Propagation properties of Helmholtz power-law nonlinear surface waves

Optical surface waves are a fundamental class of excitation in inhomogeneous nonlinear photonic systems. This type of laser light satisfies a Helmholtz-type governing equation, and is subject to certain boundary conditions. It also has an asymmetric cross-sectional shape due to abrupt changes in material properties that define the (e.g., planar) interface. The stability properties of surface-wave solution branches are notoriously difficult to predict. On the one hand, classic criteria (e.g., Vakhitov-Kolokolov) often fail; on the other hand, numerical computations have, historically, been performed only with approximated governing equations (e.g., of the Schrödinger type). In recent research endeavours [J. M. Christian et al., J. At. Mol. Opt. Phys., in press], we have derived the surface waves for a Helmholtz-type interface model with refractive-index profile that depend on the local light amplitude to an arbitrary power 0 < q < 4. This generic optical nonlinearity describes classes of semiconductors, doped filter glasses, and liquid crystals. Here, we will present the first full investigation of Helmholtz nonlinear surface waves. Exact analytical solutions will be reported, and their properties explored. Extensive simulations with the full (i.e., un-approximated) governing equation have addressed some key issues surrounding the robustness of surface-wave solutions against spontaneous instabilities. New qualitative phenomena have also been predicted when considering interactions between surface waves and (obliquely-incident) spatial solitons (that is, self-collimated laser beams). The interaction angle between the surface-wave and incoming soliton is found to play a pivotal role in determining post-collision behaviour in the system, as is the nonlinearity exponent q

Scattering of Helmholtz spatial optical solitons at material interfaces

The behaviour of light at the interface between different materials essentially defines the entire field of Optics. Indeed, the reflection and refraction properties of plane waves at the boundary between two dissimilar linear dielectrics are analysed in many classic textbooks on electromagnetism. Our research tackles geometries that involve the interplay between diffraction (linear broadening) and self-focusing (nonlinear material response) when the incident light is in the form of a spatial soliton (self-collimated, self-stabilizing optical beam). Such systems are driven and dominated by complex light-medium feedback loops. The pivotal work of Aceves and co-workers some two decades ago investigated spatial solitons impinging on the interface between Kerr-type materials. Whilst these groundbreaking studies were highly instructive, their paraxial approach restricts angles of incidence, reflection and refraction to small values. Our recent proposal of a generalised Snell law, based on analysis of a nonlinear Helmholtz equation, lifts the angular limitation inherent to paraxial theory. This generalisation comprises a single multiplicative factor that allows for both transverse effects and discontinuities in material properties. Here, we will detail our latest research into bright spatial soliton refraction. In particular, our interest lies with arbitrary-angle scattering at the planar boundary between optical materials with universal non-Kerr nonlinearities: single power-law and cubic-quintic. This is the first time that arbitrary-angle refraction phenomena have been considered within these new material contexts. The derivation of our novel Helmholtz-Snell law will be described, and simulations demonstrating excellent agreement with theoretical predictions presented

Nonparaxial refraction and giant Goos-Hänchen shifts at nonlinear optical interfaces

The scattering of spatial optical solitons (self-localizing beams of laser light) at the interface between two dissimilar materials is a problem of fundamental importance in nonlinear photonics. Theoretical analyses must take into account a highly complex interplay between diffraction, self-lensing, finite beam waists, and discontinuities in both the linear and nonlinear properties of the host medium at the boundary. Over the past three decades, various research groups worldwide have resorted to simplified mathematical descriptions based on the universal nonlinear Schrödinger equation [A.B. Aceves et al., Phys. Rev. A vol. 39, 1809 (1989)]. Our approach deploys the nonlinear Helmholtz equation [J. Sánchez-Curto et al., Phys. Rev. A vol. 85, 013836 (2012)]. We have been able to relax the strong angular constraint that is inherent to essentially all previously-published works in this arena. More specifically, we can now solve the class of problem where beam angles of incidence, reflection, and refraction may be arbitrarily large. A compact law governing arbitrary-angle refraction will be discussed. Theoretical predictions are in excellent agreement with those obtained from exhaustive numerical simulations. Striking examples will also be given of Goos-Hänchen (GH) shifts (a phenomenon whereby, close to the critical angle of incidence, the reflected beam undergoes a displacement along the interface) [F. Goos and H. Hänchen, Ann. Phys. vol. 1, 333 (1947)]. Such shifts are an inherent property of beam-interface interactions, and they can be strongly enhanced in the presence of nonlinearity. We will report what we believe to be the largest GH shifts uncovered to date

Spatial soliton refraction at cubic-quintic material interfaces

In their most general form, wave–interface problems are inherently angular in nature. For instance, the interaction between light waves and material boundaries essentially defines the entire field of optics [1]. The seminal works of Aceves et al. [2,3] considered scalar bright spatial solitons impinging on the interface between two Kerr-type media with different dielectric parameters. While these classic analyses paved the way toward understanding how self-collimated light beams behave at medium discontinuities, they suffer from a fundamental limitation: the assumption of slowly-varying wave envelopes means that, in the laboratory frame, angles of incidence, reflection and refraction (relative to the interface) must be of vanishingly small magnitude. Over the last few years, the angular restriction of conventional (paraxial) nonlinear-Schrödinger modelling has been lifted by deploying a more flexible nonlinear-Helmholtz approach [4]. This mathematical platform is ideally suited to capturing the oblique-propagation aspects of interface scenarios We will report our latest research involving arbitrary-angle soliton refraction in more general classes of cubic-quintic materials [8], for which exact analytical bright [9] and dark [10] Helmholtz solitons are now known. A novel Snell’s law will be detailed that allows for both finite-beam effects and medium mismatches. Numerical computations test analytical predictions of soliton refraction and critical angles over a wide range of parameter regimes. Qualitatively new phenomena are also uncovered by simulations in both small- and large-angle regimes

Arbitrary-angle interaction of spatial solitons with layered photonic structures

The behaviour of light at the interface between different adjoining materials underpins the entire field of Optics. In nonlinear photonics, a fundamental geometry comprises a spatial soliton incident on the planar boundary between two dissimilar Kerr-type media. Seminal analyses by Aceves et al. [Phys. Rev. A 39, 1809 (1989)] were ground-breaking and highly instructive, but they remain limited by the assumptions of the paraxial approximation. Interface geometries are, in general, intrinsically nonparaxial: angles of incidence, reflection, and refraction (measured relative to the interface in the laboratory frame) may be of arbitrary magnitude. In earlier collaborations, we derived a Snell law governing arbitrary-angle refraction of spatial solitons at the interface between different Kerr materials [Opt. Lett. 35, 1347 (2010); 32, 1127 (2007)]. Analyses were facilitated by solution of an underlying nonlinear Helmholtz equation, and they completely lifted the angular limitation that is inherent to paraxial theory. Novel material considerations have been central to our most recent studies of spatial soliton refraction. In this presentation, we extend our preliminary Kerr-based analyses to non-Kerr regimes involving optical media with cubic-quintic nonlinear susceptibilities [Opt. Quantum Electron. 11, 471 (1979)]. A key result is the derivation of a generalized Snell law, which was obtained through the deployment of exact analytical bistable Helmholtz solitons [Phys. Rev. A 76, 033833 (2007)]. Excellent agreement has been uncovered, across wide parameter ranges, between theoretical predictions and direct numerical calculations. Simulations have also identified qualitatively new phenomena, strongly dependent on the beam incidence angle, that were not captured by analysis

Spatial solitons at interfaces: nonparaxial refraction & giant Goos-Hänchen shifts

The behaviour of light at interfaces underpins, in an essential way, the entire field of optics: almost all technological device designs and architectures rely on the interplay between material mismatches (that define the interface) and the 'degree of obliqueness' of the incident, reflected, and refracted waves. The seminal works on nonlinear beam refraction [1] considered scalar bright spatial solitons impinging on the interface between two Kerr-type media with different dielectric parameters, but where all angles (relative to the interface) were constrained to be near-negligibly small. Our Group has been developing new mathematical and computational tools to describe arbitrary-angle¬ refraction of similar beams [2]. The most recent research has been considering more general material aspects (e.g., from non-Kerr nonlinearities) and also giant Goos-Hänchen (GH) shifts [2,3]. Close to a critical point (which depends upon a complicated interplay between finite-beam characteristics and material mismatches), the GH shift shows a remarkable sensitivity to incidence angle, and also to medium nonlinearity. Indeed, we believe we have uncovered GH shifts that are unprecedented in magnitude, perhaps the largest ever predicted. A universal Snell’s law describing beam refraction has also been tested directly against full numerical calculations. References: [1] A. B. Aceves, J. V. Moloney, and A. C. Newell, Phys. Rev. A 39, 1809 (1989); Phys. Rev. A 39 1828 (1989). [2] J. Sánchez-Curto, P. Chamorro-Posada, and G. S. McDonald, Opt. Lett. 32, 1126 (2007); Opt. Lett. 36, 3605 (2011). [3] F. Goos and H. Hänchen, Ann. Phys. 1, 333 (1947)

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. The use of Helmholtz-type nonparaxial models allows for a complete angular characterization of spatial solitons, which is crucial in studies of this type. Most recently, we have been investigating systematic generalizations of the Kerr response to capture higher-order medium effects. For instance, the classic power-law model [8] replaces terms at |u|^2 with the generic form |u|^q, where the exponent q assumed a continuum of values 0 < q < 4; exact analytical power-law Helmholtz solitons [9] can then be deployed as a nonlinear basis for analyzing refraction phenomena. This model plays a fundamental role in photonics, and can be used to capture, for instance, leading-order effects due to saturation of the nonlinear-optical response. In this presentation we detail our latest research, which has investigated arbitrary-angle scattering of spatial solitons at the planar boundary between two dissimilar cubic-quintic optical materials [10,11]. This is, to the best of our knowledge, the first time that refraction effects have been considered within this type of material context. The derivation of our novel Helmholtz refraction law will be discussed in detail. An illustrative selection of results will be presented, which compare theoretical predictions to full numerical computations. The level of agreement between the theoretical predictions and full numerical computations has been found to be generally excellent. Examples of the Goos-Hänchen shift (a phenomenon occurring close to the critical angle, where the trajectory of the reflected beam is displaced laterally relative to the path predicted by geometrical optics [12,13]) have also been uncovered

Arbitrary-angle scattering of spatial solitons from dielectric optical interfaces

Wave–interface phenomena appear throughout Nature and are intrinsically angular in character. For instance, the behaviour of light waves at material boundaries underpins the entire field of optics. The seminal works on nonlinear beam refraction [1] considered scalar bright spatial solitons impinging on the interface between two Kerr-type media with different dielectric parameters, but where angles of incidence, reflection and refraction (relative to the interface) were constrained to be near-negligibly small. Our Group has recently been developing new mathematical and computational tools to describe arbitrary-angle refraction of similar beams [2]. We will report our latest research involving higher-order material effects (e.g., quintic-type nonlinearity). A generalized Snell’s law describing beam refraction will be detailed and tested directly against fully-numerical calculations. New effects are also uncovered, through simulations, in both small- and large-angle regimes

Refraction at interfaces with X(5) nonlinearity: Snell’s law & Goos-Hänchen shifts

In this presentation, we give the first detailed overview of spatial soliton refraction at the planar interface between materials whose nonlinear polarization has contributions from both X(3)and X(5)susceptibilities [1]. The governing equation is of the inhomogeneous Helmholtz class with a cubic-quintic nonlinearity, and analysis is facilitated through the exact bright soliton solutions of the corresponding homogeneous problem [2]. By respecting field continuity conditions at the interface, a universal Snell’s law may be derived for describing the refractive properties of soliton beams. This compact equation contains a supplementary multiplicative factor that captures system nonlinearity, discontinuities in material properties, and finite beam waists. Extensive numerical calculations have tested analytical predictions and provided strong supporting evidence for the validity of our modelling approach across wide regions of a six-dimensional parameter space. Theoretical predictions for critical angles are generally in good agreement with simulations of beams at linear and weakly-nonlinear interfaces, and we have quantified Goos-Hänchen shifts [3,4] in such systems (see Fig. 1). References [1] K. I. Pushkarov, D. I. Pushkarov, and I. V. Tomov, Opt. Quantum Electron. 11, 471 (1979). [2] J. M. Christian, G. S. McDonald, and P. Chamorro-Posada, Phys. Rev. A 76, 033833 (2007). [3] F. Goos and H. Hänchen, Ann. Phys. 1, 333 (1947). [4] J. Sánchez-Curto, P. Chamorro-Posada, and G. S. McDonald, Opt. Lett. 36, 3605 (2011)

Helmholtz surface wave propagation along nonlinear interfaces

When two dissimilar nonlinear photonic materials are placed together, the boundary between them forms an optical interface. A light beam (typically from a laser source) may then travel along the boundary as a surface wave, remaining trapped within the vicinity of the interface and possessing a stationary (invariant) spatial profile. This type of bi-layer structure is an elementary geometry for integrated-optic device architectures. For two Kerr-type materials, surface-wave solutions to the governing equation have been known for many years [e.g., Aceves et al., Phys. Rev. A vol. 39, 1809 (1989)]. To date, many research groups worldwide have performed numerical investigations of related phenomena. A recurrent theme in the literature is the replacement of the underlying Helmholtz equation with a simpler Schrödinger-type model (by assuming slowly-varying envelopes). Hence, there has been essentially no analysis of surface waves in the Helmholtz context. In this presentation, we will give a detailed account of Helmholtz surface waves propagating along nonlinear interfaces – this is the first analysis of its kind [Christian et al., J. Atom. Mol. Opt. Phys. vol. 2012, art. no. 137967 (2012)]. Theoretical predictions of surface-wave stability (made by deploying the classic Vakhitov-Kolokolov integral criterion) are tested against fully-numerical Helmholtz-type computations using our (custom) difference-differential algorithm [Chamorro-Posada, McDonald, & New, Opt. Commun. vol. 192, 1 (2001)]. Extensive simulations have uncovered a wide range of new qualitative and quantitative phenomena in the Helmholtz regime that depend on the interplay between a set of system parameters (material mismatches, nonlinearity exponent, and the optical beam waist)