i 0 e p 0 os u 0 i # 2 Dk 2d eff e i 2 0 1 Dk 2. (15) The idler output phase will be independent of the idler intensity only if the first term on the right-hand side in Eq. (15) is also independent of the idler intensity. This requires that the quantity in brakets be zero, i.e., that os u 0 Dk 2d eff e i 0. (16) e s 0 e p 0 For any set of input intensities the idler input phase ould be adjusted to meet this ondition. However, this input phase would depend on the input intensities, violating the goal of intensity independene. Thus there is no single hoie of input idler phase that allows the output idler phase to be intensity independent. If all three waves have nonzero inident intensity, all three output phases vary with input intensity. To summarize, we have shown that, for plane waves with no linear absorption, if one wave enters the rystal

A. V. Smith and M. S. Bowers Vol. 12, No. 1/January 1995/J. Opt. So. Am. B 51 with zero intensity its output phase will be independent of the intensities of the input waves and will depend only on DkL, where L is the rystal length. The output phases of the other two waves will be intensity dependent if Dk fi 0. If there is linear absorption the phase of the wave that started with zero intensity will also beome intensity dependent for Dk fi 0, unless a speial ondition of the absorption oeffiients is met a i a s 1a p or a p a i 1a s. If the third wave enters the rystal with nonzero intensity, its input phase annot be adjusted to make its output phase intensity independent if Dk fi 0. Clearly, when Dk fi 0, the intensity independene of the output phase of one wave is satisfied only in the speial irumstane that the wave starts with zero intensity. It is also neessary that the balane of amplitudes and phases of the three interating waves as they progress through the rystal not be altered from that of plane waves interating without loss (or meeting the speial loss riterion). These results were derived for plane waves. In the remainder of this paper we onsider waves that are spatially and temporally nonuniform. In that ase, effets that upset the speial balane required for antensity-independent output phase inlude different rates of diffrative spreading for the three waves, group-veloity mismath among the waves, and walk-off among the three waves. If any of these effets is strong enough that the plane-wave approximatios invalid, and if Dk fi 0, we expet the phases of all three waves at the rystal output fae to depend on the input intensities. The phases will vary in spae and time, leading to frequeny hirps and wavefront distortions for all three waves. Furthermore, even if Dk is nominally zero, we expet walk-off to lead to intensity-dependent output phases for all three waves if the beams have diameters small enough that diffrative phase shifts are signifiant. The explanatios that the diffrative phase slippage mimis nonzero Dk. Alternatively, one an argue that small-diameter beams onsist of a sum of plane waves with a range of transverse k-vetor omponents, so Dk is not truly zero for nonlinear interations among many ontributions of the plane-wave omponents. 3. NUMERICAL MODELING WITH NONUNIFORM SPATIAL AND TEMPORAL PROFILES Our disussion of output phases for nonuniform beams was based on plane-wave interations and was by neessity qualitative in nature beause analyti solutions of the mixing equations that inlude all the effets important for nonuniform beams are not available. To ahieve quantitative results we must resort to numerial modeling when we inlude the effets of nonzero Dk in the presene of birefringene and large energy exhange among the three mixing waves. We have developed a time-dependent model of threewave mixing that numerially integrates the three wave equations with the inlusion of walk-off and diffration in the paraxial approximation. The model desribes the stepwise time evolution of eah beam on a twodimensional spatial grid of intensity and phase. We assume Gaussian spatial and temporal input profiles for the three beams. The integration of the three wave equations through the nonlinear rystal is performed by use of Fourier-transform tehniques, as for the method desribed by Dreger and MIver. 8 When nonuniform transverse profiles are onsidered and the transverse derivatives are kept in the wave equations in order to aount for walk-off and diffration, the equations that replae Eqs. (1) take the form j x, y, z, t i 2 j x, y, z, t 1 2 j x, # y, z, t z 2k j y 2 x 2 2 tan r j x, y, z, t 1 P j x, y, z, t x 2a j j x,y,z,t, (17) where j is the frequeny index and r is the walk-off angle in the x diretion. For ordinary or y-polarized light, r is zero. For extraordinary or x-polarized light, r is the walk-off angle appropriate for the rystal orientation of interest. We have ignored the small anisotropy in the diffrative term for extraordinary waves propagating in a birefringent rystal. 8 P j is the polarization term at frequeny v j and is given by P s x, y, z, t i d eff P i x, y, z, t i d eff v s n s p x, y, z, t 3 i x,y,z,t exp idkz, p x, y, z, t 3 s x,y,z,t exp idkz, P p x, y, z, t i d eff i x, y, z, t n p 3 s x,y,z,t exp 2iDkz. (18) Fourier transforming the eletri fields and polarizations in the transverse dimension, using j x, y, z, t P j x, y, z, t j s x, s y, z, t 3 exp i2p s x x 1 s y y ds x ds y, P j s x, s y, z, t 3 exp i2p s x x 1 s y y ds x ds y, (19) and substituting these definitions of j x, y, z, t and P j x, y, z, t into Eq. (17), we arrive at the following equation for the propagation of the individual spatialfrequeny omponent waves: j s x, s y, z, t z 2 i 2p 2 k j s x 2 1 s y 2 1 2ps y tan r 3 j s x,s y,z,t 1P j s x,s y,z,t. (20) This proedure results in three oupled first-order ordinary differential equations for the hange in eah spatial-frequeny omponent of the fields as they propagate through the rystal. The equations are oupled through the nonlinear interation term P j s x, s y, z, t. #

52 J. Opt. So. Am. B/Vol. 12, No. 1/January 1995 A. V. Smith and M. S. Bowers Case 1: Nonzero Phase Mismath with Birefringene The first ase is one that has been experimentally studied by Gangopadhyay et al. 10 They frequeny doubled 3.4- mj 7-ns (FWHM) pulses of 645-nm light in a 3-m-long KDP rystal. For this Type I doubling, the 645-nm light had ordinary polarization. The 322.5-nm light had extraordinary polarization and a walk-off, or birefringent, angle of 28 mrad. The input beam diameter (FWHM) was 0.25 mm. For this proess, d eff 0.32 pm V. They measured seond-harmoni onversion effiieny and phase shifts for Dk 0 and Dk 0.15 mm 21. Beause the input beam diameter is quite large, diffration is insignifiant for this ase (the Rayleigh range 11 in the rystal is 33 m). We have verified this by performing alulations with and without diffration and obtaining idential results. We also neglet group-veloity mismath as inonsequential. Figure 1 shows our alulated input fundamental [Fig. 1(a)] and output seondharmoni [Fig. 1(b)] fluene profiles for Dk 0.15 mm 21. Figure 2 displays the time development of the phase of the seond-harmoni light at three spatial grid points, again for Dk 0.15 mm 21. At these three loations the phase of the seond-harmoni wave learly varies with the input intensity of the fundamental waves. If Dk is zero or if the walk-off is artifiially set to zero we find that the phase of the seond-harmoni light is onstant in time and independent of the input fundamental intensity. These observations are onsistent with our onlusions drawn from examining the plane-wave solutions above. To haraterize the wave-front distortions at anstant of time we alulate the tilt, urvature, and residual distortion of the output waves. In our notation, x is the walk-off diretion and y is perpendiular to it. The tilt angle in the walk-off diretion, b x, is the first moment in the spatial-frequeny domain, defined by Fig. 1. Example fluene profiles for (a) the input fundamental wave and (b) the output seond-harmoni wave. The seond-harmoni wave walks off in the 1x diretion. Dk 0.15 mm 21. We model the mixing proess for eah time step by propagating half of a z step, using a Runge Kutte algorithm to numerially integrate the oupled ordinary differential equations. We then apply a fast-fouriertransform algorithm to transform the resulting spatialfrequeny fields j s x, s y, z, t into fields j x, y, z, t in x y spae. We insert these fields into Eqs. (18) to find P j x, y, z, t. We then apply the fast-fourier transform algorithm again to obtain the P j s x, s yz, z, t s, whih are used in Eq. (20) to propagate the seond half of the z step. The x y spatial grid is typially 32 3 32, and the integration of a single time slie through the rystal is performed in approximately 32 steps. The number of time slies is typially approximately 75. Run time on a Pentium PC is of the order of 1000 s. As desribed previously, 9 we have rigorously validated the model by omparison with experiments. We will fous our disussion on two examples. In the first, we model a situation whih birefringene is ombined with nonzero Dk. In the seond, birefringene is ombined with signifiant diffration but with Dk 0. b x t Z ` 2l Z ` s x j s x, s y, t j 2 ds x ds y j s x, s y, t j 2 ds x ds y, (21) where s x and s y are the transverse omponents of the spatial frequeny and Fig. 2. Phases of the output seond-harmoni optial field for Dk 0.15 mm 21 at three positions on the alulational grid. The y positions are zero, and the x positions are as labeled.

A. V. Smith and M. S. Bowers Vol. 12, No. 1/January 1995/J. Opt. So. Am. B 53 s x, s y, t x, y, t exp 2i2p s x x 1 s y y dxdy. (22) The entroid of a beam s intensity propagates at the tilt angle b x. The ylindrial urvature of the wave fronts in the plane of walk-off and perpendiular to it is haraterized by use of methods similar to those desribed by Siegman. 12 The number of waves of urvature is where x t s x 2 t lr x t, (23) # s 2 ox t R x t Z x t 1 1. l 2 s 2 (24) sx t Z x2 t The z positions of the x-dimension beam waist, Z x t, and the waist size, s ox t, are given by Z x t A x t 1 2b x t x t, 2l 2 s 2 sx t (25) s 2 ox t s 2 ox t 2 Z 2 x t l 2 s 2 sx t. (26) Z x t is measured relative to the plane where A x t and x t are speified. x t is the x position of the intensity entroid: A x t is A x t il 2p x t xj x, y, t j 2 dxdy. (27) j x, y, t j 2 dxdy # x, s y, t s x s x, s y, t 2.. ds x ds y s x ; 28 j s x, s y, t j 2 ds x ds y normalized to that for adeal Gaussian beam. An M 2 of 1 orresponds to a Gaussian beam with no phase distortions. Any amplitude or phase distortion makes M 2 larger than unity. Figure 3 ompares the output powers, tilts, urvatures, and M 2 x s for Dk 0 with those for Dk 0.15 mm 21. Figures 3(a) 3(e) show results for Dk 0, and Figs. 3(f) 3( j) show results for Dk 0.15 mm 21. Figures 3(a) and 3(f) ompare the power out of the rystal as a funtion of time for a Gaussianput pulse profile (7 ns FWHM). The alulated mixing effiienies, defined as 322.5-nm energy out divided by 645-nm energy in, are 0.56 and 0.37 for these two ases. These are larger than the measured effiienies 10 of 0.36 and 0.18. This disrepany is probably due to differenes between the experimental and model beams. Figures 3( b) and 3( g) ompare the tilts in the walk-off, or x, diretion. For Dk 0 there is no tilt of either the fundamental or the seond-harmoni beam. For Dk 0.15 mm 21 the seond-harmoni wave tilts in the diretion of walkoff, and the fundamental wave tilts by nearly an equal amount in the opposite diretion, with pronouned intensity dependene. These tilts reverse sigf the sign of Dk is reversed. Tilts in the y diretion are zero for all values of Dk. The ylindrial urvatures shown Figs. 3() and 3( h) indiate that there is no urvature for Dk 0, whereas there are intensity-dependent urvatures for Dk 0.15 mm 21. At the peak of the pulse the fundamental beam is slightly diverging at the rystal exit fae, and the seond-harmoni beam is slightly onverging. The alulated values of M 2 x are plotted in Figs. 3(d) and 3(i). For low input intensities near the beginning and the end of the pulse, M 2 x for the fundamental wave is unity, as it must be for the input Gaussian transverse s x 2 t s sx 2 t x 2 x t 2 j x, y, t j 2 dxdy, (29) j x, y, t j 2 dxdy ls x 2b x t 2 j s x, s y, t j 2 ds x ds y Z ` Z. ` (30) l 2 j s x, s y, t j 2 ds x ds y After tilt and ylindrial urvature are aounted for, the remaining distortios haraterized by two values of Siegman s 13 M 2. One, M 2 x, is alulated in the walkoff plane, and the other, M 2 y, is alulated in the plane perpendiular to walk-off. The quantity M 2 x is defined by M x 2 t 4ps ox t s sx t. (31) It is the produt of the real-spae variane and the spatialfrequeny-spae variane of intensity for an atual beam profile. The seond-harmoni wave s M 2 x is greater than 1 beause of the walk-off-indued elongation of its beam profile. Notie that the values of M 2 x inrease with intensity in eah ase and that the inrease is more pronouned for Dk fi 0. The time-varying tilts, urvatures, and residual distortions just disussed are the onsequene of intensitydependent output phases similar to those displayed in Fig. 2. Another onsequene of this time variation must be frequeny hirps and shifts. These ould be measured in two ways. One is to heterodyne the output beam with a frequeny-shifted referene beam as Gangopadhyay et al. 10 did. This an reveal phase shifts muh smaller than 1 rad over any seleted part of the beam. By its nature, this tehnique measures R ref t da 1.., that is, the eletri field weighted by the referene field and integrated over an area. In our modeling we simulate this measurement by summing the output eletri field over the spatial grid at eah time step and alulate the phase of this summed field as a funtion of time.

54 J. Opt. So. Am. B/Vol. 12, No. 1/January 1995 A. V. Smith and M. S. Bowers Fig. 3. Numerial results for frequeny doubling a 7-ns (FWHM Gaussian), 0.25-mm-diameter (FWHM lowest-order Gaussian) pulse of 645-nm light in a 3-m-long KDP rystal. The onditions are desribed for Case 1 in the text. Results for (a) (e) Dk 0, (f) (j) Dk 0.15 mm 21. In (a) and (f) the solid urve is one half of the 645-nm power and the dashed urve is the full 322.5-nm power. The tilt, urvature, and M 2 haraterize the wave-front distortions in the walk-off diretion. The heterodyne phases of the depleted fundamental and generated seond harmoni are at the exit fae of the rystal.

A. V. Smith and M. S. Bowers Vol. 12, No. 1/January 1995/J. Opt. So. Am. B 55 An alternative method of measuring the frequeny spetrum is to use a square-law-detetion apparatus suh as a spetrometer mated to antensity monitor. We simulate this measurement by alulating the Fourier time transform of the field at eah spatial grid point and summing the transforms weighted by the pulse energy at eah grid point. As we will see, these two measurement methods an produe quite different results. Using the latter method with Dk 0, we find that the fundamental and the seond-harmoni spetra are virtually idential to Fourier transforms of the power profiles shown Fig. 3(a). For D 0.15 mm 21 the time variation of the phase shown Fig. 2 suggests there will be frequeny hirps and broadening of the spetra relative to the power profile transforms. Indeed, omparing the square-law spetra with these transforms reveals that there is some broadening, but it adds less than 2 MHz to the width (FWHM) of the fundamental and seondharmoni spetra. We attribute the broadening of the seond-harmoni spetrum to a small blue shift on the leading edge of the pulse followed by an equal red shift on the trailing edge, as suggested by Fig. 2. Figures 3(e) and 3( j) display the heterodyne phases. As Fig. 3(e) shows, there are no phase shifts for Dk 0. For Dk 0.15 mm 21 the heterodyne phase of the fundamental dereases with inreasing intensity, whereas the phase of the seond-harmoni wave shifts in the opposite diretion. The maximum apparent seond-harmoni and fundamental frequeny shifts are 30 and 15 MHz, respetively. Gangopadhyay et al. 10 measured heterodyne phases of the seond-harmoni wave as a funtion of time for Dk 0 and Dk 0.15 mm 21. For Dk 0.15 mm 21 they found that the seond harmoni appeared to be red shifted by 20 MHz on the leading edge of the pulse and blue shifted by 10 MHz on the falling edge ompared with those in the Dk 0 ase. Unfortunately, the sign of Dk was not reorded, and the error bars for the measurement are nearly as large as the shifts, so detailed omparison of their measurement with our alulatios impossible. The alulated and measured heterodyne phases might be interpreted as a 30-MHz red shift followed by an equal blue shift for the seond-harmoni light and a 15-MHz blue shift followed by an equal red shift for the fundamental light. However, these onlusions would be at odds with the square-law spetra. For the seond-harmoni wave the frequeny shifts have the opposite sign from those dedued from examination of the phases of individual spatial-grid points shown Fig. 2, whih we used to alulate the square-law spetra. In addition, the frequeny shifts dedued from heterodyne phases are muh larger than those seen the square-law spetra. The ause of this apparent paradox is the ombination of beam tilt with a shift in the entroid of the beams in the walk-off dimension. Beause we assume that the detetor is very lose to the output fae of the rystal, tilt alone will not ause the alulated behavior of the heterodyne phases. It will merely weaken the heterodyne signal by produing fringes on the detetor fae where the wave interferes with the tilt-free referene wave. In our ase there is only a fration of a fringe, so this should not be amportant effet. However, the existene of a tilt toward the walk-off diretion for the seond-harmoni wave implies that the phase of this wave must inrease on sanning from 2x to 1x. As Fig. 3(g) shows, the tilt of the seond harmoni is relatively onstant throughout the pulse, so the seond-harmoni phase fronts are nearly stationary, apart from higher-order distortions. However, the entroid of the seond-harmoni beam sans from 2x to 1x as its intensity inreases. This movement of the entroid is due to fundamental wave depletion. As Fig. 1 shows, the seond-harmoni output beam is quite elongated in the walk-off diretion beause the maximum walk-off of 0.84 mm is large ompared with the input beam diameter of 0.25 mm. The part of the beam displaed the most was generated near the input fae of the rystal, whereas the part displaed the least was generated near the exit fae. For low input intensities the fundamental wave is not signifiantly depleted, so the seondharmoni wave s intensity is quite uniform over its elongated profile. Its entroid is at x 0.42 mm, i.e., shifted in the walk-off diretion by half the produt of the walkoff angle and the rystal length. As the input fundamental intensity inreases, it generates seond-harmoni light effiiently near the input fae of the rystal but beomes depleted part way through the rystal, thus generating less seond-harmoni output near the exit fae. Consequently the seond-harmoni output beam is brightest on the walk-off side. Its entroid shifts in the walk-off diretion as the fundamental intensity inreases. Beause the walk-off diretios also the diretion of inreased phase owing to the tilt of the seond-harmoni wave, and beause the heterodyne phase is weighted by the field amplitude, this ombination of nearly onstant tilt and shifting entroid explains the positive seond-harmoni-wave heterodyne phase shift with inreasing intensity. This effet is smaller for the fundamental wave beause its walk-off angle is zero so its entroid shifts muh less. The onlusios that the heterodyne measurement of the seond-harmoni wave is not neessarily a measure of the spetral shifts relevant for spetrometry. It is predominantly a measure of the tilt and the entroid shift of the wave. Indeed, the sign of the hirp measured by the heterodyne method for the seond-harmoni wave is muh larger than and opposite in diretion to that of the hirp that would be seen by a olletion of atoms. Clearly, this onlusios based on the hoie of a referene beam with zero tilt. Tilting the referene beam by 200 mrad to align it with the seond-harmoni beam would dramatially alter the expeted heterodyne phase. In addition, we alulated the heterodyne phases assuming infinite plane-wave referene beams, and the phases would be different if we used a referene beam mathed more losely in size to the atual beams. However, the onlusion remains that its sensitivity to tilt makes it diffiult to relate the heterodyne phase measurement to square-law-detetor spetral measurements. Case 2: Diffration with Birefringene This seond example of numerial modeling illustrates the effet of diffration ombined with birefringene. As disussed above, we expet diffrative phase slippage to mimi phase-veloity mismath and produe phase distortions of the generated wave even the absene of a mismath. We model the same system as in Case 1, exept that here the rystal is shortened to 2 mm and

56 J. Opt. So. Am. B/Vol. 12, No. 1/January 1995 A. V. Smith and M. S. Bowers the fundamental beam is foused to a waist of 14 mm (FWHM of intensity) at the enter of the rystal (Rayleigh range, 1 mm). This gives a onversion effiieny similar to that for Dk 0 in Case 1 and also has the same beamdiameter-to-walk-off ratio. Figure 4 summarizes the results at the exit fae of the rystal. The seond-harmoni wave tilts toward the walk-off diretion, and the fundamental wave tilts in the opposite diretion. The tilts are muh larger than for Case 1 but are still muh less than the diffrative spread of the beams. Figure 4() shows that the fundamental wave at the rystal exit fae is diverging early and late in the pulse beause it is foused at the enter of the rystal. The seond-harmoni wave is substantially less divergent. Both beams diverge less at the peak of the pulse, and the seond-harmoni wave has almost no net urvature. Figure 4(d) shows residual distortions or M 2 s omparable with those of Case 1, but here the fundamental wave is distorted more than the seond harmoni. The heterodyne phase shifts are quite small, as Fig. 4(e) shows, and the square-law spetra are notieably different from the transform of the pulse envelopes only in the far wings of the fundamental spetrum. Fig. 4. Numerial results for frequeny doubling a 7-ns (FWHM), 14-mm waist-diameter (FWHM) pulse of 645-nm light in a 2-mm-long KDP rystal with Dk 0. The onditions are desribed for Case 2 in the text. In (a) the solid urve is one half of the 645-nm power and the dashed urve is the full 322.5-nm power. The tilt, urvature, and M 2 haraterize the wave-front distortions in the walk-off diretion. The heterodyne phases of the depleted fundamental and generated seond harmoni are at the exit fae of the rystal. 4. CONCLUSIONS We have shown that in the plane-wave approximation a wave starting from zero initial intensity in sum- or differene-frequeny mixing in a quadratially nonlinear medium with nonzero Dk will suffer no intensitydependent phase shifts. This lak of sensitivity to intensity is a speial situation, however, and any upset of the balane of intensity or phase among the mixing waves is expeted to introdue intensity-sensitive phase shifts. We illustrated this by numerially modeling a ase in whih the walk-off of one wave was omparable with the beam diameters, violating the plane-wave approximation. We showed that the resulting intensity-dependent phase shifts of the seond-harmoni light produed frequeny hirps for pulsed light, and also wave-front tilts, fousing, and other distortions. We also showed that heterodyne tehniques are not a reliable means of measuring the spetral ontent of beams leaving a nonlinear mixing rystal. In a seond example we showed that diffration produes similar effets. Other situations that violate the plane-wave approximation and so an be expeted to lead to beam distortions inlude mixing with rossed beams when the rossing angle is large enough that the beams separate by an amount omparable with their diameters, group-veloity mismathes large enough to displae the pulses by an amount omparable with their durations, and linear absorption of any wave (exept in the speial ases mentioned). The resulting distortions are generally small, but they may be signifiant when beam pointing or frequeny stability is ritial. The phase distortions may also beome signifiant in optial parametri osillators, for whih phase distortions an aumulate over many transversals of the optial avity. ACKNOWLEDGMENTS We thank W. J. Alford for helpful suggestions regarding this paper. This researh is supported by U.S. Department of Energy under ontrat DE-AC04-94AL85000. REFERENCES 1. G. D. Boyd and D. A. Kleinman, Parametri interation of foused Gaussian light beams, J. Appl. Phys. 39, 3597 3639 (1968).

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