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October 1, 1995 / Vol. 20, No. 19 / OPTICS LETTERS 1961 Optical solitons supported by competing nonlinearities Alexander V. Buryak and Yuri S. Kivshar Optical Sciences Centre, Australian National University, ACT 0200, Canberra, Australia Stefano Trillo Fondazione Ugo Bordoni, Via B. Castiglione 59, 00142 Rome, Italy Received May 17, 1995 It is demonstrated that optical solitons can propagate in a dispersive (or diffractive) medium with competing quadratic [i.e., x s2d ] and cubic [i.e., x s3d ] nonlinearities. Strong interplay between the nonlinearities leads to novel effects, in particular the following: (i) stable bright solitons can still exist in a self-defocusing (owing to cubic nonlinearity) medium supported by quadratic parametric interactions and (ii) x s2d nonlinearity can lead to instabilities of x s3d solitons. 1995 Optical Society of America Recently it was demonstrated experimentally that nonlinearity-induced phase shifts can be achieved in x s2d materials (i.e., optical materials without the inversion symmetry) as the result of cascaded effects.1 This result has stimulated many efforts to analyze a variety of x s2d nonlinear effects [previously well understood only for x s3d materials] such as modulational instability, self-focusing or self-defocusing, and optical solitons. As has been already demonstrated (see, e.g., Refs. 2–4), bright solitons can exist in x s2d materials in the form of two-wave localized modes of the strongly coupled fundamental and second-harmonic fields. However, in x s2d materials there always exists next-order, x s3d nonlinearity, which, under certain conditions, might become important and strongly compete with x s2d nonlinearity (see, e.g., Refs. 5 and 6). In this Letter we discuss how the interplay of nonlinearities inf luences the existence and stability properties of bright (temporal or spatial) optical solitons. We find and investigate novel families of soliton solutions and bifurcations and also analyze the soliton stability. Physical estimates show that our results can be important for any x s2d optical material, depending on the quality of phase matching between the interacting harmonics. Considering the interaction of the fundamental sv1 vd and second sv2 2vd harmonics in a medium with both x s2d and x s3d nonlinearities described by the amplitude envelopes E1 and E2 , we obtain coupled equations in the form i ≠E1 ≠2 E1 ≠E1 1 x2 E1p E2 exps2iDkzd 1 id1 1 g1 ≠z ≠j ≠j 2 1 x3 sjE1 j2 1 rjE2 j2 E1 0 , i ≠E2 ≠E2 ≠2 E2 1 id2 1 g2 1 x2 E12 expsiDkzd ≠z ≠j ≠j 2 1 2x3 sjE1 j2 1 rjE1 j2 dE2 0 , (1) where x2 and x3 are proportional to the elements x s2d and x s3d of the quadratic and cubic susceptibility tensors, respectively; z is the propagation distance; Dk ; s2k1 2 k2 d ; 2fnsvd 2 ns2vdgvyc is the wave-vector 0146-9592/95/191961-03$6.00/0 mismatch between the harmonics fnsvd and ns2vd are the linear refractive indices of an optical medium at v and 2v]; r is the parameter of the cross-phase modulation; and the coeff icients gj s j 1, 2d characterize either the mode dispersion, for pulse propagation (temporal solitons), or the mode diffraction, gj 1y2kj , for the case of self-localized beams (spatial solitons). In the former case dj are the modal group velocities, and in the latter case sd1 2 d2 d describes the spatial walk-off effect. We are interested in stationary phase-locked wave propagation and applyp the following exact transformation: E1 sky 2sx22 dw expsibz 1 iVjd and E2 skyx2 dv expsib2 z 1 2iVjd, where b and b2 ; 2b 1 Dk are the nonlinearity-induced shifts of the propagation constant, s ; jg1 jyjg2 j, k ; b 1 d1 V 1 g1 V 2 , and V ; sd1 2 d2 dys4g2 2 2g1 d. Now the normalized equations for w and v take the form ∂ µ 2 ≠2 w ≠w p 1 x jwj 1 rjvj2 w 0 , 1r 2 w 1 vw ≠z ≠t 2 2s ∂ µ 2 2 ≠v w ≠ v is 1 s 2 2 ay 1 1 x 2sjvj2 1 rjwj2 v 0 , ≠z ≠t 2 i (2) where z kz, t sjkjyjg1 jd1/2 sj 2 nzd, n s2g2 d1 2 g1 d2 dys2g2 2 g1 d, r signskg1 d, s signskg2 d, a sb2 1 2d2 V 1 4g2 V 2 dsyk, and x kx3 yx22 . System (2) is rather complicated for analyzing all possible localized solutions. Nevertheless, we recently showed6 the existence of an exact analytical solution of these equations in the case of some special relations between the system parametersp (a 1, v and w are real, and the constraints w v 2 and r s 21 2 2s are fulf illed). Stable propagation of this bright twowave soliton supported by both nonlinearities has been shown in Ref. 6. The situation resembles the case of pure x s2d solitons for which a similar exact solution is known (see, e.g., Ref. 2). To demonstrate the existence of more general types of bright soliton in Eqs. (2) and analyze their 1995 Optical Society of America 1962 OPTICS LETTERS / Vol. 20, No. 19 / October 1, 1995 properties and stability, we consider the case s 2 and r s 11, which corresponds to spatial bright solitons. For definiteness, we select the cross-phase modulation coeff icient to be r 2 and also a 2. We expect that the general features of the stationary solutions of Eqs. (2) and their stability properties will be similar for other values of a, at least when a is not too small. [For small a the problem is more complicated; e.g., it is possible to show that, for a , acr ø 0.211 at s 2, two-wave solitons are unstable in the case of a pure x s2d nonlinearity. See Ref. 7.] We should emphasize that the general case of Eqs. (2) cannot be understood in all detail before the analysis of two limiting cases, namely, solely x s2d or x s3d nonlinearities, is completed. To find the stationary solutions of Eqs. (2) for the particular values of the parameters mentioned above we omit the derivatives in z and therefore reduce the problem to the analysis of localized solutions of the system µ ∂ 1 d2 w 2 2 2 w 1 wv 1 x 1 2jvj w 0, jwj dt 2 4 ∂ µ d2 v 1 2 2 2 v 0, w 2 2v 1 1 x 4 jvj 1 2jwj dt 2 2 (3) which can be treated as equations of motion for an effective particle in a two-dimensional potential parameterized by x. Separatrix trajectories of system (3) correspond to soliton solutions of Eqs. (2). The main types of localized solution that describe bright solitons of the lowest orders (i.e., solitons with the lowest energy) are shown in Fig. 1 by the change of the scaled (by the factor jxj) soliton energy P jxj Z Note that these nontrivial two-wave solitons exist for x 0 [the case of pure x s2d solitons3] and even for negative values of x larger than xthr ø 20.0616 [i.e., in the region of defocusing cubic nonlinearity where the standard x s3d solitons do not exist], and Fig. 2(c) presents an example of the latter solitons. The soliton shown in Fig. 2(c) can be interpreted as composed of a pair of kinks with the separation between the kinks increasing as x ! xthr . For x , xthr self-defocusing owing to x s3d nonlinearity dominates, and this can allow stable dark solitons to exist in this region. Additionally to solitons of the V and C types, system (3) has other types of soliton that, in some Fig. 1. Bifurcation diagram of the localized solutions of Eqs. (3). Stable solitons are represented by solid curves, and unstable ones are shown by dashed curves. Note the different scales for positive and negative values of x. 1` 2` sjwj2 1 4jvj2 ddt (4) with the parameter x that characterizes a balance of contributions that are due to x s2d and x s3d nonlinearities. To understand the physical origin of these localized solutions we should discuss them in more detail. First, we note that Eqs. (2) [and (3)] always have the simple solutions with w 0. For this type of solution solitons p of the the x s3d nonlinearity itself can support p ayssxd sechs a td. second harmonic: vs std These p one-wave solitons have the energy PV 4 2 saysd, and we denote them in Fig. 1 as solitons of the V type. The analysis indicates that there are many bifurcations from the V -type solitons, but the most important one, which gives birth to the branch with the lowest value of energy [Eq. (4)], starts at x xcr ø 8.76, where a nontrivial two-wave soliton appears (bifurcation point B in Fig. 1). These nontrivial solitons exist because of strong interaction between the harmonics, and we call them the solitons of the combined, or C, type. For small values of x, when x s2d nonlinearity dominates, these solitons look similar to those of a pure x s2d medium. In Fig. 2 we show the characteristic prof iles of these C-type solitons at various points of the bifurcation curve shown in Fig. 1. Fig. 2. Characteristic profiles of the two-wave solitons of the C type at (a) x 0.2 (point L in Fig. 1), (b) x 8.0 (point M in Fig. 1), (c) x 20.0615 (not shown in Fig. 1). October 1, 1995 / Vol. 20, No. 19 / OPTICS LETTERS Fig. 3. Example of the two-wave solitons of the W type at x 8.0 (point N in Fig. 1). limits, are close to the one-wave solitons of a cubic medium. An example of such solitons is shown by the two-wave solitons of the W type (see Fig. 1). The W solitons are supported mainly by the x s3d nonlinearity acting on the fundamental harmonic. This can be seen in the limit of large x when the p W solitons are described approximately as ws 2 syx sech t 1 Osx 21 d and vs Osx 21 d, so that limx!` PW 8s. However, because of parametric coupling, the second harmonic is also generated, and it has a two-hump prof ile as shown in Fig. 3. Note that Fig. 1 does not display all possible soliton families and bifurcations that exist in this model. We show only three of the most physically important soliton families of low energy P . There are many other localized solutions that describe bound states of single solitons and more complicated higher-order solitons. All these solitons are likely always to be unstable. We also carried out a numerical analysis to investigate the stability of the solitons described above. This analysis shows that the W -type solitons are unstable for any value of the parameter x of Fig. 1. The solitons of the V type are stable for x $ xcr but unstable for 0 , x , xcr . This demonstrates how the inf luence of parametric x s2d interactions leads to instabilities of the solitons that are supported mainly by x s3d nonlinearity. On the other hand, the solitons of the C type are stable in all domain of the parameters where they exist (i.e., for xthr , x , xcr ). We should note, however, that a rigorous analysis of the soliton stability is still an open problem. [For the case of a pure x s2d medium it was recently shown that the stability of bright and dark solitons depends strongly on the values of the parameters s and a (Refs. 7 and 8).] Finally, we make some physical estimates. The key parameter of our analysis is x , skcyvdx s3d yfx s2d g2 , which depends on two dimensionless (for the esu system) factors, m ; x s3d yfx s2d g2 , determined by the type of material, and h ; skcyvd lys2pzc d, a ratio of 1963 the fundamental harmonic wavelength in vacuum and the characteristic propagation scale. Typical values of x s2d and x s3d nonlinearities in crystals without the inversion symmetry are x s2d , 1027 esu units (10222 AsyV2 in mks units) and x s3d , 10212 esu units (10232 AsmyV2 in mks units). This gives the estimate m , 100. The value of h determines the importance of the competition between the nonlinearities. For the case of fixed a considered in this Letter k , Dk, which gives h , Dn ; jns2vd 2 nsvdj. As follows from our results, strong competition between x s2d and x s3d nonlinearities is expected for jxj . 1022 , which (for materials with m , 100) gives Dn . 1024 . In the opposite case, i.e., for jxj # 1022 , the inf luence of x s3d nonlinearity on solitons can be neglected, and one can readily use the standard model for x s2d solitons (see, e.g., Ref. 3). In conclusion, we have shown that stable spatial (or temporal) optical solitons can exist in a diffractive (or dispersive) medium supported by competing quadratic and cubic nonlinearities. We have found three main classes of two-wave bright soliton and have demonstrated that competition between the nonlinearities leads to novel effects. In particular, on the one hand, stable bright solitons can exist because of x s2d nonlinearity in the region where the x s3d nonlinearity is selfdefocusing and, on the other hand, the x s2d nonlinearity can make certain types of x s3d soliton unstable. We have also discussed the applicability limits of the pure x s2d parametric interaction model. References 1. R. DeSalvo, D. J. Hagan, M. Sheik-Bahae, G. Stegeman, and E. W. Van Stryland, Opt. Lett. 17, 28 (1992). 2. Y. N. Karamzin and A. P. Sukhorukov, JETP Lett. 20, 339 (1974); R. Schiek, J. Opt. Soc. Am. B 10, 1848 (1993); M. J. Werner and P. Drummond, J. Opt. Soc. Am. B 10, 2390 (1993). 3. A. V. Buryak and Yu. S. Kivshar, Opt. Lett. 19, 1612 (1994); Phys. Lett. A 197, 407 (1995). 4. L. Torner, C. R. Menyuk, and G. I. Stegeman, Opt. Lett. 19, 1615 (1994). 5. S. Trillo and S. Wabnitz, Opt. Lett. 17, 157 (1992). 6. S. Trillo, A. Buryak, and Yu. Kivshar, in Nonlinear Guided Waves and Their Applications, Vol. 6 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 36 – 38. 7. A. V. Buryak and Yu. S. Kivshar, Opt. Lett. 20, 1080 (1995); see also D. E. Pelinovsky, A. V. Buryak, and Yu. S. Kivshar, Phys. Rev. Lett. 75, 591 (1995). 8. A. V. Buryak and Yu. S. Kivshar, Phys. Rev. A 51, R41 (1995); S. Trillo and P. Ferro, Opt. Lett. 20, 438 (1995).