Helmholtz dark solitons

Abstract

A general dark-soliton solution of the Helmholtz equation (with defocusing Kerr nonlinearity) that has on- and off-axis, gray and black, paraxial and Helmholtz solitons as particular solutions, is reported. Modifications to soliton transverse velocity, width, phase period, and existence conditions are derived and explained in geometrical terms. Simulations verify analytical predictions and also demonstrate spontaneous formation of Helmholtz solitons and transparency of their interactions.

Bistable dark solitons of a cubic-quintic Helmholtz equation

“…In this paper, a Helmholtz model has been proposed for describing broad The dark solutions studied in this paper complement their bright counterparts [20], and extend our earlier analyses [32] to more general classes of nonlinear materials. Solitons, and their wave equations, are universal features across many areas of nonlinear science.…”

Nonlinear interfaces: intrinsically nonparaxial regimes and effects

“…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. When the full Helmholtz approach is used, significant differences with the predictions of NLS theory are also found at a fundamental level, for example, when analysing soliton collisions [31].…”

“…Backscattered waves are filtered out, thus avoiding an evanescent backward field, that can appear to grow in the forward direction and hence masks the contribution of the forward propagating field. This scheme has been applied to the phenomena studied in [27,28,29,30,31].…”

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

“…In contrast, paraxial diffraction equations tend to be parabolic and thus easier to solve mathematically and computationally. More subtly, the symmetric nature of nonlinear Helmholtz equations can often rule out the existence of exact solutions that are analogues of those commonly found in paraxial theory (where standard solution methods can start to break down 28,40 ). Spatial symmetry, then, has to be built into any analytical approach from the outset (e.g., solution decomposition and reduced equations).…”

“…20,21 The dark solitons studied here are non-trivial spatially-symmetric generalizations of the classic solutions derived by Krolikowski and Luther-Davies. 23 We have thus derived convenient nonlinear basis functions for analyzing dark beams in a wide range of arbitrary-angle scenarios, and where the description of a material's optical response goes beyond the traditional cubic 28 and cubic-quintic 25 power-series idealizations.…”