Non-paraxial solitons

Abstract

In this paper, we propose the use of ultranarrow soliton beams in miniaturized nonlinear optical devices. We derive a nonparaxial nonlinear Schrödinger equation and show that it has an exact non-paraxial soliton solution from which the paraxial soliton is recovered in the appropriate limit. The physical and mathematical geometry of the non-paraxial soliton is explored through the consideration of dispersion relations, rotational transformations and approximate solutions. We highlight some of the unphysical aspects of the paraxial limit and report modifications to the soliton width, the soliton area and the soliton (phase) period which result from the breakdown of the slowly varying envelope approximation

Dark solitons at nonlinear interfaces

“…Nonparaxial theory based on the nonlinear Helmholtz (NLH) equation [15,16] permits one to overcome intrinsic angular limitations of NLS descriptions. In contrast to other nonparaxial regimes [17,18], where effects have their origin in the strong focusing of high-intensity beams, we consider broad (compared to the optical wavelength) beams of moderate power.…”

Nonlinear interfaces: intrinsically nonparaxial regimes and effects

“…In absence of discontinuity, ∆ = 0 and α = 1, one recovers the NLH for a homogeneous medium [24] from (3), which can be written as [37] κ ∂ 2 u ∂ζ 2 + i ∂u ∂ζ + 1 2…”

“…In (2), κ = 1/k 2 w 2 0 is a nonparaxiality parameter [23,24] and n 01 is the linear refractive index of a first Kerr-type nonlinear material whose total refractive index is n 01 + α 1 |E| 2 where α 1 n 01 . If we now include in our analysis a second Kerr-type medium with n = n 02 + α 2 |E| 2 and consider the normalisation…”

“…This nonparaxiality is well described by the scalar nonlinear Helmholtz (NLH) equation [23,24] which has been proposed to overcome the limitations of the NLS, for instance, by arresting soliton collapse in a focusing Kerr-type medium [23] and for which exact analytical soliton solutions have been found [24,25,26]. Substantial differences with paraxial theory are not only revealed by the exact bright Kerr soliton solutions of the NLH equation but are also found in dark Kerr [27], two-component [28], boundary [29] and bistable [30] Helmholtz soliton solutions.…”

Exact analytical Helmholtz bright and dark solitons

“…Physical interpretations [6] and analytical properties [7] of Helmholtz bright solitons have been presented and allowed the development and testing of new nonparaxial beam propagation techniques [8]. To highlight the modifications to paraxial theory, here we solve the equivalent focusing NNLS [5][6][7][8]:…”