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

by Brucoli, Giovanni, PhD

Abstract (Summary)
This thesis reports a theoretical investigation of the interaction between light and metal nano-metric structures on a thick metal slab. When light is con- ned in a sub-wavelength structure, the electric eld inside such structure is attained by solving Maxwell's equations self-consistently. This can be done by means of the Green Tensor Approach (GTA). Not only does the GTA provide the eld within a sub-wavelength nano-metric structure, it also allows for an analytic study of the light scattered by the nano-structure into the surrounding space, far from the structure. Nonetheless the implementation of the Green tensor approach is rather complex, mathematically, especially if the considered scattering centers interact with a metal slab, rather than being placed in vacuum. In fact accounting for the presence of a back-ground metal slab, demands performing Sommerfeld's integrals. The rst part of this thesis is devoted to reporting the formalism of the electromagnetic Green tensor in a complete manner, for all of the considered nanooptical systems. Chapters 1-3 detail the mathematical foundations of the numerical codes by means of which the results of this thesis were attained. In Chapter 1 we dene and derive the electromagnetic Green tensor for a semispace back-ground. Our derivation is an organic elaboration of many disjointed pieces of standard material. These have been originally assembled and linked by detailed derivations, which are either briey sketched in the literature or not mentioned. In Chapter 2 we describe our implementation of the procedure to obtain numerical convergence of Sommerfeld's integrals, describing the interaction of elds with a metal plane. Part of the numerical procedure had appeared previously in the literature and so we refer to it as the standard integration technique. However after describing thoroughly our implementation of the standard technique, we also report on our modication to the standard integration technique, which speeds up computation in those cases in which surface plasmon polaritons constitute the main mechanism of light transport on the metal plane. In Chapter 3 we derive the asymptotic expressions of the Green tensor that should be used to calculate, analytically, the Poynting vector energy ux scattered by bi-dimensional nano-structures. These expressions in the considered case of bi-dimensional systems are not found in the literature, to the best of our knowledge. Furthermore we have developed a simplied formalism that deduces the elds amplitude, in the far-zone, from scalar products rather than through more tedious tensorial operations. The second part of the thesis is devoted to reporting the results we have attained for several nano-optical systems. In Chapter 4 we report our investigation of a device to launch a unidirectional stream of surface plasmon polariton, based on photonic band-gap eects. The device can achieve local light coupling into surface plasmon polariton modes. This work is original and, besides having potential technological application, it provides evidence of our theory through experimentally replicated results. In Chapter 5 we study the eect on the propagation of a surface plasmon caused by an impedance barrier. This represents a metal wire of rectangular cross-section in a thick conducting lm. By varying the in-plane direction of propagation of the surface plasmon with respect to the normal to the impedance barrier, we nd a surface plasmon analog of the Brewster angle. At such angle the incident surface plasmon is not reected by the impedance barrier. In Chapter 6 we begin a comparative study of the surface plasmon scattering by bi-dimensional protrusions (ridges) and indentations (grooves). Subwavelength protrusions and indentations of equal shape present dierent scattering coecients when their height and width are comparable. In this case, a protrusion scatters plasmons like a vertical point-dipole on a plane, while an indentation scatters like a horizontal point-dipole on a plane. We corroborate that, as previously presented in the literature with approximate methods, long and shallow asymmetrically-shaped surface defects have very similar scattering. Moreover we provide a rst principles explanation for such property. In the transition from short shallow scatterers to long shallow scatterers the radiation is explained in terms of interference between a vertical and a horizontal dipole. The results attained numerically are exact and accounted for with analytical models. In Chapter 7 we extend the comparative study of surface plasmon scattering by ridges and grooves. This time the width of the defects is xed, while their height is varied. Both individual and arrays of defects are considered, mainly in the optical regime. The width of the defects is xed, while their height is varied. It is shown that protrusions mainly reect the incident plasmons in the optical range. Indentations, however, mainly radiate the incident plasmon out of plane. An indentation produces maximum reection and out-of-plane radiation at the same wavelength, when its interaction with the incident surface plasmon is resonant. Protrusions, in general, exhibit maximum reection and radiation at dierent wavelengths. Shallow arrays of either defects produce a photonic band-gap, whose spectral width can be broadened by increasing the defects height or depth. At wavelengths inside the band-gap ridge arrays re- ect SPPs better than groove arrays, while groove arrays radiate SPPs better than ridge arrays. In Chapter 8 the scattering of an incident SPP by a three-dimensional nanoparticle is analyzed. SPP extinction spectra are calculated for gold cubic particles of various sizes placed in the vicinity of a at gold surface, as the distance of the particle from the at gold surface is varied. The results are compared with an analytical model in which the interaction of the nano-particle with the plane is approximated by that of a point-dipole.
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Bibliographical Information:

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

Document Text (Pages 1-10)


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


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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


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Ai miei genitori.


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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-


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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

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