Wave propagation in bianisotropic metamaterials: angular selective transmission

Po-Han Chang; Chih-Yu Kuo; Ruey-Lin Chern

doi:10.1364/OE.22.025710

1. Introduction

Bianisotropic media [1] are a special type of materials whose properties are characterized by the magnetoelectric as well as the permittivity and permeability tensors [2]. Due to strong modulation of wave that arises from the magnetoelectric couplings, the wave propagation in bianisotropic media is usually described by anomalous or high order dispersion relations with circularly or elliptically polarized eigenwaves [3–8]. Counterintuitive features such as negative refraction and backward wave, which are usually pertaining to metamaterials, may also appear in bianisotropic media [9–12]. In fact, the bianisotropy plays a crucial role in negative permeability and left-handed metamaterials [13].

If the diagonal elements of the magnetoelectric tensors are zero, the magnetoelectric couplings occur in mutually perpendicular directions rather than parallel directions [14]. This type of magnetoelectric coupling can be achieved, for instance, by surface currents induced on Ω-shaped metal wires [9, 15]. The magnetic fields normal to the central part of a Ω-shaped wire will give rise to electric dipoles along its two legs, and the electric fields along the legs give rise to magnetic dipoles normal to the wire. A similar coupling can also be found in other structures as split ring resonators [13, 16, 17] and fishnets [18, 19].

In bianisotropic media, the Poynting vector deviates from the wave vector, and the polarization state even changes. The former is a result of anisotropy and the letter is due to chirality. The patterns of reflection from and transmission through a bianisotropic medium may also be tailored, leading to the inversion of critical angle and Brewster angle that usually appear in hyperbolic media [20]. This feature is particularly useful when the bianisotropic media are used for angular filtering devices [21, 22]. Similar filtering effects have also been observed in metal strip gratings [23], one-dimensional photonic crystals [24], anisotropic [25], epsilon-near-zero [26, 27], and hyperbolic metamaterials [28, 29].

In this study, we investigate the basic features of wave propagation in the bianisotropic metamaterials, with emphasis on angular selective transmission. The bianisotropic media are characterized by asymmetric magnetoelectric tensors with zero diagonal elements, which can be synthesized by Ω-shaped metal wire, split ring resonators, or fishnets. The wave propagation is described by a biquadratic dispersion relation with two elliptically polarized eigenwaves. Unlike the ordinary anisotropic dielectric-magnetic materials, the bianisotropic metamaterials may possess a combination of elliptic and hyperbolic dispersions in the same eigenwave. In particular, there exist an ordinary and an inversion critical angle for a wave incident from vacuum, which is allowed to transmit through the bianisotropic medium over a range of the angle of incidence, leading to the feature of angular selective transmission. The transmitted wave can be negatively refracted, but with a different feature related to hyperbolic dispersion. The ordinary and inversion critical angles may change with the material parameters and result in a complementary type of angular selective transmission, that is, transmission is not allowed between the two critical angles. These features are illustrated with the incidence of Gaussian beams based on the Fourier integral formulation.

2. Basic equations

2.1. Dispersion relation

Bianisotropic materials are an important class of complex media [30], which are characterized by the following constitutive relations:

D = \underline{ε} E + \underline{ξ} H,

B = \underline{μ} H + \underline{ζ} E,

where ε̱, μ̱, ξ̱, and ζ̱ are in general tensors of complex quantities that depend on the frequency. Using Eqs. (1) and (2) in Maxwell’s equations: ∇ × E = iωB, ∇ × H = −iωD, and eliminating D and B, we may obtain the following separate equations for E and H fields:

\nabla \times {\underline{μ}}^{- 1} \nabla \times E - i ω \nabla \times ({\underline{μ}}^{- 1} \underline{ζ} E) + i ω \underline{ξ} {\underline{μ}}^{- 1} \nabla \times E - ω^{2} (\underline{ε} - \underline{ξ} {\underline{μ}}^{- 1} \underline{ζ}) E = 0,

\nabla \times {\underline{ε}}^{- 1} \nabla \times H - i ω \underline{ζ} {\underline{ε}}^{- 1} \nabla \times H + i ω \nabla \times ({\underline{ε}}^{- 1} \underline{ξ} H) - ω^{2} (\underline{μ} - \underline{ζ} {\underline{ε}}^{- 1} \underline{ξ}) H = 0,

which are regarded as the wave equations for bianisotropic media. Assuming that E and H are of the form e^ik·r, the wave equations are rewritten as M̱E = 0 and ṈH = 0, where

\underline{M} = (k \times \underline{I} + ω \underline{ξ}) {\underline{μ}}^{- 1} (k \times \underline{I} - ω \underline{ζ}) + ω^{2} \underline{ε},

\underline{N} = (k \times \underline{I} - ω \underline{ζ}) {\underline{ε}}^{- 1} (k \times \underline{I} + ω \underline{ξ}) + ω^{2} \underline{μ},

with I̱ being the identity tensor.

In particular, consider the bianisotropic media with the following forms of magnetoelectric tensors:

\underline{ξ} = \sqrt{ε_{0} μ_{0}} [\begin{matrix} 0 & 0 & i γ \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}], \underline{ζ} = \sqrt{ε_{0} μ_{0}} [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ - i γ & 0 & 0 \end{matrix}],

where γ is a real quantity that represents the chirality in the bianisotropic metamaterials. Note that ξ̱ and ζ̱ are asymmetric tensors with zero diagonal elements. The magnetoelectric couplings, therefore, occur in mutually perpendicular directions, which can be achieved by Ω-shaped metal wire [9, 15], split ring resonators [13, 16, 17], or fishnets [18, 19]. Let the permittivity and permeability tensors be ε̱ = ε₀ (ε_xx̂x̂ + ε_yŷŷ + ε_zẑẑ) and μ̱ = μ₀ (μ_xx̂x̂ + μ_yŷŷ + μ_zẑẑ), respectively. The existence of nontrivial solutions for E and H fields requires that |M̱| = 0 and |Ṉ| = 0, where |·| denotes the determinant. Assume that the wave vector lies on the xz plane, that is, k = (k_x, 0, k_z), without loss of generality. The condition of zero determinant gives rise to the characteristic equation:

a {k_{x}}^{4} + b {k_{z}}^{4} + c {k_{x}}^{2} {k_{z}}^{2} + d k_{0}^{2} {k_{x}}^{2} + e k_{0}^{2} {k_{z}}^{2} + f k_{0}^{4} = 0,

where k₀ = ω/c, a = μ_xε_x, b = μ_zε_z, c = ε_xμ_z + ε_zμ_x − γ², d = μ_x (ε_yγ² − ε_xε_zμ_y − ε_xε_yμ_z), e = ε_z (μ_yγ² − μ_xμ_zε_y − μ_yμ_zε_x), and f = μ_xμ_yε_yε_z (μ_zε_x − γ²). The above equation is considered a special case of the fourth-order dispersion relation for the most general bianisotropic media [31]. Equation (8) is then solved for k₀ to give the dispersion relation:

k_{0}^{2} = C_{1} k_{z}^{2} + C_{2} k_{x}^{2} \pm \sqrt{C_{3} k_{4}^{z} + C_{4} k_{x}^{4} + C_{5} k_{x}^{2} k_{z}^{2}},

where

C_{1} = \frac{1}{2} C_{0}^{- 1} ε_{z} [γ^{2} μ_{y} - (μ_{x} ε_{y} + μ_{y} ε_{x}) μ_{z}]

,

C_{2} = \frac{1}{2} C_{0}^{- 1} μ_{x} [γ^{2} ε_{y} - ε_{x} (μ_{y} ε_{z} + μ_{z} ε_{y})]

,

C_{3} = C_{0}^{- 1} ε_{z}^{2} {[γ^{2} μ_{y} + μ_{z} (μ_{x} ε_{y} - μ_{y} ε_{x})]}^{2}

,

C_{4} = C_{0}^{- 1} μ_{x}^{2} {[γ^{2} ε_{y} + ε_{x} (μ_{y} ε_{z} - μ_{z} ε_{y})]}^{2}

,

C_{5} = 2 μ_{x} ε_{z} C_{0}^{- 1} {μ_{z} ε_{x} (μ_{x} ε_{y} - μ_{y} ε_{x}) (μ_{z} ε_{y} - μ_{y} ε_{z}) - γ^{2} [μ_{y}^{2} ε_{x} ε_{z} + μ_{x} μ_{z} ε_{y}^{2} - 2 μ_{y} ε_{y} (μ_{x} ε_{z} + μ_{z} ε_{x})] - γ^{4} μ_{y} ε_{y}}

, with C₀ = μ_xμ_yε_yε_z (γ² − μ_zε_x).

For a given k₀, Eq. (8) is a biquadratic curve in the k_x-k_z plane, which is considered a combination of two standard quadratic curves of either elliptic or parabolic dispersion. As a result, the biquadratic curve may depict the elliptic-like, hyperbolic-like, and mixed-type dispersions, depending on the material parameters. Figure 1 shows three examples of the equifrequency contours for the bianisotropic media based on Eq. (9). In Fig. 1(a), both the contours with the plus sign (blue line) and minus sign (red line) are closed curves, showing the elliptic-like dispersion. In Fig. 1(b), both the contours are two-piece open curves, corresponding to the hyperbolic-like dispersion. In Fig. 1(c), one contour is represented by an ellipse and the other is by an hyperbola, showing the mixed-type dispersion.

Fig. 1 Equifrequency contours of the bianisotropic media with (a) elliptic-like dispersion, where ε_x = 2, ε_y = ε_z = 0.6, μ_x = μ_y = μ_z = 1, γ = 0.6, (b) hyperbolic-like dispersion, where ε_x = −1, ε_y = ε_z = 0.4, μ_x = −0.2, μ_y = μ_z = 0.7, γ = 0.6, and (c) mixed-type dispersion, where μ_z = 1.5, μ_y = 0.5, μ_x = −0.3, ε_x = ε_y = ε_z = 0.7, γ = 0.5. Blue and red lines are dispersion relations [Eq. (9)] with the plus and minus signs, respectively.

Download Full Size | PDF

A special type of dispersion for the bianisotropic media is shown in Fig. 2, where the two dispersion curves distort from the usual ellipses or hyperbolas and intersect at k_x = 0. This means that the transmitted wave at normal incidence will split into two waves, which will not occur for ordinary dielectric-magnetic materials (without magnetoelectric couplings). A similar splitting feature can also be observed in pseudochiral media [12]. The distinctive dispersion of the bianisotropic media is attributable to the symmetry breaking embedded in the constitutive relations [Eqs. (1) and (2)] through the magnetoelectric couplings. The chirality parameter γ in ξ̱ and ζ̱ [Eq. (7)] introduces an additional degree of freedom in the system and the degeneracy associated with the second-order dispersion for ordinary dielectric-magnetic materials is lifted. The dispersion of the bianisotropic media is therefore of higher order and hybrid in nature. This also indicates that there exist two eigenwaves with different wave vectors for a single frequency ω = k₀c.

Fig. 2 Equifrequency contours of the bianisotropic media with (a) ε_x = 2, ε_y = 1, ε_z = 0.6, μ_x = μ_y = μ_z = 1, γ = 1 and (b) ε_x = 1.5, ε = 0.5, ε_z = −0.5, μ_x = μ_y = 0.7, μ_z = 0.25, γ = 0.5.

Download Full Size | PDF

2.2. Eigenwaves

Let the xy plane be an interface between vacuum and the bianisotropic medium. For a given tangential (to the interface) wave vector components k_x ≠ 0, there exist two normal components:

k_{z}^{\pm} = {[- \frac{(c k_{x}^{2} + e k_{0}^{2}) \pm \sqrt{{(c k_{x}^{2} + e k_{0}^{2})}^{2} - 4 b (a {k_{x}}^{4} + d k_{0}^{2} {k_{x}}^{2} + f k_{0}^{4})}}{2 b}]}^{1 / 2},

where + and − denote the plus and minus modes, respectively. Each of the two vectors (k_x, 0,

k_{z}^{\pm}

) is associated with an eigenwave determined by the nullspace of M̱, given as

E^{\pm} = E_{0}^{\pm} e^{\pm}

, where

E_{0}^{+}

and

E_{0}^{-}

are the electric field magnitudes for the plus and minus modes, respectively, and

e^{\pm} = \frac{{k_{x}}^{2} - {k_{0}}^{2} μ_{y} ε_{z}}{k_{x} k_{z}^{\pm}} \hat{x} + i \frac{μ_{x} ε_{z} γ k_{0} k_{z}^{\pm}}{μ_{x} ε_{x} {k_{x}}^{2} + (γ^{2} - μ_{z} ε_{x}) ({k_{0}}^{2} μ_{x} ε_{y} - {(k_{z}^{\pm})}^{2})} \hat{y} + \hat{z}

The corresponding magnetic fields are obtained from the constitutive relation and Maxwell’s equations, given as

H^{\pm} = \frac{E_{0}^{\pm}}{η_{0}} h^{\pm}

, where

η_{0} = \sqrt{μ_{0} / ε_{0}}

and

h^{\pm} = - i \frac{({k_{x}}^{2} - {k_{0}}^{2} ε_{z} μ_{y}) (ε_{x} μ_{z} - γ^{2}) + μ_{z} ε_{z} {(k_{z}^{\pm})}^{2}}{γ μ_{x} {k_{x}}^{2}} \hat{x} - \frac{k_{0} ε_{z}}{k_{x}} \hat{y} + i \frac{ε_{z} ({(k_{z}^{\pm})}^{2} - {k_{0}}^{2} μ_{y} ε_{x}) + {k_{x}}^{2} ε_{x}}{γ k_{x} k_{z}^{\pm}} \hat{z} .

In general, the two eigenwaves in the bianisotropic medium are elliptically polarized, where the plus (or minus) mode can be either right-handed or left-handed wave, depending on the material parameters.

At the normal incidence (k_x = 0), $k_{z}^{\pm}$ is reduced to

k_{z}^{\pm} = k_{0} \sqrt{μ_{x} ε_{y} + μ_{y} ε_{x} - γ^{2} μ_{y} / μ_{z} \pm sign (ε_{z}) | μ_{x} ε_{y} - μ_{y} ε_{x} + γ^{2} μ_{y} / μ_{z} |} .

If ε_z (μ_xε_y − μ_yε_x + γ²μ_y/μ_z) > 0, Eq. (13) is further simplified to

k_{z}^{+} = k_{0} \sqrt{μ_{x} ε_{y}}, k_{z}^{-} = k_{0} \sqrt{μ_{y} (ε_{x} - γ^{2} / μ_{z})} .

The corresponding basis vectors for the eigenwaves are given by

e^{+} = (0, 1, 0), e^{-} = (1, 0, 0),

h^{+} = \sqrt{\frac{ε_{y}}{μ_{x}}} (1, 0, 0), h^{-} = (0, \frac{k_{z}^{-}}{k_{0} μ_{y}}, \frac{i γ}{μ_{z}}) .

In this case, the plus and minus modes reduce to TE and TM modes, respectively. If ε_z (μ_xε_y − μ_yε_x + γ²μ_y/μ_z) < 0, the plus and minus modes exchange accordingly.

3. Reflection and transmission coefficients

Consider a plane wave of frequency ω incident from vacuum, making an angle θ with respect to the interface normal. Let the xz plane be the plane of incidence. In vacuum, the incident and reflected wave vectors are k_i = k₀ (sinθx̂ + cosθẑ) and k_r = k₀ (sinθx̂ − cosθẑ), respectively. The incident and reflected fields are given as

E_{i} = E_{i}^{p} (cos θ \hat{x} - sin θ \hat{z}) + E_{i}^{s} \hat{y},

E_{r} = - E_{r}^{p} (cos θ \hat{x} + sin θ \hat{z}) + E_{r}^{s} \hat{y},

H_{i} = \frac{E_{i}^{p}}{η_{0}} \hat{y} + \frac{E_{i}^{s}}{η_{0}} (- cos θ \hat{x} + sin θ \hat{z}),

H_{r} = \frac{E_{r}^{p}}{η_{0}} \hat{y} + \frac{E_{r}^{s}}{η_{0}} (cos θ \hat{x} + sin θ \hat{z}),

where

E_{i, r}^{p}

and

E_{i, r}^{s}

are the electric field amplitudes for p- and s-polarized incident or reflected waves, respectively, and

η_{0} = \sqrt{μ_{0} / ε_{0}}

. In the bianisotropic medium, there are two wave vectors: (k_x, 0,

k_{z}^{\pm}

) [cf. Eq. (10)]. The transmitted fields are given as

E_{t}^{\pm} = E_{t}^{\pm} e^{\pm},

H_{t}^{\pm} = \frac{E_{t}^{\pm}}{η_{0}} h^{\pm},

where

E_{t}^{+}

and

E_{t}^{-}

are the electric field amplitudes for the plus and minus modes, respectively, and e^± and h^± are given in Eqs. (11) and (12), respectively. The continuity of tangential electric and magnetic fields at the interface gives rise to the boundary conditions:

(E_{i} + E_{r}) \times \hat{z} = (E_{t}^{+} + E_{t}^{-}) \times \hat{z},

(H_{i} + H_{r}) \times \hat{z} = (H_{t}^{+} + H_{t}^{-}) \times \hat{z},

Using Eqs. (17)–(22) in Eqs. (23) and (24), we have the solutions for

E_{r}^{p, s}

and

E_{t}^{\pm}

in terms of

E_{i}^{p, s}

as

[\begin{matrix} E_{r}^{p} \\ E_{r}^{s} \end{matrix}] = [\begin{matrix} r_{p p} & r_{p s} \\ r_{s p} & r_{s s} \end{matrix}] [\begin{matrix} E_{i}^{p} \\ E_{i}^{s} \end{matrix}],

[\begin{matrix} E_{t}^{+} \\ E_{t}^{-} \end{matrix}] = [\begin{matrix} t_{+ p} & t_{+ s} \\ t_{- p} & t_{- s} \end{matrix}] [\begin{matrix} E_{i}^{p} \\ E_{i}^{s} \end{matrix}],

where

r_{p p} = \frac{s_{x x} + cos θ (e_{x y} - h_{x y} - s_{y y} cos θ)}{s_{x x} + cos θ (e_{x y} + h_{x y} + s_{y y} cos θ)}, r_{p s} = \frac{2 s_{x y} cos θ}{s_{x x} + cos θ (e_{x y} + h_{x y} + s_{y y} cos θ)},

r_{s p} = \frac{2 s_{y x} cos θ}{s_{x x} + cos θ (e_{x y} + h_{x y} + s_{y y} cos θ)}, r_{s s} = \frac{cos θ (e_{x y} + h_{x y} + s_{y y} cos θ) - s_{x x}}{s_{x x} + cos θ (e_{x y} + h_{x y} + s_{y y} cos θ)},

t_{+ p} = \frac{2 (h_{x}^{-} - e_{y}^{-} cos θ) cos θ}{s_{x x} + cos θ (e_{x y} + h_{x y} + s_{y y} cos θ)}, t_{+ s} = \frac{2 (e_{x}^{-} + h_{y}^{-} cos θ) cos θ}{s_{x x} + cos θ (e_{x y} + h_{x y} + s_{y y} cos θ)},

t_{- p} = \frac{2 (e_{y}^{+} cos θ - h_{x}^{+}) cos θ}{s_{x x} + cos θ (e_{x y} + h_{x y} + s_{y y} cos θ)}, t_{- s} = \frac{- 2 (e_{x}^{+} + h_{y}^{+} cos θ) cos θ}{s_{x x} + cos θ (e_{x y} + h_{x y} + s_{y y} cos θ)},

with

s_{i j} = e_{i}^{+} h_{j}^{-} - e_{i}^{-} h_{j}^{+}

,

e_{x y} = e_{x}^{-} e_{y}^{+} - e_{x}^{+} e_{y}^{-}

,

h_{x y} = h_{x}^{-} h_{y}^{+} - h_{x}^{+} h_{y}^{-}

, and

e_{i}^{\pm}

and

h_{i}^{\pm}

being the i-th components of e^± and h^±, respectively. Note that r_ps is equal to r_sp, which is a consequence of symmetry [32]. They are in general not zero and therefore either a p- or s-polarized incident wave may give rise to both p- and s-polarized reflected waves. Meanwhile, either a p- or s-polarized incident wave may give rise to both right-handed and left-handed elliptically polarized transmitted waves. Note also that if γ = 0, we have from Eq. (10) that

k_{z}^{+} = \sqrt{μ_{x} (k_{0}^{2} ε_{y} - k_{x}^{2} / μ_{z})}

and

k_{z}^{-} = \sqrt{ε_{x} (k_{0}^{2} μ_{y} - k_{x}^{2} / ε_{z})}

, and

r_{p p} = \frac{ε_{x} k_{z}^{0} - k_{z}^{-}}{k_{z}^{-} + ε_{x} k_{z}^{0}}, r_{s s} = \frac{μ_{x} k_{z}^{0} - k_{z}^{+}}{k_{z}^{+} + μ_{x} k_{z}^{0}}, r_{p s} = r_{s p} = 0,

t_{p p} = \frac{2 ε_{x} k_{z}^{0}}{k_{z}^{-} + ε_{x} k_{z}^{0}}, t_{s s} = \frac{2 μ_{x} k_{z}^{0}}{k_{z}^{+} + μ_{x} k_{z}^{0}}, t_{p s} = t_{s p} = 0,

which reduce exactly to the Fresnel equations for ordinary anisotropic materials. Equations (27)–(30) are regarded as the generalized Fresnel equations for bianisotropic media. The reflection and transmission coefficients for the most general bianisotropic media can be found in Ref. [33].

At normal incidence (θ = 0), where the basis vectors e^± and h^± are given by Eqs. (15) and Eqs. (16), respectively, the reflection and transmission coefficients are simplified as

r_{p p} = \frac{η_{p} - η_{0}}{η_{p} + η_{0}}, r_{s s} = \frac{η_{s} - η_{0}}{η_{s} + η_{0}}, r_{p s} = r_{s p} = 0,

t_{p p} = \frac{2 η_{p}}{η_{p} + η_{0}}, t_{s s} = \frac{2 η_{s}}{η_{s} + η_{0}}, t_{p s} = t_{s p} = 0,

where

η_{p} = \sqrt{μ_{y} / (ε_{x} - γ^{2} / μ_{z})}

and

η_{s} = \sqrt{μ_{x} / ε_{y}}

are considered the impedances for p and s polarizations, respectively.

4. Angular selective transmission

Due to the hybrid dispersion of the bianisotropic metamaterials, the transmitted wave in this media is subject to two different types of constraints. For the elliptic (hyperbolic) type of dispersion, the wave is allowed to propagate when the angle of incidence is below (above) the critical angle. Once the two angles occur simultaneously, the transmitted wave is allowed only in a particular angular range. The transmitted wave can further be negatively refracted, owing to the anomalous dispersion in the bianisotropic media.

4.1. Ordinary and inversion critical angles

The angular range for the transmission of wave through the bianisotropic media is constrained by the critical angle at which the normal wave vector component $k_{z}^{\pm}$ becomes zero. From Eq. (10), the zeros of $k_{z}^{\pm}$ occur at two conditions: (1) $k_{x} = ω \sqrt{μ_{y} ε_{z}}$ and A₁ ± |A₁| = 0, where A₁ = (ε_yμ_z − ε_zμ_y)ε_zμ_x, and (2) $k_{x} = ω \sqrt{ε_{y} (ε_{x} μ_{z} - γ^{2}) / ε_{x}}$ and A₂ ± |A₂| = 0, where A₂ = γ²(2ε_yμ_z − ε_zμ_y + ε_yε_zμ_x/ε_x) + ε_x(ε_zμ_y − ε_yμ_z)μ_z − γ⁴ε_y/ε_x. The corresponding critical angles are given by

θ_{c_{1}} = arcsin (\frac{ω}{k_{0}} \sqrt{μ_{y} ε_{z}}),

θ_{c_{2}} = arcsin (\frac{ω}{k_{0}} \sqrt{(ε_{x} μ_{z} - γ^{2}) ε_{y} / ε_{x}}) .

For the elliptic type dispersion, the zero of

k_{z}^{\pm}

corresponds to the ordinary critical angle, beyond which the incident wave is totally reflected. For the hyperbolic type dispersion, the zero of

k_{z}^{\pm}

corresponds to the inversion critical angle, below which the incident wave is totally reflected. For the mixed type dispersion, the zero of

k_{z}^{\pm}

corresponds to either the ordinary or the inversion critical angle.

If A₁A₂ < 0, there may exist one zero of $k_{z}^{+}$ and another zero of $k_{z}^{-}$ , which means that the two critical angles belong to different eigenwaves. If A₁A₂ > 0, on the other hand, there can be two zeros of $k_{z}^{+}$ or $k_{z}^{-}$ . In this situation, the two critical angles, one being ordinary and the other being inversion, appear in the same eigenwave, leading to the angular selective transmission. Figure 3 shows two examples of the equifrequency contours for the bianisotropic media that depict the angular selective transmission. In each case, only one eigenwave is allowed to propagate ( $k_{z}^{-}$ is not a real quantity). The dispersion curves in Fig. 3(a) are two separated ellipses, with two critical angles. In this case, the ordinary critical angle θ_c₁ is larger than the inversion critical angle θ_c₂, so that the wave is allowed to transmit through the bianisotropic medium over the angular range: θ_c₂ < θ < θ_c₁.

Fig. 3 Equifrequency contours of the bianisotropic media with two critical angles, where (a) ε_x = 2, ε_y = −0.2, ε_z = 0.8, μ_x = μ_y = μ_z = 1, and γ = 1.6 and (b) ε_x = −0.5, ε_y = 0.2, ε_z = 1, μ_x = μ_y = μ_z = 0.5, and γ = 0.2. Gray circles are the equifrequency contours in vacuum.

Download Full Size | PDF

It is worthy of noting that negative refraction may occur below a threshold angle θ_NR, given by

θ_{NR} = arcsin (\sqrt{\frac{ε_{z} ρ + ε_{x}^{- 1} | γ ε_{z} (μ_{z} ε_{x} + μ_{x} ε_{z} - γ^{2}) | \sqrt{σ}}{γ^{4} - 2 γ^{2} (ε_{z} μ_{x} + ε_{x} μ_{z}) + {(ε_{z} μ_{x} - ε_{x} μ_{z})}^{2}}}),

which is determined by the condition:

\partial k_{z}^{\pm} / \partial k_{x} = 0

. If θ_c₁ < θ < θ_NR, the refracted wave orients toward the same side of the incidence wave, giving rise to negative refraction. In Fig. 3(b), the dispersion curves are a combination of an ellipse and a hyperbola, also with two critical angles. In this case, the ordinary critical angle θ_c₂ is smaller than the inversion critical angle θ_c₁, and the wave can be transmitted through the bianisotropic medium in the angular ranges: θ < θ_c₂ or θ > θ_c₁. In the latter range, the transmitted wave is also negatively refracted due to the hyperbolic dispersion in this range.

In the presence of a small to modest loss in the material parameters (as in Refs. [9, 11]), the dispersion relations may deform slightly from the lossless case. They are represented by separate equifrequency contours associated with Re[k_z] and Im[k_z], respectively, as shown in Fig. 4. Here, Im[k_z] corresponds to the decay of a transmitted wave in the medium, which is small in the allowed region and large in the forbidden region. The eigenfrequency contours with Re[k_z] (solid green lines) basically conform to the dispersion curves for the lossless case (gray lines) (cf. Fig. 3). In this circumstance, the ordinary and inversion critical angles can be evaluated by the drastic change of the dispersion curves with Re[k_z] or Im[k_z] (dashed green lines), which are close to the respective angles without loss for either the standard type of angular selective transmission [Fig. 4(a)], where θ_c₁ ≈ 63.4° and θ_c₂ ≈ 13.7°, or the complementary type [Fig. 4(b)], where θ_c₁ ≈ 45° and θ_c₂ ≈ 19.9°.

Fig. 4 Equifrequency contours of the bianisotropic media in the presence of small loss, where (a) ε_x = 2 + 0.03i, ε_y = −0.2 + 0.02i, ε_z = 0.8 + 0.05i, μ_x = μ_y = μ_z = 1 + 0.06i, and γ = 1.6 + 0.04i and (b) ε_x = −0.5 + 0.03i, ε_y = 0.2 + 0.02i, ε_z = 1 + 0.05i, μ_x = μ_y = μ_z = 0.5 + 0.04i, and γ = 0.2 + 0.04i. Solid and dashed green lines are dispersion curves associated with Re[k_z] and Im[k_z], respectively. Gray lines are dispersion curves for the lossless case (cf. Fig. 3).

Download Full Size | PDF

4.2. Gaussian beams

The angular selective transmission stated above can be illustrated with the incidence of Gaussian beams. Suppose that the xy plane is an interface between vacuum on the left (z < 0) and a bianisotropic medium on the right (z > 0), characterized by the permittivity tensor ε̱, permeability tensor μ̱, and the chirality parameter γ. Consider a Gaussian beam incident from vacuum, with the beam center making an angle θ with respect to the interface normal. Let the xz plane be the plane of incidence and k_x be the wave vector component along the interface. The Gaussian beam with the center located at x = x₀ and z = −h is well approximated by the Fourier integral on the tangential (to surface) wave vector component k_x as [34]

f (x, z) = \int_{- \infty}^{\infty} ψ (k_{x}) e^{i k_{x} x + i k_{z} z} d k_{x},

ψ (k_{x}) = \frac{w_{0}}{2 cos θ \sqrt{π}} exp [- \frac{w_{0}^{2}}{4 {cos}^{2} θ} {(k_{x} - k_{0} sin θ)}^{2} - i k_{x} x_{0} + i k_{z} h],

where k_z is the normal wave number component, and w₀ is the waist size of the Gaussian beam. Based on this formulation, the incident beams are formulated as

E_{i}^{p, s} = E_{0}^{p, s} \int_{- \infty}^{\infty} e_{i}^{p, s} ψ (k_{x}) e^{i k_{x} x + i k_{z} z} d k_{x},,

H_{i}^{p, s} = \frac{E_{0}^{p, s}}{η_{0}} \int_{- \infty}^{\infty} h_{i}^{p, s} ψ (k_{x}) e^{i k_{x} x + i k_{z} z} d k_{x},

where

E_{0}^{p, s}

are the electric fields magnitude for the p-polarized (TM) or s-polarized (TE) incident beam,

e_{i}^{p} = (k_{z} / k_{0}, 0, - k_{x} / k_{0})

,

e_{i}^{s} = (0, 1, 0)

,

h_{i}^{p} = e_{i}^{s}

, and

h_{i}^{s} = - e_{i}^{p}

.

The reflected beams are formulated as

E_{r}^{p, s} = \int_{- \infty}^{\infty} e_{r}^{p, s} R^{p, s} ψ (k_{x}) e^{i k_{x} x - i k_{z} z} d k_{x},

H_{r}^{p, s} = \frac{1}{η_{0}} \int_{- \infty}^{\infty} h_{r}^{p, s} R^{p, s} ψ (k_{x}) e^{i k_{x} x - i k_{z} z} d k_{x},

where

e_{r}^{p} = (- k_{z} / k_{0}, 0, - k_{x} / k_{0})

,

e_{r}^{s} = (0, 1, 0)

,

h_{r}^{p} = e_{r}^{s}

,

h_{r}^{s} = e_{r}^{p}

, and R^p,s are the reflection coefficients, given as

R^{p} = r_{p p} E_{0}^{p} + r_{p s} E_{0}^{s}

and

R^{s} = r_{s p} E_{0}^{p} + r_{s s} E_{0}^{s}

. Detailed formulas for r_pp, r_ps, r_sp, and r_ss are listed in Eqs. (27) and (28).

The transmitted beams in the bianisotropic medium are formulated as

E_{t}^{\pm} = \int_{- \infty}^{\infty} e_{t}^{\pm} T^{\pm} ψ (k_{x}) e^{i k_{x} x + i k_{z}^{\pm} z} d k_{x},

H_{t}^{\pm} = \frac{1}{η_{0}} \int_{- \infty}^{\infty} h_{t}^{\pm} T^{\pm} ψ (k_{x}) e^{i k_{x} x + i k_{z}^{\pm} z} d k_{x},

where

e_{t}^{\pm}

and

h_{t}^{\pm}

are given by Eqs. (11) and (12), respectively, and T^± are the transmission coefficients, given as

T^{+} = t_{+ p} E_{0}^{p} + t_{+ s} E_{0}^{s}

and

T^{-} = t_{- p} E_{0}^{p} + t_{- s} E_{0}^{s}

. Detailed formulas for t_+p, t_+s, t_−p, and t_−s are listed in Eqs. (29) and (30). After obtaining the fields of the incident, reflected, and transmitted beams, the power intensity is given as I = |〈S〉|, where

〈 S 〉 = \frac{1}{2} Re [E \times H^{*}]

is the time-averaged Poynting vector for the total electric field E and total magnetic field H in vacuum or the bianisotropic medium.

Figure 5 shows the normalized power intensity for a p-polarized (TM) Gaussian beam incident from vacuum onto a bianisotropic medium at different angles of incidence. In this configuration, there exist an ordinary critical angle θ_c₁ ≈ 18.2° and an inversion critical angle θ_c₂ ≈ 27.3°. In Fig. 5(a), where θ < θ_c₂, the incident beam is totally reflected from the interface. In Fig. 5(b), where θ_c₂ < θ < θ_c₁, the incident beam is allowed to transmit through the bianisotropic medium. Note that the transmitted beam is negatively refracted, which is consistent with the condition θ < θ_NR ≈ 23.5° [cf. Fig. 3(a)]. In Fig. 5(c), where θ > θ_c₁, the incident beam is again totally reflected.

Fig. 5 Power intensities for a p-polarized Gaussian beam incident from vacuum at (a) θ = 16° (b) θ = 20° and (c) θ = 30° onto the bianisotropic medium with ε_x = 2, ε_y = −0.35, ε_z = 0.3, μ_x = μ_y = μ_z = 0.7, and γ = 1.4. The intensities are normalized to have a maximum value of unity. White solid line denotes the interface.

Download Full Size | PDF

Another example of the angular selective transmission with a complementary feature is shown in Fig. 6. In this configuration, there exist an ordinary critical angle θ_c₂ ≈ 19.9° and an inversion critical angle θ_c₁ ≈ 26.6°. In Fig. 6(a), where θ < θ_c₂, the incident beam can be transmitted through the bianisotropic medium. In Fig. 6(b), where θ_c₂ < θ < θ_c₁, the incident beam is totally reflected. In Fig. 6(c), where θ > θ_c₁, the incident beam can be transmitted again. The transmitted beam is also negatively refracted due to the hyperbolic dispersion in this angular range [cf. Fig. 3(b)].

Fig. 6 Power intensities for a p-polarized Gaussian beam incident from vacuum at (a) θ = 15° (b) θ = 21° and (c) θ = 27° onto the bianisotropic medium with ε_x = −0.5, ε_y = 0.2, ε_z = 0.4, μ_x = μ_y = μ_z = 0.5, and γ = 0.2.

Download Full Size | PDF

5. Concluding remarks

In conclusion, we have investigated the basic features of wave propagation in bianisotropic metamaterials, which can be synthesized by Ω-shaped metal wire, split ring resonators, or fishnets. A particular emphasis is placed upon the feature of angular selective transmission, where the wave transmitted through the bianisotropic media is allowed in a certain angular region determined by an ordinary and an inversion critical angle. The bianisotropic media are therefore eligible for angular filtering devices. The angular selective feature comes from the high order dispersion described by the biquadratic curve and is characteristic of the magnetoelectric couplings pertaining to the bianisotropic metamaterials. A standard and a complementary type of angular selective transmissions are well illustrated with Gaussian beams based on Fourier integral formulation.

Acknowledgments

This work was supported in part by National Science Council of the Republic of China under Contract No. NSC 102-2221-E-002-202-MY3.

References and links

1. J. A. Kong, “Theorems of bianisotropic media,” Proc. IEEE 60, 1036–1046 (1972). [CrossRef]

2. A. Serdyukov, I. Semchenko, S. Tretyakov, and A. Sihvola, Electromagnetics of Bi-anisotropic Materials: Theory and Applications (Gordon and Breach, 2001).

3. I. Lindell and A. Viitanen, “Plane wave propagation in uniaxial bianisotropic medium,” Electron. Lett. 29, 150–152 (1993). [CrossRef]

4. S. Tretyakov and A. Sochava, “Novel uniaxial bianisotropic materials: reflection and transmission in planar structures,” Prog. Electomag. Res. 9, 157–179 (1994).

5. I. V. Semchenko, S. A. Khakhomov, S. A. Tretyakov, A. H. Sihvola, and E. A. Fedosenko, “Reflection and transmission by a uniaxially bi-anisotropic slab under normal incidence of plane waves,” J. Phys. D-Appl. Phys. 31, 2458 (1998). [CrossRef]

6. I. V. Semchenko and S. A. Khakhomov, “Artificial uniaxial bianisotropic media at oblique incidence of electromagnetic waves,” Electromagnetics 22, 71–84 (2002). [CrossRef]

7. K. Aydin, Z. Li, M. Hudlička, S. A. Tretyakov, and E. Ozbay, “Transmission characteristics of bianisotropic metamaterials based on omega shaped metallic inclusions,” New J. Phys. 9, 326 (2007). [CrossRef]

8. R.-L. Chern and P.-H. Chang, “Wave propagation in pseudochiral media: generalized Fresnel equations,” J. Opt. Soc. Am. B 30, 552–558 (2013). [CrossRef]

9. S. A. Tretyakov, C. R. Simovski, and M. Hudlička, “Bianisotropic route to the realization and matching of backward-wave metamaterial slabs,” Phys. Rev. B 75, 153104 (2007). [CrossRef]

10. X. Cheng, H. Chen, L. Ran, B.-I. Wu, T. M. Grzegorczyk, and J. A. Kong, “Negative refraction and cross polarization effects in metamaterial realized with bianisotropic S-ring resonator,” Phys. Rev. B 76, 024402 (2007). [CrossRef]

11. T. G. Mackay and A. Lakhtakia, “Negative refraction, negative phase velocity, and counterposition in bianisotropic materials and metamaterials,” Phys. Rev. B 79, 235121 (2009). [CrossRef]

12. R.-L. Chern and P.-H. Chang, “Negative refraction and backward wave in pseudochiral mediums: illustrations of Gaussian beams,” Opt. Express 21, 2657–2666 (2013). [CrossRef] [PubMed]

13. R. Marqués, F. Medina, and R. Rafii-El-Idrissi, “Role of bianisotropy in negative permeability and left-handed metamaterials,” Phys. Rev. B 65, 144440 (2002). [CrossRef]

14. S. A. Tretyakov, A. H. Sihvola, A. A. Sochava, and C. R. Simovski, “Magnetoelectric interactions in bianisotropic media,” J. Electromagn. Waves Appl. 12, 481–497 (1998). [CrossRef]

15. M. M. I. Saadoun and N. Engheta, “A reciprocal phase shifter using novel pseudochiral or Ω medium,” Microw. Opt. Technol. Lett. 5, 184–188 (1992). [CrossRef]

16. X. Chen, B.-I. Wu, J. A. Kong, and T. M. Grzegorczyk, “Retrieval of the effective constitutive parameters of bianisotropic metamaterials,” Phys. Rev. E 71, 046610 (2005). [CrossRef]

17. C. E. Kriegler, M. S. Rill, S. Linden, and M. Wegener, “Bianisotropic photonic metamaterials,” IEEE J. Sel. Top. Quantum Electron. 16, 367–375 (2010). [CrossRef]

18. Z. Ku, J. Zhang, and S. R. J. Brueck, “Bi-anisotropy of multiple-layer fishnet negative-index metamaterials due to angled sidewalls,” Opt. Express 17, 6782–6789 (2009). [CrossRef] [PubMed]

19. Z. Ku, K. M. Dani, P. C. Upadhya, and S. R. Brueck, “Bianisotropic negative-index metamaterial embedded in a symmetric medium,” J. Opt. Soc. Am. B 26, B34–B38 (2009). [CrossRef]

20. T. M. Grzegorczyk, Z. M. Thomas, and J. A. Kong, “Inversion of critical angle and Brewster angle in anisotropic left-handed metamaterials,” Appl. Phys. Lett 86, 251909 (2005). [CrossRef]

21. X.-L. Xu, B.-G. Quan, C.-Z. Gu, and L. Wang, “Bianisotropic response of microfabricated metamaterials in the terahertz region,” J. Opt. Soc. Am. B 23, 1174–1180 (2006). [CrossRef]

22. J. Ortiz, J. Baena, V. Losada, F. Medina, and J. Araque, “Spatial angular filtering by fsss made of chains of interconnected SRRs and CSRRs,” IEEE Microw. Wirel. Compon. Lett. 23, 477–479 (2013). [CrossRef]

23. D. Kinowski, M. Guglielmi, and A. Roederer, “Angular bandpass filters: an alternative viewpoint gives improved design flexibility,” IEEE Trans. Antennas Propag. 43, 390–395 (1995). [CrossRef]

24. G. Liang, P. Han, and H. Wang, “Narrow frequency and sharp angular defect mode in one-dimensional photonic crystals from a photonic heterostructure,” Opt. Lett. 29, 192–194 (2004). [CrossRef] [PubMed]

25. Y. Feng, X. Teng, J. Zhao, Y. Chen, and T. Jiang, “Anomalous reflection and refraction in anisotropic metamaterial realized by periodically loaded transmission line network,” J. Appl. Phys. 100, 114901 (2006). [CrossRef]

26. A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B 75, 155410 (2007). [CrossRef]

27. L. V. Alekseyev, E. E. Narimanov, T. Tumkur, H. Li, Y. A. Barnakov, and M. A. Noginov, “Uniaxial epsilon-near-zero metamaterial for angular filtering and polarization control,” Appl. Phys. Lett. 97, 131107 (2010). [CrossRef]

28. C. Rizza, A. Ciattoni, E. Spinozzi, and L. Columbo, “Terahertz active spatial filtering through optically tunable hyperbolic metamaterials,” Opt. Lett. 37, 3345–3347 (2012). [CrossRef]

29. A. Ciattoni and E. Spinozzi, “Optical resonances and angular filtering functionality of subwavelength hyperbolic etalons,” Optik 124, 3623–3626 (2013). [CrossRef]

30. W. S. Weiglhofer and A. Lakhtakia, Introduction to Complex Mediums for Optics and Electromagnetics (SPIE, 2003). [CrossRef]

31. R. D. Graglia, P. L. E. Uslenghi, and R. E. Zich, “Dispersion relation for bianisotropic materials and its symmetry properties,” IEEE Trans. Antennas Propag. 39, 83–90 (1991). [CrossRef]

32. L. Li, “Symmetries of cross-polarization diffraction coefficients of gratings,” J. Opt. Soc. Am. A 17, 881–887 (2000). [CrossRef]

33. R. D. Graglia, P. L. E. Uslenghi, and R. E. Zich, “Reflection and transmission for planar structures of bianisotropic media,” Electromagnetics 11, 193–208 (1991). [CrossRef]

34. B. R. Horowitz and T. Tamir, “Lateral displacement of a light beam at a dielectric interface,” J. Opt. Soc. Am. 61, 586–594 (1971). [CrossRef]

Wave propagation in bianisotropic metamaterials: angular selective transmission

Abstract

1. Introduction

2. Basic equations

2.1. Dispersion relation

2.2. Eigenwaves

3. Reflection and transmission coefficients

4. Angular selective transmission

4.1. Ordinary and inversion critical angles

4.2. Gaussian beams

5. Concluding remarks

Acknowledgments

References and links

Cited By

Figures (6)

Equations (45)

Optics Express