Nonlinear wave phenomena at optical boundaries: spatial soliton refraction

The behaviour of light at the interface between different materials underpins the entire field of Optics. Arguably the simplest manifestation of this phenomenon – the reflection and refraction characteristics of (infinitely-wide) plane waves at the boundary between two linear dielectric materials – can be found in many standard textbooks on electromagnetism (for instance, see Ref. 1). However, laser sources tend to produce collimated output in the form of a beam (whose transverse cross-section is typically “bell-shaped” and, hence, finite). When describing light beyond the plane-wave limit, a more involved and sophisticated treatment is generally required. If a beam of light is propagating in a nonlinear planar waveguide, its innate tendency to diffract (spread out) may be compensated by the self-lensing properties of the host medium (whose refractive index is intensitydependent). Such nonlinear photonic systems are driven and dominated by complex light-medium feedback loops. Under the right conditions (e.g., where the amplitude and phase of the input beam have the correct transverse distribution), the light may evolve with a stationary (invariant) intensity profile. Such self-localizing and self-stabilizing nonlinear waves are known as spatial optical solitons. The seminal analyses of spatial soliton refraction were performed by Aceves, Moloney, and Newell [2,3] more than two decades ago. They reduced the full electromagnetic complexity of the interface problem by adopting the scalar approximation, and describing the light field within an intuitive (paraxial) nonlinear Schrödinger-type framework. In this way, they provided the first description of how light beams behave when impinging on the boundary between two dissimilar Kerr-type materials. These early, ground-breaking works yielded a great deal of physical insight. However, the assumption of beam paraxiality necessarily restricts the angles of incidence, reflection, and refraction (relative to the interface) to negligibly (or near-negligibly) small values. Ideally, one would like to find a way of lifting this inherent angular limitation without forfeiting a straightforward and analytically-tractable governing equation. Recently, the seminal works of Aceves et al. [2,3] have been built upon by our Group. A new modelling formalism has been developed, based on an inhomogeneous nonlinear Helmholtz equation, whose flexibility accommodates both bright [4,5] and dark [6,7] spatial solitons at arbitrary angles of incidence, reflection, and refraction with respect to the interface: