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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 15 — Jul. 18, 2011
  • pp: 13899–13921
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Analytical study of two-dimensional degenerate metamaterial antennas

Kazuaki Sakoda and Haifeng Zhou  »View Author Affiliations


Optics Express, Vol. 19, Issue 15, pp. 13899-13921 (2011)
http://dx.doi.org/10.1364/OE.19.013899


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Abstract

Dispersion curves of metamaterial steerable antennas composed of two-dimensional arrays of metallic unit structures with the C4v and C6v 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 C4v point group and those of the E1 and E2 symmetries of the C6v 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

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

7

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

]. As a particular application, a microwave steerable antenna based on negative refraction was proposed and demonstrated [8

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]

10

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).

]. 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

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).

]. 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 ωi 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 ωu and ωl, or above the light lines (see Fig. 1), the incident wave is diffracted at a diffraction angle θ that is determind by ωi and the wave vector of the internal mode ki:
θ=cos1ckiωi,
(1)
which is measured from the direction of incidence. So, when ωi is decreased from ωu to ωl, θ changes from 0 to 180 degrees. In particular, if ki is negative, which takes place when ωi 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 fu is concave-up and the lower one denoted by fl 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 ωi is propagated in the positive k direction, it excites an eigenmode (denoted by wave vector ki) 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 ωi and ki. Thus, the upper and lower limits of working frequency are given by ωu and ωl. Reproduced with permission from Opt. Express 18, 27371 (2010). Copyright 2010 The Optical Society (OSA).

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

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]

]. 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.

Table 1. Parity of Eigenmodes on the Γ Point for a Regular Array of Metallic Unit Structure of the C 2v Symmetry

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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 σy 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 σy 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.

In the second case with vanishing slope on the Γ point, diffraction angle θ varies rapidly with incidence frequency ωi, which can be easily understood from a rapid variation of ki with ωi. 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 ωi 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, Lnm(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
×[1ɛ(r)×H(r,t)]=1c22t2H(r,t),
(3)
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

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

] 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 ɛs(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 H0(1) and H0(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:
×[1ɛs(r)×H0(1,2)(r)]=ω02c2H0(1,2)(r),
(4)
where ω 0 is the resonance angular frequency. Without loss of generality, we can assume that the two distributions are normalized as follows:
VdrH0(i)*(r)H0(j)(r)=Vδij,
(5)
where V is the volume on which we impose the periodic boundary condition. Note that H0(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 H0(1) and H0(2):
Hk(r)=1Nn,meikrnm[AH0(1)(rrnm)+BH0(2)(rrnm)],
(6)
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, C41), rotation by 180 degrees (C 2), basic mirror reflection (σx, σy), 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 (σx, σy) and (σ′d, σ″d). Lattice points are denoted by solid circles.

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

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In the case of a regular square lattice, the lattice vectors are given by
rnm=na1+ma2.
(7)
Here, a 1 and a 2 are elementary lattice vectors of the square lattice:
a1=(a0),a2=(0a),
(8)
Note that Hk(r) given by Eq. (6) satisfies the Bloch condition:
Hk(r+ai)=eikaiHk(r).
(9)

Now, we assume according to the prescription of the tight-binding approximation that Hk(r) is an eigen function of Eq. (3) and we denote its eigen angular frequency by ωk. Thus it satisfies the following equation:
Hk(r)=ωk2c2Hk(r),
(10)
where we introduced a differential operator :
H(r)=×[1ɛ(r)×H(r)].
(11)
Multiplying Eq. (10) by H0(1)*(r) and integrating over V, we obtain
VdrH0(1)*(r)Hk(r)=VNn,meikrnm[ALnm(11)+BLnm(12)],
(12)
where Lnm(ij) is defined as
Lnm(ij)=1VVdrH0(i)*(r)[H0(j)(rrnm)].
(13)

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 RC 4v is applied [14

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

]. Using this property, we can derive various relations among Lnm(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):
L0,0(11)=L0,0(22)ω02c2+M1,
(14)
L0,0(12)=L0,0(21)=0,
(15)
L±1,0(11)=L0,±1(22)M2,
(16)
L0,±1(11)=L±1,0(22)M3,
(17)
L±1,0(12)=L0,±1(12)=L±1,0(21)=L0,±1(21)=0,
(18)
L±1,±1(11)=L±1,±1(22)M4,
(19)
L1,1(12)=L1,1(12)=L1,1(12)=L1,1(12)=L1,1(21)=L1,1(21)=L1,1(21)=L1,1(21)M5
(20)
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, AVωk2/c2N. Thus, we obtain
A[ω02ωk2+c2(M1+2M2coskxa+2M3coskya+4M4coskxacoskya)]=4Bc2M5sinkxasinkya.
(21)
Similarly, multiplying Eq. (10) by H0(2)*(r) and integrating over volume V, we obtain
4Ac2M5sinkxasinkya=B[ω02ωk2+c2(M1+2M3coskxa+2M2coskya+4M4coskxacoskya.)]
(22)
By solving the secular equation derived from Eqs. (21) and (22), we finally obtain the following dispersion relation:
ωk2c2=ω02c2+M1+(M2+M3)(coskxa+coskya)+4M4coskxacoskya±[(M2M3)2(coskxacoskya)2+16M52sin2kxasin2kya]1/2.
(23)

Let us examine some special cases. First, for the Γ point (kx = ky = 0), we have a doubly degenerate eigen frequency ω Γ:
ωΓ=ω02+c2(M1+2M2+2M3+4M4).
(24)
Secondly, Eq. (23) is invariant when we exchange kx with ky, as it should be, which is a consequence of the C 4v symmetry of the square lattice. Thirdly, it has a property
ωk=ωk,
(25)
which is a consequence of the time-reversal symmetry of the wave equation [13

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

].

Next, let us examine the dispersion in the vicinity of the Γ point. For this purpose, we assume that |kxa|, |kya| ≪ 1 and keep the lowest order terms. Then, from Eq. (23), we obtain
ωk=ωΓa2c2k24ωΓ[M2+M3+4M4±2F(ϕ)],
(26)
where
k=kx2+ky2,
(27)
ϕ=tan1kykx,
(28)
F(ϕ)=(M2M32)2cos22ϕ+4M52sin22ϕ.
(29)
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 kx and ky. 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, C61), rotation by 120 degrees (C 3 and C31), rotation by 180 degrees (C 2), and two sets of three equivalent mirror reflections ((σx, σ′x, σ″x) and (σy, 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 (σx, σ′x, σ″x) and (σy, σ′y, σ″y). Seven lattice points (the origin and its nearest neighbors) are denoted by integers from 0 to 6.

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

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

]. 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

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By fully using the symmetry properties of the eigen functions, we can derive the following relations (see Appendix B):
L0(11)=L0(22)ω02c2+M1,
(31)
L0(12)=L0(21)=0,
(32)
L1(11)=L4(11)M2,
(33)
L1(22)=L4(22)M3,
(34)
L1(12)=L4(12)=L1(21)=L4(21)=0,
(35)
L2(11)=L3(11)=L5(11)=L6(11)M4,
(36)
L2(22)=L3(22)=L5(22)=L6(2)M5,
(37)
L2(12)=L3(12)=L5(12)=L6(12)=L2(21)=L3(21)=L5(21)=L6(21)M6.
(38)
We can further prove the following relations:
M4=M2+3M34,
(39)
M5=3M2+M34,
(40)
M6=3(M2M3)4.
(41)
So, the number of independent elements among the 28 Ln(ij)’s is three. The dispersion relation is given by
ωk2c2=ω02c2+M1+(M2+M3)(coskxa+2coskxa2cos3kya2)±(M2M3)[(coskxacoskxa2cos3kya2)2+3sin2kxa2sin23kya2]1/2
(42)
The dispersion curves give a degenerate eigen frequency on the Γ point:
ωΓ=ω02+c2(M1+3M2+3M3).
(43)
For small k, Eq. (42) is approximated as
ωk=ωΓ3a2c2k2M16ωΓ,
(44)
where
M={M2+3M3,3M2+M3.
(45)
So, the dispersion curves are isotropic in the vicinity of the Γ point, and they are quadratic in k.

3. Numerical results

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.

Figure 6 shows distributions of the z component of the magnetic field, Hz, 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

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]

].

Fig. 6 Distribution of Hz 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. Hz on the horizontal plane in the middle of the dielectric slab is shown.

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: (σx, σy) = (−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: (σx, σy) = (−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 Ez, which is perpendicular to the surface of the metallic unit structure, because the tangential components (Ex and Ey) 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, Hz is small. Nevertheless, we plotted Hz 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

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

]. When we denote their characters for symmetry operation R by χ (E)(R) and χ (H)(R), respectively, then
χ(H)(R)=detRχ(E)(R),
(46)
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.

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 py, 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.

In Fig. 7, the parity of the electric field with respect to the y coordinate is denoted by py. We should note that it is opposite to that of the magnetic field. Three of the four modes have the same py, 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 kx and ky are relevant to some vanishing Lnm(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 Lnm(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.

Now, let us examine symmetry reduction from C 4v to Cs. The latter is a simple point group that consists of identity (E) and a mirror reflection (σx). 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 Cs due to the lowered symmetry.

Table 4. Character Table of the Cs Point Group

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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:
×[1ɛs(r)×H1,2(r)]=ω1,22c2H1,2(r).
(47)
The Bloch function is given by the linear combination of H 1 and H 2:
Hk(r)=1Nn,meikrnm[AH1(rrnm)+BH2(rrnm)].
(48)

Because the number of symmetry operations is small for Cs, we have a smaller number of relations among Lnm(ij)’s, which are defined by the following equation in the present case:
Lnm(ij)=1VVdrHi*(r)[Hj(rrnm)].
(49)
Derivation of relations among Lnm(ij)’s is given in Appendix D. The results are summarized as follows:
L0,0(11)ω12c2+M1,L0,0(22)ω22c2+M1,
(50)
L0,0(12)=L0,0(21)=0,
(51)
L1,0(11)=L1,0(11)M2,
(52)
L1,0(22)=L1,0(22)M3,
(53)
L1,0(12)=L1,0(12)=L1,0(21)*=L1,0(21)*L1,
(54)
L0,1(11)=L0,1(11)*M3,
(55)
L0,1(22)=L0,1(22)*M2,
(56)
L0,1(12)=L0,1(12)=L0,1(21)=L0,1(21)=0.
(57)
Then, by solving the secular equation as before, we obtain the dispersion relation:
ωk2c2=12(ω12+ω22c2+M1+M1)+(M2+M3)coskxa+(M2+M3)coskya±12{[ω12ω22c2+M1M1+2(M2M3)coskxa+2(M3M2)coskya]2+16L12sin2kxa}1/2.
(58)

Now, we examine the condition for accidental degeneracy of eigen frequencies on the Γ point by an appropriate choice of device parameters. By setting kx = ky = 0 in Eq. (58), the condition for degeneracy is
ωΓω12+c2(M1+2M2+2M3)=ω22+c2(M1+2M3+2M2).
(59)
In this case, the dispersion relation is somewhat simplified:
ωk2c2=ωΓ2c2+(M2+M3)(coskxa1)+(M2+M3)(coskya1)±{[(M2M3)(coskxa1)+(M3M2)(coskya1)]2+4L12sin2kxa}1/2.
(60)
Finally, to examine its behavior in the vicinity of the Γ point, we assume |kxa|, |kya| ≪ 1 and only keep dominant terms. The result is
ωk=ωΓ±ac2kxL1ωΓ,
(61)
which has linear dependence on kx as we expected.

The reason for this feature can be understood by comparing Lnm(ij)’s between the C 4v and Cs cases. Due to lowered symmetry, L±1,0(12) and L±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

Appendix

A. E mode of C4v symmetry

A polynomial representation of the E mode is given by a pair of {x, y} [14

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

]. So, we can choose two eigen functions of the E symmetry, H0(1)(r) and H0(2)(r), such that they are transformed as the following two functions when any RC 4v is operated.
f1(r)=xandf2(r)=y.
(62)
By definition, they are transformed as
Rf1,2(r)=f1,2(R1r).
(63)
So, for example,
σxf1(r)=x=f1(r),
(64)
σxf2(r)=y=f2(r).
(65)
We write these two equations in matrix form:
σx(f1f2)=(1,00,1)(f1f2).
(66)
In this way, we can introduce the matrix representation of RC 4v:
σx:(1,00,1),σy:(1,00,1),σd:(0,11,0),σd:(0,11,0),C4:(0,11,0),C41:(0,11,0),C2:(1,00,1),E:(1,00,1).
(67)
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 H0(1) and H0(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
[σxH0(i)](r)σxH0(i)(σx1r).
(68)

Let us proceed to the derivation of relations among the Lnm(ij)’s introduced by Eq. (13). We define a differential operator by
H(r)=×[1ɛ(r)×H(r)].
(69)
Then Lnm(ij) is written as
Lnm(ij)=1VVdrH0(i)*(r)[H0(j)(rrnm)].
(70)
Now, let us change the variable of integration from r to r′ = σx r and evaluate L00(12). Since σx does not change the size of volume elements, we have
VL00(12)=VdrH0(1)*(σx1r)[H0(2)(σx1r)]=Vdr[σx1σxH0(1)*(σx1r)][σx1σxσx1σxH0(2)(σx1r)],
(71)
where we introduced three pairs of σx1 σx(≡E) for later use. Since σx1 does not change the value of inner products, we obtain
VL00(12)=Vdr[σxH0(1)*(σx1r)][σxH0(2)(σx1r)],
(72)
where ℒ′ is defined as
=σxσx1.
(73)
It is an operator in the r′ coordinate system equivalent to in the r coordinate system. We can prove ℒ′ = [13

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

], although we do not use this relation in the following. Substituting the first relation in Eq. (67) and Eq. (68), we obtain
VL00(12)=Vdr[H0(1)*(r)][H0(2)(r)]=VL00(12).
(74)
So,
L00(12)=0.
(75)
By similar calculation, we obtain
L00(21)=0.
(76)

Next, let us examine L00(11). We denote it by
L00(11)=ω02c2+M1,
(77)
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:
VL00(11)=Vdr[C4H0(1)*(C41r)][C4H0(1)(C41r)]=VdrH0(2)*(r)[H0(2)(r)]=VL00(22).
(78)
Note that =C4C41 for this case. So, we obtain
L00(11)=L00(22)ω02c2+M1.
(79)

Let us proceed to the evaluation of L10(12). Using r′ = σy r,
VL10(12)=Vdr[σyH0(1)*(σy1r)][σyH0(2)(σy1(rr10))]=Vdr[H0(1)*(r)][H0(2)(rr10)]=VL10(12).
(80)
So, we obtain
L10(12)=0.
(81)
By similar calculation, we can prove
L±1,0(12)=L0,±1(12)=L±1,0(21)=L0,±1(21)=0.
(82)

As for L10(11), changing the variable of integration from r to r′ = C 4 r,
VL10(11)=Vdr[C4H0(1)*(C41r)][C4H0(1)(C41rr10)]=VdrH0(2)*(r)[H0(2)(rr01)]=VL01(22),
(83)
where we used C41 r 01 = r 10. Using transformations by σx and C41, we can also prove
L10(11)=L1,0(11)=L0,1(22).
(84)
So, from Eqs. (83) and (84), we have
L±1,0(11)=L0,±1(22)M2.
(85)
Similarly we obtain
L0,±1(11)=L±1,0(22)M3.
(86)

For terms with (n, m) = (±1, ±1), applying all transformations of C 4v to L11(11), we can easily prove
L±1,±1(11)=L±1,±1(22)M4.
(87)
From the same calculation for L11(12), we obtain
L1,1(12)=L1,1(12)=L1,1(12)=L1,1(12)=L1,1(21)=L1,1(21)=L1,1(21)=L1,1(21)M5.
(88)
All other combinations of RC 4v and Lnm(j) yield the same relations given above.

B. E1 mode of C6v 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 RC 6v is applied [14

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

]. So, by using this property, we can derive the matrix representation of all elements of C 6v. The results are as follows:
σx:(1,00,1),σy:(1,00,1),C6:(1/2,3/23/2,1/2),C61:(1/2,3/23/2,1/2),C3:(1/2,3/23/2,1/2),C31:(1/2,3/23/2,1/2),C2:(1,00,1),E:(1,00,1).
(89)

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,
VL0(11)=VdrH0(1)*(C61r)[H0(1)(C61r)]=Vdr[C6H0(1)*(C61r)][C6H0(1)(C61r)]=V(L0(11)+3L0(21)+3L0(12)+3L0(22))4.
(90)
Using C61, we similarly obtain
L0(11)=L0(11)3L0(21)3L0(12)+3L0(22)4.
(91)
From these two equations,
L0(12)+L0(21)=0,
(92)
L0(11)=L0(22)ω02c2+M1.
(93)
Using σx for L0(12),
VL0(12)=Vdr[H0(1)*(r)][H0(2)(r)]=VL0(12).
(94)
So, combining with Eq. (92), we obtain
L0(12)=L0(21)=0.
(95)

Now we proceed to the case involving the nearest neighbor lattice points. Using C 2 for L1(11) and L1(22), we obtain,
VL1(11)=Vdr[H0(1)*(r)][H0(1)(rr4)]=VL4(11),
(96)
VL1(22)=Vdr[H0(2)*(r)][H0(2)(rr4)]=VL4(22).
(97)
So,
L1(11)=L4(11)M2,
(98)
L1(22)=L4(22)M3.
(99)
Next, by changing the variable of integration from r to C 2 r for L1(12) and L1(21), we obtain
L1(12)=L4(12), L1(21)=L4(21)
(100)
while using σx, we obtain
L1(12)=L4(12), L1(21)=L4(21)
(101)
Thus,
L1(12)=L4(12), L1(21)=L4(21)=0.
(102)
On the other hand, applying σx, C 2 and σy to L2(11) and L2(22), we obtain
L2(11)=L3(11)=L5(11)=L6(11)M4,
(103)
L2(22)=L3(22)=L5(22)=L6(22)M5.
(104)
Finally, applying σx, C 2 and σy to L2(12) and L2(21), we obtain
L2(12)=L3(12)=L5(12)=L6(12)=L2(21)=L3(21)=L5(21)=L6(21)M6.
(105)

For the six Mi’s, we can derive three relations as follows. First, by applying C61 to L2(11) ,
M4=L2(11)=1VVdrH0(1)(r)3H0(2)*(r)2[H0(1)(rr1)3H0(2)(rr1)2]=M2+3M34,
(106)
where we used Eq. (102). Similarly, applying C61 to L2(22) and L2(22), we obtain
M5=3M2+M34,
(107)
M6=3(M2M3)4.
(108)
All other combinations of RC 6v and Ln(ij) yield the same relations given above.

C. E2 mode of C6v symmetry

It is known from group theory that {2xy, x 2y 2} is a polynomial representation of the E 2 mode. We can derive the matrix representation of all elements of C 6v:
σx:(1,00,1),σy:(1,00,1),C6:(1/2,3/23/2,1/2),C61:(1/2,3/23/2,1/2),C3:(1/2,3/23/2,1/2),C31:(1/2,3/23/2,1/2),C2:(1,00,1),E:(1,00,1).
(109)
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 C31, and hence, those for C61 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. Lnm(ij) for Cs symmetry

In addition to these two relations, we can also derive two more relations using the translational symmetry and Hermitian property of operator . We denote translation by lattice vector r nm by Tnm. Then, from the translational symmetry of ɛ(r) and invariance of differential operators under any uniform translation, we have
TnmTnm1=(Tnm×Tnm1)(Tnm1ɛ(r)Tnm1)(Tnm×Tnm1)=.
(114)
On the other hand, from the symmetric form of and the periodic boundary condition, we can prove that is an Hermitian operator [13

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

]:
VdrQ1*(r)[Q2(r)]=Vdr[Q1(r)]*Q2(r),
(115)
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′ = rr 10 in the expression of L10(11):
L10(11)=1VVdrH1*(r+r10)[(T1,0T1,01)]H1(r)=1VVdr[H1(rr1,0)]*H1(r)=L1,0(11)*.
(116)
Similarly we obtain,
L10(22)=L1,0(22)*.
(117)
Combining with Eqs. (112) and (113), we can conclude that M 2 and M 3 are real numbers.

As for L10(12), similarly applying σx and the same translation, we obtain
L10(12)=L1,0(12)=L1,0(21)*.
(118)
Similarly for L10(21), we have
L10(21)=L1,0(21)=L1,0(12)*.
(119)
From these equations, we obtain
L10(12)=L1,0(12)=L1,0(21)*=L10(21)*L1.
(120)
For L01(11) and L01(22), the application of σx yields identity equations. On the other hand, by changing the variable of integration from r to rr 01, we obtain
L01(11)=L0,1(11)*M3,L01(22)=L0,1(22)*M2.
(121)
Finally, applying σx, we can prove
L01(12)=L0,1(12)=L01(21)=L0,1(21)=0.
(122)

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]

OCIS Codes
(280.5600) Remote sensing and sensors : Radar
(350.4010) Other areas of optics : Microwaves
(350.3618) Other areas of optics : Left-handed materials
(160.3918) Materials : Metamaterials
(160.5298) Materials : Photonic crystals

ToC Category:
Metamaterials

History
Original Manuscript: April 26, 2011
Revised Manuscript: June 16, 2011
Manuscript Accepted: June 17, 2011
Published: July 6, 2011

Citation
Kazuaki Sakoda and Haifeng Zhou, "Analytical study of two-dimensional degenerate metamaterial antennas," Opt. Express 19, 13899-13921 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-15-13899


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References

  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]

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