Light-matter interaction in nanostructured materials 8792062415

Citation preview

Title :

Light-matter interaction in nanostructured materials

Type :

Acade m ic disse rtationPh.d.-the sis

Participant(s):

Author: Kriste nse n, Philip Trøst (C wisno: 15120) Te chnical Unive rsity of De nm ark , De partm e nt of Photonics Engine e ring, Nanophotonics The ory and Signal Proce ssing Em ail: ptk r@fotonik .dtu.dk Supe rvisor: Mørk , Je spe r (C wisno: 6068) Te chnical Unive rsity of De nm ark , De partm e nt of Photonics Engine e ring, Nanophotonics The ory and Signal Proce ssing Em ail: je sm @fotonik .dtu.dk

Supe rvisor: Lodahl, Pe te r (C wisno: 29506) Te chnical Unive rsity of De nm ark , De partm e nt of Photonics Engine e ring, Q uantum Photonics Em ail: pe lo@fotonik .dtu.dk Abstract:

Light-m atte r inte raction in nanostructure d m ate rials is studie d the ore tically with e m phasis on spontane ous e m ission dynam ics of quantum dots in photonic crystals. The m ain topics of the work are e le ctrom agne tic scatte ring calculations,

de cay dynam ics of single quantum dots and m ultiple quantum dot dynam ics. The e le ctrom agne tic Gre e n's te nsor e nte rs naturally in calculations of light-m atte r inte raction in m ultiple scatte ring m e dia such as photonic crystals. W e pre se nt a nove l solution m e thod to the Lippm ann-Schwinge r e quation for use in e le ctric fie ld scatte ring calculations and Gre e n's te nsor calculations. The m e thod is we ll suite d for m ultiple scatte ring proble m s such as photonic crystals and m ay be applie d to proble m s with scatte re rs of arbitrary shape and non-hom oge ne ous back ground m ate rials. By the introduction of

a m e asure for the de gre e of fractional de cay we quantify to which e x te nt the e ffe ct is obse rvable in a give n m ate rial. W e focus on the case of inve rse opal photonic crystals and locate the position in the crystal whe re the effe ct is m ost pronounce d. Furthe rm ore , we quantify the influe nce of absorptive loss and give e x am ple calculations with e x pe rim e ntal param e te rs for PbSe quantum dots in Si inve rse opals showing that absorption has a lim iting but not prohibitive e ffe ct. In addition, we discuss how the re sonant nature of the phe nom e non puts rathe r se ve re re strictions on the stabilization of the syste m in possible e x pe rim e nts. Last, we e x am ine the influe nce on the de cay dynam ics from othe r quantum dots. Using a se lf-consiste nt Dyson e quation approach we de scribe how scatte ring from othe r quantum dots can be include d in the Gre e n's te nsor for a passive m ate rial syste m . W e num e rically calculate both local and non-local e le m e nts of the Gre e n's te nsor for a photonic crystallite slab and apply the m e thod for an e x am ple calculation with two quantum dots at spe cific locations in the unit ce ll. In this way it is e x plicitly shown how the de cay dynam ics of one quantum dot is qualitative ly change d by the scatte ring prope rtie s of anothe r.

Publishe d:

in se rie s: (ISBN: 87-92062-41-5) , page s: 170, 201004, Te chnical Unive rsity of De nm ark (DTU), Kgs. Lyngby, De nm ark

open in browser customize

pdfcrowd.com

Light-matter intera tion in nanostru tured materials

A dissertation submitted to the Department of Photoni s Engineering at the Te hni al University of Denmark in partial fulllment of the requirements for the degree of philosophiæ do tor

Philip Trøst Kristensen De ember 18, 2009

Light-matter intera tion in nanostru tured materials

Prefa e The resear h des ribed in this PhD thesis was arried out in the Theory and Signal Pro essing Group at DTU Fotonik in Lyngby, Denmark from O tober 2006 to De ember 2009 under the supervision of Professor Jesper Mørk and Asso iate Professor Peter Lodahl. The lose ollaboration between theoreti al and experimental a tivities at DTU Fotonik has resulted in a natural interest in spontaneous emission from quantum dots that is apparent also in this entirely theoreti al proje t. The subje ts overed over the ourse of the proje t may be divided into three main topi s; ele tromagneti s attering al ulations, de ay dynami s of single quantum dots and multiple quantum dot dynami s. At the time of writing, I feel that my study of the last of these topi s is less omplete than the rst two. Nevertheless, I nd the results interesting and I am happy to in lude them in the thesis. A number of people have helped me throughout the proje t. First and foremost I am thankful to both my supervisors. Peter for his enthusiasm and experimental inputs and for introdu ing me to the interesting eld of spontaneous emission. Jesper for sharing his impressive physi al understanding and for invaluable help and guidan e. Early in the proje t I had the pleasure of visiting the FOM Institute AMOLF in Amsterdam, the Netherlands, for a short stay. I would like to thank Dr. Femius Koenderink for wel oming me to AMOLF and for sharing his knowledge as well as his numeri al ode with me, thus enabling the LDOS al ulations of three dimensional photoni rystals in the work on fra tional de ay. Also, I would like to thank Bjarne Tromborg for many helpful dis ussions and suggestions for the fra tional de ay analysis. In the spring of 2009 I visited the resear h group of Professor Stephen Hughes at Queen's University in Kingston, Canada. I would like to thank all the group members for the good times. A spe ial thank to Steve for showing me the tools of the trade in FDTD and to the rest of the Hughes residen e; Euan, Kai and Nanae for their enormous hospitality and for many deli ious dinners and barbe ues. v

I would like to a knowledge the enjoyable working environment in the Theory and Signal Pro essing Group as well as the stimulating ollaboration with the experimental Quantum Photoni s Group. A spe ial thank to Mr. Yaohui Chen for pleasant ompany in the o e over the years. Finally, and most importantly, I am grateful to Lisbet for her help and support and for making me remember that there is more to life than work, and that one should take time o every now and then. Philip Trøst Kristensen De ember 18, 2009

vi

Abstra t Light-matter intera tion in nanostru tured materials is studied theoreti ally in relation to spontaneous emission dynami s of quantum dots in photoni rystals. We present a novel solution method to the Lippmann-S hwinger equation for use in ele tri eld s attering al ulations. The method is well suited for multiple s attering problems su h as photoni rystals and may be applied to problems with s atterers of arbitrary shape and non-homogeneous ba kground materials. The method is formulated in the general ase and details and examples are provided for the implementation in two dimensions. Appli ation of the method is illustrated by al ulating light emission from a line sour e in a nite sized photoni rystal waveguide Fra tional de ay from semi ondu tor quantum dots is investigated. By the introdu tion of a measure for the degree of fra tional de ay we quantify to whi h extent the ee t is observable in a given material. We fo us on the ase of inverse opal photoni rystals and lo ate the position in the rystal where the ee t is most pronoun ed. Furthermore, we quantify the inuen e of absorptive loss and give example al ulations with experimental parameters for PbSe quantum dots in Si inverse opals showing that absorption has a limiting but not prohibitive ee t. In addition, we dis uss how the resonant nature of the phenomenon puts rather severe restri tions on the stabilization of the system in possible experiments. Last, we examine the inuen e on the de ay dynami s of a quantum dot from other quantum dots. Using a self- onsistent Dyson equation approa h we des ribe how s attering from other quantum dots an be in luded in the Green's tensor for a passive material system. We numeri ally al ulate both lo al and non-lo al elements of the Green's tensor for a photoni rystallite slab and apply the method for an example al ulation with two quantum dots at spe i lo ations in the unit

ell. In this way it is expli itly shown how the de ay dynami s of one quantum dot is qualitatively hanged by the s attering properties of another.

vii

Resumé Lys-stof vekselvirkning i nanostrukturerede materialer studeres teoretisk med fokus på spontan emission af lys fra kvantepunkter i fotoniske krystaller. Vi præsenterer en ny metode til løsning af Lippmann-S hwinger ligningen med anvendelser i elektromagnetiske spredningsproblemer. Metoden er velegnet til at undersøge lysudbredelse i fotoniske krystaller og kan benyttes til beregning af spredning fra objekter med vilkårlig form i en inhomogen baggrund. Vi præsenterer metoden i generel form og giver detaljer og eksempler vedrørende implementeringen i to dimensioner. Anvendelse af metoden illustreres ved beregning af lysudbredelse fra en linjekilde i en fotonisk krystal-bølgeleder. Ved at indføre et mål for spaltede henfald kvanti eres det, i hvor høj grad eekten vil være mulig at observere i et givet materiale. Vi fokuserer på inverse opaler og bestemmer positionen i krystallen, hvor eekten er mest udtalt. Indydelsen af absorption i materialet analyseres, og vi giver eksempler på udregninger med eksperimentelle parametre for blyselenid kvantepunkter i inverse opaler af sili ium, der viser, at absorption har en målbar, men ikke ødelæggende indvirkning. Vi diskuterer hvordan krav til stabilisering af systemet lægger begrænsninger på mulige eksperimenter. Endeligt studeres indydelsen på spontan emission fra andre kvantepunkter i en prøve. Ved hjælp af en selvkonsistent metode baseret på Dysons ligning beskrives det, hvordan spredning fra kvantepunkter kan inkluderes i Greens tensor for et passivt materialesystem. Numerisk beregnes både lokale og ikke-lokale elementer af Greens tensor for en fotonisk krystal-membran, og der gives et eksempel på beregninger med to kvantepunkter, der ekspli it viser, at den spontane emission fra et kvantepunkt kan ændres kvalitativt på grund af spredning fra et andet kvantepunkt.

ix

List of publi ations Below we list the publi ations resulting from the work in this PhD proje t.

Journal publi ations J1 Jeppe Johansen, Søren Stobbe, Ivan S. Nikolaev, Toke Lund-Hansen, Philip

Size dependen e of the wavefun tion of self-assembled InAs quantum dots from time-resolved opti al measurements. Physi al Review B 77, 073303 (2008).

Trøst Kristensen, Jørn Mär her Hvam, Willem Vos and Peter Lodahl,

J2 Philip Trøst Kristensen, A. Femius Koenderink, Peter Lodahl, Bjarne Trom-

Fra tional de ay of quantum dots in real photoni 33, 1557-1559 (2008) .

borg, and Jesper Mørk,

rystals.

Opti s Letters

J3 Andreas Næsby Rasmussen, Troels Suhr Skovgård, Philip Trøst Kristensen and Jesper Mørk,

photon sour es.

Inuen e of pure dephasing on emission spe tra from single 78, 045802 (2008).

Physi al Review A

J4 Søren Stobbe, Jeppe Johansen, Philip Trøst Kristensen, Jørn Mär her Hvam

Frequen y dependen e of the radiative de ay rate of ex itons in self-assembled quantum dots: Experiment and theory. Physi al Review B 80, 155307 (2009.) and Peter Lodahl,

Light propagation in nite-sized photoni rystals: Multiple s attering using an ele tri eld integral equation. Journal of the Opti al So iety of Ameri a B 27, 228-237 (2010).

J5 Philip Trøst Kristensen, Peter Lodahl and Jesper Mørk,

Conferen e ontributions C1 Philip Trøst Kristensen, Bjarne Tromborg, Peter Lodahl and Jesper Mørk,

Breakdown of Wigner-Weisskopf theory for spontaneous emission: a quantative analysis. Opti al Waveguide Theory and Numeri al Modelling. Lyngby, Denmark, 2007

xi

C2 Søren Stobbe, Jeppe Johansen, Ivan S. Nikolaev, Toke Lund-Hansen, Philip Trøst Kristensen, Jørn Mär her Hvam, Willem Vos and Peter Lodahl, A

urate measurement of the transition dipole moment of self-assembled quantum dots. CLEO/Europe-IQEC. Muni h, Germany, 2007. C3 Jeppe Johansen, Søren Stobbe, Ivan S. Nikolaev, Toke Lund-Hansen, Philip Trøst Kristensen, Jørgen mär her Hvam, Willem Vos and Peter Lodahl, Quan-

tum e ien y of self-assembled quantum dots determined by a modied opti al lo al density of states. QLEO/QELS/PHAST Conferen e pro eedings, Balti-

more, Maryland, USA, 2007.

C4 Andreas Næsby Rasmussen, Troels Suhr Jørgensen, Philip Trøst Kristensen, Jesper Mørk, Inuen e of Pure Dephasing on Emission Spe tra from Quantum Dot-Cavity Systems. Coherent Opti al Te hnologies and Appli ations 2008 Summer Opti s and Photoni s Congress Abstra ts, paper JWE7, Boston, Massa husetts, USA, 2008. C5 Philip Trøst Kristensen, Femius Koenderink, Peter Lodahl, Bjarne Tromborg and Jesper Mørk, Fra tional de ay of quantum dots in photoni rystals. CLEO/QELS 2008, Te hni al Digest CD-ROM, paper QFA3, San José, California, USA, 2008. C6 Philip Trøst Kristensen, A. Femius Koenderink, Peter Lodahl, Bjarne Tromborg and Jesper Mørk, Fra tional de ay of quantum dots in photoni rystals. Danish Opti al So iety/Danish Physi al So iety annual meeting 2008, Nyborg Strand, Denmark, 2008. C7 Andreas Næsby Rasmussen, Troels Suhr Jørgensen, Philip Trøst Kristensen and Jesper Mørk, Inuen e of pure dephasing on emission spe tra from single photon sour es. Danish Opti al So iety/Danish Physi al So iety annual meeting 2008, Nyborg Strand, Denmark, 2008. C8 Troels Suhr Skovgård, Philip Trøst Kristensen, Lars Hagedorn Frandsen, Martin S hubert, Niels Gregersen and Jesper Mørk, Nonlinear dynami s in photoni rystal nano avity lasers. CLEO/Europe-EQEC 2009 Conferen e pro eedings, paper CB2.3, Muni h, Germany, 2009. C9 Philip Trøst Kristensen, Peter Lodahl and Jesper Mørk, Fast, A

urate and

Stable S attering Cal ulation Method with Appli ation to Finite Sized Photoni Crystal Waveguides. Advan es in Opti al S ien es: OSA Opti s & Pho-

toni s Congress 2009 Abstra ts, paper ITuD2, Honolulu, Hawaii, USA, 2009. xii

Popular s ien e P1 Philip Trøst Kristensen, Peter Lodahl and Jesper Mørk, de ay - or both !

To de ay or not to

Quantum me hani s of spontaneous emission

23, 1, 4-8 (2008). P2 Søren Stobbe, Philip Trøst Kristensen and Peter Lodahl,

. DOPS-Nyt

Kvanteoptik i et

Optiske Horisonter : en rejse på kommunikationsteknologiens vinger, 29-47 (COM•DTU, 2007) farvet vakuum : Anvendelser af nanoteknologi og nanofotonik.

Contributions to journal publi ations by the author All resear h in the arti les J2 and J5 was done by the author, as was the development and implementation of the ne essary numeri al ode, with the ex eption of the plane wave ode used in J2. Both arti les were written by the author and dis ussed with the o-authors. Contributions to arti les J1 and J4 were in the form of ode and al ulation methods originally developed by the author for the al ulations of lo al density-of-states and ele tron and hole wavefun tions using nite element modeling. Arti le J3 resulted from a student proje t for whi h the author a ted as o-supervisor. Contributions to this work were in the form of guidan e and help on the theory and interpretation.

xiii

Contents 1

Introdu tion

1.1 Invitation: Spontaneous emission from semi ondu tor quantum dots 1.1.1 Sample preparation and hara terization . . . . . . . . . . . . 1.1.2 Quantum dot lifetime measurements . . . . . . . . . . . . . . 1.1.3 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Controlling light emission and s attering: Quantum dots in photoni

rystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Light s attering al ulations and the ele tromagneti Green's tensor . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Photoni rystals . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Quantum opti s and the lo al density-of-states . . . . . . . . 1.2.4 Quantum dot de ay dynami s . . . . . . . . . . . . . . . . . . 1.2.5 Experiments and appli ations . . . . . . . . . . . . . . . . . . 1.3 Overview of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Theory of light-matter intera tion

2.1 Introdu tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Overview of Chapter 2 . . . . . . . . . . . . . . . . . . 2.2 Modeling light propagation and s attering . . . . . . . . . . . 2.2.1 Ele tri eld from a point sour e - the Green's tensor 2.2.2 The Lippmann-S hwinger equation . . . . . . . . . . . 2.2.3 Quantization of the ele tromagneti eld . . . . . . . . 2.2.4 Modeling photoni rystals . . . . . . . . . . . . . . . 2.2.5 An overview of existing al ulation methods . . . . . . 2.3 Quantum dot models . . . . . . . . . . . . . . . . . . . . . . . 2.4 Coupling formalisms . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Intera tion Hamiltonian . . . . . . . . . . . . . . . . . 2.4.2 Equations of motion . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

1 1 1 3 5 7 7 9 10 11 12 13

15 15 15 16 17 18 20 21 22 26 29 30 31 xv

CONTENTS 2.4.3 Quantum dot de ay dynami s in the S hrödinger pi ture . . 2.4.4 Field from quantum emitter in the Heisenberg pi ture . . . . 3

Multiple s attering al ulations using the Lippmann-S hwinger equation

3.1 Introdu tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 An example s attering problem . . . . . . . . . . . . . . . . . 3.1.2 A hybrid method . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Overview of hapter 3 . . . . . . . . . . . . . . . . . . . . . . 3.2 General formulation of the method . . . . . . . . . . . . . . . . . . . 3.2.1 Basis fun tions in dierent dimensions . . . . . . . . . . . . . 3.3 One dimensional example: The example s attering problem revisited 3.4 Implementation in two dimensions . . . . . . . . . . . . . . . . . . . 3.4.1 Self-terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 S attering terms . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Ba kground ele tri eld . . . . . . . . . . . . . . . . . . . . . 3.4.4 Exterior solution . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.5 Solution of the linear equation system . . . . . . . . . . . . . 3.5 Two dimensional examples . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Plane wave s attering from ir ular ylinders . . . . . . . . . 3.5.2 Green's tensor for a olle tion of square ylinders . . . . . . . 3.6 Error analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Inhomogeneous ba kgrounds . . . . . . . . . . . . . . . . . . . . . . . 3.7.1 Additional s attering near interfa e . . . . . . . . . . . . . . . 3.7.2 Light emission in nite sized photoni rystal waveguide . . . 3.8 Con lusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Fra tional de ay of quantum dots in photoni rystals

4.1 Introdu tion . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Overview of hapter 4 . . . . . . . . . . . . . 4.2 Cal ulation of de ay urves . . . . . . . . . . . . . . 4.2.1 Illustrative examples . . . . . . . . . . . . . . 4.2.2 A measure for the degree of fra tional de ay . 4.2.3 Estimates of the residue . . . . . . . . . . . . 4.3 Fra tional de ay in the anisotropi gap model . . . . 4.3.1 Denition of the square root . . . . . . . . . 4.3.2 Movement of poles . . . . . . . . . . . . . . . 4.3.3 Residues . . . . . . . . . . . . . . . . . . . . . 4.4 High resolution lo al density-of-states . . . . . . . . xvi

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

32 35 39 39 39 42 42 43 45 45 46 47 49 50 52 52 53 53 54 56 57 58 59 61 63 63 64 64 65 66 67 70 70 71 72 74

CONTENTS 4.4.1 Cal ulations using plane wave expansion . . . . . . . . . . . . 4.4.2 Detailed analysis of the band edge . . . . . . . . . . . . . . . 4.4.3 Inuen e of material loss . . . . . . . . . . . . . . . . . . . . 4.5 Fra tional de ay of quantum dots in inverse opals with material losses 4.5.1 Cal ulation of the spe trum . . . . . . . . . . . . . . . . . . . 4.5.2 De ay dynami s in rystals with material losses . . . . . . . . 4.6 Con lusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

75 76 79 81 81 83 86

87 87 88 89 90 90 95

Multiple quantum dots in photoni rystal slabs

5.1 Introdu tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Overview of hapter 5 . . . . . . . . . . . . . . . . . . . . . . 5.2 Finite sized photoni rystallite in diele tri slab . . . . . . . . . . . 5.2.1 Photoni rystal slab analysis . . . . . . . . . . . . . . . . . . 5.2.2 Green's tensor and the lo al density-of-states . . . . . . . . . 5.3 Additional s attering from quantum dots . . . . . . . . . . . . . . . 5.3.1 Lo al eld orre tions using a Coupled Dipole Approximation approa h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.4 De ay dynami s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.4.1 De ay dynami s in the time-domain . . . . . . . . . . . . . . 103 5.5 Con lusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 6

Con lusion

109

A

The Green's tensor in homogeneous media

111

B

Mis ellaneous ylinder fun tion results

B.1 Addition theorems for multipole expansions B.1.1 Ja obi-Anger identity . . . . . . . . B.1.2 Graf's addition theorem . . . . . . . B.2 Derivatives for ylindri al wavefun tions . . B.3 Spe ial integrals involving Bessel fun tions . C

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

113 113 113 114 114 115

. . . .

117 117 118 119 120

Pra ti al evaluation of matrix elements

C.1 Cal ulation of . . . . . . . . . . . . . . . . . . . . . . . C.2 Simpli ation of matrix element al ulations . . . . . . . . . C.2.1 A pro edure for integrating a ross a square domain . C.2.2 Example al ulations . . . . . . . . . . . . . . . . . . Iµαβ

. . . .

. . . .

. . . .

. . . .

xvii

CONTENTS

D Lo al density-of-states in homogeneous media D.1

Alternative derivation

. . . . . . . . . . . . . . . . . . . . . . . . . .

E The Coupling parameter β E.1

125 126

127

Relation to os illator strength . . . . . . . . . . . . . . . . . . . . . .

128

F Lo al density-of-states from dispersion surfa es

129

G Time dependen e of oupled quantum dot dynami s

131

G.1

The single quantum dot ase

. . . . . . . . . . . . . . . . . . . . . .

131

G.2

Two quantum dots

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

133

H Finite dieren e time-domain al ulations

137

H.1

Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

137

H.2

A

ura y

139

Bibliography

xviii

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

141

Chapter 1 Introdu tion

1.1

Invitation:

Spontaneous emission from semi-

ondu tor quantum dots

In order to introdu e some entral on epts in the theory of light-matter intera tion in mi ro- and nanostru tured media, we start out by des ribing an experiment in whi h spontaneous emission of light from InAs quantum dots (QDs) was investigated using time-resolved measurements. The QDs in the experiment were grown by mole ular beam epitaxy resulting in small protrusions of InAs atop a thin so- alled wetting layer. Both dots and wetting layer were surrounded by GaAs as illustrated in Fig. 1.1. Carrier onnement within the QDs leads to quantization of the allowed energy levels as known from atomi physi s. Therefore, for most of our purposes we

an think of the QDs as arti ial atoms with two distin t energy levels that are

ontrolled in the manufa turing pro ess. Fig. 1.1 illustrates the basi ideas of the experiment. Using a pump laser an ex itation is reated in the system. Subsequently, the system de ays to the ground state under the emission of a photon with a frequen y orresponding to the energy dieren e between the ex ited state and the ground state. Dete tion of the emitted light thus signies that a spontaneous emission event has o

urred. 1.1.1

Sample preparation and hara terization

The experiment was performed by measuring on a single sample that was spe ially prepared so that nominally identi al QDs were lo ated at dierent distan es to a silver mirror. Below, we briey dis uss details of the sample preparation and the 1

Chapter 1. Introdu tion

|ei E = ~ω |gi

Figure 1.1: Left: Sket h of a self-assembled InAs quantum dot in GaAs. As InAs (yellow atoms) is grown on top of GaAs (blue atoms), the QDs appear as small lumps of InAs on top of a thin so- alled wetting layer in order to minimize the surfa e energy. Subsequently, the QDs are overgrown with another layer of GaAs. Right: Sket h of the de ay dynami s. A pump laser reates an ex itation in the QD, and subsequently the system de ays under emission of a photon with energy E = ~ω , orresponding to the energy dieren e between the two states.

Counts

experimental setup. Starting from a GaAs wafer with embedded InAs QDs at a depth 302 nm, a total of 32 terra es were et hed into the GaAs to dierent heights above the QD layer. After the et hing pro ess, an opti ally thi k layer of silver was added to over the terra es, ee tively resulting in a silver mirror as seen from the position of the QDs. At a nal step, the thin sample (approximately 1 µm) was pla ed on a blo k of sapphire (Al2 O3 ) for support as illustrated in Fig. 1.2.

Ag GaAs QDs GaAs Height [nm℄

Al2 O3

Figure 1.2: Left: Atomi For e Mi rograph of a QD sample before overgrowth of GaAs, showing the QDs as nanometer sized bumps on the surfa e with the height distribution shown in the histogram. Figure from Ref. [1℄. Right: Sket h of a ut through the sample showing how the QDs are pla ed at dierent distan es to the silver mirror. The sample was kept in a He losed y le ryostat at onstant temperature. Within the ryostat the sample ould be translated in all three spatial dire tions, 2

Invitation: Spontaneous emission from semi ondu tor quantum dots

so that the fo us ould be hanged through the 32 dierent terra es on the sample. The system was pumped by a mode-lo ked laser sour e produ ing pulses of femtose ond width at a repetition rate of approximately 80 MHz. The emitted light was olle ted and sent through a spe trometer with a) a CCD amera for spe tral measurements or b) a sili on avalan he photodiode (APD) for time resolved measurements. Fig. 1.3 shows the spe trum of the emitted light. Apart from the pump that is learly visible at around 850 nm, we noti e that the spe trum is broadened, so that light at a ontinuum of wavelengths is emitted from the sample. This so- alled inhomogeneous broadening of the QDs arise from the experimental onditions; the pump laser ex ited a lot of dierent QDs in the sample whi h had slightly dierent sizes due to the manufa turing pro ess. Sin e the QD energy levels arise from arrier onnement, the distribution in sizes maps dire tly onto a distribution in emission energies. 1

I [a.u.℄

0.8 0.6 0.4 0.2 0 800

850

900

950

λ0 [nm℄

1000

1050

Figure 1.3: Spe trum of the QD sample.

Although the spe trum shows that light was emitted from many QDs, the mono hromator in the experimental setup enabled a spe tral ltering. This, ombined with low pump powers, resulted in the olle tion of light from QDs of approximately the same size only. Throughout the experiment, measurements were performed at a onstant wavelength of λ0 = 1020 nm, and the pump power was kept su iently low that only the lowest lying states in the QDs were ex ited. 1.1.2

Quantum dot lifetime measurements

After the initial spe tral hara terization, time orrelated measurements were arried out in order to investigate the spontaneous emission. At the repetition rate of 80 MHz, the pump laser was used to ex ite arriers in the QDs, and a timer was started. The subsequent de ay of the arriers ba k to the ground state was monitored from the emitted photon using the APD whi h stopped the timer in ase 3

Chapter 1. Introdu tion

of a dete tion event. Based on the APD dete tion events, a histogram of dete tion times was reated that we will refer to as the de ay urve. Figure 1.4 shows de ay

urves obtained from QDs at dierent distan es from the silver mirror. z = 74 nm z = 74 nm z = 167 nm

4

Counts

10

Γfast Γslow Afast Afast 2 χR

1.52 ns−1 0.07 ns−1 45.3 × 103 1.67 × 103 1.29

z = 167 nm 3

10

0

2

4

6

t [ns℄

8

10

12

Γfast Γslow Afast Afast 2 χR

0.93 ns−1 0.09 ns−1 41.4 × 103 1.91 × 103 1.19

Figure 1.4: De ay urves from QDs at emission wavelength λ0 = 1020 nm and temperature T = 45 K. The QDs are pla ed at dierent distan es from the silver mirror. Solid lines show double exponential ts to the urves with the tted parameters in the tables to the right. As with most other de ay pro esses in nature, we expe t to nd an exponential de rease, hara terized by a de ay onstant. From Fig. 1.4 we re ognize that there are (at least) two rates, namely a fast rate, Γfast , whi h is evident at times t / 6 ns, and a slow rate, Γslow , whi h be omes pronoun ed only at longer times. Therefore, we t the experimental results to a model of the type f (t) = Afast exp(−Γfast t) + Aslow exp(−Γslow t).

The table in Fig. 1.4 lists the results of the ts for the two de ay urves, in luding the goodness of t parameter χ2R [2℄. There is a pronoun ed dieren e in the fast de ay rate whi h is learly visible in the gure. Due to the sample preparation and the spe tral ltering, we know that the QDs are nominally identi al, whi h means that the dieren e in de ay urves must stem from the dieren e in distan es to the silver mirror. Moreover, these distan es are two orders of magnitude larger than the size of the QDs, whi h means that the arriers are not dire tly inuen ed by the silver mirror. This leads us to the somewhat unintuitive on lusion that it is properties of the environment that hanges the rate of de ay of the QDs. This 4

Invitation: Spontaneous emission from semi ondu tor quantum dots

phenomenon was rst pointed out by Pur ell [3℄ and is a striking feature of the quantum me hani s of spontaneous emission. 1.1.3

Interpretation

Given the solid state nature of the QDs, we expe t the total rate of de ay from a QD, ΓT , to be the sum of two ontributions, namely a radiative and a non-radiative rate ΓT = ΓR + ΓNR , (1.1) where the non-radiative de ay rate, ΓNR , represents the sum of all non-radiative de ay hannels and is assumed to be independent of the position (as long as it is su iently far away from interfa es between dierent media, su h as in the experiment). The non-radiative de ay rate is important for appli ations sin e it determines the so- alled quantum e ien y, i.e. the fra tion of initially ex ited arriers that ultimately leads to emitted photons. The experiment dire tly shows that ontrary to the non-radiative de ay rate, the radiative de ay rate ΓR may hange with position. It is reasonable to expe t that the de ay rate depends also on intrinsi properties of the QD (in the same way that the emitted eld from a lassi al dipole depends on the dipole moment), so that ΓR = αI ρx (r), (1.2) where now αI denotes intrinsi properties of the QD and ρx (r) denotes the inuen e from the surroundings. Sin e the intrinsi properties are independent of the surroundings, the radiative de ay rate an be written as ΓR = Γhom R

ρx (r) , ρhom

(1.3)

where Γhom is the referen e radiative de ay rate of the QD in the homogeneous R material des ribed by ρhom . The ratio ρx (r)/ρhom is known as the Pur ell fa tor. It des ribes how mu h faster or slower the de ay o

urs in a given environment than in the homogeneous medium. There is a subtle issue in the above interpretation. The measured de ay urves tted well to an exponential fun tion, and we therefore assigned a de ay rate to the QD. Impli itly we thus made a model of the ex itation in the QD, whereas the experiment measured the intensity of the emitted light and not the ex itation. Indeed, ea h dete tion event onsisted of the absorption of light in whi h a photon was annihilated, and an ele tron was promoted to an ex ited state in the APD semi ondu tor hip. Using a simple rate equation pi ture we an readily a

ount for the exponential de ay in the measured data. Given that the number of ex itations 5

Chapter 1. Introdu tion

of the QDs de ay exponentially, and that every radiative de ay of an ex itation results in the reation of a photon, we have the relations P˙e = −ΓT Pe P˙ph = ΓR Pe ,

where Pe (t) and Pph (t) denote the number of QD ex itations and photons, respe tively. From the rate equations it follows that the number of photons that are

reated between times t and t + ∆t is given as ∆Pph (t) = ΓR

Z

t

t+∆t

Pe (τ )dτ =

ΓR (1 − e−ΓT ∆t )e−ΓT t , ΓT

(1.4)

where we have assumed that Pe (0) = 1. Using the rate equation pi ture, we an thus argue that in the ase of an exponential de rease in the number of ex itations, the number of reated photons also de reases exponentially. The APDs measure intensity hI(t)i whi h is proportional to the number of photons per se ond impinging on the imaging hip. The measurements in Fig. 1.4 thus represents the integrated intensity, and from Eq. (1.4) we may dire tly asso iate the rate of hange in the measurement histograms with the total de ay rate ΓT . Intensity measurements are onvenient from an experimental point of view, but both Pe (t) and hI(t)i are in prin iple measurable quantities. The hoi e of whether to fo us on one or the other relates to the obje tive of the parti ular experiment or theory. For the remainder of this se tion we will fo us on the probability Pe (t) that at time t an ex itation exists within the QD, and we will refer to Pe (t) as the de ay

urve. The experiment is similar to experiments performed by Drexhage in the 1970's using dye mole ules [4℄. Whereas the fo us of the series of experiments by Drexhage was the dire t observation of the Pur ell ee t, experiments similar to the one des ribed has been arried out re ently as a means to hara terize the QDs. Sin e the radiative de ay rate is proportional to both intrinsi and extrinsi properties, one of the properties an in prin iple be measured if the other is known. For real QDs, however, there may also be a signi ant non-radiative ontribution to the de ay rate, as dis ussed above. Through systemati measurements of the de ay rate for QDs subje t to dierent values of ρx (r) one is able to separate the radiative ontribution from the non-radiative ontribution [1, 5℄. Fig. 1.5 shows the results in Ref. [5℄ in whi h measurements similar to the one des ribed were tted to a theoreti al −1 predi tion for ρx (r) in order to determine the values Γhom R = 0.95 ± 0.03 ns and −1 ΓNR = 0.11 ± 0.03ns . 6

De ay rate [ns−1 ℄

Controlling light emission and s attering: Quantum dots in photoni rystals

Distan e [nm℄ Figure 1.5: De ay rates of InAs QDs in GaAs as a fun tion of distan e to an air interfa e. The data show a lear os illatory behavior due to solid urve.

ρx (r), as ρx (r)

The dashed line shows an alternative theoreti al

shown by the

urve that is

designed to a

ount for dissipation lose to the interfa e. Figure from Ref. [5℄.

1.2

Controlling light emission and s attering: Quantum dots in photoni rystals

Based on the experiment in se tion 1.1 we will now dis uss and motivate the theoreti al work in this thesis. For now, we will not dis uss the mathemati al details but rather fo us on a qualitative introdu tion to the basi elements of light-matter intera tion. 1.2.1

Light s attering al ulations and the ele tromagneti Green's tensor

On distan es mu h larger than the wavelength, light an simply be des ribed as rays - the so- alled ray pi ture of light [6℄. This explains how, in the experiment, we

ould use lenses to fo us the emitted light onto the dete tion part of the setup. At distan es that are omparable to the wavelength, however, this pi ture is no longer valid as the wave nature of light be omes important. Indeed, the dieren e in de ay rate for QDs at dierent distan es from the silver mirror may be interpreted as an interferen e phenomenon, and we note that the variations in de ay rates in Fig. 1.5 is similar to the ee tive wavelength of light in GaAs. Fig. 1.6 shows the ele tri eld from a line sour e in air in the vi inity of two diele tri ylinders of refra tive index nR = 3.5 in air. Light is emitted from the line sour e and s atters o the ylinders. Even with only two ylinders, a omplex 7

Chapter 1. Introdu tion

interferen e pattern results. In order to properly model the propagation of light waves in mi ro-stru tured materials, one has to a

ount for s attering from all dierent obsta les in a oherent way, so that interferen e ee ts will be in luded in the des ription. In lassi al ele tromagneti theory, this is done using the ele tromagneti propagator, or Greens tensor, whi h may be interpreted as the ele tri eld at point r due to an os illating dipole at the point r′ . In a theoreti al des ription of the experiment in se tion 1.1, the Green's tensor is the mathemati al obje t that des ribes light propagation from the QDs to the dete tor. The eld in Fig. 1.6 is exa tly the Green's tensor. Given the omplexity of many mi ro-stru tures, the al ulation of the Green's tensor is a highly non-trivial task in the general ase, and usually numeri al methods are employed. 6 0.5

k0 y

5 4

0.4

3

0.3

2

0.2

1

0.1

0 0

2

4

6

8

10

k0 x Figure 1.6: Two dimensional Green's tensor des ribing propagation of light emitted from a line sour e as indi ated by the red dot. S attering due to the two ylinders results in a ompli ated interferen e pattern.

A silver mirror represents a very simple geometry in whi h to study interferen e ee ts. This is onvenient from both a theoreti al and an experimental point of view. Theoreti ally, al ulations des ribing the light s attering in ase of a simple interfa e stru ture are fast due to the symmetry of the problem. Experimentally, the symmetry relaxes the requirements on the sample preparation be ause only the QD position relative to the interfa e is important. The simple interfa e stru ture, however, limits the number of interfering waves. Other, more advan ed, stru tures exist that exploit the s attering and interferen e ee ts arising in periodi stru tures. 8

Controlling light emission and s attering: Quantum dots in photoni rystals

1.2.2

Photoni rystals

Photoni rystals (PCs) are fas inating materials that oer unique possibilities for light-propagation ontrol [7, 8, 9℄. As the name suggests, PCs are materials with a periodi stru turing on a length s ale omparable to the wavelength of light. Multiple s attering from the periodi stru ture alters the ee tive properties of the material in a ontrolled way. In order to a hieve as strong s attering as possible, PCs are typi ally made from a diele tri with a high refra tive index su h as Sili on. The multiple s attering may even lead to photoni band gaps - frequen y intervals in whi h no ele tromagneti eld is allowed to propagate in the material. Su h materials are ideally suited for the reation of opti al ir uits and even avities where light may, in prin iple, be perfe tly onned. In pra ti e, the manufa turing of full three dimensional PCs has proven to be very di ult. One type of rystal that has been su

essfully produ ed is the so alled inverse opals. Fig. 1.7 shows a sket h of the unit ell. Inverse opals are made by inltrating an f

rystal of polystyrene spheres with diele tri and subsequently evaporating the polystyrene, leaving a ompli ated periodi stru ture [10℄.

Figure 1.7: Left:

Sket h of inverse opal unit ell with two QDs pla ed at the

high-symmetry points

Γ

(in the enter of the air sphere) and

H

(between six air

spheres). Right: Sket h of a PC slab onsisting of a periodi array of air holes in a semi ondu tor membrane.

As an alternative to full three dimensional PCs, so- alled PC slabs are made from thin semi ondu tor membranes by et hing periodi arrays of holes as illustrated in Fig. 1.7. The index ontrast onnes light to in-plane motion through total internal ree tion while s attering o the periodi array of air holes ae ts the spe tral distribution of modes in the same way as a Bragg mirror, yet ee tive in all in-plane dire tions. The planar nature of these stru tures makes them ompatible with standard semi ondu tor fabri ation te hniques. In addition, the material making up the membrane may be onveniently grown using standard epitaxial growth 9

Chapter 1. Introdu tion

te hniques, thus dire tly enabling the in lusion of self-assembled QDs. From a theoreti al point of view, the onnement to in-plane motion in PC slabs means that many aspe ts of light propagation in these materials may be understood from two dimensional models. This greatly simplies the generally very omplex s attering

al ulations. This work is on erned with modeling of inverse opals and PC slabs only. Other experimental realizations of three dimensional PCs in lude woodpile

rystals [11℄ and the so- alled Yablonovite [12, 13℄. 1.2.3

Quantum opti s and the lo al density-of-states

Whereas many opti al phenomena may be adequately des ribed in a semi- lassi al model, spontaneous de ay of an initially ex ited emitter an only be fully understood in a framework in whi h both the emitter and the light eld is quantized. Early theory of spontaneous emission dates ba k to Einstein [14℄ and Dira [15℄ as well as Wigner and Weisskopf who used a full quantum opti al des ription, ombined with a novel approximation, to show how the oupling of an ex ited atom to the

ontinuum of modes in va uum naturally leads to an exponential de ay [16℄. One

an show that given an initial ondition in whi h an ele tron is in an ex ited state of an atom, the probability that at time t the ele tron is still in the ex ited state de ays exponentially with time. The al ulations by Wigner and Weisskopf an be applied also to semi ondu tor QDs to reveal an exponential de ay. In the mathemati al framework of quantum opti s the ele tromagneti modes are des ribed as harmoni os illators. Therefore, the distribution of modes at the position of the QD is of spe ial importan e as it des ribes the number of os illators with whi h the QD an intera t. The al ulated de ay rate turns out to be proportional to the number of ele tromagneti modes, and the quantum opti al theory of Wigner and Weisskopf thus seems to agree with the measurements in Figs. 1.4 and 1.5 whi h did indeed show an exponential de ay with a rate that we ould argue was proportional to properties of the environment. From the theory we an now argue that it is the ele tromagneti properties of the environment that inuen es the de ay rate. The number of ele tromagneti modes at a given position is usually referred to as the lo al density-of-states (LDOS) [17℄ and is exa tly the parameter ρx (r) that we introdu ed in Eq. (1.3). The LDOS may be al ulated analyti ally for a very limited set of material stru tures, and usually numeri al methods have to be used. Sin e the Green's tensor ontains naturally ontributions from all s attering sites in the material it

an be used also to obtain the LDOS. Fig. 1.8 shows the LDOS of a Si inverse opal at the two high-symmetry positions Γ and K . A band gap is learly visible around ωa/2πc ≈ 0.8 where the LDOS drops to zero, indi ating that no modes exist in the material at this frequen y. 10

Controlling light emission and s attering: Quantum dots in photoni rystals

3

ρx [4/3ca2 ]

2.5 2 1.5 1 0.5 0 0

0.2

0.4

0.6

0.8

1

ωa/2πc Figure 1.8: LDOS of a Si inverse opal at the two positions

H

Γ

(gray histogram) and

(blue histogram) as indi ated in Fig. 1.7. Dashed line indi ates the free spa e

LDOS,

1.2.4

ρ0 (ω) = ω 2 /3π 2 c3 .

Quantum dot de ay dynami s

Variations in the LDOS may lead to a suppression or an enhan ement of the measured de ay rate as shown expli itly in Figs. 1.4 and 1.5. It may even hange the qualitative behavior of the de ay dynami s as the de ay needs not always be exponential, and the emitter needs not always to de ay fully. Indeed, depending on the variations in the LDOS the de ay may happen in a number of fundamentally dierent ways as illustrated in Fig. 1.9. When intera ting with a ontinuum of ele tromagneti modes, the de ay is a Markovian pro ess. On e the energy is transferred from the emitter to the ele tromagneti eld it is irreversibly lost, and the emitter ends up in the ground state. A ontinuum of modes in the language of oupled os illators amounts to a LDOS that varies slowly as a fun tion of frequen y. In this ase the theory of Wigner and Weisskopf shows that the de ay urves are de aying exponentials. It follows dire tly that the rate may be enhan ed relative to the rate in the homogeneous medium by

hanging the distribution of opti al modes at the lo ation of the emitter. This is the famous Pur ell ee t. If the emitter an intera t only with a single mode of the ele tromagneti eld there will be an inter hange of energy ba k and forth between the emitter and the eld. This may happen in avities of very high quality, for example in PCs. In this

ase the LDOS will exhibit a sharp peak at the frequen y of the avity resonan e resulting in a oherent inter hange of energy between the two leading to os illations 11

Chapter 1. Introdu tion

Pe |ei

|µi

|gi

|0i t Pe

|ei

|µi

|gi

|0i t Pe

|ei

|µi

|gi

|0i t

Figure 1.9: Dierent QD de ay dynami s. Top: Exponential de ay hara teristi of a single os illator intera ting with a ontinuum of modes. Middle: Intera tion of two os illators leading to ontinuous inter hange of energy between the two known as va uum Rabi os illations. Bottom: A single os illator intera ting with a detuned

ontinuum of modes leading to fra tional de ay. The gure is inspired by a similar illustration by Weisbu h et al. [18℄. in the de ay urve known as va uum Rabi os illations. An interesting situation o

urs when the emitter is tuned spe trally very lose to the sharp edge of the band gap of a PC. In this ase the emitter may undergo a so- alled fra tional de ay in whi h the probability tends to a nite, non-zero value at long times [19, 20℄. The intera tion of the QD with modes of low (or even zero) group velo ity at the band edge leads to a situation in whi h some of the energy is preserved in the system and some is lost to the environment. This phenomenon

learly represents a very ounter intuitive example of the quantum nature of the de ay in whi h the ele tron is not fully ex ited, yet has not fully de ayed either. 1.2.5

Experiments and appli ations

On the experimental side, manipulation of the spontaneous de ay rate due to the Pur ell ee t in PCs has been shown in a number of experiments. These in lude 12

Overview of this thesis both the use of inverse opals [21℄ and PC slabs [22, 23℄. Advan es in fabri ation have enabled the study of physi s of opti al mi ro avities [24℄. The high LDOS a hievable in a small avity leads to fast de ay due to the Pur ell ee t as has been demonstrated in avities in a number of systems in luding PC slabs [25℄. The so alled strong oupling regime of an emitter intera ting with a single opti al mode has been observed (using spe tral analysis) for avities in photoni rystal slabs [26℄ as well as in other types of opti al avities [27℄. The ontrolled positioning of a mi ro avity around a single self-assembled QD in a PC slab was demonstrated in Ref. [28℄. These results have spurred a lot of theoreti al interest due to an oresonant oupling of the QD to the opti al mode in the mi ro avity, most likely due to phonon intera tions [29, 30, 31℄. Fra tional de ay has yet to be experimentally demonstrated. During this PhD proje t we have arried out quantitative al ulations in order to assess the possible demonstration of this novel quantum ee t using QDs in Si inverse opals. As for appli ations, the suppression of spontaneous emission in photoni rystals was proposed originally as a means to a hieve more e ient solar ells and semi ondu tor lasers [7, 8℄. Light-matter intera tion involving a single or a few QDs will nd appli ations in the emerging eld of quantum information te hnologies, espe ially for se ure ommuni ation [32℄ and quantum networks [33℄. Single photon sour es as well as single photon swit hes and transistors are key elements in opti al quantum networks. The use of single photon sour es for quantum ryptography has been demonstrated experimentally [34℄, and re ently the high Pur ell fa tor and the built in dire tionality of the emitted light a hievable in wave guides in photoni rystal slabs has been proposed [35, 36℄ and experimentally demonstrated [37℄ as a means to a hieve e ient and ompa t single photon sour es. Deterministi reation of entangled photon pairs are of parti ular interest for quantum ryptography [38, 39℄ as well as for quantum imaging [40, 41℄, quantum teleportation [42℄ and quantum omputing [43, 44℄. Entangled photon pairs may in prin iple be emitted from bi-ex iton de ay in QDs, and the eld has re eived attention both theoreti ally [45℄ and experimentally [46℄. 1.3

Overview of this thesis

In hapter 2 we set up the theoreti al framework that will be used throughout the thesis. Many of the results in the hapter will be well known to the reader, but are repeated here in order to present a oherent overview of the theory and to illustrate how the results relate to ea h other. For use in the modeling of nanophotoni stru tures, and in parti ular in the ontext of light-matter intera tion in photoni 13

Chapter 1. Introdu tion

rystals, we fo us on a s attering type formulation of light propagation based on the ele tri eld Green's tensor. Using the Green's tensor for the ba kground medium, one may re ast Maxwell's wave equation as a s attering problem, in whi h ase the solution is given in terms of the so- alled Lippmann-S hwinger integral equation [47℄. During the proje t we have developed a novel approa h to the numeri al solution of the LippmannS hwinger equation. The method may be applied to systems with s atterers of arbitrary shape and non-homogenous ba kground materials. In hapter 3 we formulate the method in the general ase and give details of the implementation in two dimensions as well as example al ulations. We illustrate the method by al ulating light emission from a line sour e in a nite sized photoni rystal waveguide. Depending on the obje tive, one will naturally fo us on properties of either the ex itations in the QD or the emitted eld as dis ussed in se tion 1.1.3. As noted in Ref. [48℄, this amounts to a hoi e between two pi tures in whi h to solve a quantum opti al problem of light-matter intera tion: an all-matter pi ture or an all-light pi ture. In hapter 4 we work in the all-matter pi ture and examine fra tional de ay in three-dimensional photoni rystals. Working in an all matter pi ture, the dynami s are governed by the LDOS as illustrated in se tion 1.2.4. We perform high resolution al ulations of the LDOS in a Si inverse opal in order to assess the possible realization of fra tional de ay in these systems. Furthermore, we quantify the inuen e of absorptive loss and show that it is a limiting but not prohibitive ee t. For many appli ations it is desirable to have an intera tion between the QDs and the ele tromagneti eld that is as large as possible. High-quality QDs show an impressive light-matter intera tion whi h is important for appli ations su h as single-photon sour es. Likewise, this is important for novel quantum opti s experiments aimed, for example, at the experimental demonstration of non-Markovian de ay dynami s as des ribed in se tion 1.2.4. However, if the QD intera ts strongly with light it means that it will itself a t as a s atterer for the light. This may have

onsequen es for measurements as it may hange the LDOS or even result in different de ay urves. In hapter 5 we hoose an all light approa h to study multiple QD dynami s in photoni rystal slabs. Finally, hapter 6 holds the on lusions.

14

Chapter 2 Theory of light-matter intera tion

2.1

Introdu tion

This hapter provides an overview of the theory of light-matter intera tion used for al ulations in this work. It is not the obje tive to derive the theory from rst prin iples, and we will present derivations only if they serve to larify parti ular results. Rather, we refer to the literature for the derivations and fo us on a oherent presentation of the parts of the theory important for our purpose.

2.1.1

Overview of Chapter 2

The theory divides naturally into three parts: light propagation and s attering, QD dynami s and the oupling of the two. In se tion 2.2 we dis uss the theory of light propagation and s attering based on Maxwell's equations and the ele tromagneti Green's tensor. In addition, we dis uss anoni al quantization of the eld in terms of quantum me hani al eld operators and elements of PC theory. Se tion 2.3 is on erned with QD models and a proper des ription of ex itations in the form of ex itons. Se tion 2.4 presents the typi al oupling formalism employed in the literature, and we derive and solve equations of motion for the QD dynami s as well as for the ele tri eld emitted from a QD. 15

Chapter 2. Theory of light-matter intera tion

2.2

Modeling light propagation and s attering

As we aim to model light-matter intera tion in semi ondu tors we will on ider only non-magneti materials with no free harges or urrents. Therefore, we start from Maxwell's equations in the form ∇·D = 0

(2.1a)

∇·B = 0

(2.1b)

∇×E = − ∇×H =

∂ B ∂t

∂ D, ∂t

(2.1 ) (2.1d)

where E and B are the ele tri and magneti elds, respe tively, while D and H denote the ele tri displa ement and auxiliary magneti elds, respe tively. The elds are related through the onstitutive relations D = ǫ0 E + P = ǫ0 ǫr E 1 H= B, µ0

(2.2) (2.3)

where ǫ0 and µ0 are free spa e permittivity and permeability, respe tively and ǫr is 2 , where nR the relative permittivity. For non-magneti materials we have ǫr = nR is the refra tive index. P denotes the intrinsi polarization of the passive material. In the ase of an ex itation event in a QD an ex iton is formed, and in this ase we model the polarization due to the ex iton motion as an additional extrinsi polarization as dis ussed in se tion 2.2.1. Eqs. (2.1 ) and (2.1d) together with the onstitutive relations provide the wave equation for the ele tri eld, ∇ × ∇ × E(r, t) +

ǫr ∂ E(r, t) = 0, c2 ∂t2

(2.4)

with time-harmoni solutions of the form E(r, t) = E(r)e−iωt .

(2.5)

The position dependent ele tri eld E(r) solves the ve tor Helmholtz equation ∇ × ∇ × E(r) − k02 ǫr (r)E(r) = 0,

(2.6)

where ǫr (r) is the position dependent relative permittivity and k0 = |k0 | = ω/c is the wave number in va uum. Eq. (2.6) is a generalized eigenvalue equation and we 16

Modeling light propagation and s attering refer to the ve tor eigenfun tions and orresponding frequen ies as fµ (r) and ωµ , respe tively. The ve tor eigenfun tions are normalized as Z

V

ǫr (r) fµ∗ (r) · fλ (r) dr = δµ,λ .

(2.7)

As dis ussed in se tion 1.2.3, the LDOS enters naturally in quantum opti al

al ulations of light-matter intera tion. The LDOS is a lassi al property of the ele tromagneti eld and is al ulated as a sum over all ele tromagneti modes in the system as X ρx (ω, r) = (2.8) |ex · fµ (r)|2 δ(ω − ωµ ), µ

where we have in luded a proje tion of the modes onto the dire tion ex . Sin e all al ulations in this work are based on the proje ted LDOS, Eq. (2.8), we will typi ally refer to this simply as the LDOS. In a homogeneous material the LDOS is independent of position and proje tion dire tion and depends only on frequen y as ρhom (ω) =

nR ω 2 . 3π 2 c3

This is shown expli itly in appendix D.

2.2.1 Ele tri eld from a point sour e - the Green's tensor By the introdu tion of an extrinsi polarization, Pex (r, t), the wave equation for the ele tri eld is rewritten as ∇ × ∇ × E(r, t) = −

∂2 µ0 {ǫ0 ǫr (r)E(r, t) + Pex (r, t)} , ∂t2

whi h, by transforming to the frequen y domain and rearranging, takes on a familiar form with the extrinsi polarization a ting as the sour e: ∇ × ∇ × E(r, ω) − k02 ǫr (r)E(r, ω) =

k02 Pex (r, ω). ǫ0

(2.9)

In this ase the ele tri eld may be al ulated as [49, 50℄ E(r, ω) =

Z

G(r, r′ , ω)

k02 Pex (r′ , ω)dr′ , ǫ0

(2.10)

in whi h G(r, r′ , ω) is the ele tri eld Green's tensor for the material system. The Green's tensor is the solution to the equation ∇ × ∇ × G(r, r′ , ω) − k02 ǫr (r)G(r, r′ , ω) = Iδ(r − r′ ),

(2.11) 17

Chapter 2. Theory of light-matter intera tion

subje t to the Sommerfeld radiation ondition [51℄, and diers depending on the dimensionality. A similar equation denes the magneti eld Green's tensor. We will work only with the ele tri eld version and we will refer to this as the ele tromagneti Green's tensor to emphasize the wave nature of the eld. Appendix A lists G(r, r′ , ω) for homogeneous media in dierent dimensions. Apart from homogeneous materials, losed form expressions for the Green's tensor are restri ted to a limited set of simple geometries su h as spheri ally layered stru tures [52℄ and stratied media [53, 54℄. A omprehensive theoreti al treatment of ele tri and magneti eld Green's tensors in various geometries is given by Tai [49℄. The denition of the Green's tensor in Eq. (2.11) is onsistent with that of Refs. [49, 50, 55℄. It diers by a fa tor of (−1) from the Green's tensor in Ref. [48℄ and by a fa tor of k02 from that of Refs. [56, 57℄. The interpretation of the Green's tensor be omes lear if we use Eq. (2.10) to

al ulate the eld from a point sour e, Pex (r, ω) = d(ω)δ(r − r′ ), as E(r, ω) =

Z

G(r, r′′ , ω)

= G(r, r′ , ω)

k02 d(ω)δ(r′′ − r′ )dr′′ ǫ0

k02 d(ω), ǫ0

(2.12)

whi h shows that the i'th olumn of the Green's tensor may be interpreted as the ele tri eld at r due to a point sour e at r′ oriented in the dire tion ei . For a given material system, as dened by ǫr (r), the Green's tensor may be expanded on the eigenmodes of the wave equation as [50, 58℄ G(r, r′ , ω) = c2

X µ

fµ (r)fµ∗ (r′ ) , ωµ2 − (ω + iη)2

(2.13)

where the innitesimal but positive imaginary part iη ensures ausality of the Green's tensor [48℄. Eq. (2.13), together with the identity lim Im

ǫ→0



1 2 ωµ − ω 2 − iǫ



=

π (δ(ω − ωµ ) − δ(ω + ωµ )) , 2ωµ

may be used to relate the Green's tensor to the LDOS as ρx (ω, r) = 2.2.2

2ω Im {ex G(r, r, ω) ex } . π c2

(2.14)

The Lippmann-S hwinger equation

In se tion 2.2.1, the solution to the inhomogeneous problem in Eq. (2.9) was found as an integral over the ele tromagneti Green's tensor and the sour e term. The 18

Modeling light propagation and s attering pro edure is not limited to extrinsi sour es but may equally well be applied to s attering problems in whi h the ele tri eld appears on the right hand side. In order to reformulate Eq. (2.6) as a s attering problem, we onsider the hange in permittivity, ∆ε(r) = ǫr (r) − ǫB (r), aused by the introdu tion of s attering sites into the ba kground medium des ribed by ǫB (r). With this denition, Eq. (2.6) is rewritten as ∇ × ∇ × E(r) − k02 ǫB E(r) = k02 ∆ǫ(r)E(r), (2.15) whi h is an impli it equation sin e the ele tri eld itself enters on the right hand side. The solution to Eq. (2.6) with ǫr (r) = ǫB (r) is denoted by EB (r) and represents the in oming eld. The full solution to Eq. (2.6) is the sum of the in oming eld and the s attered eld. It is given by the Lippmann-S hwinger equation [47, 55℄, Z E(r) = EB (r) +

V

GB (r, r′ ) k02 ∆ε(r′ ) E(r′ )dr′ ,

(2.16)

in whi h GB (r, r′ ) is the Green's tensor for the ba kground medium. As in Eq. (2.16), we will typi ally omit the expli it frequen y dependen e of the Green's tensor to ease notation. In two and three dimensions the real part of the Green's tensor diverges in the limit r′ = r. This means that for integrals in whi h r is inside the s attering volume (su h as in this work) an alternative formulation of the LippmannS hwinger equation must be employed in whi h the singularity is isolated in an innitesimal prin ipal volume δV and treated analyti ally [59℄. In this ase we follow Ref. [55℄ and rewrite the Lippmann-S hwinger equation as E(r) = EB (r) + lim

δV →0

Z

V −δV

GB (r, r′ ) k02 ∆ε(r′ )E(r′ )dr′ − L

∆ǫr (r) E(r), ǫB

where the ex lusion volume δV is entered on r′ = r and L is a dimensionality dependent sour e dyadi . Expressions for the sour e dyadi are listed by Yaghjian for various shapes of the ex lusion volume [59℄. We will work only with ir ular ex lusion areas in two dimensions and spheres in three dimensions for whi h we have     L2D

 1 1 =  0 2  0

0 0   1 0    0 0

and L3D

 1 1 =  0 3  0

0 0   1 0  .  0 1

The Dyson equation From Eq. (2.12) we may interpret ea h olumn in the Green's tensor as an ele tri eld. Therefore, the full Green's tensor for a given material system may be 19

Chapter 2. Theory of light-matter intera tion

al ulated from a s attering formulation similar to Eq. (2.16) known as the Dyson equation [50, 55℄, G(r, r′ ) = GB (r, r′ ) +

Z

V

GB (r, r′′ ) k02 ∆ε(r′′ ) G(r′′ , r′ )dr′′ .

(2.17)

Using the Dyson equation, known results for simple geometries may be used as ba kground Green's tensors in numeri al al ulations of the Green's tensor for more

ompli ated stru tures.

2.2.3 Quantization of the ele tromagneti eld The physi al ele tri eld is real, and in appli ations one will therefore attribute physi al signi an e only to the real part of Eq. (2.5). In quantum opti s the ele tri eld is promoted to an operator. Owing to the fundamental postulates of quantum me hani s, the operator of a dynami al variable must be Hermitian. This means that we must write the eld as a sum of positive and negative frequen y parts, ea h of the form of Eq. (2.5), with orresponding annihilation and reation operators, respe tively. There is a freedom of hoi e in the exa t form of the elds sin e any linear ombination of solutions to Eq. (2.4) will itself be a solution. In this work we use the following des ription of the ve tor potential and the ele tri and magneti elds, respe tively: A=

X εµ ωµ µ

E=−

fµ (r) aµ e−i ωµ t + fµ∗ (r) a†µ ei ωµ t




X
∂ A = i εµ fµ (r) aµ e−i ωµ t − fµ∗ (r) a†µ ei ωµ t ∂t µ

B = ∇ × A.

(2.18a) (2.18b) (2.18 )

Eqs. (2.18) are onsistent with Refs [48, 60, 61, 62, 63℄. The operators aµ and a†µ are annihilation and reation operators, respe tively, for a photon in mode µ. These satisfy bosoni ommutation relations, [aµ , a†λ ] = δµ,λ .

(2.19)

The eld distribution fun tions fµ (r) are eigenfun tions of Eq. (2.6) with eigenvalues (ωµ /c)2 and are normalized a

ording to Eq. (2.7). The eld normalization

onstant εµ is given as εµ =

20

r

~ ωµ , 2 ǫ0

(2.20)

Modeling light propagation and s attering where ~ is the redu ed Plan k onstant. The normalization is onsistent with the fundamental postulate by Plan k that the energy in a given mode of the ele tromagneti eld is found in quanta of ~ωµ . Indeed, the Hamiltonian is given as HEM = ǫ0

Z

V

ǫr (r) E† (r) · E(r) dr

(2.21)

whi h, by the use of Eqs. (2.18b) and (2.19)-(2.20), we may write as HEM =

X

=

X

ǫ0 ε2µ a†µ aµ + aµ a†µ

µ

µ

  1 ~ωµ a†µ aµ + . 2




(2.22)

The terms ~ωµ /2 are attributed to the ele tromagneti va uum and are usually omitted [60℄. 2.2.4

Modeling photoni rystals

The theory of PCs is thoroughly des ribed in the textbook by Joannopoulos et al. [64℄. In this se tion we dis uss only those elements of the theoreti al framework that are needed for the present work. PCs are hara terized by a periodi latti e with de orations in the form of a hange in permittivity. The latti e is dened by a set of basis ve tors ai whi h denes also a set of re ipro al basis ve tors bi through the relation ai · bj = 2πδij . (2.23) All latti e ve tors R and re ipro al latti e ve tors G may be expanded on the basis ve tors and re ipro al basis ve tors, respe tively, as R=

N X i

ai ai

and G =

N X

b i bi ,

i

where N denotes the dimensionality of the rystal (typi ally two or three). With this notation, the PCs of interest may be hara terized ompletely by the permittivity ǫr (r) = ǫr (r + R). Blo h's theorem [64℄ ensures that the solutions to the wave equation in PCs may be written in the form Ek (r) = eik · r uk (r),

(2.24)

in whi h uk (r + R) = uk (r) for all latti e ve tors R. Due to the Blo h form of the solution, we fo us the analysis to wave ve tors k in the rst Brillouin zone of the re ipro al latti e, dened as the set of points loser to k = 0 than to any other 21

Chapter 2. Theory of light-matter intera tion

re ipro al latti e site (the entral Voronoi ell). For any wave ve tor k′ there exists a re ipro al latti e ve tor G so that k′ = k + G with k in the rst Brillouin zone. Inserting in Eq. (2.24) we have Ek′ (r) = eik · r uk′ (r)eiG · r

(2.25)

whi h is itself a valid Blo h mode of the form in Eq. (2.24) sin e from Eq. (2.23) it follows that G · R = 2πM with integer M and hen e uk′ (r + R)ei G · (r+R) = uk′ (r)ei G · r .

Although the solutions in Eqs. (2.24) and (2.25) have the same form they are not the same solutions. Hen e, they do not ne essarily have the same frequen ies but rather belong to dierent bands of the PC band stru ture, and we shall therefore index the modes by the wave ve tor k as well as the band number n so that Eq. (2.6) takes the form ∇ × ∇ × En,k (r) =

ωn2 (k) ǫr (r)En,k (r). c2

(2.26)

Due to the symmetries of the latti e, solutions at dierent k points may have identi al frequen ies. An obvious example is time-reversal symmetry whi h holds for all photoni rystals of interest in this work. It leads to the physi ally quite reasonable ondition that waves traveling in exa t opposite dire tions will have identi al frequen ies; ωn (k) = ωn (−k) [64℄. In order to remove redundan y due to symmetry, investigations are typi ally restri ted to the boundaries of the irredu ible Brillouin zone. This is dened as the set of points in the Brillouin zone for whi h the solutions En,k (r), with orresponding eigenfrequen ies ωn (k), are not related to the solutions at other k points via any of the symmetries of the latti e. The PC band stru tures are typi ally al ulated as the eigenvalues of the ele tromagneti wave equation with periodi permittivity, ǫr (r) = ǫr (r + R), under the assumption that the solutions have Blo h form, Eq. (2.24). For al ulations on non-magneti materials it is easier to work with the wave equation for the Hk (r) eld whi h, under the assumption Hk (r) = exp(ik · r)uk (r), redu es to the form (ik + ∇) ×

ω 2 (k) 1 uk (r). (ik + ∇) × uk (r) = ǫr (r) c2

(2.27)

For a given material distribution ǫr (r), Eq. (2.27) may be dis retized and solved using any suitable numeri al method. 2.2.5

An overview of existing al ulation methods

A large number of dierent methods are being explored in the investigation of light propagation in mi rostru tured materials su h as photoni rystals. In order to re22

Modeling light propagation and s attering late the methods that we have used in this proje t to other methods in the literature, we give in this se tion an overview of relevant existing al ulation methods.

Finite Dieren e Time-Domain al ulations The Finite Dieren e Time-Domain (FDTD) method [65℄ is the workhorse of many light propagation al ulations and is regularly employed in the design of PC stru tures. Although originally developed for ele tromagneti problems it has appli ations in other areas in luding quantum me hani s [66℄. We shall fo us only on appli ations with relation to our work, namely band stru ture al ulations and al ulations of the Green's tensor and LDOS. As an alternative to the formulation in terms of an eigenvalue equation, as in Eq. (2.27), one an employ FDTD al ulations in order to tra e out the band diagram. In this approa h, so- alled Blo h boundary onditions are enfor ed on the FDTD al ulation domain in order to make the eld onform to the Blo h form, Eq. (2.24). Using a olle tion of randomly distributed broad band sour es, all relevant modes of the system are ex ited and allowed to propagate for a su iently long time that a subsequent Fourier transform will reveal only the strongest resonan es of the system. These orrespond to the eigenmodes, and the FDTD approa h thus oers an alternative to the formulations based on Eq. (2.27). In se tion 5.2 we use both plane wave al ulations and the FDTD method to al ulate the band stru ture of a photoni rystal slab. FDTD has been used for al ulations of the LDOS in photoni rystals; notably in Ref. [67℄ where the Pur ell ee t due to the mode in a PC wave guide was

al ulated using FDTD. The need for dening proper boundary onditions su h as perfe tly mat hed layers make FDTD al ulations in periodi stru tures ompli ated, and the method may be restri ted to problems with well dened modes su h as wave guides. For nite sized stru tures this problem does not arise, and FDTD has been used for the al ulation of the LDOS in photoni rystallites [68℄ and nite sized waveguides [35℄. For Cal ulations of the Green's tensor using FDTD the system is ex ited by a broad band point sour e at a given lo ation r′ and the system is evolved in time through Maxwell's equations and monitored at any position r within the al ulation domain. In this ase Eq. (2.12) establishes the onne tion between the measured eld and the sour e from whi h the Green's tensor is derived. This pro edure is used for the al ulation of the Green's tensor in a PC slab in se tion 5.2. Although it is a stable and well established method, FDTD is extremely demanding in terms of both memory and time, easily rea hing runtimes of several days for pra ti ally relevant stru tures.

23

Chapter 2. Theory of light-matter intera tion

Plane wave expansion

Plane wave expansion of Eq. (2.27) has been widely used for the study of photoni

rystals through band stru ture al ulations [69, 70℄. In plane wave al ulations, the elds uk (r) and the inverse permittivity ǫr (r)−1 are expanded on latti e plane waves as X X 1 = ηG ei G · r , uk (r) = (2.28) uG (k)ei G · r and G

ǫr (r)

G

and the a

ura y is thus set by the al ulation of the matrix elements uG and ηG as well as the number of plane waves as ontrolled by the uto in the wave ve tors G. Due to the parti ular hoi e of expansions, the plane wave method is ne for full three-dimensional photoni rystals but does lead to problems in al ulations on PC slabs where the analysis is limited to modes below the light-line [71℄. The LDOS for photoni rystals has previously been investigated using plane wave expansions [72, 73, 74℄ in whi h the modes were found as the solutions to Eq. (2.28) and summed a

ording to Eq. (2.8). Multipole methods

The Generalized Multipole Te hnique [75℄ employs an expansion of the ele tromagneti eld in terms of so- alled multipoles that are solutions to the wave equation in spheri al oordinates ( ylindri al oordinates in two dimensions). The expansion is performed in ea h region of the material, and a set of linear equations for the expansion oe ients is obtained by imposing proper boundary onditions for the elds. Sin e the basis fun tions are solutions to the wave equation, Maxwell's equations are automati ally fullled within ea h region, and an error estimate an be obtained by evaluating to what degree the boundary onditions are met. For al ulations involving large numbers of s atterers in an otherwise homogeneous ba kground, Rayleigh-multipole methods have been used for al ulations on mi rostru tured bres [76, 77, 78, 79℄ as well as photoni rystals omposed of

ylinders [80, 81℄ or spheres [82, 83℄. The use of multipole expansions for the elds ensures a signi ant redu tion in the number of basis fun tions, thus enabling al ulations on omplex stru tures of pra ti al interest. For simple s atterers, analyti al expressions for the expansion oe ients are available whi h signi antly in reases

al ulation speeds and a

ura y. Method of moments and the oupled dipole approximation

The Lippmann-S hwinger equation is an example of a volume integral equation for the ele tri eld. Various volume and surfa e integral methods exist for the ele tri and magneti elds. These methods typi ally rely on dis retization to express the 24

Modeling light propagation and s attering integrals as linear systems of equations - a pro edure known as the method of moments. In the dis retization pro ess the solution is expanded in terms of linearly independent basis fun tions whi h vary in omplexity depending on the spe i method employed [84℄. A parti ularly simple and onvenient hoi e of basis fun tion is the pulse basis fun tion, but higher order basis fun tions have been applied [85℄. The appli ation of pulse basis fun tions to the Lippmann-S hwinger equation results in what is usually termed the oupled (or dis rete) dipole approximation (CDA) [86, 87℄. The CDA allows for relatively easy implementation as well as the physi ally attra tive property that the resulting eld an be dire tly interpreted as the sum of the eld from all the individual ells in the s attering stru ture os illating as dipoles in response to the in oming eld. However, depending on the desired a

ura y and the nature of the s attering problem at hand, the required number of basis fun tions may be ome too large for pra ti al al ulations on e.g. photoni rystals. In addition, this type of dis retization may lead to stability problems whi h, for example, limit the e ien y in the ase of high refra tive index ontrasts [84℄. The CDA has previously been applied to two dimensional s attering problems in inhomogeneous ba kgrounds [88℄, and the appli ation of the CDA to the al ulation of the Green's tensor in a photoni rystal slab was reported in Ref. [89℄. In Refs [90, 91℄ an iterative s heme was suggested for the solution of the LippmannS hwinger equation whi h has been applied also for the al ulation of light propagation in nite sized photoni rystals [92℄. This elegant formulation is based on the

ombined use of the Lippmann-S hwinger equation and the Dyson equation to solve the linear equation system. However, the number of operations s ales as O(N 3 ). This means that the method is similar in performan e to most traditional solvers whereas advan ed solvers may show better performan e [84℄. Other methods

Other numeri al methods of great importan e in modeling of opti al devi es in lude the Finite Element Method (FEM) [93℄. Like FDTD, the FEM is a brute for e numeri al te hnique that an handle many dierent geometries. In addition, very e ient matrix inversion algorithms are available for the sparse matrix equation system resulting from typi al FEM dis retization. For three dimensional al ulations, however, the memory requirements for solution of the full ve tor eld problem may be ome too large for many appli ations. Last, we note that the expansion in Eq. (2.13) has been used for derivations of analyti al approximations to the Green's tensor in opti al wave guides and mi ro avities [94, 95, 96℄. 25

Chapter 2. Theory of light-matter intera tion

2.3

Quantum dot models

QDs allow for onnement of arrier motion to limited regions of the host diele tri . This leads to a quantization of the allowed energy levels as known from atomi physi s but realized in semi ondu tor materials. The energy level stru ture an therefore in prin iple be engineered by hanging the QD size and shape. In semi ondu tors, ex itations appear as the reation of an ele tron-hole pair subje t to a number of many parti le ee ts, e.g. oulomb intera tion. However, for most modeling purposes we may simply regard the QD as a two level system with a ground state and an ex ited state, similar to the ele troni states of an atom. Also, as in the ase of atoms, the QDs have an intrinsi dipole moment. Although we will ee tively model the QDs as two-level systems with parameters taken from measurements, in this se tion we will dis uss some aspe ts of QD modeling in order to illustrate the theoreti al foundations. As a starting point we onsider a perfe t semi ondu tor rystal in the ground state for whi h all states in the valen e band are lled and all states in the ondu tion band are empty. This is the so- alled Fermi va uum |F i [97℄. We des ribe the ele troni properties of the semi ondu tors lose to the band extrema in whi h the paraboli approximation is valid. The extrema are found at k = 0 for materials of interest in this work. The ele trons in the perfe t rystal are des ribed by the Hamiltonian HC whi h in ludes all many body ee ts su h as oulomb intera tion between all ele trons and nu lei. We model ex itations, su h as reated by the pump laser in the experiment in se tion 1.1, by the removal of an ele tron in the valen e band and the simultaneous reation of an ele tron in the ondu tion band, as illustrated in Fig. 2.1. Semi ondu tor theory and ee tive mass theory suggests that the missing ele tron

E

Ve Eg Vh

Figure 2.1: Left: An ex itation in the bulk semi ondu tor onsisting of an ele tron in the ondu tion band and a hole in the valen e band. Right: By the in lusion of Coulomb intera tion an ex iton is formed. In addition, the QD geometry leads to

onnement potentials Ve and Vh for the ele tron and hole, respe tively. 26

Quantum dot models in the valen e band be treated as a hole with positive harge and negative mass. The ex itation event thus reates the state |Xke ,kh i = ce†,ke c†h,kh |F i

(2.29)

where c†e,k and c†h,k denote ele tron and hole reation operators, respe tively [97℄. Being fermion operators, these satisfy anti ommutation relations, {cn,k , c†m,k′ } = δn,m δ(k − k′ ),

m, n ∈ {e, h}.

(2.30)

Due to the innite periodi latti e, the single-parti le wave fun tions may be indexed by the wave ve tor k, and owing to Blo h's theorem they may be written in the form ϕn,k (r) = hr|c†n,k |F i

eik · r = √ un,k (r), V

(2.31)

in whi h V is the volume of the rystal and un,k has the periodi ity of the latti e. The states |Xke ,kh i are eigenstates of the Hamiltonian for the perfe t rystal HC . Due to the dieren e in harge, we model the ele tron-hole system by adding an attra tive Coulomb intera tion term. To model onnement within the QD we will add also a potential in both the ondu tion band and the valen e band, as indi ated in Fig. 2.1. The additional terms in the Hamiltonian ruins the symmetry of the perfe t latti e, and as a onsequen e the eigenstates are no longer dened by a single wave ve tor. In this ase we may expand the eigenstates of the full Hamiltonian as a superposition of the ex itations in Eq. (2.29) as |Xi =

X

ke ,kh

χ(ke , kh )c†e,ke c†h,kh |F i,

(2.32)

and by proje ting onto the position eigenve tors and using the Blo h form of the solutions in the perfe t latti e we obtain the ex iton wave fun tion X(re , rh ) = hre , rh |Xi X χ(ke , kh )hre , rh |ce†,ke ch† ,kh |F i = ke ,kh

=

1 X χ(ke , kh )eike · re ue,ke (re )eikh · rh uh,kh (rh ) V ke ,kh

= χ(re , rh )ue,0 (re )uh,0 (rh ),

(2.33)

where we have evaluated the latti e-periodi fun tions un,k at k = 0, whi h is reasonable for ex itations lose to the band edge, and we have identied the sum as 27

Chapter 2. Theory of light-matter intera tion

the two-dimensional Fourier transform of the fun tion χ(ke , kh ). The two-parti le ex iton wave fun tion χ(re , rh ) solves the ee tive mass equation [98, 99℄: H(re , rh )χ(re , rh ) = Eχ(re , rh ),

(2.34)

in whi h E is the energy and H(re , rh ) = Eg +

p2e p2h q2 + + V (r ) + V (r ) − , e e h h 2m∗e 2m∗h 4πǫ0 ǫr |re − rh |

where Eg is the band gap and pn and m∗n denote momentum operators and ele tron or hole ee tive masses, respe tively. Ve and Vh are the onnement potentials for ele trons and holes, as indi ated in Fig. 2.1. The last term represents the attra tive Coulomb potential in whi h q = −1.602 × 10−19 C is the ele tron harge. Depending on the relative size of the Coulomb intera tion and the parti ular form of the onnement potentials, the solution to Eq. (2.34) takes on a number of dierent forms. In the limit of zero onnement, the solutions are Wannier ex itons for whi h the wave fun tions are given in terms of enter of mass and relative

oordinates with a hara teristi Bohr radius, a0 =

4π~2 ǫ0 ǫr , q 2 m∗R

where mR∗ = m∗e mh∗ /(me∗ + m∗h ) is the redu ed mass [98℄. For non-zero onnement potentials, an expli it model for Ve (re ) and Vh (rh ) is needed. A strong and a weak oupling regime may be dened on the basis of the onnement energy relative to the ex iton binding energy or, equivalently, the size of the QD relative to the Bohr radius [100℄. In the strong oupling regime the Coulomb intera tion may be ignored and Eq. (2.34) is separable in ele tron and hole parts that depend only on re and rh , respe tively. In this ase the solutions are produ ts of solutions to the single-parti le S hrödinger equation in the ee tive mass approximation, χ(re , rh ) = ψe (re )ψh (rh ). For most materials of interest these are relatively easy problems to solve using numeri al methods, e.g. FEM [101, 102℄. The orresponden e between results from these simple numeri al models and measurable quantities su h as energy and os illator strength has been tested with good agreement [1℄. An interesting model has been presented by Sugawara [103℄ in whi h the inplane potential of a QD is modeled with a paraboli potential and innite potential barriers are used in the out of plane dire tion. In this ase Eq. (2.34) is solvable with analyti al solutions spanning both the weak and the strong oupling regimes. 28

Coupling formalisms For most al ulations we will not be on erned with the expli it form of the ex iton state but simply dene an ex iton reation operator through the sum in Eq. (2.32) and write the ex iton state formally as |Xi = b†x |F i.

The ex iton, by denition, is an eigenstate of the S hrödinger equation HQD |Xi = ~ωx |Xi and we may onveniently write the Hamiltonian as HQD =

X

~ωx b†x bx ,

(2.35)

x

where the sum runs over all ex itons. Being omposed of ele tron and hole reation operators, the ex iton operators inherit the fermion anti ommutation relations, {bx , b†y } = δx,y . 2.4

Coupling formalisms

Having dis ussed al ulation of light propagation and s attering as well as QD models, we now turn to formalisms related to the oupling of light and matter. We shall be on erned with systems ontaining a single ex itation only. In this ase, for the oupled system, we are interested in properties onne ted to the omposite state X X |Ψi =

x

cx |x, 0i +

µ

cµ |0, µi

where |x, 0i = b†x |F i denotes the state with one ex iton in QD x and no photons and similarly |0, µi = a†µ |F i denotes the state with no ex itons and one photon in mode µ. With the above hoi e of state we are ee tively working in the one-ex itation subspa e of the entire Hilbert spa e. This potentially limits the validity of the

al ulations, so a motivation and dis ussion of the su ien y of al ulations in this subspa e is in order. The hoi e of a single ex itation in the system is reasonable for

al ulations in onne tion with experiments su h as the one des ribed in Se tion 1.1, sin e the ex itation power was kept well below one ex itation per QD. For higher ex itation powers, the probability of bi-ex iton states would be non-negligible, and we would have to expand the basis set in order to get an adequate des ription. Even with only one ex itation in the system, quantum me hani s allows for the ex itation of virtual states on short time s ales. However, the restri tion of the analysis to the one-ex itation subspa e orresponds essentially to the rotating wave approximation in whi h one keeps in the equations only terms orresponding to energy onserving transitions [104℄. 29

Chapter 2. Theory of light-matter intera tion

Physi al observables will depend on the state |Ψi as well as the time-evolution governed by the intera tion Hamiltonian. Below, we dis uss rst the intera tion Hamiltonian. Afterwards, in se tion 2.4.2, we will dis uss the derivation of the equations of motion in the S hrödinger and Heisenberg pi tures. In addition to the distin tion between S hrödinger and Heisenberg pi tures, quantum opti al al ulations of light-matter intera tion are typi ally arried out in one of two pi tures with respe t to the obje tive. One an work in an all matter pi ture in whi h the ele tromagneti eld is integrated out to reveal the dynami s of the atom or QD. Alternatively, one an work in an all light pi ture in whi h the atomi dynami s are integrated out to get the temporal behavior of the light at given positions [48℄. The two pi tures are omplementary, and the hoi e of Hamiltonian as well as the formulation of the equations of motion will depend on the obje tive. In se tion 2.4.3 we set up equations of motion for the internal QD dynami s in the S hrödinger pi ture, and in se tion 2.4.4 we onsider the eld from a quantum emitter in the Heisenberg pi ture. 2.4.1

Intera tion Hamiltonian

The intera tion Hamiltonian in quantum opti s is the subje t of some ontroversy in that it is typi ally formulated in one of two ways. The minimal oupling Hamiltonian is formulated in terms of the ve tor and s alar potentials A and Φ and reads [61℄ Z 1 1 2 1 2 (p − qA) + V (r) + qΦ + B dr ǫ0 ǫr (r)E 2 + 2m0 2 µ0 q q2 2 = HEM + HQD − (p · A + A · p) + A + qΦ, 2m0 2m0

Hint =

(2.36)

where we have identied the Hamiltonians for the un oupled ele tromagneti eld and QDs, f. Eqs. (2.22) and (2.35). Using the generalized Coulomb gauge
∇ · ǫr (r)A(r) = 0 and Φ = 0

and dropping the small q 2 A2 term, the intera tion part is usually rewritten in the form (Minimal Coupling)

min = − Hint

q p · A. m0

(2.37)

Instead of the minimal oupling Hamiltonian, the so- alled multipolar Hamiltonian is often used in quantum opti s al ulations. In this formulation the arriers

ouple to the displa ement eld D = ǫ0 ǫr (r)E + Pex [105℄, where Pex des ribes the extrinsi polarization due to the arriers in the QD, f. Eq. (2.9). In the dipole approximation, the multipolar Hamiltonian reads (Dipole) 30

dip = −qr · D/ǫ ǫ (r). Hint 0 r

(2.38)

Coupling formalisms Often the dipole Hamiltonian is expressed in terms of the ele tri eld only, thus negle ting the extrinsi polarization due to the ex itons themselves. In order to a

ount for the extrinsi polarization, a slightly dierent formulation in terms of an auxiliary eld F = D/ǫ0 ǫr (r) was used in Ref. [48℄. Starting on a fundamental level, one an show that the Lagrangians of the minimal oupling Hamiltonian and the multipolar Hamiltonian are equal up to a total time derivative. This means that although the Hamiltonians are dierent in form they must be onne ted by a anoni al transformation and thus must lead to the same physi s [61℄. However, if one makes approximations to the Hamiltonians (su h as the dipole approximation) they may lead to dierent results. This is similar to the fa t that two dierent series that give the same sum are likely to give dierent sums if they are trun ated. 2.4.2

Equations of motion

Central to quantum me hani al al ulations is the histori al and fundamental hoi e between the two dierent viewpoints of the S hrödinger pi ture, in whi h the quantum me hani al state arries the time evolution, and the Heisenberg pi ture, in whi h the time evolution is ontained in the operators [97℄. In the S hrödinger pi ture the time-evolution is governed by the S hrödinger equation i~

∂ |Ψ(t)iS = H|Ψ(t)iS , ∂t

and we write the general state as |Ψ(t)iS = =

X x

X x

cx (t)|x, 0i +

X

cµ (t)|0, µi

µ

cIx (t)e−iωx t |x, 0i +

X µ

cIµ (t)e−iωµ t |0, µi,

where we have written expli itly the time dependen e of the expansion oe ients due to the Hamiltonian H0 = HQD + HEM of the un oupled system. In the un oupled system this is the only time-dependen e and cx and cµ are onstants. Typi ally in al ulations of light-matter intera tion, a so- alled intera tion pi ture is introdu ed in whi h the state arries the time evolution due to the intera tion only and thus ignoring fast os illations due to the bare energies of the un oupled system [104℄. In the intera tion pi ture, the general state is written as   i H0 t |ΨiS |ΨiI = exp ~ X X I = cx (t)|x, 0i + cIµ (t)|0, µi x

µ

31

Chapter 2. Theory of light-matter intera tion

and evolves in time a

ording to the equation i~

∂ |ΨiI = HI |ΨiI , ∂t

where the intera tion pi ture Hamiltonian is given as HI = exp



   i i H0 t Hint exp − H0 t . ~ ~

Inserting the state in the intera tion pi ture S hrödinger equation and proje ting onto the basis states |x, 0i and |0, µi we obtain the equations of motion for the expansion oe ients iX hx, 0|HI |0, µicIµ ~ µ iX h0, µ|HI |0, xicIx . c˙Iµ (t) = − ~ x c˙Ix (t) = −

(2.39a) (2.39b)

In the Heisenberg pi ture, we solve for the time evolution of the general operator

O based on the Heisenberg equations of motion. i~

∂ O = [O, H]. ∂t

In parti ular, we may solve for the time evolution of the annihilation operators bx and aµ in whi h ase we nd i b˙ x = −iωx bx − [bx , Hint ] ~ i a˙ µ = −iωµ aµ − [aµ , Hint ], ~

(2.40a) (2.40b)

with the adjoint equations providing the time dependen e of the reation operators. 2.4.3

Quantum dot de ay dynami s in the S hrödinger pi ture

This se tion des ribes the al ulation of QD de ay dynami s in the S hrödinger pi ture. The analysis follows the general method outlined in Ref. [62℄ whi h elegantly demonstrates how the QD dynami s are governed only by the lo al ele tromagneti properties of the medium as des ribed by the LDOS. In hapter 4 we will use this general formalism for the detailed analysis of non-Markovian de ay in photoni

rystals. We onsider the de ay of a single QD that we model as a two-level system as dis ussed in se tion 2.3. For the analysis we use the minimal oupling Hamiltonian 32

Coupling formalisms in the dipole approximation where the size of the QD (or the size of the ex iton wave fun tion) is assumed to be small ompared to variations in the ve tor potential so that we may set fµ (r) = fµ (rQD ) = fµ,0 . The typi al pro edure is now to expand the momentum operator in the states |Xi and |F i as p = pF X |F ihX| + pXF |XihF | = px bx + p∗x b†x

where pij = hi|p|ji are momentum matrix elements and where we have set pXX = pF F = 0 (see below). In this way we write the intera tion Hamiltonian as Hint = −



q X εµ ∗ a†µ px bx + p∗x b†x · fµ,0 aµ + fµ,0 m0 µ ω µ

from whi h the intera tion pi ture Hamiltonian follows: HI = −



q X εµ ∗ a†µ eiωµ t . px bx e−iωx t + p∗x b†x eiωx t · fµ,0 aµ e−iωµ t + fµ,0 m0 µ ω µ

In the rotating wave approximation we drop the two fast os illating terms and keep only the two terms os illating at the angular dieren e frequen y ∆µ = ωµ −ωx to arrive at the nal form of the intera tion pi ture Hamiltonian, HI = −~

X µ

where gµ =


gµ b†x aµ e−i∆µ + gµ∗ bx a†µ ei∆µ ,

(2.41)

q p∗ p x ex · fµ,0 . m0 2~ǫ0 ωµ

As expe ted, the intera tion pi ture Hamiltonian in the rotating wave approximation involves only energy onserving terms. Therefore, if we pump the system with only one quantum of energy the dynami s are restri ted to the one-ex itation subspa e. In the above derivation we assumed that pXX = pF F = 0. Typi ally, for light-matter al ulations in atomi physi s, the symmetry of the ele tron wave fun tions are used to argue that this is the ase. The self-assembled QDs are not spheri ally symmetri , and so this argument does not hold. However, if we had retained the two additional terms, these would have resulted in terms in the intera tion pi ture Hamiltonian os illating at the angular frequen y ωµ and thus would have been dropped in the rotating wave approximation. Also, these terms would not appear if we expanded the intera tion Hamiltonian in the one-ex itation subspa e, sin e they orrespond to the pro ess of annihilation or reation of a photon without the reation or annihilation of an ex iton. 33

Chapter 2. Theory of light-matter intera tion

With the intera tion pi ture Hamiltonian, Eq. (2.41), the equations of motion take on the simple form c˙Ix = i

X

cIµ gµ e−i∆µ t

(2.42a)

µ

c˙Iµ = i cIx gµ∗ ei∆µ t ,

(2.42b)

from whi h we nd, by integration of Eq. (2.42b) and insertion into Eq. (2.42a), the equation of motion for the expansion oe ient cIx : X c˙Ix = − |gµ |2 µ

t

Z

0

cIx (t′ )e−i∆µ (t−t ) dt′ . ′

From the dis ussion in se tion 1.1 the de ay is expe ted to depend on properties of both the QD and the ele tromagneti environment. The physi al details of the

oupling is hidden in the parameter |gµ |2 . In order to make the dependen e on the LDOS expli it we follow Ref. [62℄ and add an integral over a Dira δ -fun tion to rewrite the expression as c˙Ix (t) = −α = −α

XZ µ

Z

0

∞

0

∞Z

0

t

|ex · fµ,0 )|2 ω

t

Z

0

cIx (t′ ) e−i(ω−ωx )(t−t ) dt′ δ(ω − ωµ ) dω ′

cIx (t′ ) e−i(ω−ωx )(t−t ) ′

ρx (ω, rQD ) ′ dt dω, ω

(2.43)

in whi h α = q 2 p2 /2 ~ m2 ǫ0 and ρx (ω, r) is the proje ted LDOS, Eq. (2.8). From Eq. (2.43) we an now dire tly appre iate how the LDOS governs the QD de ay dynami s, as dis ussed in se tion 1.2.4. The Wigner-Weisskopf approximation

If we assume that the LDOS varies slowly a ross the spe tral linewidth of the emitter we may evaluate it at ω = ωx and pull it outside the integral in Eq. (2.43). This assumption is valid for a large set of materials be ause of the os illatory term in the integral whi h will average to zero for frequen ies mu h larger or smaller than ω = ωx . With this assumption we lower the limit in the frequen y integral to −∞ and rewrite the integral as ρx (ωx ) c˙Ix (t) ≈ −α ωx

ρx (ωx ) = −α ωx Γ I = − cx (t), 2 34

Z

t

x

0

Z

0

cI (t′ )

t

Z

∞

′

e−i(t−t )(ω−ωx ) dωdt′

−∞

cIx (t′ ) 2πδ(t′ − t)dt′

Coupling formalisms whi h orresponds to an exponential de ay with de ay onstant given as Γ=

π q 2 p2 ρp (ωx ) . ~ m20 ǫ0 ωx

In the above integration, the δ -fun tion leads to a fa tor of 1/2 be ause it is entered at the end point of the integration interval.

Single-mode eld - Va uum Rabi os illations Opti al mi ro avities oer the possibility of enhan ing a single mode in a given frequen y interval. For su h stru tures the LDOS shows a very sharp peak and may be approximated by a Dira δ -fun tion of the form ρx (ω) = k δ(ω − ωx ),

for the ase of the emitter being perfe tly resonant with the avity mode. Substituting into Eq. (2.43), the frequen y integral is evaluated to reveal the equation: I

c˙x (t) = −κ

2

Z

0

t

cIx (t′ ) dt′ ,

where κ2 = α k/ωx . The solution is harmoni os illations of the form cIx (t) = cos(κ t) as an be veried by dire t substitution. Introdu ing a δ -fun tion in the LDOS means that the expansions of the ele tromagneti eld operators in Eqs. (2.18) redu e to a single term ea h. In this ase it would be more natural to use a JaynesCummings model [106℄ to solve the equations of motion instead of the more general method presented here. 2.4.4

Field from quantum emitter in the Heisenberg pi ture

In se tion 2.2.1 the ele tri eld from a polarization point sour e was expressed in a simple and ompa t form in Eq. (2.12) based on the ele tromagneti Green's tensor. It is tempting to ask if a similar equation holds for a quantum me hani al emitter in the form of a QD emitting a single photon if we ex hange the lassi al ele tri eld and the polarization by quantum me hani al operators. Using the dipole Hamiltonian and a harmoni os illator model for the emitter it was shown in Ref. [48℄ that the answer is positive, provided that the Green's tensor is slightly modied. In this se tion, following Refs. [57, 95℄, we dis uss the derivation in the framework of fermioni operators. Expanding the dipole operator d = qr on the ex iton states we write the dipole Hamiltonian, Eq. (2.38) in the form Hint = −i

X µ,x



∗ a†µ , εµ dx bx + dx b†x · fµ,x aµ − fµ,x

35

Chapter 2. Theory of light-matter intera tion

where we have assumed that the dipole moments are real, and where the eld modes fµ,x = fµ (rx ) are evaluated at the lo ation of QD x. In the expansion of the dipole operator we have dropped the dXX and dF F terms as they will eventually lead to dynami s that are not restri ted to the one-ex itation subspa e. Introdu ing the

oupling strength hµx = iǫµ dx · fµ,x ,

we may use Eqs. (2.40) to set up the equations of motion for the reation and annihilation operators as iX ∗ h (bx + b†x ) ~ x µx iX = i ωµ a†µ − hµx (bx + b†x ) ~ x iX (hµx aµ + h∗µx a†µ ) bzx = −iωx bx − ~ µ iX = i ωx + (hµx aµ + h∗µx a†µ ) bzx , ~ µ

a˙ µ = −iωµ aµ + a˙ †µ b˙ x b˙ †x b†x

where we have dened the operator bzx = [b†x , bx ] = b†x bx − bx b†x .

The equations of motion may be simplied onsiderably due to the restri tion of the dynami s to the one-ex itation subspa e. By expanding the operators on the states |F i, |Xi = b†x |F i and |µi = a†µ |F i we realize that within this subspa e we have the identities a†µ bzx = −a†µ

aµ bzx = −aµ ,

whi h ee tively renders the equations of motion identi al to those of the harmoni os illator model in Ref. [48℄. This is onsistent with the intuitive argument that in the limit of one ex itation the distin tion between fermions and bosons due to symmetry be omes irrelevant. Instead of working in the time domain as in se tion 2.4.3, we may follow Refs. [48, 57, 95℄ where the equations of motion are solved in the frequen y domain. The pro edure is learly explained in Ref. [48℄, and we will not repeat it here. With this approa h, the ele tri eld at position rd from an ex iton at a dierent position rx 36

Coupling formalisms is given as E(rd , ω) =

G(rd , rx , ω)Sx (ω) 1 − ex G(rx , rx , ω)ex Ux (ω)

(2.44)

= G(1) (rd , rx , ω)Sx (ω),

where Sx (ω) = ex



idx ω 2 ǫ 0 c2



b† (0) bx (0) + x ω − ωx ω + ωx



represents the sour e, f. Eq. (2.12), and Ux (ω) =



d2x ω 2 ~ǫ0 c2



2ωx 2 ωx − ω 2



is a s attering potential due to the polarizability of the ex iton itself. Comparing to Eq. (2.12) we have identied the obje t multiplying onto the sour e in Eq. (2.44) as the proper Green's tensor, G(1) (rd , r1 , ω), whi h in ludes the self intera tion from the ex iton in the QD. In Ref. [48℄ it was shown that the ele tromagneti propagator that enters the equations of motion is slightly dierent from the Green's tensor of lassi al ele tromagnetism. The two are dierent only at r = r′ , however, and furthermore only the real part is dierent. It is well known that the real part of the Green's tensor diverges in this limit. For this reason, when al ulating spontaneous emission in the dipole approximation a renormalization is usually employed [48℄. This renormalization ee tively renders the expli it fun tional form irrelevant, and in this work we therefore keep the formulation in terms of the lassi al Green's tensor G(r, r′ ). As noted in se tion 2.2.1, the Green's tensor in this work, Eq. (2.11), is dierent by a fa tor of (−1) from the Green's tensor in Ref. [48℄. For this reason we have hanged the sour e and the s attering potentials a

ordingly to keep the same notation. In hapter 5.3 we use this method to investigate s attering from multiple QDs in a photoni rystal slab.

37

Chapter 3 Multiple s attering

al ulations using the Lippmann-S hwinger equation 3.1

Introdu tion

For a proper treatment of light propagation in mi ro- and nanostru tured media su h as PCs, the full wave nature of the ele tromagneti eld needs to be taken into a

ount. In this se tion we des ribe a solution method for the Lippmann-S hwinger equation that is well suited for al ulations in nite sized PCs. We believe that the method will nd appli ations in modeling of nanophotoni stru tures and devi es su h as waveguides, jun tions and lters as well as swit hes and single photon sour es based on PCs. In order to motivate the formulation and to illustrate dieren es and similarities with respe t to existing methods, we rst onsider a small example problem. 3.1.1

An example s attering problem

We onsider the simple s attering problem in Ref. [91℄ of a plane wave in ident along the x-axis on a diele tri plate at normal in iden e, as depi ted in Fig. 3.1. Due to symmetry, the s attering of the two polarizations is identi al in this ase, and both the ree ted and the transmitted wave propagate parallel to the x-axis resulting 39

Chapter 3. Multiple s attering al ulations using the Lippmann-S hwinger equation

ǫr

x Figure 3.1: An example s attering problem. A plane wave along the

x

A exp(ikB x)

is in ident

axis onto a diele tri plate at normal in iden e..

in a one-dimensional, s alar problem. This problem is among the simplest possible s attering problems and an be solved easily using most numeri al methods.

In

parti ular, we may follow Ref. [91℄ and employ the CDA with an iterative solution s heme.

Fig.

3.2 shows the CDA solution as a histogram to emphasize the pulse

basis fun tions used in the solution.

The hosen dis retization orresponds to 20

basis fun tions per wavelength in va uum and 10 basis fun tions per wavelength in the diele tri .

2

|E|

1.5

1

0.5 PSfrag repla ements

0 0

1

0.5

x

1.5

2

Figure 3.2: Amplitude of the solution to the s attering problem in Fig. 3.1 in the

ase of a diele tri barrier at in va uum.

λ0 = 0.8 µm,

x0 = 1

with permittivity

ǫ=4

respe tively.

with a dis retization of

L = 1/2 A = 1 and

and thi kness

The amplitude and wavelength of the in ident light is

Histogram shows the solution obtained from the CDA

∆x = 0.04 µm

and the solid urve shows the analyti al

solution.

The simpli ity of the s attering problem, and the fa t that the s atterer is homogeneous, means that the s attering problem an be solved analyti ally. This is

40

Introdu tion

done by writing, in ea h se tion of the s attering stru ture, the eld in terms of the solutions to the wave equation in that material. In one dimension the solutions are forward and ba kward traveling plane waves and so we expand the eld as    A exp(ikB x) + B exp(−ikB x)    E(x) = C exp(ikR x) + D exp(−ikR x)      E exp(ikB x)

x < x0 − L/2 |x − x0 | < L/2

(3.1)

x > x0 + L/2

and mat h the elds at the interfa es a

ording to the boundary onditions. In this way a linear equation system is obtained that may be dire tly solved to yield the

oe ients en . The solid red urve in Fig. 3.2 shows the analyti al solution. The histogram style in Fig. 3.2 illustrates how the numeri al solution is never better than the sampling. In parti ular, for the simple pulse basis fun tions, the solution in ell n has the same value for all x within the ell and drops abruptly to another value in the next ell. The abrupt jumps in the numeri al solution

aused by the use of pulse basis fun tions may result in stability issues in ve torial

al ulations in two and three dimensions. The problem may be removed through the use of more sophisti ated basis fun tions whi h would ome at the expense of larger al ulation times. For the s alar problems in one dimension the method is in fa t onvergent whi h is seen dire tly from the error analysis in Ref. [91℄. The numeri al solution is al ulated rst at positions within the diele tri plate. The solution at positions outside the plate is subsequently al ulated dire tly from the Lippmann-S hwinger equation whi h is now an expli it equation. With the

hosen dis retization, the rst part of the al ulations is done using only 12 basis fun tions, whi h unders ores the utility of the integral type formulation when the s attering geometries are small. It is a striking feature of the solution in Fig. 3.2 that even with the simplest possible type of basis fun tion the solution was estimated to good a

ura y using only 12 basis fun tions. When ompared to the analyti al solution that was al ulated from a system of 4 equations this is quite impressive and unders ores the value of having to solve for the eld only within the s atterer. It is evident from Fig 3.2, however, that the dis retization is somewhat oarse and the ratio of 10 basis fun tions per wavelength is probably as low as one would typi ally go in these kinds of al ulations. In the present problem, the number of unit ells obviously s ales linearly with the number of s atterers, the size of the s atterers and the refra tive index. In higher dimensions however, the number of ells depend quadrati ally (in two dimensions) and ubi ally (in three dimensions) on the size of the s atterers, whi h may result in enormous memory requirements. The in rease in memory requirements in two and three dimensions, together with the stability problems for ve torial al ulations when expanding on pulse basis 41

Chapter 3. Multiple s attering al ulations using the Lippmann-S hwinger equation

fun tions, lead us to look for an alternative basis of fun tions. 3.1.2

A hybrid method

In this PhD proje t we have developed a multiple s attering solution method to the Lippmann-S hwinger equation. The solution method is inspired by the analysis in se tion 3.1.1 whi h shows that an integral formulation is desirable be ause it limits the al ulations to the extend of the s atterers. At the same time, the analyti al result suggests that the best expansion will be in terms of solutions to the wave equation in the dierent regions of the al ulation domain. In our approa h, the Lippmann-S hwinger equation is rst expanded on a basis of solutions to the homogeneous Helmholtz equation and solved within the s attering elements. Solutions at points outside the s attering elements are subsequently

al ulated dire tly using the Lippmann-S hwinger equation. Be ause of the integral formulation, the method may benet from known results for the Green's tensor in the ba kground material while the normal mode expansion redu es the number of basis fun tions needed, thus enabling al ulations on material systems of pra ti al relevan e. In view of the dis ussion in se tion 2.2.5 the method may be viewed as a hybrid between integral type method of moments al ulations and multiple s attering multipole methods. In parti ular, although we use an integral equation formulation, we make use of a number of theorems whi h are regularly employed in multipole methods in order to simplify the evaluation of the s attering matrix elements. Due to the hybrid formulation, a very low number of basis fun tions is needed. As an example; for a desired relative toleran e of 10−3 in PC al ulations in two dimensions, one an typi ally work with around 10 basis fun tions per s atterer as shown in se tion 3.6. In the CDA, FDTD, FEM or other brute for e numeri al te hnique, one would typi ally work with at least 10 sampling points per ee tive wavelength in both dire tions. In addition, for FDTD and FEM the full al ulation domain in luding the ba kground will need to be dis retized. For square s atterers with ea h side having the length L = 0.3λ0 , the demand for 10 sampling points per wavelength leads to approximately 100 points per s atterer for a refra tive index of n = 3. This number s ales quadrati ally (and ubi ally in three dimensions) with the refra tive index. 3.1.3

Overview of hapter 3

In se tion 3.2 we formulate the method in a general form whi h is suited for both one, two and three dimensional problems. In se tion 3.1.1 the basi ideas are illustrated by a one dimensional example before pro eeding to the two dimensional ase in 42

General formulation of the method se tion 3.4. We present details of the implementation in two dimensions and note that a similar approa h is possible for three dimensional problems as well. Se tion 3.5 presents example s attering al ulations in two dimensions and in se tion 3.6 these solutions serve as examples to dis uss a pra ti al method of evaluating the a

ura y of a given al ulation. Last, in se tion 3.7, the al ulation method is extended to inhomogeneous ba kgrounds. 3.2

General formulation of the method

We onsider s attering of mono hromati light, E(r, t) = E(r) exp(−iωt), in general systems onsisting of a nite number of pie ewise onstant diele tri and non-magneti perturbations to a pie ewise homogeneous ba kground, as illustrated in Fig. 3.3.

D1 kB

D3

D2 nA

nB

Figure 3.3: Sket h of general s attering geometry.

We will solve for the total eld, E(r), inside the s attering material only, as the solution everywhere else an be subsequently obtained by use of the LippmannS hwinger equation. The in oming eld, EB (r), is a solution to the wave equation with no s atterers and thus in general an be expanded on the solutions to the wave equation in the bulk ba kground material, φBn (r). In a similar way, the total eld inside a given s atterer may be expanded on the solutions in the bulk diele tri , φn (r). Based on these onsiderations we onstru t a basis onsisting of solutions to the s alar wave equation, ea h with support on only one of the s attering sites, ψn (r) = Kn φq (r) Sd (r)

ψ B (r) = K B φB (r) Sd (r), n

n

q

(3.2a) (3.2b)

where we have dened a ombined index n = (q, d, α), in whi h q is a parameter (or set of parameters) des ribing the solution and d denotes the parti ular subdomain 43

Chapter 3. Multiple s attering al ulations using the Lippmann-S hwinger equation

Dd of the s attering material we onsider, e.g. whi h ylinder. For onvenien e, the eld omponent α ∈ {x, y, z} is in luded in the index n as well. Kn and KnB are

normalization onstants and

  1, Sd =  0,

r ∈ Dd

otherwise

denes the support. The solutions ψn (r) have wave numbers orresponding to the index of the s attering material, kd = nd k0 , where nd denotes the refra tive index in the subdomain Dd , and ψ B (r) have wave numbers orresponding to the ba kground, kB = nB k0 . To ease the notation, we will write n in pla e of any of the indi es q, d, α as this will not lead to ambiguities. We dene an inner produ t as hψm |ψn i =

Z

∗ ψm (r) ψn (r) dr,

(3.3)

whi h is in general non-zero for m 6= n, and the basis fun tions are normalized so that hψn |ψn i = hψnB |ψnB i = 1. By expanding the ele tri elds as E(r) = EB (r) =

X n

X

en ψn (r)en

B eB n ψn (r)en ,

n

B em , where en is a unit ve tor, and proje ting onto the basis formed by ψm em and ψm the Lippmann-S hwinger equation may be rewritten in matrix form as 

1 + Lmn

in whi h

X X ∆ǫ  X 2 hψm |ψn ien = hψm |ψnB ieB Gαβ n + k0 ∆ε mn en , ǫB n n n

Gαβ mn

=

Z

V

∗ ψm (r)

lim

δV →0

Z

V −δV

′ ′ ′ GB αβ (r, r )ψn (r )dr dr,

(3.4)

(3.5)

with the dire tions α, β orresponding to the indi es m, n written expli itly for

larity. Eqs. (3.2) - (3.5) hold the omplete formulation of the method. The form of the entral matrix equation (3.4) is only slightly dierent from that of the CDA in that the left hand side is generally a matrix (although this matrix is very sparse, sin e basis fun tions belonging to dierent s attering domains are orthogonal by

onstru tion). The underlying strategy and the pra ti al implementation of the 44

One dimensional example: The example s attering problem revisited

two methods, however, are very dierent. In the CDA, the proje tion onto the basis fun tions is straight forward resulting typi ally in a very large matrix equation system that is subsequently solved by iterative methods. In the present method, it is the proje tions onto the basis fun tions that are potentially time onsuming but eventually leads to a relatively small system of equations.

3.2.1 Basis fun tions in dierent dimensions The basis fun tions are solutions to the s alar wave equation. In one dimension the basis fun tions are forward and ba kward traveling plane waves, ψn1D (x) = Kn exp(qkd x)Sd (x), (3.6a) B,1D B ψ (x) = K exp(qkB x)Sd (x), (3.6b) n

n

in whi h n = (q, d, α) with q ∈ {−1, 1}. In two dimensions we use the solutions to the wave equation in ylindri al oordinates (r,ϕ) dened with respe t to the lo al oordinate system in the subdomain Dd , ψn2D (r) = Kn Jq (kd r)eiqϕ Sd (r) (3.7a) B,2D B iqϕ ψ (3.7b) (r) = K Jq (kB r)e Sd (r), n

n

in whi h Jq with q ∈ Z denotes the Bessel fun tion of the rst kind of order q . In three dimensions we use the solutions to the wave equation in the lo al spheri al polar oordinates (r,θ,ϕ) dened in the subdomain Dd , ψn3D (r) = Kn jq (kd r)Ylq (θ, ϕ) Sd (r) (3.8a) B,3D B ψ (r) = K jq (kB r)Ylq (θ, ϕ) Sd (r), (3.8b) n

n

in whi h jq with q ∈ Z denotes the spheri al bessel fun tion of the rst kind of order q and Ylq are spheri al harmoni s [51℄. 3.3

One dimensional example: The example s attering problem revisited

In this se tion we apply the framework of se tion 3.2 to the example s attering problem in se tion 3.1.1. We solve for the ele tri eld inside the diele tri plate where the √ eld is expanded on only two plane waves of the form in Eq. (3.6a) with K = 1/ L and k1 = k2 = kR . Eq. (3.4) is a 2x2 matrix equation,  

1 ∆21

∆12 1

 

e1 e2





=

∆B 11

∆B 12

∆B

∆B

21

22

 

eB 1 eB 2





 + k02 ∆ǫ 

G11

G12

G21

G22

 

e1 e2



,

45

Chapter 3. Multiple s attering al ulations using the Lippmann-S hwinger equation

where the indi es 1 and 2 denote forward and ba kwards traveling waves, respe tively, and ∆12 =

∆∗21

1 = L

x0 +L

Z

e−2ikR x dx

x0

is non-zero for the values of kR and L in the example. Similarly, ∆Bmn is non-zero and we have for example 1 ∆B 12 =

L

x0 +L

Z

e−i(kR +kB )x dx.

x0

The proje tions GB12 of the Green's tensor onto the basis fun tion set is given in Eq. (3.5) and we get for example G12

1 = 2kB L

Z

x0 +L

x0

Z

x0 +L

e−ikR (x+y) eikB |x−y| dxdy.

x0

The integrals ∆12 , ∆Bmn and Gmn are su iently simple to allow for analyti al solutions. The in oming wave is a forward traveling wave and so we set eB1 = 1 and eB2 = 0 and solve the matrix equation√to obtain the two √ unknowns e1 and e2 . Comparing to Eq. (3.1) we must get e1 = L C and e2 = L D. Note that in one dimension the basis fun tion set ompletely spans the solution spa e, so there is no error due to trun ation. 3.4

Implementation in two dimensions

In two dimensions the light travels in the xy -plane, r = (x, y), and the ve tor equation de ouples into two independent equations for the Transverse Ele tri (TE) and the Transverse Magneti (TM) polarizations. In the ase of TE polarization, the ele tri eld is oriented in the xy -plane, whereas for TM polarization the ele tri eld is oriented along the z -axis and the s attering al ulation is essentially a s alar problem. The matrix elements need to be evaluated for basis fun tions within the same domain (self-terms) as well as dierent domains (s attering terms). The ase of a homogeneous ba kground is of spe ial interest as one will often be able to separate the ba kground Green's tensor into terms orresponding to a homogeneous ba kground and a number of additional s attering terms. Therefore, we fo us in this se tion on evaluation of the matrix elements for a homogeneous ba kground Green's tensor and return to the additional terms due to s attering in an inhomogeneous ba kground in se tion 3.7. Expli it expressions for the elements in the ba kground Green's tensor is given in Eqs. (A.3), 46

Implementation in two dimensions

A number of mathemati al results exist that an dramati ally speed up the

al ulations of the matrix elements and the implementation in general. This is dis ussed in se tions 3.4.1-3.4.4. 3.4.1

Self-terms

The evaluation of the self-terms is ompli ated by the divergen e in the Green's tensor for r′ = r and the fa t that the integrand ouples r′ and r, ee tively resulting in a four-dimensional integral. In the following, a method for the evaluation is des ribed in whi h the four-dimensional integral, Eq. (3.5), is rewritten in terms of a number of one and two dimensional integrals. This method is suitable for evaluation of general matrix elements. For spe ial s attering geometries, however, other methods may be more e ient, and we dis uss one su h ase by omparing to results from Mie s attering. r′ R y

r

x D P −D Figure 3.4:

CD

Sket h of the lo al oordinates used for al ulation of the self-term in

s attering domain

D.

Figure 3.4 shows a sket h of the lo al oordinates used for evaluation of the integral. In order to e iently treat the divergen e, for ea h r we rst integrate r′ over the entire plane P less the prin ipal volume entered on r. Subsequently, the integral for r′ ∈/ D is subtra ted. For this integral the limit δA → 0 is trivial sin e αβ αβ r′ 6= r. The matrix element is thus rewritten as Gαβ mn = Amn − Bmn , in whi h Aαβ mn =

Z

D

and αβ Bmn =

∗ ψm (r) lim

δA→0

Z

D

∗ ψm (r)

Z

Z

P −δA

P −D

′ ′ ′ GB αβ (r, r )ψn (r )dr dr

′ ′ ′ GB αβ (r, r )ψn (r )dr dr.

47

Chapter 3. Multiple s attering al ulations using the Lippmann-S hwinger equation

Using the Graf addition theorem ( f. appendix B.1), we an simplify the evaluation of Aαβ by expanding the fun tion ψn (r′ ) around the point r, so that Aαβ mn = Km Kn

XZ µ

D

Jm (kR r)Jn+µ (kR r)ei(n+µ−m)ϕ

µ −iµθR dRdr GB αβ (0, R)Jµ (kR R)(−1) e P −δA XZ = Km Kn Jm (kR r)Jn+µ (kR r)ei(n+µ−m)ϕ dr × Iµαβ ,

×

Z

µ

D

(3.9)

where kR = nd k0 and we have exploited the fa t that the integration over R = (R, θR ) is over the entire plane (less the prin ipal volume) and thus does not depend on r. The integral over R, denoted by Iµαβ above, is nonzero only for µ ∈ {−2, 0, 2} and the evaluation an be redu ed to a number of one dimensional integrals as shown in appendix C.1. In addition, we note that the simple angular dependen e of the integrand in many ases allows for a redu tion of the remaining integral over r to a sum of one dimensional integrals. In appendix C.2 we illustrate this for the

ase of regular polygons. αβ may also be substantially simplied using the Graf addition Evaluation of Bmn theorem to expand the Hankel fun tion in terms of Bessel and Hankel fun tions dened in the lo al oordinate system. The expansion diers depending on the sign of r − r′ . For r′ > r we write the integrand as αβ ∗ ′ ′ bmn (r, r′ ) = ψm (r)GB αβ (r, r )ψn (r )   X i Jµ (kB r)e−iµϕ = Km Kn Jm (kR r)e−imϕ Lαβ 4 µ ′

× Hµ (kB r′ )Jn (kR r′ )ei(n+µ)ϕ ,

(3.10)

whereas, for r′ < r we write αβ bmn (r, r′ )

= Km Kn

X

Jm (kR r)e

−imϕ

µ

αβ

L

′

× Jµ (kB r′ )Jn (kR r′ )ei(n−µ)ϕ ,

in whi h Lαβ = δα,β +

∂α ∂β 2 kB



i Hµ (kB r)eiµϕ 4



(3.11)

where δα,β is the Krone ker delta. For ir ular s atterers we always have r′ > r αβ and the expression for Bmn fa tors into a number of one dimensional integrals. αβ for non- ir ular s atterers may be onveniently Similarly, the evaluation of Bmn 48

Implementation in two dimensions

split depending on whether r′ is outside or inside the ir ums ribing ir le (denoted by CD in Fig. 3.4). In the former ase, the expression fa tors into separate integrals for r and r′ , whereas in the latter ase the two integrations are oupled. Again, the simple angular dependen e of the integrands in many ases allows for a redu tion of these integrals to a number of one and two-dimensional integrals, f. appendix C.2. Relation to Mie s attering

The s attering of an in oming plane wave o a single ir ular ylinder (or sphere) is referred to as Mie s attering. Analyti al solutions to su h problems an be obtained by expanding the elds inside and outside the ylinder on ylindri al wave fun tions and mat hing these in a

ordan e with the boundary onditions [107℄. Due to dieren es in the boundary onditions, the solutions are usually al ulated for the magneti eld in the ase of TE polarization and for the ele tri eld in the

ase of TM polarization. It is rewarding to ompare the latter with the solutions to Eq. (3.4). In the ase of a single ir ular s atterer and TM polarization, all basis fun tions with m 6= n are orthogonal under the inner produ t in Eq. (3.3). Therefore, Eq. (3.4) is diagonal and may be solved dire tly for the oe ients en : en =

hψn |ψnB ieB n . 1 − k02 ∆ǫGzz nn

Given that we already know the solution en we may solve for the matrix elements instead: en − hψn |ψnB ieB n , Gzz (3.12) nn = 2 k0 ∆ǫen

whi h provides an analyti al expression for the matrix elements, thus avoiding any numeri al quadrature. In identally, Eqs. (3.9), (3.10) and (3.12) may be used along with the Mie s attering solution to show the identity: Z

∞ 0

H0 (kB r)J0 (kR r)rdr =

−2i 2 − k2 ) , π(kR B

for kR 6= kB . 3.4.2

S attering terms

For the al ulation of s attering terms, the integration domains for r and r′ are

ompletely separated in spa e, and so the Green's tensor is well behaved at all points of interest. In this ase we employ the Graf addition theorem twi e to 49

Chapter 3. Multiple s attering al ulations using the Lippmann-S hwinger equation

r′

y′

R x′ r

L y

Dn

θ x Dm

Figure 3.5: Sket h of lo al oordinates for

r

and

r′

in two independent s atterers.

express the Hankel fun tion in terms of the distan e between the enters of the two lo al oordinate systems, as illustrated in Fig. 3.5, Gαβ mn =

iX Hµ+λ (kB L)(−1)µ ei(µ+λ)θ 4 µ,λ Z  Km Jm (km r)e−i m ϕ Lαβ Jλ (kB r)e−iλϕ dr × D Z m ′ Kn Jµ (kB r′ )Jn (kn r′ )ei (n−µ) ϕ dr′ , × Dn

(3.13)

where (L, θ) are ylindri al oordinates of O′ with respe t to O, f. appendix B.1. Eq. (3.13) shows that the s attering matrix al ulation fa tors into terms that depend only on the geometries of the individual s atterers and the distan e between them. Sin e the Hankel fun tions as well as the Bessel fun tions are well behaved at all points of interest, the integrals may be dire tly evaluated. Note that the pro edure outlined above is ompromised when L < Rm + Rn , where Rm and Rn are the radii of the ir ums ribing ir les of domains Dm and Dn , respe tively. This ould happen in the ase of lose non- ir ular s atterers. In this ase the Graf addition theorem is not valid and one an employ a strategy based on Eqs. (3.10) and (3.11) instead.

3.4.3 Ba kground ele tri eld The in ident ba kground ele tri eld, EB (r), is a solution to the wave equation without the s atterers. In the ase of a bulk ba kground, the solutions are plane waves, and the expansion in terms of ylindri al wave fun tions is readily obtained using the Ja obi-Anger identity, as dis ussed in appendix B.1. In the ase of a plane wave, A exp(i k · r), traveling at an angle θ with respe t to the x axis we 50

Implementation in two dimensions

immediately get

eB n =A

in −i n θ ikB · Rn eθ · en , e e KnB

in whi h Rn denotes the position of the s attering domain Dn and eθ is a unit ve tor in the dire tion of the ele tri eld. Instead of using plane waves as the ba kground ele tri eld we may use the

olumns of the 2D Green's tensor. These are related to the ele tri eld at r due to a line sour e at r′ [55℄. Comparing to Eq. (2.17) it is evident that the solution to the Lippmann-S hwinger equation in this ase exa tly produ es the orresponding

olumns of the full 2D Green's tensor for the s attering problem. By expanding the Green's tensor in terms of ylindri al wave fun tions, G(r, r′ ) =

X n

 ′ L Hn (kB L)e−inθ (−1)n Jn (kB r′ )einϕ ,

we may identify the expansion oe ients orresponding to the β 'th olumn as eB n =

(−1)n nβ  L Hn (kB L)e−inθ , B Kn

(3.14)

whi h is valid for L > Rn , where Rn is the radius of the ir ums ribing ir le of domain Dn , f. Fig. 3.4. The (bounded) basis fun tion set, Eqs. (3.7), is not suited to sample the Green's tensor with r inside any of the s atterers. In this ase one may possibly extend the basis fun tion set with the in lusion of fun tions of the form ψ˜n = Kn Hn (kR r) exp(inϕ). Due to the trun ation of the basis set, the divergen e in the Hankel fun tions may result in poor sampling of the Green's tensor even for r outside but lose to one of the s atterers. To get an estimate for the distan es at whi h the sampling be omes a problem we may look at the series expansions for the radial parts of the terms in the Green's tensor, f. Eqs. (A.3): i H0 (x) = 4 i H1 (x) = 4 i H2 (x) = 8

i 1 − {ln(x/2) + γ} + O(x2 ) 4 2π 1 1 1 + iπ − {ln(x/2) + γ} + + O(x2 ) 2 2πx 4π 8π 1 1 + + O(x2 ). 2 2πx 8π

(3.15)

For the TM polarization, p the absolute value of the Hankel fun tion has dropped to 1 at kB R ≈ 2 exp(−2π 1 − 1/16p − γ) ≈ 0.0031. For the TE polarization, the xy term has dropped to 1 at kB R ≈ 1/(2π − 1/4) ≈ 0.41. The same value holds for the xx and yy terms. 51

Chapter 3. Multiple s attering al ulations using the Lippmann-S hwinger equation

3.4.4

Exterior solution

The matrix equation (3.4) is solved using standard linear algebra routines to yield the solution at any point inside the s attering domains. The solution at points outside the s atterers an be subsequently obtained dire tly from the LippmannS hwinger equation whi h is now an expli it equation: E(r) = EB (r) +

Z

D

GB (r, r′ ) k02 ∆ε(r′ )

X

en ψn (r′ )en dr′ .

(3.16)

n

The sum in Eq. (3.16) runs over all basis fun tions in all s attering domains. Again, the al ulation may be simplied onsiderably by use of the Graf addition theorem to rewrite the equation in terms of the distan es from the enters of the lo al

oordinate systems. Considering for simpli ity the ase of just a single s attering domain D, the equation is rewritten as X  i L Hµ (kB L)eiµθ (−1)µ en E(r) =EB (r) + k02 ∆ε 4 µ,n Z ′ × en Kn Jn (kR r′ )Jµ (kB r′ )ei (n−µ) ϕ dr′ , D

(3.17)

where (L, θ) are now ylindri al oordinates of O′ with respe t to r. 3.4.5

Solution of the linear equation system

In the ase of pra ti ally relevant stru tures, Eq. (3.4) may onsist of several thousand oupled linear equations. In this ase a simple solution based on e.g. Gaussian elimination is impossible due to a

umulation of error, and iterative methods su h as the onjugate gradients method are usually employed. Iterative methods typi ally rely on solving the equation system Ae = eB by varying the ve tor e in order to minimize the residual ξ = |Ae − eB |.

For solution of the system of equations we have used a bi onjugate gradients stabilized method, whi h seems to be able to onverge to the desired a

ura y in most ases. For relatively small systems, however, we typi ally solve the equation system using a Moore-Penrose pseudoinverse based on singular value de omposition [108℄. After solving the system with the pseudoinverse, we al ulate the residual and nd that the pseudoinverse returns solutions with very low relative residuals, typi ally ξ ≈ 10−10 . For large systems, the solution step may be the far most expensive part of the al ulation. 52

Two dimensional examples 3.5

Two dimensional examples

To illustrate the method, we present in this se tion two example s attering problems; TE plane wave s attering from ir ular ylinders and al ulation of the TM Green's tensor for a olle tion of square ylinders. 3.5.1

Plane wave s attering from ir ular ylinders

We onsider a TE plane wave in ident on a small rystallite of air ylinders in a high-index diele tri . Be ause of the ylindri al symmetry, the self-term matrix elements are non-zero only for m − n ∈ {−2, 0, 2}. Cal ulation of the s attering matrix elements, Eq. (3.13), is also substantially simplied be ause the angular integration leads to non-zero values only for terms with µ = n and λ+m ∈ {−2, 0, 2}. Fig. 3.6 shows the absolute square of the total eld as a fun tion of position in the xy -plane. Also we show the magnitude of the Ex and the Ey omponents of the eld along the line y = 0 through the enters of three of the ylinders. Clearly, the x omponent shows a number of dis ontinuous jumps, whereas the y omponent is

ontinuous in a

ordan e with the boundary onditions. We note that the air ylinders a t to partly blo k the light, resulting in the formation of a standing wave in the upper left part of the gure. Although an innite triangular latti e of air holes will exhibit a photoni band gap at ertain frequen ies, this band gap is lo ated at higher frequen ies for the material parameters in the example [64℄. Instead, we as ribe the blo king of the light as arising from the lower average refra tive index of the rystallite relative to the ba kground. Typi ally we use the same number of basis fun tions in ea h s attering domain and for ea h polarization, so that |q| ≤ Qmax . This al ulation was performed using Qmax = 10, resulting in a matrix equation system of 294 unknowns. Using the method outlined in se tions 3.4.1 and 3.4.2 and using an absolute toleran e on the numeri al integrals of 10−6 , the average al ulation time per s attering matrix element was less than 0.1 s for the self-terms and less than 0.01 s for the s attering terms on a 2.4 GHz pro essor. Making use of the symmetry of the rystallite we redu ed the problem to the al ulation of matrix elements for s attering between 19 dierent pairs of s atterers only. In addition, the form of Eq. (3.13) suggests that for identi al s atterers the integrals a ross the domains Dm and Dn an be handled on e only and stored for use in subsequent al ulations of matrix elements for s attering between other pairs of s atterers. Using this approa h, the total time for the al ulation of all matrix elements was approximately 13 s. Due to the small size of the s attering problem, the solution of the linear equation system was handled in approximately a se ond. 53

Chapter 3. Multiple s attering al ulations using the Lippmann-S hwinger equation

3 8 2 7 6

k0 y

1

5 0 4

−1

3 2

−2 1 −3 −4

−2

0

2

4

0

2

4

|Eα |

2

1

0 −4

−2

k0 x

Figure 3.6: Example al ulation. A TE plane wave of unit amplitude, EB (r) = e exp(i nB k0 · r), is in ident from the top left on a rystallite onsisting of seven air holes (√ nd = 1) in a high-index diele tri ba kground (nB = 3.5). Parameters are k0 = ( 3/2, −1/2) and RPC = 0.3a where RPC is the radius of the ylindri al holes and a = 0.3λ0 is the distan e in between. Top: Absolute square, |E(r)|2 , of the resulting eld as a fun tion of position in the xy -plane. Bottom: Absolute value of the omponents Ex (x) (red solid line) and Ey (x) (blue dashed line) along the line y = 0. 3.5.2

Green's tensor for a olle tion of square ylinders

In this se tion we apply the framework of se tion 3.4 to non- ir ular s atterers. In addition, we use the Green's tensor for the homogeneous material as the ba kground eld, so that we may interpret the solution as the Green's tensor for the entire stru ture in luding the s atterers, as dis ussed in se tion 3.4.3. We onsider a geometry onsisting of four square diele tri rods in air. The al ulation of the matrix elements for the square s atterers is arried out in appendix C.2. Due to 54

Two dimensional examples the four-fold rotational symmetry, the self-term matrix elements are non-zero only for m − n = 4p with p ∈ Z. In Fig. 3.7 we show the real and imaginary parts of the TM Green's tensor Gzz (r, r′ ) as a fun tion of r for onstant r′ indi ated in the gure. The al ulation was performed using Qmax = 10, resulting in only 84 unknowns. We used an absolute toleran e on the numeri al integrals of 10−6 , and the average time per s attering matrix element was 0.7 s for the self-terms and 0.04 s for the s attering terms. Based on symmetry, the problem was redu ed to the al ulation of s attering matrix elements between 9 pairs of s atterers resulting in a total time for the matrix element al ulations of approximately 200 s and a solution time of less than a se ond.

0.3

1

k0 y

0.2 0.1

0

0 −0.1

−1 −3

−0.2 −2

−1

0

1

2

3

0.04 1

k0 y

0.02 0

0

−0.02 −1 −0.04 −3

−2

−1

0

1

2

3

k0 x Figure 3.7: Real (top) and imaginary (bottom) part of the total TM Green's tensor

Gzz (r, r′ ) as a

r with k0 r′ = (−1, 1/4) (as indi ated by the red dot) in a stru ture onsisting of four diele tri rods (nd = 3.5) of square ross se tion in air. Parameters are a = 2L where a is the distan e between the rods and L = λ0 /4 is fun tion of

the side length.

The real part of the Green's tensor diverges in the limit r → r′ whi h is the ase also for the results in Fig. 3.7. Indeed it must be the ase, sin e the results are obtained as the sum of the ba kground eld and the s attered eld, as dis ussed in se tion 3.4.4. The s attered eld is well behaved at all points, however, so 55

Chapter 3. Multiple s attering al ulations using the Lippmann-S hwinger equation

the divergen e in the Green's tensor stems from the divergen e in the ba kground Green's tensor only. This divergen e is logarithmi in the ase of TM polarization as dis ussed in se tion 3.4.3 whi h is the reason why it does not show up at the

hosen dis retization. The imaginary part of the Green's tensor is ontinuous at all points. In the limit r = r′ it is proportional to the LDOS, f. Eq. (2.14). 3.6

Error analysis

The numeri al error in the al ulations stems primarily from evaluation of the matrix elements and the trun ation of the basis set. After solving the linear equation system, Eq. (3.4), the a

ura y of a given solution may be estimated by substitution ba k into the Lippmann-S hwinger equation. To this end we dene the lo al error as EL (r) = |EB (r) − Enum (r) +

Z

GB (r, r′ ) k02 ∆ε(r′ ) Enum (r′ )dr′ |,

(3.18)

and we note that sin e EB (r) and GB (r, r′ ) are known analyti ally Eq. (3.18) an be used as a measure for the a

ura y of a given al ulation even if the analyti al solution is not known. Based on the lo al error, we dene the global relative error as R EL (r)dr R , EG = |EB (r)|dr

where the integrals are taken over the area of the s attering sites only. Fig. 3.8 shows the global error of the solutions to the problems in Figs. 3.6 and 3.7 as a fun tion of the number of basis fun tions used in the expansions and dependent on the error in the matrix elements. The error analysis was performed by rst al ulating the matrix elements to high pre ision using an absolute error toleran e on the numeri al integrals of 10−6 . Subsequently, for ea h value of Qmax the orresponding linear equation system, Eq. (3.4), was onstru ted, and a random omplex number of xed modulus, δGmn , was added to ea h element in the matrix of modulus larger than δGmn before solving the equation system. The analysis shows an exponential like de rease in the global error as a fun tion of the number of basis fun tions, unders oring the massive redu tion in basis fun tions due to the expansion in normal modes when ompared to onventional dis retization methods. This is the ase for the ylindri al holes in Fig. 3.6 as well as for the square rods in Fig. 3.7. The onvergen e is faster in the ase of the ylindri al holes whi h is partly be ause the basis fun tions have the same symmetry as the s atterers and partly be ause the plane wave eld is easier to approximate than the (divergent) Hankel fun tion, as noted in se tion 3.4.3. From Fig. 3.8 we argue 56

Inhomogeneous ba kgrounds

EG

10−3 10−4 10−5 10−6 10−7 Gzz PW Qmax

Figure 3.8: Global error as a fun tion of the number of basis fun tions used in the expansion of the ele tri elds ( ontrolled by Qmax ). Cir ular markers orrespond to the problem in Fig. 3.6 with dierent urves orresponding to dierent xed errors on the relevant matrix elements as indi ated. Square markers orrespond to the problem in Fig. 3.7 al ulated for the Green's tensor (Gzz ) and plane waves (P W ) as the ba kground eld. that the arti ial error on the matrix elements a ts to limit the minimum a hievable global error, and the analysis thus onrms that the global error is ontrolled by the number of basis fun tions as well as the a

ura y of the numeri al quadrature. We note that the measure in Eq. (3.18) may be viewed as a test of self- onsisten y of the method whi h is of prin ipal importan e for any solution to Eq. (2.16). From Fig. 3.8 it is evident that the measure is also of pra ti al importan e sin e, for a given toleran e on the numeri al integrals, it an be used to estimate the number of basis fun tions needed to rea h the minimum global error.

3.7

Inhomogeneous ba kgrounds

As an example of the utility of the method, we present in this se tion results for the investigation of light propagation near the edge of a nite sized two dimensional PC. We onsider a PC made from 80 ir ular rods of refra tive index nd = 3.4 in a lower-index ba kground (nB = 1.5). The ylinders are pla ed in a square latti e, and a short waveguide is reated by the omission of 4 rods along the (11) dire tion of the rystal. The waveguide along with the rystal is terminated by an interfa e to air. We fo us on TM polarized light and al ulate the Green's tensor of the system Gzz (r, r′ ). Although the integral expressions be ome larger, a similar pro edure as 57

Chapter 3. Multiple s attering al ulations using the Lippmann-S hwinger equation

the one outlined below may be used for the al ulation of TE polarized light as well as for multiple interfa es. We start by extending the formalism of the previous se tions to the ase of a non-homogeneous ba kground Green's tensor in se tion 3.7.1 and go on to show example al ulations of light emission from a nite sized PC in se tion 3.7.2. 3.7.1

Additional s attering near interfa e

For the s attering al ulations near a diele tri interfa e we use the Green's tensor for the diele tri half-spa e as the ba kground Green's tensor in Eq. (2.16). The 2D Green's tensor for general stratied media is given in Ref. [54℄ in terms of an integral in k-spa e. Below, we dis uss the al ulation of the elements Gzz mn in the spe ial ase of a single diele tri interfa e. We onsider TM polarized light in ident on an interfa e at y = 0 between two media with refra tive indi es nA and nB , respe tively. We will deal only with s atterers in the lower layer (layer B) and so, following Ref. [54℄, the 2D Green's tensor is given as ′ GB zz (r, r ) = −

y ˆy ˆ δ(R) k22 Z ∞ i 1 i kx (x−x′ ) i kB,y |y−y′ | + e e dkx 4π −∞ kB,y Z ∞ S i FBA i kx (x−x′ ) −i kB,y (y+y′ ) dkx , + e e 4π −∞ kB,y

where kl,y =

p kl2 − kx2 with kl = nl k0 , l ∈ {A, B} and

(3.19)

2 2 S = kB,y − kA,y = pkB − kx2 − pkA − kx2 FBA 2 − k2 + 2 − k2 kB,y + kA,y kB kA x x

p

p

is the Fresnel ree tion oe ient. In Eq. (3.19), the rst two terms orrespond to the Green's tensor of the homogeneous material, whereas the last term gives the ree tion o the interfa e. This means that the evaluation of the matrix element Gzz mn naturally splits into a dire t homogeneous material part and an indire t interfa e s attering part. The former is exa tly what was handled in se tions 3.4.1 and 3.4.2, so we on entrate in this se tion only on the s attering ontribution GSmn : GSmn =

i 4π

Z

∞

−∞

S (kx ) FBA kB,y (kx )

Z

Dm

Z

′

Dn

∗ ψm (r)ei kx (x−x )

× e−i kB,y (kx )(y+y ) ψn (r′ )dr′ drdkx .

58

′

(3.20)

Inhomogeneous ba kgrounds

In order to arry out the integration, we rst write (x, y) = (X, Y )+(r cos ϕ, r sin ϕ) and (x′ , y ′ ) = (X ′ , Y ′ ) + (r′ cos ϕ′ , r′ sin ϕ′ ), where (X, Y ) and (X ′ , Y ′ ) denote the absolute oordinates of the enters of the lo al oordinate systems. We then re ast the expression in terms of lo al oordinates as i GSmn = × ×

4π Z

Z

ZDm

∞

−∞

S (kx ) ′ ′ FBA ei(kx (X−X )−kB,y (kx )(Y +Y )) kB,y (kx )

Km Jm (km r)e−i m ϕ ei kB r cos(ϕ−θ(kx )) dr

Kn Jn (kn r′ )ei n ϕ ei kB r ′

′

cos(ϕ′ −θ ′ (kx ))

dr′ dkx ,

(3.21)

Dn

where we have rewritten the inner produ ts of the wave ve tors and the position ve tors in the two domains in terms of the angles between them. This angle be omes imaginary whenever kx2 > kB2 . As in the ase of the homogeneous ba kground, we are able to simplify the expression further by fa toring out the integrals over the domains Dm and Dn . To this end we use the Ja obi Anger identity, f. appendix B.1, to rewrite the matrix elements as GSmn =

i X λ+γ i 4π λ,γ Z ∞ S FBA (kx ) i(kx (X−X ′ )−kB,y (kx )(Y +Y ′ )) −i(λθ(kx )+γθ′ (kx )) × e e dkx kB,y (kx ) Z−∞ Km Jm (km r)Jλ (kB r)ei (λ−m) ϕ dr × Dm Z ′ Kn Jn (kn r′ )Jλ (kB r′ )ei (γ+n) ϕ dr′ . × (3.22) Dn

Due to the ir ular symmetry, the angular integrations over the domains Dm and Dn lead to non-zero values only for λ = m and γ = n. In these ases the radial integrals have well known analyti al values (see appendix B.3), leaving only a nal integration over kx .

3.7.2 Light emission in nite sized photoni rystal waveguide We may now ombine the s attering al ulations for a homogeneous ba kground with the pro edure in se tion 3.7.1 in order to in lude the additional s attering from an interfa e. In Fig. 3.9 we show a ontour plot of the absolute value of the Green's tensor |Gzz (r, r′ )| along with real and imaginary parts at positions along the x-axis. Results are shown for k0 r′ = (0, −7.58) in the enter of the waveguide at the lo ation of one of the missing rods. In an innite waveguide this would be the lo ation of the eld antinode of the waveguide mode. 59

Chapter 3. Multiple s attering al ulations using the Lippmann-S hwinger equation

10 −5

k0 x

8

6

0

4

5

2

−15

−10

−5

0

5

−15

−10

−5

0

5

10

Gzz

5

0

−5

k0 y

Figure 3.9: Top: Absolute value, |Gzz (r, r′ )|, of the TM Green's tensor for a nite sized PC waveguide onsisting of 80 rods of refra tive index nd = 3.4 in a ba kground with an interfa e between a low-index diele tri (nB = 1.5) and air (nA = 1). The results are al ulated as a fun tion of r with k0 r′ = (0, −7.58) (indi ated by the red dot and verti al dashed line). Bottom: Real (red) and imaginary (blue) parts of Gzz (y, r′ ) along the line x = 0. Parameters are RPC = 0.25a where RPC is the radius of the ylindri al holes and a = 0.28λ0 is the distan e in between.

The periodi Blo h-mode hara ter of the waveguide mode is evident also in the

ase of this nite waveguide, and the stru ture a ts as a resonator greatly in reasing the absolute value of the Green's tensor for positions r inside the waveguide as

ompared to the bulk medium. For r → r′ the real part of Gzz (r, r′ ) diverges. This is the ase also in Fig. 3.9, but the divergen e is too weak to show up at the

hosen dis retization. Although the nite waveguide a ts as a resonator, light an propagate out of the end fa et. Fig. 3.10 shows |Gzz (r, r′ )| at positions outside the stru ture. As noted in se tion 3.4.3, the Green's tensor is related to the ele tri 60

Con lusion eld at point r due to a line sour e at point r′ . Therefore, we may interpret the gure as the emission pattern from the sour e inside the waveguide. Due to the resonator ee t of the waveguide stru ture, the emission pattern does not show up on the s ale of the ontour plot in Fig. 3.9. 2

k0 x

−5

1.5

0

1 0.5

5

PSfrag repla ements

−15

−10

−5

0

5

10

15

20

25

k0 y Figure 3.10: Contour plot of emission pattern, but for positions outside the PC.

3.8

|Gzz (r, r′ )|,

of the system in Fig. 3.9

Con lusion

We have des ribed a pro edure for solving the Lippmann-S hwinger equation for ele tromagneti s attering in whi h the eld along with the ele tri eld Green's tensor is expanded inside ea h s atterer on a basis of solutions to the s alar Helmholtz equation. We have presented the method in a general form that is suited for both one, two and three dimensional problems, and we have shown a simple example of implementation in one dimension and provided a thorough dis ussion of the implementation in two dimensions. In this ase, the proje tion of the ele tri eld and the Green's tensor onto the basis set is fa ilitated by the use of a number of addition theorems to simplify the integral expressions. The hosen basis set ensures that all basis fun tions have the orre t wave number. This, ombined with the need for solving the system inside the s attering elements only, results in a relatively small linear equation system as ompared with other methods. Consequently, the method is fast and apable of handling large material systems su h as PCs. In two dimensions, the use of a lo al ylindri al wavefun tion basis avoids the introdu tion of  titious harges whi h may lead to instabilities for large refra tive index ontrasts in the ase of TE polarization [84℄, and the integration s heme is free of stair asing errors along the boundaries. Due to the formulation in terms of the Green's tensor for the ba kground medium there is no need for a al ulation domain, and the radiation ondition is automati ally satised as are the boundary onditions (limited only by the numeri al pre ision 61

Chapter 3. Multiple s attering al ulations using the Lippmann-S hwinger equation

hosen). The a

ura y of the method is thus limited only by the number of basis fun tions and the toleran es on the numeri al integrals employed for the evaluation of the s attering matrix elements. We have introdu ed a measure of a

ura y based on self- onsisten y that is of prin ipal as well as pra ti al importan e. On e the matrix equation has been set up, it holds all information ne essary to arry out s attering al ulations on the geometry at the hosen frequen y. It an thus be stored and used for dierent hoi es of in oming elds as well as for the al ulation of the Green's tensor between dierent points r and r′ . We have illustrated the method by two example problems, and we have shown an appli ation of the method where we have al ulated the Gzz omponent of the Green's tensor for a nite sized PC waveguide. Similar al ulations will nd appli ation in the development of nanophotoni devi es su h as in design of jun tions or avities in PCs or investigation of emission patterns from single photon sour es. Using a similar pro edure, the method may be extended to three dimensional s attering geometries, and although we have fo used on appli ations in mi ro- and nanophotoni stru tures we believe the method may be of use in other areas of ele tromagneti s attering al ulations as well.

62

Chapter 4

Fra tional de ay of quantum dots in photoni rystals 4.1

Introdu tion

Se tion 1.2.4 des ribed how variations in the LDOS may lead to qualitatively different de ay dynami s. Espe ially how a heavily modied LDOS, su h as observed in some PCs, may result in the marvelous quantum opti al phenomenon that we

alled fra tional de ay. Fra tional de ay has been investigated theoreti ally by a number of authors [19, 20, 109℄ and ontinues to be of interest [110℄. In this hapter we onsider the phenomenon in the ontext of quantum dots in photoni rystals. By a pure fra tional de ay we understand a de ay in whi h the probability amplitude for the ex ited state tends to a nite non-zero value at long times, |cx (t)|2 → k, (0 < k < 1), for t → ∞. For these solutions it is natural to dene a degree of fra tional de ay as df = 1 − k. If we in lude the ee ts of absorptive losses the de ay

urves inevitably tend to zero at long times, but nevertheless we may still meaningfully assign a degree of fra tional de ay to these solutions also. For the al ulations we employ the general approa h of Vats et. al [62℄ and use the minimal oupling Hamiltonian in the dipole approximation, as dis ussed in se tion 2.4.3. Contrary to the general treatment in Ref. [62℄, we fo us on the possible realization of fra tional de ay using spe i and realisti stru tures. In parti ular, the investigations are based on the a tual LDOS of a three dimensional PC obtained from plane wave al ulations. Due to the resonant nature of the lightmatter oupling, the qualitative nature of the dynami s are governed by the LDOS in a narrow frequen y interval around ω = ωx . At rst sight this is en ouraging for the theoreti al analysis sin e we an fo us on a narrow frequen y range of the 63

Chapter 4. Fra tional de ay of quantum dots in photoni rystals

LDOS. However, the relevant frequen y range turns out to be several orders of magnitude smaller than typi al resolutions in LDOS al ulations. To over ome this problem we use plane wave al ulations to set up an analyti al approximation to the band edge LDOS. The LDOS model is extended to in lude ee ts of absorptive losses. Absorption is shown to be a limiting fa tor and we present quantitative results showing the degree of fra tional de ay a hievable for available QDs and pra ti ally relevant material loss. 4.1.1

Overview of hapter 4

Se tion 4.2 starts out with a general dis ussion of the method and reviews some of the results from se tion 1.2.4 as illustrative examples. Motivated by the examples, it is dis ussed how a measure for the degree of fra tional de ay may be obtained from analysis of the de ay dynami s in the frequen y domain. In se tion 4.3 we employ the measure for analysis of fra tional de ay in the so- alled anisotropi band gap model. In se tion 4.4 we arry out detailed investigations of the LDOS lose to the band edge of a Si inverse opal based on an analyti al approximation to numeri al plane wave al ulations. In this way we are able to assess the validity of the anisotropi band gap model. Using perturbation theory we extend the model to in lude ee ts of absorptive loss in the LDOS. Finally, in se tion 4.5 we investigate the degree of fra tional de ay that an be obtained in Si inverse opals with absorptive losses. 4.2

Cal ulation of de ay urves

Following Ref. [62℄ the equations of motion may be solved in the frequen y domain using the initial ondition cx (t = 0) = 1. In this ase the analysis of the de ay is based on the spe trum for the expansion oe ient cx (ω), cx (ω) =

1 , α G(ω) − i (ω − ωx )

(4.1)

where α is a measure of the light-matter oupling strength, f. se tion 2.4.3, and ωx is the ex iton frequen y. The fun tion G(ω) is given for frequen ies above the integration path in the omplex plane as G(ω) = i ω

Z

0

ωC

ρx (z) dz, z 2 (ω − z)

(4.2)

where ρx (ω) denotes the LDOS, and we have introdu ed an upper uto to make the integral nite. Eq. (4.2) may be used along with the spe trum, Eq. (4.1), 64

Cal ulation of de ay urves to investigate the degree of fra tional de ay. To motivate the analysis we onsider rst the ase of an integration path along the real axis in whi h ase we obtain the important limiting form G(ω) = GR (ω) + iGI (ω),

(4.3)

where the two fun tions GR (ω) and GI (ω) are dened as GR (ω) = π

ρ(ω) ωZ

GI (ω) = ω P

(4.4)

ωC

0

ρx (z) dz, z 2 (ω − z)

(4.5)

with P denoting the Cau hy prin ipal value of the integral. 4.2.1

Illustrative examples

With ρx (ω) = 0 we have G(ω) = 0 and the spe trum has a simple pole at ω0 = ωx whi h, upon transforming ba k to the time domain, results in the onstant fun tion |cx (t)|2 = 1. This has the dire t physi al interpretation that if the only de ay me hanism is through spontaneous emission, and if no opti al modes are available, then no de ay an o

ur. Exponential de ay

For ρ(ω) > 0 the ee t of G(ω) is to shift the pole away from the real axis and into the fourth quadrant of the omplex ω plane. For weak light-matter intera tion, the spe trum will have a single pole, ω0 , leading to an exponential de ay. If the LDOS varies su iently slowly we an estimate the position of the pole from Eq. (4.3) by evaluating G(ω) at ω = ωx . This is the pole approximation, and in this ase we nd ω0 ≈ ωx + αGI (ωx ) − iαGR (ωx ), (4.6) so we may write the spe trum as

cx (ω) =

i . ω − ω0

The time-domain dynami s may now be readily found through the inverse Fourier transform, Z ∞ i 1 e−iωt dω = e−iω0 t , t > 0, cx (t) = (4.7) 2π

−∞

ω − ω0

where we have losed the integration path in the lower half of the omplex plane. The physi ally observable de ay urves are given as |cx (t)|2 denoting the probability 65

Chapter 4. Fra tional de ay of quantum dots in photoni rystals

that an ex itation exists in the QD at time t. From the above analysis we nd that for a slowly varying LDOS the de ay rate is given as Γ = 2πα

ρx (ωx ) . ωx

Pure fra tional de ay For the probability amplitude to go to a nite onstant value there will need to be at least one pole on the real axis. From Eq. (4.7) we realize that any pole with a non-zero imaginary part will lead to a de aying exponential fun tion and so will not meet the riterion we have set up for a pure fra tional de ay. From Eq. (4.4) it is evident that if the pole is real then ρ(ω0 ) = 0. This means, that a ne essary

ondition for pure fra tional de ay to o

ur is for the LDOS to be zero at least for one frequen y. The real part of the pole is determined by ωx and the fun tion GR (ω), whi h will move the pole in the dire tion of the negative real axis (the Lamb shift). In the general ase, and in parti ular for fra tional de ay al ulations, the pole approximation is not valid, so Eq. (4.6) annot be used. From the spe trum, Eq. (4.1), we see that ρx (ω0 ) = 0 at whatever frequen y solves the equation ω0 − ωx − αGR (ω0 ) = 0. 4.2.2

A measure for the degree of fra tional de ay

The analysis in se tion 4.2.1 illustrates how a pole in the spe trum results in a de reasing exponential term. The de ay rate is proportional to the imaginary part of the pole, and in the limiting ase of pure fra tional de ay this may be zero. We write a general pole term as a−1 , ω − ω0

and we note that the absolute square of the residue, |a−1 |2 , represents the relative magnitude of the pole term and is equal to the value of the exponential part at t = 0. In the ase of slow variations in the LDOS we have |a−1 |2 = 1 and the spe trum onsists only of a single pole term. This is hara teristi of the weak

oupling Pur ell regime. For stru tures in whi h the LDOS show large variations around the emitter frequen y the single pole approximation is no longer valid. In this ase we an usually still nd a pole, but there may also be other ontributions to the spe trum. Therefore, in general the spe trum may be written as a pole term and a rest term as cx (ω) =

66

a−1 + c′ (ω). ω − ω0

Cal ulation of de ay urves If |a−1 |2 < 1 the rest term must be non-zero in order to meet the initial ondition |cx (t = 0)|2 = 1. Consequently, this signies a deviation from the Pur ell regime and we dene df = 1 − |a−1 |2

as a measure of the degree of fra tional de ay. This measure is reasonable when des ribing fra tional de ay urves sin e it measures the deviation from the exponential de ay of the Pur ell regime. For the pure fra tional de ay |a−1 |2 gives exa tly the limiting value of the de ay urves for long times. The measure is onvenient from a mathemati al point of view also, sin e it an be al ulated simply from the derivative of the denominator in Eq. (4.1) at the lo ation of the pole. The denominator, Φ(ω), is expanded in a Taylor series around the pole as
Φ(ω) = αG(ω) − i(ω − ωx ) = Φ′ (ω0 )(ω − ω0 ) + O (ω − ω0 )2 ,

where we have used the notation

f ′ (ω0 ) =

d f (ω) . dω ω=ω0

From the Taylor series it follows that we an rewrite the spe trum in the form of a pole term and a non-zero rest term as 1 1 1/Φ′ (ω0 ) + ′ − ′ 2 ω − ω0 Φ (ω0 )(ω − ω0 ) + O ((ω − ω0 ) ) Φ (ω0 )(ω − ω0 )
′ O (ω − ω0 )2 1/Φ (ω0 ) = . + 2 ω − ω0 (Φ′ (ω0 )(ω − ω0 )) + O ((ω − ω0 )3 )

cx (ω) =

In pra ti e, we rst al ulate the pole term and then simply subtra t this from the spe trum to get the rest term. 4.2.3

Estimates of the residue

In the Pur ell regime, the term αG(ω) is small ompared to the ex iton frequen y ωx . Indeed, from the analysis of exponential de ay in se tion 4.2.1 it follows that the de ay rate is simply Γ = 2 α GR . With ωx ≈ 1015 s−1 and typi al values for Γ in the range of ns−1 (as seen in Fig 1.4) we have αGR (ωx )/ωx ≈ 10−6 . Therefore, in order to estimate the residue of the pole term we may use, as a rst approximation, the derivative of Φ(ωx ). With this, the derivative of the denominator is given as ′ Φ′ (ωx ) = αGR (ωx ) + i(αGI′ (ωx − 1)),

and the absolute square of the residue readily follows as ′ |a−1 |2 = (αGR (ωx ))2 + (αG′I (ωx ) − 1)2


−1

.

(4.8) 67

Chapter 4. Fra tional de ay of quantum dots in photoni rystals

We are mainly interested in material systems for whi h the de ay dynami s deviate from the Pur ell regime. Due to the formulation in terms of an integral, the fun tion GI is somewhat ompli ated to work with. The fun tion GR , on the other hand, is a lot simpler and is basi ally the s aled LDOS. In addition, both terms in Eq. (4.8) are positive, so provided that G′R (ωx ) = GR′ (ω0 ) we may state (4.9)

′ αGR (ωx ) > 1

as a su ient (but not ne essary) ondition that the dynami s deviate from the Pur ell regime. This represents a mathemati al formulation of the argument used in se tion 2.4.3 to justify the Wigner-Weisskopf approximation; namely that the LDOS should vary little over the spe tral linewidth of the emitter. At this point it is illustrative to introdu e a s aled (and dimensionless) LDOS through the substitution ρx (ω) =

ωx2 ρ˜x (˜ ω) = ρ0 (ωx )˜ ρx (˜ ω ), 3π 2 c3

where ρ0 (ω) denotes the DOS of free spa e, and where we have introdu ed a s aled (and dimensionless) frequen y through the substitution ω = ωx ω ˜.

In terms of the s aled quantities we may write the ondition in Eq. (4.9) as βπ

d d˜ ω



ρ˜x (˜ ω) ω ˜



= βπ



ρ˜′x (˜ ω ) ρ˜x (˜ ω) − ω ˜ ω ˜2



> 1,

(4.10)

where the dimensionless oupling strength beta is dened as β=

p2 q 2 , 6~m2 π 2 ǫ0 c3

and is related dire tly to the de ay rate in free spa e as Γ0 = 2πωx β.

Experimental values for β range from β ≈ 10−8 for InAs QDs [111℄ to β ≈ 6 × 10−8 for PbSe QDs [112℄ with so- alled interfa e defe t QDs possibly rea hing values of β ≈ 10−6 [113℄. The oupling strength β arises naturally when re asting the expression for the spe trum, Eqs. (4.1) and (4.2) in dimensionless form. However, a large variety of dierent oupling parameters are used by dierent authors. Appendix E provides a dis ussion of the relation of β to the os illator strength. The se ond term in Eq. (4.10) is typi ally vanishing and we will ignore it, leaving the approximate

ondition ρ˜x (˜ ω) β (4.11) & 1. ω ˜

68

Cal ulation of de ay urves We are thus left with the onvenient guideline as to the kind of material systems that are likely to show fra tional de ay dynami s: • The LDOS should be zero at least at one frequen y, so that the we an hope to nd g(ω0 ) = 0 and hen e a real pole. • The relative slope of the LDOS should be on the order of β −1 so that deviations

from the Pur ell regime will o

ur.

This is in a

ordan e with the hoi es of LDOS used in Refs. [19, 20, 109℄. Obvious

andidates for material systems are three dimensional PCs whi h may show a full photoni band gap and for whi h the slope of the LDOS may be very large. Dierent models for the LDOS has been used in the literature, in luding both isotropi and anisotropi gap models [114℄ in whi h the LDOS at the band edge takes the forms (Isotropi gap) (Anisotropi gap)

Kiso ω − ωBE √ BE ρx (ω) = KBE ω − ωBE , ρiso x (ω) = √

with Kiso and KBE being material onstants of dimensions [Kiso ] = s1/2 /m3 and [KBE ] = s3/2 /m3 . The validity of both LDOS models have been questioned in the literature. Notably in Ref. [115℄, where the authors use dense sampling of the LDOS to on lude that the anisotropi model an be valid only in a region below ∆ω PC = 0.005 (2πc/a) from the band edge. From the above analysis, however, we realize that this frequen y range should be ompared to the light-matter oupling strength. For the Si inverse opal (ǫSi = 11.76), the upper band edge is lo ated at ωBE = 0.8163 (2πc/a). If we hoose to s ale the photoni rystal so that the band edge is at the emitter frequen y we therefore have aωx = 0.8163. 2πc

This means, that in the s aled frequen ies, ω˜ = ω/ωx , the restri tion on the validity of the anisotropi model, as stated in Ref. [115℄, reads ∆˜ ω = 0.005

2πc ≈ 6 × 10−3 , aωx

whi h is more than 4 orders of magnitude larger than the intrinsi bandwidth set by β . In se tion 4.4 we show that for most spatial positions in the latti e, the limiting form of the LDOS in high-index inverse opals is indeed given by the anisotropi gap model. At this point, however, it is illustrative rst to apply the analysis to this model. 69

Chapter 4. Fra tional de ay of quantum dots in photoni rystals

4.3

Fra tional de ay in the anisotropi gap model

In this se tion the analysis in se tion 4.2.2 is used to investigate fra tional de ay in the anisotropi gap model. For onvenien e in the numeri al modeling we will often use the s aled frequen ies ω˜ = ω/ωx. The spe trum, Eq. (4.1), is given as c˜e (˜ ω) =

1 , ˜ ω ) − i (˜ β G(˜ ω − 1)

˜ ω ), upon insertion of the anisotropi band gap model for in whi h the fun tion G(˜ the LDOS, is given for frequen ies above the integration path as √ ˜ BE z˜ − ω ˜ BE K d˜ z z˜2 (˜ ω − z˜) ω ˜ BE   p p π ˜ BE ω √ ˜ + ω ˜ + 2 ω ˜ − ω ˜ ω ˜ −2˜ ω = iK BE BE BE 2˜ ω2 ω ˜ BE p 2 p π ˜ BE ω √ = −i K ˜ ω ˜ BE − ω ˜− ω ˜ BE 2 2˜ ω ω ˜ BE

˜ ω) = i ω G(˜ ˜

Z

∞

(4.12)

with the s aled material onstant dened through the relation ρ0 (ωx ) ˜ KBE . KBE = √ ωx

For the al ulations we will normally have ω˜ BE ≈ 1. The integral representation of ˜ ω ) is valid only for frequen ies above the integration path, but using Eq. (4.12) G(˜ we may dire tly make an analyti al ontinuation to the entire omplex plane. In this way we are able to al ulate the pole term and the residue to arbitrary a

ura y. By denition, the atomi frequen y is at ω˜ = 1, so detuning of the emitter with respe t to the band edge is handled by hanging the value of ω˜ BE . To this end we write ω˜ BE = 1 + ∆, in whi h ∆ is the detuning. For large positive detunings the band edge is at mu h higher frequen ies than the emitter frequen y (the emitter is deep within the band gap), and very little or no de ay at all is expe ted. For large negative detunings the emitter frequen y is deep within the ontinuum of modes above the band edge and one will expe t the Wigner-Weisskopf approximation to hold, leading to exponential de ay.

4.3.1 Denition of the square root There is a subtlety in Eq. (4.12) be ause the square root is multiple valued with values in either of two Riemann sheets as illustrated in Fig. 4.1. The bran h ut is dened by the integration path whi h will typi ally be along the real axis from ω ˜ BE to innity. It is possible, however, to dene a new square root in whi h the 70

Fra tional de ay in the anisotropi gap model

phase φ of the omplex number Z = |Z| exp(iφ) is hosen to lie in a given interval. Irrespe tive√of the denition of the square root, the se ond Riemann sheet is found by using − ω˜ BE − ω˜ in Eq. (4.12).

Figure 4.1: Sket h of the Riemann surfa e for the square root fun tion. The fun tion is multiple valued with values in either of two Riemann sheets.

For a given point

on the surfa e, only by ir ling the enter twi e do we arrive the same point.

Fig. 4.2 shows the absolute value of the denominator in Eq. (4.1) al ulated ˜ BE = 10 and on the two dierent Riemann sheets for the ase of β = 5.5×10−8, K −7 ∆ = 8.645×10 . We have rotated the bran h ut from the real axis into the lower half of the omplex plane. This means, that one pole is found on the rst Riemann sheet and one is found on the se ond Riemann sheet. Note that the fun tion is

ontinuous a ross the bran h ut in the sense that as one traverses the bran h ut one should ontinue onto the other Riemann sheet as illustrated in Fig. 4.1. 4.3.2

Movement of poles

The pole dening the de ay pro ess is lo ated below (or potentially on) the real axis. Fig. 4.3 shows the movement of the poles in the omplex ω˜ plane as the detuning is varied. For large negative detunings the pole of interest is situated to the right of and below the band edge in the omplex plane. As the detuning is in reased the pole wanders towards the band edge, in the pro ess inevitably rossing over to the se ond Riemann sheet. At a riti al detuning, ∆0 , the two ( omplex

onjugate) poles ollide on the real axis and start to move in opposite dire tions. For β = 5.5×10−8 and KBE = 10 the riti al detuning is at ∆0 = −8.6394 × 10−7 . At another riti al detuning one of the poles rea hes the band edge frequen y and

rosses over on e again to the rst Riemann sheet. The gray shaded area in the right gure indi ates the interval in whi h the real part of the pole is less than 71

Chapter 4. Fra tional de ay of quantum dots in photoni rystals

−9

−9

x 10

x 10

1

1 −21

0.5

−21 0.5

0

−23

−0.5

−1 −5

0

5

ω ˜R − ω ˜ BE

10

15 −10 x 10

−22

ω ˜I

ω ˜I

−22 0

−24

−0.5

−25

−1

−23

−24

−25 −5

0

5

ω ˜R − ω ˜ BE

10

15 −10 x 10

Figure 4.2: Logarithm of the absolute value of the denominator in the spe trum

al ulated on the rst and on the se ond Riemann sheet, respe tively, in the neighborhood of the lo ation of the poles. The bran h ut is hosen to lie in the lower half of the omplex plane as indi ated by the red line, thus exposing the pole just below the real axis. the band edge. If the bran h ut was hosen to be verti ally downwards from the band edge, this would indi ate the interval in whi h both poles are on the se ond Riemann sheet. For onvenien e, we shall refer to the pole in the upper half of the omplex plane as pole 1 and the other pole as pole 2. At the riti al detuning we an no longer distinguish the two, but nevertheless we shall refer to the pole traveling in the dire tion of positive frequen ies as pole 2, sin e this is the pole that we are ultimately interested in. 4.3.3

Residues

When the pole is on the rst Riemann sheet the absolute value of the residue and the imaginary part of the pole denes the long time behavior of the temporal solution. In parti ular, if the pole has a non-zero imaginary part the temporal solution will eventually de ay to zero. This is the ase for ∆ − ∆0 < 0 as seen in gure 4.3, right. On the other hand, if the pole is lo ated on the real axis, the temporal solution will tend to a non-zero stable value given as the absolute square of the residue. This is fra tional de ay and will happen at detunings above the gray shaded area in the gure. For detunings in the region of the gray shaded area there are no poles on the rst Riemann sheet, and hen e the pole does not dene the long time behavior 72

Fra tional de ay in the anisotropi gap model −12

−12

3

x 10

x 10

ω ˜R − ω ˜ BE

1

2

ω ˜I

0 0

ω ˜

ω ˜I

1

−1

Pole 1

−2

Pole 2

−3 −4

−3

−2

−1

−1

ω ˜R − ω ˜ BE

0

1 −12

x 10

−2 −2

−1

0

1

∆ − ∆0

2

3

4 −12

x 10

Figure 4.3: As the detuning is varied, the poles of the spe trum move around in the omplex plane. Left: Movement of poles in either of the two Riemann sheets,

orresponding to the two dierent signs of the square root. Right: Position of the pole of interest that was originally in the rst Riemann sheet below the real axis. The gure shows the real and the imaginary part, respe tively, as a fun tion of the detuning ∆. ∆0 = −8.6394 × 10−7 denotes the riti al detuning. of the temporal solution. Fig. 4.4, left shows the absolute value of the residues of the two poles as a fun tion of the detuning ∆. Again, the gray shaded area denes the qualitatively dierent regimes. At large negative detunings the poles are omplex onjugates and the absolute values of the residues are equal. As the poles ollide on the real axis the residues seem to diverge. As noted above, this has no impli ations for the temporal solutions as both poles are on the se ond Riemann sheet. At a ertain riti al detuning one pole rosses over to the rst Riemann sheet and the de ay be omes fra tional. At this detuning the absolute value of the residue is zero ( orresponding to a full de ay at long times) but in reases with in reasing detuning. Fig 4.4, right shows three examples of temporal solutions at detunings above the riti al detuning where the pole reenters the rst Riemann sheet and fra tional de ay o

urs. From Fig. 4.4 it is seen that the detuning an always be varied to a hieve an arbitrarily large a degree of fra tional de ay. This is onsistent with the analysis in se tions 4.2.3 from whi h we know that non-Markovian dynami s are expe ted whenever the relative slope of the LDOS is larger than the intrinsi bandwidth set by β . In the al ulations we have modelled the LDOS using a square root for whi h the relative slope diverges. Therefore, in prin iple we an always tune the frequen y to obtain as large a degree of fra tional de ay as wanted. We note that the region of fra tional de ay, as dened by |a−1 |2 < 1, extends for several de ades of ∆ − ∆0 , 73

Chapter 4. Fra tional de ay of quantum dots in photoni rystals

5

1

Pole 1

4

0.8

|a−1 |2

Pole 2 2 1 0 −2

0.6

∆ − ∆0 =1×10−11

0.4

∆ − ∆0 =4×10−12

0.2

∆ − ∆0 =2×10−12

|cx |2

3

−1

0

1

∆ − ∆0

2

3

4 −12

x 10

0 0

2

4

6

Γ0 t

8

10 5 x 10

Figure 4.4: Left: Absolute value of the residues of the two poles as a fun tion of detuning, ∆. Right: Examples of temporal solutions at three dierent detunings as fun tion of normalized time Γ0 t, where Γ0 is the de ay rate in va uum. Dashed lines show the part of the solution orresponding to the pole term. slowly approa hing the limit |a−1 |2 = 1. In pra ti e, a relative detuning of ∆ = 10−12 means that the system will have to be stabilized to within ∆ω ≈ 103 s−1 whi h is several orders of magnitude ner than any available spe trometer. This apparent problem is linked to the formulation in whi h there seems to be only one material parameter of importan e, namely the produ t βKBE . In real samples there will likely be other important parameters dening the de ay dynami s, and the obtainable degree of fra tional de ay will depend on the size of βKBE relative to these parameters. One su h parameter is the material absorption, as we will dis uss in se tions 4.4 and 4.5. When material losses are in luded in the model there will be an optimum detuning for whi h the degree of fra tional de ay is largest. In this ase the optimum detuning is found within a frequen y interval dened by the size of the light-matter oupling strength relative to the absorption. 4.4

High resolution lo al density-of-states

We now aim at applying the riterion in Eq. (4.11) to evaluate the possibility of non-Markovian de ay of QDs in three dimensional PCs. To this end we must be able to assess whether or not the relative slope of the LDOS an rea h values on the order of β −1 . This in turn puts rather severe onstraints on the al ulation methods employed, be ause the LDOS will need to be sampled at a mu h higher resolution than what is typi al for su h al ulations. Fig. 4.5 shows the band diagram for a lose pa ked inverse opal photoni rystal made from Sili on (ǫr = 11.76). All 74

High resolution lo al density-of-states

plane wave al ulations in this se tion were performed using ode based on Ref. [74℄, kindly shared by Dr. Femius Koenderink. The band diagram shows a band gap with the upper and lower band edges dened by the W and X points in the irredu ible Brillouin zone, respe tively. Based on the band diagram, and the dis ussion of a suitable LDOS for the observation of fra tional de ay in Se tion 4.2.3, we realize that inverse opals are promising andidates for a model system showing fra tional de ay. 1

L

0.8

U

ωa/2πc

Γ X

0.6

W

K

0.4

0.2

0

X

U

L

Γ

X

Figure 4.5: Band stru ture of a lose pa ked Si (ǫr

W

K

= 11.76) inverse opal along with

a sket h of the Brillouin zone. The irredu ible Brillouin zone is indi ated in the sket h along with the high-symmetry points.

In this se tion we rst dis uss typi al al ulations of the LDOS in photoni

rystals using plane wave expansion. In order to evaluate the possibility of nonMarkovian de ay we fo us on the upper band edge and set up an analyti al expression for the LDOS lose to the band edge based on an expansion of the relevant dispersion surfa es in powers of the k ve tor. Based on the model LDOS we next use perturbation theory to in lude the ee ts of material absorption whi h a ts to broaden the features of the LDOS. 4.4.1

Cal ulations using plane wave expansion

The LDOS is typi ally al ulated using a plane wave expansion te hnique [72, 73, 74℄. In this way the eigenmodes are found along with the orresponding eigenfrequen ies and are summed a

ording to Eq. (2.8). The al ulations result in histograms whi h onverge to the true LDOS only in the limit of vanishing binω . In the ase of perfe t sampling in k-spa e the al ulations result in the width, ∆˜ 75

Chapter 4. Fra tional de ay of quantum dots in photoni rystals

ωi ), dis retely sampled fun tion h(˜ h(˜ ωi ) =

1 ∆˜ ω

Z

ω ˜ i +∆˜ ω

ρ˜(˜ ω ′ ) d˜ ω′.

(4.13)

ω ˜i

In pra ti e, however, we al ulate only the solutions at a nite number of k points. The binwidth, ∆˜ ω , and the sampling in k-spa e are intimately onne ted, so that ∆˜ ω ≈ ∆k|∇˜ ω (k)|,

in whi h |∇˜ ω (k)| represents the group velo ity [74℄. This means that smaller binwidth ne essarily omes at the pri e of ner sampling in k-spa e and hen e longer

al ulation times. ωi ) at the Γ and√ P points of a lose Fig. 4.6 shows an example of the fun tion h(˜ pa ked Si inverse opal (hole radius per latti e onstant R/a = 1/ 8 ≈ 0.3536). The binwidth in the al ulations was ∆˜ ω = 0.005 and the histograms were al ulated using 725 plane waves and 26670 k points in half of the rst Brillouin zone. The ωi ) lose small binwidth was hosen in order to illustrate the limiting form of h(˜ to the band edge as seen in the insets. Despite the large number of k points, the overall appearan e of the LDOS histograms is somewhat rugged and irregular spikes indi ate that the number of data points (one for ea h band at ea h k point) is almost too low for the hosen binwidth. For the same reason we were for ed to lower the number of plane waves for these al ulations - the analysis in se tion 4.4.2 is arried out using 1243 plane waves. The appearan e of the LDOS histograms

an be smoothened by using a larger binwidth or a larger number of k points at the prize of longer omputation times as dis ussed above. Although the number of k points is large, it is by no means unique in the literature. For omparison, the results in Ref. [74℄ were obtained using 145708 k points and a binwidth of ∆ω = 0.01(2πc/a) (roughly equivalent also to ∆˜ ω = 0.01). We note that hosen binwidth is omparable to the binwidth in Ref. [115℄ in whi h the anisotropi band gap model was questioned based on the appearan e of the LDOS histograms lose to the band edge. From the insets in Fig. 4.6 we see that the LDOS at the Γ point does in fa t follow a square root behavior lose to the gap, whereas the LDOS at the P point does not. 4.4.2

Detailed analysis of the band edge

As dis ussed in se tion 4.2.3, the binwidth ∆˜ ω = 0.005 is four orders of magnitude too large to sample the slope of the LDOS that is required for non-Markovian de ay to be observable. Therefore, in this se tion we take a semi-analyti al approa h in whi h the numeri al dispersion urves lose to the band edges are approximated by 76

High resolution lo al density-of-states

4

4 1

1

0.5

0.5

2

h

h

h

3

h

3

0 0.98

1

1.02

0 0

0 1

ω ˜

1

0.2

0.4

2

1.04

0.6

0.8

1

0 0

1.2

ω ˜

1.02

1.04

ω ˜

1

0.2

0.4

0.6

0.8

1

1.2

ω ˜

Figure 4.6: Examples of dis rete samplings h(˜ ωi ) of the LDOS of the Si inverse opal at the Γ (left) and P (right) points in the Wigner-Seitz ell. In both histograms the band gap is learly visible just below ω˜ = 1. Dashed urve indi ates the s aled LDOS in free spa e ρ˜0 = ω˜ 2 . In both gures, the inset shows a zoom in at values

lose to the upper band gap edge. The bla k urve in the left inset shows the limiting square root behavior of the LDOS lose to the band edge. analyti al fun tions and integrated to get an analyti al expression for the LDOS. The LDOS is given originally as the sum in Eq. (2.8) whi h may be written, in the limit of large quantization volume, as an integral, ρx (r, ω) =

X p

V (2π)3

Z

|ex · fµ (r)|2 δ(ω − ωk ) dk,

where the sum is now over the two polarizations p only. Owing to the periodi stru ture of the material, the frequen ies are dened by k ve tors in the rst Brillouin zone only and may be ordered in bands of index n. Considering the ontribution from just a single band, one may use standard methods from solid state physi s [116, 117℄ to rewrite the integral as ρx (r, ω) =

X p

V (2π)3

Z

Sn (ω)

|ex · fµ (r)|2 dS, |∇ωn (k)|

(4.14)

in whi h the integral is now over the surfa e of onstant frequen y in the n'th band. Appendix D.1 illustrates the pro edure for the homogeneous medium. Close to the band edge, the LDOS of the Si inverse opal is dened only by the ninth band. Further, the band edge is dened only by the high-symmetry point X in k-spa e, whi h means that the dispersion surfa e is an ellipsoid. By making a suitable expansion of the integrand in Eq. (4.14) in terms of k we may evaluate the resulting integral analyti ally to obtain the orre t behavior of the LDOS. The 77

Chapter 4. Fra tional de ay of quantum dots in photoni rystals

integral is arried out over the dispersion surfa e in the rst Brillouin zone. However, one annot restri t the analysis to the irredu ible Brillouin zone be ause the proje ted mode fun tions, |ex · fµ (r)|, are not invariant under the symmetry operations that applies to the underlying latti e [73℄. Details of the al ulations may be found in Appendix F. By expanding the dispersion surfa e in a oordinate system

entered on the X point as ω = ω0 +


1 ωxx kx2 + ωyy ky2 + ωzz kz2 2

it is shown that to lowest order in ω the limiting form of the LDOS at the band edge is given by the anisotropi gap model as Ix ρBE (r, ω) = x

(2π)2

s

2 2 ω ωxx zz

√

√ ω − ωBE = KBE ω − ωBE ,

(4.15)

where ωxx and ωzz denote, respe tively, the urvatures of the dispersion surfa e in the dire tions parallel and perpendi ular to the Brillouin zone surfa e and where Ix = V

6 X

n=1

|ex · fµ (r, Xn )|2

denotes the sum of the proje ted eld strength at the 6 X points in the Brillouin zone. For non-vanishing Ix the band edge is seen to follow the square-root behavior of the anisotropi gap model. The solid urve in the inset of Fig. 4.6 (left) illustrates how the analyti al approximation onforms to the purely numeri al histogram lose the band edge. Fig. 4.7 shows the values of KBE along lines between symmetry points of the Wigner-Seitz ell of a Si inverse opal with hole radius per latti e onstant of R/a = 0.3436. The analyti al approa h allows for the use of only 5 k points in ea h dire tion for the determination of the urvature and we have used 1243 plane waves to ensure onvergen e. The maximum value of KBE is found at the H point (blue sphere in the sket h of the inverse opal unit ell in Fig. 1.7). This value may be further in reased by redu ing the radius of the air spheres as shown by the dashed

urve in Fig. 4.7. The large value of KBE at the H point makes this the position in the rystal where non-Markovian ee ts are most pronoun ed. The radius of the air spheres R = 0.3436 a used in the al ulation of Fig. 4.7. is slightly less than that of the lose pa ked rystal. For the lose pa ked rystal the air spheres tou h at the N points. This means that the thi kness of the diele tri vanishes at these points, resulting in a divergen e of the ele tri eld omponent parallel to the surfa es of the spheres. The ee t is visible already at the hosen value of R/a as a pronoun ed in rease of the value of KBE for the x polarization. 78

High resolution lo al density-of-states

R/a 0.34

0.3434

0.3468

0.3502

0.3536

x ˆ y ˆ/ ˆ z

15

ˆ z

H 10

˜ BE K

P x ˆ

5

b

N y ˆ

0

H

P

Γ

N

H

Figure 4.7: The parameter KBE (in units of ρ0 (ωx )/ωx1/2 ) as a fun tion of position in the Wigner-Seitz ell (shown in the inset) for the three prin ipal emitter orientations. Grey dash-dotted line shows values of KBE at the H point for dierent R/a (top s ale). In addition to the numeri al problems at the N point, we experien ed some trouble

onverging the results for the z polarized emitter at positions along the Γ− H route. Sin e these positions are less relevant for the dis ussion in this work we have left them out.

4.4.3 Inuen e of material loss In order to introdu e loss in the material, a small imaginary part is added to the permittivity, ǫR → ǫR + i ǫI ,

whi h in turn leads to a small (imaginary) shift in frequen y. For small losses we use rst order perturbation theory [64, 118, 119℄ to write ω = ω (0) − iδ , where ω (0) is the frequen y in the absen e of losses and δ=

ω (0) hEµ |i ǫI |Eµ iC ω (0) ǫI = f, 2 hEµ |i ǫR (r)|Eµ iV 2 ǫR

where subs ript C denotes the volume of the lossy material only, and subs ript V denotes the entire volume. The parameter f is given as f=

hEµ |ǫR (r)|Eµ iC