Analytical study of two-dimensional degenerate metamaterial antennas

Kazuaki Sakoda; Haifeng Zhou

doi:10.1364/OE.19.013899

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_u and ω_l, or above the light lines (see Fig. 1), the incident wave is diffracted at a diffraction angle θ that is determind by ω_i and the wave vector of the internal mode k_i:

θ = {cos}^{- 1} \frac{c k_{i}}{ω_{i}},

which is measured from the direction of incidence. So, when ω_i is decreased from ω_u to ω_l, θ changes from 0 to 180 degrees. In particular, if k_i is negative, which takes place when ω_i is in the frequency range of the lower dispersion curve, negative diffraction occurs.

Fig. 1 Conceptual dispersion curves of a metamaterial steerable antenna. The vertical axis is the angular frequency ω of the electromagnetic field normalized by the light velocity in free space c and the lattice constant of the regular array of unit structures a. The horizontal axis is the wave vector in the first Brillouin zone. The dispersion consists of two curves: The upper one denoted by f_u is concave-up and the lower one denoted by f_l is concave-down. They touch each other on the Γ point (k = 0), which can be realized by accidental degeneracy of eigen frequencies due to appropriate combination of device parameters or by deterministic degeneracy caused by spatial symmetry of device structure. When an incident wave with angular frequency ω_i is propagated in the positive k direction, it excites an eigenmode (denoted by wave vector k_i) with a positive group velocity, which is given by ∂ω/∂k. So, only eigenmodes on dispersion branches with positive slopes denoted by the blue color are excited. If the angular frequency of the excited mode is located above the light lines, which are given by ω = ±ck, the incident wave is leaky and diffracted in the direction determined by ω_i and k_i. Thus, the upper and lower limits of working frequency are given by ω_u and ω_l. Reproduced with permission from Opt. Express 18, 27371 (2010). Copyright 2010 The Optical Society (OSA).

Download Full Size | PDF

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_y). Because the total system also had the 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

ω_{- k} = ω_{k},

which is derived from time-reversal symmetry of Maxwell’s wave equation in the absence of static magnetic field, we could conclude that dispersion curves were quadratic in k and their slopes were vanishing on the Γ point.

Table 1. Parity of Eigenmodes on the Γ Point for a Regular Array of Metallic Unit Structure of the C _2v Symmetry

View Table | View all tables in this article

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_y parities, the dispersion curves are quadratic in k and have vanishing slopes around the Γ point as was shown previously in Fig. 1.

Fig. 2 Illustration of dispersion curves in the case of accidental degeneracy, which can be realized by A ₁ and B ₁ modes, or by A ₂ and B ₂ modes on the Γ point of one-dimensional metamaterial steerable antennas composed of regular arrays of unit structures of the C _2v symmetry.

Download Full Size | PDF

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_i with ω_i. 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 θ with ω_i 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.

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 μm, 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, $L_{n m}^{(i j)}$ , 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

\nabla \times [\frac{1}{ɛ (r)} \nabla \times H (r, t)] = - \frac{1}{c^{2}} \frac{\partial^{2}}{\partial t^{2}} H (r, t),

where 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 $H_{0}^{(1)}$ and $H_{0}^{(2)}$ . 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:

\nabla \times [\frac{1}{ɛ_{s} (r)} \nabla \times H_{0}^{(1, 2)} (r)] = \frac{ω_{0}^{2}}{c^{2}} H_{0}^{(1, 2)} (r),

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

\int_{V} d r H_{0}^{(i) *} (r) \cdot H_{0}^{(j)} (r) = V δ_{i j},

where V is the volume on which we impose the periodic boundary condition. Note that

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 $H_{0}^{(1)}$ and $H_{0}^{(2)}$ :

H_{k} (r) = \frac{1}{N} \sum_{n, m} e^{i k \cdot r_{n m}} [A H_{0}^{(1)} (r - r_{n m}) + B H_{0}^{(2)} (r - r_{n m})],

where k is the wave vector in the first Brillouin zone, N is the number of unit structures in volume V, and r _nm is the lattice vector. 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_y), and diagonal mirror reflection (σ′_d, σ″_d).

Fig. 3 Symmetry operations of the C _4v group. There are two sets of two equivalent mirror reflections, which are denoted by (σ_x, σ_y) and (σ′_d, σ″_d). Lattice points are denoted by solid circles.

Download Full Size | PDF

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.

Table 2. Character Table of the C _4v Point Group

View Table | View all tables in this article

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

r_{n m} = n a_{1} + m a_{2} .

Here, a ₁ and a ₂ are elementary lattice vectors of the square lattice:

a_{1} = (\begin{matrix} a \\ 0 \end{matrix}), a_{2} = (\begin{matrix} 0 \\ a \end{matrix}),

Note that H_k(r) given by Eq. (6) satisfies the Bloch condition:

H_{k} (r + a_{i}) = e^{i k \cdot a_{i}} H_{k} (r) .

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_k. Thus it satisfies the following equation:

ℒ H_{k} (r) = \frac{ω_{k}^{2}}{c^{2}} H_{k} (r),

where we introduced a differential operator ℒ:

ℒ H (r) = \nabla \times [\frac{1}{ɛ (r)} \nabla \times H (r)] .

Multiplying Eq. (10) by

H_{0}^{(1) *} (r)

and integrating over V, we obtain

\int_{V} d r H_{0}^{(1) *} (r) \cdot ℒ H_{k} (r) = \frac{V}{N} \sum_{n, m} e^{i k \cdot r_{n m}} [A L_{n m}^{(11)} + B L_{n m}^{(12)}],

where

L_{n m}^{(i j)}

is defined as

L_{n m}^{(i j)} = \frac{1}{V} \int_{V} d r H_{0}^{(i) *} (r) \cdot [ℒ H_{0}^{(j)} (r - r_{n m})] .

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 $L_{n m}^{(i j)}$ ’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):

L_{0, 0}^{(11)} = L_{0, 0}^{(22)} \equiv \frac{ω_{0}^{2}}{c^{2}} + M_{1},

L_{0, 0}^{(12)} = L_{0, 0}^{(21)} = 0,

L_{\pm 1, 0}^{(11)} = L_{0, \pm 1}^{(22)} \equiv M_{2},

L_{0, \pm 1}^{(11)} = L_{\pm 1, 0}^{(22)} \equiv M_{3},

L_{\pm 1, 0}^{(12)} = L_{0, \pm 1}^{(12)} = L_{\pm 1, 0}^{(21)} = L_{0, \pm 1}^{(21)} = 0,

L_{\pm 1, \pm 1}^{(11)} = L_{\pm 1, \pm 1}^{(22)} \equiv M_{4},

\begin{array}{l} L_{1, 1}^{(12)} & = L_{- 1, - 1}^{(12)} = - L_{1, - 1}^{(12)} = - L_{- 1, 1}^{(12)} \\ = L_{1, 1}^{(21)} = L_{- 1, - 1}^{(21)} = - L_{1, - 1}^{(21)} = - L_{- 1, 1}^{(21)} \equiv M_{5} \end{array}

Because essential features of dispersion curves, especially those around the Γ point, can be described well with a small number of 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, $A V ω_{k}^{2} / c^{2} N$ . Thus, we obtain

\begin{array}{l} A [ω_{0}^{2} - ω_{k}^{2} + c^{2} (M_{1} + 2 M_{2} cos k_{x} a + 2 M_{3} cos k_{y} a + 4 M_{4} cos k_{x} a cos k_{y} a)] \\ = & 4 B c^{2} M_{5} sin k_{x} a sin k_{y} a . \end{array}

Similarly, multiplying Eq. (10) by

H_{0}^{(2) *} (r)

and integrating over volume V, we obtain

\begin{array}{l} 4 A c^{2} M_{5} sin k_{x} a sin k_{y} a \\ = & B [ω_{0}^{2} - ω_{k}^{2} + c^{2} (M_{1} + 2 M_{3} cos k_{x} a + 2 M_{2} cos k_{y} a + 4 M_{4} cos k_{x} a cos k_{y} a .)] \end{array}

By solving the secular equation derived from Eqs. (21) and (22), we finally obtain the following dispersion relation:

\begin{array}{l} \frac{ω_{k}^{2}}{c^{2}} & = & \frac{ω_{0}^{2}}{c^{2}} + M_{1} + (M_{2} + M_{3}) (cos k_{x} a + cos k_{y} a) + 4 M_{4} cos k_{x} a cos k_{y} a \\ \pm {[{(M_{2} - M_{3})}^{2} {(cos k_{x} a - cos k_{y} a)}^{2} + 16 M_{5}^{2} {sin}^{2} k_{x} a {sin}^{2} k_{y} a]}^{1 / 2} . \end{array}

Let us examine some special cases. First, for the Γ point (k_x = k_y = 0), we have a doubly degenerate eigen frequency ω _Γ:

ω_{Γ} = \sqrt{ω_{0}^{2} + c^{2} (M_{1} + 2 M_{2} + 2 M_{3} + 4 M_{4}) .}

Secondly, Eq. (23) is invariant when we exchange k_x with k_y, as it should be, which is a consequence of the C _4v symmetry of the square lattice. Thirdly, it has a property

ω_{- k} = ω_{k},

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

ω_{k} = ω_{Γ} - \frac{a^{2} c^{2} k^{2}}{4 ω_{Γ}} [M_{2} + M_{3} + 4 M_{4} \pm 2 F (ϕ)],

where

k = \sqrt{k_{x}^{2} + k_{y}^{2}},

ϕ = {tan}^{- 1} \frac{k_{y}}{k_{x}},

F (ϕ) = \sqrt{{(\frac{M_{2} - M_{3}}{2})}^{2} {cos}^{2} 2 ϕ + 4 M_{5}^{2} {sin}^{2} 2 ϕ} .

So, the dispersion curves have a quadratic dependence on the wave vector in the vicinity of the Γ point. Their effective mass, which is inversely proportional to the second-order derivative of ω _k with respect to the wave vector, is periodic in ϕ with a period of 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_y. 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.

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″_x) and (σ_y, σ′_y, σ″_y)).

Fig. 4 Symmetry operations of the C _6v group. There are two sets of three equivalent mirror reflections that are denoted by (σ_x, σ′_x, σ″_x) and (σ_y, σ′_y, σ″_y). Seven lattice points (the origin and its nearest neighbors) are denoted by integers from 0 to 6.

Download Full Size | PDF

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.

Table 3. Character Table of the C _6v Point Group

View Table | View all tables in this article

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

a_{1} = (\begin{matrix} a \\ 0 \end{matrix}), a_{2} = (\begin{matrix} a / 2 \\ \sqrt{3} a / 2 \end{matrix}) .

Then, we can write down the Bloch function as before. By similar calculations as in the previous section, we can derive a secular equation to determine the dispersion relation. It is sufficient to take into consideration terms up to the nearest neighbor lattice points in the calculation of

L_{n m}^{(i j)}

in order to obtain the same accuracy as Eq. (23). Thus, it is convenient to label the origin and the nearest neighbors by integers from 0 to 6 as shown in Fig. 4.

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

L_{0}^{(11)} = L_{0}^{(22)} \equiv \frac{ω_{0}^{2}}{c^{2}} + M_{1},

L_{0}^{(12)} = L_{0}^{(21)} = 0,

L_{1}^{(11)} = L_{4}^{(11)} \equiv M_{2},

L_{1}^{(22)} = L_{4}^{(22)} \equiv M_{3},

L_{1}^{(12)} = L_{4}^{(12)} = L_{1}^{(21)} = L_{4}^{(21)} = 0,

L_{2}^{(11)} = L_{3}^{(11)} = L_{5}^{(11)} = L_{6}^{(11)} \equiv M_{4},

L_{2}^{(22)} = L_{3}^{(22)} = L_{5}^{(22)} = L_{6}^{(2)} \equiv M_{5},

\begin{array}{l} L_{2}^{(12)} & = - L_{3}^{(12)} = L_{5}^{(12)} = - L_{6}^{(12)} \\ = L_{2}^{(21)} = - L_{3}^{(21)} = L_{5}^{(21)} = - L_{6}^{(21)} \equiv M_{6} . \end{array}

We can further prove the following relations:

M_{4} = \frac{M_{2} + 3 M_{3}}{4},

M_{5} = \frac{3 M_{2} + M_{3}}{4},

M_{6} = \frac{\sqrt{3} (M_{2} - M_{3})}{4} .

So, the number of independent elements among the 28

L_{n}^{(i j)}

’s is three. The dispersion relation is given by

\begin{array}{l} \frac{ω_{k}^{2}}{c^{2}} & = & \frac{ω_{0}^{2}}{c^{2}} + M_{1} + (M_{2} + M_{3}) (cos k_{x} a + 2 cos \frac{k_{x} a}{2} cos \frac{\sqrt{3} k_{y} a}{2}) \\ \pm (M_{2} - M_{3}) {[{(cos k_{x} a - cos \frac{k_{x} a}{2} cos \frac{\sqrt{3} k_{y} a}{2})}^{2} + 3 {sin}^{2} \frac{k_{x} a}{2} {sin}^{2} \frac{\sqrt{3} k_{y} a}{2}]}^{1 / 2} \end{array}

The dispersion curves give a degenerate eigen frequency on the Γ point:

ω_{Γ} = \sqrt{ω_{0}^{2} + c^{2} (M_{1} + 3 M_{2} + 3 M_{3})} .

For small k, Eq. (42) is approximated as

ω_{k} = ω_{Γ} - \frac{3 a^{2} c^{2} k^{2} M}{16 ω_{Γ}},

where

M = {\begin{matrix} M_{2} + 3 M_{3}, \\ 3 M_{2} + M_{3} . \end{matrix}

So, the dispersion curves are isotropic in the vicinity of the Γ point, and they are quadratic in 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.

Fig. 5 Illustration of metallic unit structures of (a) square and (b) hexagonal symmetries. These unit structures were assumed to be fabricated on a dielectric slab with a ground electrode on its back surface. Based on the device design of Ref. [8], the dielectric constant and thickness of the slab were assumed to be 2.2 and 0.127 mm, respectively. For calculating the dispersion relation, (c) a periodic square array of the unit structure shown in (a) was assumed, whose lattice constant a was 0.6 mm.

Download Full Size | PDF

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].

Fig. 6 Distribution of H_z of resonance states of a unit structure. (a), (b) E mode at 158 GHz, (c), (d) E ₁ mode at 177 GHz, and (e), (f) E ₂ mode at 295 GHz. H_z on the horizontal plane in the middle of the dielectric slab is shown.

Download Full Size | PDF

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

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_y) = (−1, −1) for (e) and (1, 1) for (f).

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_x and E_y) 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, H_z is small. Nevertheless, we plotted H_z in Fig. 6 in order to provide a graphic description consistent with the analytical calculations given in Section 2.

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

χ^{(H)} (R) = det R χ^{(E)} (R),

where detR is the determinant of the transformation matrix for R. detR 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.

Fig. 7 Dispersion curves of the regular square array of unit structures illustrated in Fig. 5(c). In the analyzed frequency range from 100 to 250 GHz, there are four electromagnetic modes: (1) two modes originating from a degenerate E resonance state shown in Fig. 6(a) and 6(b), (2) one mode originating from a non-degenerate B ₂ resonance state, and (3) one mode that has the character of the lowest TM waveguide mode of the dielectric slab whose original dispersion is very close to the light line given by ω = ck. The parity of the electric field with respect to the y coordinate is denoted by p_y, which the reader should note is opposite to that of the magnetic field. Because modes with the same parity mix with each other when their dispersion curves come close, they show apparent anti-crossing behaviors.

Download Full Size | PDF

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

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_x and k_y are relevant to some vanishing $L_{n m}^{(i j)}$ ’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 $L_{n m}^{(i j)}$ ’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.

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_x). Its characters are given in Table 4. It is known by group theory that the E mode of C _4v splits into an A mode and a B mode of C_s due to the lowered symmetry.

Table 4. Character Table of the C_s Point Group

View Table | View all tables in this article

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:

\nabla \times [\frac{1}{ɛ_{s} (r)} \nabla \times H_{1, 2} (r)] = \frac{ω_{1, 2}^{2}}{c^{2}} H_{1, 2} (r) .

The Bloch function is given by the linear combination of H ₁ and H ₂:

H_{k} (r) = \frac{1}{N} \sum_{n, m} e^{i k \cdot r_{n m}} [A H_{1} (r - r_{n m}) + B H_{2} (r - r_{n m})] .

Because the number of symmetry operations is small for C_s, we have a smaller number of relations among $L_{n m}^{(i j)}$ ’s, which are defined by the following equation in the present case:

L_{n m}^{(i j)} = \frac{1}{V} \int_{V} d r H_{i}^{*} (r) \cdot [ℒ H_{j} (r - r_{n m})] .

Derivation of relations among

L_{n m}^{(i j)}

’s is given in Appendix D. The results are summarized as follows:

L_{0, 0}^{(11)} \equiv \frac{ω_{1}^{2}}{c^{2}} + {M'}_{1}, L_{0, 0}^{(22)} \equiv \frac{ω_{2}^{2}}{c^{2}} + {M ″}_{1},

L_{0, 0}^{(12)} = L_{0, 0}^{(21)} = 0,

L_{1, 0}^{(11)} = L_{- 1, 0}^{(11)} \equiv M_{2},

L_{1, 0}^{(22)} = L_{- 1, 0}^{(22)} \equiv M_{3},

L_{1, 0}^{(12)} = - L_{- 1, 0}^{(12)} = L_{- 1, 0}^{(21) *} = - L_{1, 0}^{(21) *} \equiv L_{1},

L_{0, 1}^{(11)} = L_{0, - 1}^{(11) *} \equiv {M^{'}}_{3},

L_{0, 1}^{(22)} = L_{0, - 1}^{(22) *} \equiv {M^{'}}_{2},

L_{0, 1}^{(12)} = L_{0, - 1}^{(12)} = L_{0, 1}^{(21)} = L_{0, - 1}^{(21)} = 0.

Then, by solving the secular equation as before, we obtain the dispersion relation:

\begin{array}{l} \frac{ω_{k}^{2}}{c^{2}} & = & \frac{1}{2} (\frac{ω_{1}^{2} + ω_{2}^{2}}{c^{2}} + {M^{'}}_{1} + {M^{″}}_{1}) + (M_{2} + M_{3}) cos k_{x} a + ({M^{'}}_{2} + {M^{'}}_{3}) cos k_{y} a \\ \pm \frac{1}{2} {[\frac{ω_{1}^{2} - ω_{2}^{2}}{c^{2}} + {M^{'}}_{1} - {M^{″}}_{1} + 2 (M_{2} - M_{3}) cos k_{x} a \\ {+ 2 ({M^{'}}_{3} - {M^{″}}_{2}) cos k_{y} a]}^{2} + 16 L_{1}^{2} {sin}^{2} k_{x} a}^{1 / 2} . \end{array}

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

\begin{array}{l} ω_{Γ} & \equiv & ω_{1}^{2} + c^{2} ({M^{'}}_{1} + 2 M_{2} + 2 {M^{'}}_{3}) \\ = & ω_{2}^{2} + c^{2} ({M^{″}}_{1} + 2 M_{3} + 2 {M^{'}}_{2}) . \end{array}

In this case, the dispersion relation is somewhat simplified:

\begin{array}{l} \frac{ω_{k}^{2}}{c^{2}} & = & \frac{ω_{Γ}^{2}}{c^{2}} + (M_{2} + M_{3}) (cos k_{x} a - 1) + ({M^{'}}_{2} + {M^{'}}_{3}) (cos k_{y} a - 1) \\ \pm {{[(M_{2} - M_{3}) (cos k_{x} a - 1) + ({M^{'}}_{3} - {M^{'}}_{2}) (cos k_{y} a - 1)]}^{2} + 4 L_{1}^{2} {sin}^{2} k_{x} a}^{1 / 2} . \end{array}

Finally, to examine its behavior in the vicinity of the Γ point, we assume |k_xa|, |k_ya| ≪ 1 and only keep dominant terms. The result is

ω_{k} = ω_{Γ} \pm \frac{a c^{2} k_{x} L_{1}}{ω_{Γ}},

which has linear dependence on k_x as we expected.

The reason for this feature can be understood by comparing $L_{n m}^{(i j)}$ ’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, $L_{n m}^{(i j)}$ , 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.

Appendix

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, $H_{0}^{(1)} (r)$ and $H_{0}^{(2)} (r)$ , such that they are transformed as the following two functions when any R ∈ C _4v is operated.

f_{1} (r) = x and f_{2} (r) = y .

By definition, they are transformed as

R f_{1, 2} (r) = f_{1, 2} (R^{- 1} r) .

So, for example,

σ_{x} f_{1} (r) = - x = - f_{1} (r),

σ_{x} f_{2} (r) = y = f_{2} (r) .

We write these two equations in matrix form:

σ_{x} (\begin{matrix} f_{1} \\ f_{2} \end{matrix}) = (\begin{matrix} - 1, & 0 \\ 0, & 1 \end{matrix}) (\begin{matrix} f_{1} \\ f_{2} \end{matrix}) .

In this way, we can introduce the matrix representation of R ∈ C _4v:

\begin{array}{l} σ_{x} : (\begin{matrix} - 1, & 0 \\ 0, & 1 \end{matrix}), & σ_{y} : (\begin{matrix} 1, & 0 \\ 0, & - 1 \end{matrix}), \\ {σ^{'}}_{d} : (\begin{matrix} 0, & 1 \\ 1, & 0 \end{matrix}), & {σ^{″}}_{d} : (\begin{matrix} 0, & - 1 \\ - 1, & 0 \end{matrix}), \\ C_{4} : (\begin{matrix} 0, & 1 \\ - 1, & 0 \end{matrix}), & C_{4}^{- 1} : (\begin{matrix} 0, & - 1 \\ 1, & 0 \end{matrix}), \\ C_{2} : (\begin{matrix} - 1, & 0 \\ 0, & - 1 \end{matrix}), & E : (\begin{matrix} 1, & 0 \\ 0, & 1 \end{matrix}) . \end{array}

It is easily confirmed that this matrix representation gives the correct values of characters for the E representation in Table 2, which are the trace (sum of the diagonal elements) of the above matrices.

Because $H_{0}^{(1)}$ and $H_{0}^{(2)}$ 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

[σ_{x} H_{0}^{(i)}] (r) \equiv σ_{x} H_{0}^{(i)} (σ_{x}^{- 1} r) .

Let us proceed to the derivation of relations among the $L_{n m}^{(i j)}$ ’s introduced by Eq. (13). We define a differential operator ℒ by

ℒ H (r) = \nabla \times [\frac{1}{ɛ (r)} \nabla \times H (r)] .

Then

L_{n m}^{(i j)}

is written as

L_{n m}^{(i j)} = \frac{1}{V} \int_{V} d r H_{0}^{(i) *} (r) \cdot [ℒ H_{0}^{(j)} (r - r_{n m})] .

Now, let us change the variable of integration from r to r′ = σ_x r and evaluate

L_{00}^{(12)}

. Since σ_x does not change the size of volume elements, we have

\begin{array}{l} V L_{00}^{(12)} & = & \int_{V} d r' H_{0}^{(1) *} (σ_{x}^{- 1} r') \cdot [ℒ H_{0}^{(2)} (σ_{x}^{- 1} r')] \\ = & \int_{V} d r' [σ_{x}^{- 1} σ_{x} H_{0}^{(1) *} (σ_{x}^{- 1} r')] \cdot [σ_{x}^{- 1} σ_{x} ℒ σ_{x}^{- 1} σ_{x} H_{0}^{(2)} (σ_{x}^{- 1} r')], \end{array}

where we introduced three pairs of

σ_{x}^{- 1}

σ_x(≡E) for later use. Since

σ_{x}^{- 1}

does not change the value of inner products, we obtain

V L_{00}^{(12)} = \int_{V} d r' [σ_{x} H_{0}^{(1) *} (σ_{x}^{- 1} r')] \cdot [ℒ^{'} σ_{x} H_{0}^{(2)} (σ_{x}^{- 1} r')],

where ℒ′ is defined as

ℒ^{'} = σ_{x} ℒ σ_{x}^{- 1} .

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

\begin{array}{l} V L_{00}^{(12)} & = & \int_{V} d r' [- H_{0}^{(1) *} (r')] \cdot [ℒ^{'} H_{0}^{(2)} (r')] \\ = & - V L_{00}^{(12)} . \end{array}

So,

L_{00}^{(12)} = 0.

By similar calculation, we obtain

L_{00}^{(21)} = 0.

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

L_{00}^{(11)} = \frac{ω_{0}^{2}}{c^{2}} + M_{1},

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:

\begin{array}{l} V L_{00}^{(11)} & = & \int_{V} d r' [C_{4} H_{0}^{(1) *} (C_{4}^{- 1} r')] \cdot [ℒ' C_{4} H_{0}^{(1)} (C_{4}^{- 1} r')] \\ = & \int_{V} d r' H_{0}^{(2) *} (r') \cdot [ℒ' H_{0}^{(2)} (r')] \\ = & V L_{00}^{(22)} . \end{array}

Note that

ℒ' = C_{4} ℒ C_{4}^{- 1}

for this case. So, we obtain

L_{00}^{(11)} = L_{00}^{(22)} \equiv \frac{ω_{0}^{2}}{c^{2}} + M_{1} .

Let us proceed to the evaluation of $L_{10}^{(12)}$ . Using r′ = σ_y r,

\begin{array}{l} V L_{10}^{(12)} & = & \int_{V} d r' [σ_{y} H_{0}^{(1) *} (σ_{y}^{- 1} r')] \\ \cdot [ℒ' σ_{y} H_{0}^{(2)} (σ_{y}^{- 1} (r' - r_{10}))] \\ = & \int_{V} d r' [H_{0}^{(1) *} (r')] \cdot [- ℒ' H_{0}^{(2)} (r' - r_{10})] \\ = & - V L_{10}^{(12)} . \end{array}

So, we obtain

L_{10}^{(12)} = 0.

By similar calculation, we can prove

L_{\pm 1, 0}^{(12)} = L_{0, \pm 1}^{(12)} = L_{\pm 1, 0}^{(21)} = L_{0, \pm 1}^{(21)} = 0 .

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

\begin{array}{l} V L_{10}^{(11)} & = & \int_{V} d r' [C_{4} H_{0}^{(1) *} (C_{4}^{- 1} r')] \cdot [ℒ' C_{4} H_{0}^{(1)} (C_{4}^{- 1} r' - r_{10})] \\ = & \int_{V} d r' H_{0}^{(2) *} (r') \cdot [ℒ' H_{0}^{(2)} (r' - r_{01})] \\ = & V L_{01}^{(22)}, \end{array}

where we used

C_{4}^{- 1}

r ₀₁ = r ₁₀. Using transformations by σ_x and

C_{4}^{- 1}

, we can also prove

L_{10}^{(11)} = L_{- 1, 0}^{(11)} = L_{0, - 1}^{(22)} .

So, from Eqs. (83) and (84), we have

L_{\pm 1, 0}^{(11)} = L_{0, \pm 1}^{(22)} \equiv M_{2} .

Similarly we obtain

L_{0, \pm 1}^{(11)} = L_{\pm 1, 0}^{(22)} \equiv M_{3} .

For terms with (n, m) = (±1, ±1), applying all transformations of C _4v to $L_{11}^{(11)}$ , we can easily prove

L_{\pm 1, \pm 1}^{(11)} = L_{\pm 1, \pm 1}^{(22)} \equiv M_{4} .

From the same calculation for

L_{11}^{(12)}

, we obtain

\begin{array}{l} L_{1, 1}^{(12)} = L_{- 1, - 1}^{(12)} = - L_{1, - 1}^{(12)} = - L_{- 1, 1}^{(12)} = L_{1, 1}^{(21)} \\ = & L_{- 1, - 1}^{(21)} = - L_{1, - 1}^{(21)} = - L_{- 1, 1}^{(21)} \equiv M_{5} . \end{array}

All other combinations of R ∈ C _4v and

L_{n m}^{(j)}

yield the same relations given above.

B. E₁ 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:

\begin{array}{l} σ_{x} : (\begin{matrix} - 1, & 0 \\ 0, & 1 \end{matrix}), & σ_{y} : (\begin{matrix} 1, & 0 \\ 0, & - 1 \end{matrix}), \\ C_{6} : (\begin{matrix} 1 / 2, & \sqrt{3} / 2 \\ - \sqrt{3} / 2, & 1 / 2 \end{matrix}), & C_{6}^{- 1} : (\begin{matrix} 1 / 2, & - \sqrt{3} / 2 \\ \sqrt{3} / 2, & 1 / 2 \end{matrix}), \\ C_{3} : (\begin{matrix} - 1 / 2, & \sqrt{3} / 2 \\ - \sqrt{3} / 2, & - 1 / 2 \end{matrix}), & C_{3}^{- 1} : (\begin{matrix} - 1 / 2, & - \sqrt{3} / 2 \\ \sqrt{3} / 2, & - 1 / 2 \end{matrix}), \\ C_{2} : (\begin{matrix} - 1, & 0 \\ 0, & - 1 \end{matrix}), & E : (\begin{matrix} 1, & 0 \\ 0, & 1 \end{matrix}) . \end{array}

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,

\begin{array}{l} V L_{0}^{(11)} & = & \int_{V} d r' H_{0}^{(1) *} (C_{6}^{- 1} r') \cdot [ℒ H_{0}^{(1)} (C_{6}^{- 1} r')] \\ = & \int_{V} d r' [C_{6} H_{0}^{(1) *} (C_{6}^{- 1} r')] \cdot [ℒ' C_{6} H_{0}^{(1)} (C_{6}^{- 1} r')] \\ = & \frac{V (L_{0}^{(11)} + \sqrt{3} L_{0}^{(21)} + \sqrt{3} L_{0}^{(12)} + 3 L_{0}^{(22)})}{4} . \end{array}

Using

C_{6}^{- 1}

, we similarly obtain

L_{0}^{(11)} = \frac{L_{0}^{(11)} - \sqrt{3} L_{0}^{(21)} - \sqrt{3} L_{0}^{(12)} + 3 L_{0}^{(22)}}{4} .

From these two equations,

L_{0}^{(12)} + L_{0}^{(21)} = 0,

L_{0}^{(11)} = L_{0}^{(22)} \equiv \frac{ω_{0}^{2}}{c^{2}} + M_{1} .

Using σ_x for

L_{0}^{(12)}

,

\begin{array}{l} V L_{0}^{(12)} & = & \int_{V} d r' [- H_{0}^{(1) *} (r')] \cdot [ℒ' H_{0}^{(2)} (r')] \\ = & - V L_{0}^{(12)} . \end{array}

So, combining with Eq. (92), we obtain

L_{0}^{(12)} = L_{0}^{(21)} = 0.

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,

\begin{array}{l} V L_{1}^{(11)} & = & \int_{V} d r' [- H_{0}^{(1) *} (r')] \cdot [- ℒ' H_{0}^{(1)} (r' - r_{4})] \\ = & V L_{4}^{(11)}, \end{array}

\begin{array}{l} V L_{1}^{(22)} & = & \int_{V} d r' [- H_{0}^{(2) *} (r')] \cdot [- ℒ' H_{0}^{(2)} (r' - r_{4})] \\ = & V L_{4}^{(22)} . \end{array}

So,

L_{1}^{(11)} = L_{4}^{(11)} \equiv M_{2},

L_{1}^{(22)} = L_{4}^{(22)} \equiv M_{3} .

Next, by changing the variable of integration from r to C ₂ r for

L_{1}^{(12)}

and

L_{1}^{(21)}

, we obtain

L_{1}^{(12)} = L_{4}^{(12)}, L_{1}^{(21)} = L_{4}^{(21)}

while using σ_x, we obtain

L_{1}^{(12)} = - L_{4}^{(12)}, L_{1}^{(21)} = - L_{4}^{(21)}

Thus,

L_{1}^{(12)} = L_{4}^{(12)}, L_{1}^{(21)} = L_{4}^{(21)} = 0.

On the other hand, applying σ_x, C ₂ and σ_y to

L_{2}^{(11)}

and

L_{2}^{(22)}

, we obtain

L_{2}^{(11)} = L_{3}^{(11)} = L_{5}^{(11)} = L_{6}^{(11)} \equiv M_{4},

L_{2}^{(22)} = L_{3}^{(22)} = L_{5}^{(22)} = L_{6}^{(22)} \equiv M_{5} .

Finally, applying σ_x, C ₂ and σ_y to

L_{2}^{(12)}

and

L_{2}^{(21)}

, we obtain

L_{2}^{(12)} = - L_{3}^{(12)} = L_{5}^{(12)} = - L_{6}^{(12)} = L_{2}^{(21)} = - L_{3}^{(21)} = L_{5}^{(21)} = - L_{6}^{(21)} \equiv M_{6} .

For the six M_i’s, we can derive three relations as follows. First, by applying $C_{6}^{- 1}$ to $L_{2}^{(11)}$ ,

\begin{array}{l} M_{4} & = & L_{2}^{(11)} \\ = & \frac{1}{V} \int_{V} d r' \frac{H_{0}^{(1)} (r') - \sqrt{3} H_{0}^{(2) *} (r')}{2} \cdot [ℒ' \frac{H_{0}^{(1)} (r' - r_{1}) - \sqrt{3} H_{0}^{(2)} (r' - r_{1})}{2}] \\ = & \frac{M_{2} + 3 M_{3}}{4}, \end{array}

where we used Eq. (102). Similarly, applying

C_{6}^{- 1}

to

L_{2}^{(22)}

and

L_{2}^{(22)}

, we obtain

M_{5} = \frac{3 M_{2} + M_{3}}{4},

M_{6} = \frac{\sqrt{3} (M_{2} - M_{3})}{4} .

All other combinations of R ∈ C _6v and

L_{n}^{(i j)}

yield the same relations given above.

C. E₂ mode of C_6v symmetry

It is known from group theory that {2xy, 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:

\begin{array}{l} σ_{x} : (\begin{matrix} - 1, & 0 \\ 0, & 1 \end{matrix}), & σ_{y} : (\begin{matrix} - 1, & 0 \\ 0, & 1 \end{matrix}), \\ C_{6} : (\begin{matrix} - 1 / 2, & - \sqrt{3} / 2 \\ \sqrt{3} / 2, & - 1 / 2 \end{matrix}), & C_{6}^{- 1} : (\begin{matrix} - 1 / 2, & \sqrt{3} / 2 \\ - \sqrt{3} / 2, & - 1 / 2 \end{matrix}), \\ C_{3} : (\begin{matrix} - 1 / 2, & \sqrt{3} / 2 \\ - \sqrt{3} / 2, & - 1 / 2 \end{matrix}), & C_{3}^{- 1} : (\begin{matrix} - 1 / 2, & - \sqrt{3} / 2 \\ \sqrt{3} / 2, & - 1 / 2 \end{matrix}), \\ C_{2} : (\begin{matrix} 1, & 0 \\ 0, & 1 \end{matrix}), & E : (\begin{matrix} 1, & 0 \\ 0, & 1 \end{matrix}) . \end{array}

A unique feature of this matrix representation is that the matrix of the 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. $L_{n m}^{(i j)}$ for C_s symmetry

Because $L_{n m}^{(i j)}$ 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

L_{00}^{(11)} \equiv \frac{ω_{1}^{2}}{c^{2}} + M_{1}^{'}, L_{00}^{(22)} \equiv \frac{ω_{2}^{2}}{c^{2}} + M_{1}^{″},

where ω ₁ and ω ₂ are eigen frequencies for an isolated unit metallic structure described by dielectric constant ɛ_s(r). Applying σ_x to

L_{00}^{(12)}

and

L_{00}^{(21)}

as we did in previous sections, we can prove

L_{00}^{(12)} = L_{00}^{(21)} = 0.

Next, applying σ_x to

L_{10}^{(11)}

and

L_{10}^{(22)}

, we obtain

L_{10}^{(11)} = L_{- 1, 0}^{(11)} \equiv M_{2},

L_{10}^{(22)} = L_{- 1, 0}^{(22)} \equiv M_{3} .

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_nm. Then, from the translational symmetry of ɛ(r) and invariance of differential operators under any uniform translation, we have

\begin{array}{l} T_{nm} ℒ T_{nm}^{- 1} & = & (T_{nm} \nabla \times T_{nm}^{- 1}) (T_{nm} \frac{1}{ɛ (r)} T_{nm}^{- 1}) (T_{nm} \nabla \times T_{nm}^{- 1}) \\ = & ℒ . \end{array}

On the other hand, from the symmetric form of ℒ and the periodic boundary condition, we can prove that ℒ is an Hermitian operator [13]:

\int_{V} d r Q_{1}^{*} (r) \cdot [ℒ Q_{2} (r)] = \int_{V} d r {[ℒ Q_{1} (r)]}^{*} \cdot Q_{2} (r),

where 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 $L_{10}^{(11)}$ :

\begin{array}{l} L_{10}^{(11)} & = & \frac{1}{V} \int_{V} d r' H_{1}^{*} (r' + r_{10}) \cdot [(T_{- 1, 0} ℒ T_{- 1, 0}^{- 1})] H_{1} (r') \\ = & \frac{1}{V} \int_{V} d r' {[ℒ H_{1} (r' - r_{- 1, 0})]}^{*} \cdot H_{1} (r') \\ = & L_{- 1, 0}^{(11) *} . \end{array}

Similarly we obtain,

L_{10}^{(22)} = L_{- 1, 0}^{(22) *} .

Combining with Eqs. (112) and (113), we can conclude that M ₂ and M ₃ are real numbers.

As for $L_{10}^{(12)}$ , similarly applying σ_x and the same translation, we obtain

L_{10}^{(12)} = - L_{- 1, 0}^{(12)} = L_{- 1, 0}^{(21) *} .

Similarly for

L_{10}^{(21)}

, we have

L_{10}^{(21)} = - L_{- 1, 0}^{(21)} = L_{- 1, 0}^{(12) *} .

From these equations, we obtain

L_{10}^{(12)} = - L_{- 1, 0}^{(12)} = L_{- 1,0}^{(21) *} = - L_{10}^{(21) *} \equiv L_{1} .

For

L_{01}^{(11)}

and

L_{01}^{(22)}

, the application of σ_x yields identity equations. On the other hand, by changing the variable of integration from r to r – r ₀₁, we obtain

L_{01}^{(11)} = L_{0, - 1}^{(11) *} \equiv M_{3}^{'}, L_{01}^{(22)} = L_{0, - 1}^{(22) *} \equiv M_{2}^{'} .

Finally, applying σ_x, we can prove

L_{01}^{(12)} = L_{0, - 1}^{(12)} = L_{01}^{(21)} = L_{0, - 1}^{(21)} = 0.

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).

References and links

1. V. G. Veselago, “Electrodynamics of substances with simultaneously negative values of sigma and mu,” Sov. Phys. Usp. 10, 509–514 (1968). [CrossRef]

2. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006). [CrossRef] [PubMed]

3. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000). [CrossRef] [PubMed]

4. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001). [CrossRef] [PubMed]

5. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006). [CrossRef] [PubMed]

6. D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).

7. S. A. Ramakrishna and T. M. Grzegorczyk, Physics and Applications of Negative Refractive Index Materials (SPIE Press, 2008). [CrossRef]

8. S. Matsuzawa, K. Sato, Y. Inoue, and T. Nomura, “W-band steerable composite right/left-handed leaky wave antenna for automotive applications,” IEICE Trans. Electron. E89-C, 1337–1344 (2006). [CrossRef]

9. A. Grbic and G. V. Eleftheriades, “Experimental verification of backward-wave radiation from a negative refractive index metamaterial,” J. Appl. Phys. 92, 5930–5935 (2002). [CrossRef]

10. C. Caloz and T. Ito, “Application of the transmission line theory of left-handed (LH) materials to the realization of a microstrip LH line,” IEEE-AP-S Int. Symp. Dig. 2, 412–415 (2002).

11. S. Tokoro, K. Kuroda, A. Kawakubo, K. Fujita, and H. Fujinami, “Electronically scanned millimeter-wave radar for pre-crush safety and adaptive cruise control system,” Proc. IEEE Intelligent Vehicles Symp. , 304–309 (2003).

12. K. Sakoda and H.-F. Zhou, “Role of structural electromagnetic resonances in a steerable left-handed antenna,” Opt. Express 18, 27371–27386 (2010). [CrossRef]

13. K. Sakoda, Optical Properties of Photonic Crystals, 2nd Ed. (Springer-Verlag, 2004).

14. T. Inui, Y. Tanabe, and Y. Onodera, Group Theory and its Applications in Physics (Springer, 1990). [CrossRef]

15. A. Taflove, Computational Electrodynamics (Artech House, 1995).

16. D. M. Sullivan, Electromagnetic Simulation Using the FDTD Method (IEEE Press, 2000). [CrossRef]

Analytical study of two-dimensional degenerate metamaterial antennas

Abstract

1. Introduction

2. Tight-binding calculation

2.1. Square lattice

2.2. Triangular lattice

3. Numerical results

4. Controlled symmetry reduction

5. Conclusion

Appendix

A. E mode of C_4v symmetry

B. E₁ mode of C_6v symmetry

C. E₂ mode of C_6v symmetry

D. $L_{n m}^{(i j)}$ for C_s symmetry

Acknowledgments

References and links

Cited By

Figures (7)

Tables (4)

Equations (122)

Optics Express

C _4v	E	2C ₄	C ₂	2σ_v	2σ_d
A ₁	1	1	1	1	1
A ₂	1	1	1	−1	−1
B ₁	1	−1	1	1	−1
B ₂	1	−1	1	−1	1
E	2	0	−2	0	0

Abstract

1. Introduction

2. Tight-binding calculation

2.1. Square lattice

2.2. Triangular lattice

3. Numerical results

4. Controlled symmetry reduction

5. Conclusion

Appendix

A. E mode of C4v symmetry

B. E1 mode of C6v symmetry

C. E2 mode of C6v symmetry

D. Lnm(ij) for Cs symmetry

Acknowledgments

References and links

Cited By

Figures (7)

Tables (4)

Equations (122)

Optics Express

A. E mode of C_4v symmetry

B. E₁ mode of C_6v symmetry

C. E₂ mode of C_6v symmetry

D. $L_{n m}^{(i j)}$ for C_s symmetry