G. S. McDonald

Pattern formation in a passive Kerr cavity

Analytic and numerical investigations of a cavity containing a Kerr medium are reported. The mean field equation with plane-wave excitation and diffraction is assumed. Stable hexagons are dominant close to threshold for a self-focusing medium. Bistable switching frustrates pattern formation for a self-defocusing medium. Under appropriate parametric conditions that we identify, there is coexistence of a homogeneous stationary solution, of a hexagonal pattern solution and of a large (in principle infinite) number of localized structure solutions which connect the homogeneous and hexagonal state. Further above threshold, the hexagons show defects, and then break up with apparent turbulence. For Gaussian beam excitation, the different symmetry leads to polygon formation for narrow beams, but quasihexagonal structures appear for broader beams.

Nondiffracting beams: travelling, standing, rotating and spiral waves

A reformulation of nondiffracting beams, based on more general (travelling wave) solutions of the nonparaxial wave equation, is presented. Zero order nondiffracting beams are found to be radial standing waves arising from counterpropagating zero order Hankel waves of the first and second kind, while higher order nondiffracting beams are formed from counter-rotating spiral waves which are described by Hankel functions of the corresponding order. The resulting physical picture is more general than the well-known integral representation of Bessel functions and we expect it to have implications for studies of the applications of nondiffracting beams. Generic descriptions of the transverse profiles of the electric field, applicable to experimental configurations for realising nondiffracting beams, follow directly from this formulation. Finally, the existence of classes of periodically nondiffracting beams, possessing finite angular momentum and having the characteristics of rotating and spiral waves, is predicted.

Fractal modes in unstable resonators

One of the simplest optical systems, consisting of two mirrors facing each other to form a resonator, turns out to have a surprising property. Stable resonators, in which the paths of the rays are confined between the two mirrors, have a well known mode structure (hermite–gaussian), but the nature of the modes that can occur in unstable reson-ant cavities (from which the rays ultimately escape) are harder to calculate, particularly for real three-dimensional situations. Here we show that these peculiar eigenmodes of unstable resonators are fractals, a finding that may lead to a better understanding of phenomena such as chaotic scattering and pattern formation. Our discovery may have practical application to lasers based on unstable resonators.

Non-paraxial beam propagation methods

Exact analytical results are employed in the testing of split-step and finite difference approaches to the numerical solution of the non-paraxial non-linear Schrodinger equation. It is shown that conventional split-step schemes can lead to spurious oscillations in the solution and that fully finite difference descriptions may require prohibitive discretisation densities. Two new non-paraxial beam propagation methods, that overcome these difficulties, are reported. A modified split-step method and a difference-differential equation method are described and their predictions are validated using dispersion relations, an energy flow conservation relation and exact solutions. To conclude, results concerning 2D (transverse) beam self-focusing, for which no exact analytical solutions exist, are presented.

Non-paraxial solitons

In this paper, we propose the use of ultranarrow soliton beams in miniaturized nonlinear optical devices. We derive a nonparaxial nonlinear Schrodinger equation and show that it has an exact non-paraxial soliton solution from which the paraxial soliton is recovered in the appropriate limit. The physical and mathematical geometry of the non-paraxial soliton is explored through the consideration of dispersion relations, rotational transformations and approximate solutions. We highlight some of the unphysical aspects of the paraxial limit and report modifications to the soliton width, the soliton area and the soliton (phase) period which result from the breakdown of the slowly varying envelope approximation.

Optical vortices in beam propagation through a self-defocussing medium

We present numerical simulations and analysis of optical vortex propagation in a 3D self-defocussing medium. On plane wave background initial conditions composed of points and lines of phase discontinuity are shown, in some cases, to evolve towards a regular crystal of fully nonlinear vortices. In other cases the solution more resembles a vortex gas. For gaussian beams, we have derived a simple set of ordinary differential equations which are shown to accurately describe the motion of a single vortex while the background beam undergoes significant spreading.

Helmholtz solitons at nonlinear interfaces.

The evolution of Helmholtz solitons at the interface separating two Kerr-type media is described by means of a generalized nonlinear Helmholtz equation. The bright soliton solution for this equation and arbitrary media properties are presented. Numerical simulations show that, when there is only mismatch in the linear refractive index, Helmholtz solitons behave according to Snells law. Also, the general reflection and refraction properties of optical solitons show features that cannot be captured in the paraxial theory.

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.

Dark spatial soliton break-up in the transverse plane

We report on numerical simulations investigating the instabilities which arise when dark soliton solutions of the two-dimensional (2D) nonlinear Schrodinger equation are allowed to propagate in a 3D self-defocussing medium. Firstly, propagation of fully 3D gaussians beams is studied. Then, on plane background, small sinusoidal and random perturbations are considered. We demonstrate break-up of the dark soliton profile into patterns of dark spots which are identified as phase singularities.

Exact soliton solutions of the nonlinear Helmholtz equation: communication

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