PLASMONICS NANOFOCUSING BASED ON LONGRANGE SURFACE PLASMON POLARITONS

Table of ContentsPreface1Abstract2List of Figures5List of Symbols7List of Abbreviations8Introduction9Chapter 1 Basics of Surface Plasmon Polaritons111.1Maxwell’s Equations111.2 Surface Plasmon Polaritons at Metal-Insulator Interfaces131.2.1 The Wave Equation of TM Mode and TE Mode131.2.2 Surface Plasmon Polaritons at a Single Interface171.2.3 Multilayer Systems and the Long Ranging Modes191.3 Excitation Methods231.3.1 Prism Coupling231.3.2 Grating Coupling251.3.3 End-fire Coupling Excitation27Chapter 2 Demand for nanofocusing, an idea and supporting results292.1 The Increasing Demand of Nanofocusing and an Idea292.2 Supporting Results302.2.1 Extended Long Range Surface Plasmon Polaritons302.2.2 Apertureless Nanotip312.2.3 Plasmonic Nanofocusing Based on a Metal Coated Axicon Prism32Chapter 3 FDTD simulation353.1 FDTD Simulation353.2 Numerical Dispersion413.3 Numerical Stability – The Courant-Friedrichs-Lewy Stability Criterion423.4 Perfect Electric Conductors433.5 Dielectric-Dielectric Interface433.6 Terminating The Simulation Domain46Chapter 4 Waveguide structure, simulation results and discussions534.1 Centre symmetric longrange plasmonics waveguide.534.2 Asymmetric MDM waveguide for long-range surface wave excitation.544.3 Combination of centre symmetric MDM and two side-asymmetric plasmonic waveguide.564.4 Waveguide Structure594.5 Simulation results614.3 The Dependence of the Field Magnification Factor and the Beam Side on Parameters67Conclusion71References72 List of FiguresFigure 1.1 Definition of a planar wave guide geometry..15Figure 1.2 Geometry for SPP propagation at a single interface between a metal and a dielectric.18Figure 1.3 Geometry of a three-layer system consisting of a thin layer I20Figure 1.4 Dispersion relation of the coupled odd and even modes.22Figure 1.5 Prism coupling to SPPs using attenuated total internal reflection24Figure 1.6 Prism coupling and SPP dispersion.24Figure 1.7 Phase-matching of light to SPPs using a grating.26Figure 1.8 End-fire coupling technique for exciting SPPs.27Figure 2.1 Scheme of the analyzed apertureless SNOM probe [8].32Figure 2.2 Schematic for the localization of photons by a metal-coated axicon prism [10]33Figure 2.3 Intensity distributions on a gold-coated axicon prism for radially polarized incident light [10].34Figure 3.1 Discretization of the model into cubes and the position of field.37Figure 3.2 Basic flows for implementation of Yee FDTD scheme.38Figure 3.3 The three-dimensional computational region for FDTD simulation.41Figure 3.4 An example with PEC on the top surface of Cube (i, j, k).43Figure 3.5 Four adjacent cubes with different permittivity and conductivity.44Figure 3.6 A closed loop C crossing four cubes with different permittivity and conductivity.44Figure 3.7 Boundary E field component between two layers of different dielectrics.52Figure 4.1 Centre symmetric waveguide structure53Figure 4.2 A 3D view of longrange surface plasmon polariton excited in a MDM structure.54Figure 4.3 A 2D view of longrange surface plasmon excited in a MDM structure.54Figure 4.4 An asymmetric plasmonic waveguide structure.55Figure 4.5 A 3D view of LRSPPs excitation in an asymmetric structure.55Figure 4.6 A 2D view of LRSPPs excitation in an asymmetric structure.56Figure 4.6 A combination of a symmetric structure with two asymmetric waveguides.56Figure 4.7 Electric field distribution at x-z plane.57Figure 4.8 A 2D view of the electric field distribution at x-z plane.57Figure 4.9a Hy at the x-z plane.58Figure 4.9b Magnetic field distribution at x-z plane.58Figure 4.11 Schematic of the simulated model with an incident wavelength of 1550 nm.59Figure 4.12 Electric field distribution at x-z plane..61Figure 4.12a’ A 3D view of the electric field distribution at x-z plane.61Figure 4.12b Electric field distribution at y-z plane..62Figure 4.12c Poynting vector in z-direction.63Figure 4.12d Magnetic field profile of the guided LRSPPs mode.63Figure 4.12e Magnetic field distribution at the x-z plane.64Figure 4.12f A 3D view of Hy.64Figure 4.12g Electric field (upper) and phase (lower) distribution at the end of the structure.65Figure 4.13 Distribution of electric field at 5nm far from the apex.66Figure 4.14 Electric field profiles of Ez and Ex components at the apex.66Figure 4.15 Dependence of magnification factor and beam side on the outer cladding thickness67Figure 4.16 Dependence of magnification factor and beam side on the exponential coefficient.68Figure 4.17 Dependence of magnification factor and beam side on the tip width.69Figure 4.18 Dependence of magnification factor and beam side on the central film thickness.70List of SymbolsSymbolsExplanationEElectric FieldHMagnetic FieldBMagnetic Flux DensityDDielectric DisplacementJCurrent DensityJextExternal Current Densityλ_0Excitation wavelengthεDielectric ConstantμRelative PermeabilityρCharge Densityρ_extExternal Charge DensityσConductivityχDielectric Susceptibilityk0Wave vectorβPropagation Constant List of AbbreviationsAbbreviationOriginal termSPPsSurface Plasmon PolaritonsLRSPPsLong Range Surface Plasmon PolaritonsFDTDFinite Difference Time DomainBEMBoundary Element MethodFEMFinite Element MethodSNOMsScanning Near-Field Optical Microscopes IntroductionSurface plasmon polaritons (SPPs) are tranverse magnetic polarized optical surface waves that propagate along an interface between a dielectric and a conductor and it exponentially decays in the perpendicular direction. These electromagnetic excitations are excited by coupling of an incident electromagnetic field to oscillations of the conductor’s free electrons. In a metal slab comprised of a sufficiently thin metal film embedded in dielectrics, bound SPPs modes at the upper metal- dielectric interfaces couples to that of the lower one forming two bound super-modes. One of these coupled modes has lower attenuation as the metal film thickness is reduced. It is called long- range surface plasmon polaritons (LRSPPs).In recent explosive progress in nanometric optics, the strong concentration of optical energy, which has been based on the great localization of surface plasmon waves in nanostructures, provides people with a great ability to manipulate substance at nanoscale. It was proposed that smoothly tapered metal plasmonic waveguides focused light energy at its tip due to resonant properties of metal nanoparticles [9-22]. Furthermore, conical dielectric waveguide structures with metal-coating have asserted their ability in nanofocusing with great field enhancement and ultra-high energy confinement based on the constructive interference of surface waves. And in most cases, a transverse magnetic radially polarized optical wave was used due to surface plasmon polaritons excitation and the convergence of photons at the tips of the structures. Recently, some scientits have proposed an excitation method for the localization of photons at the apex of a gold- coated axicon prism. An enhanced spot was generated 5 nm below the apex with a magnification factor of 120 times and confined within 35 nm for an incident wavelength of 632.8 nm. Even though the results are quite impressed, that method consumes high excitation energy and provides quite low field enhancement and large beam side. This thesis proposes another tapered dielectric plasmonic waveguide structure for the convergence of long-range surface plasmon polaritons at the tip of the waveguide in the near infrared (λ0 = 1550 nm). It requires low excitation energy but provides with extreme high field enhancement and small beam side at 5 nm far from the apex.The research bases on the results of a number of computational simulations. Currently, there are some methods to solve electromagnetic problems such as BEM, FDTD. Because of the advantages in near field observation, FDTD is chose to implement the simulations, with a tool named OptiFDTD. In this thesis, all the simulations were executed at 2D level. This thesis consists of 4 chapters, in which the first chapter indicates basic literature of surface plasmon polaritons and methods of excitation. Whereas, chapter 2 illustrates demand for nanofocusing and shows some external results, which support the final design. Next, chapter 3 shows some basic literature of finite difference time domain method used in simulations. Finally, chapter 4 shows output data of the simulation and discussions about these results.

HANOI UNIVERSITY OF SCIENCE AND TECHNOLOGY

SCHOOL OF ELECTRONICS AND TELECOMMUNICATIONS

GRADUATE THESIS

TOPIC:

PLASMONICS NANOFOCUSING BASED ON

LONGRANGE SURFACE PLASMON

POLARITONS

Student: NGUYỄN NGỌC AN

Class No.8 – K51 Supervisor: Professor ĐÀO NGỌC CHIẾN, Ph.D

Hanoi, May 2011

HANOI UNIVERSITY OF SCIENCE AND TECHNOLOGY

SCHOOL OF ELECTRONICS AND TELECOMMUNICATIONS

GRADUATE THESIS

TOPIC:

PLASMONICS NANOFOCUSING BASED ON

LONGRANGE SURFACE PLASMON

Student: NGUYỄN NGỌC AN

Class No.8 – K51 Supervisor: Professor ĐÀO NGỌC CHIẾN, Ph.D

Hanoi, May 2011

PURPOSE OF GRADUATE THESIS

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Preface

The research presented in this thesis aims to discover the responses of gold to the excitation in a proposed waveguide structure based on a phenomenon excavated in the time of the Holy Roman Empire It is also the result of intensive efforts to create

a potential application for those properties on the current developing field of nanometric optics

This thesis has been completed in one year after I came to the Computational Electromagnetic R&D Laboratory By that time, I have worked with a number of great people who contributed in many ways to the research I would like to thank all

of them for helping and inspiring me during my graduate study

At the first place, I especially want to thank my supervisor, Prof Dao Ngoc Chien, for his essential guidance and endless encouragements during my research and study

at Hanoi University of Science and Technology His perpetual energy and enthusiasm in research had motivated all of his students, including me In addition,

he was always accessible and willing to help his students with their research in various ways As a result, research life became smooth and rewarding for me

I gratefully acknowledge Hoang Van Son for his advice and important supports which made the research much easier I would like to give him the deep thanks for his kind consideration on me

All my lab buddies at the Computational Electromagnetic R&D Laboratory made it

a convivial place to work In particular, I would like to thank Quang Ngoc Hieu and Nguyen Cong Anh for their constructive comments and helps All other folks had inspired me in research and life through our interactions during the long hours in the lab

Finally, my deepest gratitude goes to my family for their eternal love and support throughout my life

Abstract

In this research, I propose an excitation method for the convergence of range surface plasmon polaritons at the tip of a 2D tapered waveguide in the near infrared (λ0 = 1550 nm) The waveguide structure comprises of three thin metal films

long-of finite width which are embedded in dielectrics in order to create both asymmetric and symmetric metal slab waveguides which support long-range surface plasmon polartitons (LRSPPs)

The 20 nm width Au film with complex permittivity εr,Au,20 nm = -111.2558 - j17.46 and two different dielectrics, of which the refractive indices are n1 = 1.453 and n2 =

4 respectively, are used to implement the structure Based on the end-fire coupling technique with an incident plane wave source, long range surface plasmon polaritons are excited and propagate along the thin Au films then converge drastically at the tip

of the tapered waveguide by constructive interferences causing an extreme field enhancement The simulation was implemented using finite difference time domain method

Table of Contents

Preface 1

Abstract 2

List of Figures 5

List of Symbols 7

List of Abbreviations 8

Introduction 9

Chapter 1 Basics of Surface Plasmon Polaritons 11

1.1 Maxwell‟s Equations 11

1.2 Surface Plasmon Polaritons at Metal-Insulator Interfaces 13

1.2.1 The Wave Equation of TM Mode and TE Mode 13

1.2.2 Surface Plasmon Polaritons at a Single Interface 17

1.2.3 Multilayer Systems and the Long Ranging Modes 19

1.3 Excitation Methods 23

1.3.1 Prism Coupling 23

1.3.2 Grating Coupling 25

1.3.3 End-fire Coupling Excitation 27

Chapter 2 Demand for nanofocusing, an idea and supporting results 29

2.1 The Increasing Demand of Nanofocusing and an Idea 29

2.2 Supporting Results 30

2.2.1 Extended Long Range Surface Plasmon Polaritons 30

2.2.2 Apertureless Nanotip 31

2.2.3 Plasmonic Nanofocusing Based on a Metal Coated Axicon Prism 32

Chapter 3 FDTD simulation 35

3.1 FDTD Simulation 35

3.2 Numerical Dispersion 41

3.3 Numerical Stability – The Courant-Friedrichs-Lewy Stability Criterion 42

3.4 Perfect Electric Conductors 43

3.5 Dielectric-Dielectric Interface 43

3.6 Terminating The Simulation Domain 46

Chapter 4 Waveguide structure, simulation results and discussions 53

4.1 Centre symmetric longrange plasmonics waveguide .53

4.2 Asymmetric MDM waveguide for long-range surface wave excitation .54

4.3 Combination of centre symmetric MDM and two side-asymmetric plasmonic waveguide 56

4.4 Waveguide Structure 59

4.5 Simulation results 61

4.3 The Dependence of the Field Magnification Factor and the Beam Side on Parameters 67

Conclusion 71

References 72

List of Figures

Figure 1.1 Definition of a planar wave guide geometry .15

Figure 1.2 Geometry for SPP propagation at a single interface between a metal and a dielectric .18

Figure 1.3 Geometry of a three-layer system consisting of a thin layer I 20

Figure 1.4 Dispersion relation of the coupled odd and even modes .22

Figure 1.5 Prism coupling to SPPs using attenuated total internal reflection 24

Figure 1.6 Prism coupling and SPP dispersion .24

Figure 1.7 Phase-matching of light to SPPs using a grating .26

Figure 1.8 End-fire coupling technique for exciting SPPs .27

Figure 2.1 Scheme of the analyzed apertureless SNOM probe [8] .32

Figure 2.2 Schematic for the localization of photons by a metal-coated axicon prism [10] 33

Figure 2.3 Intensity distributions on a gold-coated axicon prism for radially polarized incident light [10] .34

Figure 3.1 Discretization of the model into cubes and the position of field .37

Figure 3.2 Basic flows for implementation of Yee FDTD scheme .38

Figure 3.3 The three-dimensional computational region for FDTD simulation .41

Figure 3.4 An example with PEC on the top surface of Cube (i, j, k) .43

Figure 3.5 Four adjacent cubes with different permittivity and conductivity .44

Figure 3.6 A closed loop C crossing four cubes with different permittivity and conductivity .44

Figure 3.7 Boundary E field component between two layers of different dielectrics .52

Figure 4.1 Centre symmetric waveguide structure 53

Figure 4.2 A 3D view of longrange surface plasmon polariton excited in a MDM structure .54

Figure 4.3 A 2D view of longrange surface plasmon excited in a MDM structure .54

Figure 4.4 An asymmetric plasmonic waveguide structure .55

Figure 4.5 A 3D view of LRSPPs excitation in an asymmetric structure .55

Figure 4.6 A 2D view of LRSPPs excitation in an asymmetric structure .56

Figure 4.6 A combination of a symmetric structure with two asymmetric waveguides .56

Figure 4.7 Electric field distribution at x-z plane 57

Figure 4.8 A 2D view of the electric field distribution at x-z plane 57

Figure 4.9a H y at the x-z plane .58

Figure 4.9b Magnetic field distribution at x-z plane .58

Figure 4.11 Schematic of the simulated model with an incident wavelength of 1550 nm .59

Figure 4.12 Electric field distribution at x-z plane .61

Figure 4.12a‟ A 3D view of the electric field distribution at x-z plane .61

Figure 4.12b Electric field distribution at y-z plane .62

Figure 4.12c Poynting vector in z-direction .63

Figure 4.12d Magnetic field profile of the guided LRSPPs mode .63

Figure 4.12e Magnetic field distribution at the x-z plane .64

Figure 4.12f A 3D view of H y 64

Figure 4.12g Electric field (upper) and phase (lower) distribution at the end of the structure .65

Figure 4.13 Distribution of electric field at 5nm far from the apex .66

Figure 4.14 Electric field profiles of Ez and Ex components at the apex .66

Figure 4.15 Dependence of magnification factor and beam side on the outer cladding thickness 67

Figure 4.16 Dependence of magnification factor and beam side on the exponential coefficient 68

Figure 4.17 Dependence of magnification factor and beam side on the tip width .69

Figure 4.18 Dependence of magnification factor and beam side on the central film thickness .70

List of Abbreviations

Abbreviation Original term

Introduction

Surface plasmon polaritons (SPPs) are tranverse magnetic polarized optical surface waves that propagate along an interface between a dielectric and a conductor and it exponentially decays in the perpendicular direction These electromagnetic excitations are excited by coupling of an incident electromagnetic field to oscillations of the conductor‟s free electrons In a metal slab comprised of a sufficiently thin metal film embedded in dielectrics, bound SPPs modes at the upper metal- dielectric interfaces couples to that of the lower one forming two bound super-modes One of these coupled modes has lower attenuation as the metal film thickness is reduced It is called long- range surface plasmon polaritons (LRSPPs)

In recent explosive progress in nanometric optics, the strong concentration of optical energy, which has been based on the great localization of surface plasmon waves in nanostructures, provides people with a great ability to manipulate substance at nanoscale It was proposed that smoothly tapered metal plasmonic waveguides focused light energy at its tip due to resonant properties of metal nanoparticles [9-22] Furthermore, conical dielectric waveguide structures with metal-coating have asserted their ability in nanofocusing with great field enhancement and ultra-high energy confinement based on the constructive interference of surface waves And in most cases, a transverse magnetic radially polarized optical wave was used due to surface plasmon polaritons excitation and the convergence of photons at the tips of the structures

Recently, some scientits have proposed an excitation method for the localization of photons at the apex of a gold- coated axicon prism An enhanced spot was generated 5 nm below the apex with a magnification factor of 120 times and confined within 35 nm for an incident wavelength of 632.8 nm Even though the results are quite impressed, that method consumes high excitation energy and provides quite low field enhancement and large beam side

This thesis proposes another tapered dielectric plasmonic waveguide structure for

the convergence of long-range surface plasmon polaritons at the tip of the waveguide in the near infrared (λ0 = 1550 nm) It requires low excitation energy but provides with extreme high field enhancement and small beam side at 5 nm far from the apex

The research bases on the results of a number of computational simulations Currently, there are some methods to solve electromagnetic problems such as BEM, FDTD Because of the advantages in near field observation, FDTD is chose to implement the simulations, with a tool named OptiFDTD In this thesis, all the simulations were executed at 2D level

This thesis consists of 4 chapters, in which the first chapter indicates basic literature

of surface plasmon polaritons and methods of excitation Whereas, chapter 2 illustrates demand for nanofocusing and shows some external results, which support the final design Next, chapter 3 shows some basic literature of finite difference time domain method used in simulations Finally, chapter 4 shows output data of the simulation and discussions about these results

Chapter 1

Basics of Surface Plasmon Polaritons

This chapter presents some principles of surface waves, especially long range

surface plasmon polaritons and excitation methods.

1.1 Maxwell’s Equations

It is widely known that for frequencies up to the visible part of the spectrum metals are highly reflective and do not allow electromagnetic waves to propagate through them Metals have been employed as cladding layers for the construction of waveguides and resonators for electromagnetic radiation at microwave and far-infrared frequencies for a long time

At low frequencies, the perfect conductor approximation of infinite or fixed finite conductivity is valid for most purposes, because only a negligible fraction of the impinging electromagnetic waves penetrates into the metal At higher frequencies towards the near-infrared and visible part of the spectrum, field penetration increases remarkably, causing increased dissipation Finally, at ultra violet frequencies, metals express dielectric character and allow the propagation of electromagnetic waves, albeit with varying degrees of attenuation depending on the details of the electronic band structure For noble metals such as gold or silver on the other hand, transitions between electronic bands lead to strong absorption at these frequencies

The interaction of metals with electromagnetic field can be understood in a classical way based on the macroscopic Maxwell equations [1] Maxwell‟s equations of macroscopic electromagnetism are taken in the following form:

induction or magnetic flux density) with the external charge and current densities

ρext and Jext[1]

These macroscopic equations are not explained normally via dividing the total

charge and current densities ρtot and Jtot into free and bound sets, which is an arbitrary division and can (especially in the case of metallic interfaces) confuse the application of the boundary condition for the dielectric displacement Instead, we

distinguish between external (ρext, Jext) and internal (ρ,J) charge and current

densities, so that in total ρtot = ρext + ρ and Jtot = Jext + J The external set drives the

system, while the internal set responds to the external stimuli The four macroscopic

fields are further linked via the polarization P and magne-tization M by

consider a magnetic response represented by M, but can limit our description to

electric polarization effects P describes the electric dipole moment per unit volume

inside the material, caused by the alignment of microscopic dipoles with the electric field It is related to the internal charge density via ∇ · 𝑷 = −𝜌 Charge conservation (∇ · 𝑱 = −𝜕𝜌/𝜕𝑡) further requires that the internal charge and current densities are linked via

𝑱 = 𝜕𝑷

The strong advantage of this approach is that the macroscopic electric field includes all polarization effects: in other words, both the external and the induced fields are absorbed into it This can be shown via inserting (1.2a) into (1.1c), leading to

ε is called the dielectric constant or relative permittivity and μ = 1 is the relative

permeability of the nonmagnetic medium The linear relationship (1.5a) between D and E is often also implicitly defined using the dielectric susceptibility χ which describes the linear relationship between P and E via

1.2 Surface Plasmon Polaritons at Metal-Insulator Interfaces

Surface plasmon polaritons are electromagnetic excitations propagating at the interface between a dielectric and a conductor, evanescently confined in the perpendicular direction These electro-magnetic surface waves arise via the coupling of the electromagnetic fields to oscillations of the conductor‟s electron plasma This section describes the fundamentals of surface plasmon polaritons both

at single, flat interfaces and in metal- dielectric multilayer structures

1.2.1 The Wave Equation of TM Mode and TE Mode

In order to investigate the physical properties of surface plasmon polaritons (SPPs), Maxwell‟s equations (1.1) needs to be applied to the flat interface between a conductor and a dielectric It is an advantage to cast the equations first in a general

form applicable to the guiding of electromagnetic waves, the wave equation In the absence of external charge and current densities, the curl equations (1.1c, 1.1d) can

be combined to yield

∇ × ∇ × 𝑬 = − µ0𝜕

2𝑫

Using the identities ∇×∇× E ≡ ∇ (∇· E) −∇2E as well as ∇· (εE) ≡ E ·∇ε + ε∇· E,

and remembering that due to the absence of external stimuli ∇· D = 0, (1.8) can be rewritten as

∇(−1

ε𝑬 ∇𝜀) − ∇2𝑬 = − µ0ε0ε∂

2𝑬

For negligible variation of the dielectric profile ε = ε(r) over distances on the order

of one optical wavelength, (1.9) simplifies to the central equation of electromagnetic wave theory,

To cast (1.10) in a form suitable for the description of confined propagating waves,

we proceed in two steps First, we assume in all generality a harmonic time

dependence E(r,t) = E(r)e−iωt of the electric field Inserted into (1.10), this yields

where k0 = ω/c is the wave vector of the propagating wave in vacuum Equation (1.11) is known as the Helmholtz equation

Next, the propagation geometry need to be defined We assume for simplicity a

one-dimensional problem ε depends only on one spatial coordinate Specifically, the

waves propagate along the x-direction of a Cartesian coordinate system, and show

no spatial variation in the perpendicular, in-plane y-direction (see Fig.2.1); therefore

ε = ε(z) Applied to electromagnetic surface problems, the plane z = 0 coincides

with the interface sustaining the propagating waves, which can now be described as

E(x,y,z) = E(z)e iβx

Figure 1.1 Definition of a planar wave guide geometry The waves propagate along the x-direction

in a cartesian coordinate system

The complex parameter β = kx is called the propagation constant of the traveling waves and corresponds to the component of the wave vector in the direction of propagation Inserting this expression into (1.11) yields the desired form of the wave equation

∂2𝑬(𝑧)

Likewise, a similar equation exists for the magnetic field H

Equation (1.12) is the beginning point for the general analysis of guided electromagnetic modes in waveguides In order to use the wave equation for calculate the spatial field profile and dispersion of propagating waves, we now need

to find explicit expressions for the different field components of E and H This can

be achieved in a direct way using the curl equations (1.1c, 1.1d)

For harmonic time dependence (𝜕/𝜕𝑡 = - iω), propagation along the x-direction (𝜕/𝜕𝑥 =- iβ) and homogeneity in the y-direction (𝜕/𝜕𝑦 = 0), we arrive at the following set of equations

iβH y = - iωε 0 εE z (1.13f)

It can easily be shown that this system allows two sets of self-consistent solutions with different polarization properties of the propagating waves The first set are the

transverse magnetic (TM or p) modes, where only the field components Ex, Ez and

Hy are non zero, and the second set the transverse electric (TE or s) modes, with

only Hx, Hz and Ey being non zero

For TM modes, the system of governing equations (1.13) reduces to

𝜕2𝑬(𝑦)

1.2.2 Surface Plasmon Polaritons at a Single Interface

The most simple geometry sustaining SPPs is that of a single, flat interface (Fig.1.2) between a dielectric, non-absorbing half space (z> 0) with positive real dielectric constant ε2 and an adjacent conducting half space (z< 0) described via a dielectric function ε1(ω) The requirement of metallic character implies that Re [ε1] < 0 For metals this condition is fulfilled at frequencies below the bulk plasmon frequency

E x (z)= -iA 1 (ωε 0 ε 1 ) -1 k 1 e iβx e k 1 z (1.17b)

E z (z)=- A 1 β(ωε 0 ε 1 ) -1 eiβxek 1 z (1.17c) for z < 0

ki ≡ kz,i (i = 1, 2) is the component of the wave vector perpendicular to the interface

in the two media Its reciprocal value, 𝐳 = 1/ |k z|, defines the evanescent decay length of the fields perpendicular to the interface, which quantifies the confinement

of the wave

Continuity of Hy and εi Ez at the interface requires that A1 = A2 and

Figure 1.2 Geometry for SPP propagation at a single interface between a metal and a dielectric

Note that with the convention of the signs in the exponents in (1.16, 1.17), confinement to the surface demands Re [ε1] < 0 if ε2 > 0 – the surface waves exist only at interfaces between materials with opposite signs of the real part of their dielectric permittivity, i.e between a conductor and an insulator The expression for

Hy further has to fulfill the wave equation (1.14c), yielding

H x (z)= -iA 2 (ωµ 0 ) -1 k 2 e iβx e -k 2 z (1.21b)

H z (z)= A 2 β(ωµ 0 ) -1 eiβxe-k 2 z (1.21c) for z > 0 and

H x (z)= iA 1 (ωµ 0 ) -1 k 1 e iβx e k 1 z (1.22b)

H z (z)= A 1 β(ωµ 0 ) -1 eiβxek 1 z (1.22c)

for z < 0 Continuity of Ey and Hx at the interface leads to the condition

Since confinement to the surface requires Re [k1] > 0 and Re [k2] > 0, this condition

is only fulfilled if A1 = 0, so that also A2 = A1 = 0.Thus, no surface modes exist for

TE polarization Surface plasmon polaritons only exist for TM polarization

1.2.3 Multilayer Systems and the Long Ranging Modes

Multilayer systems consist of alternating conducting and dielectric thin films In such a system, each single interface can sustain bound SPPs When the separation between adjacent interfaces is comparable to or smaller than the decay length z of the interface mode, interactions between SPPs give rise to coupled modes In order

to investigate the general properties of coupled SPPs, we will focus on two specific three layer systems of the geometry illustrated in Fig.1.3: Firstly, a thin metallic layer (I) sandwiched between two infinitely thick dielectric claddings (II, III), an insulator/metal/insulator (IMI) hetero structure, and secondly a thin dielectric core layer (I) sandwiched between two metallic claddings (II, III), a metal/insulator/metal (MIM) hetero structure Since we are here only interested in the lowest- order bound modes, we start with a general description of TM modes that are non-oscillatory in the z-direction normal to the interfaces using (1.14) For z>a, the field components are

Figure 1.3 Geometry of a three-layer system consisting of a thin layer I

sandwiched between two infinite half spaces II and III

Thus, we demand that the fields decay exponentially in the claddings (II) and (III) Note that for simplicity as before we denote the component of the wave vector perpendicular to the interfaces simply as ki ≡ kz,i In the core region −a<z<a, the modes localized at the bottom and top interface couple, yielding

H y = Ce iβx e k 1 z + De iβx e -k 1 z (1.26a)

E x = -iC(ωε 0 ε 1 ) -1 k 1 e iβx e k 1 z + iC(ωε 0 ε 1 ) -1 k 1 e iβx e -k 1 z (1.26b)

for i = 1, 2, 3 Solving this system of linear equations results in an implicit

expression for the dispersion relation linking β and ω via

are equal in terms of their dielectric response, i.e ε 2 = ε 3 and thus k 2 = k 3 In this case, the dispersion relation (1.30) can be split into a pair of equations, namely

tanh𝑘1𝑎 = −𝑘2𝜀1

tanh𝑘1𝑎 = −𝑘1𝜀2

It can be shown that equation (1.31a) describes modes of odd vector parity (Ex (z) is

odd, Hy(z) and Ez(z) are even functions), while (1.31b) describes modes of even

vector parity (Ex(z) is even function, Hy(z) and Ez(z) are odd)

The dispersion relations (1.31a, 1.31) can now be applied to IMI and MIM structures to investigate the properties of the coupled SPP modes in these two systems We first start with the IMI geometry-a thin metallic film of thickness 2a sandwiched between two insulating layers In this case ε1 = ε1(ω) represents the dielectric function of the metal, and ε2 the positive, real dielectric constant of the insulating sub-and superstrates As an example, Fig.1.4 shows the dispersion relations of the odd and even modes (1.31a, 1.31b) for an air/silver/air geometry for two different thicknesses of the silver thin film For simplicity, here the dielectric function of silver is approximated via a Drude model [12] with negligible damping

(ε(ω) real and ε(ω) = ω 2 p / ω 2

), so that Im[β]= 0

Figure 1.4 Dispersion relation of the coupled odd and even modes for an air/silver/air multilayer with a metal core of thickness 100nm (dashed gray curves) and 50 nm (dashed black curves)

As can be seen, the odd modes have frequencies ω+ higher than the respective frequencies for a single interface SPP, and the even modes lower frequencies ω−

For large wave vectors β (which are only achievable if Im[ε(ω)] = 0), the limiting

absorptive metals described via a complex ε(ω), this implies a drastically increased

SPP propagation length, called long range surface plasmon polaritons [2] The even modes show the reverse behavior-their confinement to the metal increases with decreasing metal film thickness, resulting in a decline in propagation length

To summarize, we see that surface plasmon polaritons can be excited by TM polarized wave on the surface between a dielectric and a conductor In a metal slab waveguide, when the gap between the two metal/dielectric interfaces is thin enough, surface waves at both side couple together creating supermodes, with the odd mode called long range surface plasmon polaritons [3]

1.3 Excitation Methods

Surface plasmon polaritons propagating at the interface between a conductor and a dielectric are essentially two-dimensional electromagnetic waves Confinement is

achieved because the propagation constant β is greater than the wave vector k in the

dielectric, causing to evanescent decay on both sides of the interface The SPP dispersion curve therefore lies to the right of the light line of the dielectric (given by

ω = ck), and excitation by three-dimensional light beams is not possible unless special techniques for phase-matching are employed Alternatively, thin film geometries such as insulator-metal-insulator hetero structures sustaining weakly confined SPPs are amenable to end-fire coupling, relying on spatial mode-matching rather than phase-matching

This section reviews the most common techniques for SPP excitation Various optical techniques for phase-matching such as prism and grating coupling as well as excitation using highly focused beams will be presented Wave vectors in excess of

|k| can also be achieved using illumination in the near-field, making use of evanescent waves in the immediate vicinity of a sub-wavelength aperture The section closes with a brief look at the excitation of SPPs using optical fiber tapers or end-fire excitation This allows coupling of SPPs to modes in conventional dielectric waveguides

1.3.1 Prism Coupling

Surface plasmon polaritons on a flat metal/dielectric interface cannot be excited

directly by light beams since β>k, where k is the wave vector of light on the

dielectric side of the interface Therefore, the projection along the interface of the momentum kx = k sin θ of photons impinging under an angle θ to the surface normal

is always smaller than the SPP propagation constant β, even at grazing incidence,

prohibiting phase-matching

Figure 1.5 Prism coupling to SPPs using attenuated total internal reflection in the Kretschmann

(left) and Otto (right) configuration.

However, phase-matching to SPPs can be achieved in a three-layer system consisting of a thin metal film sandwitched between two insulators of different dielectric constants For simplicity, we will take one of the insulators to be air (ε = 1) A beam reflected at the interface between the insulat or of higher dielectric constant ε, usually in the form of a prism (see Fig.1.5), and the metal will have an in-plane momentum kx = k 𝜀 sin θ, which is sufficient to excite SPPs at the interface between the metal and the lower-index dielectric, in this case at the metal/air interface

Figure 1.6 Prism coupling and SPP dispersion

This way, SPPs with propagation constants β between the light lines of air and the

higher index dielectric can be excited (Fig.1.6) SPP excitation manifests itself as a

minimum in the reflected beam intensity Note that phase-matching to SPPs at the prism/metal interface cannot be achieved, since the respective SPP dispersion lies outside the prism light cone (Fig.1.6).

This coupling scheme-also known as attenuated total internal reflection- therefore involves tunneling of the field of the excitation beam to the metal/air interface where SPP excitation takes place Two different geometries for prism coupling are possible, depicted in Fig.1.5 The most common configuration is the Kretschmann method, in which a thin metal film is evaporated on top of a glass prism Photons from a beam impinging from the glass side at an angle greater than the critical angle

of total internal reflection tunnel through the metal film and excite SPPs at the metal/air interface Another geometry is the Otto configuration, in which the prism

is separated from the metal film by a thin air gap Total internal reflection takes place at the prism/air interface, exciting SPPs via tunneling to the air/metal interface This configuration is preferable when direct contact with the metal surface

is undesirable, for example for studies of surface quality We want to stress that

SPPs excited using phase-matching via β = k 𝜀 sin θ are inherently leaky waves, i.e they lose energy not only due to the inherent absorption inside the metal, but also due to leakage of radiation into the prism: the excited propagation constants lie within the prism light cone (Fig.1.6) The minimum in the intensity of the reflected beam is due to destructive interference between this leakage radiation and the reflected part of the excitation beam For an optimum metal film thickness, the destructive interference can be perfect, providing a zero in the reflected beam intensity, so that leakage radiation cannot be detected

1.3.2 Grating Coupling

The mismatch in wave vector between the in-plane momentum kx = k sin θ of

impinging photons and β can also be overcome by patterning the metal surface with

a shallow grating of grooves or holes with lattice constant a For the simple dimensional grating of grooves depicted in Fig.1.7, phase-matching takes place whenever the condition

one-𝛽 = 𝑘 𝑠𝑖𝑛 𝜃 ± 𝑣 𝑔 (1.33)

is fulfilled, where g = 2π a is the reciprocal vector of the grating, and ν =(1, 2, 3 )

As with prism coupling, excitation of SPPs is detected as a minimum in the reflected light

Figure 1.7 Phase-matching of light to SPPs using a grating

The reverse process can also take place: SPPs propagating along a surface modulated with a grating can couple to light and thus radiate The gratings need not

be milled directly into the metal surface, but can also consist of dielectric material

By designing the shape of the grating, the propagation direction of SPPs can be influenced and even focusing can be achieved

For one-dimensional gratings, significant changes to the SPP dispersion relation occur if the gratings are sufficiently deep so that the modulation can no longer be treated as a small perturbation of the flat interface Appreciable band gaps appear already for a groove depth on the order of 20nm for metallic gratings For even larger depths, localized modes inside the grooves lead to distortions of the first higher-order band folded back at the Brillouin zone boundary, enabling coupling even for short pitches a < λ/2 upon normal incidence due to a lowering in frequency

of the modified SPP dispersion curve

More generally, SPPs can also be excited on films in areas with random surface roughness or manufactured localized scatterers Momentum components∆kx are provided via scattering, so that the phase-matching condition

can be fulfilled The efficiency of coupling can be assessed by for example measuring the leakage radiation into a glass prism situated underneath the metal film

1.3.3 End-fire Coupling Excitation

While the other optical excitation schemes are suitable for the investigation of SPP propagation and functional plasmonic structures in proof of concept characterizations, practical applications of SPPs in integrated photonic circuits will require high efficiency (and ideally high-bandwidth) coupling schemes Preferably, the plasmonic components should allow efficient matching with conventional dielectric optical waveguides and fibers, which would in such a scenario be used to channel energy over large distances to plasmon waveguides and cavities The latter will then enable high-confinement guiding and localized field enhancement, for example for the routing of radiation to single molecules [3][15]

Figure 1.8 End-fire coupling technique for exciting SPPs

One such coupling scheme is end-fire coupling, in which a free-space optical beam

is focused on the end-facet of the desired waveguide Rather than relying on

phase-matching, this scheme operates via matching the spatial field distribution of the

waveguide as much as possible by adjusting the beam width For SPPs propagating

at a single interface, coupling efficiencies up to 90% was demonstrated using this

technique In contrast to prism coupling, end-fire excitation allows for the excitation

of truly bound modes that do not radiate into the substrate End-fire coupling is also

particularly useful and efficient for exciting the long-ranging SPP mode propagating

along thin metal films embedded in a symmetric dielectric host Due to the

delocalized nature of this mode, spatial mode matching works especially well in this

case

Naturally however, for geometries showing field-localization below the diffraction

limit such as metal/insulator/metal waveguides with a deep sub-wavelength

dielectric core, the overlap between the excitation beam and the coupled SPP mode

is very small, leading to low excitation efficiencies For SPPs with larger

confinement, a convenient interfacing scheme makes use of optical fiber tapers

brought into the immediate vicinity of the waveguide to enable phase matched

power transfer via evanescent coupling

In conclusion, we see that surface plasmon polaritons can be excited by TM

polarized wave on the surface between a dielectric and a conductor In a metal slab

waveguide, when the gap between the two metal/dielectric interfaces is thin enough,

surface waves at both side couple together creating supermodes, with the odd mode

called long range surface plasmon polaritons Finally, end-fire coupling techniques

emerges as the most effective one among LRSPPs excitation methods

2.1 The Increasing Demand of Nanofocusing and an Idea

Vigorous interest in LRSPPs has stimulated a large number of studies over three decades showing the LRSPP in diverse phenomena, including nonlinear interactions, molecular scattering, fluorescence, surface-enhanced Raman spectroscopy, transmission through opaque metal films and emission extraction, amplification and lasing, surface characterization, metal roughness and islandization, optical interconnects and integrated structures, gratings, thermo-, electro-and magneto-optics, and (bio) chemical sensing [7,14,15]

Progress in nano-optics is connected with increasing interest in the underestimated role, up to not long ago, of evanescent fields and SPPs in imaging and energy guiding along metal nanostructures Local electromagnetic field enhancement due

to SPPs decides about resolution of nanoscale investigations of biomedical and chemical molecules and surface enhanced Roman scattering This progress is conditioned by the development of several nanotechnologies that allow fabrication

of metal and dielectric nanostructures as well as patterning, etching and drilling techniques with nanometer accuracy

The use of evanescent waves decides that the classical diffraction limit of resolution

to about half a wavelength does not restrict scanning near-field and tunneling optical microscopes [8, 9, 10, 11, 14, 15] To a great extent it is possible, due to advantageous use of SPPs in light transmission and technical means, to concentrate

optical energy to nanometer size spots Efforts to develop easy to fabricate nanotips for scanning near-field optical microscopes (SNOMs) continue since the mid-eighties of the previous century In near-field microscopy the narrower is the confinement of scanning light the better is the resolution For SNOMs with aperture probes a practical limit of the resolution is the sum of the diameter of the aperture of

a tapered-fiber metal-coated tip and twice the skin depth of the metal coating, what corresponds to light penetration into metal layers on both sides of the aperture Understanding of charge densities and current distributions on the metal rim of the aperture of a tip suggested that a way to improve resolution of scanning microscopes is possible due to enhancement of energy throughput of aperture probes Several methods aiming at this were recently proposed, e.g Another approach to SNOM resolution improvement is the use of apertureless metallic or metal-coated tips for nanofocusing of light, which with efficient light coupling may offer better resolution

Thus, actually, there is an increasing demand of focusing optical radiation energy at nanoscale which is also the central formidable problem of nanometric optics

2.2.1 Extended Long Range Surface Plasmon Polaritons

Based on what mentioned above, surface plasmon waves can only be supported along the boundary of two optical materials with opposite signs of the real parts of the dielectric constants, such as a dielectric and a metal Surface plasmon waves attenuate rapidly during the propagation due to the intrinsic electron oscillation damping loss in the metal

A thin metal film embedded in a homogeneous dielectric medium with a symmetric structure supports two guided plasmon wave modes [4] One mode, which has the symmetric field profile with respect to the center of the metal layer, is called the symmetric mode, and has has a relatively long propagation distance compared with the surface plasmon wave along the surface of the bulk metal in a dielectric medium This symmetric surface plasmon wave mode is also called the long range surface plasmon (LRSP) mode

The fundamental reasons that thin metal films (or strips) in symmetric structures can support long range plasmon waves are: (1) most energy is located in the surrounding dielectric material for thin metal films; (2) the surface plasmon waves associated with two boundaries travel with the same velocity due to the symmetry

of the guide structure

Although long range plasmon wave modes can be supported by thin metal films in homogeneous dielectric media with symmetric structures, the long range modes still attenuate due to the electron oscillation damping loss in the metal To increase the travel distance of the long range plasmon mode, a simple strategy is to reduce the metal layer thickness [6] But it is experimentally difficult to deposit homogeneous metal film of less than 15 nm thickness because metals, such as gold, typically form nanoscale islands in the initial deposition process [23,24]

In a brilliant attempt to extend the propagation length of LRSPPs, Junpeng Guo and Ronen Adato [5] found that if we place intermediate dielectric layers with different dielectric constant on both sides of the metal film, the propagation distance of the symmetric long range mode can be extended several orders of magnitude Also we found that when the intermediate dielectric layer thickness approaches certain critical thickness, the propagation distance of one long range goes to infinite

2.2.2 Apertureless Nanotip

Tomasz J Antosiewicz, Piotr Wróbel, and Tomasz Szoplik [7] introduced a novel structure of an apertureless tip: a tapered dielectric fiber with metal coating and dielectric nanocladding (DMD) in 2009 Fig.2.1 shows schematically and not to

scale such a tip: a dielectric core of radius r is tapered at α = 20° half-angle, has a silver coating of thickness d, and a very thin w = 5 nm outer dielectric layer Photon

to plasmon coupling occurs in the tapered part, where metal is sandwiched between two dielectrics In this region asymmetric plasmon modes are generated at both interfaces At a certain core diameter the core mode ceases, but the plasmon at the outer metal surface and polariton in the cladding propagate towards the apex Their wavelength and velocity decrease and evanescent field concentration increases

Figure 2.1 Scheme of the analyzed apertureless SNOM probe [8]

The role of the outer dielectric nanocladding is twofold Chemically, it protects against oxidation of silver, which is used for its low loss plasmon propagation Electromagnetically, it modifies the coupling conditions of light modes propagating

in the core to SPP modes at the metal-nanocladding interface The spectral location

of maximum enhancement at the tip apex can be tuned by choosing the value of nanocladding dielectric constant Moreover, it decreases the plasmon wavelength, what increases the optical distance over which plasmons accumulate and in effect enhances evanescent field concentration Efficient coupling of internal illumination

to outer silver-dielectric interface is due to Laguerre-Gauss mode with radial- polarization guided in the fiber core

2.2.3 Plasmonic Nanofocusing Based on a Metal Coated Axicon Prism

Explosive reseach on this field show that light energy is strongly concentrated by local surface plasmon resonance and a nanofocused spot is generated by

constructive interference of surface plasmon polaritons in conically tapered metal- coated dielectric waveguides [8, 9, 10, 11] Recently, Keisuke Kato, Atsushi Ono, Wataru Inami, and Yoshimasa Kawata have proposed an excitation method for the localization of photons at the apex of a gold- coated axicon prism The cone angle

of the prism and the metallic film thickness are designed to match the excitation conditions for surface plasmons [9] The plasmons propagate along the sides of the prism and converge at its apex

Figure 2.2 Schematic for the localization of photons by a metal-coated axicon prism And Model of the device as simulated by an FDTD algorithm, for He-Ne laser excitation at a

wavelength of 632.8 nm [10]

The resulting nanofocusing was investigated by simulating the intensity distributions around the apex of the prism using a finite-difference time-domain algorithm For incident radial-polarization, a localized and field enhanced spot is generated by the constructive interference of surface plasmons Even though the results are quite impressed, that method consumes high excitation energy and provides quite low field enhancement and large beam side

Figure 2.3 Intensity distributions on a gold-coated axicon prism for radially polarized

incident light [10]

in which (a) shows a vertical (x-z) cross section including the central axis of the prism Field enhancements are observed on the side surface due to excitations of surface plasmons (b) shows enlarged image showing a hot spot generated at the apex (c) presents a horizontal (x-y) cross section 5 nm below the apex, corresponding to the dashed line in (b) (d) illustrates intensity profile along line a-a‟ in (c) The FWHM of 35 nm is obtained The scale bar in each figure is 200 nm long

In order to improve the problem, I propose a special plasmonic waveguide configuration that needs lower input energy (λ0 = 1550 nm) but provides higher throughput, extreme field enhancement and a very narrow penetrating beam (FWHM < 30 nm) based on long- range surface plasmon polaritons