## Abstract

Dispersion curves of metamaterial steerable antennas composed of two-dimensional arrays of metallic unit structures with the *C*
_{4v} and *C*
_{6v} symmetries are calculated both qualitatively by the tight-binding approximation and quantitatively by the finite-difference time-domain method. Special attention is given to the case of eigenmodes of the *E* symmetry of the *C*
_{4v} point group and those of the *E*
_{1} and *E*
_{2} symmetries of the *C*
_{6v} point group, since they are doubly degenerate on the Γ point of the Brillouin zone so that they naturally satisfy the steerability condition. We show that their dispersion curves have quadratic dependence on the wave vector in the vicinity of the Γ point. To get a linear dispersion, which is advantageous for steerable antennas, we propose a method of *controlled symmetry reduction*. The present theory is an extension of our previous one [Opt. Express **18**, 27371 (2010)] to two-dimensional systems, for which we can achieve the deterministic degeneracy due to symmetry and the *controlled symmetry reduction* becomes available. This design of metamaterial steerable antennas is advantageous in the optical frequency.

© 2011 OSA

## 1. Introduction

Negative refraction realized by left-handed materials, or metamaterials, has attracted a great deal of technological attention [1–7]. As a particular application, a microwave steerable antenna based on negative refraction was proposed and demonstrated [8–10]. Microwave antennas of this kind are important, since they may be used for automotive radar sensors for adaptive cruise control and pre-crash safety systems [11]. In this paper, we present an analytical method to investigate the properties of dispersion curves relevant to the problem of the steerable antenna based on the group-theoretical treatment of the electromagnetic transfer integral that appear in the tight-binding approximation.

The idea of steerable antennas is illustrated in Fig. 1. We assume that an incident wave with angular frequency *ω _{i}* is propagated into a one- or two-dimensional regular array of metallic unit structures that have two dispersion curves in the relevant frequency range. Two important characters for beam steerability are (1) the upper dispersion curve is concave-up and the lower one is concave-down and (2) the two curves touch each other on the Γ point of the Brillouin zone. Then, the incident wave excites an internal eigenmode with a positive group velocity (measured in the direction of incidence). If the frequency of the internal eigenmode is located between

*ω*and

_{u}*ω*, or above the light lines (see Fig. 1), the incident wave is diffracted at a diffraction angle

_{l}*θ*that is determind by

*ω*and the wave vector of the internal mode

_{i}*k*:

_{i}*ω*is decreased from

_{i}*ω*to

_{u}*ω*,

_{l}*θ*changes from 0 to 180 degrees. In particular, if

*k*is negative, which takes place when

_{i}*ω*is in the frequency range of the lower dispersion curve, negative diffraction occurs.

_{i}The degeneracy of eigen frequencies on the Γ point can be realized by two ways: accidental degeneracy due to appropriate combination of device parameters, and deterministic degeneracy due to spatial symmetry of the device structure.

The former case for a one-dimensional regular array of metallic unit structures was examined using tight-binding approximation in our recent paper [12]. It was found that (1) the tight-binding picture, which is based on the localized nature of electromagnetic resonance states of the single unit structure, gives a qualitatively correct description of the metamaterial steerable antenna, (2) the sign of the slope of the dispersion curves around the Γ point depends on the spatial symmetry of the resonance states, (3) when the symmetry of the resonance states satisfies certain conditions, the two dispersion curves are linear and have finite slopes in the vicinity of the Γ point, and (4) otherwise they are quadratic in *k*.

Since item (3) is a main motivation of and closely related to the present study, let us describe it in some detail here. In Ref. [12], we examined a one-dimensional regular array of metallic unit structures of the *C*
_{2v} (rectangular) symmetry, which were fabricated on a dielectric plate [8]. As a result, electromagnetic resonance states localized on a unit structure were characterized by two inplane parities (*σ _{x}* and

*σ*). Because the total system also had the

_{y}*C*

_{2v}symmetry, we could show that eigenmodes on the Γ point are also characterized by these two parities, which are shown in Table 1. Note that we only have one-dimensional representations because of low symmetry of the

*C*

_{2v}point group, that is, all eigenmodes on the Γ point are non-degenerate. Their degeneracy can only be brought about by accidental degeneracy due to appropriate combination of device parameters. In the case of non-degeneracy, combined with a general property of

*k*and their slopes were vanishing on the Γ point.

On the other hand, there are two cases for accidental degeneracy. In the first case, which is given by combinations of {*A*
_{1}, *B*
_{1}} or {*A*
_{2}, *B*
_{2}} symmetries where the two modes in the braces have the same *σ _{y}* parity, dispersion curves are linear in

*k*and have the slopes of the same magnitude but with different signs in the vicinity of the Γ point. Dispersion curves of this case are illustrated in Fig. 2. In the second case, which is given by combinations of {

*A*

_{1},

*A*

_{2}} or {

*B*

_{1},

*B*

_{2}} symmetries where the two modes in the braces have different

*σ*parities, the dispersion curves are quadratic in

_{y}*k*and have vanishing slopes around the Γ point as was shown previously in Fig. 1.

In the second case with vanishing slope on the Γ point, diffraction angle *θ* varies rapidly with incidence frequency *ω _{i}*, which can be easily understood from a rapid variation of

*k*with

_{i}*ω*. This rapid variation results in a bad tunability of the diffraction angle, which is an undesirable feature of steerable antennas. On the other hand, if the slopes of the two dispersion curves are finite and have the same magnitude on the Γ point as shown in Fig. 2, variation of

_{i}*θ*with

*ω*is more constant, which results in a better tunability. Therefore, realization of linear dispersion has been one of the main issues of metamaterial steerable antennas.

_{i}In the microwave region, the design of metamaterial antennas has a large capability, so it might not be too difficult to adjust the device structure. However, in the optical frequencies, the size of the unit structure is less than 1 *μ*m, so that the antennas have to be fabricated by sophisticated lithographical methods like FIB (focused ion beam) milling and EB (electron beam) lithography. In this case, it may be difficult to adjust the device structure to achieve accidental degeneracy.

Thus in this paper, we examine the possibility of using deterministic degeneracy due to structural symmetry. Then we propose a method of controlled symmetry reduction, which is easier to achieve because the original two frequencies are degenerate, so the amount of frequency tuning is small, and it may be attained by electro-optic effect, for example.

This paper is organized as follows. In Section 2, we formulate the tight-binding approximation for two-dimensional regular arrays of metallic unit structures by fully using the localized nature and spatial symmetry of electromagnetic resonance states. We derive dispersion curves and discuss their behavior in the vicinity of the Γ point. As particular applications of the tight-binding formulation, we examine square and triangular lattices. In Section 3, we give some numerical results of the resonance states and dispersion curves to graphically illustrate the qualitative description given in Section 2. In Section 4, we describe the idea of controlled symmetry reduction and show how linear *k* dependence is realized. A brief summary is given in Section 5. Massive calculations on mutual relations among electromagnetic transfer integrals,
${L}_{nm}^{(ij)}$, are given in Appendex A to Appendix D.

## 2. Tight-binding calculation

As we mentioned in Section 1, we deal with two-dimensional regular arrays of metallic unit structures and calculate their electromagnetic dispersion curves. In this section, we present them by analytical calculation based on the tight-binding approximation, and discuss their properties qualitatively.

The wave equation for the magnetic field **H**(**r**, *t*) is given by

*ɛ*(

**r**) is the position-dependent dielectric constant of the system and

*c*is the light velocity in free space. The magnetic permeability was assumed to be unity, since we do not deal with magnetic materials. We impose the periodic boundary condition on

**H**to make our problem well-defined [13] and assume, as an approximation to extract basic features of our problem, that

*ɛ*(

**r**) is real. Thus, Eq. (3) leads to eigen value problems.

For a single unit structure described by dielectric constant *ɛ _{s}*(

**r**), we assume a doubly degenerate resonant state due to the geometrical symmetry of the unit structure, and denote its two magnetic field distributions by ${\mathbf{\text{H}}}_{0}^{(1)}$ and ${\mathbf{\text{H}}}_{0}^{(2)}$. This situation can be realized by eigenmodes of the

*E*symmetry of the

*C*

_{4v}point group and those of the

*E*

_{1}and

*E*

_{2}symmetries of the

*C*

_{6v}point group, as we describe in detail later. Thus, the two distributions satisfy the following eigen value equation:

*ω*

_{0}is the resonance angular frequency. Without loss of generality, we can assume that the two distributions are normalized as follows:

*V*is the volume on which we impose the periodic boundary condition. Note that ${\mathbf{\text{H}}}_{0}^{(i)}$ (

*i*= 1, 2) is dimensionless by this definition.

For the regular array of metallic unit structures, the magnetic field is described by a Bloch function due to the periodicity of the system. If there is no other resonance state in the vicinity of *ω*
_{0}, we can assume in the tight-binding picture that the Bloch wave function is a linear combination of
${\mathbf{\text{H}}}_{0}^{(1)}$ and
${\mathbf{\text{H}}}_{0}^{(2)}$:

**k**is the wave vector in the first Brillouin zone,

*N*is the number of unit structures in volume

*V*, and

**r**

*is the lattice vector.*

_{nm}*n*and

*m*are integers to denote the two-dimensional lattice points.

Since we are interested in the case that the two bands are naturally degenerate on the Γ point of the Brillouin zone, we deal with regular metallic arrays of the *C*
_{4v} and *C*
_{6v} symmetries in the following, since these two symmetries allow the symmetry-induced degeneracy.

#### 2.1. Square lattice

First, we examine the *C*
_{4v} symmetry whose symmetry operations are illustrated in Fig. 3. They consist of identity operation (*E*), rotation by 90 degrees (*C*
_{4},
${C}_{4}^{-1}$), rotation by 180 degrees (*C*
_{2}), basic mirror reflection (*σ _{x}*,

*σ*), and diagonal mirror reflection (

_{y}*σ′*,

_{d}*σ″*).

_{d}If the unit structure also has the symmetry of a regular square, or the *C*
_{4v} symmetry, the whole system of the square lattice has the *C*
_{4v} symmetry. Then, the Bloch states on the Γ point are classified into four one-dimensional representations (*A*
_{1}, *A*
_{2}, *B*
_{1}, and *B*
_{2}) and one two-dimensional representation (*E*). The first four are non-degenerate, while the last is doubly degenerate. Their characters are listed in Table 2. In the rest of this section, we only consider the *E* mode.

In the case of a regular square lattice, the lattice vectors are given by

**a**

_{1}and

**a**

_{2}are elementary lattice vectors of the square lattice:

**H**(

_{k}**r**) given by Eq. (6) satisfies the Bloch condition:

Now, we assume according to the prescription of the tight-binding approximation that **H _{k}**(

**r**) is an eigen function of Eq. (3) and we denote its eigen angular frequency by

*ω*. Thus it satisfies the following equation:

_{k}*ℒ*:

*V*, we obtain

It is known from group theory that the two eigen functions of the *E* mode can be assumed to transform like the two coordinates, *x* and *y*, when any symmetry operation *R* ∈ *C*
_{4v} is applied [14]. Using this property, we can derive various relations among
${L}_{nm}^{(ij)}$’s. Details of the calculations are given in Appendix A. As a result, we can prove that there are only five independent elements among 36 integrals for the origin ((*n*, *m*) = (0, 0)), the nearest neighbor ((*n*, *m*) = (±1, 0), (0, ±1)), and the second nearest neighbor ((*n*, *m*) = (±1, ±1)) lattice points in Eq. (13):

*n*and

*m*in Eq. (6), we only consider contributions from these lattice points in the following.

As for the contribution from the right-hand side of Eq. (10), we only keep its most dominant term, $AV\hspace{0.17em}{\omega}_{\mathbf{\text{k}}}^{2}/{c}^{2}\hspace{0.17em}N$. Thus, we obtain

*V*, we obtain

Let us examine some special cases. First, for the Γ point (*k _{x}* =

*k*= 0), we have a doubly degenerate eigen frequency

_{y}*ω*

_{Γ}:

*k*with

_{x}*k*, as it should be, which is a consequence of the

_{y}*C*

_{4v}symmetry of the square lattice. Thirdly, it has a property which is a consequence of the time-reversal symmetry of the wave equation [13].

Next, let us examine the dispersion in the vicinity of the Γ point. For this purpose, we assume that |*k _{x}a*|, |

*k*| ≪ 1 and keep the lowest order terms. Then, from Eq. (23), we obtain

_{y}a*ω*

**with respect to the wave vector, is periodic in ϕ with a period of**

_{k}*π*/2.

Finally, let us comment on the influence of truncation of contributions up to terms from the second-nearest neighbors in Eqs. (21) and (22). Those contributions from the third-nearest and farther lattice points give higher order terms with respect to *k _{x}* and

*k*. So, as far as the behavior of dispersion curves in the vicinity of the Γ point is concerned, they do not bring about a qualitative change.

_{y}#### 2.2. Triangular lattice

In this section, we examine the triangular lattice with the *C*
_{6v} symmetry whose symmetry operations are illustrated in Fig. 4. They consist of identity operation (*E*), rotation by 60 degrees (*C*
_{6},
${C}_{6}^{-1}$), rotation by 120 degrees (*C*
_{3} and
${C}_{3}^{-1}$), rotation by 180 degrees (*C*
_{2}), and two sets of three equivalent mirror reflections ((*σ _{x}*,

*σ′*,

_{x}*σ″*) and (

_{x}*σ*,

_{y}*σ′*,

_{y}*σ″*)).

_{y}If the unit structure also has the symmetry of a regular hexagon, or the *C*
_{6v} symmetry, the whole system of the triangular lattice has the *C*
_{6v} symmetry. Then, the Bloch states on the Γ point are classified into four one-dimensional representations (*A*
_{1}, *A*
_{2}, *B*
_{1}, and *B*
_{2}) and two two-dimensional representations (*E*
_{1} and *E*
_{2}) [14]. The former four are non-degenerate and the latter two are doubly degenerate. Their characters are listed in Table 3. In the rest of this section, we only consider the *E*
_{1} and *E*
_{2} modes.

In the case of the regular triangular lattice, the elementary lattice vectors are given by

By fully using the symmetry properties of the eigen functions, we can derive the following relations (see Appendix B):

*k*, Eq. (42) is approximated as

*k*.

## 3. Numerical results

We calculated the field distribution of structural electromagnetic resonances of metallic unit structures and dispersion relations of their periodic arrays by the FDTD (finite-difference time-domain) method [15, 16]. The details of the method of calculation are described in Ref. [12]. In the present study, we assumed two metallic unit structures of the square and hexagonal symmetries as shown in Fig. 5 to graphically illustrate our analytical calculation presented in the previous sections. According to the antenna design demonstrated in Ref. [8], these structures were assumed to be fabricated on a dielectric slab with a ground electrode on the back surface. Since the antenna was originally designed to operate in a microwave frequency around 76 GHz (freespace wavelength = 3.95 mm), the size of the unit structures was chosen as of the order of 1 mm. To calculate the dispersion relation, regular square and triangular arrays of the unit structures were assumed.

Figure 6 shows distributions of the *z* component of the magnetic field, *H _{z}*, of resonance states. The resonance states were identified by finding peaks (resonance frequencies) in the Fourier transform of temporal variation of the electromagnetic field after pulsed excitation, whereas their field distributions were obtained by CW (continuous wave) excitation at the resonance frequencies [12].

Figures 6(a) and 6(b) show two distributions of an *E* mode found at 158 GHz for the unit structure of the square symmetry given in Fig. 5(a). These two distributions were obtained by imposing the following boundary conditions on the electromagnetic field according to the symmetry of corresponding polynomial representations of the *E* mode given in Appendix A: (*σ _{x}*,

*σ*) = (−1, 1) for (a) and (1, −1) for (b). As is apparent, one distribution is obtained by rotating the other by 90 degrees.

_{y}Figures 6(c) and 6(d) show the distributions of an *E*
_{1} mode found at 177 GHz for the unit structure of the hexagonal symmetry given in Fig. 5(b). In this case, we imposed the same boundary conditions as the *E* mode according to the symmetry of the polynomial representations of the *E*
_{1} mode given in Appendix B. Note that one distribution is *not* obtained by rotating the other for this case. Finally, Figs. 6(e) and 6(f) are the distributions of an *E*
_{2} mode found at 295 GHz. To obtain these distributions, we imposed the following boundary conditions according to the symmetry of the polynomial representations of the *E*
_{2} mode given in Appendix C: (*σ _{x}*,

*σ*) = (−1, −1) for (e) and (1, 1) for (f).

_{y}Let us make three comments about Fig. 6 here. First, the dominant component among the three electric field components is *E _{z}*, which is perpendicular to the surface of the metallic unit structure, because the tangential components (

*E*and

_{x}*E*) are small due to their continuity across the surface and the small electric field inside the metal caused by the large conductivity. So, the electromagnetic field has mostly a TM (transverse magnetic) character, and therefore,

_{y}*H*is small. Nevertheless, we plotted

_{z}*H*in Fig. 6 in order to provide a graphic description consistent with the analytical calculations given in Section 2.

_{z}Second, the symmetry of the electric field is generally different from that of the magnetic field [13]. When we denote their characters for symmetry operation *R* by *χ*
^{(E)}(*R*) and *χ*
^{(H)}(*R*), respectively, then

*R*is the determinant of the transformation matrix for

*R*. det

*R*is equal to 1 for proper transformations like rotations and is equal to −1 for improper transformations like mirror reflections. This difference should be treated appropriately when we impose symmetric and antisymmetric boundary conditions on the electromagnetic field to extract resonances and field distributions of particular symmetries.

Thirdly, the resonance frequencies of the *E*, *E*
_{1}, and *E*
_{2} modes shown in Fig. 6 are considerably larger than those values obtained by our previous analysis of a similar structure [12], since the unit structures are smaller in the present analysis. We chose these sizes to have the same lattice constant, 0.6 mm, as before. Because the resonance frequency is inversely proportional to the device size, we may choose a larger lattice constant and device size if we need smaller frequencies.

Next, let us examine the dispersion curves. As an example, Fig. 7 shows the dispersion curves of the regular square array of unit structures illustrated in Fig. 5(c). Among the four modes identified by numerical calculations, two modes originate from the *E* resonance state shown in Figs. 6(a) and 6(b). They are degenerate on the Γ point as we expected. Their dispersion is consistent with the quadratic dependence on the wave vector in the vicinity of the Γ point, which was predicted by the tight-binding calculation in Section 2.1. In addition to these modes, we also have one mode originating from a non-degenerate *B*
_{2} resonance state and another mode that has a character of the lowest TM mode of the dielectric slab, whose original dispersion curve is located very close to the light line given by *ω* = *ck*.

In Fig. 7, the parity of the *electric* field with respect to the *y* coordinate is denoted by *p _{y}*. We should note that it is opposite to that of the magnetic field. Three of the four modes have the same

*p*, so that they mix with each other when their dispersion curves come close and show apparent anti-crossing behaviors.

_{y}## 4. Controlled symmetry reduction

As we showed in Section 2 and Section 3, we can realize dispersion curves that are degenerate on the Γ point due to the spatial symmetry of the system. However, their dependence on the wave vector is quadratic in the vicinity of the Γ point, which may be an undesirable feature for the beam steering application.

The absence of *k*-linear terms in the Taylor expansion of *ω*
** _{k}** with respect to

*k*and

_{x}*k*are relevant to some vanishing ${L}_{nm}^{(ij)}$’s due to symmetry, as we will describe in the following. So, we may get a linear dispersion in the vicinity of the Γ point by intentional reduction of symmetry to yield non-vanishing ${L}_{nm}^{(ij)}$’s. Of course, this symmetry reduction results in non-degeneracy on the Γ point. So, in order to make the two frequencies coincide, we further have to make a fine adjustment of sample parameters, which is possible by using the electro-optic effect for example. We call this method “controlled symmetry reduction”. In this section, we examine its basic features with the square lattice as an example.

_{y}Now, let us examine symmetry reduction from *C*
_{4v} to *C _{s}*. The latter is a simple point group that consists of identity (

*E*) and a mirror reflection (

*σ*). Its characters are given in Table 4. It is known by group theory that the

_{x}*E*mode of

*C*

_{4v}splits into an

*A*mode and a

*B*mode of

*C*due to the lowered symmetry.

_{s}Even when the symmetry of the square lattice is reduced, the two split eigen frequencies are expected to be close to each other if the modification of the lattice structure is small. We assume that frequency separation with other resonant states, if any, is large compared with the splitting of the *E* mode, and construct a Bloch function with the split *E* mode functions alone. We denote them by **H**
_{1} (*A* mode) and **H**
_{2} (*B* mode). They satisfy Eq. (4) as before but with different eigen frequencies:

**H**

_{1}and

**H**

_{2}:

Because the number of symmetry operations is small for *C _{s}*, we have a smaller number of relations among
${L}_{nm}^{(ij)}$’s, which are defined by the following equation in the present case:

Now, we examine the condition for accidental degeneracy of eigen frequencies on the Γ point by an appropriate choice of device parameters. By setting *k _{x}* =

*k*= 0 in Eq. (58), the condition for degeneracy is

_{y}*k*|, |

_{x}a*k*| ≪ 1 and only keep dominant terms. The result is

_{y}a*k*as we expected.

_{x}The reason for this feature can be understood by comparing
${L}_{nm}^{(ij)}$’s between the *C*
_{4v} and *C _{s}* cases. Due to lowered symmetry,
${L}_{\pm 1,0}^{(12)}$ and
${L}_{\pm 1,0}^{(21)}$ are non-zero for

*Cs*, while they are vanishing for

*C*

_{4v}. These terms, which are represented by

*L*

_{1}in Eqs. (58), (60), and (61), apparently give the

*k*-linear term.

This linear *k* dependence is desirable for application to steerable antennas as we mentioned in Section 1. Compared with the case of purely accidental degeneracy, the two frequencies on the Γ point are close to each other due to their original degeneracy in the case of controlled symmetry reduction. So, we may expect that their frequency tuning can be attained relatively easily.

## 5. Conclusion

We calculated dispersion curves of degenerate metamaterial antennas by tight-binding approximation based on structural electromagnetic resonances of metallic unit structures. Using the properties of the electromagnetic transfer integral,
${L}_{nm}^{(ij)}$, which were clarified by the group-theoretical treatment, we calculated the behavior of dispersion curves in the vicinity of the Γ point of the Brilluin zone. In the square lattice of the *C*
_{4v} symmetry, the *E* modes are doubly degenerate on the Γ point and their dispersion shows quadratic dependence on the wave vector. In the triangular lattice of the *C*
_{6v} symmetry, dispersion curves of the *E*
_{1} and *E*
_{2} modes, which are also doubly degenerate on the Γ point, are isotropic and have quadratic dependence on the wave vector around the Γ point. To achieve linear *k* dependence, which is desirable for application to steerable antennas, we proposed a method of *controlled symmetry reduction*. As an example, we examined the case of symmetry reduction from *C*
_{4v} to *C _{s}* and derived the dispersion relation of the latter, which showed linear

*k*dependence as we expected. To verify the analytical calculation, we presented some numerical results of the resonant states of unit structures of square and hexagonal symmetries together with dispersion curves of a square lattice. The theory given in this paper is an extension of our previous one [12] to two-dimensional systems, for which we can achieve the deterministic degeneracy due to symmetry on the Γ point and the

*controlled symmetry reduction*becomes available.

## A. *E* mode of *C*_{4v} symmetry

A polynomial representation of the *E* mode is given by a pair of {*x*, *y*} [14]. So, we can choose two eigen functions of the *E* symmetry,
${\mathbf{\text{H}}}_{0}^{(1)}(\mathbf{\text{r}})$ and
${\mathbf{\text{H}}}_{0}^{(2)}(\mathbf{\text{r}})$, such that they are transformed as the following two functions when any *R* ∈ *C*
_{4v} is operated.

*R*∈

*C*

_{4v}:

*E*representation in Table 2, which are the trace (sum of the diagonal elements) of the above matrices.

Because
${\mathbf{\text{H}}}_{0}^{(1)}$ and
${\mathbf{\text{H}}}_{0}^{(2)}$ are also an *E* representation, they transform like *f*
_{1} and *f*
_{2}. But we should note that there is a difference due to the vector nature of the former. Their transformation is defined as

Let us proceed to the derivation of relations among the
${L}_{nm}^{(ij)}$’s introduced by Eq. (13). We define a differential operator *ℒ* by

**r**to

**r′**=

*σ*

_{x}**r**and evaluate ${L}_{00}^{(12)}$. Since

*σ*does not change the size of volume elements, we have

_{x}*σ*(≡

_{x}*E*) for later use. Since ${\sigma}_{x}^{-1}$ does not change the value of inner products, we obtain

*ℒ′*is defined as It is an operator in the

**r′**coordinate system equivalent to

*ℒ*in the

**r**coordinate system. We can prove

*ℒ′*=

*ℒ*[13], although we do not use this relation in the following. Substituting the first relation in Eq. (67) and Eq. (68), we obtain

Next, let us examine ${L}_{00}^{(11)}$. We denote it by

where the first term on the right-hand side is its original value for an isolated unit structure. Now, we change the variable of integration from**r**to

**r′**=

*C*

_{4}

**r**:

Let us proceed to the evaluation of
${L}_{10}^{(12)}$. Using **r′** = *σ _{y}*

**r**,

As for
${L}_{10}^{(11)}$, changing the variable of integration from **r** to **r′** = *C*
_{4}
**r**,

**r**

_{01}=

**r**

_{10}. Using transformations by

*σ*and ${C}_{4}^{-1}$, we can also prove So, from Eqs. (83) and (84), we have Similarly we obtain

_{x}For terms with (*n, m*) = (±1, ±1), applying all transformations of *C*
_{4v} to
${L}_{11}^{(11)}$, we can easily prove

*R*∈

*C*

_{4v}and ${L}_{nm}^{(j)}$ yield the same relations given above.

## B. *E*_{1} mode of *C*_{6v} symmetry

As the *E* mode of the *C*
_{4v} symmetry, it is known from group theory that two eigen functions of the *E*
_{1} mode of the *C*
_{6v} point group can be assumed to transform like the *x* and *y* coordinates when any symmetry operation *R* ∈ *C*
_{6v} is applied [14]. So, by using this property, we can derive the matrix representation of all elements of *C*
_{6v}. The results are as follows:

Since the essential features of the dispersion curves around the Γ point can be clarified by a summation over a relatively small number of lattice points in Eq. (6), we only consider the lattice points on the origin and the nearest neighbors and denote them by integers from 0 to 6 as shown in Fig. 4.

Let us start with the case of *n* = 0. By changing the variable of integration from **r** to **r′** = *C*
_{6}
**r**, we obtain,

*σ*for ${L}_{0}^{(12)}$,

_{x}Now we proceed to the case involving the nearest neighbor lattice points. Using *C*
_{2} for
${L}_{1}^{(11)}$ and
${L}_{1}^{(22)}$, we obtain,

**r**to

*C*

_{2}

**r**for ${L}_{1}^{(12)}$ and ${L}_{1}^{(21)}$, we obtain while using

*σ*, we obtain Thus, On the other hand, applying

_{x}*σ*,

_{x}*C*

_{2}and

*σ*to ${L}_{2}^{(11)}$ and ${L}_{2}^{(22)}$, we obtain Finally, applying

_{y}*σ*,

_{x}*C*

_{2}and

*σ*to ${L}_{2}^{(12)}$ and ${L}_{2}^{(21)}$, we obtain

_{y}For the six *M _{i}*’s, we can derive three relations as follows. First, by applying
${C}_{6}^{-1}$ to
${L}_{2}^{(11)}$
,

*R*∈

*C*

_{6v}and ${L}_{n}^{(ij)}$ yield the same relations given above.

## C. *E*_{2} mode of *C*_{6v} symmetry

It is known from group theory that {2*xy*, *x*
^{2} – *y*
^{2}} is a polynomial representation of the *E*
_{2} mode. We can derive the matrix representation of all elements of *C*
_{6v}:

*C*

_{2}rotation is an idendity matrix. As a consequence, the product of matrices for

*C*

_{6}and

*C*

_{3}is also an identity matrix, so that matrices for

*C*

_{6}and ${C}_{3}^{-1}$, and hence, those for ${C}_{6}^{-1}$ and

*C*

_{3}are the same, respectively.

Following the same procedure as we did for the *E*
_{1} mode in Appendix B, we can derive various relations. The results are completely the same as the case of the *E*
_{1} mode shown in Eqs. (31)–(38).

## D. ${L}_{nm}^{(ij)}$ for *C*_{s} symmetry

_{s}

Because
${L}_{nm}^{(ij)}$ integrals for the origin and the nearest neighbors are necessary in Section 4, we only consider (*n*, *m*) = (0, 0), (±1, 0), and (0, ±1). Since wave functions **H**
_{1} and **H**
_{2} are mutually different due to the symmetry reduction from *C*
_{4v} to *C _{s}*,
${L}_{00}^{(11)}$ and
${L}_{00}^{(22)}$ are generally different from each other. So, we denote them by

*ω*

_{1}and

*ω*

_{2}are eigen frequencies for an isolated unit metallic structure described by dielectric constant

*ɛ*(

_{s}**r**). Applying

*σ*to ${L}_{00}^{(12)}$ and ${L}_{00}^{(21)}$ as we did in previous sections, we can prove Next, applying

_{x}*σ*to ${L}_{10}^{(11)}$ and ${L}_{10}^{(22)}$, we obtain

_{x}In addition to these two relations, we can also derive two more relations using the translational symmetry and Hermitian property of operator *ℒ*. We denote translation by lattice vector **r**
* _{nm}* by

*T*. Then, from the translational symmetry of

_{nm}*ɛ*(

**r**) and invariance of differential operators under any uniform translation, we have

*ℒ*and the periodic boundary condition, we can prove that

*ℒ*is an Hermitian operator [13]:

**Q**

_{1}and

**Q**

_{2}are arbitrary vector functions that satisfy the periodic boundary condition.

Now, we change the variable of integration from **r** to **r′** = **r** – **r**
_{10} in the expression of
${L}_{10}^{(11)}$:

*M*

_{2}and

*M*

_{3}are real numbers.

As for
${L}_{10}^{(12)}$, similarly applying *σ _{x}* and the same translation, we obtain

*σ*yields identity equations. On the other hand, by changing the variable of integration from

_{x}**r**to

**r**–

**r**

_{01}, we obtain

*σ*, we can prove

_{x}## Acknowledgments

This study was supported by a Grant-in-Aid for Scientific Research on Innovative Areas from the Japanese Ministry of Education, Culture, Sports and Technology (Grant number 22109007).

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