This book covers computation of the bandstructure of photonic crystals to detect and forecast transmissions, to find new materials with prescribed band gaps, to recover crystal properties from scattered data and study the behaviour of wave packets, and more.

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.

Contains a general introduction to the so-called Floquet-Bloch theory, which provides analytical tools to investigate the spectrum of periodic differential operators. Describes the numerical technique to solve Maxwell eigenvalue problems to compute the bandstructure of 3D photonic crystals. Introduces the theory of direct and inverse scattering and applies it to periodic media. Investigation of nonlinear optical effects in wave guides that can be described by the nonlinear Schrödinger equation. Includes supplementary material: sn.pub/extras

Klappentext
This volume collects a series of lectures which provide an introduction to the mathematical background needed for the modeling and simulation of light, in particular in periodic media, and for its applications in optical devices.The 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.

Inhalt
Preface.- 1 Introduction.- 2 Photonic bandstructure calculations.- 3 On the spectra of periodic differential operators.- 4 An introduction to direct and inverse scattering theory.- 5 Nonlinear optics.- Bibliography.