Surface Plasmon Polaritons: Fundamentals and Applications Robin Cole School of Physics and Astronomy University of Southampton Southampton SO17 1BJ, UK September 11, 2006
Abstract Surface Plasmon Polaritons (SPPs) are of interest to a broad range of disciplines, including physics, chemistry and biology. In particular the science of Surface Enhanced Raman Spectroscopy (SERS) requires understanding in all three disciplines and is the subject of an interdisciplinary project at the University of Southampton. SERS is a highly sensitive technique for studying the Raman signal from molecules attached to a surface, in the presence of a strong electric field. For my research I am investigating the relationship between a nanostructured geometry, the electric fields in the structure, and the SERS signal from that structure. Hopefully the insights gained will allow us to engineer nanostructured surfaces for specific SERS applications, as well as elucidating the fundamental physics of electric fields in nanostructures.
Contents 1 Introduction
2
2 Surface Plasmon Polaritons
3
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2.2
Basic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2.3
Current Research into Plasmonic Surfaces . . . . . . . . . . . . . . . . . .
5
2.4
Experiments Undertaken . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
3 Void Theory
9
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2
Spherical Geometry Field Solutions . . . . . . . . . . . . . . . . . . . . . . 10
3.3
Truncated Void Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4 Conclusions 4.1
9
16
Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . 16
A Easily-coupled Whispering Gallery Plasmons in Embedded Nanospheres 19
1
Chapter 1 Introduction In recent years Surface Plasmon Polaritons (SPPs) have been the subject of intense research in a broad range of disciplines, including physics, chemistry and biology. One example of the science of SPPs bringing together the different disciplines is in the application of SPPs to Surface Enhanced Raman Spectroscopy (SERS). At the University of Southampton there are well established interdisciplinary collaborations, particularly between the Schools of Physics and Chemistry, which have a shared interest in the field of nanostructured gold surfaces applied to SERS. Scientists in the Electrochemistry group have manufactured unique surfaces and demonstrated their potential as substrates for SERS. However the optimisation of these surfaces can only progress with a detailed understanding of the physical processes occurring at the nanostructure surface. Physicists in the Quantum Light and Matter (QLM) group have been investigating the electric fields present in these nanostructures. The first eight months of my PhD have been spent investigating the optical properties of nanostructured surfaces which contain spherical cavities, which we call ‘nanodishes’. In this report I present the background theory supporting my work, details of the key experiments undertaken (which have led to a Phys. Rev. Lett. publication), and discuss the future direction of my research.
2
Chapter 2 Surface Plasmon Polaritons 2.1
Introduction
SPPs were first discovered by in 1902 by R.W.Wood as he investigated the light reflected from a diffraction grating. He observed drops in the reflected light at specific wavelengths, and this phenomenon came to be known as Woods anomaly [1]. SPPs were first given a theoretical description in the 1940s by U.Fano. He showed that in a real (i.e. not perfect) metal, light can be coupled into a surface mode which propagates along the surface of the metal, causing the drop in reflected light. A full theory describing SPPs was developed in the 1950s and 60s with important contributions from R.H.Ritchie [2] and H.Raether [3] among others.
2.2
Basic Theory
A coherent fluctuation of surface charges on a metal boundary is called a surface plasma oscillation. A longitudinal Surface Plasma Oscillation coupled to an EM field is called a Surface Plasmon Polariton, illustrated in Fig2.1a. Here I consider an SPP at a dielectric to metal boundary. To describe the behaviour of SPPs at such a boundary, it is necessary to solve Maxwells equations with appropriate boundary conditions, i.e. that the normal component of the electric displacement Dz be continuous at the boundary. The equations show that the interface can maintain spatially decaying Z components of the E-field in both media. In the metal free electrons undergo longitudinal oscillations, with associated
3
Z
E
(b ) Z
D i elec tr i c
(c )
X
Metal
ω=c k
ωp √2 EZ
S P P
Frequency
( a)
kx
Figure 2.1: ( a) the SPPs are a coherent fluctuation in electron density in the surface of a metal, with an E-field normal to the surface, (b) the field above the metal is evanescent and therefore non-radiative, (c) the dispersion relation of SPPs lies to the right of the light line, which represents a free space photon of the same frequency. E-field, which penetrate to the skin depth of the metal. Above the metal there is an evanescent field with maximum field at Z = 0 (Fig.2.1,b). Since TE light has no Z component of E-field it cannot couple to SPPs and is not considered. Solving the appropriate equations yields the SPP dispersion relation ksp :
ksp
ω = c
d m d + m
1/2
(2.1)
For ksp to have a real component, the permittivity of the metal, |m | > |d |, and m < 0, which is satisfied for metals which have m both negative and complex. The imaginary component of m determines the frequency dependent absorption in the metal, which in turn determines the SPP propagation length δsp , which is also dependent on surface roughness. The SPP dispersion relation is shown in Fig.2.1c. ksp lies to the right of the light line ( k0 =
ω ), c
and tends toward
ω √p , 2
the limiting
frequency is that of a 2D plasma. The SPP therefore cannot be coupled to by free space photons, and this momentum mismatch must be overcome to couple to the SPP. There are three main methods to do this [4], these are; prism coupling, scattering from a defect on a surface and scattering from a periodically corrugated metal surface, traditionally a diffraction grating. The prism coupling method has been used extensively to characterise SPPs, such as determining propagation lengths. However using the third method allows 4
ω 3
cK0sinθi n 1
4
G
Ki n
cK0sin(90) S P P 2
KS P
Figure 2.2: Incoming light of wavevector K0 sin θ0 , at 1 the photon is given momentum G and transformed to Ksp at 2. The reverse process occurs at 3 and Ksp decays to light at 4 via G. direct excitation of SPPs and places fewer limitations on the experimental conditions. A periodic corrugation, such as that of a diffraction grating, will scatter a photon, incident at some angle θ0 , giving it extra momentum, nG, where G = 2π/a is the grating wavevector and a is the period of the grating. By conservation of momentum;
ksp =
ω sinθ0 ± nG c
(2.2)
Therefore tuning of θ0 and a allows direct coupling to the SPP. Also the periodicity of the surface allows the reverse process and the coupling of non-radiative SPPs back to free space photons, illustrated in Fig2.2.
2.3
Current Research into Plasmonic Surfaces
There has been much interest in SPPs on nanostructured surfaces in recent years, enabled by advances in nanofabrication techniques, motivated in part by the discovery of a ‘2D plasmonic band gap’, and of ‘extraordinary’ transmission. The SPP band gap was first demonstrated on a 1D grating by R.H.Ritchie in 1968 [5] and on a 2D surface by W.L.Barnes in 1996 [6], who showed that if the period a of a corrugated surface is half that of the effective wavelength of the SPP, scattering forms two standing waves of the 5
same plasmon wavelength, but with different energies, corresponding to a low frequency mode in n1 and a high frequency mode in n2 . It is possible that in the future such surfaces could be used to create components of an SPP-photonic circuit. Also of interest is the more highly enhanced optical fields at the band edges, which suggest SERS applications. ‘Extraordinary’ transmission was first demonstrated by T.Ebbesen [7] who postulates that the phenomena is due to plasmonic standing waves formed on the patterned nanostructures. Ebbesen discovered that for a suitably patterned surface optical transmission through sub-wavelength holes is far higher than predicted by classical diffraction theory. Ebbesen explains that on these surfaces, standing wave plasmons form field intensity maxima at locations which coincide with the sub-wavelength apertures, producing an evanescent field that can tunnel through the hole. The small evanescent field on the far side of the film then scatters away from the hole and eventually couples back to a free space photon. If the metal film is thin enough the tunnelling may become resonant due to interactions between SPPs on either side of the film, further increasing transmission [8]. In both of these high profile examples the nanostructured surfaces were produced by ‘top down’ techniques, producing highly controlled surfaces, but at the expense of scalability, both physical area and volume production. The nanostructured surfaces I have studied are produced by a ‘nanocasting’ process, described in Ref[9], which can rapidly produce large surface area samples, for a number of materials (Au, Ag, CdTe etc), and for a range of sample pitches from 400nm to several µm. These samples are graded in thickness so that a single sample contains several geometries, illustrated in Fig2.3 [10]. The normalised thickness is defined as t¯ =
tvoid dvoid
where dvoid is the diameter of
the void cavity (or sphere) and tvoid the grown film thickness. The experiments I have undertaken reveal complex SPP modes. To help interpret our data we classify the SPPs observed as either localized plasmons, e.g. on spheres or in voids, commonly known as Mie plasmons, which are confined in spatial extent and feel only an local effective E-field, or delocalized plasmons, which we call ‘Bragg’ plasmons since they are coupled to by Bragg diffraction, which propagate over longer distances and 6
(a)
(b )
(c )
Figure 2.3: The nanostructured surfaces I have studied contain a spectrum of geometries from (a) shallow dishes which resemble a diffraction grating, to (b) a high aspect ratio grating which supports both SPPs and Mie plasmons, and (c) with embedded microspheres. feel an effective E-field dependent on the surface morphology. For thin samples (t¯ ≈ 0.2) our samples behave like a thin 2D grating, supporting SPPs. The SPPs are modelled using a weak scattering approximation, the dispersions of which follow the six-fold symmetry of the sample surface. The energy of the SPPs is therefore a function of the incident light wavelength, angle, and azimuthal orientation with respect to the sample, as well as the pitch a of the sample. The SPPs are described by Bloch waves (a plane wave modulated by a periodic potential, or ‘Bloch envelope’). The superposition of the SPP modes gives the optical field distribution on the surface, and we postulate that the nature of these fields determines the coupling to the localized plasmon modes. However all localized modes are not coupled into with equal strength, incident angle and polarisation dependence, and current calculations are only beginning to be able to take into consideration all these factors, and to date only for fully embedded spherical cavities. The latest theoretical models consider the full coupling between Mie plasmons and SPPs, and are discussed in section 3.
2.4
Experiments Undertaken
To date the bulk of my time has been spent on reflectivity measurements of gold films with embedded spheres, described in the Phys. Rev. Lett. appended to this report. The paper covers the details of the experiments and the theory developed to interpret our results. Here I list the experiments undertaken. 7
• Reflectivity measurements using the automated goniometer rig, in energy, incident angles θ and φ, TM and TE, for graded samples of spheres embedded in gold, for sphere sizes from 400nm diameter to 1.6µm. • Reflectivity measurements using a microscope to identify the spectra from single 1.6µm voids. • Reflectivity measurements on traditionally non-plasmonic surfaces, including CdTe, IrO and Pd. So far I have demonstrated the active electrochemical tuning of the absorption of IrO structured surfaces. • SEM study of the surface morphology to callibrate t¯ and to measure the dimensions. • SERS and fluorescence measurements on Pd and CdTe void samples. In summary I have planned and implemented a series of reflectivity measurements using the automated goniometer rig. This work has paved the way for further reflectivity measurements on new materials. I have also learned to perform more complicated SERS measurements, and to use the microscope and SEM for detailed studies.
8
Chapter 3 Void Theory 3.1
Introduction
There is currently intense interest in characterising the electric fields in nanoscale structures and analytical solutions have been found in many cases. These include homogenous spheres, multilayered spheres, ellipsoids, infinately long circular cylinders, infinately long elliptical cylinders and spheres with a eccentrically located spherical inclusion [11]. These solutions are examples of generalized Lorenz-Mie theory (GLMT), theories based on the well known Mie theory. Mie theory gives the analytical solution to Maxwells equations governing the scattering of a plane wave from a spherical particle, in the limit of a sphere which is small compared to the wavelength of the incident plane wave, first published in 1908. The theory describes the incident plane wave and the scattered field in spherical coordinates and provides equations, spherical Bessel and Hankel functions, describing the surface mode resonant frequencies in terms of a quantum numbers n, l , m, analogous to the atomic orbitals description of a hydrogen atom [12]. This analogy has suggested the description of microspheres as ‘plasmonic atoms’, and coupled microspheres as ‘plasmonic molecules’, discussed in the appended paper. The Mie solutions are also valid for a single spherical dielectric cavity within an infinite expanse of metal, and the mode energies predicted agree with experimental data for spherical voids with t¯ ≈ 1 [10]. There are also theoretical studies of spherical shells, buried arrays of spheres, and recently, truncated spherical cavities. Understanding the coupling to the modes is of key importance to optimise the structure geometry for real applications, for example to maximise the field 9
(a)
h b
a
l i g ht
(b )
θ
(c ) s
d s
t
r
d
l i g ht r
q Figure 3.1: (a) a nano-metallic shell, having external and internal radii a and b respectively, and shell thickness h, (b) side view of a nanoporous metal film containing a hexagonal array of spherical voids, suspended in vacuum, diameter d and separation s, nanovoids under the surface of a bulk metal were also studied, (c) a single truncated spherical void, thickness t, rim rounding radius r , gold thickness q, and incident angle θ. enhancement in a open cavity for SERS applications.
3.2
Spherical Geometry Field Solutions
We currently have a collaboration with the theorists F.J.Garcia de Abajo and T.V.Teperik, who have studied various structures containing some form of spherical geometry. In chronological order, the first geometry considered was that of a spherical nanoshell, illustrated in Fig3.1a [13]. The nanoshell work applied Mie theory to calculate the mode frequencies and the decay time of the modes (which is proportional to the local field enhancement), for various shell thicknesses h, and a number of sphere radii a. It was found that the energy of the voidlike (mostly bound to the inner surface) and spherelike (bound to the outer-surface) plasmon modes do not change dramatically with shell thickness h, but that the radiative decay rates of these modes do change dramatically with h. For nanoshells the total decay rate is determined by the combination of radiative and electronic (disipative) contributions. It was shown that for shell thicknesses h thinner than the skin depth δ, the voidlike 10
plasmon modes have lifetimes of order a few femtoseconds, and are more radiative than spherelike plasmons. For h > δ the voidlike plasmon mode lifetime is of the order of tens of femtoseconds, with ultrahigh local-field enhancements. For spherelike plasmons the lifetime decreases with decreasing shell thickness h as the electronic relaxation contribution increases, but for h ≥ δ the spherelike plasmons are overwhelmingly radiative. Therefore optimisation of the local field enhancement inside the void, at a specific wavelength for example, is not only achieved by choice of a but by consideration of the relationship between a, and δ. Taking this into account, a maximum field enhancement |f | of 60 for the l = 1 voidlike mode is predicted for a gold shell with h = 1.5×δ, when γ r = γd (i.e. radiative and dissipative broadening are equal). Also the decay rate α for voidlike plasmons decreases with increasing shell radius a, and so the local field enhancement increases with increasing a. The second geometry considered was a planar metal surface containing a 2D hexagonal array of embedded spherical cavities, illustrated in Fig 3.1b. Early work [14] showed that there is almost total absorption of light at the plasma resonance if the void spacing and depth of voids beneath the metal surface (both h) are a specific distance, which is less than the skin depth δ, for optimal parameters. More recently the coupling between Mie plasmons and SPPs in these structures has been investigated [15]. For a regular flat metal surface, SPPs cannot be excited. Due to the highly radiative nature of void plasmons (for s < δ) these modes are strongly coupled to. If the spacing of the voids is comparable to the SPP wavelength then the voids act as a coupling element, and propagating SPPs can also be excited. Such a structure would provide a flat surface which supports SPPs, making it a desirable substrate for many sensing applications. It is therefore possible to bring the SPP and void plasmon into resonance by engineering the structure parameters, or changing the angle of incident light. Where the two types of plasmon cross, ‘strong coupling’ behaviour is observed. An example of this is shown in Fig3.2, which shows the simulated reflectivity spectra, normalized to the reflectivity of a planar metal surface, for TM light, incident at azimuthal angle φ = 30◦ to the ΓM plane, on a flat gold surface with embedded 600nm diameter spherical cavities. The distance 11
Figure 3.2: (a) shows the simulated absorption spectra, Mie modes and SPPs are observed, green represents maximum absorption. (b) shows the normal-to-surface E-field due to the two SPPs on the metal surface in real space, with the location of the voids shown. (c) shows the scattering of the incident light represented in k-space, the incident light E p has a strong component in the q11 and q0−1 directions and strongly excites these plasmons. from the planar metal surface to the top of the voids is 5nm, much less than the skin depth δ (25nm for gold). In Fig3.2a the Mie plasmon modes l = 1 and l = 2 are clearly visible, and are very close in energy to the values predicted for a single void in bulk gold. Higher order Mie modes, which are more pinned to the surface of the void, are slightly red-shifted, due to interactions with neighbouring voids, and the close proximity to the metal surface. The dispersion of the SPPs closely match theoretical predictions of an ‘empty lattice approximation’, which calculates point scattering within the first Brillouin zone and zone folds the bands [16]. At φ = 30◦ the q11 and q0−1 modes are degenerate, due to the symmetry of the surface, so only four SPPs are observed. The Mie plasmons are much stronger than the SPPs since they are radiative, and easily coupled to. However the SPPs are coupled to by the buried voids, which provide a low aspect ratio, and so the SPPs are relatively weak. To help understand the coupling between the Mie and SPPs it is necessary to consider the charge distribution on the structure surface. The dipole moment (i.e. the charge distribution) of the Mie modes follows the external field (in angle and phase), whereas the SPP dipole moment follows their in-plane wavevector. We postulate that when the dipole moment of the Mie mode, and the surface charge distribution due to the
12
SPP match (i.e. are in phase), then the two modes form a lower energy ‘bonding’ state, the lower energy are of the mixed Mie-SPP arm,, and when the charges are out of ‘phase’, they are in an anti-bonding state, the higher energy arm. However the exact nature of this interaction is unclear and further research is necessary. Whilst this theoretical data has been compared with experimental results, there is not exact agreement, due to imperfections in the samples, such as the corrugated surface and ‘windows’ between the voids. Also this model is only capable of modelling fully encapsulated voids, whereas our samples always contain a small ‘window’ at the surface. It has recently become possible to model a fully truncated isolated void, and this study has been a key element of my research.
3.3
Truncated Void Modelling
The analytical solutions derived using Mie theory require that the cavity be spherically symmetric, which the truncated voids are not. However it is possible to computationally model this structure, employing a ‘boundary element method’ to calculate the electric field intensity inside the structure, described in Ref[17] and [18]. In this model the Efield is expressed in terms of the charges and currents on the structure surface. The boundary conditions provide a set of linear integral equations, with charges and currents as the unknowns. These equations are solved by discretizing the integrals. The structure must be axially-symmetric, but accepts a range of incident angles and polarizations. The frequency-dependent complex dielectric function must be input in a text file. An illustration of the structures I have been investigating is given in Fig3.1c. I have been investigating the truncated void for void diameters from 200nm to 2µm, normalised thicknesses t¯ from 0.125 to 1.03, incident angle from θ = 0◦ to θ = 75◦ , TM and TE polarizations, and a range of rim rounding radii r , from 10nm to 40nm. The thickness of metal used q, and the number of surface points are sufficient that the calculations are convergent. Each simulation requires an input text file, which defines the structure geometry, the incident light, and the data grid to be output, which can be a line (e.g. along
13
(a)
(b )
!
%& "# $
(c ) )
8 ,) )
! ;?
=>= ;<
9:
8 +) )
!
8 *) )
8 ( ) )
+ ) )
* ) )
( ) )
' ) ) ) ' , ) ) - . / 01 02 3 4 56 2 7
' + ) )
' * ) )
' ( ) )
Figure 3.3: (a) shows a plot of the field intensity round the vertical circumference of the void, in wavelength. (b) shows the field intensity along the vertical Z azis, and (c) shows a cross-section of the void at the wavelength corresponding to the Mie l = 2 mode. the central Z axis), a single plane or a 3D volume matrix. Examples of the simulation output are shown in Fig3.3. The model confirms the Mie mode energies for isolated encapsulated voids, and shows the evolution of these modes with truncation, for example Fig3.3c shows the l = 2 mode, for t¯=0.95. Of particular significance is the important role which the rim of the void plays in determining the energy and linewidth broadening of the Mie modes. It is clear that there are bonding/anti-bonding states between the Mie mode and the rim modes, which results in a splitting of the Mie modes, again particular to l . The effect of the rim-rounding will be investigated, but a direct comparison with experiment will be difficult since this parameter cannot be tuned in our samples. However this rim mode also appears to be instrumental in determining the angular coupling to the Mie modes. It has been shown that for completely encapsulated voids, the dipole moment of the Mie modes follows the external field, but for truncated structures, the bonding with the rim mode can ‘pin’ a Mie mode at a particular dipole orientation, over a range of incident angles. This may partly explain the angular dependence of the coupling to the Whispering Gallery Modes (WGMs) of the nanospheres. It is clear that the Mie modes scale with void diameter for visible wavelengths in gold, and further investigation is required to determine whether the rim modes scale similarly with circumference, or some other parameter. Although the program can only model isolated voids, and therefore does not consider the interaction between neighbouring voids, it will help to develop an
14
intuitive understanding of the interaction between neighbouring voids. Also since the dielectric function is input directly, it will also be possible to model the field in other materials, e.g. Pd.
15
Chapter 4 Conclusions 4.1
Conclusions and Future Work
In this report I have mainly focussed on the theoretical aspect of my research, since my experimental work is described in detail in the appended paper. However planning future experimental investigations will require a good understanding of the fundamental physics of these structures, so the theoretical element of my work is crucial. It is my feeling that the fields in the isolated truncated structures can be fully characterized in the near future, to such an extent that an intuitive understanding of the multi-void structure can be developed, whilst we are still unable to model these extended structures. As stated in the abstract, one of the motivations behind this research is to advance the science of SERS, and I anticipate performing angle resolved SERS measurements for comparison with theoretical predictions of the field theory. The theoretical background for SERS is well established and covered in Ref[19]. I anticipate that the understanding gained from the theoretical and experimental work, combined with advances in the fabrication techniques, will yield greatly optimised substrates for SERS in the near future. I will also investigate the linear properties semiconductor-metal nanostructures, as I have started to do with the CdTe structures.
16
Bibliography [1] R.W.Wood, Phil. Mag, 4, 396, (1902). [2] R.H.Ritchie, Phys. Rev. Letts, 106, 874-881 (1957). [3] H.Raether, ‘Surface Plasmons on Smooth and Rough Surfaces and on Gratings’(ed. Hohler, G.), Springer, Berlin (1988). [4] W.L.Barnes, A.Dereux, and T.W.Ebbesen, Nature, 424, 824-830 (2003). [5] R.H.Ritchie, E.T.Arakawa, E.T.Cowan, and J.J.Hamm, Phys. Rev. Letts, 21, 15301533 (1968). [6] S.C.Kitson, W.L.Barnes, and J.R.Sambles, Phys. Rev. Letts, 77, 2670-2673 (1996). [7] T.W.Ebbesen, H.J.Lezec, H.J.Ghaemi, H.F.Thio, and T.Wolff, Nature, 391, 667-669 (1998). [8] A.Degiron, H.J.Lezec, W.L.Barnes, and T.W.Ebbesen, Appl. Phys. Lett, 81, 43274329 (2002). [9] P.N.Bartlett, J.J.Baumberg, S.Coyle, and M.Abdesalam, Faraday Discuss. Chem. Soc, 125, 117 (2004). [10] T.A. Kelf, Y. Sugawara, R.M. Cole, J.J. Baumberg, M. Abdelsalam, S. Cintra, S. Mahajan, A.E. Russell, and P.N. Bartlett, submitted to PRB (2006). [11] G.Gouesbet, and G.Grehan, Pure Appl. Opt, 1, 706-712 (1999).
17
[12] A.D.Boardman, ’Electromagnetic Surface Modes’, J.Wiley and Sons, New York (1982). [13] T.V.Teperik, V.V.Popov, and F.J.Garcia de Abajo, Phys. Rev. B, 69, 155402 (2004). [14] T.V.Teperik, V.V.Popov, and F.J.Garcia de Abajo, Phys. Rev. B, 71, 085408 (2005). [15] T.V.Teperik, V.V.Popov, and F.J.Garcia de Abajo, Optics Express, 14, 1965-1972 (2006). [16] C.Kittel, ‘Introduction to Solid State Physics’, J.Wiley and Sons, New York (1996). [17] F.J.Garcia de Abajo, and A.Howie, Phys. Rev. Letts, 80, 5180-5183 (1998). [18] N.Ida, ‘Numerical Modelling for Electromagnetic Non-destructive Evaluation’, Chapman and Hall, New York (1995). [19] A.Campion, and P.Kambhampati, Chem. Soc. Rev, 27, 241 (1998).
18
Appendix A Easily-coupled Whispering Gallery Plasmons in Embedded Nanospheres
19
