Effective lensing effects in parametric frequency conversion

Transcription

1 852 J. Opt. Soc. Am. B/ Vol. 19, No. 4/ April 2002 Conti et al. Effective lensing effects in parametric frequency conversion C. Conti and S. Trillo Department of Engineering, University of Ferrara, Via Saragat 1, Ferrara, Italy, and Istituto Nazionale di Fisica della Materia, RM3, Via della Vasca Navale 84, Rome, Italy P. Di Trapani, J. Kilius,* A. Bramati, S. Minardi, and W. Chinaglia Istituto Nazionale di Fisica della Materia and Department of Chemical, Physical and Mathemetical Sciences, University of Insubria, Via Lucini 3, Como, Italy G. Valiulis Department of Quantum Electronics, Building 3, Vilnius University, Sauletekio Avenue 9, 2040, Vilnius, Lithuania Received June 4, 2001; revised manuscript received October 23, 2001 We show that, in the high wave-vector-mismatch (cascading) limit, the well-known paraxial description of parametric frequency conversion in quadratic media entails effective lensing effects, which can have a selffocusing or a self-defocusing nature, critically depending on the mismatch sign, the selected wave, and the launching condition (second-harmonic generation or downconversion). Numerical and experimental evidence of this behavior is reported Optical Society of America OCIS codes: , , , INTRODUCTION It is well known that one can compensate for diffraction of electromagnetic wave packets by focusing Kerr nonlinearities to yield self-trapped beams or spatial solitons. 1 In the paraxial limit and two transverse dimensions this balance is marginal and separates catastrophic blow-up instability (collapse) and diffraction (see, e.g., Ref. 2 and references therein). Spatial solitons, however, are sustained also by several other nonlinear processes. 3 Two major recent achievements in this area have shown that stable soliton propagation occurs in photorefractive materials (as a result of saturating Kerr-like nonlinearities) 4,5 and in parametric frequency conversion sustained by (2) nonlinearities. Parametric solitons constitute an active area of research (see Refs. 3, 6, and 7 for reviews), brought to maturity by the pioneering observation of solitons in second-harmonic generation (SHG) in 2 1 (Ref. 8) and 1 1 (Ref. 9) dimensions. Interestingly enough, these successful experiments were performed only recently, in spite of the fact that isolated theoretical studies date back to the 1970s. 10 Observations of other intriguing phenomena have now widened the perspectives of this topical area. Among these, it is worth mentioning nondegenerate (i.e., three-wave) solitons, 11 spatial locking in optical parametric amplification (OPA) or downconversion from quantum noise 12 or from a finite seed, 13,14 multibeam generation by means of transverse modulational instability, 15,16 temporal 17 and spatiotemporal trapping, 18,19 solitons in periodically poled materials, 20 Bessel-like or vortex beams in SHG and OPA, and pattern formation in optical parametric oscillators. 25 The interest in (2) solitons has also stimulated a deep theoretical understanding of general stability properties of optical solitons Parametric solitons originate from defeating diffraction by means of nonlinear phase shifts associated with parametric conversion. In the limit of large wave-vector mismatches, SHG behaves as an effective self-induced Kerr effect (i.e., intensity-dependent index change) for the fundamental frequency (FF) beam 31 as a result of the cascaded processes of upconversion (or degenerate sumfrequency generation) 2 and of downconversion (or degenerate difference-frequency generation) 2. This so-called cascading effect is relevant for device applications 6 and arises also through other (2) phenomena such as optical rectification. 32 In SHG, whenever phase mismatch k 2k 1 k 2 2k( ) k(2 ) is positive, the effective nonlinear index change at the FF is of the self-focusing type. In other words, it is such as to induce a local (concave) wave-front curvature that counteracts the (convex) curvature associated with diffraction (incidentally, at high enough intensities, the curvature is dominated by the nonlinearity, thus causing the shrinking of beam diameter that constitutes the common experimental evidence of self-focusing behavior). In this regime the model for paraxial propagation of the two interacting beams reduces to a selfconsistent focusing nonlinear Schrödinger equation (NLSE) for the FF envelope 33,34 whose simplest derivation assumes that the second-harmonic (SH) beam adiabatically follows the beam at the FF (see also Refs. 6 and 7). Yet this does not explain why parametric solitons do exist and can be generated by means of SHG also for k 0, where, according to the same NLSE picture, the FF /2002/ $ Optical Society of America

2 Conti et al. Vol. 19, No. 4/April 2002/J. Opt. Soc. Am. B 853 beam would experience self-defocusing action, that is, a nonlinear wave-front curvature of the same sign as of the diffractive curvature, that leads to enforced beam spreading. Moreover, the analysis that yields the NLSE cannot account for the features of OPA, where it is natural to assume that the leading role is played by the SH beam. The aim of this research is to propose an improved description of cascading that permits one to understand how the sign of phase mismatch determines the self-focusing or self-defocusing nature of the effective nonlinearities, depending in a critical way on the dominant process through the launching conditions (SHG or OPA) and even on the specific beam that one looks at. The outcome of our analysis shows that the dynamics of both beams in OPA are dominated by effective cross-induced lensing effects, which are of a focusing nature for k 0, at variance with the SHG case for which the prevailing nonlinear term is of a self-induced nature and leads to selffocusing for k0. Moreover, we find unexpectedly that in the SHG case, no matter what the mismatch sign is, the two beam envelopes experience opposite focusing action. We also report numerical and experimental evidence for this property. 2. MULTISCALE EXPANSION We start from the following conservative (Hamiltonian) coupled-mode system, which is obtained from Maxwell equations when two-wave parametric mixing of the wave packets u 1 (x, y, z) at the FF and u 2 (x, y, z) at the SH is considered in the paraxial and lossless limits without walk-off effects (for its derivation see, e.g., Ref. 35): i u 1 z u 1 u 2 u 1 * exp i kz 0, i u 2 z u 2 exp i kz 0. (1) 2 In Eqs. (1), 2 2 x 2 y is the transverse Laplacian, and we have introduced longitudinal and transverse dimensionless scaled variables z Z/Z d and (x, y) (X, Y)/r 0 (r x 2 y 2 is the radial coordinate), respectively, where Z d k 1 r 2 0 is the diffraction length that pertains to reference spot size r 0 (twice the Rayleigh range for a Gaussian beam of waist r 0 ). The scaled slowly varying amplitudes in Eqs. (1) are u 1 Z d E 1 and u 2 Z d 1 E 2, where E 1,2 (X, Y, Z) 2 are real-world intensities in watts per square centimeter and 1 k /(n 2 1 n 2 ) 1/2 (2) ( ; 2, ), 2 k /(n 2 1 n 2 ) 1/2 (2) (2 ;, ) are nonlinear coefficients expressed in terms of vacuum impedance 0 and effective element (2) of the second-order susceptibility tensor, evaluated at the frequencies involved in the mixing. We have also set m k 1 /k m, m 1, 2, and we can reasonably assume in what follows that 1 1 and 2 1/2. Finally, in Eqs. (1) the only external parameter is the normalized wave-vector mismatch k kz d 2k 0 (n 1 n 2 )Z d, where n m stands for the linear index at frequency m 0. A widely used tool for isolating the evolutions of the field envelopes at different orders in a suitably small u 1 2 quantity is the multiscale method. 36,37 We apply such a technique by taking k 1, without any further assumption on its sign and the relative magnitude of the two interacting fields. This permits us to deal with a unified analysis of SHG and OPA. To this end, the two fields are assumed to depend on both a fast z/ and slow longitudinal scales z n n z, with n 0, 1, 2,... For our purpose it is not necessary to expand the transverse variables. However, it is crucial to expand the two interacting fields in power series of (asymptotic expansion) as u 1 A A 1 2 A 2..., u 2 B B 1 2 B 2..., (2) where all the field amplitudes on the right-hand sides of Eqs. (2) depend on (x, y,, z 0, z 1, z 2,...). We then proceed in a standard way by substituting these expansions into Eqs. (1), grouping terms of the same order in and exploiting the constraints that arise from elimination of secular growing terms. 36,37 In terms of new variables, the linear operators L m i( / z) ( m /2) 2, m 1,2, that appear in Eqs. (1) take the form L m i L 0 m L 1 2 L 2..., (3) where the zero-order [L (0) m, m 1, 2] and higher-order operators [L (n), n 1] read as L 0 m i m z , L n i. 2 2 x y z n (4) By substituting Eqs. (2) and (3) into Eqs. (1), we find that at order O(1/ ) A 0, B 0; (5) that is, leading-order terms A and B do not depend on the fast longitudinal scale. This implies that the dynamics can be described by two slowly varying functions (A and B) with superimposed rapidly oscillating perturbations. At next order O(1) we find that i A 1 L 1 0 A A*B exp i 0, i B 1 L 2 0 B 1 2 A2 exp i 0. (6) According to Eqs. (4), the following constraint must be imposed to eliminate secular growing terms in Eqs. (6): L 0 1 A L 0 2 B 0. (7) Therefore the integration of Eqs. (6) yields the following expression for the first-order corrections: A 1 A*B exp i a 1 x, y, z n n 0, B A2 exp i b 1 x, y, z n n 0. (8) Note that we have retained integration constant terms in Eqs. (8) because they can be useful in matching appropriate initial conditions. At order O( ) we obtain [employing also Eqs. (8)]

3 854 J. Opt. Soc. Am. B/ Vol. 19, No. 4/ April 2002 Conti et al. i A 2 L 1 0 A*B exp i A*b 1 exp i Ba 1 * exp i L 1 0 a 1 L 1 A 1 /2 A 2 A B 2 A, i B 2 1 /2 L 2 0 A 2 exp i Aa 1 exp i L 0 2 b 1 L 1 B A 2 B. (9) To eliminate secular terms we must set the -independent right-hand sides of Eqs. (9) to zero; thus the second-order corrections are given by A 2 L 1 0 A*B exp i A*b 1 exp i B 2 L 2 0 A2 a 1 *B exp i a 2, exp i Aa 1 exp i b 2, (10) 2 where (a 2, b 2 ) (a 2, b 2 )(x, y, z n n 0) and the following relations hold: L 1 A 1 /2 A 2 A B 2 A L 1 0 a 1, L 1 B A 2 B L 0 2 b 1. (11) Because the left-hand sides of Eqs. (11) do not depend on x, y, and z 0 [see Eqs. (7)], they must be set to zero, and thus we are left with the following system: and the relations L 1 A 1 /2 A 2 A B 2 A 0, L 1 B A 2 B 0, (12) L 0 1 a 1 0, L 0 2 b 1 0. (13) Returning to original variables by employing Eqs. (3), (5), and (7), we obtain from Eqs. (12) the main outcome of our analysis, namely, that the leading-order envelopes A and B obey the following reduced set of equations: i A z A 1 2 k A 2 A 1 k B 2 A 0, i B z B 1 k A 2 B 0. (14) Conversely, from Eqs. (13) we obtain that the evolution of a 1 and b 1 is governed by the linear equations i a 1 z a 1 0, i b 1 z b 1 0. (15) Our final expressions (at first order in k 1 ) for the fields are u 1 A 1 k BA* exp i kz a 1, u 2 B 1 A2 k 2 exp i kz b 1. (16) Therefore, according to our analysis, the evolution of the two fields u 1,2 is entirely determined [up to the second order in expansion (2)] by the dynamics of the two leadingorder envelopes A and B, which obey the reduced coupledmode system [Eqs. (14)] and linear wave components a 1 and b 1. The initial conditions for Eqs. (1) determine those for Eqs. (14) and (15). For example, in the case of unseeded SHG we have u 1 x, y, z 0 F x, y, u 2 x, y, z 0 0, (17) and, comparing powers of k 1 in Eqs. (16), we have A(x, y, z 0) F(x, y), B(x, y, z 0) a 1 (x, y, z 0) 0, and b 1 (x, y, z 0) F 2 (x, y)/2. In this case the asymptotic model [Eqs. (14)] reduces to the wellknown self-consistent NLSE for the FF beam u 1 A: i A z A 1 2 k A 2 A 0. (18) Consistently with previous results, a self-trapped (soliton) solution A(x, y, z) u 1 (x, y, z) of Eq. (18), which exists for k 0, is accompanied by a trapped SH component u 2 u 2 1 exp(i kz)/(2 k), whereas b 1 represent linear waves (radiation) generated in the process of SH generation. Before discussing the physics behind the lensing terms in Eqs. (14), let us first point out the links of our results to those that previously appeared in the literature. The most comprehensive multiscale derivation of a reduced model that describes cascading beyond the NLSE was presented by Kalocsai and Haus. 38 They start, however, from the nonlinear Helmholtz (second-order in z) equation. Therefore this approach is not suited for establishing a link with the usual paraxial model [Eqs. (1)] that is commonly employed to model propagation and describe parametric solitons. Moreover, in Ref. 38 the authors report having derived a system that contains a self-action term at the SH that arises from coupling to higher harmonics. As a result, it is not straightforward from the model in Ref. 38 to understand the role of the phasemismatch sign. Another strictly related result was reported by Clausen et al. in Ref. 39. In that analysis the effective Kerr terms are due to higher-order mismatched Fourier coefficients of the spatially periodic quadratic nonlinearity (physically corresponding to a residual phase shift) that are present when one is considering quasiphase-matched materials. These terms have similar form because they entail cross-phase modulation for both beams and self-phase modulation of a different sign for the FF beam. These results suggest that the effective nonlinear terms of mismatched modes that are mixed parametrically through (2) always have the universal form given in Eqs. (14). There are, however, important differences between the case treated here and the quasiphase-matched grating dealt with in Ref. 39. In quasiphase matching, the effective Kerr terms appear as a correction to the leading-order quadratic nonlinear terms that are phase matched through the first spatial harmonic of the grating; hence their effect can be expected to be weak in general. In our (homogeneous) case, however, they are the leading-order terms, which have strong measurable (see below) effects on the beam focusing dynam-

4 Conti et al. Vol. 19, No. 4/April 2002/J. Opt. Soc. Am. B 855 ics. Furthermore, in quasi-phase matching their sign is not coincident with the sign of k but rather depends on the quasi-phase-matched period. Equations (14) are more general than the NLSE [Eq. (18)] description of the high-mismatch or cascading limit. The only nonlinear term in common between these two models is the self-lensing term A 2 A, which describes the nonlinear phase shift acquired during sequences of cascaded photon processes, each involving upconversion to 2 followed by backconversion at. This two-step process leads to an effective cubic susceptibility (3) eff ( ;,, ) (2) ( ; 2, ) (2) (2 ;, ), which is characteristic of the effective four-photon interaction. Here the inverse order of the quadratic susceptibility elements corresponds to that of the relative photon process. However, our model [Eqs. (14)] shows that other two-cascaded processes are inherent in twowave mixing and become relevant when the SH beam is sufficiently intense (i.e., the number of 2 photons is large). The first process involves downconversion followed by upconversion (i.e., as above in reverse order), which leads to a cross-induced phase shift at the SH (term A 2 B) associated with the effective four-photon process 2 2 and the susceptibility (3) eff (2 ;,,2 ) (2) (2 ;, ) (2) ( ; 2, ). The second process constitutes two successive differencefrequency generation processes 2, which are responsible for the cross-induced phase shift at the FF (term B 2 A) through the effective process 2 2, which is ruled by cubic susceptibility (3) eff ( ;2, 2, ) (2) ( ;2, ) (2) ( ;2, ). To give a more pictorial description of these processes, we can summarize them all in a generalized qualitative scheme for cascading, as shown in Fig. 1. Here the labels and 2 identify photons at the FF and the SH, respectively, whereas DFG (SFG) stands for difference- (sum-) frequency generation. As shown, the initial (at the left) and the final (at the right) states are the same, thus justifying that, when they are viewed globally, the processes involve only phase shifts without energy coupling (i.e., net conversion). In our scheme, the phase shifts experienced by the two beams are associated with the three possible combinations of cascaded processes in Fig. 1, as follows: SFG DFG, self-induced lensing at the FF; DFG SFG, cross-induced lensing at the SH; and DFG DFG, crossinduced lensing at the FF. It is important to note that the sequence SFG SFG does not make any contribution to phase shifting because, unlike the three cases listed, the second (upconversion) process does not involve any photon arising from the first (upconversion) process. However, a self-induced phase shift at the SH could origi- Fig. 1. Schematic of cascading by two-wave mixing of photons at frequencies and 2. DFG and SFG, (degenerate) downconversion and upconversion, respectively. nate from upconversion followed by backconversion at 2. This is, however, a second-order process with respect to two-wave mixing. It involves the mixing of three beams, and neither the original model [Eqs. (1)] nor (a fortiori) its reduced version [Eqs. (14)] could ever account for it. To the best of our knowledge, the observation of fourth-harmonic generation by means of SHG in transparent crystals has never been reported, and the determination of its role (if any) needs further accurate experimental studies. Going back to Eqs. (14), because nonlinear coupling occurs only through phase terms the individual powers of the two beams are conserved, at variance with Eqs. (1), for which it is only the total power that is constant along z. This is an ultimate consequence of the fact that Eqs. (14) provide a description of the evolution averaged over many period of the (weak) conversion and backconversion processes that occur over a faster spatial scale, in turn dictated by the mismatch. The advantage of Eqs. (14) is that they allow for gaining a simple insight into the basic features of the effective lensing effects. In Eqs. (14) the self-induced and cross-induced (at both the FF and the SH) effective nonlinear lensing effects have opposite signs for any given mismatch k. Therefore the consequence of the overall focusing or defocusing nature of the process is affected not only by the mismatch sign but also by the relative strength of the two fields. Let us consider the various features of SHG and OPA. As we discussed above by specializing Eqs. (14) to SHG launching conditions, i.e., B 0, we recover the wellknown self-consistent NLSE for the FF beam envelope. Clearly, the effective self-action of the FF beam dominates; it is focusing for k 0. Importantly, however, under more-general dynamic evolution, a small, generated SH beam (i.e., B 0) experiences opposite action with respect to the FF beam (i.e., self-defocusing for k 0) owing to the cross-induced phase curvature of different sign in Eqs. (14). In OPA, launching conditions require a small seed at the FF ( B 2 A 2 ). Therefore it is clear that the FF self-action in Eqs. (14) is negligible. Here, unlike the NLSE limit of SHG, a similar self-consistent equation cannot be derived, because of the cross-induced origin of nonlinear phase shifts. To describe the dynamics of the mixing process and its solitons, we must retain crossinduced terms that are, unlike SHG, both focusing for k NUMERICAL RESULTS To confirm the validity of our multiscale expansion we performed a series of numerical experiments with the original model [Eq. (1)], taking for the sake of simplicity the 1 1 dimensional case (i.e., y 0). As an example we show below four cases that correspond to the possible combinations of launching conditions (seeded OPA and SHG) and choice of mismatch sign with k 100. In all cases the energy conversion is weak. Let us consider first the seeded SHG case, A 2 B 2, obtained by launching the input Gaussian fields u 1 x, z 0 A 0 exp x/x 0 2,

5 856 J. Opt. Soc. Am. B/ Vol. 19, No. 4/ April 2002 Conti et al. where we have chosen A and B 0 1, and input beam widths x 0 8 such that the effect of nonlinear terms in Eqs. (14) prevails over the diffractive spreading. Figures 2(a) and 2(b) display the evolution of the FWHM of the two beams. As shown in Fig. 2(a), a positive mismatch results in self-focusing for the FF beam and in selfdefocusing for the SH beam, in agreement with what is expected from Eqs. (14) with a negligible cross-phase modulation term for A ( A 2 B 2 ). When the sign of the mismatch is reversed [Fig. 2(b)], the observed trend is opposite, again in agreement with our reduced model [Eq. (14)]. We then considered the OPA regime A 2 B 2 by setting A 0 1 and B in the initial conditions [Eqs. (19)]. In this case, the self-lensing term for the FF beam is negligible, and, on the basis of the reduced model [Eq. (14)], the two harmonics are expected to undergo either defocusing or focusing behavior, depending on the sign of the mismatch. Such behavior is well reproduced by the numerical results shown in Figs. 2(c) and 2(d) for positive and negative mismatches, respectively. In the four cases shown in Fig. 2 note the rapid oscillations related to conversion and backconversion processes, which are superimposed onto the slow-scale dynamics correctly captured by our multiple-scales approach. Moreover, whenever the input SH intensity is weak (seeded SHG), the SH and FF beams follow opposite spreading behavior, irrespective of the mismatch sign. It is worth noting that this type of dynamics cannot be explained on the basis of the NLSE model [Eqs. (18)], which would predict the same behavior for the two beams. In this case the SH beam does not follow the FF beam adiabatically, in spite of the fact that one operates with a large mismatch and small SHG efficiency. In fact, the adiabatic following argument and the NLSE limit can be reasonably invoked only to approximate stationary or nearly stationary conditions (e.g., soliton formation in which both beams are trapped), whereas they could provide an erroneous picture of lensing effects under dynamic conditions. Fig. 2. Beam widths versus propagation distance as obtained from numerical integration of Eqs. (1), with y 0, in the limit of large absolute wave-vector mismatch ( k 100) in four qualitatively distinct cases: (a) SHG, positive mismatch; (b) SHG, negative mismatch; (c) OPA, positive mismatch; (d) OPA, negative mismatch. Dashed lines, initial width x 0 8 of both FF and SH beams. u 2 x, z 0 B 0 exp x/x 0 2, (19) 4. MEASUREMENTS Next, we investigated experimentally the beam spreading in a SHG experiment performed with a laser source delivering 1.5-ps pulses (measured by autocorrelation) at 0 2 /k nm, in a 3-cm-long lithium triborate crystal [ (2) (2 ;, ) (2) ( ; 2, ) 0.85 pm/v and W 1/2, with n 1 n 2 1.6] with temperature-tuned noncritical phase matching. The measured phase-matching temperature, corresponding to peak SHG conversion for a wide FF beam, was 159 C. A nearly Gaussian input pump beam with a FWHM diameter of 45 m was focused on the input face of the crystal, and its energy was measured by a calibrated energy meter. The output beam was imaged onto a CCD camera, where the beam profiles and diameters at both FF and SH were measured by means of suitable filters. For low-power operating conditions, we measured a diffracted output FWHM diameter d m. To demonstrate directly the self-focusing and selfdefocusing nature of the effective nonlinear lensing effect, we performed measurements of SHG from an input FF beam with fixed diameter (d 44 m). Its input energy was set to E 1 J, corresponding to the formation of a parametric soliton near phase matching. Then the output beam diameters of the FF and SH beams were recorded against mismatch k, as shown in Fig. 3, in a range of 100 cm 1, corresponding to temperatures of C. For large absolute mismatches (say, k 40 cm 1 ), diffraction of the FF beam strongly dominated SHG, and the two beam diameters approached the linear limit, shown as a dashed line in Fig. 3. Here we did not readjust the input energy to form a soliton when the absolute mismatch was changed. Thus, for intermediate mismatches, the nonlinearity did not completely balance diffraction, and the beams experienced significant width variations. Remarkably, the two beams ex- Fig. 3. Output beam FWHM diameters d of the FF and SH beams measured as a function of wave-vector mismatch k in a SHG experiment with fixed input energy and input beam diameter. Dashed line, linear limit of FF diffractive spreading.

6 Conti et al. Vol. 19, No. 4/April 2002/J. Opt. Soc. Am. B 857 hibited strongly asymmetric behavior against reversal of mismatch. For k 0 the FF beam spread well above the linear limit following the defocusing nature of the effective nonlinearity dominated by self-phase modulation. Conversely, for k 0, it was the SH beam that experienced a significant broadening above the linear limit, in agreement with the defocusing nature of the crossinduced lensing effect, as correctly displayed in Eqs. (14). This is, to the best of our knowledge, the only compelling evidence that cascading is a subtle limit with opposite features that depend on which harmonics is considered. Further experimental evidence of asymmetry of the soliton formation threshold in the OPA relative to that in SHG will be presented elsewhere LINK WITH PARAMETRIC SOLITONS To establish further how nonlinear lensing in Eqs. (14) affects self-trapping ruled by the original model, we studied the entire family of radially symmetric (i.e., dependent on r x 2 y 2 ) nodeless soliton solutions, u 1 u 10 (r)exp(i z) and u 2 u 20 (r)exp(2i z i kz), of Eqs. (1), where is the nonlinear phase shift (propagation constant) of the soliton. The existence of such solutions was originally studied by Torner et al. 41 and Buryak et al. 42 ; the stability of the solutions was assessed by Pelinovski et al. 26 and later extended to the type II case. 27 Bright solitons exist for any propagation constant 0 when k 0, and only for k/2 when k 0, becoming infinitely wide close to the existence threshold k/2. In the whole region of existence of the plane k,, we found the soliton profiles u 10 (r), u 20 (r) numerically by means of a relaxation technique. The relative harmonic content of the soliton can be characterized by the power imbalance Q 2 /Q 1, where Q m Q m (, k) u m0 2 /(3 m)dxdy gives the parametric dependence of the FF (m 1) and the SH (m 2) powers. One can characterize the entire soliton family by listing values of Q 2 /Q 1 versus k for constant. More physically, we show in Fig. 4 such curves at constant total power Q Q(, k) Q 1 Q 2. Power Q is indeed a quantity that turns out to be directly controllable in the experiments (and invariant in propagation along with the Hamiltonian ), unlike phase shift, which depends on Q in a nontrivial way. From Fig. 4 we clearly see that harmonic imbalance Q 2 /Q 1 is highly asymmetrical, becoming small in the region of positive mismatches and large in the region of negative mismatches. Furthermore, for k 0 the curves decrease smoothly, whereas for large negative k they rise abruptly beyond a characteristic value of mismatch. From this property, it is now clear physically why solitons exist also for large negative k. According to Eqs. (14), the lensing effect (in this case, of cross-induced origin) experienced by the dominant (SH) soliton component is of the self-focusing type and hence can balance diffraction. On this basis we can conclude that, in the cascading limit, solitons can be viewed as being due to the usual balance between diffraction and effective focusing nonlinearities, no matter what the mismatch sign. The positive sign of the effective nonlinear term, however, must be correctly related to the dominant beam, i.e., the SH in OPA and the FF in SHG. Yet there is an important difference between the two regions that pertains to different signs of mismatch. For k 0, the low value of Q 2 /Q 1 permits easy generation (i.e., with a lower threshold) of solitons in a SHG arrangement because the propagation-invariant soliton solutions are better matched by the launching conditions. In this case the generated SH tends to defocus [see Eqs. (14)], thereby representing, in two transverse dimensions, a stabilizing mechanism against collapse of the soliton. Indeed, solitons are always stable for k 0 and can be generated for arbitrarily large mismatches and with arbitrarily low SH power content (in this case, the higher k, the higher the required power Q; see Fig. 4). Conversely, for k 0, solitons are more easily formed in OPA because of their large Q 2 /Q 1 ratio. However, in OPA this stabilizing mechanism is absent [both harmonics follow a self-focusing trend; see Eqs. (14)], and instabilities cannot be ruled out. Indeed, for large enough k the solitons become bistable, that is, in our picture two solutions with different amounts of imbalance exist for the same value of mismatch and the same total power Q (a stricter definition of bistability refers to two solitons with different values of the Hamiltonian and the same power, as turns out to be the case here). The solutions with larger values of Q 2 /Q 1 belong to an unstable branch (dq/d 0, shown as bold branches of the multivalued curves in Fig. 4) and cannot be observed. The instability mechanisms involve either decay of these solitons into diffractive waves or oscillation about a nearby stable solution. 26,27 Although it is true that the instability domain involves only a tiny region of the parameter plane (, k) close to the existence threshold, 26,27 our accurate numerical results show (see Fig. 4) how abrupt and dramatic is the increase of imbalance Q 2 /Q 1 (or SH power fraction f 2 Q 2 /Q; not shown) that occurs in this region. Our relaxation code has allowed us to find easily (unstable) solutions with a SH power fraction f 2 up to 99%. For example, in Fig. 5 we compare the intensity profiles of an unstable and a stable soliton, obtained for a mismatch k 10. In spite of the fact that the two solutions possess the same total power (Q 378) but different values of the Hamiltonian, the unstable solution exhibits a dramatic enhancement of SH soliton content [ f in Fig. 5(a); f in Fig. 5(b)] and a substantial broadening of the spatial pro- Fig. 4. SH to FF power imbalance Q 2 /Q 1 of the parametric solitons versus mismatch k for several values of total soliton power Q Q 1 Q 2. Unstable soliton branches are plotted as bold curves.

7 858 J. Opt. Soc. Am. B/ Vol. 19, No. 4/ April 2002 Conti et al. *Permanent address, Department of Quantum Electronics, Building 3 Sauletekio, Vilnius University, Avenue 9, 2040, Vilnius, Lithuania. Fig. 5. Soliton intensity versus x (profile sections at y 0: SH field u 20 (x, y 0) 2, thinner curves; FF field u 10 (x, y 0) 2, thicker curves;) for k 10 and total power Q 378: (a) unstable case, 5.01; (b) stable case, Note the different horizontal and vertical scales. files. Because of this property (the soliton tends to become a plane wave at the SH as its existence threshold is approached), the instability of parametric solitons can be considered a decay instability of wave packets with strong SH content, thus suggesting its similarity to the wellknown parametric decay instability of a SH plane-wave eigenmode. Whereas the latter occurs in a range of mismatches symmetrically located about phase matching, 43,44 the focusing character of the effective lensing effect pushes this instability in the k 0 domain. 6. CONCLUSIONS In summary, starting from the usual paraxial model for parametric mixing of two beams in a quadratic medium, we have discussed an improved description of cascading. Our results show that the effective (cubic) lensing terms that originate from parametric conversion and backconversion can have self-focusing or self-defocusing natures, depending on the sign of mismatch, the selected wave, and the dominant process (SHG or OPA), through the interplay or self- and cross-induced lensing contributions. Although our analysis was carried out for the spatial (diffractive) case, it is clear that our results apply to the temporal (dispersive) case as well, with obvious changes for the physical meaning of the effective Kerr nonlinearities. Our approach can be also generalized to deal with nondegenerate (in frequency, i.e., 1 2 3, or polarization) three-wave mixing. In this case, we conjecture that the simple description of cascading based on cross-phase modulation for the two low-frequency beams 45 could be improved along similar lines. ACKNOWLEDGMENTS This study is dedicated to the memory of one of the authors, our young and talented friend Jonas Kilius, called to timeless life by a sad destiny before this paper was published. We acknowledge useful discussions with O. Bang, A. Buryak, C. De Angelis, Y. S. Kivshar, G. I. Stegeman, and L. Torner. This research was funded by the Istituto Nazionale di Física della Materia, Italy [Progretto Avanzato o Iniziativa di Sezione (PAIS) project 1999] and UNESCO project Uvo-Roste S. Trillo s address is strillo@ing.unife.it. REFERENCES 1. R. Y. Chiao, E. Garmire, and C. H. Townes, Self-trapping of optical beams, Phys. Rev. Lett. 13, (1964). 2. L. 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