Spatiotemporal solitons in inhomogeneous nonlinear media

Ž . We investigate the possibility of forming spatiotemporal solitons optical bullets in inhomogeneous, dispersive nonlinear media using a graded-index Kerr medium as an example. We use a variational approach to solve the multidimensional, inhomogeneous, nonlinear Schrodinger equation and show that spatiotemporal solitons can be stabilized under certain ̈ conditions. We verify their existence by means of a full numerical analysis and show that such solitons should be observable experimentally. q 2000 Published by Elsevier Science B.V. All rights reserved. PACS: 42.65.-k; 42.65.Jx; 42.65.S; 42.65.Tg; 42.82.Dp Optical solitons have been shown to form and w x propagate inside a nonlinear Kerr medium 1–3 . They are called temporal or spatial solitons depending on whether their shape remains intact in time or in one space dimension. Mathematically, wave propagation in a Kerr medium is governed by the nonlinŽ . ear Schrodinger equation NLSE , which can be ̈ Ž . solved exactly in 1q1 -dimensions by using the w x inverse scattering method 1–3 . Whether an optical soliton can be confined simultaneously in space and time is a question that has attracted considerable w x attention in recent years 4–15 , and the term optical bullet has been coined for such a soliton. Among other applications, optical bullets are useful for allw x optical digital logic 9 . Most of previous work has focused on solitons formed in a self-focusing medium exhibiting anomalous dispersion. In this paper, we ) Corresponding author. E-mail: [email protected] consider pulse propagation in a general nonlinear, dispersive medium with special emphasis on a selfdefocusing medium with normal dispersion. An example of such a medium occurs for visible or nearinfrared light propagating inside a semiconductor. To counteract the beam spreading due to self-defocusing, we assume that the refractive index is spatially nonuniform as in a graded-index medium. An example of such a nonlinear medium is provided by the semiconductor-doped graded-index silica fiber. We analyze the possibility of formation of stable spatiotemporal solitons in an inhomogeneous Kerr medium, for which the refractive index is of the form 2 2 < < 2 n r ,v sn v qn x qy qn E , 1 Ž . Ž . Ž . Ž . 0 1 2 Ž . where the homogeneous part n v takes into ac0 count chromatic dispersion, n is the nonlinear pa2 rameter responsible for self-focusing or self-defocusing, and n governs the change in refractive index in 1 0030-4018r00r$ see front matter q 2000 Published by Elsevier Science B.V. All rights reserved. Ž . PII: S0030-4018 00 00727-6 ( ) S. RaghaÕan, G.P. AgrawalrOptics Communications 18