QUEENSLAND UNIVERSITY OF TECHNOLOGY

SCHOOL OF PHYSICAL AND CHEMICAL SCIENCES

APPLIED OPTICS AND NANOTECHNOLOGY PROGRAM

Theoretical and Numerical Investigation of Plasmon Nanofocusing in Metallic Tapered

Rods and Grooves  

Submitted by Michael Werner Vogel, Dipl.-Ing. (FH), to the School of Physical

and Chemical Sciences, Queensland University of Technology, in partial

fulfilment of the requirements of the degree of Doctor of Philosophy.

2009

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Key Words

Near field optics, nano-optics, plasmonics, surface plasmons, localised

surface plasmons, film plasmons, gap plasmons, nanofocusing, adiabatic

nanofocusing, non-adiabatic nanofocusing, local field enhancement, metallic

V-groove, metal tapered rod.

 

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Abstract

Effective focusing of electromagnetic (EM) energy to nanoscale regions is

one of the major challenges in nano-photonics and plasmonics. The strong

localization of the optical energy into regions much smaller than allowed by

the diffraction limit, also called nanofocusing, offers promising applications in

nano-sensor technology, nanofabrication, near-field optics or spectroscopy.

One of the most promising solutions to the problem of efficient nanofocusing

is related to surface plasmon propagation in metallic structures. Metallic

tapered rods, commonly used as probes in near field microscopy and

spectroscopy, are of a particular interest. They can provide very strong EM

field enhancement at the tip due to surface plasmons (SP’s) propagating

towards the tip of the tapered metal rod. A large number of studies have

been devoted to the manufacturing process of tapered rods or tapered fibers

coated by a metal film. On the other hand, structures such as metallic V-

grooves or metal wedges can also provide strong electric field enhancements

but manufacturing of these structures is still a challenge. It has been shown,

however, that the attainable electric field enhancement at the apex in the V-

groove is higher than at the tip of a metal tapered rod when the dissipation

level in the metal is strong. Metallic V-grooves also have very promising

characteristics as plasmonic waveguides.

This thesis will present a thorough theoretical and numerical investigation

of nanofocusing during plasmon propagation along a metal tapered rod and

into a metallic V-groove. Optimal structural parameters including optimal

taper angle, taper length and shape of the taper are determined in order to

achieve maximum field enhancement factors at the tip of the nanofocusing

structure.

An analytical investigation of plasmon nanofocusing by metal tapered rods

is carried out by means of the geometric optics approximation (GOA), which

is also called adiabatic nanofocusing. However, GOA is applicable only for

analysing tapered structures with small taper angles and without considering

a terminating tip structure in order to neglect reflections. Rigorous numerical

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methods are employed for analysing non-adiabatic nanofocusing, by tapered

rod and V-grooves with larger taper angles and with a rounded tip. These

structures cannot be studied by analytical methods due to the presence of

reflected waves from the taper section, the tip and also from (artificial)

computational boundaries. A new method is introduced to combine the

advantages of GOA and rigorous numerical methods in order to reduce

significantly the use of computational resources and yet achieve accurate

results for the analysis of large tapered structures, within reasonable

calculation time.

Detailed comparison between GOA and rigorous numerical methods will

be carried out in order to find the critical taper angle of the tapered structures

at which GOA is still applicable. It will be demonstrated that optimal taper

angles, at which maximum field enhancements occur, coincide with the

critical angles, at which GOA is still applicable. It will be shown that the

applicability of GOA can be substantially expanded to include structures

which could be analysed previously by numerical methods only.

The influence of the rounded tip, the taper angle and the role of dissipation

onto the plasmon field distribution along the tapered rod and near the tip will

be analysed analytically and numerically in detail. It will be demonstrated that

electric field enhancement factors of up to ~ 2500 within nanoscale regions

are predicted. These are sufficient, for instance, to detect single molecules

using surface enhanced Raman spectroscopy (SERS) with the tip of a

tapered rod, an approach also known as tip enhanced Raman spectroscopy

or TERS.

The results obtained in this project will be important for applications for

which strong local field enhancement factors are crucial for the performance

of devices such as near field microscopes or spectroscopy. The optimal

design of nanofocusing structures, at which the delivery of electromagnetic

energy to the nanometer region is most efficient, will lead to new applications

in near field sensors, near field measuring technology, or generation of

nanometer sized energy sources. This includes: applications in tip enhanced

Raman spectroscopy (TERS); manipulation of nanoparticles and molecules;

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efficient coupling of optical energy into and out of plasmonic circuits; second

harmonic generation in non-linear optics; or delivery of energy to quantum

dots, for instance, for quantum computations.

 

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Glossary of Terms and Abbreviations

ATR Attenuated Total Reflection

CPU Computer Processing Unit

DLM Drude-Lorentz Model

EELS Electron Energy Loss Spectrum

EIM Effective Index Method

EM Electromagnetic

FDTD Finite-Difference Time-Domain

FEM Finite Element Method

FIB Focused Ion Beam

FTIR Frustrated Total Internal Reflection

GDT Green’s dyadic technique

GOA Geometric Optics Approximation

IMI Insulator-Metal-Insulator

LRSPP Long Range Surface Plasmon-Polariton

LSP Localized Surface Plasmon

MIM Metal-Insulator-Metal

MoM Method of Moments

MOSFET Metal-Oxide Semiconductor Field-Effect Transistor

NFM Near Field Microscope

NSOM/SNOM Near-field Scanning Optical Microscope

PSTM Photon Scanning Tunnelling Microscope

SERS Surface Enhanced Raman Spectroscopy

SP Surface Plasmon

SPP Surface Plasmon Polaritons

SRR Split Ring Resonators

TE Transverse Electric

TEM Transmission Electron Microscope

TERS Tip Enhanced Raman Spectroscopy

TM Transverse Magnetic

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List of Publications and Manuscripts

Refereed publications

A1. Michael W. Vogel, Dmitri K. Gramotnev, “Adiabatic Nano-focusing in metal tapered

rod in the presence of dissipation”, Physics Letters A, 363, 507-511 (2007)

A2. Dmitri K. Gramotnev, David F. P. Pile, Michael W. Vogel, Xiang Zhang, “Local

Electric Field Enhancement during Nano-focusing of Plasmons by a Tapered Gap”,

Physics Review B, 103, 034304 (2008).

A3. Dmitri K. Gramotnev, Michael W. Vogel, Mark Stockman, “Optimized non-adiabatic

nano-focusing of plasmons by tapered metal rods”, Journal of Applied Physics, 104, 034311(2008)

A4. Michael W. Vogel, Dmitri K. Gramotnev, “Optimization of plasmon nano-focusing in

tapered metal rods”, Journal of Nanophotonics, Vol. 2, 021852 (21. Nov 2008)

A5. Michael W. Vogel, Dmitri K. Gramotnev, “Shape effects on adiabatic nano-focusing

of localized plasmons in tapered metal rods”, to be submitted to Journal of Applied

Physics

Conference publications

1. Michael W. Vogel, Dmitri K Gramotnev. “Does Adiabatic Nano-Focusing in Metallic

Tips Really Exist?” Australasian Conf. on Optics, Lasers and Spectroscopy

(ACOLS), Rotorua, New Zealand (2005), Poster.

2. Michael W. Vogel, Dmitri K. Gramotnev. “Adiabatic Nano-Focusing in Metallic Nano-

Structures in the Presence of Dissipation”, ETOPIM 7, Sydney, Australia (2006).

Oral Presentation.

3. Michael W. Vogel, Dmitri K. Gramotnev. “Analyses of adiabatic nano focusing in

metal tips and nano-holes in the presence of dissipation.”, 17th Australian Institute of

Physics Congress – AOS, Brisbane, Australia (2006).

4. Michael W. Vogel, Dmitri K. Gramotnev, " Adiabatic Nano-Focusing in Metal Tips in

the Presence of Dissipation ", Surface Plasmon Photonics 3, p.168, Dijon, France

(2007).

5. Michael W. Vogel, Dmitri K. Gramotnev. “Adiabatic Nano-Focusing in Metal Tips with

different profiles”, Surface Plasmon Photonics 3, p.168, Dijon, France (2007).

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6. Michael W. Vogel, Dmitri K. Gramotnev. “Non-Adiabatic Nano-Focusing in Metallic

Rods”, Oral Presentation, Surface Plasmon Photonics 3, p.168, Dijon, France

(2007).

7. Michael W. Vogel, Dmitri K. Gramotnev. “Excitation of Surface Plasmons in Metal

Tapered Rods using Hollow Waveguides”, 18th Australian Institute of Physics

Congress – AOS, Adelaide, Australia (2008).

8. Michael W. Vogel, Dmitri K. Gramotnev. “Adiabatic Nano-Focusing in Metal Tips with

different profiles”, 18th Australian Institute of Physics Congress – AOS, Adelaide,

Australia (2008)

Invited Talks (Other than Conference)

1. Michael W. Vogel. “Plasmonics Research at QUT”, invited talk at the Max Planck

Institute for Polymer Research, Mainz, Germany (June 26, 2007).

 

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Statement of Original Authorship

The research contained in this thesis has not been previously submitted to

meet requirements for an award at this or any other higher education

institution. To the best of my knowledge and belief, the thesis contains no

material previously published or written by another person except where due

reference is made.

Signature

Date

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Acknowledgement

Writing a PhD thesis is a very solitary process, I am sure that everybody who

went through such an ordeal understands what I am talking about. On the

other hand, it is the most gratifying moment when you have finished your

work. I would never have come so far without the love and the

encouragement I got (and still get) from Judith.

My gratitude also goes to my supervisors, Dmitri (Благодари многих) and

Esa (Kiitos paljon!) for their support.

Without the support from the high performance computational group (HPC)

this project would not have been possible, thanks a lot to Mark Berry and the

whole team who showed me how to reduce the runtime of my MATLAB(C)

program from 35 hours to 1.5 micro seconds.

Finally I would like to thank Birgit Alves-Stein and Tanya Cairns who were

forced to proof read my thesis and compelled to give me valuable comments.

 

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http://mexiko.pauker.at/pauker/DE_DE/FI/wb/?x=Kiitoshttp://mexiko.pauker.at/pauker/DE_DE/FI/wb/?x=paljon

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Table of Contents

Key Words ..................................................................................................... iii 

Abstract ........................................................................................................... v 

Glossary of Terms and Abbreviations ............................................................ ix 

List of Publications and Manuscripts .............................................................. xi 

Statement of Original Authorship ................................................................. xiii 

Acknowledgement ......................................................................................... xv 

Table of Contents ........................................................................................ xvii 

Table of Figures ........................................................................................... xix 

List of Tables ............................................................................................... xxiii

 

1.  Introduction .......................................................................................... 1 

1.1.  Research Problem ..................................................................... 1 1.2.  Research Background ............................................................... 8 1.3.  Aim of the project and linking the papers ................................. 11 

2.  Theory and Background information ................................................ 17 

2.1.  Introduction .............................................................................. 17 2.2.  Maxwell’s Equations and Wave Equation ................................ 21 2.3.  Constitutional Parameters ....................................................... 23 

2.3.1.    Drude-Lorentz model .................................................. 25 2.3.2.    Non-local Effects and Loss Mechanism ...................... 26 

2.4.  Surface Plasmons on a Flat Surface ....................................... 28 

2.5.  Surface plasmons in Three Layer System ............................... 33 

2.5.1.    MIM-Structure (Metal Gap) .......................................... 33 2.5.2.    IMI-Structure (Metal Film) ............................................ 36 

2.6.  Excitation of Surface Plasmons ............................................... 41 

2.6.1.    Excitation by Electrons ................................................ 41 2.6.2.    Excitation by Photons .................................................. 42 

2.6.2.1. Otto and Kretschmann Configuration ................ 42 2.6.2.2. Grating and Surface Structures ......................... 44 2.6.2.3. Tip of Near Field Microscopes ........................... 45 

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2.7.  Detection of Surface Plasmons by Near Field Microscopes .... 47 

2.8.  Surface Plasmons on Cylindrical Surfaces ............................. 53 

2.8.1.    Field Structure and Dispersion Relation ...................... 57 2.8.2.    Analyses of the Dispersion Relation ........................... 63 

2.9.  Nanofocusing in Tapered Rods and Grooves ......................... 66 

2.9.1.    Methods of Analysis .................................................... 70 

2.9.1.1. Staircase Approximation ................................ 70 2.9.1.2. Geometric Optics Approximation ................... 72 2.9.1.3. Intensity Distribution with GOA ...................... 72 2.9.1.4. Numerical methods ........................................ 75 

3.  Adiabatic Nano-focusing of Plasmons by Metallic Tapered Rods in the Presence of Dissipation ............................................................ 79 

4.  Local Electric Field Enhancement during Nanofocusing of Plasmons by a Tapered Gap ........................................................... 87 

5.  Shape Effects on Adiabatic Nano-focusing of Localised Plasmons in Tapered Metal Rods ..................................................................... 95 

6.  Optimized Nonadiabatic Nanofocusing of Plasmons by Tapered Metal Rods ...................................................................................... 129 

7.  Optimisation of Plasmon Nano-focusing in Tapered Metal Rods 139 

8.  Conclusion........................................................................................ 159

 

Appendix A: Derivation Dispersion Relation: Flat Metal Surface ................ 169 

Appendix B: Determination of Incident Wave from Fabry-Perot Pattern ..... 171 

Appendix C: Optimal Length of Tapered Rod ............................................. 173 

Appendix D: Determination Intensity Distribution in GOA ........................... 177

References ................................................................................................. 179 

 

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Table of Figures

Figure 2.1: 3D-model of a surface plasmon propagating along a flat metal

surface in the x-direction. Schematically a snap shot of the Hy distribution

(TM-mode) is shown. The relative permittivity εd are for the dielectric material

and εm for the metal, respectively. The evanescent waves in y-direction are

indicated by the dash-dotted line. ................................................................ 29 

Figure 2.2: Propagation length for SP’s on a flat surface for gold, silver and

aluminium. The permittivities of Au, Ag and Al were taken from Palik [129]. 31 

Figure 2.3: Dispersion relation of SP’s on a flat Ag surface, with the

permittivity of Ag modelled by DLM, λp is the corresponding plasmon

wavelength and εd = 1 is the permittivity of the adjacent dielectric material.

See text for description of the curves. .......................................................... 32 

Figure 2.4: MIM structure with (a) anti-symmetric and (b) symmetric

magnetic field distribution with respect to the middle planeThe field

distribution is also indicated by the dash-dotted black line at an arbitrary

cross section.The metal gap width h is assumed to be 500 nm. .................. 34 

Figure 2.5: Dispersion relation qx(h), h being the gap width, of a MIM (Ag-

Vacuum-Ag) plasmon waveguide for symmetric magnetic field Hy (anti-

symmetric surface charge distribution) and anti-symmetric magnetic field Hy

(symmetric charge distribution). The dash-dotted line indicates the threshold

wave number; below the value qx / k0 = 1 no gap plasmons exist. ............... 36 

Figure 2.6: 3D image of SP’s propagating along a metal film (IMI structure)

with (a) anti-symmetric and (b) symmetric magnetic field distribution with

respect to the middle plane. The field distribution is also indicated by the

dash-dotted black line at an arbitrary cross section. The grey box indicates

the metal film which is aligned along the xy-plane. ...................................... 37 

Figure 2.7: Dispersion relation qx(h), h being the film thickness, of a IMI

(Vacuum-Ag-Vacuum) plasmon waveguide for symmetric magnetic field Hy

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(anti-symmetric surface charge distribution) and anti-symmetric magnetic

field Hy (symmetric charge distribution). ....................................................... 38 

Figure 2.8: Otto-Raether configuration: A laser beam is coupled into a prism

at the critical angle θc. Due to total internal reflection an evanescent wave

penetrates the metal surface and triggers surface plasmon propagation. The

reflected beam is measured by a photodiode. In the diagram the photo

current i vs. Angle of incident θ is shown schematically. The dip indicates an

energy transfer to excite SP’s at the critical angle θc. ................................... 43 

Figure 2.9: Kretschmann geometry: Similarly to Otto-Raether configuration a

laser beam is coupled into the prism at the critical angle θc. The evanescent

wave penetrates through the thin metal film, attached at the base of the

prism, and triggers SP propagation at the outward surface of the metal. As in

Figure 2.8, the dip in the photocurrent indicates the transfer of energy to the

SP’s at the critical angle θc. .......................................................................... 43 

Figure 2.10: Excitation of SP’s by means of a 1D diffraction grating. .......... 44 

Figure 2.11: Basic approach in near field microscopy (NFOM/NSOM). The

yellow area indicates the near field region, or the evanescent field. ............. 49 

Figure 2.12: 3 D Model of a metal cylinder or wire with permittivity εm

surrounded by a dielectric εd. The cylinder axis is aligned parallel to the z-

axis of a cylindrical coordinate system. The surface plasmon propagation is

assumed along the cylindrical surface in z-direction. .................................... 57 

Figure 2.13: Dispersion relation for the fundamental (n = 0) and the first two

hybrid plasmon modes HE1 and HE2 (corresponding to n=1 and n=2) on a

silver cylinder (εm = -16.102+ 1.076i at λvac = 632.8 nm) surrounded by air (εd

= 1). The circles indicate the common radius r = 500 nm at which the electric

field distribution is determined (see text below and Figure 2.14). ................ 59 

Figure 2.14: Normalized electric field E distribution for a metallic cylinder

with a radius r = 500 nm for a) TM-plasmons b) HE1- and c) HE2-plasmons.

The parameters correspond to those in Fig. 2.13. ........................................ 61 

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Table 1: EM field components of TM-plasmons in cylindrical coordinates. . 62 

Figure 2.15: Modified Bessel functions of the first (I) and second (K) kind of

the zero and first order. ................................................................................ 64 

Figure 2.16: Wave number of TM-plasmons propagating along tapered gold

and a silver wire. Curve 1 corresponds to silver and curve 2 to gold ........... 65 

Figure 2.17: Tapered rod design with parameters. The rounded tip is

considered only within non-adiabatic or rigorous numerical calculations.

However, the same structures can be considered for adiabatic nanofocusing,

one has to replace the half sphere by an infinitely long wire with radius equal

the exit radius to avoid any reflections. ........................................................ 66 

Figure 2.18: Staircase approximation of a tapered structure. The cone, the

wedge or the V-groove (shown as cross section along the symmetry axis z) is

decomposed into a consecutive row of basic structures with thickness t and

with sufficiently small distances dz between adjacent structures. ................ 71 

Figure 2.19: Intensity distribution for gold tapered rod with different taper

angles β=2α, initial radius r1 = 600 nm and exit radius r2 = 5 nm. ................ 73 

Figure 2.20: Adiabatic parameter versus distance from the tip of a metal

tapered rod for different taper angles with the same parameters as in Figure

2.19. The gray shadowed area indicates the region of δ at which GOA is

increasingly breached. ................................................................................. 74 

Figure 2.21: Phase velocity versus distance from the tip of a metal tapered

rod for different taper angles. The parameters correspond to those in Figure

2.19. ............................................................................................................. 75 

Figure 2.22: COMSOL model setup. Magnification of the tip area a) node

distribution for FEM and b) Plot normalized electric field as surface plot,

magnetic field H as contour plot. Taper angle β = 20o and tip radius rtip = 5

nm, c) schematics of the computational window .......................................... 76 

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Figure A.1: Electric field distribution at the interface of the metal and the

dielectric. The different curves stem from the phase shift, see text. ........... 171 

Figure B.1: Tapered rod with rounded tip .................................................. 173 

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List of Tables

Table 1: EM field components of TM-plasmons in cylindrical coordinates. . 62 

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1. Introduction

1.1. Research Problem

Currently, microelectronics and photonics are among the fundamental

technologies on which modern science and engineering is based. Around 25

years ago, the US government initiated a study to identify and define

important scientific and technological projects for the near future, also called

great challenges [1]. Among the issues identified are: long term weather

prediction; evaluating the effects of climate change; deciphering and

cataloguing the human DNA (human genome project); analyses of air

turbulences for prediction of tornadoes; computational ocean science for

tsunami warning; and speech recognition. Due to the complexity of the

problems, the study of these challenges relies almost entirely on high-

performance computation. In order to increase computational power, the data

processing speed has to be increased by either further miniaturizing

electronic and optical devices or increasing the processing speed, or clock

rate.

A modern single CPU (Computer Processing Unit), commonly used in

personal computers or other microelectronic devices, contains more than 200

million basic transistors, typically metal-oxide semiconductor field-effect

transistors or MOSFET’s, within an area of ~ 800 mm2 [2, 3]. Although this

enormous number of transistors, packed into such small areas, clearly

highlights the advantage of microelectronics, there is one problem related to

the space between the n-junction (electrons) and the p-junction (holes) in

MOSFET’s. This distance becomes so small that, quantum mechanically,

electrons could tunnel across the potential barrier between them, and yield

an unwanted electron current, also known as the tunneling current [2].

In addition to tunneling currents, parasitic inductance due to the

heterostructure of a basic transistor, is also present [2, 4]. The insulating

oxide layer has to be very thin, less than 1 nm, in order to reduce the

inductance [2]. This measurement, however, increases the tunnel current,

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because the decrease of the width of the potential wall would lead to an

increase of the probability that electrons penetrate the barrier [2]. Other

limiting factors for miniaturization are the interaction of electrons with an

interface as well as with other electrons, the practical problem of granularity

of semiconductor dopants, not to mention the challenges related to thermal

effects [2, 5].

Increasing the clock rate by keeping the characteristic sizes of basic field

transistors constant brings along the problem of the transit time for an

electron from the source to the hole. This characteristic transit time,

determined by structure and material used would be much larger than one

period of the oscillation, which limits the operational frequency in

microelectronics to the GHz region (~109 Hz) [4].

In summary, all these problems, as illustrated above, are among the

reasons why currently the computer industry is shifting towards parallel

processing and multiprocessor techniques.

Photonic devices, on the other hand, operate at much higher frequencies,

theoretically increasing the speed of information processing in (hypothetical)

optical computers by a factor of ~ 105 or more, compared to conventional

computers based on microelectronics. Photonic devices, such as optical

fibers, couplers and switches have already been successfully incorporated

into telecommunication and computer networks [6]. In fact, photonic elements

today are the backbone of the worldwide communication system because

they are insensitive to electromagnetic (EM) fields, experience less heat

production and therefore lower losses during operation, and possess

transmission bandwidths at almost the speed of light because they operate at

optical frequencies (~1014 Hz) [6].

The advantages of photonic devices over microelectronic devices arise

from the fact that the carriers of information are photons rather than electrons

[6, 7]. Unfortunately, photonic devices are much larger than their

microelectronic counterparts (by at least 10 times) because of the diffraction

limit, theoretically limiting the characteristic minimum sizes of photonic

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devices (such as the core diameter of an optical fiber) to around half the

wavelength [6-9]. How the diffraction limit effects ordinary photonic devices

can be demonstrated by two well known examples, namely a conventional

microscope and an optical fiber used, for instance, for long signal

transmissions.

The resolution δ of a conventional microscope is determined by Abbè’s

criterion [10]

           θλδsinn2

=                   (1.1)

where λ is the operational wavelength, n is the refractive index and θ is the

aperture angle. The term n sinθ is also known as the numerical aperture or

NA [6, 8, 9, 11, 12]. According to (1.1) the resolution can be improved by

increasing the numerical aperture or decreasing the wavelength. Increasing

the numerical aperture can be achieved by employing special objectives to

increase the aperture angle [11, 12], or by applying immersion oil to increase

the refractive index [9, 12]. However, as previously mentioned, the best

resolution achievable with conventional optical microscopes is in the range of

half the operational wavelength. It is mainly the lack of materials available

(natural or synthetic) with a high enough refractive index n that inhibits the

achievements nanometer resolution with common optical instruments.

The transmission efficiency of a multimode optical fiber decreases rapidly

with decreasing core diameter until a cut-off is reached. The cut-off diameter

for a multimode fiber, at which no propagating waveguide mode is supported,

is typically in the region of tens of micrometers at optical frequencies [13, 14].

For a single (fundamental) mode fiber, which has no cut-off diameter, the

electric field also becomes less and less confined in the fiber with decreasing

core diameter, with the result that the evanescent field (due to total reflection

at the interface core-cladding) spreads out further from the cladding into the

surrounding area and potentially interferes with other optical devices nearby.

Therefore to prevent cross talk the optical cables have to be coated with an

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extra layer or be separated by a larger distance with the result of further

increase of the typical dimensions of the device [13, 14]. It is evident that

photonics based on conventional optical techniques faces a fundamental

miniaturization problem and it seems imperative to develop new methods and

other technologies to further reduce the characteristic size of optical devices

and components [15, 16].

On the other hand, microelectronics has already reached a high degree of

miniaturization, but faces the problem of manufacturing at the nanometer

scale, not to mention the intrinsic problems related to heat or interference

with external EM fields [17]. An enormous synergy effect could be obtained

with respect to increasing computational power by combining the advantages

of microelectronics (high degree of miniaturization) with photonics (operating

at high frequency). It should be mentioned though that the first steps toward

this goal have been already completed by combining silicon (a typical

material used in microelectronics) with photonics and investigating the

possibility of chip-to-chip or even inter-chip communication [15, 18, 19].

However, the essential problem that remains is the size mismatch between

photonic and microelectronic devices [15, 16].

One possible and promising solution is to employ EM surface waves on

metallic interfaces, called surface plasmons (SP’s) or just plasmons. Surface

plasmon research, also known as plasmonics, is a new branch of

nanophotonics or nano-optics [7, 19, 20]. Plasmonic components and

devices are based on the propagation of SP’s on artificially created metallic

structures and may be one of the future approaches to resolve the problem of

how to merge microelectronics and photonics. This is because they combine

the benefits of very strong electromagnetic field localization at the boundary

of the metal and the dielectric within the operation at optical frequencies. The

strong localisation of SP’s is due to the opposite signs of the real part of the

dielectric permittivities of both media in contact. At optical frequencies the

dielectric permittivity of metals, such as silver and gold or aluminium, is

negative which renders them as promising materials for future plasmonic

devices.

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Plasmonic Devices

Surface plasmons (SP’s) are collective and coherent oscillations of the

conduction electrons coupled to photons at the surface of a metal. Once

excited they propagate along the interface between the metal and the

adjacent dielectric [7, 11, 16, 17, 19-34]. One of the unique properties of

SP’s, as mentioned in the previous section, is the strong localization at the

interface, which renders SP’s very sensitive to any change of the surface

condition. This sensitivity has been exploited in sensors in a large variety of

applications in chemistry, medicine and biology [35-37]. SP’s are also ideal

candidates to design real “flat” optics, that is, where the propagation of light is

confined to the surface of a metal. This could pave the way to the ability to

shrink photonic devices and components down to sub-wavelength

dimensions [17, 24, 26, 33, 38-40].

Merging plasmonics with microelectronics is a major long-term goal in

science and engineering due to the huge potential benefits, as indicated in

the previous section. Hence, SP based circuits have to perform the same

tasks as their electronic counterparts [15, 41]. For instance, plasmonic

circuits have to be able to a) support the propagation of SP’s around sharp

corners and bends without significant losses b) split one plasmon stream into

two and vice versa c) couple and switch and d) focus SP’s into nanometer

regions, to name just few possible options. A plasmonic circuit must also be

capable of coupling light into and recover light from the plasmonic circuit with

high coupling and decoupling efficiency as well as interconnecting with

common microelectronic circuits [15, 22, 41]. However, owing to the

presence of the metal in all potential plasmonic devices, dissipation is one of

the intrinsic problems of plasmonics which limits the propagation distances

and hence limits miniaturisation of plasmon devices [20, 22, 28, 33, 42].

It should also be mentioned that SP’s not only exist on metals but also on

doped semiconductors [43]. However, this is beyond the scope of this thesis,

for an detailed overview of SP’s on semiconductors see, for instance, the

extensive review from Kushwaha [43].

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Over the last two decades several metallic structures to guide SP’s, also

called plasmonic waveguides, have been suggested, including a thin metal

film sandwiched between two dielectrics [44-46], metal stripes embedded in

dielectrics [47-49], nano-chains [50-59], a V-groove in a metal [60-66], a

metal wedge [67, 68], or metal wires [69-75]. In order to compare the

characteristics (and performance) of different plasmonic waveguides and

devices, criteria have to be defined, such as confinement of the electric field

within or close to the waveguide, propagation distance and also the

possibility of single mode operation. Other more practical figures of merit are

the efficiency of transmission around sharp bends and corners, the tolerance

to imperfections or even the complexity of the manufacturing process for

plasmon waveguides [76-78].

6 

 

Localized Surface Plasmons and Nanofocusing

SP’s are not only bound to flat surfaces but can also propagate along

interfaces of curved surfaces, such as metal cylindrical wires. They can even

be confined to tiny metal structures, called nanoparticles. In this case, the

conducting electrons in the nanoparticles do not propagate but oscillate in

resonance with the external EM field. The first rigorous analytical study of

surface waves on metallic spheres was carried out by G. Mie (1869-1957)

[79], as he studied scattering of an EM wave by small metallic spheres. SP’s

excited on nanoparticles (sometimes called Mie-resonance [17, 80]) are also

known as localized surface plasmons (LSP’s), to distinguish them from

ordinary SP’s. In a sense they can be considered as non-propagating SP’s.

The dispersion relation of LSP’s depends strongly on the form and shape of

the nanoparticles and the surrounding environment, which renders LSP’s

very sensitive to any changes of the environmental condition. This sensitivity

has been exploited, for instance, in bio sensors [23, 33, 81]. Their resonance

spectrum also exhibits strong peaks, indicating that LSP’s can only be

excited at certain frequencies [23, 33]. One has only to match the resonance

frequency in order to excite LSP’s whereas to excite SP’s both the resonance

frequency and the wave number have to match [23, 33].

The strong localization effect, due to the resonance effect of the EM field

in close proximity of the nanoparticles, has been utilized in research areas

such as non-linear optics where the high energy density near the

nanoparticles can trigger, for instance, second harmonic generation [82, 83].

Another important application of LSP’s is related to Raman spectroscopy

where the Raman signal is strongly enhanced due to the presence of

roughness of the metal surface, a metal tip (also known as tip enhanced

Raman spectroscopy or TERS) [84, 85] or nanoparticles close to the probe

being examined [85-89], hence the name surface enhanced Raman

spectroscopy (SERS). The overall SERS enhancement factor is proportional

to the fourth power of the local electric field [82, 85]. A local electric field

enhancement factor of ~1000 would enhance the SERS signal by a factor of

~1012 which would be sufficient to detect single molecules [87, 90].

7 

 

The SP’s, propagating along metal tapered rods or other confining metal

structures, such as a wedge or a V-groove, may experience a decrease of

both the phase and group velocity. In that event the plasmon wave number

increases. This is equivalent to the transformation of (propagating) SP’s to

(non-propagating) LSP’s and the optical energy may become concentrated at

the tip of the structure, or focused, when the effect of dissipation is

sufficiently weak [91-95]. The effect of the concentration of plasmon energy

down to the nanometer region, much smaller than allowed by diffraction limit,

is known in the literature as superfocusing [91, 94, 96, 97] or nanofocusing

[92, 93, 95, 98-100]. Nanofocusing by a solid metal tapered rod has been

studied analytically [91, 95, 96], numerically [98, 101] and verified

experimentally [102]. Other metallic structures have also been investigated

including nanoparticles [53, 54], metal wedges [67, 93, 97, 103] or a metallic

V-groove [92, 97, 99, 104].

1.2. Research Background

As indicated in the preceding section, efficient delivery of optical energy to

the nanoscale region is crucial for the performance of sensors employed in

nanophotonics and spectroscopy. However, the generation of nanometer

sized energy sources is a considerable challenge. Several ways have been

suggested to overcome this difficulty including a laser beam focused close to

the tip region [105], an approach also known as tip enhanced microscopy

(see chapter 2.6). Unfortunately it suffers from poor efficiency, and since the

laser beam heats up the tip, damage to the probe or even the sample can

occur [12, 106]. Another approach consists of guiding light through a tapered

optical fiber coated with a metal film and terminated by an aperture.

However, this design essentially experiences the problems of low throughput

and emergence of heat [11, 27, 107-112]. On the other hand, tapered rods

are probably the most common component in near field optics, compared to

more complex structures capable of nanofocusing, including metallic wedges,

metallic V-grooves and gaps. They are made of either solid metals or

dielectrics, uncoated or coated with a metal film. The manufacturing process

8 

 

is well understood and high precision tapered rods and tips of any shape can

be produced in great quantities [11, 113].

SP propagation along metal tapered structures like a metal rod or a

metallic V-groove may overcome the problems related to efficient delivery of

optical energy to the nanoscale region. As mentioned previously,

Babadjanyan et al. [91] and Stockman [95] demonstrated that the SP’s

propagating towards the tip of a metallic tapered rod may lead to a strong EM

field enhancement (nanofocusing) when dissipation is sufficiently weak.

Babadjanyan and colleagues analysed plasmon focusing by a tapered rod by

means of the wave equation in spherical coordinates in combination with

separation of variables [91]. This could only be done under very restrictive

conditions which are applicable only in close proximity of the tip, as pointed

out by Kurihara et al. [96]. Stockman [95] studied plasmon nanofocusing by a

metal tapered rod on the base of the geometric optics approximation (GOA)

or adiabatic theory, also referred to in quantum mechanics as WKB-method.

GOA is applicable only under the condition that the tapered structure does

not vary significantly within one wavelength. Stockman [95] also

demonstrated that the applicability condition of GOA includes the whole

tapered rod including the tip itself, when the taper angle is sufficiently small.

The effect of nanofocusing under GOA is also called adiabatic nanofocusing

[95, 114]. As demonstrated, the attainable plasmon field enhancement

factors were in the range of ~ 1000 for a silver tapered rod with a taper angle

α ~ 2.3O and taper length ~2.5 μm. However, the effect of dissipation and

taper angle on the efficiency of nanofocusing has not been discussed. No

attempt had been made to determine the limits of the applicability condition of

GOA.

Adiabatic nanofocusing during plasmon propagation by a V-groove in

metal has been first reported by Gramotnev [92]. He demonstrated within

GOA that the magnetic field at the tip of the groove remains finite even in the

absence of dissipation. The analysis of the dependence of dissipation and

taper angle on the maximum local field enhancement has also been carried

out [92]. In addition, a comparison study of adiabatic nanofocusing and

9 

 

numerical has been conducted by Pile and Gramotnev [99]. However, the

field enhancement factors have been determined for the magnetic field only.

No comparison study with other structures, such as a tapered rod or a metal

wedge has been conducted.

Issa and Guckenberger [98] studied numerically nanofocusing (also called

non-adiabatic nanofocusing) by metal tapered rods attached to a long

straight cylindrical wire and terminated by a spherical tip of radius rtip = 5 nm.

The length of the wire was chosen in such a way in order to avoid any

interference from reflections originating from the initial computational

boundary and the tip region. Optimal taper angles at which the plasmon field

enhancement is maximal were found, which did not depend on the radius of

the cylindrical wire [98]. It has been demonstrated that the plasmon field

enhancement increases with increasing tip length. Although an optimal angle

has been found, no attempt had been made to determine other structural

parameters such as an optimal taper length. The dependence of the tip

radius on maximum field enhancement has not been determined and no

detailed analysis of the field structure near the tip has been conducted.

In summary, no thorough investigation into the effects of different levels of

dissipation, taper angles and shape variations on plasmon nanofocusing and

plasmon field distribution by metal tapered rods has been conducted. The

influence of the tip geometry on the maximal achievable field enhancement

factors as well as the field structure around the tip has not been considered.

No detailed analysis has been carried out to determine the limits of the

applicability condition of the adiabatic approximation (GOA). For a metallic

V-groove no electric field enhancement factors have been determined and no

comparison with a metal tapered rod has been done.

10 

 

1.3. Aim of the project and linking the papers

As emphasized in the preceding section, the propagation of SP’s along

tapered metallic structures may lead to a strong field enhancement at the tip

of the structure which can be used to deliver electromagnetic energy to the

nanometer region. The strong field enhancement, due to the strong

localisation of the plasmon field near the tip, is crucial and determines, for

instance, the resolution of near field microscopes [11, 12, 115]. It is also of

particular importance in SERS or TERS, because the sensitivity of SERS,

along with the distance sample to probe, depends mainly on the strength of

the local electric field [82, 85].

The overall goal of this PhD project was to analyse in detail plasmon

nanofocusing by metal tapered rods and metallic V-grooves and determine,

theoretically and numerically, structural conditions, such as optimal taper

angles, taper length or ideal shape, under which maximum plasmon field

enhancement (nano-focusing) can be achieved.

With these topics in mind, the project was divided into two parts. In the first

part, the adiabatic approximation (GOA) has been employed to study

analytically plasmon nanofocusing by a tapered rod and a metallic V-groove.

However, this approach is applicable, as emphasized in the previous section,

only for small taper angles; hence the effect of reflection from the tapered

section and from the tip is not taken into account.

In particular, the specific topics investigated in the first part were:

• Analysis of the effects of different levels of dissipation and taper

angles on the efficiency of plasmon nanofocusing by metal tapered

rods. Determination and interpretation of the different

characteristics of the intensity distribution (chapter 3).

Methods and outcomes:

The optimal parameters, such as optimal taper lengths and taper

angles at which the plasmon field enhancement at the tip is

maximal, were determined and discussed. The different plasmon

11 

 

intensity distributions along the tapered rod, between two fixed

radii, were analysed in detail by means of GOA. It will be shown

that three distinctive intensity distribution patterns exist, owing to

the presence of two critical taper angles, or equivalently, two critical

dissipation levels.

• Detailed analysis of the electric field enhancement in a metallic V-

groove. Comparison study of plasmon field enhancement during

nanofocusing by a tapered rod and a V-groove in a metal (chapter

4). Comparison of the results achieved by GOA and rigorous

numerical methods.

Methods and outcomes:

Following recent reports on adiabatic nanofocusing by a metallic

V-groove [92, 99], a comparison study with a metal tapered rod was

conducted, to determine the characteristics of the EM field in the

V-groove and the plasmon field enhancement. It will be shown that

the attainable field enhancement at the tip of a V-groove in the

presence of strong dissipation in the metal is higher than compared

to a tapered rod having the same structural parameters, such as

taper angle or taper length. A comparison study of the results

achieved by GOA and rigorous numerical methods (FEM) has been

carried out to determine the limits of GOA for plasmon

nanofocusing by a metallic V-groove.

• Analyses of the shape variation of the tapered rod on nanofocusing

(chapter 5).

Methods and outcomes:

A detailed analysis, based on GOA, on the effects of shape

deviations from the ideal conical geometry on the plasmon field

enhancement at the tip as well as the intensity distribution along the

rods, is presented. Two basic structures were considered in detail,

a convex (curved outwards) and a concave (curved inwards)

tapered rod. It will be demonstrated that, in general, the attainable

12 

 

plasmon field enhancement by a concave tapered rod is

significantly higher compared to a convex tapered rod or even a

conical tapered rod.

In the second part, non-adiabatic nanofocusing by a tapered rod

terminated by a spherical tip and with large taper angles is studied in detail.

Owing to the presence of the a) tapered section and b) a rounded and closed

tip, parts of the plasmon wave are reflected back and interfere with the

forward propagating (towards the tip) plasmon wave. This leads to potentially

complex interference patterns, also called Fabry-Perot resonance, which

cannot be treated by analytical methods. Therefore, in order to study

nanofocusing accurately, rigorous numerical methods have to be employed.

In practice, many different kinds of numerical techniques for analysing

electromagnetic fields have been applied, such as finite difference time

domain (FDTD), methods of moments (MOM) and finite element methods

(FEM). Within this thesis, however, the numerical simulations for tapered

rods have been carried out exclusively by means of FEM. To this end, a

commercial software package COMSOL© and in-house programs, mainly

developed in MATLAB©, have been employed.

As mentioned in the previous section, Issa and Guckenberger [98] studied

numerically (with COMSOL©) non-adiabatic nanofocusing by considering

tapered rods attached to long cylindrical wires. Optimal taper angles at which

the plasmon field enhancement is maximal were found, which did not depend

on the radius of the cylindrical wire. However, the dependence of

nanofocusing on the tip radius has not been taken into account. It has been

demonstrated, though, that the plasmon field enhancement increases with

increasing tip length. This result was found for short tapered rods only and so

the analysis was limited to a small section near the tip. It is not clear if the

increase would continue with further increase in the taper length. Of course

the field enhancement will not increase infinitely but it is of interest whether a

correlation between optimal taper angle and optimal taper length exists.

In response to these questions, the topics in the second part were:

13 

 

• Determination of the optimal taper angle and an optimal taper

length at which the plasmon field enhancement is maximal (chapter

6 and chapter 7).

Methods and outcomes:

Non-adiabatic nanofocusing by metal tapered rods was studied by

means of FEM. The existence of an optimal taper angle, as

concluded by [98, 101], and an optimal tip length at which the

plasmon field enhancement at the tip is maximal was found. An

equation to determine the optimal taper length, based on energy

conservation consideration, has been derived. The plasmon field

enhancement is determined by the ratio of normalized electric field

amplitude at the top of the rounded tip to the normalized electric

field at the launching point. To this end, a new approach was

introduced, which combines GOA and numerical analysis in order

to study large tapered structures within reasonable time and

reducing significantly the use of computational resources. A new

method was also introduced to extract the contribution of the

forward propagating plasmon wave from the interference pattern

due to Fabry-Perot resonance.

• Analysing the dependency of the tip radius on plasmon field

enhancement and determining the characteristics of the EM field in

the proximity of the spherical tip region (chapter 6 and chapter 7).

Methods and outcomes:

The achievable plasmon field enhancement was calculated for

different metals and different tip radii. The attainable plasmon field

enhancement strongly depends on the radius of the rounded tip.

The maximum field enhancement factor, as predicted in the

numerical calculations, are in the range of ~2500, which is sufficient

for detecting single molecules with SERS. The field distribution

along the rounded tip, the effect of the tip radius and the presence

of different metals on to the field enhancement factors has been

studied in detail.

14 

 

• Investigation of the validity of the adiabatic theory for nanofocusing

by a metal tapered rod. Determining the accurate applicability

condition for adiabatic nanofocusing by a tapered rod (chapter 6

and chapter 7).

Methods and outcomes:

In chapter 4, for a V-groove in a metal, the critical taper angle, at

which the plasmon field enhancement calculated by means of the

adiabatic theory starts to deviate from the results obtained by

rigorous numerical methods, is in the range of 12-14o, for gold at

wavelength λvac = 632.8 nm. No such comparison had been done

for tapered rods. Therefore, a detailed investigation of the

applicability condition for different metals and different wavelengths

was carried out. Surprisingly, the critical angle was found to be

much larger than for the groove in the metal. This increases the

number of possible applications of the adiabatic theory in the

context of plasmon nanofocusing by metal tapered rods. As another

key result, it has been shown that the applicability condition for

adiabatic nanofocusing by metal tapered rods, as applied in [95], is

too strict. As a consequence, a new modified applicability condition

for adiabatic nanofocusing by metal tapered rods has been

suggested.

 

15 

 

16 

 

2. Theory and Background information

2.1. Introduction

Robert W. Wood (1868-1955) was the first scientist who accurately described

the effects of surface plasmons (SP’s) as he examined the spectra of an

incandescent lamp with a new metallic grating. In 1902, he wrote in his report

[116]: “When the light is incident on the opposite side of the normal from the

spectrum we find the red and orange extremely brilliant up to a certain wave-

length, where the intensity suddenly drops almost to zero, the fall occurring,

as I have said, within a range not greater than the distance between the D

lines. A change of wave-length of 1/1000 is then sufficient to cause the

illumination in the spectrum to change from a maximum to a minimum. The

theory of the diffraction-grating, as it stands at the present time, appeared to

me to be wholly inadequate to explain this most extraordinary distribution of

light, and I accordingly endeavoured to find out if possible the necessary

modifications which must be introduced.”

Wood expected a smooth and slow change in the intensity distribution but

instead he observed sharp and narrow bright and dark bands. He also

noticed that this unexpected behaviour occurs only when the incident wave is

s-polarized, that is, when the only magnetic field component of the incident

wave is parallel to the grating. Woods could not explain the results he

obtained within the framework of the existing theory and described them as

anomalies. Ever since, this phenomena is referred to as Wood’s anomalies

[28, 117, 118]. Fano [117] and later Hessel and Oliner [118] were among the

first to develop a theory to explain Wood’s anomalies by coupling of light into

EM surface waves, mediated by the metallic grating structure.

According to the free electron model, a metal consists of a lattice formed

by positive ions and electrons from the valence band which move through the

body , only weakly bound to the ion cores, similar to a gas [25, 119-121].

The coherent oscillations of the valence (or conducting) electrons are called

bulk (volume) plasmons or SP’s, depending on whether the plasmon

17 

 

oscillation takes place inside or at the surface of the metal. Bulk plasmons

had been first scrutinized in detail in the early 1950’s by Pines and Bohm as

they studied the effect of electrons passing through the metal [122-125]. In

1957, SP’s were first predicted by Ritchie [21] to explain unexpected low

frequency losses (below the bulk plasmon frequency ωp, see also section

2.3.1) during a study of the angle-energy distribution of electrons passing

through thin metal films. Two years later, SP’s had been verified

experimentally by Powell and Swan [126] by measuring the energy

distribution of electrons passing through an aluminium film.

In the literature, SP’s are sometimes called surface plasmon polaritons

(SPP) which reflects the hybrid nature of being a mixture of plasma

oscillations and photons, but essentially SP’s and SPP’s are identical [28,

119]. The EM field, associated with SP’s, which propagates along the

interface between the metal and the dielectric, is highly localized and decays

rapidly into both media, like a surface wave [16, 20, 24, 26, 28, 29, 33, 42].

The EM surface wave has both, longitudinal and transversal components,

whereas surface plasmons are purely longitudinal surface charge density

waves [119]. Because of their wavelike propagation, SP’s can be described

by Maxwell’s equations and the EM wave equation. It should be mentioned,

however, that in the beginning of plasmonics research most of the results

including the dispersion relation or the response of metals to the external

applied field, were derived by applying Bloch’s equations, because of the free

electron model used to describe the metal [21, 127, 128].

Maxwell’s equations are introduced in section 2.2, followed by a brief

discussion of the constitutive relations in section 2.3, in context of the

interaction of light and matter in regions smaller than allowed by diffraction

limit. At this length scale, metallic structures, such as nanoparticles or

nanorods, have an impact on the permittivity ε, permeability μ or conductivity

δ. Within section 2.3, the Drude-Lorentz model is introduced, a

representation of the electric response of metals to an applied EM field which

is commonly applied in plasmonics and near-field optics, followed by a brief

discussion about loss mechanisms.

18 

 

In section 2.4, a detailed analysis of the propagation of SP’s on a flat

metal surface is presented. Starting from Helmholtz’s equation and applying

the appropriate boundary conditions, the dispersion relation for the plasmon

wave number q(ω), with ω being the angular frequency, is derived and the

main characteristics of the solutions are discussed. As a major result, it will

be demonstrated in section 2.4, that only TM-plasmons can propagate along

the metal surface. In the optical and near infrared frequency regime the real

part of the complex relative permittivity of most of the metals, such as silver,

gold, aluminium and copper, takes on negative values which is the basic

requirement for the existence of SP’s [129].

SP propagation on a metal film bound by a dielectric on each side; also

called IMI (insulator-metal-insulator) and within a metal gap filled with a

dielectric, called MIM (metal-insulator-metal) is studied in section 2.5. In

these configurations coupling of SP’s for sufficiently small film or gap widths,

give rise to additional plasmon modes, called film and gap plasmons. As it

will be shown, these gap (film) plasmons have either a symmetric or an anti-

symmetric EM field distribution across the gap (film) with respect to the

middle plane. The dispersion relations of all possible modes are analysed

and the results in relation with nanofocusing are discussed.

In section 2.6 an overview of the different excitation methods of SP’s is

presented, followed by an outline of the tools and sensors used to detect and

visualise SP’s, such as fluorescent molecules or near field microscopes, in

section 2.7.

SP propagation along a metal cylindrical wire is studied in detail in section

2.8. Starting from the wave equation the dispersion relation q(r) is derived,

with r being the local radius, followed by a detailed investigation of the most

important solutions (modes). Contrary to a flat metal surface, cylindrical

structures not only support fundamental TM-plasmon modes but higher order

modes (hybrid modes) as well. However, all modes except of the

fundamental TM-mode experience a cut-off when the diameter decreases. As

it will be shown in section 2.9, TM-plasmons mode can be employed for

nanofocusing.

19 

 

Flat metal surfaces, IMI, MIM and cylinders constitute the building blocks

for the theoretical and numerical analyses of adiabatic nanofocusing of SP’s

by a tapered rod and a metallic V-groove. A detailed analysis of

nanofocusing by those structures, including analytical and numerical

methods applied within this thesis, is presented in section 2.9 which will

conclude the literature review.

 

20 

 

2.2. Maxwell’s Equations and Wave Equation

SP’s are EM surface waves which propagate along the interface of a metal-

dielectric system. They can be analysed analytically by means of the

Maxwell’s equations. The amplitude of the SP is maximal at the interface and

decays exponentially into both media; thus they are strongly localised at the

interface at optical frequencies [16, 20, 24, 26, 28, 29, 33, 42].

In SI-units, which are adopted throughout this thesis, the equations for the

electric field strength E 1 (or simply electric field) and the magnetic field

strength H (magnetic field) are given in differential forms as

t∂∂

−=×∇ E B (2.1)

and

t∂

∂+=×∇

DJH (2.2)

where B is the magnetic flux density, D the electric displacement and J is the

density of the free currents [130]. The magnetic field H and the magnetic flux

density B, as well as the electric field E and the displacement D are related

by the constitutive equations

HB r0μtμ= (2.3)

ED r0εt

ε= (2.4)

Where μr and εr denote the relative permeability and permittivity tensor,

respectively. µ0 2 and ε0 3 are the permeability and permittivity in vacuum,

respectively. Metals considered in this thesis are isotropic, linear and

                                                            

1 Vectors are denoted by bold letters

2  ε0 ≈ 8.854·10-12 F/m 

3  μ0≈ 4π·10-7 H/m

21 

 

homogeneous; hence equations (2.3) and (2.4) are simple proportional

relations. The constitutional parameters, however, may still be dependent on

the applied frequency but for the sake of simple notation, the arguments are

not shown.

By combining (2.1), (2.2) and (2.4) the wave equation for the electric field E

can be derived as

tr00 ∂

∂−=×∇×∇ E εεμ E . (2.5)

A plane wave solution has the form

(2.6) )t(ie rk0EE

⋅−= ω

where k denotes the wave vector, ω the angular frequency and E0 the

amplitude of the electric field, the wave equation (2.5) can be expressed as

(2.7) ( ) EkEEkkEkk ),(kk r202 ωε=−⋅=××

with k0 = ω/c with c being the speed of light. In the last step the vector

relation x x A = ( ·A) - 2A has been applied. For transversal waves

(k⋅E = 0), (2.7) is reduced to the well known Helmholtz Equation

0k 22 =+∇ EE . (2.8)

For longitudinal waves (k⋅E = kE) the wave equation for any arbitrary electric

field is reduced to

(2.9) 0),(0),(k rr2

0 =⇒= kkE ωεωε

Equation (2.10) implies that at the zeros of εr(ω,k) only, longitudinal waves

can propagate in the metal.

 

22 

 

2.3. Constitutional Parameters

The relative permittivity εr in (2.5) describes the response of matter to an

applied external electric field E, likewise the relative permeability μr in (2.4) describes the magnetic response to the applied magnetic field H. In the high-

frequency regime of optics and plasmonics, however, the magnetic response

can be neglected for two reasons. Firstly, the magnetic domains are too large

and therefore too inert to follow the rapidly varying EM field [114]. Secondly,

as mentioned in the introduction of this chapter, noble metals such as Gold

and Silver are isotropic and diamagnetic by nature; whereas aluminium (as

another important metal) is a paramagnetic material [25]. Hence, the relative

permeability μr reduces to 1 in the optical frequency range for all metals

considered in this thesis. Nevertheless it is worth noting that Ishikawa and

colleagues have shown theoretically that the relative permeability μr can

indeed take on negative values at optical frequencies for artificial nano-sized

structures, composed of split ring resonators (SRR) [131]. It should also be

mentioned that external magnetic fields have been applied in the context to

plasmonics, for instance, to study the effect on surface plasmon propagation

in semiconductors, also called magnetoplasmons. An extensive overview of

this topic, which is beyond the scope of this thesis, has been given, for

instance, by Kushwaha [43].

The electrical properties of metals, in contrast to the magnetic properties,

play a crucial role in the response to optical waves. This is because electrons

can react almost instantly to any changes of an external EM field, especially

in metals where the valence electrons are nearly free to move throughout the

whole lattice structure.

The characteristic sizes of metallic structures studied in nano-photonics

and plasmonics are in the nanoscale region. At this length scale the effective

constitutive parameters become size dependent. This is the subject of a

whole new and rapidly growing scientific research area dealing with left-hand

materials and metamaterials [132]. As a simple definition, metamaterials are

designed materials that gain their EM properties not only by the material

23 

 

properties they are composed of, but also by the structural composition at the

nanoscale. The potential applications of metamaterials are staggering. Just

recently, for instance, it has been demonstrated that an object surrounded by

a cylindrical shell, composed of SRR’s, is rendered invisible [133]. The

incoming EM wave is routed around the shell and no scattering occurs from

the object inside [133]. Another interesting example is the possibility of

designing a perfect lens in the form of a simple slab made of negative

refractive index material [134], where both the permittivity and the

permeability are simultaneously negative [135], a concept originally

envisioned by Veselago [136].

Photonic crystals are another class of artificial material, composed of

periodically built up dielectrics or metal-dielectrics in 1D-, 2D- or 3D-structure

[20, 137-140], though, the 2D- and 3D-structure are still difficult to engineer

[140]. EM waves can propagate in photonic crystals with designed gaps or

along designed defects at the surface [138, 140] and it has also been

demonstrated, for instance, that light can be guided around bends with low

losses [22, 141]. However, the propagation of EM waves along those

manufactured defects is still subject to the diffraction limit, therefore photonic

crystals are not applicable for the purpose of nanofocusing or sub

wavelength guiding. Plasmon nanofocusing by structures made of artificially

designed materials, however, due of its complexity, is beyond the scope of

this PhD-project and cannot be considered.

Over the past century several models to describe the dielectric properties

of metals been developed. One of the first models, the Drude-Lorentz model

(DLM), is still commonly applied in plasmonics and nanophotonics, because

at optical frequencies it describes sufficiently well the dielectric response of

noble metals, despite the simple equation [11, 21, 28, 33, 142]. Noble metals

play an important role in plasmonics, because the dielectric permittivity of

gold or silver, for instance, is negative at frequencies within the visible

spectrum. As a consequence it is possible to excite SP’s at optical

frequencies. This will be the subject of the next section.

24 

 

2.3.1. Drude-Lorentz model

Around 100 years ago, Paul Drude (1863-1906) developed a simple

description of the dielectric response of a metal, based on the free electron

model, in order to explain the high electric conduction and thermal

conductivity of metals [143]. Later improved by Hendrik Lorentz (1853-1928),

the Drude-Lorentz Model (DLM) essentially treats the electrons as a gas

inside the metal, only weakly bound to the positive ions, which are fixed in a

rigid lattice structure. Starting from a simple model of an electron exerted to

an applied external electric field with a restoring force originating from the

positive background, the relative permittivity εr(ω) can be expressed as

γωω

ωωε

i1)( 2

pr −

−=2

(2.10)

where ωp is the bulk plasma frequency, and γ is the damping frequency,

which accounts for scattering of the electrons by other electrons or the lattice

[11, 25, 33, 42, 114, 119, 120, 142]. For a weakly damped system (γ 0)

the relative permittivity takes on a particularly simple form:

2p

r 1)(2

ωω

ωε −= (2.11)

According to (2.11), a metal becomes transparent (wave propagates insides

the metal) when the applied frequency ω is larger than the bulk plasma

frequency ωp. The metal becomes impenetrable (the metal acts as a

reflector) when ω becomes larger than ωp, as the wave is damped inside the

metal and no wave propagation occurs. When ω is equal to the bulk plasma

frequency ωp, or equivalently εr becomes zero, however, longitudinal EM

waves can propagate inside the metal, as mentioned in the previous section.

The bulk plasma frequency can be intuitively understood by considering

the following scenario: In equilibrium, the free electron density is uniform

across the whole metal surface, with a fixed and positively charged

background. A slight disturbance causes an inhomogeneous surface charge

distribution which generates a restoring force. However, due to the non

25 

 

uniform charge distribution an electric field is generated in which the

electrons gain additional momentum. As a result the electrons move towards

the equilibrium position but overshoot due to inertia of the electrons, hence

an oscillation of the electron gas with a natural frequency ωp follows, damped

by internal frictions, collisions and other mechanisms [144].

The DLM predicts sufficiently well the dielectric response of noble metals

such as gold or silver to an applied EM field at optical and near infrared

frequencies [42]. However, for completeness it should be mentioned, that the

DLM does not take quantum mechanical phenomena, such as inter- and

intra-band transitions, into account. These effects have an impact on the

dielectric response, and in particular on dissipation of the metal [25, 119-

121].

2.3.2. Non-local Effects and Loss Mechanism

The complex problem of dissipation and dielectric response of metals has

been addressed by many researchers [25, 114, 120, 122-125, 127, 128, 130,

145, 146]. Landau damping (damping of longitudinal waves in plasma), inter-

and intra-band transitions and the anomalous skin effect are examples of

non-local loss mechanisms which have to be considered when the plasmon

wavelength becomes very short. This is especially important when the SP’s

approaches the tip of a tapered metallic structure. The wave number

increases anomalously near the tip, hence both the phase and group velocity

decrease and the plasmon wavelength shortens [91, 92, 94, 95]. This will be

demonstrated in section 2.9, when nanofocusing by metal tapered rods and

metallic V-grooves is discussed.

When the size of the sample or structure becomes comparable with the

mean-free path of an electron, spatial dispersion has to be taken into

account. In this case the permittivity ε becomes size and wave vector

dependent [114]. In conjunction with nanofocusing by tapered rods,

Stockman [95] discussed qualitatively the effect of spatial dispersion,

whereas Ruppin [100] demonstrated, analytically and numerically, that spatial

dispersion has to be considered only when the local diameter of a tapered

26 

 

rod becomes much smaller than the plasmon wavelength. To this end, he

derived a modified dispersion relation q(r), owing to the fact that at the

interface an interaction of the transverse surface wave with the longitudinal

bulk plasmons takes place. The intensity of the SP field at a diameter

of ~1 nm, calculated without spatial dispersion, would be more than twice as

large [100]. However, this result has been achieved only for a silver tapered

rod of one plasmon wavelength (λvac = 632.8 nm).

Within this PhD thesis, the response of a metal to an applied EM field, as

well as all loss mechanisms mentioned above, are represented by the

complex dielectric permittivity, as published by Palik [129]. The real part of

the permittivity describes the propagation of SP’s and the imaginary part is

linked to dissipation. A minimum diameter of 4 nm as a lower limit in the

investigations of nanofocusing in metallic structures within this thesis has

also been applied in order to avoid any effects from spatial dispersion, as

suggested by Stockman [95].

27 

 

2.4. Surface Plasmons on a Flat Surface

A single flat metal surface is probably the most basic metal structure to

describe SP propagation. Nevertheless, some important results can be

derived with this simple configuration. Within this section, it is assumed that

the interface, which separates the metal and the dielectric, is placed on the

x-z plane of a Cartesian coordinate system. It is also assumed that the metal

occupies the region y < 0 and the SP’s propagate along the x-axis. It can be

demonstrated that only TM-plasmons can propagate along the surface [28,

33, 42]. This is due to the fact that the relative permeability μr is close to unity

for most natural metals at optical frequencies, for details see Appendix A.

There is alternative explanation derived from the boundary condition

⋅− nDD md = δ)( (2.12)

where Dd and Dm are the dielectric displacement in the dielectric and

metal, respectively and n is the normal vector at the interface. As a

consequence of (2.12) only the components of E in the plane of the incident

wave can induce longitudinal surface charge density oscillations in the

direction of propagation (x-axis) [29]. The only magnetic field component left

is Hy, hence only TM-plasmon can propagate along the surface of the metal.

The EM field associated with TM-plasmons, also decay exponentially away

from the interface into both media, see also Figure 2.1. Therefore, the following solution for Hy can be assumed:

( ))xxqt(j −ω

( )

metal,0yyexpeAH my −⋅⋅= α (2.13b)

where A and B are amplitudes to be determined by the boundary conditions,

αm and αd are the reciprocal (positive and real) penetration depths into the

28 

 

metal and the dielectric, respectively. The TM-plasmon wave number qx is

complex because of the presence of a metal with the permittivity

εm = εm′ + iεm′′. Consequently, the amplitude of the SP’s decreases with

increasing propagation distance and eventually dissipates. The penetration

depth is defined as the distance from the interface metal-dielectric at which

the amplitude of Hy is reduced by e-1:

md

20

2x

md kq εα −= (2.14)

Figure 2.1: 3D-model of a surface plasmon propagating along a flat metal surface in the x-direction. Schematically a snap shot of the Hy distribution (TM-mode) is shown. The relative permittivity εd are for the dielectric material and εm for the metal, respectively. The evanescent waves in y-direction are indicated by the dash-dotted line.

 

The penetration depths and the permittivities are also related by (see

Appendix A for derivation):

d

m'

m

d

εα

εα

=− (2.15)

where εd and εm′ are the real permittivities of the dielectric material and the

metal, respectively. The permittivity of the dielectric is assumed lossless;

hence εd is real throughout this thesis. From (2.14) it can also be concluded

29 

 

that the penetration depth of the plasmon into the metal is much smaller than

into the dielectric, as indicated in Figure 2.1.

The real part of the permittivity of the metal, εm′, must take on negative

values to satisfy (2.15), because the penetration depths, by definition, are

positive real numbers. The real part of the complex wave number can then

be written as [20, 26, 28, 29, 42]

''k'qmd

md0x εε

εε+

= . (2.16)

In order to obtain a positive wave number the condition for the permittivity is

│εm│> εd. The imaginary part of the plasmon wave number is

2m

m2

3

md

md0x '

'''

'2k''q

εε

εεεε

⎟⎟⎠

⎞⎜⎜⎝

⎛+

= . (2.17)

The propagation distance of SP’s along the surface is limited, owing to losses

in the metal, which is represented by the imaginary part of the permittivity

εm′′. The propagation length L is defined as [28, 33]

1−x )''q2(L = . (2.18)

Inserting (2.17) into (2.18) yields

'')'('

k1L

m

2m

2

d

md

0 εε

εεε

⎟⎟⎠

⎞⎜⎜⎝

⎛ +=

13

. (2.19)

The propagation distance L on a flat metal surface for gold, silver and

aluminium as determined by (2.19) is depicted in Figure 2.2. For optical wavelengths in the range of 500-700 nm, silver has the largest propagation

30 

 

distance followed by aluminium and gold. The strongly increased propagation

distance towards the infrared region is of importance for the applicability and

possible implementation of plasmonic devices in telecommunication

technology [66, 147, 148].

Figure 2.2: Propagation length for SP’s on a flat surface for gold, silver and aluminium. The permittivities of Au, Ag and Al were taken from Palik [129].

Returning to dispersion relation (2.16), the solutions for the real part of the

plasmon wave number qx for TM-plasmons on a surface of silver, with the

permittivity of the metal described by DLM (2.12) surrounded by vacuum is

shown in Figure 2.3. For small wave numbers the dispersion curve of the

TM-plasmons (Figure 2.3, curve 1) asymptotically approaches the dispersion curve of the bulk wave (light-line) in the adjacent dielectric (Figure 2.3, curve 3). For large wave numbers it approaches the SP condition ωsp=ωp/(εd+1)1/2,

which also corresponds to the condition εd = -εm(ω) (Figure 2.3, curve 2). One important result is that curve 3 does not intersect curve 1, which

represents the dispersion relation of the bulk wave in the adjacent dielectric.

Consequently, light waves cannot excite SP’s on a flat metal surface or SP’s

cannot radiate into light waves without appropriate devices, hence the name

non-radiative SP. This is a consequence of Snell’s law, which imposes the

condition that the component of the wave vector parallel at the interface is

continuous. In other words, the wave vector of the surface plasmon and the

wave vector of the bulk wave do not match at the interface; once the SP’s are

coupled into a smooth metal surface, they are trapped [20].

31 

 

Figure 2.3: Dispersion relation of SP’s on a flat Ag surface, with

the permittivity of Ag modelled by DLM, λp is the corresponding

plasmon wavelength and εd = 1 is the permittivity of the adjacent

dielectric material. See text for description of the curves.

Between the bulk plasma frequency ωp and the surface plasmon

frequency ωsp, no solution corresponding to SP’s exist. This can also be

shown by substituting (2.11) into (2.16). This frequency gap is also known as

the plasmon gap [42]. For applied frequencies higher than the bulk frequency

ωp another solution exist, at which the plasmon wave radiates in the metal

(Figure 2.3, curve 4) [28, 42]. However, these modes are not confined to the interface and are of no interest for this project.

32 

 

2.5. Surface plasmons in Three Layer System

An important extension of the simple metal surface is a three layer system

sometimes also called heterostructure [46], where each of the layers has an

infinite extension in two dimensions. Two basic heterostructures can be

distinguished, a dielectric gap in a metal, or MIM (metal-insulator-metal)

system and a metal film surrounded by two dielectrics, or IMI (insulator-

metal-insulator) system. The method of analyses for SP’s is essentially the

same as shown in the previous chapter for a single flat surface. However,

because of the additional interface the dispersion relation becomes more

complex, owing to the fact that SP’s on each interface couple and form new

additional modes when the thickness of the metal film or the gap width

becomes sufficiently small. The SP’s on each interface can be excited in

phase and in anti-phase by means of, for instance, the Kretschmann

configuration, see also section 2.6.2. The new plasmon modes are also

called film plasmons (for IMI) or gap plasmons (for MIM) [92, 93].

In this section it is assumed that the structure extends infinitely in the

x- and y- direction and the plasmon propagation is along the x-axis. In view of

the fact that only TM-plasmons can propagate, the distinction of the different

plasmon modes is determined by the distribution of the sole magnetic field

component Hy across the gap or metal film [92, 93].

2.5.1. MIM-Structure (Metal Gap)

Two infinite metal planes separated by a gap filled with a dielectric or vacuum

in which a film plasmon propagates constitute a MIM- (metal-insulator-metal)

or simply a slot waveguide [149]. Economou [44] analysed in detail, for the

first time, the dispersion relation q(ω) by means of the DLM. Dionne and

colleagues [149] extended the investigation to SP’s propagation in a Ag-

SiO2-Ag p