Jm Christian

Fresnel diffraction and fractal patterns from polygonal apertures

Two compact analytical descriptions of Fresnel diffraction patterns from polygonal apertures under uniform illumination are detailed. In particular, a simple expression for the diffracted field from constituent edges is derived. These results have fundamental importance as well as specific applications, and they promise new physical insights into diffraction-related phenomena. The usefulness of the formulations is illuminated in the context of a virtual source theory that accounts for two transverse dimensions. This application permits calculation of fractal unstable-resonator modes of arbitrary order and unprecedented accuracy.

Helmholtz algebraic solitons

We report, to the best of our knowledge, the first exact analytical algebraic solitons of a generalized cubic-quintic Helmholtz equation. This class of governing equation plays a key role in photonics modelling, allowing a full description of the propagation and interaction of broad scalar beams. New conservation laws are presented, and the recovery of paraxial results is discussed in detail. The stability properties of the new solitons are investigated by combining semi-analytical methods and computer simulations. In particular, new general stability regimes are reported for algebraic bright solitons.

Exact dipole solitary wave solution in metamaterials with higher-order dispersion

Abstract We present an exact dipole solitary wave solution in a mutual modulation form of bright and dark solitons for a higher-order nonlinear Schrödinger equation with third- and fourth-order dispersion in metamaterials (MMs) using an ansatz method. Based on the Drude model, the formation conditions, existence regions and propagation properties are discussed. The results reveal that the solitary wave may exist in a few parameter regions of MMs, different from those in optical fibres, and its propagation properties can be controlled by adjusting the frequency of incident waves in each existence region.

Spontaneous spatial fractal pattern formation in dispersive systems

We report spontaneous spatial optical fractal patterns in a ring cavity containing a thin slice of diffusive Kerr-type material. The Turing threshold instability condition is derived through linear analysis, and static patterns are found to be described by spectra with multiple-minimum characteristics. These theoretical predictions are subsequently verified through numerical simulations with both one and two transverse dimensions. Our findings support that a proposed fractal-generating criterion for nonlinear wave-based systems with thin-slice host media can have independence with respect to both system geometry and nonlinearity. We conclude by detailing further potential research directions and possible applications of fractal light sources.

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

We present, to the best of our knowledge, the first exact dark spatial solitons of a nonlinear Helmholtz equation with a self-defocusing saturable refractive-index model. These solutions capture oblique (arbitrary-angle) propagation in both the forward and backward directions, and they can also exhibit a bistability characteristic. A detailed derivation is presented, obtained by combining coordinate transformations and direct-integration methods, and the corresponding solutions of paraxial theory are recovered asymptotically as a subset. Simulations examine the robustness of the new Helmholtz solitons, with stationary states emerging from a range of perturbed input beams.

SPONTANEOUS SPATIAL FRACTAL PATTERN FORMATION IN ABSORPTIVE SYSTEMS

We predict, for the first time to our knowledge, that purely-absorptive nonlinearity can support spontaneous spatial fractal pattern formation. A passive optical ring cavity with a thin slice of saturable absorber is analyzed. Linear stability analysis yields threshold curves for Turing (static) instabilities with features proposed as characteristics of potential fractal pattern formation. Numerical simulations of the fully-nonlinear dynamics, with both one and two transverse dimensions, confirm theoretical predictions.

Helmholtz solitons in optical materials with a dual power-law refractive index

A nonlinear Helmholtz equation is proposed for modelling scalar optical beams in uniform planar waveguides whose refractive index exhibits a purely-focusing dual power-law dependence on the electric field amplitude. Two families of exact analytical solitons, describing forward- and backward-propagating beams, are derived. These solutions are physically and mathematically distinct from those recently discovered for related nonlinearities. The geometry of the new solitons is examined, conservation laws are reported, and classic paraxial predictions are recovered in a simultaneous multiple limit. Conventional semi-analytical techniques assist in studying the stability of these nonparaxial solitons, whose propagation properties are investigated through extensive simulations.

Optical soliton pulses with relativistic characteristics

The slowly varying envelope approximation and the ensuing Galilean boost to a local time frame are near-universal features of conventional scalar pulse models. Here, we will give an overview of our recent progress with a new approach to nonlinear pulse modelling, which is based on a Helmholtz-type formalism.

Korteweg-de Vries description of Helmholtz-Kerr dark solitons

A wide variety of different physical systems can be described by a relatively small set of universal equations. For example, small-amplitude nonlinear Schrodinger dark solitons can be described by a Korteweg-de Vries (KdV) equation. Reductive perturbation theory, based on linear boosts and Gallilean transformations, is often employed to establish connections to and between such universal equations. Here, a novel analytical approach reveals that the evolution of small-amplitude Helmholtz–Kerr dark solitons is also governed by a KdV equation. This broadens the class of nonlinear systems that are known to possess KdV soliton solutions, and provides a framework for perturbative analyses when propagation angles are not negligibly small. The derivation of this KdV equation involves an element that appears new to weakly nonlinear analyses, since transformations are required to preserve the rotational symmetry inherent to Helmholtz-type equations.

From Maxwell's equations to new families of Helmholtz solitons

In this presentation, we give an overview of new results in Helmholtz soliton theory. Firstly, fundamental considerations are made in terms of new contexts for Helmholtz solitons that arise directly from Maxwells equations. We will then explore applications involving a variety of different material interfaces and the role of Helmholtz solitons in these configurations. Finally, specific new families of solutions arising from the generalisation of the Manakov equation will be reported.