## 1.

## Introduction

The modest aim of this communication is to theoretically show that a homogeneous anisotropic dielectric slab with topologically insulating surface states (TISS)^{1}^{,}^{2} reflects and transmits light in such a way as to exhibit asymmetry with respect to reversal of projection of the propagation direction of the incident plane wave on the illuminated surface of the slab. We refer to this phenomenon as left/right asymmetry.

A homogeneous anisotropic material is characterized by a frequency-dependent relative permittivity dyadic $\underset{\_}{\underset{\_}{\u03f5}}$. Suppose that this material occupies the region ${\mathcal{V}}_{\mathrm{in}}$ bounded by the surface $\mathcal{S}$, which separates ${\mathcal{V}}_{\mathrm{in}}$ from the vacuous region ${\mathcal{V}}_{\mathrm{out}}$. If the anisotropic material possesses TISS, then the boundary conditions^{3}

## (1)

$$\begin{array}{l}\widehat{\mathbf{n}}({\mathbf{r}}_{S})\times [{\mathbf{E}}_{\mathrm{out}}({\mathbf{r}}_{S})-{\mathbf{E}}_{\mathrm{in}}({\mathbf{r}}_{S})]=\mathbf{0}\\ \widehat{\mathbf{n}}({\mathbf{r}}_{S})\times [{\mathbf{H}}_{\mathrm{out}}({\mathbf{r}}_{S})-{\mathbf{H}}_{\mathrm{in}}({\mathbf{r}}_{S})]=-\tilde{\gamma}\widehat{\mathbf{n}}({\mathbf{r}}_{S})\times {\mathbf{E}}_{\mathrm{in}}({\mathbf{r}}_{S})\end{array}\},\phantom{\rule[-0.0ex]{1em}{0.0ex}}{\mathbf{r}}_{S}\in \mathcal{S},$$^{4}5.6.

^{–}

^{7}as well as experimentally,

^{2}

^{,}

^{8}this communication is possibly the first report of optical scattering by an anisotropic dielectric material with TISS. The existence of such materials is deemed possible because the isotropic materials with TISS are chalcogenides,

^{1}

^{,}

^{2}columnar thin films (CTFs) of other chalcogenides have been fabricated,

^{9}and CTFs function as anisotropic dielectric materials at sufficiently low frequencies.

^{10}Furthermore, periodically multilayered composite materials

^{11}

^{,}

^{12}comprising laminas of an isotropic topological insulator and some other material should function as effectively anisotropic continuums at sufficiently low frequencies.

^{13}

We have found that the TISS induce the exhibition of left/right asymmetry in reflection and transmission by a homogeneous anisotropic dielectric slab. This asymmetry could be exploited for one-way optical devices. The boundary-value problem of reflection and transmission of an obliquely incident plane wave by a homogeneous anisotropic dielectric slab with TISS is described and solved in Sec. 2. Illustrative numerical results are presented and discussed in Sec. 3.

The free-space wavenumber, the free-space wavelength, and the intrinsic impedance of free space are denoted by ${k}_{0}=\omega \sqrt{{\u03f5}_{0}{\mu}_{0}}$, ${\lambda}_{0}=2\pi /{k}_{0}$, and ${\eta}_{0}=\sqrt{{\mu}_{0}/{\u03f5}_{0}}$, respectively, with ${\mu}_{0}$ and ${\u03f5}_{0}$ being the permeability and permittivity of free space. We denote the fine structure constant by $\tilde{\alpha}=({q}_{e}^{2}{\eta}_{0})/2\tilde{h}$, where ${q}_{e}$ is the quantum of charge and $\tilde{h}$ is the Planck constant. Vectors are in boldface, dyadics are underlined twice, column vectors are in boldface and enclosed within square brackets, while matrixes are underlined twice and similarly bracketed. Cartesian unit vectors are identified as ${\widehat{\mathbf{u}}}_{x}$, ${\widehat{\mathbf{u}}}_{y}$, and ${\widehat{\mathbf{u}}}_{z}$.

## 2.

## Theory

Suppose that the regions ${\mathcal{V}}_{\mathrm{in}}=\{(x,y,z):z\in (0,L)\}$ and ${\mathcal{V}}_{\mathrm{out}}=\{(x,y,z):z\notin [0,L]\}$ are separated by the surface $\mathcal{S}=\{(x,y,z):z\in \{0,L\}\}$.

A plane wave, propagating in the half-space $z<0$ at an angle $\theta \in [0,\pi /2)$ to the $z$-axis and at an angle $\psi \in [\mathrm{0,2}\pi )$ to the $x$-axis in the $xy$ plane, is incident on the slab, as shown in Fig. 1. The electromagnetic field phasors associated with the incident plane wave are represented as

## (2)

$$\begin{array}{l}{\mathbf{E}}_{\mathrm{inc}}(r)=({a}_{s}\mathbf{s}+{a}_{p}{\mathbf{p}}_{+})\mathrm{exp}[i\kappa (x\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\psi +y\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\psi )+i{k}_{0}z\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta ]\\ {\mathbf{H}}_{\mathrm{inc}}(r)=\frac{1}{{\eta}_{0}}({a}_{s}{\mathbf{p}}_{+}-{a}_{p}\mathbf{s})\mathrm{exp}[i\kappa (x\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\psi +y\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\psi )+i{k}_{0}z\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta ]\end{array}\},\phantom{\rule[-0.0ex]{1em}{0.0ex}}z<0.$$## (3)

$$\begin{array}{l}\kappa ={k}_{0}\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\theta ,\phantom{\rule[-0.0ex]{1em}{0.0ex}}\mathbf{s}=-{\widehat{\mathbf{u}}}_{x}\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\psi +{\widehat{\mathbf{u}}}_{y}\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\psi \\ {\mathbf{p}}_{\pm}=\mp ({\widehat{\mathbf{u}}}_{x}\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\psi +{\widehat{\mathbf{u}}}_{y}\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\psi )\mathrm{cos}\text{\hspace{0.17em}}\theta +{\widehat{\mathbf{u}}}_{z}\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\theta \end{array}\}.$$The reflected electromagnetic field phasors are expressed as

## (4)

$$\begin{array}{l}{\mathbf{E}}_{\mathrm{ref}}(r)=({r}_{s}\mathbf{s}+{r}_{p}{\mathbf{p}}_{-})\mathrm{exp}[i\kappa (x\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\psi +y\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\psi )-i{k}_{0}z\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta ]\\ {\mathbf{H}}_{\mathrm{ref}}(r)=\frac{1}{{\eta}_{0}}({r}_{s}{\mathbf{p}}_{-}-{r}_{p}\mathbf{s})\mathrm{exp}[i\kappa (x\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\psi +y\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\psi )-i{k}_{0}z\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta ]\end{array}\},\phantom{\rule[-0.0ex]{1em}{0.0ex}}z<0,$$## (5)

$$\begin{array}{l}{\mathbf{E}}_{\mathrm{tr}}(r)=({t}_{s}\mathbf{s}+{t}_{p}{\mathbf{p}}_{+})\mathrm{exp}[i\kappa (x\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\psi +y\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\psi )+i{k}_{0}(z-L)\mathrm{cos}\text{\hspace{0.17em}}\theta ]\\ {\mathbf{H}}_{\mathrm{tr}}(r)=\frac{1}{{\eta}_{0}}({t}_{s}{\mathbf{p}}_{+}-{t}_{p}\mathbf{s})\mathrm{exp}[i\kappa (x\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\psi +y\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\psi )+i{k}_{0}(z-L)\mathrm{cos}\text{\hspace{0.17em}}\theta ]\end{array}\},\phantom{\rule[-0.0ex]{1em}{0.0ex}}z>L.$$The frequency-domain electromagnetic constitutive relations of the homogeneous anisotropic dielectric material in ${\mathcal{V}}_{\mathrm{in}}$ can be written as^{10}

## (6)

$$\mathbf{D}(\mathbf{r})={\u03f5}_{0}\underset{\_}{\underset{\_}{\u03f5}}\xb7\mathbf{E}(\mathbf{r}),\phantom{\rule[-0.0ex]{1em}{0.0ex}}\mathbf{B}(\mathbf{r})={\mu}_{0}\mathbf{H}(\mathbf{r}),\phantom{\rule[-0.0ex]{1em}{0.0ex}}z\in (0,L),$$## (7)

$$\underset{\_}{\underset{\_}{\u03f5}}={\underset{\_}{\underset{\_}{S}}}_{y}\xb7({\u03f5}_{a}{\widehat{\mathbf{u}}}_{z}{\widehat{\mathbf{u}}}_{z}+{\u03f5}_{b}{\widehat{\mathbf{u}}}_{x}{\widehat{\mathbf{u}}}_{x}+{\u03f5}_{c}{\widehat{\mathbf{u}}}_{y}{\widehat{\mathbf{u}}}_{y})\xb7{\underset{\_}{\underset{\_}{S}}}_{y}^{T},$$## (8)

$${\underset{\_}{\underset{\_}{S}}}_{y}=({\widehat{\mathbf{u}}}_{x}{\widehat{\mathbf{u}}}_{x}+{\widehat{\mathbf{u}}}_{z}{\widehat{\mathbf{u}}}_{z})\mathrm{cos}\text{\hspace{0.17em}}\chi +({\widehat{\mathbf{u}}}_{z}{\widehat{\mathbf{u}}}_{x}-{\widehat{\mathbf{u}}}_{x}{\widehat{\mathbf{u}}}_{z})\mathrm{sin}\text{\hspace{0.17em}}\chi +{\widehat{\mathbf{u}}}_{y}{\widehat{\mathbf{u}}}_{y}$$^{14}or the manufactured CTFs.

^{10}

In ${\mathcal{V}}_{\mathrm{in}}$, the electric and magnetic field phasors can be represented as^{15}

## (9)

$$\begin{array}{l}\mathbf{E}(\mathbf{r})=\mathbf{e}(z)\mathrm{exp}[i\kappa (x\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\psi +y\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\psi )]\\ \mathbf{H}(\mathbf{r})=\mathbf{h}(z)\mathrm{exp}[i\kappa (x\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\psi +y\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\psi )]\end{array}\},$$## (10)

$$\frac{d}{dz}[\mathbf{f}(z)]=i[\underset{\_}{\underset{\_}{P}}]\xb7[\mathbf{f}(z)],\phantom{\rule[-0.0ex]{2em}{0.0ex}}z\in {\mathcal{V}}_{\mathrm{in}},$$## (12)

$$[\underset{\_}{\underset{\_}{P}}]=\omega \left[\begin{array}{cccc}0& 0& 0& {\mu}_{0}\\ 0& 0& -{\mu}_{0}& 0\\ 0& -{\u03f5}_{0}{\u03f5}_{c}& 0& 0\\ {\u03f5}_{0}{\u03f5}_{d}& 0& 0& 0\end{array}\right]+\kappa \frac{{\u03f5}_{d}({\u03f5}_{a}-{\u03f5}_{b})}{2{\u03f5}_{a}{\u03f5}_{b}}\mathrm{sin}(2\chi )\left[\begin{array}{cccc}\mathrm{cos}\text{\hspace{0.17em}}\psi & 0& 0& 0\\ \mathrm{sin}\text{\hspace{0.17em}}\psi & 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& -\mathrm{sin}\text{\hspace{0.17em}}\psi & \mathrm{cos}\text{\hspace{0.17em}}\psi \end{array}\right]\phantom{\rule{0ex}{0ex}}+\frac{{\kappa}^{2}}{\omega {\u03f5}_{0}}\frac{{\u03f5}_{d}}{{\u03f5}_{a}{\u03f5}_{b}}\left[\begin{array}{cccc}0& 0& \mathrm{cos}\text{\hspace{0.17em}}\psi \text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\psi & -{\mathrm{cos}}^{2}\text{\hspace{0.17em}}\psi \\ 0& 0& {\mathrm{sin}}^{2}\text{\hspace{0.17em}}\psi & -\mathrm{cos}\text{\hspace{0.17em}}\psi \text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\psi \\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]\phantom{\rule{0ex}{0ex}}+\frac{{\kappa}^{2}}{\omega {\mu}_{0}}\left[\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ -\mathrm{cos}\text{\hspace{0.17em}}\psi \text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\psi & {\mathrm{cos}}^{2}\text{\hspace{0.17em}}\psi & 0& 0\\ -{\mathrm{sin}}^{2}\text{\hspace{0.17em}}\psi & \mathrm{cos}\text{\hspace{0.17em}}\psi \text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\psi & 0& 0\end{array}\right],$$## (13)

$${\u03f5}_{d}=\frac{{\u03f5}_{a}{\u03f5}_{b}}{{\u03f5}_{a}\text{\hspace{0.17em}}{\mathrm{cos}}^{2}\text{\hspace{0.17em}}\chi +{\u03f5}_{b}\text{\hspace{0.17em}}{\mathrm{sin}}^{2}\text{\hspace{0.17em}}\chi}.$$Equation (10) has the following straightforward solution:

## (14)

$$[\mathbf{f}({L}^{-})]=\mathrm{exp}\{i[\underset{\_}{\underset{\_}{P}}]L\}\xb7[\mathbf{f}({0}^{+})],$$## (17)

$$[\underset{\_}{\underset{\_}{V}}]=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ -\tilde{\gamma}& 0& 1& 0\\ 0& -\tilde{\gamma}& 0& 1\end{array}\right].$$## (18)

$$[\mathbf{f}({L}^{+})]=[\underset{\_}{\underset{\_}{V}}]\xb7\mathrm{exp}\{i[\underset{\_}{\underset{\_}{P}}]L\}\xb7{[\underset{\_}{\underset{\_}{V}}]}^{-1}\xb7[\mathbf{f}({0}^{-})].$$But the elements of $[\mathbf{f}({0}^{-})]$ are known by virtue of Eqs. (2) and (4), and those of $[\mathbf{f}({L}^{+})]$ by virtue of Eq. (5). Accordingly, Eq. (18) may be written as

## (19)

$$\left[\begin{array}{c}{t}_{s}\\ {t}_{p}\\ 0\\ 0\end{array}\right]={[\underset{\_}{\underset{\_}{K}}]}^{-1}\xb7[\underset{\_}{\underset{\_}{V}}]\xb7\mathrm{exp}\{i[\underset{\_}{\underset{\_}{P}}]L\}\xb7{[\underset{\_}{\underset{\_}{V}}]}^{-1}\xb7[\underset{\_}{\underset{\_}{K}}]\xb7\left[\begin{array}{c}{a}_{s}\\ {a}_{p}\\ {r}_{s}\\ {r}_{p}\end{array}\right],$$## (20)

$$[\underset{\_}{\underset{\_}{K}}]=\left[\begin{array}{cccc}-\mathrm{sin}\text{\hspace{0.17em}}\psi & -\mathrm{cos}\text{\hspace{0.17em}}\psi \text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta & -\mathrm{sin}\text{\hspace{0.17em}}\psi & \mathrm{cos}\text{\hspace{0.17em}}\psi \text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta \\ \mathrm{cos}\text{\hspace{0.17em}}\psi & -\mathrm{sin}\text{\hspace{0.17em}}\psi \text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta & \mathrm{cos}\text{\hspace{0.17em}}\psi & \mathrm{sin}\text{\hspace{0.17em}}\psi \text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta \\ -\left(\frac{1}{{\eta}_{0}}\right)\mathrm{cos}\text{\hspace{0.17em}}\psi \text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta & \left(\frac{1}{{\eta}_{0}}\right)\mathrm{sin}\text{\hspace{0.17em}}\psi & \left(\frac{1}{{\eta}_{0}}\right)\mathrm{cos}\text{\hspace{0.17em}}\psi \text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta & \left(\frac{1}{{\eta}_{0}}\right)\mathrm{sin}\text{\hspace{0.17em}}\psi \\ -\left(\frac{1}{{\eta}_{0}}\right)\mathrm{sin}\text{\hspace{0.17em}}\psi \text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta & -\left(\frac{1}{{\eta}_{0}}\right)\mathrm{cos}\text{\hspace{0.17em}}\psi & \left(\frac{1}{{\eta}_{0}}\right)\mathrm{sin}\text{\hspace{0.17em}}\psi \text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta & -\left(\frac{1}{{\eta}_{0}}\right)\mathrm{cos}\text{\hspace{0.17em}}\psi \end{array}\right].$$The solution of Eq. (19) yields the reflection and transmission coefficients that appear as the elements of the $2\times 2$ matrixes in the following relations:

## (21)

$$\left[\begin{array}{c}{r}_{s}\\ {r}_{p}\end{array}\right]=\left[\begin{array}{cc}{r}_{ss}& {r}_{sp}\\ {r}_{ps}& {r}_{pp}\end{array}\right]\left[\begin{array}{c}{a}_{s}\\ {a}_{p}\end{array}\right],\phantom{\rule[-0.0ex]{2em}{0.0ex}}\left[\begin{array}{c}{t}_{s}\\ {t}_{p}\end{array}\right]=\left[\begin{array}{cc}{t}_{ss}& {t}_{sp}\\ {t}_{ps}& {t}_{pp}\end{array}\right]\left[\begin{array}{c}{a}_{s}\\ {a}_{p}\end{array}\right].$$## 3.

## Numerical Results and Discussion

Let the left side of the $xy$ plane be specified by $\psi \in [0,\pi ]$ and the right side by $\psi \in [\pi ,2\pi ]$. In order to delineate the characteristics of and the factors responsible for left/right asymmetry of reflection and transmission, we need to consider four distinct cases as follows.

Case I: Suppose that ${\u03f5}_{a}={\u03f5}_{b}={\u03f5}_{c}$ and $\tilde{\gamma}=0$. Then the material in ${\mathcal{V}}_{\mathrm{in}}$ is a homogeneous isotropic dielectric material and the TISS are absent. The boundary-value problem then turns into a textbook reflection/transmission problem.^{16} None of the four reflectances (${R}_{ss}$, ${R}_{ps}$, ${R}_{pp}$, and ${R}_{ss}$) and the four transmittances (${T}_{ss}$, ${T}_{ps}$, ${T}_{pp}$, and ${T}_{ss}$) then depend on $\psi $. Furthermore, the cross-polarized remittances are null valued. In other words, the following relationships hold:

## (22)

$$\begin{array}{ll}{R}_{ss}(\theta ,\psi )={R}_{ss}(\theta ,0),& {R}_{pp}(\theta ,\psi )={R}_{pp}(\theta ,0)\\ {T}_{ss}(\theta ,\psi )={T}_{ss}(\theta ,0),& {T}_{pp}(\theta ,\psi )={T}_{pp}(\theta ,0)\\ {R}_{ps}(\theta ,\psi )={R}_{sp}(\theta ,\psi )\equiv 0,& {T}_{ps}(\theta ,\psi )={T}_{sp}(\theta ,\psi )\equiv 0\end{array}\}.$$Case II: Suppose next that ${\u03f5}_{a}={\u03f5}_{b}={\u03f5}_{c}$ but $\tilde{\gamma}\ne 0$. Then the material in ${\mathcal{V}}_{\mathrm{in}}$ is a homogeneous isotropic dielectric material with TISS. None of the eight remittances (${R}_{ss}$, and so on, and ${T}_{ss}$, and so on) then depend on $\psi $, and the cross-polarized remittances are not identically zero. Analysis of numerical results reveals that the following relationships hold:

## (23)

$$\begin{array}{ll}{R}_{ss}(\theta ,\psi )={R}_{ss}(\theta ,0),& {R}_{pp}(\theta ,\psi )={R}_{pp}(\theta ,0)\\ {T}_{ss}(\theta ,\psi )={T}_{ss}(\theta ,0),& {T}_{pp}(\theta ,\psi )={T}_{pp}(\theta ,0)\\ {R}_{ps}(\theta ,\psi )={R}_{sp}(\theta ,\psi )\not\equiv 0,& {T}_{ps}(\theta ,\psi )={T}_{sp}(\theta ,\psi )\not\equiv 0\end{array}\}.$$Case III: Suppose that ${\u03f5}_{a}$, ${\u03f5}_{b}$, and ${\u03f5}_{c}$ are all dissimilar, but $\tilde{\gamma}=0$. Then the material in ${\mathcal{V}}_{\mathrm{in}}$ is a homogeneous anisotropic dielectric material and the TISS are absent. Calculations then show the following symmetries:

## (24)

$$\begin{array}{ll}{R}_{ss}(\theta ,\psi )={R}_{ss}(\theta ,\psi +\pi ),& {R}_{pp}(\theta ,\psi )={R}_{pp}(\theta ,\psi +\pi )\\ {R}_{ps}(\theta ,\psi )={R}_{sp}(\theta ,\psi +\pi )\not\equiv 0,& {T}_{ps}(\theta ,\psi )={T}_{sp}(\theta ,\psi )\not\equiv 0\end{array}\}.$$## (25)

$$\begin{array}{ll}{R}_{ps}(\theta ,\psi )\ne {R}_{ps}(\theta ,\psi +\pi ),& {R}_{sp}(\theta ,\psi )\ne {R}_{sp}(\theta ,\psi +\pi )\\ {T}_{ss}(\theta ,\psi )\ne {T}_{ss}(\theta ,\psi +\pi ),& {T}_{pp}(\theta ,\psi )\ne {T}_{pp}(\theta ,\psi +\pi )\\ {T}_{ps}(\theta ,\psi )\ne {T}_{ps}(\theta ,\psi +\pi ),& {T}_{sp}(\theta ,\psi )\ne {T}_{sp}(\theta ,\psi +\pi )\end{array}\}.$$Case IV: Finally, ${\u03f5}_{a}$, ${\u03f5}_{b}$, and ${\u03f5}_{c}$ are all dissimilar and $\tilde{\gamma}\ne 0$, so that the material in ${\mathcal{V}}_{\mathrm{in}}$ is a homogeneous anisotropic dielectric material with TISS. All eight remittances then depend on $\psi $, the cross-polarized remittances are not identically zero, and only one relationship can be found.

All eight remittances exhibit left/right asymmetry, i.e.,## (27)

$$\begin{array}{ll}{R}_{ss}(\theta ,\psi )\ne {R}_{ss}(\theta ,\psi +\pi ),& {R}_{pp}(\theta ,\psi )\ne {R}_{pp}(\theta ,\psi +\pi )\\ {R}_{ps}(\theta ,\psi )\ne {R}_{ps}(\theta ,\psi +\pi ),& {R}_{sp}(\theta ,\psi )\ne {R}_{sp}(\theta ,\psi +\pi )\\ {T}_{ss}(\theta ,\psi )\ne {T}_{ss}(\theta ,\psi +\pi ),& {T}_{pp}(\theta ,\psi )\ne {T}_{pp}(\theta ,\psi +\pi )\\ {T}_{ps}(\theta ,\psi )\ne {T}_{ps}(\theta ,\psi +\pi ),& {T}_{sp}(\theta ,\psi )\ne {T}_{sp}(\theta ,\psi +\pi )\end{array}\}.$$The complete left/right asymmetry that arises for case IV is illustrated in Figs. 2 and 3, wherein all reflectances and transmittances, respectively, are plotted as functions of the incidence angles $\theta \in [0,\pi /2)$ and $\psi \in [\mathrm{0,2}\pi )$. For these representative calculations, we chose ${\u03f5}_{a}=2.14$, ${\u03f5}_{b}=3.67$, ${\u03f5}_{c}=2.83$, $\chi =38\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{deg}$, $\tilde{\gamma}=100\tilde{\alpha}/{\eta}_{0}$, and $L=1.4{\lambda}_{0}$. The chosen values of ${\u03f5}_{a,b,c}$ emerged from a homogenization model for dielectric CTFs,^{15} whereas $\tilde{\gamma}$^{17}18.^{–}^{19} was chosen to clearly highlight left/right asymmetry, in the absence of any experimental data for anisotropic topological insulators.

The inequalities in Eq. (27) are readily observed in the two figures. The left/right asymmetry is most easily discernible in the plots of ${R}_{ss}$ (Fig. 2) and ${T}_{pp}$ (Fig. 3), but can be identified in the plots of the remaining six remittances too for mid-range values of $\theta $.

In further numerical calculations (not presented here), the left/right asymmetry was found to be even more conspicuous for various remittances, when the magnitude of $\tilde{\gamma}$ was increased. Intrinsic topological insulators are characterized by $\tilde{\gamma}=\pm \tilde{\alpha}/{\eta}_{0}$,^{1} but a very thin coating of a magnetic material can be used to realize $\tilde{\gamma}=(2q+1)\tilde{\alpha}/{\eta}_{0}$, $q\in \{0,\pm 1,\pm 2,\pm 3,\dots \}$.^{17}^{,}^{18} Values of $q$ other than $-1$ and 0 can also be obtained by immersing a topological insulator in a magnetostatic field.^{19}

Practically oriented research on topological insulators is embryonic though steady progress is being made in the identification of several relevant materials.^{2}^{,}^{20} As stated in Sec. 1, attention is chiefly being given to isotropic topological insulators, although the fabrication of anisotropic topological insulators appears possible. The exploitation of left/right asymmetry theoretically shown here to be possible with anisotropic topological insulators is promising for one-way optical devices, which could reduce backscattering noise^{21} in optical communication networks, microscopy, and tomography, for example. But high magnitudes of ${\eta}_{0}\tilde{\gamma}/\tilde{\alpha}$ would be needed for practical implementation.

## Acknowledgments

A.L. is grateful to the Charles Godfrey Binder Endowment at Penn State for the ongoing support of his research. T.G.M. acknowledges the support of EPSRC Grant EP/M018075/1.

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## Biography

**Akhlesh Lakhtakia** received degrees from the Banaras Hindu University and the University of Utah. He is the Charles Godfrey Binder Professor of Engineering Science and Mechanics at the Pennsylvania State University. His research interests include surface multiplasmonics, biorepli-cation, forensic science, solar energy, sculptured thin films, and mimumes. He is a fellow of OSA, SPIE, IoP, AAAS, APS, and IEEE. He received the 2010 SPIE Technical Achievement Award and the 2016 Walston Chubb Award for Innovation.

**Tom G. Mackay** is a reader in the School of Mathematics at the University of Edinburgh and an adjunct professor in the Department of Engineering Science and Mechanics at the Pennsylvania State University. He is a graduate of the Universities of Edinburgh, Glasgow, and Strathclyde, and a fellow of the Institute of Physics (UK) and SPIE. His research interests include the electromagnetic theory of novel and complex materials, including homogenized composite materials.