# Theory on light scattering by metal nano-structures, by way of the Green tensor method

Advisor:Luis Martìn Moreno

School:Universidad de Zaragoza

School Location:Spain

Source Type:Doctoral Dissertation

Keywords:Surface Plasmon, Nano-Optics, Green Tensor, Sommerfeld Integrals ,Ridges and Grooves

ISBN:

Date of Publication:05/10/2010

Theory on light scattering by metal nano-structures,

by way of the Green tensor method

Departamento de Física de la Materia Condensada

Instituto de Ciencia de Materiales de Aragón

CSIC-Universidad de Zaragoza

DOCTORAL THESIS

Theory on light scattering by

metal nano-structures,

by way of the Green tensor

method.

Giovanni Brucoli

Thesis advisor:

Luis Martín Moreno

Zaragoza, Mayo 2010

Ai miei genitori.

Contents

Acknowledgements

Introduction

xiii

xv

1 De nition and derivation the Green Tensor 1

1.1 Electromagnetic Green Tensor . . . . . . . . . . . . . . . . . . . 1

1.2 Free-space Green's tensor spatial and spectral representations . 4

1.3 The Green Tensor Divergence . . . . . . . . . . . . . . . . . . . 10

1.4 Depolarizing eld and Depolarization dyadic . . . . . . . . . . . 11

1.5 The Electromagnetic Lippmann-Schwinger equation in a homogenous

background . . . . . . . . . . . . . . . . . . . . . . . 14

1.6 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.7 Lipmann-Schwinger Equation in a the presentence of an interface 17

1.8 The Green Tensor for a semi-space: The Method of scattering

superposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.9 2D Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.10 Homogeneous medium Dyadic Green`s Function . . . . . . . . 24

1.11 Depolarization Dyadic in 2D . . . . . . . . . . . . . . . . . . . . 26

1.12 2D Discretized Lippmann-Schwinger and M . . . . . . . . . . . 27

1.13 2D Green tensors of the Interface . . . . . . . . . . . . . . . . 29

2 Numerical Computation of Sommerfeld's Integrals 33

2.1 3D Sommerfeld Integrals . . . . . . . . . . . . . . . . . . . . . . 33

2.2 Standard Integration Technique R_{∥ }< λ . . . . . . . . . . . . . 38

2.2.1 Solution Scheme . . . . . . . . . . . . . . . . . . . . . . 38

2.2.2 The topology of Sommerfeld's Integrands . . . . . . . . 39

2.2.3 Path 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.2.4 Path 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.2.5 Path 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.3 Modi ed Integration technique for large values of R_{∥ }. . . . . . 46

2.4 2D Sommerfeld Integrals . . . . . . . . . . . . . . . . . . . . . 49

3 Asymptotic Expressions for the 2D Green tensor in the far-

x Contents

eld 53

3.1 The eld outside the region of the defect . . . . . . . . . . . . 53

3.2 The Transmitted 2D Green Tensor . . . . . . . . . . . . . . . 55

3.3 The Surface Plasmon Field in the 2D Green Tensor . . . . . . 58

3.3.1 Surface Plasmon Polariton Mode . . . . . . . . . . . . . 59

3.4 Emission of Surface Plasmon Polaritons . . . . . . . . . . . . . 61

3.4.1 2D Source above the Metal . . . . . . . . . . . . . . . . 61

3.4.2 Source below the Metal . . . . . . . . . . . . . . . . . . 62

3.4.3 3D source Analog . . . . . . . . . . . . . . . . . . . . . . 63

3.5 The Far-Field Scattered by 2D Systems . . . . . . . . . . . . . 64

3.5.1 Direct GT . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.5.2 Re ected GT source upon the metal . . . . . . . . . . 66

3.5.3 Refracted GT (Source in the metal) . . . . . . . . . . . 66

3.6 Far Fields at oblique incidence . . . . . . . . . . . . . . . . . . 67

3.7 Radiative Energy at Oblique Incidence . . . . . . . . . . . . . 69

3.8 The Extinction Coe cient in 2D . . . . . . . . . . . . . . . . . 71

3.9 Transmission Re ection and Radiation for 2D systems . . . . . 71

4 Device for launching surface plasmon polaritons 73

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.2 Local surface plasmon polariton excitation on ridges . . . . . . 76

4.3 Numerical Results and Experimental Agreement . . . . . . . . 79

4.3.1 Wavelength dependence of coupling e ciency . . . . . . 79

4.3.2 Dependence on geometrical parameters of ridge . . . . . 81

4.3.3 Optimum wavelength for excitation on ridges . . . . . . 81

4.3.4 Directionality of SPP excitation on periodic sets of ridges 86

4.4 E cient unidirectional ridge excitation of surface plasmons . . 88

4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5 Scattering of surface plasmon polaritons by 2D impedance barrier

93

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.2 Scattering of surface plasmon polaritons by impedance barriers:

dependence on angle of incidence . . . . . . . . . . . . . . . . . 94

5.2.1 The theoretical methods . . . . . . . . . . . . . . . . . . 94

5.2.2 The scattering system . . . . . . . . . . . . . . . . . . . 95

5.2.3 Solutions and Results: The surface plasmons Brewster

angle analog . . . . . . . . . . . . . . . . . . . . . . . . 96

5.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

6 Comparative study of surface plasmon scattering by shallow

ridges and grooves 101