Wave propagation in periodic media has attracted much attention from the scientists for a very long time. Forward or backward propagation regimes, refraction and diffraction properties, all these subjects are theoretically formalized and experimentally investigated since many centuries… Nevertheless, a renewed interest for this domain exists for a few decades with the emergence of the field of metamaterials and/or photonic crystals. The origin of this renewal can be found in the theoretical works of V. Veselago in the late 1960’s, followed by the proposals of E. Yablonovitch for photonic crystals in the 1980’s and of J. Pendry for meta-materials in the 1990’s. The promise is the opportunity to design original devices built with artificial materials, metallic or dielectric, with extraordinary propagation properties (that is to say not found in nature). The most popular new effect is undoubtedly the “negative refraction” that can be obtained either by using specific propagation regimes in periodic structures or by creating double negative media in terms of permittivity and permeability. Such a concept has allowed generalizing the famous Snell-Descartes law for refraction at the interface of two propagating media to any arbitrary values of the refractive index, positive or negative, greater or lower than unity… Moreover, as it can be shown that most of the concepts do not depend on wavelength scale, they can be potentially applied from microwave to optical waves. The main limiting factor is the patterning scale of the supporting artificial materials which is proportional to the targeted wavelength of operation. Fortunately, owing to the progress in fabrication technology down to the nanometer scale, it is now possible to bring to reality most of the theoretical predictions.
At present, this research field is very attractive as shown by the huge literature devoted to meta-materials or photonic crystals. Perfect lenses, hyperlenses, cloaking or invisibility devices are widely investigated, exploiting the fact that intrinsic materials parameters as refraction index, impedance, permittivity and permeability can be locally modulated to any positive, near zero or negative values. Open theoretical problems also re-appear following some “advanced” proposals and “lively” debates occasionally occur within the scientific community. As the field gains in maturity, devices become more complex and new modeling approaches are now required to interpret and understand properly the underlying effects. That is why the field becomes more and more multidisciplinary with mathematicians, numericians and physicists for theory and concepts, applied physicists for fabrication and measurements in connection with chemists, telecommunication engineers or biologists for original applications.
The book edited by Prof. Matthias Ehrhardt provides some particularly interesting keys to enter in this vast and exciting research domain. By focusing on specific advanced subjects related to wave propagation in periodic media from the different viewpoints of analysis, numerical techniques and practical applications with chapters written by experts in their respective fields, the reader will find an overview of state-of-the-art research results over a wide range of approaches from theory and concepts to real devices. As such, the volume will be very useful to Ph.D. students and lecturers of computational physics and numerical mathematics, and also to applied by physicists searching for accurate theoretical models to support their tremendous imagination.
Olivier Vanbésien, Professor
IEMN, University of Lille
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.
List of Contributors
Bergische Universitet Wuppertal
Full Professor, Institut Elie Cartan Nancy (IECN), Université Henri Poincaré Nancy 1
Full Professor, A.F. Ioffe Physico-Tecnical Institute
Full Professor, Bergische UniversitätWuppertal, Fachbereich C Mathematik und Naturwissenschaften
Lehrstuhl für Angewandte Mathematik und Numerische Analysis
Assistant Researcher, INRIA POEMS Project and Applied Mathematics Department, ENSTA
Full Professor, University of Liège, Department of Electrical Engineering and Computer Science
Head of INRIA POEMS Project and Full Professor, Applied Mathematics Department, ENSTA
Full Professor, Lehrstuhl für Angewandte Mathematik II, Universität Erlangen-Nürnberg
Assistant Researcher, INRIA POEMS Project, Domaine de Voluceau - Rocquencourt
Le Chesnay Cedex
Full Professor, Massey University, Albany, Auckland, NZ Institute for Advanced study
Full Professor, Massey University,
NZ Institute for Advanced study,
V.A. Fock Institute for Physics of St.-Petersburg University
Assistant Researcher, INRIA (CORIDA Team ) & Institut Elie Cartan Nancy (IECN)
Université Henri Poincaré Nancy 1,
Vandoeuvre-lÅ´ls- Nancy Cedex
Head of MBS Department
University of West Bohemia, Research Centre New Technologies
Lehrstuhl für Angewandte Mathematik II, Universität Erlangen-Nürnberg
Stephen P. Shipman
Associate Professor, Department of Mathematics
Louisiana State University
Associate Professor, Department of Mathematics
Southern Methodist University, Dallas
Full Professor, Institut d'Electronique, de Microélectronique et de Nanotechnologie
Université des Sciences et Technologies de Lille
Associate Professor, Department of Mathematical Sciences