Symmetric and antisymmetric resonances in a pair of metal-dielectric nanoshells: tunability and closed-form formulas

Resonances of symmetric and antisymmetric polarization states in tightly coupled nanoshell particles made of either a metallic core and a dielectric shell or, vice versa, a dielectric core and a metallic shell were analyzed at optical frequencies. The investigation was performed by using the single dipole approximation (SDA) with all the dynamical retarded field terms included. Furthermore, analytic formulas for the four possible resonances were derived for the first time by retaining only the static (non-retarded) term in the dipolar field expression. The image principle was used to distinguish a priori between symmetric and antisymmetric modes and for full-wave simulations performed to confirm the identification of resonances achieved by the SDA. It was observed that the resonance frequencies of a pair of nanoshells can be tuned over a wide range of wavelength/frequencies by varying the relative dimensions of the core and shell. This makes this kind of particle pairs suited very well to be adopted either as constituents of metamaterials or to enhance local fields when operating frequencies range from the visible to the infrared spectral regions.


INTRODUCTION
The physical mechanism underlying the enhancement of the electromagnetic field intensities around subwavelength-size metal particles and their increased absorption and scattering cross sections is the coupling between the photons of incident light at certain frequencies and resonant collective oscillations of free electrons at the metal surface, referred to as localized surface plasmon resonances [1,2]. Surface-enhanced Raman scattering (SERS) is an important example in which metallic nanoparticles could be employed because of the strong enhancement of the scattered field [3,4].
The plasmonic properties of a nanostructure depend dramatically on its size and shape. Plasmonic pairs of various nanoparticles have been studied experimentally and theoretically, with the general observation that for incident light polarized along the dimer (pair) axis, a redshift of the longitudinal plasmon resonance occurs with decreasing nanoparticle separation [5,6]. Pairs of plasmonic nanoparticles constitute a simple system to achieve strong enhancements of near fields or scattering at optical frequencies, that could also be employed in SERS. Plasmon theory applied to nanoparticle pairs has led to the understanding of pair plasmon modes as symmetric and antisymmetric hybridized modes of the characteristic plasmons of the constituent nanoparticles [7]. The use of nanosphere pairs and rings has been also proposed to create artificial magnetism [8][9][10][11][12][13][14], similarly to what has been proposed based on pairs of rods [15][16][17]. A particularly useful nanoparticle is the metal-dielectric nanoshell [1,18]. A dielectric core-metal shell nanoparticle is a composite nanoparticle whose optical resonance can be designed in a controlled manner [18,19]. By varying the relative dimensions of the core and shell, the optical resonance of such a nanoparticle can be varied over hundreds of nanometers in wavelength, across the visible and into the infrared region of the spectrum. For example, the tunability of a nanoshell resonance is useful for SERS applications [20][21][22]. Moreover, using nanoshells with active materials could result in a reduction of the losses in pairs of plasmonic nanoparticles [23]. Whereas plasmonic properties of individual nanoshell particles are well understood, few studies have been performed on the optical properties of nanoshell pairs [20,[24][25][26][27].
In this paper, we show how tuning of both symmetric and antisymmetric resonance frequencies in pairs of electromagnetically coupled nanoparticles can be achieved by exploiting the additional degree of freedom provided by using metal-dielectric nanoshells, in contrast to using single-metal nanoparticles. To this aim, the nanoparticles are modeled by using the single dipole approach (SDA) [1], which has been summarized in Ref. 28. We also provide closed-form formulas representing the four resonance frequencies in a pair of nanoshells by combining the SDA with a quasistatic approximation for the field emitted by the dipole, valid for small and very close nanoshells. Finally, the excitation of the resonant modes identified by the SDA by an illuminating wave field is discussed presenting some fullwave simulation results. Figure 1 shows the two considered nanoshell pair configurations, where 1 r and 2 r are the core and shell radii, respectively, and d is the center-to-center distance between the two particles. The nanoshells are made up of either a metal core and a dielectric shell ( Fig. 1(a)) or a dielectric core and a metal shell ( Fig. 1(b)).

STATEMENT OF THE PROBLEM
The purpose of this paper is to characterize the resonant behavior of these two structures by varying the structure parameters described above, stressing resonances' tunability.

FORMULATION
In this section, we derive two scalar equations for predicting the resonance modes of a pair of nanoshells, resulting into simple closed-form expressions for the resonance frequencies associated to the structures. These expressions also permit to account for the presence of a host medium with permittivity different from that of free space.

Electromagnetic modes of two coupled metallic spheres
The electric field at the position 2 r (ˆ/ 2 d = z in Fig. 1) generated by a dipole with dipole moment 1 p placed at the position 1 r (ˆ/ 2 d = −z in Fig. 1) can be expressed by means of the dyadic Green's function (electromagnetic fields are assumed to be time harmonic with an i t e − ω time variation) where the expression of the dyadic Green's function is given in Refs. 28 and 29. Note that because of reciprocity 2 1 1 2 ( , ) ( , ) = G r r G r r . When a subwavelength metallic nanoshell is immersed in an electric field, its induced dipole moment p is given by where E is the local electric field at the particle location. For nanosphere shells, the static expression for the inverse polarizability is given by [1,22,30] where h ε is the relative permittivity of the host medium, 1 ε is the relative permittivity of the core (of radius 1 r ), and 2 ε is the relative permittivity of the shell (of external radius 2 r ). Furthermore, where 0 k denotes the free space wavenumber. The last imaginary term in (2) has been introduced to account for particle radiation [1,28]. The Drude model is used to describe the metal permittivity where p ω is the plasma angular frequency, γ the damping term, and ∞ ε is a "high frequency" permittivity determined to match experimental data. Here, for silver we use the values provided in Refs. 8 and 31: 5 ∞ ε = , 16 1.37 10 p ω = × rad/s, and 12 1 27.3 10 s − γ = × .
We aim at finding the eigenvalues and eigenvectors of the two-sphere system ( Fig. 1) by solving the coupled equations with the information that Er . Combining these equations with (4), and following the procedure outlined in Ref. 14, we obtain two scalar equations, one for the transverse polarization (electric dipole moment orthogonal to the axis of the pair, z ) and one for the longitudinal polarization (electric dipole moment along the axis of the pair).
The resonances for the two transverse antisymmetric/symmetric polarizations are found by solving numerically where the " + " sign is for the antisymmetric mode ( 2 1 = − p p ), whereas the " − " sign is for the symmetric one ( 2 1 = p p ). Analogously, the resonances for the longitudinal antisymmetric/symmetric polarizations are found from 3 3 The eigenvectors (5), with the " + " (antisymmetric) and " − " (symmetric) sign, respectively. The two eigenfrequencies 1 f and 2 f associated to the two longitudinal (parallel to the pair axis) resonances in Figs. 2(d) and 2(a) are solutions of equation (6), with the " + " (antisymmetric) and " − " (symmetric) sign, respectively.

Quasistatic (QS) field approximation for resonance modes of two metallic nanospheres
Here we are interested in deriving the analytical expressions for the resonances associated to a nanoshell pair. If the particles are small compared to the free space wavelength, it is possible to replace some electrodynamic terms by electrostatic ones. In SDA calculations, this is equivalent to setting 0 h k = in (2), (5), and (6) (see also Ref. 28) but keeping the correct frequency-dependent dielectric constant m ε in (3). This approximation applies specifically to the dynamics of a single particle, but it is here used in combination with the SDA method to describe the electromagnetic behaviour of a pair of nanoshells [32,33]. Therefore, after neglecting the dynamic terms (in other words we keep only the by taking into account that p γ << ω , (5) and (6) are solved analytically for the resonant angular frequencies leading to the expressions for transverse and longitudinal polarizations for the two cases of metal core-dielectric shell and dielectric core-metal shell.

Metal core -dielectric shell
For the transverse polarization, the procedure described above leads to where ( ) 3 2 r d δ = , and the upper/lower sign applies to the antisymmetric/symmetric transverse polarization. The formula coefficients are For the longitudinal polarization instead, the closed-form expression for the antisymmetric (upper sign) and symmetric (lower sign) resonances is: Note that expression (10) is the same as (7)

RESULTS
We show the resonant frequencies for the dynamic, fully retarded case ("Dyn", continuous lines in Figs

Analysis of resonant frequencies
The real and imaginary parts of the four resonance frequencies of a pair of nanoshells are plotted versus the ratio of the core and shell radii 1 2 r r ρ = in Fig. 3 for the metallic coredielectric shell case, and in Fig. 4 for the dielectric core-metallic shell one, keeping the nanoshells center-to-center distance at 125 d = nm and the shell radius at 2 50 r = nm. . This is because radiation losses for symmetric modes are significantly larger since the pair of nanoshells radiates like a dipole, whereas the two radiation contributions associated to the two antisymmetric (opposite) dipoles tend to cancel out. In other words, the transverse resonance for the antisymmetric pair tends to radiate like a magnetic dipole [11,14] plus an electric quadrupole [11]. Indeed, the antisymmetric mode is associated to a current loop (magnetic dipole) and it is a classic result of antenna theory that a small loop radiates less than a small dipole of same current and dimension. Note that the transverse antisymmetric resonance frequency 1 t f is smaller than the symmetric one 2 t f , for all cases considered. Also, note that the longitudinal symmetric resonance frequency 2 f is smaller than the antisymmetric one 1 f , except for very small values of 2 r d τ = . And in general (but not always), the resonances are ordered as 2 f It is also noted that at small ratios 2 r d τ = the real parts of the four resonance frequencies tend to be similar, and close to the resonance of a single particle, due to weaker coupling.
The results obtained by including in the calculations all the dynamic field components of the dipole radiation (5) and (6), which are denoted as "Dyn", are compared with those computed by the closed-form expressions (7) and (10)−(12) derived by retaining only the quasistatic (denoted as QS) dipolar field terms as described in Sec. 3.2.
(a) (b) Fig. 3. Real and imaginary parts of the four resonance frequencies versus ratio of radii, for a sphere distance d = 125 nm (r 2 = 50 nm) for the metallic core-dielectric shell structure. "Dyn" means that all dipolar dynamic field components are considered; "QS" means that only the static component 3 1 d of the dipolar field is considered.
(a) (b) Fig. 4. Real and imaginary parts of the four resonance frequencies versus ratio of radii, for a sphere distance d = 125 nm (r 2 = 50 nm) for the dielectric core-metallic shell structure.
From Fig. 3 one can notice that the dynamic and quasistatic solutions agree well for small values of the ratio of radii (with fixed 125 d = nm) for the case in Fig. 1(a), and for high values of the ratio of radii (with fixed 125 d = nm) for the case in Fig. 1(b). In general, the QS approximation is less accurate than the fully dynamic solution because it neglects the effect of retardation in the coupling mechanism between the two nanoshells and the 1 d and 2 1 d field terms. Therefore, the stronger the coupling between the two nanoshells is, the less accurate the QS results are. To illustrate this, in Figs. 5 and 6 we show that for larger shell radius r 2 (and therefore for larger scattered field) the QS approximation tends to diverge from the dynamic solution because it cannot correctly describe the interaction between the particles. On the other hand, in Figs. 7 and 8 (that show the resonant frequencies for varying 2 r d τ = , at a fixed ratio of the core and shell radii 1 2 0.8 r r ρ = = ), the "QS" and "Dyn" solutions, though in general agreement with each other, show a small difference which is more or less independent from the distance d. This is because though the QS approximation for small d is expected to be better than for large d, the amount of coupling between the nanoshells is inversely proportional to d, and the influence of the coupling on the estimation of the resonant frequencies affects the overall approximation. for fixed shell radius r 2 = 40 nm (i.e., the distance d between the nanoshells is variable) and 1 2 0.8 r r ρ = = for the metallic core-dielectric shell structure.
based on the static dipole expressions and the static polarizability expression (2), which yields 1 564.5 t f = THz and 2 577.2 t f = THz. From these comparisons we can conclude that though the SDA, with the static or quasi-static polarizability, fails to predict the exact resonant frequencies, it still provides useful insight in the coupling mechanism of nanoshell pairs. Indeed, by using a more accurate expression for the polarizability based on the Lorenz-Mie theory [1], we expect that the accuracy of SDA could be greatly improved [28,35]. The electric field at the center of each nanoshell induced by a single normally incident plane wave polarized along y, is shown in Fig. 9(b), by using the SDA and CST. The field 2 E at the center of the upper nanoshell exhibits two peaks, associated to the antisymmetric and symmetric modes, whereas the field 1 E at the center lower nanoshell exhibits only the strongest peak associated to the antisymmetric mode. In fact, the superposition of the antisymmetric and symmetric resonant modes with similar amplitudes, because their peaks are located at close frequencies for the considered configuration, and opposite phases, make the field at the lower nanoshell almost completely vanishes in the vicinity of the higher frequency (symmetric) resonance. The electric and magnetic field distributions induced in a pair of nanoshells by a single normally incident plane wave polarized along y at frequencies f = 547 THz and f = 565 THz, close to the transverse antisymmetric and symmetric resonances, respectively, are illustrated in Fig. 10, where the arrow plots representing the magnitude of the electric field vector are superimposed to the color maps of the magnetic field intensity. The shell radius is r 2 =35 nm, the ratio of the radii 1 2 0.8 r r ρ = = and the center-to-center distance is d =100 nm. The direction of the electric field is consistent with the orientation of the dipole eigenvectors for the transverse resonance modes deduced by the SDA in Sec. 3.1. Consistently with Fig. 9(b), close to the transverse antisymmetric resonance (Fig. 10(a)) the magnitude of the electric field at the lower nanoshell appears to be comparable with that at the upper nanoshell, whereas at the symmetric resonance frequency (Fig. 10(b)) it is considerably smaller.  Fig. 10. Electric (y-z plane) and magnetic (along x) field distributions for a pair of silver nanoshells, with shell outer radius r 2 =35 nm, ratio of the radii 1 2 0.8 r r ρ = = and center-to-center distance d =100 nm, for the dielectric core-metallic shell structure. (a) Antisymmetric resonance mode excited by a single plane wave coming from +z at f = 547 THz (note the strong magnetic field between the two spheres); (b) symmetric resonance mode excited by a single plane wave at f = 565 THz.

CONCLUSION
In this paper we have shown how pairs of tightly coupled nanoshells offer the possibility of wide control of resonance frequencies and so their tunability, achievable by varying geometrical parameters. We have derived closed-form formulas for predicting resonance frequencies, based on a quasi-static approximation, and we have compared these results to the solution comprehensive of all retarded field terms in the single dipole approximation. Moreover, these structures are shown to be able to support an antisymmetric mode that can be interpreted as an effective magnetic dipole; therefore, they could be used for providing artificial magnetism in metamaterials that may support backward propagation or have equivalent high/low characteristic wave impedance.