Exponential Asymptotics and Spectral Theory for Curved Optical Waveguides

Abstract

The purpose of this paper is to present recent work on a class of eigenvalue problems for ordinary differential equations arising in optical tunnelling from a weakly-guiding fibre, and to discuss some generalisations which are of mathematical rather than physical interest. It is based on publications with colleagues at Dublin City University and with Dr. R. B. Paris of Dundee Institute of Technology. These seven papers fall into three distinct groups. The first group ([16], [5]) consists of one-dimensional models which seek to provide a deeper understanding of the mechanisms described by Kath and Kriegsmann [9] in determining radiation losses in bent fibre-optic waveguides. The second group considers related eigenvalue problems for singularly-perturbed ordinary differential equations ([4], [11]) and their connection to the theory of resonances in quantum mechanics [10]. The physical problem in the above papers is essentially one of radiation damping, a difficult area of transcendental asymptotics which has yet to be given a satisfactory general treatment, even for ordinary differential equations. For another non-quantum mechanics application, arising in the theory of surface waves trapped by round islands of small seabed slope, the reader is referred to Lozano and Meyer [12]. The third group ([17], [18]) is concerned with obtaining exponentially-improved asymptotic expansions for special functions which appear in the eigenvalue relations above.