Colin Pask

Gaussian and equivalent-step-index approximations for nonlinear waveguides

The field of the fundamental waveguide mode, when it propagates at extremely high intensities or when the core or cladding material has a large nonlinear coefficient, may be quite significantly distorted from that of the corresponding linear mode. We derive a variational formulation of the scalar wave equation for waveguides with arbitrary nonlinearity in the core and show that this formulation can be used to find simple, but accurate, analytical approximations for these nonlinear fields. In particular, we find Gaussian and equivalent-step-index approximations for a variety of planar waveguide and optical-fiber structures and show how they can be used to calculate quantities such as the effective area and group-velocity dispersion.

Variational and finite element analyses of nonlinear strip optical waveguides

Abstract A new, iterative finite element scheme for solving nonlinear wave guiding problems is proposed. Unlike previous numerical methods, this approach gives both the stable and unstable stationary solutions of a nonlinear strip waveguide. We use a double gaussian trial function to solve the same problem by variational principles. This approximation gives very good results and provides a simple model to characterize nonlinear stationary waves. The influence of waveguide parameter on nonlinear behaviour is discussed. We also apply both the variational and finite methods to analyze the stripe surface waves at a linear-nonlinear interface.

Theoretical study of spatial solitons in planar waveguides

We investigate some of the theoretical notions underlying the observation of spatial solitons in nonlinear planar waveguides. We find that, when the beam is confined in one dimension principally by the action of a linear refractive-index profile, the nonlinear behavior of the beam in the orthogonal dimension is governed by the usual nonlinear Schrodinger equation with parameters modified by the linear waveguide modal properties. When either the power or the nonlinearity of the material is high, nonlinearity affects both dimensions, and a form of three-dimensional self-trapping begins to occur. A simple variational approximation gives an accurate picture of what is happening in this regime.

Wave propagation in a graded-index taper

We present a model of a graded-index taper for which the field solutions can be obtained directly by separation of variables. Those fields tightly concentrated about the axis of the taper are given very accurately by remarkably simple expressions which clearly illustrate the influence of the tapering. Frequent comparisons with a geometrical optics analysis demonstrate the link between the ray and field approaches and also assist in the physical interpretation of several results. Examples of the application of these field solutions are also described.

Variational approximations and mode stability in planar nonlinear waveguides

In a previously published paper we used a variational method with simple trial functions to find accurate approximations for the modes of waveguides in which the nonlinearity is confined mainly to the core. Systems in which the nonlinearity occurs in the cladding present more difficulties because the solution at a particular power may not be unique. There may be two, or more, nonlinear modes with quite different intensity profiles and stability properties corresponding to the one linear mode. We apply our variational method to planar waveguides with this more general distribution of nonlinear material and also investigate the relationship between the stationary properties of our solutions and modal stability.

Simple model for spatial optical solitons in planar waveguides

We use a variational method to develop a simple model for the steady-state behavior of self-trapped beams that have recently been observed in nonlinear planar waveguides. The model allows us to study the transition from two-dimensional to three-dimensional self-trapping as power levels increase and also to investigate the effect of saturation on these beams.

The missing wave momentum mystery

The usual suggestion for the longitudinally propagating momentum carried by a transverse wave on a string is shown to lead to paradoxes. Numerical simulations provide clues for resolving these paradoxes. The usual formula for wave momentum should be changed by a factor of 2 and the involvement of the cogenerated longitudinal waves is shown to be of crucial importance.

Optical properties of retinal photoreceptors and the Campbell effect

In 1958, Campbell observed that certain artificial pupil displacements could considerably change acuity (measured by viewing gratings) while others had very little effect. He sought an explanation of the small retinal contribution to those effects that was consistent with the Stiles-Crawford effect. This paper suggests an explanation that satisfies that requirement using a waveguide model of the retinal cones. We show that the waveguiding properties of the receptors make them sensitive to obliquely incident exciting waves and this provides some support for the hypothesis that both the Stiles-Crawford and Campbell effects are manifestations of the same underlying waveguide nature of the receptors.

Interplay of grating and nonlinearity in mode coupling

The effect of a uniform grating on the evolution of the orthogonal polarization modes in a nonlinear birefringent waveguide is examined for the first time to our knowledge. The problem involves the interplay of the linear coupling that is due to the grating, the linear birefringence, and the nonlinear mode coupling. After examining the basic formulation and methods of solution, we present a series of results demonstrating a variety of nonlinear phenomena, such as bistability. Other two-mode systems are predicted to behave in a similar way. The possible implementation of three logic gates from one device configuration is discussed.

Mathematics and the science of analogies

Analogies have been used to great effect in the physical sciences. A brief review is given of the relevant points about the role of mathematics in physics and a simple framework in which analogies are viewed in terms of mechanism and descriptive, usually mathematical, components. The main body of the paper analyzes cases, principally in wave theory and electromagnetism, to emphasize three points: the importance of the mathematical description component in powerful analogies; the possibly destructive aspect of the mechanism component of analogies; and the underlying reasons for the choice of particular analogies.