Photonic Crystals: Mathematical Analysis and Numerical Approximation

verfasst von: Willy Dörfler, Armin Lechleiter, Michael Plum, Guido Schneider, Christian Wieners

Verlag: Springer Basel

Buchreihe : Oberwolfach Seminars

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book concentrates on the mathematics of photonic crystals, which form an important class of physical structures investigated in nanotechnology. Photonic crystals are materials which are composed of two or more different dielectrics or metals, and which exhibit a spatially periodic structure, typically at the length scale of hundred nanometers.

In the mathematical analysis and the numerical simulation of the partial differential equations describing nanostructures, several mathematical difficulties arise, e. g., the appropriate treatment of nonlinearities, simultaneous occurrence of continuous and discrete spectrum, multiple scales in space and time, and the ill-posedness of these problems.

This volume collects a series of lectures which introduce into the mathematical background needed for the modeling and simulation of light, in particular in periodic media, and for its applications in optical devices.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract

In this introduction we will present both the main physical concepts of electromagnetism (Section 1.1.1) and mathematical basic tools (Section 1.2) that are needed for the topics discussed here.

Willy Dörfler, Armin Lechleiter, Michael Plum, Guido Schneider, Christian Wieners

Chapter 2. Photonic bandstructure calculations

Abstract

We recall that for the time-harmonic case with frequency ω we derived for the spatially varying part of the electric field $x\longmapsto E(x)$ the equation (1.17),

$$\nabla \times (\frac{1}{\mu} \nabla \times E) = \omega^2\varepsilon E$$

with the additional constraint

$$\nabla. (\varepsilon E) = 0.$$

For the material parameters we set $\mu = \mu_0 \mu_r$ and $\varepsilon = \varepsilon_0 \varepsilon_r.$ Moreover, we assume here that Ω is bounded in $\mathbb{R}^3$ and we impose the boundary condition (1.20), n × u = 0, of a perfect conductor on $\partial\Omega.$

Willy Dörfler, Armin Lechleiter, Michael Plum, Guido Schneider, Christian Wieners

Chapter 3. On the spectra of periodic differential operators

Abstract

The main mathematical tool for treating spectral problems for differential operators with periodic coefficients is the so–called Floquet–Bloch theory.

Willy Dörfler, Armin Lechleiter, Michael Plum, Guido Schneider, Christian Wieners

Chapter 4. An introduction to direct and inverse scattering theory

Abstract

Scattering theory deals with perturbation of waves by obstacles. Our viewpoint on wave propagation is therefore somewhat different compared to the one taken in the previous chapters, particularly since we are here interested in wave propagating through unbounded domains. Such problems occur quite naturally in scientific and engineering applications.

Willy Dörfler, Armin Lechleiter, Michael Plum, Guido Schneider, Christian Wieners

Chapter 5. The role of the nonlinear Schrödinger equation in nonlinear optics

Abstract

We explain the role of the Nonlinear Schrödinger (NLS) equation as an amplitude equation in nonlinear optics. The NLS equation is a universal amplitude equation which can be derived via multiple scaling analysis in order to describe slow modulations in time and space of the envelope of a spatially and temporarily oscillating wave packet. It turned out to be a very successful model in nonlinear optics. Here we explain its justification by approximation theorems and its role as amplitude equation in some problems of nonlinear optics.

Willy Dörfler, Armin Lechleiter, Michael Plum, Guido Schneider, Christian Wieners

Backmatter

Titel: Photonic Crystals: Mathematical Analysis and Numerical Approximation
verfasst von: Willy Dörfler
Armin Lechleiter
Michael Plum
Guido Schneider
Christian Wieners
Verlag: Springer Basel
Electronic ISBN: 978-3-0348-0113-3
Print ISBN: 978-3-0348-0112-6
DOI: https://doi.org/10.1007/978-3-0348-0113-3