Photonic Crystals: Mathematical Analysis and Numerical Approximation
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
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Willy Dörfler Armin Lechleiter Michael Plum Guido Schneider Christian Wieners
Photonic Crystals: Mathematical Analysis and Numerical Approximation
Willy Dörfler Institut für Angewandte und Numerische Mathematik 2 Karlsruher Institut für Technologie 76128 Karlsruhe Germany [email protected]
Armin Lechleiter INRIA Saclay and CMAP Ecole Polytechnique Route de Saclay 91128 Palaiseau Cedex France [email protected]
Michael Plum Institut für Analysis Karlsruher Institut für Technologie 76128 Karlsruhe Germany [email protected]
Guido Schneider Fachbereich Mathematik Universität Stuttgart Pfaffenwaldring 57 70569 Stuttgart Germany [email protected]
Christian Wieners Institut für Angewandte und Numerische Mathematik 3 Karlsruher Institut für Technologie 76128 Karlsruhe Germany [email protected]
2010 Mathematics Subject Classification: 35B, 35Q, 35R, 78A
ISBN 978-3-0348-0112-6 DOI 10.1007/978-3-0348-0113-3
e-ISBN 978-3-0348-0113-3
Library of Congress Control Number: 2011926985 © Springer Basel AG 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the right of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use, permission of the copyright owner must be obtained. Cover design: deblik Printed on acid-free paper Springer Basel AG is part of Springer Science+Business Media www.birkhauser-science.com
Contents Preface
vii
1 Introduction 1.1 The Maxwell Equations . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Some tools from analysis . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 1 13 19
2 Photonic bandstructure calculations 2.1 Approximation of Maxwell eigenvalues and eigenfunctions . . . . . 2.2 Calculation of the photonic band structure . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23 23 40 58
3 On the spectra of periodic differential operators 3.1 The spectrum of selfadjoint operators . . . . . . . . 3.2 Periodic differential operators . . . . . . . . . . . . . 3.3 Fundamental domain of periodicity and the Brillouin 3.4 Bloch waves, Floquet transformation . . . . . . . . . 3.5 Completeness of the Bloch waves . . . . . . . . . . . 3.6 The spectrum of A . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . .
63 64 65 67 68 70 71 76
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4 An introduction to direct and inverse scattering theory 79 4.1 Scattering of Time-harmonic Waves . . . . . . . . . . . . . . . . . 81 4.2 Time-Harmonic Inverse Wave Scattering . . . . . . . . . . . . . . . 106 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 5 The 5.1 5.2 5.3 5.4 5.5
role of the n
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