Solitons due to second harmonic generation

It is shown that mutual trapping of the fundamental and second-harmonic waves in a dispersive medium with quadratic nonlinearity supports a variety of solitary waves including bright and dark solitons, solitons with trapped radiation, stable bound states of two (or more) solitons. The results are demonstrated for parametric interactions in optical ,$*I materials but can be generalized to other physical systems. Intense study of nonlinear effects in optics has offered new facilities for all-optical signal processing as well as optical communications. It is usually believed that one of the typical nonlinear effects, setf-focusing of light beams, is possible only due to the intensity-dependent nonlinear refractive index of a dielectric medium with a third-order (cubic) nonlinear response, i.e. in the so-called xC3) medium. In other nonlinear problems cubic nonlinearities are known to be responsible for self-modulation and stable propagation of solitons, e.g. pulse solitons in (silica glass) optical fibers [ 11. At the same time, second-order (or quadratic) nonlinearities of the socalled ~1~) materials are usually discussed in the theory of the second-hamonic generation (SHG) (see, e.g., Ref. [2] ). These two important nonlinear phenomena, self-focusing (or self-modulation) and second-harmonic generation, are well known and investigated in many branches of physics where nonlinearities become important (see, for example, the case of surface acoustic waves [ 31). As a matter of fact, they usually manifest themselves independently Elsevier Science B.V. SW103759601(94)00989-9 and seem to be hardly related to each other. At the same time, it is known that optical xC2) materials provide one of the fastest electronic nonlinearities among those which are available at the moment. This stimulates efforts to increase indirectly effective xc3) nonlinearities taking advantage of cascaded secondorder effects [4]. Recent progress in materials with high second-order nonlinearities (see, e.g., Ref. [5] to cite a few), including polymeric electro-optical waveguides, is a good background for analysing nonlinearity-induced effects which can be achieved due to cascading. In particular, it was shown that such effects related to xc31 materials as nonlinear phase shift [ 61 and self-diffraction [ 71 can, be enhanced by cascaded processes. The cascaded nonlinearities can also support the intensity-dependent light propagation in the form of parametric self-guided waves [8-121 which can be analyzed by means of the multi-scale asymptotic technique leading, for a large phase mismatch, to an effective nonlinear Schrtjdinger (NLS) equation. However, many features of nonlinear effects in dispersive xC2) materials, including modulational 408 A.V. Buryak, Y.S. Kivshar/Physics Letters A 197 (1995) 407-412 instability, self-focusing and solitons often cannot be described, even approximately, by a single NLS equation. This Letter aims to present a theory of two-wave (bright and dark) solitary waves in quadratic nonlinear media. The theory applies to parametric interactions between the fundamental and second harmonics in optical xc2) materials but can be generalized to other physical situations. Considering interaction of the first (wt = w) and second (~2 = 20) harmonics in a dielectric medium with xc2) nonlinear susceptibility, we assume their amplitude envelopes El and E2 slowly varying and derive from Maxwell’s equations the system of two nonlinear equations coupled through components ,$j of the nonlinear susceptibility tensor, a-f.5 i--j-z+ iSt %+Yt@El % as2 + xt E; E2 eiAkz = 0, =2 dE2 a2E2 i+ if&-+ Yza2 &? as2 + x2Ef eFiAkz = 0, (1) where xt 3 (~T~~/~~c~)x(~)(w;~w, -w) and ~2 = ( 8sw2/k2c2)X(2)( 2w; w, o), z is the propagation distance, and Ak G 2kl k2 is the wave vector mismatch between the harmonics. System ( 1) is derived assuming El N EZ N E (but other harmonics are of higher order in E). Thus, z + EZ and 6 + 46 are “slow” variables, and this also requires 81 82 N fi and Ak N e. System ( 1) describes two different physical situations. In the first, spatial case, 61 = & = 0, 5 stands for the transverse coordinate, and Yj = 1/2kj (j = 1,2), SO that Eqs. ( 1) take into account the effect of diffraction. In the second, temporal case, 5 stands for time, Sj = dkj/dUj and yj = -~G”kj/$ describe group velocities and group-velocity dispersions, respectively. We are interested below in stationary solutions of Hqs. (l), when the walk-off effect and the wavevector mismatch are compensated due to nonlinearityinduced phase-locking, so that we apply the exact transformations E2=uK X1 exp(i&z + 2i@), (2) where ,Br and P2 s 281 Ak are the nonlinearityinduced shifts of the propagation constant, G = 1Yt1/1Y21, K = Pt + 610 + yl.n’, and L.J = (61 62) /2( 2Y2 Yt ) . Now the equations for w and u take the form .a~ a2w laf+ra72-W+W*u = 0, io-!E+sa2U_~~+Lw2 al a+ 2 = 0, where t = KZ, r = (IKl/lY11)“~(6 YZ), v = (2~24 r1~2>/(2~2 rl), r = sign(Kyl), s = sign(Kyz), and a = (/$ +2&L?+ 4y&)a/~. Eqs. (3) are the exact reduction of Eqs. ( 1) and they form the fundamental system to describe parametric interactions in dispersive/diffractive media with quadratic nonlinearity. For large a, when the derivatives in the equation for u can be neglected, we obtain u M w2/2cy and Eqs. (3) can be formally reduced to a single NLS equation for w which possesses bright (r = + 1) or dark (r = 1) solitons. However, this reduction is not satisfactory either physically or mathematically because (i) for large LY the intermode interaction can involve more than two harmonics, and (ii) formal localized solutions of the effective NLS equation can be nonstationary for Eqs. (3) (i.e. single solitons radiate) or they can be unstable due to paramettic modulational instability. Conditions for self-focusing (and localized soliton solutions) due to quadratic nonlinear%& are obtained by analyzing the modulational instability of tbe continuous wave (cw) solution of Hqs. (3), wi = 2a!, uo = 1. Analysis of the linear stability of this cw solution against modulations N exp( iql + ipr) reveals two branches for the dispersion relation, &(p2) = (Al f jjA: A;>/K (4) where At = 2~~ua + i(sp’ + (u) + $rp2a2(2 + rp*) andA~=4Cu2+rp2(2+rp2)(SP2+~)2-40(sp2+ a) ( 1 + rp2). In the limit of small p2 we have the following expansions, qf w -yp2, $ x q$+,up2, where d = a(4a + a)/a2, y = 2(2s + rcr)/(4u + (Y), and ,u = 2(r + w/d) + K. These two spectrum branches resemble “optical” and “acoustic” modes of a diatomic lattice and they exist due to relative and colA.V. Buryak, Y.S. Kivshar/Physics Letters A 197 (1995) 407-412 409 Bright solitons (r=+l, s=+l)