Optical properties of a plasmonic nano-antenna: an analytical approach

The optical properties of a plasmonic nano-antenna made of two metallic nanospheroids (prolate or oblate) were investigated analytically in quasi-static approximation. It is shown that in clusters of two nanospheroids, three types of plasmonic modes can be present. Two of them can be effectively excited by a plane electromagnetic wave, while the third one can be effectively excited only by a nanolocalized light source (an atom, a molecule or a quantum dot) placed in the gap between the nanoparticles. Analytical expressions for the absorption cross-section, the enhancement of local fields and the radiative decay rate of an excited atom placed near such a nano-antenna are presented and analyzed.


Introduction
Very recently, quite a number of works have been devoted to the study of the optical properties of single nanoparticles and their clusters. Special attention is paid to metal nanoparticles with the help of which it is possible to enhance electric fields at frequencies of localized plasmon resonances [1][2][3]. On the basis of this effect, a variety of possible applications were considered. The most developed is the use of large local fields near a rough surface to increase surfaceenhanced Raman scattering (SERS) [4]. Modification of fluorescence by means of nanoparticles of different shapes is the basis for the creation of nanobiosensors [5][6][7][8][9], nano-antennas [10][11][12][13][14], devices for the decoding of DNA structure [15], etc.
In this paper, we present the results of an analytical study of the optical properties of clusters of two metallic prolate or oblate spheroidal nanoparticles. Such clusters are investigated both experimentally and numerically and form the basis for various possible applications, including nanosensors, nano-antennas and plasmon waveguides [39][40][41][42][43][44][45][46]. In principle, the optical properties of such two-spheroid clusters can be investigated analytically by full analogy to two-sphere clusters [19], [47][48][49]. However, as far as we know, there is only one analytical investigation of the optical properties of two-nanospheroid clusters carried out with the help of a plasmon hybridization method [50]. In this paper, the interaction between unperturbed plasmonic modes of prolate spheroids was calculated by numerical integration. Here, we continue the investigation of this system with the help of a new translational addition theorem [51]. It allows us to find an analytical description of the interaction between unperturbed plasmonic modes and to derive the solution for the cluster of two spheroidal nanoparticles, placed in an arbitrary external field. The geometry of the considered problem is shown in figure 1. For simplicity, we will consider that the cluster consists of two equal nanospheroids made of a material with dielectric constant ε and placed in a vacuum.
Significant attention will be paid to the case of nearly touching and strongly interacting spheroids since this is the case that seems to be most interesting for applications, because a substantial enhancement of electric fields occurs there. The opposite case of weakly interacting spheroids can be easily treated with the approximation of spheroids by point dipoles with corresponding polarizabilities [3].
For an illustration of the analytical results obtained, we will consider the case of two identical (prolate or oblate) nanospheroids made of silver [52]. We suppose that the largest size of the nanospheroid is equal to 30 nm and the aspect ratio of the spheroid is taken to be equal to 0.6.
The rest of this paper is organized as follows. In section 2, free plasmon oscillations of a two-nanospheroid cluster are investigated. The results of this section reveal the underlying physics and are necessary for interpreting the results of other sections. In section 3, we will consider the optical properties of a two-nanospheroid cluster placed in the field of a plane electromagnetic wave. Here, we find out the absorption cross-section and the factor of local field enhancement. In section 4, the objects of examination are the optical properties of a twonanospheroid cluster placed in the field of a radiating atom or a molecule, whose decay rates are calculated here.

Plasmon oscillations in a cluster of two nanospheroids
It is well known that all optical properties of nanoparticles can be derived from their plasmonic spectra, i.e. from the related plasmon eigenvalues ε ν and eigenfunctions e ν and h ν , which are 4 solutions of the sourceless Maxwell equations [53], whereε = ε ν inside the nanoparticle andε = 1 outside it, ω is the frequency of electromagnetic oscillations and v c is the speed of light in vacuum. As a result, the electric field in the presence of any nanoparticle can be presented in the following form [53], where ε(ω) describes the dependence of dielectric permittivity of the nanoparticle's specific material on frequency ω, E 0 is the excitation field and ν is a vector index that defines the specific plasmonic mode. From (2), it is possible to find the optical properties of a nanoparticle or a cluster of nanoparticles. So, to understand very complicated optical properties of a twonanospheroid cluster, we should first investigate the plasmonic spectrum of this system. To study the plasmon oscillations and other optical properties of clusters of two nanospheroids, it is enough to solve the quasi-static equations, div(εe ν ) = 0, rot e ν = 0, which can be reduced to solutions of the Laplace equations by substituting e ν = −∇ϕ ν , ϕ in ν = 0, inside the nanoparticle, ϕ out ν = 0, outside the nanoparticle, at the surface of the nanoparticle.
In (4), ϕ in ν and ϕ out ν are the potentials of plasmonic eigenfunctions inside and outside the nanoparticle correspondingly, and ∂ϕ ν ∂n S denotes the normal derivative at the nanoparticles' surface S. The last equation in (4) provides continuity of the normal components of electrical induction. Note that in this case there is no need to find magnetic fields for the description of plasmonic oscillations.
The systems of equations obtained in such a way have nontrivial solutions only for some negative values of permittivity ε ν defining the frequency of plasmon oscillations [2,3]. In the case of the Drude theory, ε(ω) = 1 − ω 2 pl /ω 2 , the frequency of plasmon oscillations can be found from the expression where ω pl is the bulk plasmon frequency of a metal from which the nanoparticles are made. Our approach allows us to investigate arbitrary spheroids, but for simplicity in the present 5 section we examine the equations for the plasmon oscillations in a cluster of two identical metal nanospheroids.
In the case of a two-nanospheroid cluster, we will look for a solution as follows. The total potential outside the spheroids will be the sum of their partial potentials (we will omit the mode index ν further) [20,54], while the potentials inside each nanospheroid will be denoted by ϕ in j ( j = 1, 2). To find ϕ out and ϕ in 1 , ϕ in 2 , it is natural to use spheroidal coordinates. In the case of a prolate nanospheroid, the relation between the Cartesian and the spheroidal coordinates where f = √ c 2 − a 2 is half of the focal distance in a prolate spheroid (a < c) whose surface is set by the equation (x 2 + y 2 )/a 2 + z 2 /c 2 = 1. In the case of an oblate spheroid (a > c), the relation between the Cartesian and the spheroidal coordinates (0 ξ < ∞, −1 η 1, 0 φ 2π) has the following form [55], where f = √ a 2 − c 2 is half of the focal distance in the oblate spheroid. Let us note that this expression can be obtained from (7) by the substitutions ξ → iξ and f → −i f . Further, we will use this formal replacement since it is fundamental and allows us to find a solution for oblate spheroids if the solution for prolate spheroid geometry is known [54][55][56][57].

Plasmon oscillations in a cluster of two identical prolate nanospheroids
To find plasmonic spectra of a two-nanospheroid cluster, it is natural to use two local systems of spheroidal coordinates (ξ j , η j , φ j , j = 1, 2), the origins o j of which are placed at the centers of corresponding nanospheroids and separated from each other by the distance l (see figure 1(a)). The coordinates (and all other values) related to the first or second nanospheroid will be denoted by the index '1' or '2', respectively. The potential inside the jth nanospheroid can be presented in the following form [58] ( j = 1, 2), where P m n (η) is an associated Legendre function [59] defined in the region −1 η 1 and P m n (ξ ) is an associated Legendre function [59] defined in a complex plane with the branch cut from −∞ to +1. The partial potential outside the jth nanospheroid can be presented [58] as ( j = 1, 2) where Q m n (ξ ) is an associated Legendre function of the second kind [59] defined in a complex plane with the branch cut from −∞ to +1.
By construction, the potentials (9) and (10) are solutions of the Laplace equation [58]. So, to find a solution of (4), one should use only the boundary conditions where ξ 0 = c/ √ c 2 − a 2 = c/ f are local radial coordinates defining the surfaces of the nanospheroids and ε is the permittivity of materials from which the nanoparticles are made. To reduce the boundary conditions (11) to a system of linear equations, we apply the translational addition theorem to the wave functions of the prolate nanospheroid [51]. In the case of two identical coaxial nanospheroids, this theorem gives ( j, s = 1, 2, j = s, where Applying the boundary conditions (11) and the theorem (12), one can obtain the following system of equations (n = 0, 1, 2, . . . ; m = 0, 1, 2, . . . , n), When deriving (14), we made use of the fact that, for identical nanospheroids, S (2) mnmq = (−1) n+q S (1) mnmq (see (13)) and take S (0) mnmq = (−1) m+n S (1) mnmq . The system of equations for D ( j) mn is identical to (14) and gives no additional information for plasmonic spectra of coaxial spheroids. So, we will not consider it further.
As it results from the symmetry of the considered cluster and the system (14), there are two independent types of solutions (plasmonic modes) with opposite parity. To select these modes, 7 one should choose C (1) mn = ±(−1) m+n C (2) mn in (14). As a result, we shall obtain the following system of equations for the modes with definite parity, where '+' and '−' signs correspond to modes that are symmetric (+) or antisymmetric (−) relative to z → −z transformation. It is important to note that separation of the spectra into symmetric and antisymmetric plasmon modes is possible only in the case when there is a plane of symmetry. When m is even, antisymmetric modes have nonzero dipole moment and they are 'bright' modes. In contrast, symmetric modes have zero dipole moment and are 'dark' modes when m is even. In the case of odd m, the 'bright' and 'dark' modes correspond to the symmetric and antisymmetric modes, respectively. One can expect that the antisymmetric mode m = 0 will have the largest polarizability and thus will be the 'brightest' one for the excitation of our cluster with a longitudinally (along the z-axis) polarized plane wave.
To study plasmon oscillations in clusters of two prolate spheroidal nanoparticles, we have solved the eigenvalue problems (15) numerically. In figure 2, the normalized plasmon frequency ω/ω pl of a cluster of two prolate nanospheroids (see figure 1(a) for the geometry), corresponding to the first four plasmon modes, is shown as a function of normalized distances l/2c between the nanoparticles' centers. Eigenvalues ε have been obtained as a nontrivial solution of the equation system (15) in the case of an axis-symmetric problem (m = 0). Then, the obtained solutions have been substituted into (5) to obtain the plasmon oscillation frequency.
In figure 2, one can observe that plasmon frequencies of a cluster of two prolate nanospheroids tend to plasmon frequencies of a single nanospheroid (see figure 2(c)) if the distances between the nanospheroids are large enough. When the width of the gap between the nanospheroids tends to zero, the solutions of the equations (15) behave very differently. For symmetric modes (figure 2(a)), there are two branches: T-modes and M-modes. Modes of 'T' type can be obtained by the method of hybridization of plasmon modes of a single prolate nanospheroid [50]. When the width of the gap between the nanoparticles is decreasing to zero, normalized plasmonic frequencies of T-modes tend to various values in the range from 0 to 1/ √ 2, in analogy to a two-sphere cluster [21,60]. T-modes with higher indices (not shown for clarity) will concentrate near ω/ω pl = 1/ √ 2. In figure 2(a), one can also see that at very short distances between the nanospheroids (l/2c < 1.1), a new type of plasmonic modes (M-modes) appears. M-modes are characterized by strong spatial localization in the gap between the nanoparticles. As a result, they can be effectively excited only by a strongly nonuniform electric field of the molecule or the quantum dot. Values of plasmonic frequencies of these modes lie in the range ω pl / √ 2 < ω < ω pl . As the gap width decreases to zero, the plasmon frequency of M-modes tends to the bulk plasmon frequency ω pl .
In figure 2(b), nontrivial solutions of the equation system (15) for the antisymmetric potential in an axial-symmetric case (m = 0) are shown. In analogy to a two-sphere cluster, we will call these modes L-modes (longitudinal) because they are 'bright' only for longitudinal excitation. These modes can be described by the hybridization method of plasmon oscillations of single nanospheroids forming the considered cluster [50]. As the width of the gap between shows plasmon frequencies of the single prolate nanospheroid as a function of the inverse aspect ratio c/a. The vertical line corresponds to a/c = 0.6 and allows us to select asymptotic values for panels (a) and (b).
prolate nanospheroids decreases to zero, normalized plasmon frequencies of these modes tend to zero as it also takes place in the case of spherical nanoparticles [21,60]. Plasmonic frequencies of L-modes of higher orders (not shown) tend to ω pl / √ 2, and concentration of an infinite number of L-modes occurs near this value.
In figure 3, the distribution of a surface charge of plasmonic modes of the lowest order in clusters of two identical prolate nanospheroids is shown. It is seen in this figure that the T-and M-modes have symmetric distribution of the surface charge in contrast to the antisymmetric L = 1 mode. This behavior, of course, is in agreement with the symmetry of the equations (15). Another interesting feature is that the surface charge of T-modes is distributed over the surface of all of the nanoparticles for any distances between them, while for M-and L-modes it is concentrated near the gap between the nanospheroids if the distance between them is sufficiently small. It is interesting to note also that the surface charge of M-modes is more concentrated in comparison with that of L-modes. Indeed, due to an electroneutrality requirement, the total surface charge on each nanospheroid should be equal to zero. Here, both positive and negative charges of the M-modes are localized near the gap between the nanoparticles so that in the rest of the nanoparticles the charge is almost equal to zero, as is clearly seen in figure 3(a). At the same time, in the case of L-modes for each of the nanospheroids, near the gap a charge of only one sign is concentrated, and a charge of the opposite sign is distributed with small magnitude over the remaining surface of the nanoparticles. Therefore, strictly speaking, the surface charge in an L-mode is distributed over the entire surface of the cluster of nanoparticles although it is not clearly seen at small distances between the nanoparticles (see figure 3(a)). As the distance increases, the charge distribution changes in the cluster: it spreads over the nanoparticles' surface, tending in the limit to a distribution corresponding to single prolate nanospheroids (see figure 3(c)).

Plasmon oscillations in a cluster of two identical oblate nanospheroids
In this geometry, one should also use local systems of coordinates (ξ j , η j , φ j , j = 1, 2) that are connected to each nanospheroid, have origins o j in their centers and are separated from each other by the distance l (see figure 1(b)). Now, the electric potential inside the jth nanospheroid can be presented in the form ( j = 1, 2) and the partial potential outside the jth oblate nanospheroid will look like ( j = 1, 2) The total potential outside the nanospheroids will be expressed by (6). As boundary conditions for the potential, (11), In the case of oblate nanospheroids, the addition translation theorem has the following form [51] ( j, s = 1, 2, j = s), where in which and δ 0 p is a Kronecker delta symbol. Now, substituting (16) and (17) into (11) and making use of the translational addition theorem (18), we obtain the following system of equations (n = 0, 1, 2, . . . ; m = 0, 1, 2, . . . , n), By deriving (21), we take into account the fact that M (2) mnpq = (−1) m+ p M (1) mnpq (see (19)) and denote M (0) mnpq = (−1) m M (1) mnpq . The system of equations for D ( j) mn is analogous to (21), and we will not analyze it here.
Owing to the symmetry of a cluster of two identical oblate nanospheroids, there are two types of plasmon oscillations: symmetric and antisymmetric, relative to the symmetry plane. To select these modes, we take C (1) mn = ±(−1) m C (2) mn . As a result, we shall obtain the following system of equations, where the '+' and '−' signs correspond to modes that are symmetric (+) or antisymmetric (−) relative to the x → −x transformation. In figure 4, the dependence of normalized plasmon frequencies ω/ω pl = 1/ √ 1 − ε of a cluster of two identical oblate nanospheroids on normalized distances l/2a between the nanoparticles' centers is shown for the first four plasmon modes. Eigenvalues ε were obtained as a solution of the equations systems (22).
One can see in figure 4 that in clusters of two oblate nanospheroids, modes of 'T', 'M' and 'L' types, which are analogous to T-, M-and L-modes of a cluster made of two prolate spheroids (see figure 2), can exist. The T-and M-modes are the solutions of the system (22) with '+' sign, whereas the L-modes are the solutions of the system (22) with '−' sign. T-and L-modes can be derived by the method of hybridization of plasmonic modes of two oblate nanospheroids and their plasmonic frequencies are lying in the range 0 < ω < ω pl / √ 2. An infinite number of plasmonic frequencies of higher L-and T-modes lie near ω pl / √ 2. When the width of the gap decreases to 0, the ratio ω/ω pl for T-modes tends to various values in the range from 0 to 1/ √ 2, whereas plasmonic frequencies of L-modes approach zero in analogy to L-modes in a cluster of two spherical nanoparticles [21,60]. Plasmonic frequencies of strongly localized M-modes (figure 4(a)) lie in the range ω pl / √ 2 < ω < ω pl , as it happens in a cluster of two spherical nanoparticles [21,60]. As the width of the gap between oblate nanospheroids decreases to zero, plasmon frequencies of M-modes tend to bulk plasmon frequency ω pl , analogous to the case of a two-sphere cluster [21,60]. For large distances between the spheroids, M-modes disappear, and the plasmon frequencies of L-and T-modes of a cluster of two oblate nanospheroids tend to plasmonic frequencies of a single spheroid (see figure 4(c)) and can be found by means of a self-consistent model with approximation of spheroids by anisotropic point dipoles.
In figure 5, the distribution of a surface charge of plasmon modes of lower order in a cluster of two identical oblate nanospheroids is shown. One can see in this figure that the charge distribution is symmetric in T-and M-modes, while in L-mode it is antisymmetric, in agreement with the definition of these modes. For small distances between nanospheroids, charges in M-and L-modes are strongly localized near the gap. In contrast, when the distance between the spheroids increases, the charge distribution tends to a symmetric or antisymmetric combination of a surface charge in a single oblate nanospheroid (see figure 5(c)). Thus, in a cluster of two oblate or prolate spheroidal nanoparticles, fundamental symmetric and antisymmetric plasmon modes of 'T', 'M' and 'L' types can be excited, and it is these modes that define all the optical properties of a two-nanospheroid cluster.

A cluster of two metal nanospheroids in the field of a plane electromagnetic wave
In this section, we will consider a two-spheroid cluster in a uniform electric field with the potential where the time factor e −iωt is omitted. This case corresponds to a plane wave incidence and is important for the transformation of far fields into near fields, the enhancement of electric fields

A cluster of two prolate nanospheroids
Here, we will also use local systems of spheroidal coordinates, the origins of which are placed in the nanospheroids' centers (see figure 1(a)). The potential inside the jth nanospheroid again can be presented as a series in spheroidal harmonics (9), while the potential outside the nanospheroids should now be presented in the form where ϕ out 1 and ϕ out 2 are contributions from the first and second nanospheroids (see (10)), and ϕ 0 is the potential of the external electric field (23).
The electric potential of the incident plane wave (23) in local coordinates of the jth ( j = 1, 2) prolate nanospheroid looks like Making use of the boundary conditions (11) and the translational addition theorem (12), one can obtain the following system of equations for the coefficients C (1) mn and C (2) mn that define the outside field (see (10) and (24)) (n = 0, 1, 2, . . . ; m = 0, 1, 2, . . . , n), where ( j = 1, 2) The equation system for coefficients D (1) mn and D (2) mn (n, m = 1, 2, 3, . . . ) can be obtained from (26) by substituting b ( mn . It should be noted here that due to axial symmetry of the considered cluster, the system of equations (26) allow one to find the coefficients C ( j) mn and D ( j) mn for given order m, while degree n runs over n = m, m + 1, m + 2, . . . , m + N , where N is a large number that defines the accuracy of the solution.
The induced dipole moment of a cluster of two prolate nanospheroids, placed in the field of a plane electromagnetic wave, can be calculated in analogy to a single prolate nanospheroid [32], that is, by finding far-field asymptotes of the potential (24). As a result, for the dipole moment induced in the jth nanospheroid ( j = 1, 2), we have and the total dipole moment of the cluster will be the sum of the momenta (28). An absorption cross-section can be easily found if the dipole momenta (28) are known [61], where d = d (1) + d (2) denotes the dipole momentum of the whole system, and the asterisk denotes the operation of complex conjugation.
In figure 6, the absorption cross-section of a cluster of two identical prolate nanospheroids made from silver is shown as a function of wavelength. For longitudinal (z) polarization, the cross-section has two peaks that correspond to longitudinal plasmonic oscillations with L = 1, 2 (see figure 2(b)). It is very important that both of the peaks are split substantially relative to the case of a single spheroid (the 'z' dashed curve) due to a strong interaction between the nanospheroids. In contrast, for transversal (x or y) polarization one can see only one peak due to the excitation of the symmetrical T = 1 mode, and this peak is shifted just slightly relative to the single spheroid resonance (the 'y' dashed curve). This means that transversal (x or y polarization) excitation of a two-spheroid cluster induces only a weak interaction between the nanospheroids (see the dispersion curves for T-modes in figure 2(a)). Owing to this weak interaction, the absorption cross-section is approximately equal to double of a single spheroid.
It should be noted that in figure 6 the maxima of absorption, corresponding to plasmon oscillations of M-type that should lie in the interval ω pl / √ 2 < ω < ω pl , which corresponds to 326 < λ < 337 nm for silver [52], are not visible. This is related to the fact that M-modes interact with a homogeneous electric field weakly and can be effectively excited only by a source of radiation that is nonuniform in comparison with the size of the gap between nanoparticles [60] (see figure 3).
To control the correctness and accuracy of our analytical calculations, we have also carried out finite element simulation of this system with Comsol Multiphysics® software. The results of this simulation are shown by circles in figure 6. One can see that there is fine agreement between the analytical and pure numerical calculations. This fact confirms the correctness and accuracy of both of the approaches.

A cluster of two oblate nanospheroids
The case of two oblate nanospheroids is in many aspects similar to the case of two prolate nanospheroids considered above. So let us again choose local systems of coordinates that are connected to each of the nanospheroids and have origins in their centers (see figure 1(b)). The potential inside the jth nanospheroid can be presented again as a series in spheroidal harmonics (16), whereas the potential outside the oblate spheroids can be presented in the form where ϕ out 1 and ϕ out 2 are contributions from the first and second nanospheroids (see (17)) and ϕ 0 is the potential of the external electric field (23). In the local coordinates of the jth ( j = 1, 2) oblate nanospheroid, it looks like Making use of the boundary conditions (11) with ξ 0 = c/ √ a 2 − c 2 = c/ f , and the translational addition theorem (18), we shall obtain the following systems of equations (n = 0, 1, 2, . . . ; m = 0, 1, 2, . . . , n), where ( j = 1, 2) The equation system for coefficients D (1) mn and D (2) mn (n, m = 1, 2, 3, . . . ) can be obtained from (32) by substituting b (1) mn = b (2) mn = iδ m1 δ n1 f E 0y instead of a ( j) mn , and N ( j) mnpq instead of M ( j) mnpq . Apparently, the equations (32) have a more complicated structure than (26) because now, due to the lack of axial symmetry, one cannot split the system of equations into systems with fixed order m of the Legendre function.
For the calculation of the absorption cross-section of a cluster in the field of a plane electromagnetic wave, one can again use (29), where dipole momenta of each spheroid can be expressed by the next way ( j = 1, 2), In figure 7, the absorption cross-section of a cluster of two identical oblate nanospheroids made from silver is shown as a function of wavelength. For longitudinal (x) polarization, the crosssection has two peaks, which correspond to antisymmetric plasmonic oscillations with L = 1, 2 (see figure 4(b)). It is very important that now only one peak (L = 1) is shifted substantially For transversal (y) polarization, one can see only one peak owing to the excitation of the symmetrical T = 1 mode, and this peak is only slightly shifted relative to the single spheroid resonance (the dashed curve). This means that transversal (y polarization) excitation of a twospheroid cluster results in only a weak interaction between the nanospheroids (see the dispersion curves for the T-modes in figure 4(a)). Due to this weak interaction, the absorption cross-section for this polarization is approximately equal to the doubled cross-section of a single spheroid. It is also interesting that the plasmonic frequency of the L = 2 mode is very close to the plasmonic frequency of the T = 1 mode. This fact can be easily understood from the analysis of figure 4. Indeed, when the width of the gap tends to zero, the plasmonic frequency of L = 2 modes also decreases to zero, while the plasmonic frequency of the T = 1 mode increases slightly. So, at some point these modes will intersect and have the same frequency, and we observe this situation in figure 7.
It should be noted that in figure 7 the maxima of absorption corresponding to plasmon oscillations of M-type are again not visible. It is related to the fact that M-modes interact with a homogeneous electric field weakly and can be effectively excited only by a source of radiation that is nonuniform in comparison with the size of the gap between nanoparticles [60] (see figure 3).
To control the correctness and accuracy of our analytical calculations for a cluster of two oblate spheroids, we have also carried out finite element simulation of this system with Comsol Multiphysics® software. The results of this simulation are shown by circles in figure 7. One can see that there is fine agreement between the analytical and pure numerical calculations. This fact confirms the correctness and accuracy of both the approaches again.

Enhancement of local fields
The most important characteristic of nanoparticle clusters is the incident field enhancement factor in the gap between nanoparticles. This is the characteristic that allows us to determine the excitation rate of molecules near nanoparticles or the intensity of SERS [4]. Moreover, achieving high values of this factor is the main goal of optical nano-antenna development.
The distribution of squared electric field for the L = 1 resonance in a cluster of two prolate spheroids is shown in figure 8, which shows that, indeed, maximal field enhancement takes place in the gap between the nanoparticles on their surfaces. The field maxima are also present in the outer side of the cluster; however, field amplitude is essentially less there. According to general theorems for harmonic functions, the field maximum can be reached only on the region boundaries. In our case, the field maxima are reached in those points of the spheroids' surface where the distance between the spheroids is minimal.
Using (10) and (17), one can find explicit expressions for the field enhancement factor G. For clusters of two identical prolate spheroids in the considered configuration ( figure 1(a)), one can obtain the following expression for the field maximum in the case of an incident field polarized along the z-axis, where In the most interesting case of a small gap and strongly prolate spheroids, ξ 1 , ξ 2 ≈ 1, and one may use the asymptotic form dQ n (ξ ) dξ | ξ ≈1 ≈ − 1 2(ξ −1) . As a result, the field enhancement factor takes the form In the case of clusters of two identical oblate spheroids ( figure 1(b)) and incident field polarized along the x-axis, we obtain where ξ 1 = a 2 / f 2 − 1 and ξ 2 = (l − a) 2 / f 2 − 1; C mn = −(−1) m C (1) mn = C (2) mn . In figure 9, the dependence of squared electric field enhancement (35) and (37) for clusters of two identical silver nanospheroids on the wavelength is shown. Comparing peak positions with the dispersion curves in figures 2 and 4, one can come to a conclusion that only 'L' type plasmon modes are excited in the clusters for the considered configurations of nanospheroids and incident electromagnetic wave polarizations (along the line joining the nanoparticles' centers). In particular, the excitation of the L = 1 and L = 2 modes is noticeable. At that, the position of squared field enhancement peaks agrees with the maxima of absorption cross-section shown in figures 6 and 7 by the solid lines z and x correspondingly. It should be mentioned that the value of squared field enhancement near a cluster of two nanospheroids can reach values of up to 10 6 . In the case of single nanoparticles, this value is almost two orders lower than that of clusters (cf solid and dashed curves in figure 9). This fact determines the greater attractiveness of metal nanoparticle clusters in comparison to single nanoparticles for the investigation of SERS and SEF. Note that the obtained great values of the field enhancement factor can be slightly lower in practice, since for small particles and for small gaps between them, nonlocal and other effects not considered in this research become essential.

A cluster of two nanospheroids in the field of a radiating atom
In the previous section, we considered the case of a nano-antenna placed in the field of a plane wave. However, highly nonuniform optical fields occur very often in the nano-environment. For example, such fields arise when a plasmonic nano-antenna is excited by an atom or a molecule or any other nanolocalized source of light. So, in this section, we will consider the important case of a two-nanospheroid cluster in the field of electric dipole sources. The excitation of the cluster by magnetic dipole and electric quadrupole sources can be analyzed analogously.

A cluster of two prolate nanospheroids
The case of two prolate nano-spheroids in the field of a dipole source of radiation can be considered in perfect analogy to the case of the same cluster in a uniform field. One should again look for solutions in the form (9) and (10) and then apply the boundary conditions (11). The only difference is that now the external potential is the potential ϕ 0 of the dipole that has the following form in the jth local system of coordinates of a prolate spheroid [58] ( j = 1, 2), In (38), d 0 denotes the dipole momentum of a source placed at r , ∇ j is a gradient over r in local coordinates and sin(mφ j ) , are expansion coefficients of the unit charge potential in local coordinates of a prolate spheroid. As a result of applying the boundary conditions, one can obtain a system of equations for the unknown coefficients C ( j) mn , D ( j) mn in (10). The new system can be easily derived from (26) if one makes the following replacement for the coefficients a ( j) mn and b ( j) mn , where α ( j) mn and β ( j) mn are defined by (39). After the systems for coefficients C ( j) mn and D ( j) mn have been solved with taking into account (40), one can find the total induced dipole moment of both the prolate nanospheroids, Knowing the dipole momenta (41), it is easy to find (see e.g. [62]) the radiative decay rate of an excited atom placed near the cluster of two prolate nanospheroids, where P rad is the radiation power at frequency ω andhω is the emitted photon energy. The radiative decay rate is a very important characteristic in applications such as SERS, SEF, nanolasers and so on. To characterize the radiative decay rate, it is natural to normalize it to the radiative decay rate of a dipole in free space, γ 0 = P rad 0 hω = ω 3 3hv 3 c |d 0 | 2 . In figure 10, the normalized radiative decay rate of a dipole source placed at the middle point of the gap is shown. As is clearly seen in figure 10(a), if the distance between the prolate nanospheroids is small (figure 10(a), curves α and δ), the dipole source with a moment oriented perpendicular to the cluster's axis of rotation can excite both symmetrical T-and M-modes. This fact contrasts with the case of the excitation of the same cluster with a plane wave, when M-modes with peaks located in the region of λ < 337 nm (see figure 2(a)) are not excited. When the distance between the nanospheroids increases (see figure 10(a)), the peak corresponding to M-modes shifts to λ ≈ 337 nm (ω ≈ ω pl / √ 2) and then disappears. After that point, only the peaks corresponding to plasmonic T-modes can be observed. Of course, this picture is in agreement with the behavior of the plasmonic M-modes shown in figure 2(a). We also note that for large enough distances between the nanospheroids (see figure 10(a), curve γ ), the self-consistent model [3,22], in which nanoparticles are replaced by point dipoles with corresponding polarizabilities [3,35], can be effectively used for the calculation of the radiative decay rate (dashed curve).
When the dipole moment of a source is oriented along the axis of symmetry ( figure 10(b)), only antisymmetric L-modes can be excited owing to symmetry reasons. From figure 10(b), one can also see that for small enough distances between the spheroids there are two plasmonic modes (L = 1, 2) that interact with the dipole source. When the distance between the spheroids diminishes, right peaks of radiation power shift towards long wavelengths. At large distances between the nanospheroids, there is only one maximum corresponding to the L = 1 plasmonic mode (see figure 10(b), curve γ ). In this case, the radiative decay rate of a dipole placed near a Figure 10. Normalized radiative decay rate of a dipole placed at the middle point between two identical prolate nanospheroids made from silver as a function of the wavelength. The dipole source moment is oriented along the x or y axes (a) and along the z axis (b). The large semi-axes of the nanospheroids are c = 15 nm, the aspect ratios are a/c = 0.6. The curves α, β, γ and δ correspond to l/2c = 1.05, 1.1, 1.3 and 1.03, respectively. The asymptotic expression obtained by approximation of the spheroids by point dipoles (l/2c = 1.3) is shown by the dashed curve.
cluster of two prolate nanospheroids can be calculated also by making use of the self-consistent analytical model in which the spheroids are approximated by point dipoles (see the dashed curve in figure 10(b)).

A cluster of two oblate nanospheroids
The case of two oblate nanospheroids in the field of a dipole source of radiation can be considered in perfect analogy to a case of the same cluster in a uniform field. One should again look for solutions in the forms (16) and (17) and then apply the boundary conditions (11) with ξ 0 = c/ √ a 2 − c 2 = c/ f . The only difference is that now the external potential is the potential ϕ 0 of the dipole that has the following form in the jth local system of coordinates of an oblate spheroid [58] ( j = 1, 2), Q m n (iξ j )P m n (η j ) (d 0 ∇ j )γ ( j) mn cos(mφ j )

Conclusion
Thus, in the present work, the optical properties of clusters made of two metal nanospheroids are considered theoretically, and analytical results are obtained. Plasmonic eigenoscillations were analyzed in detail, and it was found that in a cluster of two prolate or oblate nanospheroids there can be three types of plasmon modes. Two of them (low frequency, 0 < ω < ω pl / √ 2, L-and T-modes) can be effectively excited by a plane electromagnetic wave, while the third type (high frequency, ω pl / √ 2 < ω < ω pl , M-modes) can be excited only by a strongly nonuniform field of a nanolocalized source of light (a molecule, a quantum dot) located in the gap between two adjacent nanoparticles.
We have also investigated the excitation of a nano-antenna made from two silver nanospheroids by the fields of a plane wave and an electric dipole. The results of these investigations allow us to obtain the absorption cross-section of the nano-antenna as a function of wavelength for various polarizations of an incident plane electromagnetic wave and to attribute all of the observable peaks to the excitation of corresponding plasmonic modes. We also analyzed the radiative decay rate (or local density of state) of an excited atom placed in the gap between nanospheroids and attributed all observable peaks to the excitation of corresponding plasmonic modes.
The obtained analytical results can be used in many applications based on plasmonic nanoantennas or the enhancement of local fields (SERS, SEF, nanolasers, nano-optical circuits and so on). In addition, our results are very important for controlling the accuracy of different computational software programs that have no a priori test of accuracy.