1. Introduction

Topological insulators (TIs) are a class of materials that have recently opened up interesting scenarios for a variety of applications requiring unidirectional wave propagation at terahertz frequencies [1,2]. From the classical point of view, a TI is a macroscopically isotropic material with topologically insulating surface states quantified by a surface admittance $\gamma _\textrm {TI}$. This surface admittance mediates the discontinuity in the tangential component of the magnetic field across the surface of the TI [3].

When a linearly polarized plane wave impinges on the surface of a planar TI layer, with the incidence direction making an angle $\theta \in [0,\pi /2)$ with respect to the normal to the surface and an angle $\phi \in [0,2\pi )$ with respect to a fixed axis in that surface, depolarization does occur due to $\gamma _\textrm {TI}\ne 0$, but all co-polarized and cross-polarized remittances (i.e., reflectances and transmittances) are independent of $\phi$ [4]. However, when the TI layer is partnered by a layer of an anisotropic dielectric material, then the remittances for a specific direction $\left \{\theta ,\phi \right \}$ can differ from those for $\left \{\theta ,\phi +\pi \right \}$ [4,5]. This phenomenon is called left/right asymmetry. If strong enough, left/right reflection asymmetry could enable one-way optical devices that can reduce back-scattering noise as well as instabilities in optical communication networks; help efficiently deliver internet at ultrahigh baud rates through lighting fixtures; and sharpen two-dimensional and three-dimensional images for microscopy, tomography, process control, and surgery.

Left/right asymmetry can be enhanced by increasing the magnitude of the surface admittance of the TI material. However, $\gamma _\textrm {TI}$ is quantized in terms of the fine-structure constant [6] and typically, just one quantum is available although two magnetic modalities to increase to three quanta have been investigated [7]. But even a few quanta are insufficient for exploiting left/right asymmetry in practical applications.

A possible alternative way for enhancing left/right asymmetry is to resort to periodic multilayers comprising alternating layers of a TI and an anisotropic dielectric material [5]. Recently, we calculated the reflectances and transmittances of a periodic multilayer in which the anisotropic dielectric material is a columnar thin film (CTF) [8]. A CTF is an ensemble of inclined parallel nanocolumns grown onto a planar substrate using physical vapor deposition techniques such as thermal evaporation, electron-beam evaporation, and sputtering [9,10]. CTFs are macroscopically modeled as homogeneous biaxial anisotropic materials with orthorhombic symmetry [8].

In order to understand the radiation characteristics of a finite source in the vicinity of materials with topologically insulating surface states, for this paper we considered radiation by a vertical electric dipole placed above a periodic multilayer comprising alternating layers of a TI and a CTF. The multilayer was taken to have a finite number of unit cells deposited over a silicon substrate. The far-zone electric field of the vertical electric dipole was calculated in the half space above the periodic multilayer. The left/right asymmetry of the radiated electric field was tested by varying both the magnitude of the surface admittance of the TI and the number of unit cells in the multilayer.

An $\exp ({-i\omega t})$ dependence on time $t$ is assumed here, with $\omega$ and $i$ denoting the angular frequency and the imaginary unit, respectively. The free-space wavelength, the free-space wavenumber, and the intrinsic impedance of free space are denoted by $\lambda _{\scriptscriptstyle 0}=2\pi / k_{\scriptscriptstyle 0}$, $k_{\scriptscriptstyle 0}=\omega \sqrt { \varepsilon _{\scriptscriptstyle 0} \mu _{\scriptscriptstyle 0}}$, and $\eta _{\scriptscriptstyle 0}=\sqrt { \mu _{\scriptscriptstyle 0}/ \varepsilon _{\scriptscriptstyle 0}}$, respectively, where $\varepsilon _{\scriptscriptstyle 0}$ is the permittivity and $\mu _{\scriptscriptstyle 0}$ is the permeability of free space. The speed of light is denoted by $c_{\scriptscriptstyle 0}=1/\sqrt { \varepsilon _{\scriptscriptstyle 0} \mu _{\scriptscriptstyle 0}}$.

2. Theory

As illustrated in Fig. 1, the half-spaces $z>0$ and $z<-L_\Sigma$ are vacuous. An electric dipole of moment $\textbf {p}= p_{\textrm {z}} {\hat {\textbf {u}}}_{\textrm {z}}$ is placed at the point $P_0(0,0, z_0)$, $z_0>0$, above the uppermost surface of a periodic multilayer deposited on a dielectric substrate. The unit cell of the periodic multilayer is formed by a TI layer of thickness $L_\textrm {TI}$ and a CTF of thicknesses $L_\textrm {CTF}$. The thickness of the unit cell $\Lambda = L_\textrm {TI} + L_\textrm {CTF}$ is the period of the multilayer. The multilayer comprising $N$ unit cells sits atop an isotropic dielectric layer with scalar relative permittivity $\varepsilon _\textrm {sub}$ and thickness $L_\textrm {sub}$; thus, $L_\Sigma =N\Lambda + L_\textrm {sub}$.

 

Fig. 1. Schematic of the analyzed problem.

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The TI is modeled as an isotropic dielectric layer with scalar relative permittivity $\varepsilon _\textrm {TI}$ and surface admittance $\gamma _\textrm {TI}$. Intrinsic TIs have $\gamma _\textrm {TI}=\pm \alpha / \eta _{\scriptscriptstyle 0}$ where $\alpha =(q_e^{2}/\hbar c_{\scriptscriptstyle 0})/4\pi \varepsilon _{\scriptscriptstyle 0}$ is the (dimensionless) fine structure constant, $q_e$ is the quantum of charge, and $\hbar$ is the reduced Planck constant. The normalized surface admittance $\overline {\gamma }_\textrm {TI}= \gamma _\textrm {TI} \eta _{\scriptscriptstyle 0}/\alpha$ is thus $\pm 1$. The magnitude of $\overline {\gamma }_\textrm {TI}$ can be increased in unit increments up to $3$ either by application of a static magnetic field or by coating with a film of a magnetic material [7].

The CTF is macroscopically characterized by the relative permittivity matrix [8]

The electric field at a point $P(x,y,z)\equiv P(r,\theta ,\phi$) in the half-space $z>0$ can be expressed as the sum of a primary field and the field reflected by the periodic multilayer and the substrate. The primary field is the one that would exist if all matter in the region $0>z>-L_\Sigma$ were to be replaced by the medium occupying the half-spaces $z>0$ and $z<-L_\Sigma$. The primary field can be expressed as a spectrum of plane waves [11,12], i.e.,

At large distances from the dipole (i.e., $r \gg \lambda _{\scriptscriptstyle 0}$), the integral can be analytically determined by considering that only the plane wave propagating along the observation direction contributes to the field. Since $x_0= y_0=0$, for $z> z_0$ the far-zone primary field simplifies to

If a plane wave were to be incident on the periodic multilayer, it will be partially reflected into the half-space $z>0$ as a plane wave. To determine the reflected plane wave, the matrix $\underline {\underline {\textbf {M}}}$ in Eq. (3) has to be decomposed in two matrices $\underline {\underline {\textbf {M}}}^{s}$ and $\underline {\underline {\textbf {M}}}^{p}$ corresponding to the $s$- and $p$- polarization states of the incident plane wave, respectively. After rearranging the detailed expressions of such matrices reported elsewhere [13], the essence of the plane-wave expansion of the incident field can be expressed through the vector function

Correspondingly, the plane-wave expansion of the reflected field requires the vector function

Finally, the total electric field can be written as

3. Numerical results and discussion

All calculations were carried out at ${ \lambda _{\scriptscriptstyle 0}}=4.5 \ \mu$m which is a value consistent with a TI bandgap not exceeding 300 meV [7]. We set ${ z_0}=100 { \lambda _{\scriptscriptstyle 0}}$ and calculated the total electric field on a finite portion of the plane $z=1000 { z_0}$ which was mapped by varying the direction angles $\theta$ and $\phi$ in the ranges $\left (0^{\circ },75^{\circ } \right )$ and $\left ( 0^{\circ },360^{\circ }\right )$, respectively. Periodic multilayers with $N \in \left \lbrace 1,10,20,30,60\right \rbrace$ were considered. A CTF made by evaporating Ta$_2$O$_5$ was chosen with $\chi = 48.50^{\circ }$, $\varepsilon _\textrm {a}=2.2532$, $\varepsilon _\textrm {b}=2.7737$, $\varepsilon _\textrm {c}=2.5475$, $L_\textrm {CTF}={ \lambda _{\scriptscriptstyle 0}}/4\sqrt { \varepsilon _\textrm {a}}=749.5$ nm, $\varepsilon _\textrm {TI}=3$, and $L_\textrm {TI}={ \lambda _{\scriptscriptstyle 0}}/4\sqrt { \varepsilon _\textrm {TI}}=649.5$ nm. We also chose a $5$-$\mu$m-thick layer of silicon as the substrate ($\varepsilon _\textrm {sub}=11.68$). For the TI layers, the following three different values of the normalized surface admittance were considered: $\overline {\gamma }_\textrm {TI} \in \left \lbrace 1,2,3\right \rbrace$.

When $N=1$ and $\overline {\gamma }_\textrm {TI}=1$, the left/right asymmetry is extremely weak for $E_{\theta }$ with $|{\Delta E_{\theta }}|_ \textrm {max}=2.1\times 10^{-4}$, and it is slightly stronger for $E_{\phi }$ with $|{\Delta E_{\phi }}|_\textrm {max}=0.014$. The asymmetry increases slightly as $\overline {\gamma }_\textrm {TI}$ increases; thus, $|{\Delta E_{\theta }}|_\textrm {max}=6.3\times 10^{-4}$ and $|{\Delta E_{\phi }}|_\textrm {max}=0.028$ for $\overline {\gamma }_\textrm {TI}=3$.

When the number $N$ of unit cells increases, left/right asymmetry appears more markedly for both $E_{\theta }$ and $E_{\phi }$. Figures 2–4 present density plots in the $\theta$-$\phi$ plane of the amplitudes of the asymmetry functions $|{\Delta E_{\theta }}|$ and $|{\Delta E_{\phi }}|$ for $\overline {\gamma }_\textrm {TI}=1,2$ and $3$, when $N=30$. These plots were drawn for $\theta \in \left [0^{\circ },75^{\circ } \right ]$ and $\phi \in \left [0^{\circ },180^{\circ } \right ]$.

 

Fig. 2. Asymmetry functions for $\theta \in \left [0^{\circ },75^{\circ } \right ]$ and $\phi \in \left [0^{\circ },180^{\circ } \right ]$, when $N=30$ and $\overline {\gamma }_\textrm {TI}=1$.

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Fig. 3. Same as Fig. 2 but for $\overline {\gamma }_\textrm {TI}=2$.

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Fig. 4. Same as Fig. 2 but for $\overline {\gamma }_\textrm {TI}=3$.

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Figures 2–4 clearly demonstrate that the left/right asymmetry of $E_{\theta }$ occurs in a narrow $\theta$-range located at $\theta \sim 35^{\circ }$. In contrast, the left/right asymmetry of $E_{\phi }$ is weaker but occurs in a wide $\theta$-range that extends from about $35^{\circ }$ to $75^{\circ }$. The three figures also demonstrate that the variation of the left/right asymmetry of $E_{\theta }$ in the $\theta$-$\phi$ plane does not change much, except in amplitude, with $\overline {\gamma }_\textrm {TI}$.

Plots of the left/right asymmetry functions for $N\ne 30$ present substantially similar features. A deeper analysis of the data for $\overline {\gamma }_\textrm {TI}\in [1,3]$ shows that:

A graphical presentation of these remarks is given in Fig. 5, wherein $|{\Delta E_{\theta }}|_\textrm {max}$ and $|{\Delta E_{\phi }}|_\textrm {max}$ are plotted as functions of $N$ for the three considered values of $\overline {\gamma }_\textrm {TI}$. These values along with the direction $\left (\theta ,\phi \right )$ at which they occur are also summarized in Table  1. For a comparison, asymmetry data of the multilayers with no insulating surface states ($\overline {\gamma }_\textrm {TI}=0$) are also reported in Table  1. The following conclusions were drawn from these data.

 

Fig. 5. Maximum values of left/right asymmetry functions in relation to the number $N$ of unit cells and the normalized surface admittance $\overline {\gamma }_\textrm {TI}$. Black diamonds: $\overline {\gamma }_\textrm {TI}=0$; red stars: $\overline {\gamma }_\textrm {TI}=1$; blue circles: $\overline {\gamma }_\textrm {TI}=2$; magenta triangles: $\overline {\gamma }_\textrm {TI}=3$.

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Table 1. Maximum values of the left/right asymmetry functions $|{\Delta E_{\theta }}|$ and $|{\Delta E_{\phi }}|$ and the direction $(\theta ,\phi )$ at which they occur.

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It is worth noting that the asymmetry of $|\Delta E_\phi |$ is due to the asymmetry of $| r_\textrm {sp}|$, as is clear from Eq. (17). Figure 6 shows the reflectances $| r_\textrm {sp}|^{2}$ and $| r_\textrm {pp}|^{2}$ as functions of $\theta$ and $\phi$ for $\bar{\gamma}_{\mathrm{TI}}=2$. Figure 7 shows the same reflectances as functions of $\theta$ for $\phi =120^{\circ }$ and $\phi =300^{\circ }$. The figures highlight that (i) $| r_\textrm {pp}|$ is close to unity whereas (ii) $| r_\textrm {sp}|$ assumes very small values and its asymmetry is negligible, when $\theta \lesssim 35^{\circ }$.

 

Fig. 6. Reflectances $| r_\textrm {sp}|^{2}$ and $| r_\textrm {pp}|^{2}$ as functions of $\theta$ and $\phi$ for $\bar{\gamma}_{\mathrm{TI}}=2$.

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Fig. 7. Reflectances as functions of $\theta$ and $\phi$ for $\bar{\gamma}_{\mathrm{TI}}=2$. Black curves: $\phi =120^{\circ }$; red curves: $\phi =300^{\circ }$.

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4. Concluding remarks

The problem of a vertical electric dipole radiating above a periodic multilayer whose unit cell comprises a TI layer and a CTF was solved in order to investigate the left/right asymmetry of the total electric field in the far zone in the half-space containing the dipole. The total field can be decomposed into a primary field and the field reflected jointly by the periodic multilayer and the substrate. The total field exhibits left/right asymmetry arising from the left/right asymmetry of the reflected field.

While the primary field has only a symmetric $p$-polarized component (the $E_{\theta }$ component) in the far zone, the total field has both $p$- and $s$-polarized components (the $E_{\theta }$ and $E_{\phi }$ components, respectively) affected by left/right asymmetry, depending on the observation direction specified by $\left ( \theta ,\phi \right )$. The left/right asymmetry of $E_{\phi }$ can be related to the presence of both the CTFs and the TI layers and occurs on a wide range of the polar observation angle. The left/right asymmetry of $E_{\theta }$ is entirely due to the TI layers and occurs on a narrow range of the polar observation angle. Numerical results show that, for presently available values of the magnitude of the surface admittance $\gamma _\textrm {TI}$ quantitating the topologically insulating surface states, significant left/right asymmetry occurs if the number of unit cells in the periodic TI/CTF multilayer is high enough.