## Abstract

Plasmonic nanoparticle clusters are widely considered experimentally and numerically. In the clusters consisting of one central particle and *N* satellite particles, not only the magnetic modes but also the toroidal modes can exist. Here, the eigenmodes of such clusters and the corresponding excitation efficiency under the illumination of a plane wave are studied analytically by using the eigen-decomposition method. The angular dependence of the optical response of these clusters is clearly demonstrated. The behavior of excitation efficiency is dependent on both the value and the parity of *N*, the number of satellite particles. Our results may provide a guide for the selective excitation of plasmonic modes in the plasmonic nanoparticle clusters.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Plasmonic nanoparticles can support strong localized surface plasmon resonances and hence enhance the interaction of light and metallic nanoparticles. They have attracted wide attention in manipulating the electromagnetic fields at the nanoscale [1,2]. The plasmonic nanoparticles can find applications in near-field spectroscopy [3,4], nanoantennas [5], biochemical sensors [6] and near-field imaging [7]. The clusters of nanoparticles can provide much more degrees of freedom in tailoring optical field. Coupling between nanoparticles can give rise to interesting collective behavior, such as strong magnetic response at optical frequencies [8] and Fano resonances [9,10]. The collective plasmonic modes can be used to generate toroidal dipole resonance [11,12], engineer negative permeability [13,14], realize directional scattering [15,16] and so on.

Due to the complexity of the plasmonic nanoparticle clusters, their optical properties were studied experimentally and numerically in most cases. Only when the clusters possess high symmetry such as a circular array, the optical response becomes analytically solvable based on an eigen-decomposition method [17–20]. If a nanoparticle is introduced at the center of the circular array, the existence of the central particle can have a significant effect on the optical response of the cluster [21]. At the same time, such clusters can support novel optical resonance of the toroidal dipole, a newly discovered member of the electromagnetic multipole family. Subwavelength structures supporting resonant toroidal modes can be applied in high-Q factor sensing [22], polarization control [23], negative refraction [24], enhanced optical nonlinearity [25] and lasing [26]. Furthermore, previous researches [11,16] show that the optical response of such nanoclusters exhibit prominent angular dependence on the external excitation. It is found that the particle number *N*, both the value and the parity play an important role in this angular dependence. All these results of the optical response of such clusters are clearly demonstrated but the corresponding physical mechanism is worth further study.

In this paper, the plasmonic nanoparticle clusters containing one central particle and *N* satellite nanoparticles are systematically investigated, and an analytical estimation of the excitation efficiency of the plasmonic modes is conducted. Due to the subwavelength size of the plasmonic nanoparticles, the dipole approximation method is adopted in this work. Based on the coupled-dipole equation [17,18,20], an analytical form for the eigenmodes of the clusters are obtained. According to the polarization of the central particle, the eigenmodes can be divided into two classes. In the first class, the polarization for the central particle vanishes and it is called the anti-symmetrical modes with zero total electric dipole moment [17]. The second class includes two type of modes with the central particle polarized nearly parallel or anti-parallel with the satellite ones, respectively. Then the coupling of these eigenmodes to an out-of-plane polarized plane wave are studied analytically. The extinction cross section can be obtained through the coupling of each eigenmode. The role of the particle number *N* and its parity in the optical response is studied based on the response of each eigenmode. In the end, numerically simulated near-field patterns of the clusters excited by an external field are presented to further confirm the theoretical analysis. The angular dependence of the clusters considered here can be applied in the selective excitation of plasmonic modes [27,28] and find applications in related metamaterials [29].

## 2. Eigenmode analysis

A plasmonic nanoparticle cluster under consideration contains a central nanosphere and *N* satellite nanospheres, as shown in Fig. 1 schematically. All of the particles lie in the *x*-*y* plane, and the radii of the central sphere and surrounding spheres are *r*_{c} and *r*_{s}, respectively. The permittivity of plasmonic nanospheres is described by the Drude model $\varepsilon (\omega ) = 1 - {\omega ^2}/(\omega _\textrm{p}^2 + i\gamma \omega )$, where *γ* is the electron scattering rate and *ω* _{p} is the plasma frequency. The central sphere is located at the origin, and the surrounding spheres are distributed uniformly on a circle with radius *R*_{0}. Therefore, the position vector **r*** _{m}* (

*m*= 1, 2, …,

*N*) for each surrounding particle can be expressed as

As the radii of the nanospheres are much smaller than the incident wavelength and the separation between nanospheres is larger than 3*r*_{s} [30], the plasmonic nanoparticles can be treated as electric point dipoles. The induced electric dipole moment **p*** _{m}* (

*m*= 1, 2, …,

*N*+ 1, with

*N*+ 1 corresponding to the central sphere) can be determined by the coupled dipole equation [20]:

*α*(

_{m}*ω*) = (3

*ic*

^{3}/2

*ω*

^{3})

*a*

_{1}(

*ω*) is the polarizability of the

*m*-th nanoparticle,

*a*

_{1}(

*ω*) is the electric term of the Mie’s coefficients [31], ${\textbf E}_m^{\textrm{ext}}$ represents the external electric field on the

*m*-th nanosphere, and ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over {\textbf {G}}} }({{\textbf r}_m} - {{\textbf r}_n})$ is the Green function with ${G_{uv}}(\textbf{r}) = k_0^3[{A({k_0}r){\delta_{uv}} + B({k_0}r){{{r_u}{r_v}} \mathord{\left/ {\vphantom {{{r_u}{r_v}} {{r^2}}}} \right.} {{r^2}}}} ]$, $A(x) = ({{x^{ - 1}} + i{x^{ - 2}} - {x^{ - 3}}} ){e^{ix}}$, $B(x) = ({ - {x^{ - 1}} - 3i{x^{ - 2}} + 3{x^{ - 3}}} ){e^{ix}}$.

The coupled dipole Eq. (2) can be reformulated to a matrix form:

with |**P**〉 = (

**P**

_{1},

**P**

_{2},…,

**P**

_{N}_{+1})

^{T}, $|{{{\textbf E}^{\textrm{ext}}}} \rangle = {({\textbf E}_1^{\textrm{ext}},{\textbf E}_2^{\textrm{ext}},\ldots ,{\textbf E}_{N + 1}^{\textrm{ext}})^\textrm{T}}$, ${\textbf M}(\omega ) = {\boldsymbol{\alpha }^{ - 1}}(\omega ) - {\textbf G}(\omega )$, and ${\boldsymbol{\alpha }^{ - 1}} = (\alpha _1^{ - 1},\alpha _2^{ - 1},\ldots ,\alpha _{N + 1}^{ - 1})$. The response of the cluster to external optical field can be analyzed by a dynamic eigenmode decomposition method [17,18]. The eigenmodes of the system $|{{{\textbf p}^j}} \rangle$ can be obtained by solving the eigenvalue problem: $\textbf{M}|{\textbf P} \rangle \textrm{ = }{\lambda ^j}|{\textbf P} \rangle$, where λ

*is the eigenvalue of*

^{j}**M**. And the solution of Eq. (3) can be expressed as the linear combination of the eigenmodes [17]:

**M**. As

**M**is a non-Hermitian and symmetric matrix, $\left\langle {{{\bar{{\textbf p}}}^j}} \right|$ has the same entries as that of $|{{{\textbf p}^j}} \rangle$. Note that Eq. (4) is valid when $\left\langle {{{\bar{{\textbf p}}}^j}} \right|{{{\textbf p}^j}} \rangle \ne 0 $ and $\left\langle {{{\bar{{\textbf p}}}^j}} \right|{{{\textbf p}^j}} \rangle = 0 $ for

*i*≠

*j*.

**(1) Eigenmodes and eigenvalues**

In this study a *z*-polarized plane wave ${\textbf{E}_{inc}}\textrm{ = }{\textbf{E}_0}{e^{i{k_0}(\cos [{\phi _i}]x + \sin [{\phi _i}]y)}}$ is employed. As shown in Fig. 1, the plane wave propagates in the *x*-*y* plane and the incident angle is *ϕ _{i}*. In this case all the particles polarize in the

*z*direction and hence only the modes with out-of-plane polarization are considered here. Make use of the discrete rotational symmetry of the system (see the Appendix), the analytical results for the

*N*+ 1 eigenmodes can be obtained as follows:

For *j* = 1, 2, …, *N *− 1,

*α*

_{s}is the polarizability of a single satellite sphere, and

*D*= 2

_{m }*R*

_{0}sin(

*m*π/

*N*).

For *j* = *N* and *N *+ 1,

*p*

_{c,N},

*p*

_{c,N+1}are the dipole moment of the central particle for the

*N-*th and (

*N*+ 1)-th mode, respectively. The corresponding eigenvalues are

*α*

_{c}is the polarizability of central sphere.

It must be pointed out that the *j*-th and (*N − j*)-th modes in Eq. (5) are degenerated and they are not orthogonal to each other. For the convenience of further discussion, orthogonal eigenmodes are constructed by linear combination of the non-orthogonal modes. The orthogonalized eigenmodes of the first *N-*1 modes are as follows: for *j* = 1, 2, …, Int[(*N *− 1)/2],

*j*= Int[(

*N*-1)/2] + 1, …,

*N*− 1,

*N*/2)-th mode for an even

*N*,

The above discussion indicates that the *N *+ 1 eigenmodes can be divided into two classes according to the polarization of the central particle. In the first *N *− 1 eigenmodes the central particle has a vanishing polarization and the total dipole moment of the cluster is zero. They are called the anti-symmetrical modes in Ref. [17]. For the last two eigenmodes described by Eq. (7), the polarization of the central particle is nearly parallel or anti-parallel with the surrounding ones and the total dipole moment is non-zero. For the mode corresponding to *j* = *N*, all the surrounding particles oscillate in phase while the central one is nearly out of phase with respect to the surrounding ones. In this case, a toroidal dipole mode can be formed and become a dominant term of the electromagnetic multipoles [11,12]. While for the mode correspond to *j* = *N *+ 1, the central particle is nearly in phase with the surrounding ones and thus a strong electric dipole can be formed.

Based on the analytical expressions of the eigenmodes, the optical response of the clusters under illumination of the external electromagnetic filed can be calculated and understood clearly. In the following text, the excitation efficiency and the extinction cross section of the clusters are calculated and the corresponding angular dependence are discussed.

**(2) Excitation efficiency**

According to Eq. (4), the excitation efficiency of these modes by the external field can be expressed with ${{\left\langle {{{\bar{{\textbf p}}}^j}} \right|{{{\textbf E}^{\textrm{ext}}}} \rangle } \mathord{\left/ {\vphantom {{\left\langle {{{\bar{{\textbf p}}}^j}} \right|{{{\textbf E}^{\textrm{ext}}}} \rangle } {\left\langle {{{\bar{{\textbf p}}}^j}} \right|{{{\textbf p}^j}} \rangle }}} \right. } {\left\langle {{{\bar{{\textbf p}}}^j}} \right|{{{\textbf p}^j}} \rangle }}$. In the following text, we use the normalized $|{{{\textbf p}^j}} \rangle$ defined by ${{|{{{\textbf p}^j}} \rangle } \mathord{\left/ {\vphantom {{|{{{\textbf p}^j}} \rangle } {\sqrt {\left\langle {{{\bar{{\textbf p}}}^j}} \right|{{{\textbf p}^j}} \rangle } }}} \right. } {\sqrt {\left\langle {{{\bar{{\textbf p}}}^j}} \right|{{{\textbf p}^j}} \rangle } }}$ for simplicity.

Firstly, the excitation of the *N*-th mode (*j* = *N*) and (*N *+ 1)-th modes (*j* = *N *+ 1) can be directly derived as:

*J*(

_{n}*x*) diminishes quickly with the increase of

*n*for

*x*∼1, it is sufficient in the summation to only take the terms of

*l*= −1, 0, 1 into account. Then we have

For the antisymmetric modes corresponding to *j* = 1, 2, …, *N *− 1, the corresponding coupling efficiency can be obtained similarly. Still, only the dominant terms correspond to *l* = 0 and 1 in the Bessel functions are considered in the summation.

For *j* = 1, 2, …, Int[(*N *− 1)/2],

*j*= Int[(

*N*− 1)/2] + 1,…,

*N*− 1,

*j*=

*N*/2 of even

*N*, the non-degenerated mode, the coupling efficiency reads:

**(3) Extinction cross section**

Based on the optical theorem, the extinction cross section of the nanosphere clusters can be expressed in terms of the eigenmodes as [10]:

## 3. Results and discussion

In the following discussion, the clusters with *N _{ }=_{ }*2, 3, 4, 5 are considered as examples. Here the geometric parameters are set as:

*r*

_{c }=

_{ }23 nm,

*r*

_{s }=

_{ }20 nm and

*R*

_{0}=

_{ }60 nm. Without loss of generality, we fix η

*ω*

_{p }=

_{ }6.18 eV and

*γ*=

_{ }0, and the background is the free space.

The resonant properties of the eigenmodes of the system can be indicated by the inverse of eigenvalues 1/λ* ^{j}* since it stands for the effective mode polarizability of the

*j*-th eigenmode [20]. The imaginary part of 1/λ

*, as a function of frequency, gives both the resonant frequency and the extinction information of the corresponding eigenmode. The imaginary parts of 1/λ*

^{j}*for the eigenmodes of clusters are illustrated in Figs. 2(a1)–2(d1). It can be seen that for the anti-symmetrical modes, the linewidth decreases as the corresponding mode order approaches*

^{j}*N*/2. The previous study showed that the mode with

*j*= Int[(

*N*-1)/2] has the highest Q-factor for the circular array of plasmonic nanoparticles [20]. However, with the introduction of a central nanoparticle, the

*N*-th mode, in which all the satellite particles oscillate in phase but the central particle oscillate out-of-phase with satellite ones, possesses a higher Q-factor than the one with

*j*= Int[(

*N*-1)/2]. The Q-factors of all the modes increase with the increase of

*N*. For example, the Q-factors of the

*N*-th mode and the anti-symmetrical mode increase from 342 and 57 for

*N*= 2 to 1235 and 740 for

*N*= 5, respectively.

The σ_{e} of the clusters, including the total σ_{e} and that for each single eigenmode, are calculated according to Eq. (17). Figures 2(a2)–2(d2) and Figs. 2(a3)–2(d3) show the σ_{e} of the clusters at two different incident angles. It can be seen that the resonant peaks of σ_{e} for the single modes coincide with the corresponding peaks of Im[1/λ* ^{j}*]. The analytical results for the total cross sections can be confirmed by the numerical simulations. As shown in Figs. 2(a2)–2(d2) and Figs. 2(a3)–2(d3), the numerical results agree well with the analytical ones. Figures 2(a2)–2(d2) and Figs. 2(a3)–2(d3) show that the σ

_{e}of the clusters, both the total σ

_{e}and the σ

_{e}for each single mode, can exhibit angular dependence. In Figs. 2(a4)–2(d4), the total σ

_{e}of the clusters are shown for three incident angles. It can be seen that the angular dependence relies on the parity of

*N*: σ

_{e}are angular sensitive for clusters with an odd

*N*while insensitive for clusters with an even

*N*.

One can find a clue to the angular dependence of the σ_{e} by studying the coupling efficiency of the eigenmodes with the external electromagnetic filed. Figures 3(a)–3(d) give the magnitude of the coupling efficiency of each eigenmodes, $\left|{\left\langle {{{\bar{\textrm{p}}}^j}} \right|{{\textrm{E}^{\textrm{ext}}}} \rangle } \right|$. For the anti-symmetrical modes, the coupling efficiency exhibits a prominent angular dependence for all the clusters. While for the *N*-th mode and the (*N *+ 1)-th modes, the angular dependence is much weaker than that of the anti-symmetrical modes. In fact, for these the two modes, only the cluster with *N* = 2 exhibits a strong angular dependence. Mathematically, according to Eq. (14)-(16), all the terms are dependent on the incident angle for the anti-symmetrical modes. While for the *N*-th and (*N *+ 1)-th modes, Eq. (13) has angular independent terms of *J*_{0}(*x*) and *p*_{c,N} or *p*_{c,N+1}, and the angular dependent term *J _{N}*(

*x*) diminishes quickly with the increase of

*N*since

*x*∼1for the clusters considered here. This difference of angular dependence for the optical response can also be explained by the symmetry of the eigenmodes. For the

*N*-th and (

*N*+ 1)-th modes, all the surrounding particles are in phase while in the anti-symmetrical modes there are phase differences between the surrounding particles. In other words, the

*N*-th and (

*N*+ 1)-th modes have higher symmetry than the anti-symmetrical modes, hence they exhibit a much weaker angular dependence.

On the other hand, only the σ_{e} of clusters with an even *N* exhibit strong angular dependence according to Figs. 2(a4)-(d4), although all the anti-symmetrical modes exhibit strong angular dependence. By examining the angular behavior for the anti-symmetrical modes shown in Fig. 3, one can see that the coupling efficiency of the *j*-th and the (*N − j*)-th mode basically exhibits an opposite angular dependence, and so do the corresponding σ_{e}. Thus the sum of the σ_{e} of these two degenerate modes exhibits a weak angular dependence, as indicated in Figs. 3(f)–3(h). However, for clusters with an even *N*, there is a non-degenerate mode corresponding to *j* = *N*/2, which is very sensitive to the incident angle both for the coupling efficiency and the σ_{e}. Since the σ_{e} of other modes (σ* _{N}*, σ

_{N}_{+1}, σ

*+ σ*

_{j}

_{N}_{−}

*) are angular insensitive, the (*

_{j}*N*/2)-th mode contributes the most to the angular dependence in the cluster with an even

*N*.

The angular dependence for the coupling efficiency of the anti-symmetrical modes can also be interpreted by the symmetry of these modes. Figure 4 shows the anti-symmetrical modes of the cluster with *N* = 4 under the illumination of an external plane wave. For the (*N*/2)-th mode (*j* = 2) the neighboring particles are out of phase, then it is an even mode with respect to the incident direction of *ϕ _{i}*=0, while an odd mode with respect to the incident direction of

*ϕ*= π/

_{i}*N*. As the odd mode cannot be excited, thus the corresponding coupling efficiency drops to zero when

*ϕ*varies from 0 to π/4. For the degenerate modes of

_{i}*j*= 1 and 3, mode of

*j*= 1 can be excited but the mode of

*j*= 3 cannot be excited when

*ϕ*=0; while mode of

_{i}*j*= 3 can be excited but the mode of

*j*= 1 cannot be excited when

*ϕ*= π/2. As the degenerate modes appear in pairs, the total extinction cross section of these two modes exhibit weak angular dependence.

_{i}The optical response of the clusters can also be reflected by the corresponding near-field patterns. Figure 5 shows the simulated field patterns corresponding to the first two resonant peaks (resonance of *j *=* N* and *j *= 1 modes, respectively) of σ_{e} for the clusters with *N* = 2 and 3. For the cluster with *N* = 2, both two field patterns exhibit apparent angular dependence. The first field pattern corresponding to *ϕ _{i }*= 0 is characterized by a magnetic field vortex localized in the cluster, which is the characteristic of the toroidal dipole [32]. A typical model of a toroidal dipole is a torus on which the currents circulate along the meridians. Considering the polarization configuration of the

*N*-th mode in which the center particle polarizes nearly π radians out of phase with the surrounding ones, one can see that this configuration is similar to a torus with meridian currents since the particles (dipoles) can be regarded as discretized current elements [12]. Consequently, a significant toroidal dipole moment can be supported here. For the field pattern corresponding to

*ϕ*= π/2, although a magnetic field vortex still can be seen, the field pattern is quite different from that in the case of

_{i}*ϕ*= 0. According to σ

_{i}_{e}indicated by Fig. 2(a3), the filed pattern is a combination of the

*N*-th and (

*N*+ 1)-th modes. For the second resonant peak, the field patterns corresponding to

*ϕ*= 0 and

_{i}*ϕ*= π/2 are asymmetric and symmetric with respect to the

_{i}*y*axis, respectively. For

*ϕ*= 0, superposition of the eigenmodes of

_{i}*j*= 1 and

*j*= 3 exists and the mode with

*j*= 3 is dominant according to Figs. 2(a2)–2(a3). For

*ϕ*= π/2, it is almost totally come from the eigenmode of

_{i}*j*= 3 in which all the particles oscillate almost in phase.

For the cluster with *N _{ }=*

_{ }3, the field patterns at the first resonant peak for different incident angles are similar to each other in spite of a rotation of 2π/3, which is consistent with the corresponding σ

_{e}shown in Figs. 2(b2) and 2(b3). The field patterns corresponding to the second resonant peak show a larger difference. Compared to the case of

*ϕ*= 0 in which the mode of

_{i}*j*= 1 dominates, in the case of

*ϕ*= π/2 the eigenmode of

_{i}*j*= 2 dominates and the mode of

*j*= 1 also plays an important role, hence a larger difference appears in the near-field patterns. Nevertheless, the σ

_{e}corresponding to the two degenerate modes (

*j*= 1, 2) have the same line shape and the total σ

_{e}exhibits no angular dependence.

Another point worth noting is that σ_{e} coming from the *N*-th mode can be negative in certain frequency range, as shown in Fig. 2. For system considered here there is no gain and hence the total σ_{e} cannot be negative. The existence of negative σ_{e} for the *N*-th mode indicates that such mode cannot be excited individually, and it is always accompanied by other modes to make sure that the total σ_{e} is always positive. The following discussion will show that the negative σ_{e} of the *N*-th mode comes from the projection of the external field onto the eigenmode.

If the external filed coincides with a certain eigenmode, for example, $|{{{\textbf E}^{\textrm{ext}}}} \rangle = |{{{\textbf E}^{\textrm{ext},j}}} \rangle = {\lambda ^j}|{{{\textbf p}^j}} \rangle$, according to Eq. (17), the corresponding σ_{e} is: $\sigma _e^j = \frac{{4\pi k}}{{{{|{{{\textbf E}_0}} |}^2}}}{\mathop{\rm Im}\nolimits} \left( {\frac{1}{{{\lambda^j}}}{{|{{{\textbf E}^{\textrm{ext},j}}} |}^2}} \right)$. As Im(1/λ* ^{j}*) is positive for the eigenmodes considered here, the corresponding σ

_{e}is always positive. In the general case, for a given eigenmode, the external filed can be written as: $|{{{\textbf E}^{\textrm{ext}}}} \rangle = {c^j}|{{{\textbf E}^{\textrm{ext},j}}} \rangle + |{{{\textbf E}^{\textrm{ext},\textrm{other}}}} \rangle$. Then $\left\langle {{{\bar{{\textbf p}}}^j}} \right|{{{\textbf E}^{\textrm{ext}}}} \rangle = \left\langle {{{\bar{{\textbf p}}}^j}} \right|{c^j}{\lambda ^j}|{{{\textbf p}^j}} \rangle = {c^j}{\lambda ^j}\left\langle {{{\bar{{\textbf p}}}^j}} \right|{{{\textbf p}^j}} \rangle$, and $\left\langle {{{\textbf E}^{\textrm{ext}}}} \right.|{{{\textbf p}^j}} \rangle = {c^{j \ast }}\left\langle {{{\textbf E}^{\textrm{ext},j}}} \right.|{{{\textbf p}^j}} \rangle + \left\langle {{{\textbf E}^{\textrm{ext},\textrm{other}}}} \right.|{{{\textbf p}^j}} \rangle = {c^{j \ast }}{\alpha ^j}\left\langle {{{\textbf E}^{\textrm{ext},j}}} \right.|{{{\textbf E}^{\textrm{ext},j}}} \rangle + \left\langle {{{\textbf E}^{\textrm{ext},\textrm{other}}}} \right.|{{{\textbf p}^j}} \rangle$, where the asterisk denotes the complex conjugate. Note that $\left\langle {{{\textbf E}^{\textrm{ext},\textrm{other}}}} \right.|{{{\textbf p}^j}} \rangle \ne \textrm{0}$ generally. By defining ${\gamma ^j} = \frac{{\left\langle {{{\textbf E}^{\textrm{ext}}}} \right.|{{{\textbf p}^j}} \rangle }}{{{c^{j \ast }}{\lambda ^{j \ast }}\left\langle {{{\textbf p}^j}} \right.|{{{\textbf p}^j}} \rangle }}$, the extinction cross section is

As ${|{{c^j}} |^2}{|{{{\textbf E}^j}} |^2}$ is a non-negative, the sign of σ_{e} is determined by ${\mathop{\rm Im}\nolimits} ({\gamma ^j}{\alpha ^j})$, a quantity dependent on the external field. As an example, Fig. 6 gives the ${\mathop{\rm Im}\nolimits} ({\gamma ^j}{\alpha ^j})$ for the plane wave source as a function of frequency for the cluster with *N* = 2. ${\mathop{\rm Im}\nolimits} ({\gamma ^j}{\alpha ^j})$ is positive for modes with *j* = 1 and 3, while for the *N*-th mode it can be either positive or negative. As for comparison, σ_{e} of the *N*-th mode are presented in Figs. 6(c) and 6(d) from which it can be seen that the sign of σ_{e} is consistent with that of ${\mathop{\rm Im}\nolimits} ({\gamma ^j}{\alpha ^j})$.

The analysis above indicates that the negative σ_{e} comes from the fact that $\left\langle {{{\textbf E}^{\textrm{ext},\textrm{other}}}} \right.|{{{\textbf p}^j}} \rangle \ne \textrm{0}$, which is the result of the non-Hermiticity of the matrix **M**. The left eigenvector $\left\langle {{{\bar{{\textbf p}}}^j}} \right|$ of **M** is not Hermitian conjugate to the right eigenvector $|{{{\textbf p}^j}} \rangle$, thus $\left\langle {{{\textbf E}^{\textrm{ext},\textrm{other}}}} \right.|{{{\textbf p}^j}} \rangle \ne \left\langle {{{\bar{{\textbf p}}}^j}} \right|{{{\textbf E}^{\textrm{ext,other}}}} \rangle = 0$. Of course, the value of the projection $\left\langle {{{\textbf E}^{\textrm{ext},\textrm{other}}}} \right.|{{{\textbf p}^j}} \rangle$ and hence ${\mathop{\rm Im}\nolimits} ({\gamma ^j}{\alpha ^j})$ depend on the specific form of the eigenmode. For the systems considered here, only the σ_{e} corresponding to the *N*-th mode can be negative.

## 4. Conclusion

In summary, analytical solutions for the eigenmodes of the plasmonic nano-particle clusters are obtained by using the eigen-decomposition method. The optical response of such clusters under the illumination of a plane wave are analyzed analytically based on the response of each single eigenmode. For all the eigenmodes, the *N*-th mode has the largest Q factor while the (*N *+ 1)-th mode has the smallest Q factor. The excitation of these modes and the corresponding extinction cross sections can be angular dependent. Both the satellite particle number *N* and its parity play significant roles in the angular dependence. On the whole, the angular dependence decrease with the increase of *N*. when *N *> 2, the angular dependence is relatively weak. In clusters with even numbered *N*, there is a non-degenerate anti-symmetric mode (*j* = *N*/2) which is very sensitive to the incident angle. As a result, the extinction cross section for clusters with even numbered *N* is much more sensitive to the incident angle than that of the clusters with odd numbered *N*. The extinction cross section for the *N*-th mode can be negative in some frequency range. The negative extinction cross section results from the projection of the external field to the eigenmode of the system which is basically an eigenvector of a non-Hermitian matrix. This work provides an example of analytical investigation for the electromagnetic response of nano-particle arrays with rotational symmetry and can be applied in the analysis of related metamaterials.

## Appendix: Derivation of the analytical solution of the eigen-problem

Taking advantage of the discrete rotational symmetry of the system, the eigen-problem $\textbf{M}|{\textbf P} \rangle \textrm{ = }{\lambda ^j}|{\textbf P} \rangle$ can be solved analytically. Consider the following (*N *+ 1)×(*N *+ 1) permutation matrix **T**(*n*), where *n *= 1, 2,…, *N*, *N *+ 1,

**T**(

*n*)

**MT**(

*n*)

^{−1}=

**M**. That is,

**M**and

**T**(

*n*) commute, then the eigenvectors of

**M**can be constructed by the linear combination of the eigenvectors of

**T**(

*n*).

The eigenvalues and eigenvectors of **T**(*n*) are given by $t_n^j$ and ${{\textbf v}^j}$, respectively, with

*j*= 1, 2, …,

*N*,

*j*=

*N*+ 1,

The eigenvectors **p*** ^{j}* of

**M**can be expressed as follows: for

*j*= 1, 2, …,

*N*− 1,

*j*=

*N*,

*N*+ 1, the two eigenmodes of

**T**(

*n*) are degenerated, and the eigenvector of

**M**can be constructed with the linear combination of Eqs. (20) and (21):

Substituting the eigenvectors into equation $\textbf{M}{\textbf P}\textrm{ = }{\lambda ^j}{\textbf P}$, the eigenvalues and eigenvectors can be obtained as follows: for *j* = 1, 2, …, *N *− 1, the eigenvalues and eigenvectors are:

*j*=

*N*,

*N*+ 1, the eigenvalues and eigenvectors are:

*D*= 2

_{m}*R*

_{0}sin[

*m*π/

*N*].

## Funding

National Natural Science Foundation of China (11804288, 91750102); Projects of President Foundation of Chongqing University (2019CDXZWL002).

## Disclosures

The authors declare no conflicts of interest.

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