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  • Quantum optics and multiplescattering in dielectrics

  • Cover: Picture by Yvonne Grootenboer

    Printed by PrintPartners Ipskamp B.V., Enschede.

    ISBN: 90-9016950-4

  • Quantum optics and multiplescattering in dielectrics

    ACADEMISCH PROEFSCHRIFT

    ter verkrijging van de graad van doctoraan de Universiteit van Amsterdam,op gezag van de Rector Magnificus

    prof. mr. P.F. van der Heijdenten overstaan van een door het college voor promotiesingestelde commissie, in het openbaar te verdedigen

    in de Aula der Universiteitop woensdag 11 juni 2003, te 12:00 uur

    door

    Cornelis Martijn Wubsgeboren te Hoogeveen

  • Promotiecommissie:

    Promotor Prof. dr. A. LagendijkCo-promotor Dr. L.G. Suttorp

    Overige leden Prof. dr. ir. F.A. BaisProf. dr. P.L. KnightProf. dr. G. NienhuisDr. R. SprikDr. B.A. van TiggelenProf. dr. W.L. Vos

    Faculteit der Natuurwetenschappen, Wiskunde en Informatica

    The work described in this thesis is part of the research program of theStichting Fundamenteel Onderzoek der Materie (FOM),

    which is financially supported by theNederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO).

    The major part of the work was carried out at the

    Van der Waals-Zeeman InstituutUniversity of Amsterdam

    Valckenierstraat 65NL-1018 XE Amsterdam

    The Netherlands.

    The work was completed and this thesis was written at the

    Complex Photonic Systems GroupFaculty of Science and Technology

    University of TwenteP.O. Box 217

    NL-7500 AE EnschedeThe Netherlands,

    where a limited number of copies of this thesis is available.

  • Liggen in de zon

    Ik hoor het licht het zonlicht pizzicatode warmte spreekt weer tegen mijn gezichtik lig weer dat gaat zo maar niet dat gaat zoik lig weer monomaan weer monodwaas van licht.

    Ik lig languit lig in mijn huid te zingenlig zacht te zingen antwoord op het lichtlig dwaas zo dwaas niet buiten mensen dingente zingen van het licht dat om en op mij ligt.

    Ik lig hier duidelijk zeer zuidelijk lig zonderte weten hoe of wat ik lig alleen maar stilik weet alleen het licht van wonder boven wonderik weet alleen maar alles wat ik weten wil.

    Hans AndreusUit: Muziek voor kijkdieren (1951)

    Aan mijn ouders

  • Contents

    1 Introduction: quantum optics of photonic media 111.1 Light in free space and in dielectrics . . . . . . . . . . . . . . . . . . . . 111.2 Classical and quantum optics . . . . . . . . . . . . . . . . . . . . . . . . 121.3 Elements of quantum optics in dielectrics . . . . . . . . . . . . . . . . . 131.4 Spontaneous emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.5 Photonic media and photonic band gaps . . . . . . . . . . . . . . . . . . 161.6 Multiple light scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.7 True modes of a beam splitter . . . . . . . . . . . . . . . . . . . . . . . . 191.8 Outlook into this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

    2 Scalar waves in finite crystals of plane scatterers 252.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.2 Multiple-scattering theory for scalar waves . . . . . . . . . . . . . . . . . 272.3 T-matrix formalism for plane scatterers . . . . . . . . . . . . . . . . . . . 30

    2.3.1 General properties of the single plane scatterer . . . . . . . . . . 302.3.2 N plane scatterers . . . . . . . . . . . . . . . . . . . . . . . . . . 312.3.3 Model for the single plane scatterer . . . . . . . . . . . . . . . . 33

    2.4 Optical modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.4.1 Propagating modes . . . . . . . . . . . . . . . . . . . . . . . . . 362.4.2 Guided modes . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

    2.5 Local optical density of states . . . . . . . . . . . . . . . . . . . . . . . . 402.6 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 45

    3 Spontaneous emission of vector waves in crystals of plane scatterers 473.1 Multiple-scattering theory for vector waves . . . . . . . . . . . . . . . . 473.2 Plane scatterers for vector waves . . . . . . . . . . . . . . . . . . . . . . 49

    3.2.1 Dyadic Green function in plane representation . . . . . . . . . . 503.2.2 Attempt to define a T-matrix . . . . . . . . . . . . . . . . . . . . 513.2.3 Regularization of the Green function . . . . . . . . . . . . . . . . 523.2.4 T-matrix of a plane for vector waves . . . . . . . . . . . . . . . . 543.2.5 Transmission and energy conservation . . . . . . . . . . . . . . . 563.2.6 T-matrix for N planes . . . . . . . . . . . . . . . . . . . . . . . . 573.2.7 A model for the optical potential . . . . . . . . . . . . . . . . . . 58

    7

  • 8 Contents

    3.3 Optical modes and omnidirectional mirrors . . . . . . . . . . . . . . . . 593.3.1 Propagating modes . . . . . . . . . . . . . . . . . . . . . . . . . 593.3.2 Guided modes . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

    3.4 Spontaneous emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.4.1 General theory applied to planes . . . . . . . . . . . . . . . . . . 643.4.2 Spontaneous emission near one plane scatterer . . . . . . . . . . 653.4.3 Spontaneous emission near a ten-plane scatterer . . . . . . . . . . 68

    3.5 Radiative line shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.6 Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . 71

    4 Multipole interaction between atoms and their photonic environment 734.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.2 Inhomogeneous dielectric without guest atoms . . . . . . . . . . . . . . . 74

    4.2.1 Classical Lagrangian and Hamiltonian . . . . . . . . . . . . . . . 744.2.2 Complete sets and quantum Hamiltonian . . . . . . . . . . . . . 76

    4.3 Inhomogeneous dielectric with guest atoms . . . . . . . . . . . . . . . . 804.3.1 Choice of suitable Lagrangian . . . . . . . . . . . . . . . . . . . 814.3.2 Fixing the gauge . . . . . . . . . . . . . . . . . . . . . . . . . . 82

    4.4 The quantum multipolar interaction Hamiltonian . . . . . . . . . . . . . 844.4.1 Polarization, magnetization and displacement fields . . . . . . . . 844.4.2 Classical multipolar Lagrangian and Hamiltonian . . . . . . . . . 854.4.3 In need of a local-field model . . . . . . . . . . . . . . . . . . . 874.4.4 Quantum multipolar interaction Hamiltonian . . . . . . . . . . . 894.4.5 Dipole approximation . . . . . . . . . . . . . . . . . . . . . . . 90

    4.5 Inhomogeneous magnetic media . . . . . . . . . . . . . . . . . . . . . . 914.6 Dipole-coupling controversy . . . . . . . . . . . . . . . . . . . . . . . . 924.7 Summary and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 93

    5 Point scatterers and quantum optics in inhomogeneous dielectrics 955.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955.2 Atoms as point sources and as point scatterers . . . . . . . . . . . . . . . 96

    5.2.1 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . 975.2.2 Integrating out atomic dynamics . . . . . . . . . . . . . . . . . . 985.2.3 Volume-integrated dipole field . . . . . . . . . . . . . . . . . . . 102

    5.3 Single-atom properties altered by the medium . . . . . . . . . . . . . . . 1035.3.1 Light emitted by a point source . . . . . . . . . . . . . . . . . . 1045.3.2 Light scattered by a point scatterer . . . . . . . . . . . . . . . . . 1065.3.3 Single atom as a point source in homogeneous dielectric . . . . 107

    5.4 Several atoms as point sources and scatterers . . . . . . . . . . . . . . . 1105.5 Two-atom superradiance in inhomogeneous medium . . . . . . . . . . . 112

    5.5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1125.5.2 Calculation of the two-atom source Field . . . . . . . . . . . . . 1135.5.3 Effects of two-atom superradiance . . . . . . . . . . . . . . . . . 115

    5.6 Two-atom superradiance in homogeneous dielectrics . . . . . . . . . . 1165.7 Discussion: superradiance in photonic crystals . . . . . . . . . . . . . . . 118

  • CONTENTS 9

    5.8 Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . 120

    6 Transient QED effects in absorbing dielectrics 1236.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1236.2 The model and solutions of the equations of motion . . . . . . . . . . . . 1256.3 Short-time limit: sum rules . . . . . . . . . . . . . . . . . . . . . . . . . 1286.4 Long-time limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

    6.4.1 Field and medium operators . . . . . . . . . . . . . . . . . . . . 1316.4.2 Relation with phenomenological theories . . . . . . . . . . . . . 132

    6.5 Model dielectric functions . . . . . . . . . . . . . . . . . . . . . . . . . 1346.5.1 The Lorentz oscillator model . . . . . . . . . . . . . . . . . . . . 1356.5.2 The point-scattering model . . . . . . . . . . . . . . . . . . . . . 136

    6.6 Spontaneous emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1386.7 Discussion and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 142

    A Analytical expression for T-matrix of N-plane crystal 145

    B Functional differentiation after choosing a gauge 149B.1 Two definitions of functional derivatives . . . . . . . . . . . . . . . . . . 149B.2 Simple rules to compute constrained functional

    derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150B.3 Functional derivatives of the minimal-coupling

    Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

    C Dyadic Green and delta functions 153

    D Laplace operators and time-depen