1. Introduction

Chiral materials have been intensively investigated in the past decade due to their exotic properties which are not exhibited by conventional isotropic media, such as the optical activity and the circular dichroism [1]. Recently, Pendry has proved that negative refraction can be realized using an isotropic chiral material, which has aroused great interest in scientific communities [2]. In an earlier paper, Tretyakov et al. have analyzed the possibility on existence of negative refraction in chiral nihility [3]. The negative phase velocity (NPV) in Faraday chiral medium was investigated by Mackay and Lakhtakia in Ref. [4], where the condition for NPV in isotropic chiral medium was obtained implicitly. In the isotropic chiral medium, when the chirality becomes sufficiently large, it has been shown that both negative refraction and backward propagation could be supported simultaneously for one of internal eigenwaves [5]. This may help to achieve super-resolution imaging. Further investigations on negative refraction in chiral medium could be found in some recent literatures [6,7].

In a recent work, the refraction and propagation properties of plane waves which are incident from free space to a uniaxially anisotropic chiral medium have been investigated [8], where the chirality appears only in one direction. It is shown that negative refraction or backward propagation can be supported in one of the eigenwaves, or negative refraction and backward waves are possible in two eigenwaves separately. However, the negative refraction and backward waves cannot exist simultaneously in the uniaxially anisotropic chiral medium [8].

In this paper, we will study the refraction and propagation properties of plane waves in biaxially anisotropic chiral media, where the chirality appears in two directions. Such biaxially chiral media can be realized by putting chiral elements pointing to two directions into a host anisotropic medium. We will prove that negative refractions or/and backward waves can be supported separately or simultaneously in the biaxially chiral media. We remark that a similar topic has been investigated by Mackay and Lakhtakia in a recent paper [9] on the negative phase velocity. However, the constitutive relations of the chiral media we are considering are different from those in Ref. [9], and therefore achieve different conclusions.

2. General Formulations

The problem geometry is shown in Fig. 1, where the whole space is divided into two regions. Region 0 is free space, while Region 1 is occupied by a chiral medium whose constitutive relations are given by

where I̿ _t =x̂x̂+yŷŷ, κ is the chirality factor, and ε ₀ and µ ₀ are permittivity and permeability of free space. Clearly, both the permittivity and permeability are uniaxial tensors which are ε _t and µ _t in the x and y directions and ε _z and µ _z in the z direction, while the chirality appears in y and z directions. Hence the overall medium behaves a biaxial property. Such a kind of chiral medium can be easily constructed by embedding chiral objects like wire helices or Möbius strips regularly in two directions in the host dielectric.

Consider an arbitrarily polarized plane wave incident from free space to the chiral medium at an oblique angle θ _i . Assume that the plane of incidence is within the yoz plane so that k̄ _i =k _y ŷ+k _0z ẑ and the electric field is written as ${\overline{E}}_i=\overline{E}{e}^{i({\overline{k}}_i\cdot \overline{r}-\omega t)}$. Substituting the constitutive relations into Maxwell equations, we will obtain a system of coupled ordinary differential equations for tangential electric and magnetic fields as [10]

where

in which p=$k_y^2$ /ω ²(ε _z µ _z -κ ² ε ₀ µ ₀) and $\alpha =i\kappa \sqrt{{\epsilon }_0{\mu }_0}$. For plane waves in the biaxially chiral medium, we have ∂/∂zĒ (H̄)=ik _1z Ē(H̄). From Eq. (3), it is clear that the following relation always holds

where λ is the eigenvalue of matrix M̿. From Eqs. (5) and (6), the dispersion relations for eigenwaves within the biaxially chiral medium can be obtained after some lengthy algebra

where

in which a, b and c are constants related to medium parameters:

Here, q=ε _t µ _t +ε _z µ _z -ε _z µ _t -ε _t µ _z . When the chirality disappears, i.e., κ=0, Eq. (7) obviously turns into the following form

which are consistent with the dispersion relations for TM and TE waves in the normal anisotropic medium [3,5].

 

Fig. 1. A plane wave incident from free space to a biaxially anisotropic chiral medium.

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After some lengthy mathematic manipulations, we can obtain two eigenwaves in the biaxially chiral medium which are represented in terms of two vectors

where A is the amplitude of vectors, and

in which

From Eqs. (17)–(21), we can easily obtain the components of the Poynting vector S _y and S _1z in the chiral medium as

For nonmagnetic chiral medium with µ _t =µ _z =µ ₀, the eigenvectors of electric and magnetic fields are simplified as

in which

Then the Poynting-vector components in y and z directions can be obtained in simple forms as

where

From the basic electromagnetic theory, the power within the biaxially chiral medium must always flow in the positive z direction to satisfy the radiation condition. On the other hand, the transverse components of incident and refracted wave vectors must keep identical according to the boundary condition and the phase matching. When S _y has the same sign with k _y , a positive refraction occurs at the medium interface. On the contrary, an opposite sign of k _y and S _y implies that a negative refraction is supported on the boundary. Similarly, when S _1z has the same sign with k _1z, the eigenwaves within the biaxially chiral medium are propagating away from the medium interface. That is to say, they are travelling forwardly due to the positive longitudinal wavenumber k _1z. When they have the opposite signs, the eigenwaves will then travel toward the medium interface, and they are actually backward waves.

From Eqs. (16)–(34), the refraction and propagation properties of eigenwaves in the chiral medium are actually quite complicated. In the following analysis, we will focus on how such properties are affected by the variance of medium parameters. For simplicity, we assume κ>0 in the following discussions. For the case when κ<0, similar conclusions may be drawn as we can see later.

3. Theoretical Analysis and Numerical Simulations

First we consider a case when the host medium is a nonmagnetic isotropic dielectric with ε _t =ε _z >0 and µ _t =µ _z =µ ₀ . The corresponding dispersion equation is easily obtained from Eqs. (7)–(13) as

where ± stands for two eigenwaves, called as p ⁺ and p ^- waves. Clearly, when the chirality satisfies the condition $\kappa <\sqrt{{\epsilon }_t⁄{\epsilon }_0}$, both p ⁺ and p ^- waves exist in the chiral medium with a circular dispersion relation. When $\kappa >\sqrt{{\epsilon }_t⁄{\epsilon }_0}$, however, the p ^- wave becomes evanescent and cannot be supported by the chiral medium. Generally, the corresponding y and z components of the Poynting vector in the chiral half space are expressed as

Obviously, for the p ⁺ wave, S _y and S _1z always have the same sign with k _y and k _1z for all possible κ, indicating that neither negative refraction nor backward wave exists in such a chiral medium. For the p ^- wave, S _y and S _1z are still parallel with k _y and k _1z, respectively, when $\kappa <\sqrt{{\epsilon }_t⁄{\epsilon }_0}$. Again, the anomalous refraction and propagation do not exist. When κ is greater than $\sqrt{{\epsilon }_t⁄{\epsilon }_0}$, the p ^- wave can be refracted negatively at the interface. However it becomes evanescent within the chiral medium, as we have mentioned earlier.

The refraction and propagation properties of the two eigenwaves are demonstrated in Fig. 2, where the directions of Poynting vector S̄ and wave vector k̄ for incident and refracted waves have been plotted with $\kappa <\sqrt{{\epsilon }_t⁄{\epsilon }_0}$ and $\kappa >\sqrt{{\epsilon }_t⁄{\epsilon }_0}$, respectively. Note that the energy velocity is parallel to the phase velocity in Fig. 2 since the refraction angles for phase and energy flow satisfy tan^-1 S _1z/S _y =tan^-1 k _1z/k _y . As a consequence, we conclude that neither negative refraction nor backward wave exists in the propagating eigenwaves if the host medium is the nonmagnetic isotropic dielectric.

 

Fig. 2. Refractions of phase and energy flow for p ⁺ and p ^- waves on the interface of free space and the biaxially chiral medium. (a) $\kappa <\sqrt{{\epsilon }_t⁄{\epsilon }_0}$. (b) $\kappa >\sqrt{{\epsilon }_t⁄{\epsilon }_0}$.

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When the host medium is no longer an isotropic dielectric (ε _t ≠ε _z ) but it still keeps nonmagnetic (µ _t =µ _z =µ ₀), the dispersion equation becomes much more complicated. From Eq. (9), f may be imaginary since ap ²+bp+c may be less than zero with the change of incident angles. Therefore the transverse wavenumbers of the eigenwaves in the biaxially chiral medium may be complex, implying that such a wave would decay exponentially when propagating in the biaxially chiral medium. Moreover, the right-hand side of Eq. (7) may also be negative even when f>0 is satisfied, and the wave becomes totally evanescent.

The Poynting vector formed by S _y and S _1z may become complex from Eqs. (30) and (31), whose real part represents the electromagnetic power going away from the interface and imaginary part stands for the interchange of electric and magnetic energies. Since the transverse wavenumber k _y is continuous at the interface between free space and the chiral medium and has a positive real value as we have assumed in Fig. 1, the refraction property is only dependent on the sign of Re(γ) based on Eq. (30). When Re(γ)>0, the power flow S _y has the same direction as the transverse wavenumber k _y , which shows positive refraction at the medium interface and vice versa. Based on Eq. (31), however, the propagation properties of eigenwaves in the biaxially chiral medium are determined by Re(τ)·L(k _1z). Here, L is a function of k _1z which keeps zero if the real part of k _1z is zero and keeps unity otherwise. The reason we introduce the function L is because k _1z may become pure imaginary as we have mentioned earlier. It is meaningless to discuss the backward propagation for evanescent waves.

In order to understand the refraction and propagation properties of eigenwaves in the biaxially chiral medium better, we plot the curves for signs of Re(γ) and Re(τ)·L(k _1z) with respect to incident angles θ _i numerically. For simplicity, we define a new function as

and study the properties of α(γ) and α(τ)·L(k _1z) for different incident angles.

 

Fig. 3. The refraction and propagation properties of eigenwaves in the biaxially chiral medium with respect to incident angles, where ε _t =ε _z =2ε ₀ and µ _t =µ _z =µ ₀. (a) α(γ) for κ=0.2, (b) α(τ)·L(k _1z) for κ=0.2, (c) α(γ) for κ=1.5, (d) α(τ)·L(k _1z) for κ=1.5.

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When we choose ε _t =ε _z =2ε ₀ and κ=0.2, the corresponding curves of α(γ) and α(τ)·L(k _1z) are shown in Figs. 3(a) and 3(b). Obviously, both p ⁺ and p ^- waves are positively refracted at the interface and propagating forwardly in the biaxially chiral medium. When the chirality increases to κ=1.5, the α(γ) and α(τ) ·L(k _1z) curves are illustrated in Figs. 3(c) and 3(d), where the p ⁺ wave still keeps positive refraction and forward propagation. For the p ^- wave, however, it becomes evanescent since the real part of the Poynting vector S _1z turns to be zero although α(γ)<0 in Fig. 3(c). In this case, the p ^- wave are totally travelling along the interface. Numerical results are consistent with the theoretical analysis earlier.

Next we consider a case when the host medium is anisotropic, where all elements of the permittivity tensor are positive. When we choose ε _t =8ε ₀, ε _z =2ε₀, and κ=0.3, the curves of α(γ) and α(τ)·L(k _1z) are illustrated in Figs. 4(a) and 4(b). In such a case, both eigenwaves have positive refraction and forward propagation like that in Figs. 3(a) and 3(b). However, when κ becomes larger (κ=1.5), negative refractions may occur on the medium interface for both p ⁺ and p ^- waves, as illustrated in Fig. 4(c). From Fig. 4(c), we also notice that the p ^- wave changes to positive refraction as the incident angle θ _i is greater than 70°. The two eigenwaves are still forward waves under such a case, as shown in Fig. 4(d).

We remark that backward waves can also be supported using such an anisotropic dielectric as the host medium if we choose the dielectric parameters appropriately, which are not shown here for space reason. Hence the refraction and propagation properties are significantly different from those we have discussed in the earlier case.

 

Fig. 4. The refraction and propagation properties of eigenwaves in the biaxially chiral medium with respect to incident angles, where ε _t =8ε ₀, ε _z =2ε ₀, and µ _t =µ _z =µ ₀. (a) α(γ) for κ=0.3, (b) α(τ)·L(k _1z) for κ=0.3, (c) α(γ) for κ=1.5, (d) α(τ)·L(k _1z) for κ=1.5.

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When the host medium is selected as an anisotropic electric plasma, the refraction and propagation properties of the two eigenwaves in the biaxially chiral medium are of great interests. We have computed α(γ) and α(τ)·L(k _1z) at different incident angles, as demonstrated in Fig. 5. When ε _t =-2ε ₀, ε _z =ε ₀, and κ=0.3, the value of α(γ) is always equal to one for both p ⁺ and p ^- waves. That is to say, both eigenwaves are positively refracted at the medium interface, as shown in Fig. 5(a). Furthermore, from Fig. 5(b), the p ⁺ wave are backward wave when θ _i <35° or θ _i >75°. In the middle region of 35°<θ _i <75°, the p ⁺ wave is totally evanescent since the power flow along the z direction is always zero. For the p ^- wave, it is backward wave when the incident angle is less than 35°, and becomes evanescent for larger incident angles.

When κ increases to 1.5, both eigenwaves are refracted negatively as we can see from Fig. 5(c). These eigenwaves travel in the biaxially chiral medium backwardly under when the incident angle is less than 57° and become evanescent at larger incident angles, as shown in Fig. 5(d). We remark that the negative refraction and backward wave are supported simultaneously in such a case, which helps to realize the superlens effect.

For the electric plasma along the z direction, e.g. ε _t =2ε ₀, ε _z =-ε ₀, and κ=0.3, the p ⁺ wave experiences negative refraction and forward propagation, while the p ^- wave has negative refraction only when θ _i <23° and keeps forward propagation, as illustrated in Figs. 6(a) and 6(b). When κ increases to 0.8, only the refraction properties of the two eigenwaves vary with the change of incident angles (see Fig. 6(c)), while both eigenwaves keep forward propagation (see Fig. 6(d)).

 

Fig. 5. The refraction and propagation properties of eigenwaves in the biaxially chiral medium with respect to incident angles, where ε _t =-2ε ₀, ε _z =ε ₀, and µ _t =µ _z =µ ₀. (a) α(γ) for κ=0.3, (b) α(τ)·L(k _1z) for κ=0.3, (c) α(γ) for κ=1.5, (d) α(τ)·L(k _1z) for κ=1.5.

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Finally we consider a more general case where the chiral medium is both electrically and magnetically anisotropic, i. e., ε _t ≠ε _z , µ _t ≠µ _z . The propagation and refraction properties of eigenwaves in the chiral medium have been illustrated in Fig. 7, where we choose ε _t =2ε ₀, ε _z =-ε ₀, µ _t =2µ ₀, µ _z =µ ₀, κ=0.3 and κ=0.8, respectively. From Figs. 7(a) and 7(c), it is clear that there exist transition angles for both eigenwaves to turn from positive (negative) refraction to negative (positive) refraction. But they remain forward propagation in the chiral half space, as illustrated in Figs. 7(a) and 7(d). In comparison to Fig. 6, we find that the magnetic anisotropy mainly affects the refraction behaviors of the eigenwaves.

4. Conclusions

In conclusions, we have analyzed the refraction and propagation properties of a plane wave incident from free space into a biaxially anisotropic chiral medium. When the host medium is the conventional isotropic dielectric, we have proved that no negative refractions or/and backward waves exist for the propagating eigenwaves. When the host medium changes to the anisotropic dielectric or electric plasma, both negative refractions and backward waves may be supported. In some cases, they can even exist simultaneously.

Although the numerical results we have presented here show the possibility of negative refractions and backward waves in the biaxially chiral medium, we remark that the conclusion may not apply to the chiral materials with arbitrary parameters.

 

Fig. 6. The refraction and propagation properties of eigenwaves in the biaxially chiral medium with respect to incident angles, where ε _t =2ε ₀, ε _z =-ε ₀, and µ _t =µ _z =µ ₀. (a) α(γ) for κ=0.3, (b) α(τ)·L(k _1z) for κ=0.3, (c) α(γ) for κ=0.8, (d) α(τ)·L(k _1z) for κ=0.8.

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Fig. 7. The refraction and propagation properties of eigenwaves in the biaxially chiral medium with respect to incident angles, where ε _t =2ε ₀, ε _z =-ε ₀, and µ _t =2µ ₀, µ _z =µ ₀. (a) α(γ) for κ=0.3, (b) α(τ)·L(k _1z) for κ=0.3, (c) α(γ) for κ=0.8, (d) α(τ)·L(k _1z) for κ=0.8.

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