Negative refraction and backward wave in pseudochiral mediums: illustrations of Gaussian beams

Ruey-Lin Chern; Po-Han Chang

doi:10.1364/OE.21.002657

1. Introduction

Pseudochiral mediums are a special type of bianisotropic materials [1], whose properties are characterized by the magnetoelectric tensors [2]. The chirality parameters are purely imaginary as in chiral mediums. Their diagonal elements, however, are zero as in ordinary dielectrics. Therefore, the magnetoelectric couplings in pseudochiral mediums occur at mutually perpendicular directions rather than the parallel direction [3]. This feature is related to some optical properties that are different from either isotropic chiral or ordinary dielectric materials. In practice, the underlying magnetoelectric couplings in pseudochiral mediums can be achieved by surface currents induced on Ω-shaped metal wires [4, 5].

Due to symmetry breaking intrinsic in bianisotropic mediums, there exit two eigenwaves with elliptically polarizations. These waves are considered the most general form of eigenwaves in complex mediums [6]. From the perspective of constitutive relations, the chirality parameters in pseudochiral mediums play dual roles: magnetoelectric coupling and anisotropy. The former is the cross coupling effect that the polarization (magnetization) in the medium can be induced by the magnetic (electric) field. The latter is a directionally dependent property that causes the electric field no longer perpendicular to the wave vector, or the wave vector no longer parallel to the Poynting vector. The wave propagation in pseudochiral mediums thus behave in part as isotropic chiral mediums and in part as anisotropic dielectrics. In particular, negative refraction and backward wave that occur in anisotropic [7–9], biisotropic (or isotropic chiral) [10–12], and bianisotropic (or anisotropic chiral) materials [13, 14] may appear as well in pseudochiral mediums.

In this study, we investigate the phenomena of negative refraction and backward wave in pseudochiral mediums. In order to characterize these features in a more concise manner, we study the propagation of Gaussian beams based on Fourier integral formulations [15]. The eigenwaves and Fresnel equations for the reflection and transmission coefficients are employed to formulate the beams in terms of their spectrsal components. In practice, the resulting infinite integrals are performed by the Fourier transforms and accelerated by the FFT algorithm. Due to the magnetoelectric couplings in pseudochiral mediums, the transmitted waves are split into two waves with different handednesses. For a beam incident from vacuum onto a pseudochiral medium, negative refraction may occur for the right-handed beam, whereas backward wave may appear for the left-handed beam. These features are manifest on the power flow of beam centers in space and the movement of beam wavefronts in time.

2. Basic equations

2.1. Eigenwaves

Bianisotropic materials are an important class of complex mediums [6], which are characterized by the following constitutive relations:

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

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

where ε, μ, ξ, and ζ are in general tensors of complex quantities. In particular, the magneto-electric tensors for pseudochiral mediums have the following forms:

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

with γ being a real quantity. Consider the case where the anisotropy of the permittivity and permeability be small compared with the effect of chirality, and the material parameters be approximated by ε = εI and μ = μI, where I is the identity tensor. This is the case when the magnetoelectric coupling is much more significant than the dielectric-magnetic property of the medium. The dispersion relation for pseudochiral mediums is given as (see Appendix A for details)

ω = \frac{1}{ρ \sqrt{ε μ}} \sqrt{| k |^{2} \pm 2 \tilde{γ} | k_{x} k_{z} |} .

where

\tilde{γ} = | γ | / \sqrt{ε μ}

and

ρ = \sqrt{1 - {\tilde{γ}}^{2}}

. Assume that ε > 0, μ > 0, and γ² < ε μ. Therefore, 0 < γ̃ < 1 and 0 < ρ < 1. For a single frequency ω, there exist two wave numbers k^± = |k^±| in the pseudochiral medium.

Let the xy plane be an interface between vacuum and a pseudochiral medium. Consider a wave incident from vacuum with the wave vector lying on the xz plane. For a given tangential wave vector component k_x, there are two solutions for the normal wave vector component [cf. Eq. (4)]:

k_{z}^{\pm} = ρ \sqrt{h_{0}^{2} - k_{x}^{2}} \pm \tilde{γ} | k_{x} |,

where

h_{0} = ω \sqrt{ε μ}

. The eigenwaves in the pseudochiral medium are given as (see Appendix A for details)

E^{\pm} = E_{0}^{\pm} (\frac{h_{z}}{h_{0}}, \pm i ρ, - \frac{h_{z}}{h_{0}} σ^{\pm}), H^{\pm} = \mp \frac{i}{η} E^{\pm},

where

E_{0}^{+}

and

E_{0}^{-}

are the electric field magnitudes for right-handed elliptically polarized (REP) and left-handed elliptically polarized (LEP) waves, respectively,

h_{z} = \sqrt{h_{0}^{2} - k_{x}^{2}}

,

η = \sqrt{μ / ε}

, and

σ^{\pm} = \frac{k_{x}}{k_{z}^{\pm}} (1 - \frac{{\tilde{γ}}^{2} h_{0}^{2}}{h_{z}^{2}}) \mp sign (k_{x}) \tilde{γ} (1 - \frac{k_{x}^{2}}{h_{z}^{2}}) .

The time-averaged Poynting vectors,

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

, where * denotes the complex conjugate, are given as

〈 S^{\pm} 〉 = \frac{ρ h_{z}}{η h_{0}} {(E_{0}^{\pm})}^{2} (σ^{\pm} \hat{x} + \hat{z}) .

The angles of wave vectors k^± and Poynting vectors 〈S^±〉 with respect to the interface normal (the z axis) are given, respectively, as

θ^{\pm} = arccos (\frac{k_{z}^{\pm}}{k^{\pm}}), ϕ^{\pm} = arctan (σ^{\pm}) .

For a wave incident from vacuum onto a pseudochiral medium at an angle of incidence θ, negative refraction will occur for the REP wave (ϕ⁺ < 0°) when θ < θ_NR, where

θ_{NR} = arcsin (\frac{\tilde{γ} h_{0}}{k_{0}})

is the threshold angle for negative refraction. Note that when |k_x| = γ̃h₀, the time-averaged Poynting vector 〈S⁺〉 is directed to the interface normal (σ⁺ = 0) [cf. Eq. (8)]. On the other hand, backward wave will occur for the LEP wave (θ⁻ > 90°) when θ > θ_BW, where

θ_{BW} = arcsin (\frac{ρ h_{0}}{k_{0}})

is the threshold angle for backward wave. Note that when |k_x| = ρh₀, the wave vector k⁻ is oriented along the interface (

k_{z}^{-}

= 0) [cf. Eq. (5)].

2.2. Gaussian beams

Suppose that the xy plane is an interface between vacuum on the left (z < 0) and a pseudochiral medium on the right (z > 0), characterized by the permittivity ε, permeability μ, 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, as schematically shown in Fig. 1. 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 k_x as [15]

f (x, z) = \int_{- \infty}^{\infty} ψ (k_{x}) e^{{ik}_{x} x + {ik}_{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} - {ik}_{x} x_{0} + {ik}_{z} h],

where

k_{z} = \sqrt{k_{0}^{2} - k_{x}^{2}}

is the wave number component normal to the interface, 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^{{ik}_{x} x + {ik}_{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^{{ik}_{x} x + {ik}_{z} z} d k_{x},

where

E_{0}^{p, s}

are the electric field magnitudes for the p- or s-polarized incident waves,

η_{0} = \sqrt{\frac{μ_{0}}{ε_{0}}}

,

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

,

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

and

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

,

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

are the basis vectors of the incident electric and magnetic fields, respectively.

Fig. 1 Schematic diagram of the wave vectors and Poynting vectors for a Gaussian beam incident from vacuum (z < 0) onto a pseudochiral medium (z > 0) characterized by ε, μ, and γ, where (a) θ < θ_NR, θ < θ_BW and (b) θ_BW< θ < θ_NR.

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The reflected beams are formulated as

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

where

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

,

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

and

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

,

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

are the basis vectors of the reflected electric and magnetic fields, respectively, 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 Appendix B.

In the pseudochiral medium, there are two wave numbers: k^± [cf. Eq. (4)], corresponding to the REP and LEP waves. The transmitted beams are formulated as

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

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

where

η = \sqrt{\frac{μ}{ε}}

,

e_{t}^{\pm} = (\frac{h_{z}}{h_{0}}, \pm i ρ, - \frac{h_{z}}{h_{0}} σ^{\pm})

and

h_{t}^{\pm} = \mp i e_{t}^{\pm}

are the basis vectors of the transmitted electric and magnetic fields, respectively [cf. Eq. (6)], 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 Appendix B.

If the incidence beams are circularly polarized, we have

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

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

where

E_{0}^{+}

and

E_{0}^{-}

are the electric field magnitudes for the right-handed circularly polarized (RCP) and left-handed circularly polarized (LCP) incident waves, respectively, and

e_{i}^{\pm} = (\frac{k_{z}}{k_{0}}, \pm i, - \frac{k_{x}}{k_{0}})

and

h_{i}^{\pm} = \mp i e_{i}^{\pm}

are the basis vectors of the incident electric and magnetic fields, respectively. The corresponding reflected beams are formulated as

E_{r}^{\pm} = \int_{- \infty}^{\infty} e_{r}^{\pm} R^{\pm} ψ (k_{x}) e^{{ik}_{x} x - {ik}_{z} z} d k_{x},

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

where

e_{r}^{\pm} = (\frac{k_{z}}{k_{0}}, \mp i, \frac{k_{x}}{k_{0}})

and

h_{r}^{\pm} = \mp i e_{r}^{\pm}

are the basis vectors of the reflected electric and magnetic fields, respectively, and R^± are the reflection coefficients, given as

R^{+} = r_{+ +} E_{0}^{+} + r_{+ -} E_{0}^{-}

and

R^{-} = r_{- +} E_{0}^{+} + r_{- -} E_{0}^{-}

. Detailed formulas for r₊₊, r_+−,r₋₊, and r₋₋ are listed in Appendix B. The corresponding transmitted beams have the same forms as Eqs. (18) and (19), except that the transmission coefficients are changed to

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

and

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

. Detailed formulas for t₊₊, t₊₋, t₋₊, and t₋₋ are listed in Appendix B.

2.3. Fourier transforms

The Gaussian beams formulated as Fourier integrals in Eqs. (14)–(23) can be carried out by performing the Fourier transform (FT) and inverse Fourier transform (IFT) as

f (x, z) = IFT [FT [f_{0} (x)] ϕ (k_{x}) e^{i q_{z} z}],

where

f_{0} (x) = exp [- \frac{{(x - x_{0})}^{2} {cos}^{2} θ}{w_{0}^{2}} + {ik}_{0} sin θ (x - x_{0}) + {ik}_{z} h]

is the field distribution of the Gaussian beam at the interface (z = 0), ϕ (k_x) is the function of k_x that contains the components of basis vectors (

e_{i, r}^{p, s}

,

e_{i, r, t}^{\pm}

,

h_{i, r}^{p, s}

,

h_{i, r, t}^{\pm}

) and the reflection and transmission coefficients (R^p,s, R^±, T^±), q_z = k_z (−k_z) for the incident (reflected) beam, and

q_{z} = k_{z}^{\pm}

for the transmitted beam. FT and IFT are defined, respectively, as

FT [f (x)] = F (k_{x}) \equiv \frac{1}{\sqrt{2 π}} \int_{- \infty}^{\infty} f (x) e^{- {ik}_{x} x} d x,

IFT [F (k_{x})] = f (x) \equiv \frac{1}{\sqrt{2 π}} \int_{- \infty}^{\infty} F (k_{x}) e^{{ik}_{x} x} d k_{x},

In Eq. (24), FT[f₀ (x)] is to obtain the spectral components of the fields at the interface, and IFT is to restore these components to the physical quantity f (x,z), taking into account the wave propagation in the z direction and the influence of the interface (reflection and transmission). In practice, the calculations of FT and IFT can be significantly accelerated by employing the FFT algorithm.

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 pseudochiral medium.

3. Results and discussion

Figure 2(a) shows the normalized power intensity for a p-polarized (TM) Gaussian beam incident from vacuum onto a pseudochiral medium with ε/ε₀ = ε_r = 2, μ/μ₀ = μ_r = 1, and γ = 0.2. The incident beam center makes an angle θ = 20° with respect to the interface normal (the z axis). There are two transmitted beams in the pseudochiral medium, one is the REP wave with a smaller angle of the beam center, and the other is the LEP wave with a larger angle. The beam centers coincide with the time-averaged Poynting vectors 〈S^±〉 [cf. Eq. (8)], with ϕ⁺ ≈ 6° and ϕ⁻ ≈ 21.2°. Note that in this case, θ > θ_NR ≈ 11.5° [cf. Eq. (10)] and ordinary refraction is expected to occur. As the chirality parameter γ increases, the REP beam tends to move toward the interface, whereas the LEP beam tends to move away from it. In this configuration, the reflection coefficients r_pp and r_sp are relatively small. Compared to the incident beam, the intensity of the reflected beam is insignificant.

Fig. 2 Power intensities for a p-polarized (TM) Gaussian beam incident from vacuum at θ = 20° onto a pseudochiral medium with ε_r = 2, μ_r = 1, and (a) γ = 0.2 (b) γ = 0.8. The intensities are normalized to have a maximum value of unity. White solid line denotes the interface. Black dashed lines indicate the beam centers.

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In Fig. 2(b), the chirality parameter is increased to γ = 0.8, with the other material parameters unchanged. The REP (lower) beam is located on the same side of the incident beam, with a negative angle of refraction: ϕ⁺ ≈ −19.8°. Note that in this case, θ < θ_NR ≈ 53.1° [cf. Eq. (10)] and negative refraction is expected to occur. The LEP (upper) beam has an angle of refraction: ϕ⁻ ≈ 37.6°, which is larger than that in Fig. 2(a). For a s-polarized incident Gaussian beam, the features stated above are similar.

Figure 3(a) shows the incidence of a RCP Gaussian beam from vacuum at θ = 20° onto a pseudochiral medium with the same material parameters as in Fig. 2(b). In this case, the transmission is dominated by the REP beam, which is negatively refracted with the same ϕ⁺ as in Fig. 2(b). Compared to the REP beam, the power intensity of the LEP beam is not significant (0.037%). In Fig. 3(b), the incident beam is changed to a LCP Gaussian beam. The transmission, on the other hand, is dominated by the LEP beam, which is positively refracted with the same ϕ⁻ as in Fig. 2(b). Compared to the LEP beam, the power intensity of the REP beam is not significant (0.037%).

Fig. 3 Power intensities for a (a) RCP and (b) LCP Gaussian beam incident from vacuum at θ = 20° onto a pseudochiral medium with the same material parameters as in Fig. 2(b).

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A special feature of the Gaussian beam incident from vacuum at θ = 0° onto a pseudochiral medium is shown in Fig. 4(a). The transmitted wave is split into two beams even at normal incidence. This feature does not appear in either isotropic chiral or anisotropic dielectric materials, and can be manifest on the time-averaged Poynting vectors 〈S^±〉 at k_x = 0. In this situation, σ^± = ∓γ̃ [cf. Eq. (7)] and ϕ^± = arctan(∓γ̃) [cf. Eq. (9)]. In the presence of chirality parameter γ̃, ϕ⁺ ≠ ϕ⁻ at θ = 0°. The effect of γ̃ on ϕ^± is plotted in Fig. 4(b). The splitting angle between the two transmitted beams increases with γ̃. This feature is further analyzed by examining the basis wave functions that compose the Gaussian beams:

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

Using Eq. (5) for

k_{z}^{\pm}

, the above equation can be integrated to give

f^{\pm} (x, z) = \frac{w_{0}}{2 w} {e^{- \frac{{(x + \tilde{γ} z)}^{2}}{w^{2}}} [1 \pm i erfi (\frac{x + \tilde{γ} z}{w})] + e^{- \frac{{(x - \tilde{γ} z)}^{2}}{w^{2}}} [1 \mp i erfi (\frac{x - \tilde{γ} z}{w})]} e^{i ρ h_{0} z},

where

w = \sqrt{w_{0}^{2} + i 2 ρ z / h_{0}}

and erfi (x) = erf (ix)/i is the imaginary error function. In Eq. (29), f^± (x,z) attain their local maximal values along x = ±γ̃z, which are the locations of beam centers at θ = 0°.

Fig. 4 (a) Power intensities for a p-polarized Gaussian beam incident from vacuum at θ = 0° onto the pseudochiral medium with ε_r = 1, μ_r = 1, and γ̃ = 0.6. (b) Effect of the chirality parameter γ̃ on the angles of Poynting vectors for REP and LEP waves.

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Finally, an instance of backward wave is shown in Fig. 5 ( Media 1) for a s-polarized Gaussian beam incident from vacuum at θ = 25° onto the pseudochiral medium with ε_r = 1, μ_r = 1, and γ̃ = 0.95. In this case, the angles of wave vectors with respect to the interface normal for the REP and LEP waves are θ⁺ ≈ 31.7° and θ⁻ ≈ 105.7°, respectively. The latter represents a backward wave. The wavefronts of the LEP (upper) beam move toward the interface rather than away from it, which is indicated by the orientation of k⁻. The angles of time-averaged Poynting vectors with respect to the interface normal for the REP and LEP waves are ϕ⁺ ≈ −38.8° and ϕ⁻ ≈ 47.6°, respectively. The former corresponds to negative refraction. Note that the threshold angles are θ_NR ≈ 71.8° and θ_BW ≈ 18.2° [cf. Eqs. (10) and (11)]. Accordingly, both negative refraction and backward wave occur in the same configuration.

Fig. 5 Electric field (E_y) for a s-polarized Gaussian beam incident from vacuum at θ = 25° onto a pseudochiral medium with ε_r = 1, μ_r = 1, and γ̃ = 0.95 ( Media 1).

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4. Concluding remarks

In conclusion, we have investigated the phenomena of negative refraction and backward wave in pseudochiral mediums, with illustrations of Gaussian beams. The physical origin of these features comes from the symmetry breaking intrinsic in pseudochiral mediums. There exist two elliptically polarized eigenwaves, with the wave vectors no longer parallel to the Poynting vectors. The right- and left-handed waves behave in a very different manner, which is more evident as the chirality parameter increases. These features are well illustrated with Gaussian beams based on Fourier integral formulations for the incident, reflected, and transmitted waves.

Appendix

A. Dispersion relation and eigenwaves

Using the constitutive relations (1) and (2) in Maxwell’s equations: ∇ ×E = iωB, ∇ × H = −iωD and eliminate 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 mediums. Assume that E and H are of the form e^ik·r, the wave equations are rewritten as ME = 0 and NH = 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 I being the identity tensor. The existence of nontrivial solutions for E and H fields requires that |M| = 0 and |N| = 0. Using k = (k_x, k_y, k_z) and the chirality parameters (3), the zero determinant gives rise to the characteristic equation:

ε^{2} μ^{2} ρ^{4} ω^{4} - 2 ε μ ρ^{2} | k |^{2} ω^{2} + | k |^{4} - 4 {\tilde{γ}}^{2} k_{x}^{2} k_{z}^{2} = 0,

where

\tilde{γ} = | γ | / \sqrt{ε μ}

and

ρ = \sqrt{1 - {\tilde{γ}}^{2}}

. The above equation is then solved to give the dispersion relation:

ω = \frac{1}{ρ \sqrt{ε μ}} \sqrt{| k |^{2} \pm 2 \tilde{γ} | k_{x} k_{z} |} .

The nullspace of M or N gives the eigenwaves as

E^{\pm} = (\frac{h_{z}}{h_{0}}, \pm i ρ, - \frac{h_{z}}{h_{0}} σ^{\pm}), H^{\pm} = \mp \frac{i}{η} E^{\pm},

where

η = \sqrt{μ / ε}

,

h_{0} = ω \sqrt{ε μ}

,

h_{z} = \sqrt{h_{0}^{2} - k_{x}^{2}}

, and

σ^{\pm} = \frac{k_{x}}{k_{z}^{\pm}} (1 - \frac{{\tilde{γ}}^{2} h_{0}^{2}}{h_{z}^{2}}) \mp sign (k_{x}) \tilde{γ} (1 - \frac{k_{x}^{2}}{h_{z}^{2}}) .

B. Reflection and transmission coefficients

Consider a planar interface between vacuum and a pseudochiral medium. The continuity of the tangential electric and magnetic fields at the interface gives rise to the following relations between the incident, reflected, and transmitted waves 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{α - ρ β}{α + ρ β}, r_{p s} = 0, r_{s p} = 0, r_{s s} = \frac{ρ - α B}{ρ + α β},

t_{+ p} = \frac{+}{α + ρ β}, t_{+ s} = - \frac{i}{ρ + α β}, t_{- p} = \frac{1}{α + ρ β}, t_{- s} = - \frac{i}{ρ + α β} .

with

α = \frac{cos θ^{0}}{cos θ}

,

β = \frac{η_{1}}{η_{2}}

, and

cos θ^{0} = \frac{h_{z}}{h_{0}}

. If the incident wave is circularly polarized, the relations between the incident, reflected, and transmitted waves are given by

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

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

where

r_{+ +} = r_{- -} = \frac{ρ α (1 - β^{2})}{(α + ρ β) (ρ + α β)}, r_{+ -} = r_{- +} = \frac{β (α^{2} - ρ^{2})}{(α + ρ β) (ρ + α β)},

t_{+ +} = t_{- -} = \frac{1}{α + ρ β} + \frac{1}{ρ + α β}, t_{+ -} = t_{- +} = \frac{1}{α + ρ β} - \frac{1}{ρ + α β} .

Acknowledgments

This work was supported in part by National Science Council of the Republic of China under Contract No. NSC 99-2221-E-002-121-MY3.

References and links

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Negative refraction and backward wave in pseudochiral mediums: illustrations of Gaussian beams

Abstract

1. Introduction

2. Basic equations

2.1. Eigenwaves

2.2. Gaussian beams

2.3. Fourier transforms

3. Results and discussion

4. Concluding remarks

Appendix

A. Dispersion relation and eigenwaves

B. Reflection and transmission coefficients

Acknowledgments

References and links

Supplementary Material (1)

Cited By

Figures (5)

Equations (45)

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