Left/right asymmetry in Dyakonov–Tamm-wave propagation guided by a topological insulator and a structurally chiral material

Dyakonov–Tamm waves are electromagnetic surface waves whose propagation is guided by the planar interface of two dielectric materials, one of which is isotropic and homogeneous whereas the second is anisotropic and periodically nonhomogeneous normal to the interface plane [1]. In contrast, the second partnering material must be isotropic for Tamm-wave propagation [2–5], whereas that material must be homogeneous for Dyakonov-wave propagation [6–8]. All three types of surface waves propagate ideally without attenuation, unlike surface-plasmon-polariton waves [9], as all three exist in all-dielectric metamaterial architectures [10]. Dyakonov and Dyakonov–Tamm waves offer different phase speeds in different directions of propagation in the interface plane, which makes them more attractive than Tamm waves for communication. But, while the allowed directions of propagation of Dyakonov waves are confined to two minute angular sectors (typically, each less than 1° in width [8, 11]) in the interface plane, Dyakonov–Tamm waves were theoretically predicted not to suffer from that restriction. The existence of these waves has been confirmed recently in two distinct experimental configurations [12, 13].

Theoretical investigation [14] has recently shown that left/right asymmetry can be introduced in Dyakonov-wave propagation by

• (i)  
  endowing the isotropic, homogeneous, dielectric partnering material with topologically insulating surface states (TISS) [15–17] and
• (ii)  
  choosing the anisotropic, homogeneous, dielectric partnering material to possess orthorhombic crystallographic symmetry such that no more than one of the three eigenvectors of its relative permittivity dyadic lies in the interface plane.

Then, the Dyakonov wave propagating coparallel to a vector ${\bf{u}}$ in the interface plane has a different phase speed and different spatial profile as compared to the Dyakonov wave which propagates antiparallel to ${\bf{u}}$. Indeed, it may be possible for a Dyakonov wave to propagate coparallel to ${\bf{u}}$ but for no Dyakonov to propagate antiparallel to ${\bf{u}}$. We refer to this asymmetry with respect to interchanging the direction of surface-wave propagation as left/right asymmetry. The exploitation of left/right asymmetry is promising for one-way optical devices, which could reduce backscattering noise [18] in optical communication networks, microscopy, and tomography, for example. Let us note here that left/right asymmetry is not exhibited when the TISS are replaced by ordinary surface conducting states [19, 20].

Although the incorporation of an isotropic topological insulator (TI) as a partnering material [16, 21] introduces left/right asymmetry in surface-wave propagation, the angular sectors of allowable propagation remain minute in extent [14]. With the aim of widening those angular sectors, we decided to make the anisotropic partnering material periodically nonhomogeneous in the direction normal to the interface plane. Specifically, we chose that partnering material to be a structurally chiral material (SCM) [1]—exemplified by chiral smectic liquid crystals [22] and chiral sculptured thin films [23]—the other partnering material being an isotropic TI [24, 25].

The plan of this paper is as follows. Section 2 contains a formulation of the canonical problem for Dyakonov–Tamm-wave propagation guided by the planar interface of an isotropic TI and an SCM. In the canonical problem, all space is partitioned into two half spaces, one of which is occupied by one partnering material and the second by the other partnering material. Although practically unimplementable in the strict sense, the canonical problem lies at the heart of practically implementable configurations such as the prism-coupled, grating-coupled, and waveguide-coupled configurations [26, 27]. Numerical results are provided and discussed in section 3.

An ${\rm{\exp }}(-{\rm{i}}\omega t)$ dependence on time t is implicit, with ω denoting the angular frequency and ${\rm{i}}=\sqrt{-1}$. The free-space wavenumber, the free-space wavelength, and the intrinsic impedance of free space are denoted by ${k}_{0}=\omega \sqrt{{\varepsilon }_{0}{\mu }_{0}}$, ${\lambda }_{0}=2\pi /{k}_{0}$, and ${\eta }_{0}=\sqrt{{\mu }_{0}/{\varepsilon }_{0}}$, respectively, with ${\varepsilon }_{0}$ and ${\mu }_{0}$ being the permeability and permittivity of free space. The speed of light in free space is denoted by ${c}_{0}=1/\sqrt{{\varepsilon }_{0}{\mu }_{0}}$. Vectors are in boldface; dyadics are underlined twice; Cartesian unit vectors are identified as ${\hat{{\bf{u}}}}_{x},$ ${\hat{{\bf{u}}}}_{y},$ and ${\hat{{\bf{u}}}}_{z};$ column vectors are in boldface and enclosed with square brackets; and matrixes are underlined twice and enclosed with square brackets.

A schematic of the boundary-value problem for the propagation of the Dyakonov–Tamm wave is provided in figure 1. The half-space $z\lt 0$ is occupied by an isotropic TI with a relative permittivity ${\varepsilon }_{{\rm{TI}}}={n}_{{\rm{TI}}}^{2}$ and a surface admittance ${\gamma }_{{\rm{TI}}}$ which quantifies the TISS [16] that arise in consequence of a geometric phase that cannot be gauged away in a cyclic system [28, 29]. Alternatively, the half-space $z\lt 0$ can be modeled as being occupied by an isotropic, nonreciprocal, achiral, nonmagnetic material with relative permittivity ${\varepsilon }_{{\rm{TI}}}$ and Tellegen parameter ${\gamma }_{{\rm{TI}}};$ but we prefer the former description because it brings out the presence of TISS very clearly [20] and conforms to the Post constraint [21].

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The half-space $z\gt 0$ is occupied by an SCM whose permittivity dyadic is given by

$$\begin{eqnarray}&&\,{\underline{\underline{\varepsilon }}}_{{\rm{SCM}}}(z)={\varepsilon }_{0}\,{\underline{\underline{S}}}_{z}(z)\cdot {\underline{\underline{S}}}_{y}(\chi )\cdot {\underline{\underline{\varepsilon }}}_{{\rm{ref}}}^{\circ }\cdot {\underline{\underline{S}}}_{y}^{-1}(\chi )\cdot {\underline{\underline{S}}}_{z}^{-1}(z).\end{eqnarray} \tag{ 1 }$$

Here, the dyadics

$$\begin{eqnarray}&&\left.\begin{array}{l}\,{\underline{\underline{S}}}_{z}(z)=({\hat{{\bf{u}}}}_{x}{\hat{{\bf{u}}}}_{x}+{\hat{{\bf{u}}}}_{y}{\hat{{\bf{u}}}}_{y}){\rm{\cos }}(\pi z/{\rm{\Omega }})\\ \qquad +h\,({\hat{{\bf{u}}}}_{y}{\hat{{\bf{u}}}}_{x}-{\hat{{\bf{u}}}}_{x}{\hat{{\bf{u}}}}_{y}){\rm{\sin }}(\pi z/{\rm{\Omega }})+{\hat{{\bf{u}}}}_{z}{\hat{{\bf{u}}}}_{z}\\ \,{\underline{\underline{S}}}_{y}(\chi )=({\hat{{\bf{u}}}}_{x}{\hat{{\bf{u}}}}_{x}+{\hat{{\bf{u}}}}_{z}{\hat{{\bf{u}}}}_{z}){\rm{\cos }}\chi \\ \qquad +({\hat{{\bf{u}}}}_{z}{\hat{{\bf{u}}}}_{x}-{\hat{{\bf{u}}}}_{x}{\hat{{\bf{u}}}}_{z}){\rm{\sin }}\chi +{\hat{{\bf{u}}}}_{y}{\hat{{\bf{u}}}}_{y}\\ \,{\underline{\underline{\varepsilon }}}_{{\rm{ref}}}^{\circ }={\varepsilon }_{a}\,{\hat{{\bf{u}}}}_{z}{\hat{{\bf{u}}}}_{z}+{\varepsilon }_{b}\,{\hat{{\bf{u}}}}_{x}{\hat{{\bf{u}}}}_{x}+{\varepsilon }_{c}\,{\hat{{\bf{u}}}}_{y}{\hat{{\bf{u}}}}_{y}\end{array}\right\};\end{eqnarray} \tag{ 2 }$$

h = 1 for structural right-handedness and $h=-1$ for structural left-handedness; $2{\rm{\Omega }}$ is the structural period of the SCM along the z axis; and $\chi \in (0,\pi /2]$. Whereas ${\varepsilon }_{a}={\varepsilon }_{c}\ne {\varepsilon }_{b}$ and $\chi =0$ for cholesteric liquid crystals, ${\varepsilon }_{a}\ne {\varepsilon }_{b}\ne {\varepsilon }_{c}$ and $\chi \in (0,\pi /2]$ for chiral smectic liquid crystals [22] and chiral sculptured thin films [23]. Both partnering materials are assumed to be nonmagnetic.

2.1. Field representations

We consider the Dyakonov–Tamm wave to be propagating parallel to the unit vector ${\hat{{\bf{u}}}}_{{\rm{prop}}}={\hat{{\bf{u}}}}_{x}{\rm{\cos }}\psi +{\hat{{\bf{u}}}}_{y}{\rm{\sin }}\psi$, $\psi \in [0^\circ ,360^\circ )$, in the xy plane and decaying far away from the interface z = 0. With q as the wavenumber of the Dyakonov–Tamm wave, the electric and magnetic phasors can be represented everywhere by

$$\begin{eqnarray}&&\left.\begin{array}{l}{\bf{E}}({\bf{r}})={\bf{e}}(z)\,{\rm{\exp }}({\rm{i}}{q}{\hat{{\bf{u}}}}_{{\rm{prop}}}\cdot {\bf{r}})\\ {\bf{H}}({\bf{r}})={\bf{h}}(z)\,{\rm{\exp }}({\rm{i}}{q}{\hat{{\bf{u}}}}_{{\rm{prop}}}\cdot {\bf{r}})\end{array}\right\}.\end{eqnarray} \tag{ 3 }$$

In the region $z\lt 0$, the field phasors may be written as [21, 27]

$$\begin{eqnarray}&&{\bf{e}}(z)=[{A}_{1}(-{\hat{{\bf{u}}}}_{x}{\rm{\sin }}\psi +{\hat{{\bf{u}}}}_{y}{\rm{\cos }}\psi )\\ &&\quad +\left.{A}_{2}\left(\displaystyle \frac{{\alpha }_{{\rm{TI}}}}{{k}_{0}}\,{\hat{{\bf{u}}}}_{{\rm{prop}}}+\displaystyle \frac{q}{{k}_{0}}\,{\hat{{\bf{u}}}}_{z}\right)\right]\\ &&\quad \times \ {\rm{\exp }}(-{\rm{i}}{\alpha }_{{\rm{TI}}}z),\quad z\lt 0\end{eqnarray} \tag{ 4 }$$

and

$$\begin{eqnarray}{\bf{h}}(z) & = & {\eta }_{0}^{-1}\left[{A}_{1}\left(\displaystyle \frac{{\alpha }_{{\rm{TI}}}}{{k}_{0}}\,{\hat{{\bf{u}}}}_{{\rm{prop}}}+\displaystyle \frac{q}{{k}_{0}}\,{\hat{{\bf{u}}}}_{z}\right)\right.\\ & & -{A}_{2}\,{n}_{{\rm{TI}}}^{2}(-{\hat{{\bf{u}}}}_{x}{\rm{\sin }}\psi +{\hat{{\bf{u}}}}_{y}{\rm{\cos }}\psi )]\\ & & \times {\rm{\exp }}(-{\rm{i}}{\alpha }_{{\rm{TI}}}z),\quad z\lt 0,\end{eqnarray} \tag{ 5 }$$

where A₁ and A₂ are unknown scalars, q is positive and real-valued for unattenuated propagation in the xy plane, ${q}^{2}+{\alpha }_{{\rm{TI}}}^{2}={k}_{0}^{2}{\varepsilon }_{{\rm{TI}}}$, and ${\rm{Im}}({\alpha }_{{\rm{TI}}})\gt 0$ for attenuation as z→ −$\infty$.

The field representation in the region $z\gt 0$ requires the formulation of the column vector [23, 27]

$$\begin{eqnarray}&&[{\bf{f}}(z)]={[{e}_{x}(z){e}_{y}(z){h}_{x}(z){h}_{y}(z)]}^{T}\,\end{eqnarray} \tag{ 6 }$$

which satisfies the matrix differential equation [1]

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}}{{\rm{d}}{z}}[{\bf{f}}(z)]={\rm{i}}\left[\underline{\underline{P}}\left(h\displaystyle \frac{\pi z}{{\rm{\Omega }}},\psi \right)\right]\cdot [{\bf{f}}(z)],\quad z\gt 0,\end{eqnarray} \tag{ 7 }$$

where the 4 × 4 matrix

$$\begin{eqnarray} [{\underset{\raise{2pt}=}{P}}(\zeta ,\psi )]\,&=&\omega \left[\begin{array}{cccc}0 & 0 & 0 & {\mu }_{0}\\ 0 & 0 & -{\mu }_{0} & 0\\ {\varepsilon }_{0}({\varepsilon }_{c}-{\varepsilon }_{d}){\rm{\cos }}\zeta \,{\rm{\sin }}\zeta & -{\varepsilon }_{0}({\varepsilon }_{c}{{\rm{\cos }}}^{2}\zeta +{\varepsilon }_{d}{{\rm{\sin }}}^{2}\zeta ) & 0 & 0\\ {\varepsilon }_{0}({\varepsilon }_{c}{{\rm{\sin }}}^{2}\zeta +{\varepsilon }_{d}{{\rm{\cos }}}^{2}\zeta ) & -{\varepsilon }_{0}({\varepsilon }_{c}-{\varepsilon }_{d}){\rm{\cos }}\zeta \,{\rm{\sin }}\zeta & 0 & 0\end{array}\right]\\&&+q\,\displaystyle \frac{{\varepsilon }_{d}({\varepsilon }_{a}-{\varepsilon }_{b})}{{\varepsilon }_{a}\,{\varepsilon }_{b}}\,{\rm{\sin }}\chi {\rm{\cos }}\chi\times \,\left[\begin{array}{cccc}{\rm{\cos }}\zeta {\rm{\cos }}\psi & {\rm{\sin }}\zeta {\rm{\cos }}\psi & 0 & 0\\ {\rm{\cos }}\zeta {\rm{\sin }}\psi & {\rm{\sin }}\zeta {\rm{\sin }}\psi & 0 & 0\\ 0 & 0 & {\rm{\sin }}\zeta {\rm{\sin }}\psi & -{\rm{\sin }}\zeta {\rm{\cos }}\psi \\ 0 & 0 & -{\rm{\cos }}\zeta {\rm{\sin }}\psi & {\rm{\cos }}\zeta {\rm{\cos }}\psi \end{array}\right]\\&&+\displaystyle \frac{{q}^{2}}{\omega {\varepsilon }_{0}}\,\displaystyle \frac{{\varepsilon }_{d}}{{\varepsilon }_{a}\,{\varepsilon }_{b}}\left[\begin{array}{cccc}0 & 0 & {\rm{\cos }}\psi {\rm{\sin }}\psi & -{{\rm{\cos }}}^{2}\psi \\ 0 & 0 & {{\rm{\sin }}}^{2}\psi & -{\rm{\cos }}\psi {\rm{\sin }}\psi \\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\end{array}\right]\\ & & +\displaystyle \frac{{q}^{2}}{\omega {\mu }_{0}}\left[\begin{array}{cccc}0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ -{\rm{\cos }}\psi {\rm{\sin }}\psi & {{\rm{\cos }}}^{2}\psi & 0 & 0\\ -{{\rm{\sin }}}^{2}\psi & {\rm{\cos }}\psi {\rm{\sin }}\psi & 0 & 0\end{array}\right]\,\end{eqnarray} \tag{ 8 }$$

and the scalar

$$\begin{eqnarray}&&{\varepsilon }_{d}=\displaystyle \frac{{\varepsilon }_{a}{\varepsilon }_{b}}{{\varepsilon }_{a}\,{{\rm{\cos }}}^{2}\chi +{\varepsilon }_{b}\,{{\rm{\sin }}}^{2}\chi }.\end{eqnarray} \tag{ 9 }$$

Equation (7) has to be solved numerically in order to determine the matrix $[\underline{\underline{Q}}]$ that appears in the relation

$$\begin{eqnarray}&&[{\bf{f}}(2{\rm{\Omega }})]=[\underset{\_}{\underset{\_}{Q}}]\cdot [{\bf{f}}(0+)]\end{eqnarray} \tag{ 10 }$$

to characterize the optical response of one period of the SCM.

By virtue of the Floquet theory [30], we can define a matrix $[\underline{\underline{\tilde{Q}}}]$ such that

$$\begin{eqnarray}&&[\underline{\underline{Q}}]={\rm{\exp }}\{{\rm{i}}2{\rm{\Omega }}[\underline{\underline{\tilde{Q}}}]\}.\end{eqnarray} \tag{ 11 }$$

Both $[\underline{\underline{Q}}]$ and $[\underline{\underline{\tilde{Q}}}]$ share the same eigenvectors, and their eigenvalues are also related. Let ${[{\bf{t}}]}^{(n)}$, $(n=1,2,3,4)$, be the eigenvector corresponding to the nth eigenvalue ${\sigma }_{n}$ of $[\underline{\underline{Q}}];$ then, the corresponding eigenvalue ${\alpha }_{n}$ of $[\underline{\underline{\tilde{Q}}}]$ is given by

$$\begin{eqnarray}&&{\alpha }_{n}=-{\rm{i}}\displaystyle \frac{{\rm{ln}}{\sigma }_{n}}{2{\rm{\Omega }}}.\end{eqnarray} \tag{ 12 }$$

2.2. Dispersion equation

For the Dyakonov–Tamm wave to propagate parallel to ${\hat{{\bf{u}}}}_{{\rm{prop}}},$ we must ensure that ${\rm{Im}}({\alpha }_{\mathrm{1,2}})\gt 0$, and set

$$\begin{eqnarray}&&[{\bf{f}}({0}^{+})]=[\,{[{\bf{t}}]}^{(1)}\quad {[{\bf{t}}]}^{(2)}\,]\cdot \left[\begin{array}{c}{B}_{1}\\ {B}_{2}\end{array}\right],\end{eqnarray} \tag{ 13 }$$

where B₁ and B₂ are unknown scalars, and $[{\bf{f}}({z}^{\pm })]$ stands for ${{\rm{lim}}}_{\delta \to 0}\,[{\bf{f}}(z\pm \delta )]$ with $\delta \nless 0$. The other two eigenvalues of $[\underline{\underline{\tilde{Q}}}]$ describe waves that amplify as $z\to \infty$ and cannot therefore contribute to the Dyakonov–Tamm wave. At the same time

$$\begin{eqnarray}[{\bf{f}}({0}^{-})]=\left[\begin{array}{cc}-{\rm{\sin }}\psi & \tfrac{{\alpha }_{{\rm{TI}}}}{{k}_{0}}{\rm{\cos }}\psi \\ {\rm{\cos }}\psi & \tfrac{{\alpha }_{{\rm{TI}}}}{{k}_{0}}{\rm{\sin }}\psi \\ \tfrac{{\alpha }_{{\rm{TI}}}}{{k}_{0}}\,{\eta }_{0}^{-1}{\rm{\cos }}\psi & {n}_{{\rm{TI}}}^{2}\,{\eta }_{0}^{-1}{\rm{\sin }}\psi \\ \tfrac{{\alpha }_{{\rm{TI}}}}{{k}_{0}}\,{\eta }_{0}^{-1}{\rm{\sin }}\psi & -{n}_{{\rm{TI}}}^{2}\,{\eta }_{0}^{-1}{\rm{\cos }}\psi \end{array}\right]\cdot \left[\begin{array}{c}{A}_{1}\\ {A}_{2}\end{array}\right],\end{eqnarray} \tag{ 14 }$$

by virtue of equations (4) and (5).

Whereas the tangential component of the electric field phasor is continuous across the plane z = 0, the existence of the protected TISS on the boundary of the TI implies a discontinuity in the tangential component of the magnetic field phasor across the same plane [14, 20, 21]. Accordingly

$$\begin{eqnarray}\left[\begin{array}{cccc}1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ -{\gamma }_{{\rm{TI}}} & 0 & 1 & 0\\ 0 & -{\gamma }_{{\rm{TI}}} & 0 & 1\end{array}\right]\cdot [{\bf{f}}({0}^{-})]=[{\bf{f}}({0}^{+})],\end{eqnarray} \tag{ 15 }$$

which may be rearranged as

$$\begin{eqnarray}&&[\underline{\underline{M}}]\cdot \left[\begin{array}{c}{A}_{1}\\ {A}_{2}\\ {B}_{1}\\ {B}_{2}\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\\ 0\end{array}\right].\end{eqnarray} \tag{ 16 }$$

For a nontrivial solution, the 4 × 4 matrix $[\underline{\underline{M}}]$ must be singular, so that

$$\begin{eqnarray}&&{\rm{\det }}\,[\underline{\underline{M}}]=0\end{eqnarray} \tag{ 17 }$$

is the dispersion equation for the Dyakonov–Tamm wave.

We numerically solved the dispersion equation to obtain the normalized wavenumbers $\tilde{q}=q/{k}_{0}$ of the Dyakonov–Tamm waves. Knowing q, we can calculate the phase speed ${v}_{{\rm{ph}}}={c}_{0}/\tilde{q}$ of the Dyakonov–Tamm wave. The spatial profile of the rate of energy flow associated with a Dyakonov–Tamm wave is provided via the time-averaged Poynting vector ${\bf{P}}({\bf{r}})\equiv {\bf{P}}(x,y,z)=(1/2){\rm{Re}}[{\bf{e}}(z)\times {{\bf{h}}}^{* }(z)]$, where the asterisk denotes the complex conjugate.

For all numerical results reported here, we fixed ${\lambda }_{0}=633\,{\rm{nm}}$ and ${\rm{\Omega }}=197\,{\rm{nm}}$. For definiteness, the SCM was taken to be a chiral sculptured thin film, which comprises an array of parallel nanohelixes that rise at an angle χ to the interface plane by means of a vapor deposition process [23]. In accordance with empirical relationships determined for a columnar thin film of patinal titanium oxide produced by directing the vapor flux at an angle ${\chi }_{v}$ onto a rotating substrate, the principal relative permittivities are [31]

$$\begin{eqnarray}\left.\begin{array}{rcl}{\varepsilon }_{a} & = & {\left[1.0443+2.7394\displaystyle \frac{{\chi }_{v}}{90}-1.3697{\left(\displaystyle \frac{{\chi }_{v}}{90}\right)}^{2}\right]}^{2}\\ {\varepsilon }_{b} & = & {\left[1.6765+1.5649\displaystyle \frac{{\chi }_{v}}{90}-0.7825{\left(\displaystyle \frac{{\chi }_{v}}{90}\right)}^{2}\right]}^{2}\\ {\varepsilon }_{c} & = & {\left[1.3586+2.1109\displaystyle \frac{{\chi }_{v}}{90}-1.0554{\left(\displaystyle \frac{{\chi }_{v}}{90}\right)}^{2}\right]}^{2}\end{array}\right\}\end{eqnarray} \tag{ 18 }$$

with ${\chi }_{v}$ being in degree, and the angle

$$\begin{eqnarray}&&\chi ={\rm{\arctan }}(2.8818\,{\rm{\tan }}\,{\chi }_{v}).\end{eqnarray} \tag{ 19 }$$

We considered ${\chi }_{v}=\{7.2^\circ ,15.0^\circ ,19.1^\circ \}$ along with ${n}_{{\rm{TI}}}=\{1.64,1.7,1.8\}$, but kept $\bar{\gamma }={\eta }_{0}{\gamma }_{{\rm{TI}}}{\widetilde{\alpha }}^{-1}$ variable, where $\widetilde{\alpha }=7.297352566\times {10}^{-3}$ is the fine structure constant [32]. The direction of propagation was also varied in the xy plane, i.e., $\psi \in [0^\circ ,360^\circ )$. All calculations were restricted to ${n}_{{\rm{TI}}}\lt \tilde{q}\leqslant 3$ to avoid computational instabilities that emerged for $\tilde{q}\gt 3$, whereas $\tilde{q}$ must be greater than ${n}_{{\rm{TI}}}$ to ensure that ${\alpha }_{{\rm{TI}}}^{2}={k}_{0}^{2}{n}_{{\rm{TI}}}^{2}-{q}^{2}\lt 0$.

In figure 2(a), $\tilde{q}$ is displayed as a function of both ψ and $\bar{\gamma }$ when ${\chi }_{v}=7.2^\circ$ and ${n}_{{\rm{TI}}}=1.64$. Solutions to the dispersion equation (17) were found in the range $1.6405\leqslant \tilde{q}\leqslant 1.6435$ corresponding to a normalized phase speed ${v}_{{\rm{ph}}}/{c}_{0}$ in the range $0.6085\leqslant {v}_{{\rm{ph}}}/{c}_{0}\leqslant 0.6096$. Dyakonov–Tamm-wave propagation is exhibited for $\psi \subseteq [0^\circ ,16^\circ ]\cup [92^\circ ,196^\circ ]\ \cup [272^\circ ,360^\circ ]$, the widths of the angular sectors available for Dyakonov–Tamm-wave propagation being large in comparison to the $\lesssim 1^\circ$ widths of angular sectors for Dyakonov-wave propagation [14].

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The same is true in figure 2(b) for ${\chi }_{v}=15^\circ$ and ${n}_{{\rm{TI}}}=1.7$, and in figure 2(c) for ${\chi }_{v}=19.1^\circ$ and ${n}_{{\rm{TI}}}=1.8$. In figure 2(b), the solutions cover the range $1.7394\leqslant \tilde{q}\leqslant 1.7402$, corresponding to normalized phase speeds in the range $0.5746\leqslant {v}_{{\rm{ph}}}/{c}_{0}\leqslant 0.5749$. In figure 2(c), the solutions cover the range $1.8193\leqslant \tilde{q}\leqslant 1.8264$, corresponding to normalized phase speeds in the range $0.5475\leqslant {v}_{{\rm{ph}}}/{c}_{0}\leqslant 0.5497$.

All three panels in figure 2 indicate that the propagation of Dyakonov–Tamm waves, if it can occur for chosen values of the pair $\{{\chi }_{v},{n}_{{\rm{TI}}}\}$, is possible in two non-overlapping regions in the $\psi \bar{\gamma }$ plane. Each region is bounded by an elliptical contour represented parametrically as the ellipse $\{\psi (\vartheta ),\bar{\gamma }(\vartheta )\}$, with

$$\begin{eqnarray}\begin{array}{rcl} & & \left.\begin{array}{rcl}\psi (\vartheta ) & = & {\psi }_{C}+180^\circ {\ell }+{\psi }_{D}{\rm{\cos }}\vartheta ({\rm{mod}}\,\,360^\circ )\\ \bar{\gamma }(\vartheta ) & = & {(-1)}^{{\ell }}\,{\bar{\gamma }}_{C}+{\bar{\gamma }}_{D}{\rm{\sin }}\vartheta \end{array}\right\}\\ & & \qquad \quad \vartheta \in [0^\circ ,360^\circ ],\quad {\ell }\in \{1,2\}.\end{array}\end{eqnarray} \tag{ 20 }$$

The region described by ${\ell }=1$ is located roughly in the center of the $\psi \bar{\gamma }$ plane in each panel, while the region described by ${\ell }=2$ is split into two parts because $\psi (\vartheta )$ is cyclic with period 360°. The center of the ellipse is located at $\{{\psi }_{C}+180^\circ {\ell },{(-1)}^{{\ell }}\,{\bar{\gamma }}_{C}\}$ for ${\ell }\in \{1,2\}$, the projection of each ellipse on the ψ axis is $2{\psi }_{D}$, and the projection of each ellipse on the $\bar{\gamma }$ axis is $2{\bar{\gamma }}_{D}$. Values of the parameters ${\psi }_{C,D}$ and ${\bar{\gamma }}_{C,D}$ for all three panels are provided in table 1.

Table 1.  Parameters in equation (20) delineating the regions in the $\psi \bar{\gamma }$ plane in which the propagation of Dyakonov–Tamm waves is allowed in figures 2(a)–(c).

Figure 2	${\chi }_{v}$	${n}_{{\rm{TI}}}$	${\psi }_{C}$	${\psi }_{D}$	${\bar{\gamma }}_{C}$	${\bar{\gamma }}_{D}$
(a)	7.2°	1.64	−38°	52°	80	450
(b)	15.0°	1.70	−18°	21°	248	145
(c)	19.1°	1.80	−16°	53°	415	395

Figure 2 clearly shows that Dyakonov–Tamm waves are allowed for both positive and negative values of the surface admittance ${\gamma }_{{\rm{TI}}}$. However, the angular sectors (on the ψ axis) are different for ${\gamma }_{{\rm{TI}}}\gt 0$ than for ${\gamma }_{{\rm{TI}}}\lt 0$.

Whereas Dyakonov–Tamm-wave propagation is possible for ${\gamma }_{{\rm{TI}}}=0$ in figure 2(a), that is not true in figures 2(b) and (c). Therefore, the incorporation of the protected TISS with an appropriate value of ${\gamma }_{{\rm{TI}}}$ can trigger the excitation of Dyakonov–Tamm waves.

Left/right asymmetry is evident in all three panels in figure 2. If Dyakonov–Tamm waves are allowed to propagate in the two directions indicated by $\psi \in [0^\circ ,180^\circ ]$ and $\psi +180^\circ$ for a specific value of ${\gamma }_{{\rm{TI}}}\ne 0$, the two Dyakonov–Tamm waves have different phase speeds (and, therefore, other characteristics). More significantly, figure 2 makes it clear that if a Dyakonov–Tamm wave can propagate in the direction indicated by $\psi \in [0^\circ ,360^\circ$), there is also the likelihood that no Dyakonov–Tamm wave can propagate in the direction indicated by $\psi \pm 180^\circ$. Cross hatching in all three panels highlights the values of ψ for which Dyakonov–Tamm-wave propagation is possible but not for $\psi \pm 180^\circ$. These regions of total left/right asymmetry are very attractive for one-way devices, although they require high values of $| \bar{\gamma }|$ [17].

Further insights into the nature of these Dyakonov–Tamm waves may be gained by considering the spatial profiles of the Cartesian components of the electric and magnetic field phasors, along with those of the corresponding time-averaged Poynting vector. To allow for a consistent comparison, the amplitude of the time-averaged Poynting vector was constrained as ${\hat{{\bf{u}}}}_{{\rm{prop}}}\cdot {\bf{P}}(0,0,{0}^{-})=1$ W m⁻². Then, by virtue of equations (4) and (5), we get

$$\begin{eqnarray}&&| {A}_{1}{| }^{2}=\displaystyle \frac{2{\eta }_{0}}{\tilde{q}}-{\varepsilon }_{{\rm{TI}}}| {A}_{2}{| }^{2}\end{eqnarray} \tag{ 21 }$$

which allows all coefficients in the column vector on the left side of equation (16) to be specified.

The Cartesian components of ${\bf{P}}(0,0,z)$ are plotted versus z in figure 3 for the Dyakonov–Tamm wave excited when ${\chi }_{v}=7.2^\circ$, ${n}_{{\rm{TI}}}=1.64$, and $(\psi ,\bar{\gamma })\in \{(148^\circ ,\pm 100),(328^\circ ,\pm 100)\}$. A comparison of figures 3(a) and (b) reveals that, for $\psi =148^\circ$, the power density is slightly more confined to the TI when ${\gamma }_{{\rm{TI}}}$ is positive while for negative ${\gamma }_{{\rm{TI}}}$ the confinement is slightly greater to the SCM. In contrast, a comparison of figures 3(c) and (d) reveals that, for $\psi =328^\circ$, the power density is slightly more confined to the TI for negative ${\gamma }_{{\rm{TI}}}$ while the confinement is slightly greater to the SCM for positive ${\gamma }_{{\rm{TI}}}$. After noting that $328^\circ =148^\circ +180^\circ$, left/right asymmetry becomes evident on comparing figures 3(a) and (c) and/or comparing figures 3(b) and (d). The signs of P_x and P_y are reversed when together ${\hat{{\bf{u}}}}_{{\rm{prop}}}$ changes to $-{\hat{{\bf{u}}}}_{{\rm{prop}}}$ and ${\gamma }_{{\rm{TI}}}$ changes to $-{\gamma }_{{\rm{TI}}}$, as may be appreciated after comparing figures 3(b) and (c) and/or comparing figures 3(a) and (d).

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Figures 4 and 5 show the magnitudes of the Cartesian components of ${\bf{E}}(0,0,z)$ and ${\bf{H}}(0,0,z)$ respectively, plotted against z for the same values of ${\chi }_{v}$, ${n}_{{\rm{TI}}},$ ψ, and $\bar{\gamma }$ as used for figure 3. All Cartesian components decay exponentially inside the TI as z→ −$\infty$, consistently with equations (4) and (5). The periodic undulations of the Cartesian components inside the SCM dampen as $z\to \infty$, in accord with the periodic nonhomogeneity of the SCM [27], as is warranted by Floquet theory [30].

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The spatial profiles of the Cartesian components of the electric and magnetic field phasors in figures 4 and 5 vary relatively little in the SCM when either ${\hat{{\bf{u}}}}_{{\rm{prop}}}$ changes to $-{\hat{{\bf{u}}}}_{{\rm{prop}}}$ or ${\gamma }_{{\rm{TI}}}$ changes to $-{\gamma }_{{\rm{TI}}}$, but more substantial variations are observed in the TI, especially for E_z, H_x, and H_y. Left/right asymmetry is obvious, e.g., on comparing figures 4(a) and (c) or comparing figures 5(b) and (d). Finally, the spatial profiles of the fields remain unchanged when ${\hat{{\bf{u}}}}_{{\rm{prop}}}$ changes to $-{\hat{{\bf{u}}}}_{{\rm{prop}}}$ and ${\gamma }_{{\rm{TI}}}$ changes to $-{\gamma }_{{\rm{TI}}}$, as may be appreciated by comparing figures 4(b) and (c) and/or comparing figures 5(a) and (d).

Similar conclusions can be drawn looking at the spatial profiles for a case of total left/right asymmetry, in which Dyakonov–Tamm-wave propagation is possible for some ψ but not for $\psi \pm 180^\circ$. As an example, figure 6 provides the spatial profiles of magnitudes of the Cartesian components of ${\bf{P}}(0,0,z)$, ${\bf{E}}(0,0,z)$, and ${\bf{H}}(0,0,z)$ of the Dyakonov–Tamm wave guided by the TI/SCM interface when ${\chi }_{v}=15^\circ$, ${n}_{{\rm{TI}}}=1.70$, $\psi =162^\circ$, and $\bar{\gamma }=-248$. Dyakonov-wave propagation is not possible for $\psi =342^\circ$, other parameters remaining unchanged, according to figure 2(b). The spatial profiles in figure 6 are very similar to those in figures 3(b), 4(b), and 5(b), for which $\psi \lt 180^\circ$ and $\bar{\gamma }\lt 0$.

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A discussion of the discontinuities and continuities of the Cartesian components of the electric and magnetic field phasors across the plane z = 0 is in order. Both E_x and E_y must be continuous while E_z must be discontinuous, according to the standard boundary conditions of electromagnetics [33]. The plots in figures 4 and 6(b) are in accord with these constraints. As both partnering materials have been taken to be non-magnetic, H_z must be continuous across the plane z = 0 [33]. The plots in figures 5 and 6(c) show this continuity. The existence of the protected TISS must make H_x and H_y discontinuous across the plane z = 0, which is evident in figures 5 and 6(c). The continuities of E_x and E_y and the discontinuities of H_x and H_y were, of course, incorporated via equation (15).

We formulated and solved the boundary-value problem for electromagnetic surface waves guided by the planar interface of an SCM and a TI, both materials assumed to be nonmagnetic. The protected TISS on the interface were quantitated through a surface admittance ${\gamma }_{{\rm{TI}}}$. Our numerical investigation demonstrated that the phase speeds and the spatial profiles of Dyakonov–Tamm waves are significantly affected by ${\gamma }_{{\rm{TI}}}$. A left/right asymmetry is exhibited whereby the phase speed and electromagnetic field profiles for a Dyakonov–Tamm wave that propagates co-parallel to a vector ${\bf{u}}$ in the interface plane are generally different to those for a Dyakonov–Tamm wave that propagates anti-parallel to ${\bf{u}}$. Even more importantly, the existence of a Dyakonov–Tamm wave that propagates co-parallel to a vector ${\bf{u}}$ in the interface plane does not imply the existence of a Dyakonov–Tamm wave that propagates anti-parallel to ${\bf{u}}$. The left/right asymmetry, which vanishes if the surface admittance vanishes, is highly attractive for one-way on-chip optical communication.

TGM acknowledges the support of EPSRC Grant EP/M018075/1. A Lakhtakia thanks the Charles Godfrey Binder Endowment at the Pennsylvania State University for ongoing support of his research.