Cathodoluminescence Brief Overview

UNIVERSITY OF ILLINOIS, URBANA CHAMPAIGN

Cathodoluminescence Spectroscopy

by

Anshuman Kumar

Final Report for the requirements of the course

Optical Spectroscopy (PHYS-552)

in the

Department of Physics

December 2010

http://illinois.edu

http://physics.illinois.edu/

Abstract

This report discusses some of the basic aspects of Cathodoluminiscence(CL) Spec-

troscopy, with special reference to plasmonics. CL setup offers several advantages. The

multimode imaging capabilities of the electron microscope enable the correlation of op-

tical properties (via cathodoluminescence) with surface morphology (secondary electron

mode) at the nanometre scale. But perhaps most intriguingly, the small beam can probe

a single selected nanostructure.

The report is divided into three chapters and a bibliography. Chapter one discusses a

general overview of CL and provides the mathematical background of the field of a mov-

ing charge in presence of an interface. Chapter two gives some idea of the experimental

setup involved, without going into much detail. Chapter three lists two applications of

CL: one involving a study of plasmon modes in a annular shaped nanoresonator; the

second application deals with probing the gap plasmon modes in a metal-sphere & metal

film configuration. In both these examples, I haven’t delved into the theory of plasmon-

ics itself, but just attempted to highlight how CL gives us a means to probe some of

these resonances, by varying excitation location.

Contents

Abstract i

List of Figures iii

1 Introduction: Overview, History and Mathematical Formulation 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.3 Principles of the technique . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.4 Mathematical Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4.1 Transition Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4.2 Surface Plasmons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Experimental Setup 8

3 Examples from Literature 10

3.1 Plasmonic Modes of Annular Nanoresonators Imaged by Spectrally Re-solved Cathodoluminescence[1] . . . . . . . . . . . . . . . . . . . . . . . . 10

3.1.1 Fabrication of Nanoresonators . . . . . . . . . . . . . . . . . . . . . 10

3.1.2 CL measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.2 Gap and Mie Plasmons in Individual Silver Nanospheres near a SilverSurface[2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2.1 What we expect? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2.2 Fabrication of the nanostructures . . . . . . . . . . . . . . . . . . . 13

3.2.3 CL measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Bibliography 17

ii

List of Figures

1.1 Cherenkov radiation glowing from a spent fuel core of the High FluxIsotope Reactor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Evanescent character of of the electromagnetic field produced by a fastelectron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1 CL setup incorporated in a Scanning Electron Microscope . . . . . . . . . 8

3.1 Panchromatic CL imaging of Ag annular nanoresonators . . . . . . . . . 11

3.2 Spectrally resolved imaging of plasmonic modes in an Ag annular nanores-onator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.3 Excitation of gap and Mie plasmons in supported silver nanospheres . . . 14

3.4 Schematic representation of the charge distribution in gap and Mie plas-mons excited in silver spheres supported on silver for different beam con-figurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.5 Angular and position dependence of the emission from gap and dipoleMie plasmons in supported spheres . . . . . . . . . . . . . . . . . . . . . . 16

iii

Chapter 1

Introduction: Overview, History

and Mathematical Formulation

1.1 Introduction

Cathodoluminescence(CL)[3][4] is basically the emission of photons from a specimen

stimulated using an electron beam. A common example of the CL process is the screen

of a television CRT monitor. Here, the electron beam generated by a cathode ray

tube, impacts a luminiscent material causing it to emit light. Today, CL is used in the

fields of geology, minerology, materials science and fundamental experimental physics.

Usually these are used, in conjunction with a scanning electron microscope, to examine

the internal structure of semiconductors, rocks, ceramics, glass and to study surface

plasmon modes in nanostructures.

This configuration of CL setup offers several advantages. The multimode imaging ca-

pabilities of the electron microscope enable the correlation of optical properties (via

cathodoluminescence) with surface morphology (secondary electron mode) at the nanome-

tre scale. The broad energy range of the electrons can excite wide-bandgap materials,

such as diamond- or gallium-nitride-based structures that are not easily excited by con-

ventional optical means. But perhaps most intriguingly, the small beam can probe a

single selected nanostructure.

1.2 History

Although the phenomenon of luminescence was recognized as early as the seventeenth

century, systematic observations and discussio of cathodoluminiscence did not take place

1

Chapter 1. Introduction: Overview, History and Mathematical Formulation 2

until around 1960’s.

Early CL studies were carried out using a CL microscope, which is basically a pet-

rographic microscope to which a cathode gun is attached. Subsequently, the scanning

electron microscope has been utilized to generate high resolution and high magnification

CL images.

CL was mostly employed for determining characteristics of geological materials. How-

ever, with the advent of nanotechnology, CL measurements are now routinely used to

investigate localized surface plasmons excited by electron impact. Here, the photon

emission of a metal nanostructure is induced via a high energy electron beam and col-

lected using a suitable detection pathway. The advantage of this technique is that by

scanning the electron beam over the particle surface, spatial profile of these modes can

be efficiently mapped out by light collection at particular resonance wavelengths . This

technique is also found to work for propogating Surface Plasmon Polaritons.

1.3 Principles of the technique

The main emission processes involved in CL are Cherenkov radiation and transition

radiation. It turns out that each of these processes is coherent with the field generated

by the incoming electron. Besides, although the generation of SPPs can be considered as

an indirect process (2-step), it is still coherent emission. This kind of coherence between

different sources of emission can result in interesting interference phenomenon, which

can be used for easy characterization.

However, there are also incoherent emission channels associated with electron-hole pair

generated which eventually recombine and emit. But in metals, which are the major

focus of this report, electronic relaxation proceeds much faster than radiative recombi-

nation, hence one can safely consider the incoherent contribution to CL to be a minor

contribution only.

Now we provide a brief overview of the emission processes involved.

Cherenkov Radiation

When a charged particle passes through a medium with speed greater than the speed of

light in that medium, it emits Cherenkov Radiation. The basic idea here is that these

charged particlespolarize the molecules of the medium, which return to their ground

state rapidly resulting in an emission of radiation. A classic example of this is the

characteristic ”blue-glow” of nuclear reactors. See for instance, figure 1.1.


Figure 1.1: Cherenkov radiation glowing from a spent fuel core of the High FluxIsotope Reactor[Wikipedia].

However, this emission is not important for metals, since the disruption in the local

electromagnetic field can be restored without emission of a photon.

Transition Radiation

This kind of emission occurs when a charged particle passes through a boundary between

two media of different dielectric constants. It is usually a result of the time dependent

variation of the dipole moment formed by the incident charged particle and its image

charge inside the other dielectric. This radiation can be explained by noting that since

the electric field of the particle is different in each medium, the particle has to get rid

of the difference when it crosses the boundary. This results in a net energy loss of the

particle.

In view of the above two phenomena, if metals are bombarded with an electron beam,

the excitation of bulk or surface plasmons can ocur if the moving charge couples to

the free electrons in the metal. This effect becomes apparent in Electron Energy Loss

Experiments (EELS)

1.4 Mathematical Theory

We know that an electron moving with a constant velocity in free space does not result

in a radiation. However, in the presence of a boundary of another material, it induces

a polarization charge, which together with the original electron can be considered to

be a dipole. In a metal, there are two channels available for the dipole to decay: (a)

transition radiation emission into the far field and (b) generation of surface plasmons.

We look at both these processes in the following subsections[5].


1.4.1 Transition Radiation

Consider an electron moving with a velocity v along the z axis. This electron is incident

from the lower half space of z < 0, which is considered vacuum (ε1 = 1) and corsses the

inrface with the upper half consisting of a dielectric of permittivity given by (ε1 = ε)

at time t=0. The electromagnetic fields are expected to satisfy Maxwell’s Equations (’j’

identifies the medium 1,2):

∇ ·H = 0 (1.1)

∇×E = −1

c

∂H

∂t(1.2)

∇ · εjE = 4πρ (1.3)

∇×H = −εjc

∂E

∂t+

4π

cj (1.4)

The charge and current densities associated with the moving electron can be written as:

ρ(z, t) = −eδ(z − vt) (1.5)

j(z, t) = −evδ(z − vt) (1.6)

To solve these equations, we use the method of fourier transforms:

H(r, t) =1

(2π)4

∫dωe−iωt

∫d3qH(q, z, ω)eiq·r (1.7)

The fourier transforms of the sources are found to be:

ρ(k, ω) = − e

(2π)δ(ω − k · v) (1.8)

j(k, ω) = vρ(k, ω) (1.9)


The next task is to to decompose the fields H into a a bulk component and an induced

component near interfaces. The solution for the bulk part is relatively straight-forward

and one gets:

Hbulkj (Q, z, ω) =

4πieQ

ceiωz/v

t

k2j − q2(1.10)

where t = z × Q and Q is the in plane momentum vector.

Performing an inverse fourier transform of the above equation over momentum, we

obtain:

Hbulkj (r, ω) = −2eω

vcγeiωz/vK1(

ωρ

γv)φ (1.11)

where ρ is the perpendicular distance form the electron trajectory, γ is the lorentz

contraction factor, φ is the azimuthal angle and K1 is the modified Bessel function

of second kind. From this equation, one can see that the moving electron acts as a

broadband source of ”electromagnetic field”. However, due to the nature of the Bessel

function, the field decays away from the electron trajectory. I have attached a figure1.2

from a paper which uses different variables for these quantities but shows the ideas well.

Figure 1.2: Evanescent character of of the electromagnetic field produced by a fastelectron[3]


There are some important observations related to this bulk field. The field diverges

at the location of the electron trajectory. Hence it means that the resolution of the

experiments is only limited by the size of the beam spot. However we must also consider

the response of teh material as delocalized, which reduces the actual resolution.

Next, we turn our attention to the induced field, which is evaluated to be

Hindj (Q, z, ω) = 2πekjsje

ikzj |z|αj t (1.12)

where kzj =√k2j −Q2, s1 = −1 and s2 = 1 and αj ’s are the boundary coefficients

calculated using appropriate boundary conditions on parellel components of the electric

and magnetic fields. The actual expressions are:

α1(Q) =2Qi/c

kz1ε2 + kz2ε1[−ω/vε2 + kz2ε1

q2 − k21− −ω/vε1 + kz2ε1

q2 − k22] (1.13)

α2(Q) =2Qi/c

kz1ε2 + kz2ε1[ω/vε2 + kz1ε2

q2 − k21− −ω/vε1 + kz1ε2

q2 − k22] (1.14)

Using the above two expressions, we get the expression for the induced field as

Hindj (r, ω) = −iφesj

∫ ∞0

QdQαjeikzj |z|J1(QR) (1.15)

where J1 is the first order Bessel function.

Since in experiments, we observe the CL signal in the far field, far away from the actual

impact point of the electron, it is clear that the field detected there cannot arise from

the evanescent bulk fields. Hence it is the induced fields that we observe.

1.4.2 Surface Plasmons

An electron incident on a metal surface also excites SPPs. The results of the previous

section can be used to derive the plasmon generation rate. The wave-vector condition

describing SPPs is given by:

kz1ε2 + kz2ε1 = 0 (1.16)

Using a Taylor expansion around the plasmon wave-vector, Qp one gets:

Hindj (ω) = πeQpAje

−ikzj |z|H11 (QpR)φ (1.17)


where H11 is the first Hankel function and Aj = αj(Q−Qp)

Chapter 2

Experimental Setup

As discussed in the previous chapter, CL measurements rely on the detection of radia-

tion emitted when a sample is bombarded by an electron beam. In order to carry out

spectroscopy, the emitted radiation has to be spectrally resolved. On the other hand,

for imaging, one requires spatial resolution of the position from which the emission orig-

inates. However, as proved in the previous section, the spatial resolution of the CL is

determined by the excitation source and not detection.

A schematic of the CL setup incorporated in a Scanning Electron Microscope is shown

in figure 2.1.

Figure 2.1: CL setup incorporated in a Scanning Electron Microscope.

For excitation, a focussed electron beam of a SEM is used. The accelerating voltage

of the beam is tunable between 1-30keV. Beam current is a function of aperture size

and voltage, and can be varied between several pA to tens of nA. Beam currents in the

range of tens of nA provide an electron impact rate which is significantly slower than the

electron relaxation processes in most noble metals important for plasmonics. Hence is it

8

Chapter 2. Experimental Setup 9

safe to consider single electron excitations. We can also accurately position the electron

beam to an accuracy of tens of nanometres over the sample.

The emitted light is detected using a parabolic mirror which is placed a few millimetres

above the sample in such a way that it’s focal point coincides iwth the sample. The

size of this focus is usually in tens of microns. The mirror is arranged to as to have

a large acceptance angle. The light collected from the focal point is reflected as a

parellel beam through a holllow waveguide tube and focussed onto the entrance slits of

a monochromator.

Chapter 3

Examples from Literature

In this chapter, we briefly discuss two examples, from recent literature, of the application

of cathodoluminescence.

3.1 Plasmonic Modes of Annular Nanoresonators Imaged

by Spectrally Resolved Cathodoluminescence[1]

In this study, the authors excite plasmonic modes in engineered annular nanoresonators

on Ag and Au surfaces, using a highly localized electron beam source and use spectrally

resolved CL imaging to probe the plasmon field intensity as a function of excitation

position. The details of the experiment are given below.

3.1.1 Fabrication of Nanoresonators

Nanoresonators are fabricated on Ag surface. The Ag structures were prepared by

evaporation on a quartz substrate and Focussed Ion Beam was used to patter the annular

rings. Each annular resonator has a central plateau and five concentric rings separated

by grooves 50nm deep with varying grating ring period and center diameter.

3.1.2 CL measurement

CL measurements were performed using a field emission SEM operating at 30keV, which

used a usual mirror based detection system. The electron beam spot size was 5nm (which

is also the limit of spatial resolution). To get spectrally resolved images, the emitted

light, after passing through the mirror, is sent to grating monochromator, which is set to

10

Chapter 3. Examples from Literature 11

a specific wavelength, each time. Secondary electron and CL images are simultaneously

recorded. In order to get panchromatic images, the emitted light is directly focussed on

the broadband PMT detection system.

Figure 3.1 shows the panchromatic CL images of nano-resonators in Ag with 315nm

grating period and three different center diameters. These images represent the radiation

collected from the entire resonator as a function of the electron beam excitation position

on the structure. Bright regions correspond to greater emitted photon intensity. From

this figure, it is clear that the high intensity is observed for excitation near at the edges of

the center and of the concentric rings (The locally increased emission inside the grooves

is attributed to scattering from roughness in the polycrystalline Ag film). The CL profile

clearly shows peaks in emission when the electron beam dwells near an edge. An overall

decay in emission intensity is observed as the electron beam moves outward from the

center. Thus, we see that a higher emitted photon intensity is obtained for electron

beam excitation in the center of the structure, indicating that more efficient excitation

and/or more efficient outcoupling occurs in this region. This effect was justified using

simulations, the description of which is not within the scope of the present paper.

Figure 3.1: Panchromatic CL imaging of Ag annular nanoresonators with 315 nmperiod and center sizes of (a) 620 nm, (b) 1.07 m, and (c) 1.70 m. (d) SEM imagetaken concurrently with panchromatic CL image of structure in (c). (e) Line profilesfrom regions indicated by the dashed line in (c) and (d) illustrating strong emissionwhen the electron beam is positioned at an edge and decreasing intensity as the beam

moves outward from the center[1].


Another study that they carried out was to experimentally probe the plasmonic modes,

as shown in figure 3.2. At = 350 nm, which is very close to the Ag surface plasmon

resonance, nearly uniform emission occurs for excitation anywhere in the structure. Near

resonance, surface plasmon propagation lengths are very short and thus no resonator

modes can build up. Several different modes are observed at longer wavelengths, as

illustrated in (c) part of Figure 3.2. At = 700 nm, CL data show bright emission for

excitation near the edges of the center plateau, but uniform emission from the rest of

the structure. In figure 3.2, the first subscript ’s’ of Ms,n describes the symmetry of the

mode: s=0 means that the field profile has a node at the centre of the plateau and s=1

stands for an antinode. Numerical methods like FDTD and BEM, given in the figure,

are common in the parlance of plasmonics people and the interested reader is requested

to read the relevant references provided to figure out the meanings of these terms.

Figure 3.2: Spectrally resolved imaging of plasmonic modes in an Ag annular nanores-onator with 620 nm center diameter and 315 nm grating period. (a) Spectrally resolvedCL images at the indicated wavelengths (b) SEM image of nanoresonator indicatingthe scan region for the CL images in (a). (c) Line profiles of modes M0,0,M1,0, andM0,1

from finite element (FDTD) simulated time-averaged electric field intensity, probabil-ity of CL emission from BEM simulations, and spectrally resolved CL images at the

indicated wavelengths. The corresponding surface topography is shown in gray[1].

In summary, the above application demonstrates high-resolution spectrally resolved CL

imaging as a powerful tool to reveal plasmonic modes in Ag annular nanoresonators.

Such a study of plasmonic modes excitation via excitation at precise (highly resolved)

excitation regions on the nanostructure, is a novelty of the CL method.


3.2 Gap andMie Plasmons in Individual Silver Nanospheres

near a Silver Surface[2]

In this study, CL is used to study the plasmons confined in the gap between silver

nanospheres and silver planar surfaces using angle and space resolved CL. Plasmons

in individual nanoparticles are excited by an electron beam, giving rise to light emis-

sion that is analyzed as a function of photon-energy, emission direction, and position

of the beam spot. The gap plasmons obtained by bringing a nanoparticle close to a

metal surface have been recently used to produce controlled 1010 enhancement factors

in surface-enhanced Raman scattering (SERS). However, gap plasmons are extremely

sensitive to the distance between metals, and therefore their analysis requires studying

individual structures. In view of what we discussed above, CL seems to be the obvious

choice for such a study.

3.2.1 What we expect?

The gap is a region of huge induced-charge pileup, so it has a large influence on the

plasmons of the nanoparticles. The emergence of a gap mode with m = 0 azimuthal

symmetry relative to the rotational axis of the particle-surface system results in the

aforesaid large pileup of induced charge near the gap, which is compensated by opposite

charges distributed over the rest of the particle. But the sphere can also support Mie

modes that are almost unperturbed with respect to the isolated particle, and in particu-

lar, the sphere dipole parallel to the surface remains rather unaffected by the interaction

with the substrate. We want to use CL to study these properties.

3.2.2 Fabrication of the nanostructures

Silver particles of size 50-600nm were obtained by evaporating silver in Ar atmosphere.

The silver film is deposited on either evaporated silver films or TEM carbon films.

3.2.3 CL measurement

Unlike the previous case, here angle resolved spectral CL is measured. In order to do

this, a TEM is used and ellipsoidal and parabolic mirrors are used for light collection.

A CCD camera provides an image of the mirror. The emission angle can be found out

from the position in the mirror image. Here the beam energy is 200keV and the beam

size is about 10nm.


Our main experimental results are summarized in Figure 3.3. A collection of CL spectra

from individual spheres are shown in Figure 3.3(a)(c) for different particle sizes. Figure

3.3(a),(b) corresponds to the emission recorded when the electron beam is passing nearly

grazing with respect to the particle surface, as shown in the upper insets. The spectra

are displays a number of maxima, the position of which is nearly insensitive to the

substrate, as deduced from the fact that their energies are almost the same when the

particles are deposited on either a carbon film or a silver substrate.

Figure 3.3: Excitation of gap and Mie plasmons in supported silver nanospheres. Mieplasmons are resolved in the cathodoluminescence emission when the spheres are excitedby a grazing electron beam, as observed in the spectra measured for particles of differentsize deposited on either (a) a 10 nm carbon film or (b) an optically thick silver surface.Gap plasmons are observed at lower energies in silver particles deposited on silver whenthe beam spot is close to the center of the spheres, as shown in (c). The size-dependentmeasured energies of both multipolar Mie plasmons (extracted from (a) and (b)) andgap plasmons (taken from (c)) are represented in (d) (symbols) and compared to thecalculated cathodolulminescence intensity from self-standing particles under grazingincidence (background density plot). The size-dependence of the gap plasmon obtainedfrom a simple analytical model and from full numerical simulations is shown as pink andyellow solid curves, respectively. The emission is collected over the upper hemisphere(backward emission) in all cases. The arrows in (ac) indicate the position of dipole

(red), quadrupole (green), hexapole (blue), and gap (black) plasmons[2].

From the plots, one can infer that the interaction between Mie modes and the substrate

is weak for grazing trajectories( since, as aforesaid, the position of maxima are nearly

independent of substrate). The only significant effect of the substrate is a small red


shift of the modes with a silver substrate as compared to the modes observed with

a thin carbon film, which seems to originate in the attractive interaction with image

charges.

However, we have an interesting observation when the beam spot is positioned at the

center of the particles which are deposited on the silver substrate, there is a new peak

which appears at low energies. This new mode disappears in case of carbon film sub-

strate. This is attributed to the gap mode, it being highly sensitive to the substrate.

Remarkably, the substrate plays a dominant role in this case, even though the external

excitation produced by the electron is initially located far from the gap region (actually,

the electron is likely to undergo strong collisions with silver atoms for diameters above

100 nm).

A physical picture of what is happening is presented via simulations in figure 3.4.

Figure 3.4: Schematic representation of the charge distribution in gap and Mie plas-mons excited in silver spheres supported on silver for different beam configurations(lower part), alongside the calculated electric near-field intensity component corre-

sponding to the frequencies of these modes for 140 nm particles (upper part)[2].

In order to confirm this picture, we can look at the angular distribution of emitted light

as shown in figure 3.5. The dipole picture clearly explains figure 3.5 in view of the model

of fig 3.4.

In conclusion, using the power of high spatial resolution offered by CL imaging, we could

study the Mie and Gap plasmons in this configuration. The present study demonstrates

that spectral CL provides the necessary power for a more detailed investigation of the gap


Figure 3.5: Angular and position dependence of the emission from gap and dipoleMie plasmons in supported spheres. A dipole Mie plasmon is observed at 2.5 eV in (a)and (b), which show the measured light intensity as a function of emission angle andenergy for an 140 nm silver sphere supported on a carbon film and a silver substrate, re-spectively, and excited by a grazing electron beam and a central trajectory, respectively(see insets of Figure 1a,c). An additional gap plasmon feature is observed at 1.4 eV in(b). The beam-position dependence of the emission with the silver substrate is shownin (c). The angular patterns of emission for dipole and gap plasmons are represented in(d) (symbols), as compared to the calculated emission from both a dipole placed rightat a silver planar substrate and oriented normal to it (red curved) and a dipole parallel

to the substrate at a distance of 70 nm (i.e., the sphere radius; blue curve)[2].

mode as a function of spacing, which should be of great importance for understanding

the mechanisms involved in ultrasensitive analysis based on plasmon confinement.

Bibliography

[1] Carrie E. Hofmann, Ernst Jan R. Vesseur, Luke A. Sweatlock, Henri J. Lezec,

F. Javier Garca de Abajo, Albert Polman, and Harry A. Atwater. Plasmonic

modes of annular nanoresonators imaged by spectrally resolved cathodolumines-

cence. Nano Letters, 7(12):3612–3617, 2007. doi: 10.1021/nl071789f. URL

http://pubs.acs.org/doi/abs/10.1021/nl071789f.

[2] N. Yamamoto, S. Ohtani, and F. Javier Garcia de Abajo. Gap and mie plasmons

in individual silver nanospheres near a silver surface. Nano Letters, 0(0), 0. doi:

10.1021/nl102862x. URL http://pubs.acs.org/doi/abs/10.1021/nl102862x.

[3] F. J. Garcıa de Abajo. Optical excitations in electron microscopy. Rev. Mod. Phys.,

82(1):209–275, Feb 2010. doi: 10.1103/RevModPhys.82.209.

[4] Martin Kuttge. Cathodoluminescence plasmon microscopy. Ph.D. thesis, March

2009.

[5] J.D. Jackson. Classical electrodynamics. 2007.

17

http://pubs.acs.org/doi/abs/10.1021/nl071789f

http://pubs.acs.org/doi/abs/10.1021/nl102862x

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