Optical waveguide theory. Mathematical models, spectral theory and numerical analysis (Q2122568)

In this textbook, the authors consider eigenvalue problems of partial differential equations (PDEs), which model optical waveguides. Thus wave equations and Maxwell's equations, which are PDEs of hyperbolic type, are investigated. The authors derive a variational formulation for each problem using Sobolev spaces. The spectrum of each resulting linear operator is analysed. The authors conclude the existence of real and complex waves. In some cases, the asymptotic distribution of the eigenvalues or the completeness of the system of eigenfunctions is examined. A minor part of the textbook deals with a numerical simulation of the eigenvalue problems. The authors reduce the PDE problem to a problem with an ordinary differential equation (ODE) of second order for a tangential component only. A collection of initial value problems of the ODE is solved to determine propagation constants. The textbook includes six chapters. In Chapter 1, a short introduction is given. The Chapters 2--5 discuss four types of problems: shielded waveguide, planar waveguide, waveguides of circular cross section, and open waveguides. Therein, several subsections address associated subtypes. Most subsections have the same structure: problem definition, Sobolev spaces and variational formulation, spectrum of the problem and its properties. Conclusions are summarised in Chapter 6. The authors present the properties of the investigated spectra using theorems and lemmata, where always a proof is included. Some numerical results are illustrated by figures in two of the chapters. Each chapter exhibits a list of references at its end. Exercises are not contained in this textbook. The mathematical contents of this contribution are based on previous publications of the authors in journals and conference proceedings. This textbook is dedicated to mathematicians working in electromagnetic theory as well as engineers with deeper knowledge in mathematics.