Asymptotic analysis of array-guided waves

Abstract

We develop and apply computational and analytical techniques to study wave-like propagation and resonant effects in periodic and quasi-periodic systems. Two themes that unify the content herein are the guidance and confinement of energy using periodic structures, and the utility of asymptotic analysis to aid computation and produce results that lend physical insight to the problems in question.

In the first research chapter, we develop the method of high-frequency homogenisation (HFH) for electromagnetic waves in dielectric media, and apply this to the example of a planar array of dielectric spheres. The theory conveniently describes a range of dynamic effects, including effectively anisotropic behaviour in certain frequency regimes.

In the second research chapter, we apply the HFH method to a cylindrical Bragg fibre, and use this to set up an effective eigenvalue problem in which the quasi-periodic system representing the fibre cladding is represented by a single continuous Bessel-like equation. We compare the results with those of direct numerical simulations and discuss how the theory could be developed to aid the study of photonic crystal cavities or fibres.

In the remaining chapters, we consider the complex resonances of structures with angular periodicity. We demonstrate the emergence of quasi-normal modes with high Q-factors for the Helmholtz equation in such domains, and explore some of their properties using multiple scale analysis. In the final two chapters, we focus on a particular subset of these domains, and using matched asymptotic expansions show that the Q-factors for certain solutions depend exponentially on the number of inclusions arranged in a circular ring. Finally, we extend this analysis to flexural waves in thin elastic plates, and discuss the possibility of structured-ring resonators based on these solutions.