Preface

This is the first volume of a new e-book series that is devoted to very recent research trends in computational physics. Hereby, it focuses on the computational perspectives of current physical challenges, publishing new numerical techniques for the solution of mathematical equations including chapters describing certain real-world applications concisely. The goal of this series is to emphasize especially approaches that are of interdisciplinary nature.

Nowadays, with the help of powerful computers and sophisticated methods of numerical mathematics it is possible to simulate many ultramodern devices, e.g. photonic crystal structures, semiconductor nanostructures or fuel cell stacks devices, thus preventing expensive and longstanding design and optimization in the laboratories. This first volume treats both, mathematical analysis of periodic structure problems and state-of-the art numerical techniques, like frequency domain methods , beam propagation methods and eigenmode expansion methods. Several chapters are devoted to concrete applications of periodic media simulations, e.g. in optical applications these periodic media, have the special capability to select the ranges of frequencies of the waves that are allowed to pass or blocked in the waveguide and act as an efficient frequency filter.

This book consists of 8 invited chapters that are structured in the three parts analysis, numerical techniques and finally practical applications. In the first part analysis we deal with the analysis of periodic square lattices constructed of rhomboidal quantum wells interacting via narrow links. An accurate analysis of Bloch waves, based on DN–maps of the quantum wells is presented. In Chapters 2–3 so–called open periodic waveguides are investigated. First, the problem of resonant enhancement of fields in the waveguide and anomalous transmission of energy across it due to the interaction between guided electromagnetic or acoustic modes is considered. This mechanism for resonant scattering is studied analytically using the Floquet-Bloch decomposition of the periodic differential operator underlying the waveguide structure. Secondly, in Chapter 3, the author uses the spectrum of the Helmholtz operator on an infinite strip with quasiperiodic boundary conditions to describe the propagation of electromagnetic waves in dielectric slab waveguides with periodic corrugations. Hereby, the typical ingredients like guided modes, radiation modes and leaky modes are explained in detail. Furthermore, methods are presented to compute guided and leaky modes by matching the Dirichlet-to-Neumann operator on the corresponding interfaces.

This topic is a good bridge to the second part, the numerical techniques, consisting of Chapters 4–6. In Chapter 4 the authors review several numerical approaches for solving high-frequency scattering problems most particularly, hereby focusing on the multiple scattering problem where rays are multiply bounced by a collection of separate objects.

The next chapter describes a new efficient numerical method to simulate time harmonic wave propagation in infinite periodic media including a local perturbation. Here, the main challenge is the confinement of computations to a bounded region enclosing the perturbation using so–called Dirichlet-to-Neumann (DtN) operators. In Chapter 6 the authors explain how to solve problems with periodic coefficients of periodic geometry efficiently, if they are defined on an unbounded (or very large) domain. Hereby, the usual strategy is ii to introduce so-called artificial boundaries and impose suitable boundary conditions that mimic the perfect absorption of waves traveling out of the computational domain through the artificial boundaries.

In the last part we present some application, illustrating the impact of the mathematical ideas. In Chapter 7 several potential applications of negative refraction in artificial periodic media in the wavelength range from microwaves down to optics. Both, physical concepts to create such an abnormal propagation regime and practical examples of real devices, like a photonic crystal slab for optical waves, are presented. Finally, Chapter 8 considers electromagnetic waves propagating through periodically heterogeneous layer, involving dielectrics and conductors with the goal to obtain homogenized transmission conditions and to determine the optimal structure of the periodic cells with respect to desired material properties as in meta-materials. The influence of the material composition of the layer influences the reflection and transmission of the scattered fields is discussed concisely.

We would like to thank Prof. Olivier Vanbésien for writing the foreword and providing the figures for the title page and Bentham Science Publishers, particularly Manager Bushra Siddiqui, for their support and efforts.