Nonparaxial spatial solitons and propagation-invariant pattern solutions in optical Kerr media

Abstract

We investigate nonlinear propagation in the presence of the optical Kerr effect by relying on a rigorous generalization of the standard parabolic equation that includes nonparaxial and vectorial terms. We show that, in the ͑1 1 1͒-D case, both soliton and propagation-invariant pattern solutions exist (while the standard hyperbolic-secant function is not a solution). © 2004 Optical Society of America OCIS codes: 190.0190, 190.3270, 190.5530. Monochromatic optical propagation in the presence of a refractive-index distribution n͑r͒ n 0 1 dn͑r͒ is usually described by the scalar parabolic equation. This is a by-product of the Helmholtz equationand the scalar parabolic equation is derived from Eq.(1) in the paraxial approximation. More precisely, after writing the electric f ield as E͑r, t͒ A͑r Ќ , z͒exp͑ikz 2 ivt͒, where v is the angular frequency, k ͑v͞c͒n 0 , and introducing d as a typical length scale of variations in n͑r͒, we assume that the parameter h ͑Dn͞n͒ ͑l͞d͒ , , 1, which expresses the smallness of the relative variation of the refractive index over a wavelength scale. Besides, if the f ield is initially (approximately) transversely polarized, we neglect its longitudinal component A z with respect to the transverse one A Ќ ͑r Ќ , z͒, and we can derive, using the slowly varying approximation, the standard scalar parabolic equation:We can then automatically satisfy the divergence equation = ? ͑eE͒ 0 (where e e 0 n 2 ) by using it to derive the (small) longitudinal component A z . If the smallness parameter h becomes comparable to 1, then the approach presented above, which represents only the lowest-order approximation in h, fails, and higher-order terms have to be added to account for nonparaxial contributions. The recent progress in nanotechnology and the possibility of fabricating optical structures with subwavelength features are compelling reasons for generalizing Eq. (2) to the nonparaxial regime. This can be done rigorously if, after splitting the f ield into a transverse and a longitudinal part, one is able to derive an equation for the transverse part alone to first order in ≠͞≠z that contains terms to all orders in h. This task was recently carried out in the general case (linear or nonlinear) of a tensorial refractive-index distribution, $ n ͑r͒, in a manner that fully preserved the vectorial nature of the problem.

Nonparaxial dark solitons in optical Kerr media

“…3 These solitons, besides being reduced to standard paraxial solitons in the limit of small normalized peak intensity u 2 0 , as expected, present new and interesting features. In particular, the soliton width turns out to be practically independent of the peak intensity for u 2 0 .…”

“…The analytical approach presented below and in Ref. 3 would allow, if they were applied to those equations, one to prove the existence of spatial nonparaxial solitons of the same form as our solitons but numerically different in amplitude, width, and nonlinear phase.…”

“…The bright-soliton case was discussed in Ref. 3, where the existence of bright solitons has been proved for g . 0 (no bright soliton exists for g , 0).…”

Bistable Helmholtz dark spatial optical solitons in materials with self-defocusing saturable nonlinearity

“…46,47 Hence, the polarization-scrambling term ∇(∇ · E) can be safely neglected [14][15][16]48,49 and attention is paid exclusively to scalar diffraction. 26,41 We consider a TE-polarized cw beam, as represented bỹ…”

Nonlinear interfaces: intrinsically nonparaxial regimes and effects

“…For ultra-narrow beams, a full vectorial analysis starting from the Maxwell equations can also be necessary [17,18,19] to include the tensorial refractive index dependence. Solutions to these equations in the form of bright [20] and dark [21,22] nonparaxial solitons have been reported and analysed.…”