Exact soliton solutions of the nonlinear Helmholtz equation: communication

Abstract

Exact analytical soliton solutions of the nonlinear Helmholtz equation are reported. A lucid generalization of paraxial soliton theory that incorporates nonparaxial effects is found.

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

“…The spatial asymptotics of these solitons are mapped onto a plane-wave field using similar techniques to those established in Ref. 25, and geometrical transformations 26 are deployed to obtain more general off-axis solutions. A bistability characteristic is also discussed.…”

“…Helmholtz soliton theory then provides a compact and elegant mathematical formalism for modelling oblique-evolution properties of scalar beams at arbitrary angles (in the laboratory frame) with respect to the longitudinal reference direction. 26 Indeed, off-axis considerations are elementary for even the simplest configurations: (i) interacting beams 40 and (ii) reflection / refraction of bright 42,43 and dark 44,45 solitons at interfaces between dissimilar materials. The Helmholtz angular type of nonparaxiality thus underpins, in principle, almost every conceivable optical device design and architecture.…”

Nonlinear interfaces: intrinsically nonparaxial regimes and effects

“…By phase-matching of exact soliton solutions [26] for ξ < 0 and ξ ≥ 0, one obtains a generalised Snell's law [37] that governs the evolution of beams at boundary separating two Kerr focusing media γn 01 cos(θ i ) = n 02 cos(θ t ).…”

“…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.…”

Helmholtz solitons in non-Kerr media

“…Models based on the NLH equation are suitable for describing accurately the angular aspects of wave propagation [2]. Since the assumption of beam paraxiality is omitted, such descriptions can support both travelling-and standing-wave solutions.…”