## Abstract

In this paper, we analyze a cylindrical waveguide consisting of two layers of bianisotropic material with anti-symmetric magnetoelectric coupling tensors. The analysis is carried out in terms of pseudo-electric and pseudo-magnetic fields which satisfy Maxwells’ equations with gyrotropic permittivity and permeability tensors. We show that the rotationally symmetric modes of the waveguide are unidirectional with transverse pseudo-electric and transverse pseudo-magnetic modes propagating in opposite directions. These modes are surface waves whose electromagnetic field is concentrated near the interface between the two anisotropic materials. They follow the contour of the interface even in the case of sharp discontinuities and pass through an obstacle without backscattering if the obstacle does not change the polarization of the wave. Higher-order modes of the waveguide are also investigated. Although these modes are hybrid modes and not, strictly speaking, unidirectional, they practically behave as the rotationally symmetric mode.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Unidirectional waves have attracted much attention in recent years as they provide a means for lossless transmission of signals. Their immunity to reflection from barriers and imperfections leads to a variety of applications including the formation of hotspots [1,2], directional couplers [3], and optical diodes [4].

It is well-known that gyrotropic materials can be used to realize unidirectional wave-guiding structures [5–7]. These materials exhibit a non-reciprocal electromagnetic response that is reflected in their asymmetric permeability or permittivity tensors. In waveguides containing gyrotropic media, the propagation constant may depend on the direction of propagation. In some cases, propagation takes place in one direction only. A classic example of such a unidirectional waveguide is a grounded ferromagnetic or ferrite layer at microwave frequencies [8,9]. At terahertz and optical frequencies, magneto-optic materials with asymmetric permittivity tensors [10] can be used to realize unidirectional waves but their non-reciprocal behavior is weak, reducing their practicality. Recently, a cylindrical waveguide was proposed that consists of a metal wire enclosed by a hollow cylinder of a magneto-optical material [11]. This structure was shown to support one-way (unidirectional) surface modes. The unidirectional surface waves at the interface between a hyperbolic-gyromagnetic material and a dielectric or negative-index medium have been studied in [12] and their immunity to disorder has been demonstrated by investigating their propagation around sharp corners. The direction of propagation can be reversed by changing the sign of the gyrotropic parameter.

Recently, it has been demonstrated that unidirectional waves can propagate on the surface of photonic topological insulators (PTI’s) [13–19], built from reciprocal materials. PTI’s are artificial periodic structures that have a topologically non-trivial dispersion band and prohibit the propagation of waves in their bulk. In [20], a metawaveguide behaving as a PTI with complete topological band gap has been proposed. The waveguide consists of an interface between two periodic arrays of metallic cylinders attached to one of the two confining metal plates with opposite signs of the bianisotropy [21]. This structure topologically supports surface waves guiding along sharp discontinuities without reflections. This phenomenon is, however, not restricted to periodic structures. Even on the surface of homogeneous reciprocal media, surface waves may propagate that are immune to scattering from imperfections. In [22], topologically protected surface waves at an interface between a hyperbolic chiral metamaterial and vacuum have been investigated. They are robust against disorder and can be used for one-way propagation in photonic integrated circuits. By changing the sign of the chirality parameter, the direction of propagation is reversed. In [23] it was shown that the surface of a homogeneous bianisotropic material [24–26] with an anti-symmetric magnetoelectric coupling tensor also supports unidirectional waves. To understand this behavior a simple transformation was proposed in [23] which maps a bianisotropic material with an anti-symmetric magnetoelectric coupling tensor onto a medium with gyrotropic permittivity and permeability tensors that operate on pseudo-electric and magnetic fields.

Although [23] demonstrated the propagation of unidirectional waves on the interface between bianisotropic and conventional half-spaces, a more realistic concept for a three-dimensional unidirectional waveguide involving homogeneous media is still lacking. In this paper, we propose and analyze a cylindrical unidirectional waveguide that comprises two bianisotropic materials with opposite magnetoelectric coupling constants. We show that the rotationally symmetric modes of this waveguide are of either the transverse pseudo-electric (Te) or transverse pseudo-magnetic (Tm) type (spin polarizations). Moreover, they are unidirectional for each type of polarization, with Te and Tm modes propagating in opposite directions. These results are analytically obtained and validated by comparison with full-wave electromagnetic simulations. Using numerical calculations we show that the spin-polarized, lowest order modes are not reflected at waveguide terminations. They also circumvent imperfections provided the latter do not flip the spin polarization. The proposed waveguide can be realized by bianisotropic materials comprising metallic omega particles [13,27] or split-ring resonators [28] that are reciprocal and do not violate time-reversal symmetry. Bianisotropic mediums can also be implemented through photonic metamaterials [29], consisting of plasmonic [30,31] or dieletrics [32,33] components.

The remaining part of this paper is organized as follows. In section 2, the mapping presented in [23] is reviewed and it is used to obtain the equations for longitudinal components of electric and magnetic fields inside the waveguide in section 3. Section 4 investigates the propagation of rotationally-symmetric surface waves at the cylindrical interface between two bianisotropic mediums with opposite magnetoelectric coupling constants and the results are verified using full-wave simulations. These simulations demonstrate that the unidirectional waves can propagate through obstacles without any reflections. In section 5, we analyze the higher-order modes and their dispersion diagram and calculate their field profiles. section 6 shows how the desired medium can be realized using $\Omega$-particles. The paper will be concluded in section 7.

## 2. Field equations in bi-anisotropic media with antisymmetric magneto-electric coupling tensors

The constitutive relations between the electric ($\mathbf {D}$) and magnetic ($\mathbf {B}$) flux densities and the electric ($\mathbf {E}$) and magnetic ($\mathbf {H}$) fields in a general bianisotropic medium can be expressed as

Here $i,k=x,y,z$, ${\bar {\bar {\mathcal {{\epsilon }}}}}$ and ${\bar {\bar {\mathcal {{\mu }}}}}$ are the permittivity and permeability tensors, respectively. ${\bar {\bar {\mathcal {{\chi }}}}}$ and ${\bar {\bar {\mathcal {{\chi }}}}}'$ denote the magnetoelectric coupling tensors. In a reciprocal medium It was shown in [23] that if (i) ${\bar {\bar {\mathcal {{\epsilon }}}}}$ and ${\bar {\bar {\mathcal {{\mu }}}}}$ satisfy everywhere inside the medium with $\zeta$ a constant reference impedance, and (ii) ${\bar {\bar {\mathcal {{\chi }}}}}$ is antisymmetric, i.e. , then one can introduce the pseudo-electric and pseudo-magnetic fields [23] that satisfy Maxwell’s equations in a fictitious medium without magneto-electric coupling, but with the effective, gyrotropic permittivity and permeability tensors Using the above formulation, it was shown in [23] that the interface between two bianisotropic half-spaces, whose magneto-electric coupling coefficients vanish except for ${\chi }_{yz}=-{\chi }_{zy}=\chi$, supports unidirectional surface waves if $\chi$ takes opposite signs in the two media [see Fig. 1]. The waves are of the Te or Tm type in which either the pseudo-electric or pseudo-magnetic field has no component in the direction of propagation (along $z$). They propagate along the interface in opposite directions and their associated electromagnetic field decays exponentially away from the interface.## 3. Bianisotropic cylindrical waveguide

A configuration comprising two infinite media hardly qualifies as a practical waveguide due to its infinite cross section. As a natural generalization of [23], in this work we consider the waveguide depicted in Fig. 2. It comprises a bi-anisotropic cylinder with radius $\rho _0$ that is enclosed by another bi-anisotropic, cylindrical layer. Here, however, the constitutive relations (1) and (2) relate cylindrical, rather than Cartesian field components at every point inside the bi-anisotropic media, i.e., $i,k=\rho ,\phi ,z$ in (1) and (2). We assume diagonal, uniaxial permittivity and permeability tensors in this representation,

In order to analyze the waveguide of Fig. 2, we look for solutions of Maxwell’s equations that propagate with a propagation constant $\beta$ in the $z$ direction in each region. Given the cylindrical symmetry of the structure, we express the pseudo-fields as

## 4. Unidirectional rotationally symmetric ($m=0$) mode

Equations for $e_\phi ^{m}(\rho )$ and $h_\phi ^{m}(\rho )$ are coupled for all values of $m$ except for $m=0$. In the case of $m=0$, where the fields are independent of $\phi$, two separate sets of equations are obtained. The first set of equations ($\mathrm {Te}_{0}$ mode), involves the field components $e^{0}_{\phi }(\rho )$, $h^{0}_{\rho }(\rho )$, and $h_{z}^{0}(\rho )$:

We shall take the parameters $\nu$ and $\nu _\phi$ to be positive. However, $\nu$ and $\chi$ will be chosen such $\nu _\perp <0$ [see(21)]. As a result, $\nu _\phi \nu _\perp$ becomes negative, leading to a real-valued $\kappa$ in (32). The behaviour of $M_{a,b}(u)$ and $W_{a,b}(u)$ resembles that of the modified Bessel function $I(u)$ and $K(u)$, respectively, for real arguments [35]. Therefore, $M_{a,b}(\kappa \rho ), W_{a,b}(\kappa \rho )$, behave like exponentially decaying or growing functions with increasing values of $\rho$.

Let us first, for simplicity, assume that the outer bi-anisotropic layer is infinitely thick. The sign of $\chi$ is taken to be positive for $\rho >\rho _0$ and negative for $\rho <\rho _0$ [see Fig. 2]. The function $W_{a,b}(\kappa \rho )$ diverges at $\rho =0$, whereas $M_{a,b}(\kappa \rho )$ diverges as $\rho \rightarrow \infty$. Starting with the $\mathrm {Te}_{0}$ mode, we express $e_{\phi }$ as

By demanding $e_{\phi }$ and $h_{z}$ to be continuous across $\rho =\rho _{0}$, and using the relations [35]:

The analysis of the $\mathrm {Tm}_{0}$ mode is nearly identical. One has

The dispersion diagram of unidirectional $\mathrm {Te}_{0}$ and $\mathrm {Tm}_{0}$ modes is shown in Fig. 3 for $\rho _0=150\,\mu {m}$, $\nu =1.3613\sqrt {\mu _0\epsilon _0}$, $\nu _\phi =\sqrt {\mu _0\epsilon _0}$ and $\chi =2.1029\sqrt {\mu _0\epsilon _0}$. The parameters of the bi-anisotropic media are assumed to be frequency-independent, for simplicity. As shown in Fig. 3, the $\mathrm {Te}_{0}$ mode can only propagate in the $+z$ direction ($\beta >0$), whereas the $\mathrm {Tm}_{0}$ mode can only propagate in the $-z$ direction ($\beta <0$). Both modes are, therefore, unidirectional. Figure 4 shows $e_\phi$ in $\mathrm {Te}_{0}$ mode and $h_\phi$ in $\mathrm {Tm}_{0}$ mode as a function of radial coordinate $\rho$ at $f=1\,\mathrm {THz}$. The surface wave nature of these unidirectional modes is clearly visible as the field decays away from the interface at $\rho =\rho _0$.

In order to validate our analytical results, full-wave simulations have been carried out using COMSOL Multiphysics. It is worth mentioning that direct inclusion of bi-anisotropic materials is usually not allowed in available commercial electromagnetic software. Nevertheless, by modification of the built-in constitutive relations in COMSOL Multiphysics we were able to run the simulations in the 2D axisymmetric module. As the first simulation, a source consisting of a ring of electric current is placed at the center of waveguide. In order to verify the dispersion diagrams obtained, an infinitely long waveguide must, in principle, be considered. Since in the actual simulation we were restricted to waveguides of finite length, loss was gradually added to the bianisotropic material in order to let the excited waves decay before reaching the terminations, thereby discarding the termination effects at the first instance. The parameters used in the simulation were $\nu =1.3613\left (1-jf(z)\right )\sqrt {\mu _0\epsilon _0}$, $\nu _\phi =\left (1-jf(z)\right )\sqrt {\mu _0\epsilon _0}$ and $\chi =2.1029\left (1-jf(z)\right )\sqrt {\mu _0\epsilon _0}$ where the function $f(z)$ is zero up to a specific point along the guide after which it gradually increases. The external radius of the outer layer of the waveguide is taken to be $450\,\mu {m}$ in COMSOL simulations. The waveguide is assumed to be surrounded by air. Figure 5 shows the real part of $e_\phi$ ($\mathrm {Te}_{0}$ field) and $h_\phi$ ($\mathrm {Tm}_{0}$ field) at $f=1\,\mathrm {THz}$. Obviously, Te and Tm modes can only propagate in a single direction as expected. Due to the loss added, their amplitude is gradually reduced until the waves totally vanish. Furthermore, the dispersion diagram and mode profile of Te and Tm polarization obtained using COMSOL have been plotted in Fig. 3 and Fig. 4. Good agreement is observed between analytical and simulated results. Take note that although we took the thickness of the second bianisotropic layer to be infinite in our analytical calculations, the results do not significantly change if this assumption is dropped provided that the outer layer is thick enough, as demonstrated by COMSOL simulations. This is because the waves considered are surface waves that decay away from the interface between the two bi-anisotropic layers.

As the second test, a rotationally symmetric obstacle was placed on the propagation path of unidirectional waves as shown in Fig. 6. The obstacle must satisfy the condition (4) in order to prevent the so-called spin-flip between Te and Tm modes (Te mode scattered into Tm mode and vice versa). As shown in Fig. 6(a), the unidirectional surface waves simply pass through a dielectric obstacle ($\epsilon _0$, $\mu _0$) without any reflection. Besides, these waves can follow the contour of the interface even in case of sharp discontinuities as shown in Fig. 6(b). In order to investigate the behaviour of the electromagnetic field near the obstacle in more detail, we have plotted the amplitude of $e_\phi$ as function of $z$ near the interface for the two cases considered above in Fig. 7. The field amplitude rises sharply near the obstacle which points to accumulation of electromagnetic energy. This phenomenon, which is caused by impossibility of reflection, is also observed for unidirectional waves in gyrotropic structures [36].

Finally, it is interesting to investigate a waveguide consisting of only one bianisotropic cylinder with radius $\rho _0$ surrounded by a homogeneous medium characterized by $\epsilon _d$ and $\mu _d$. The dispersion equations of this waveguide can be obtained using the same technique described above. For the $\mathrm {Te}_{0}$ mode, one has

## 5. Higher-order waveguide modes

The previous section was mainly concerned with the zero-order, unidirectonal modes of the waveguide of Fig. 2. In what follows we present numerical results obtained for the higher-order modes. Unlike the zero-order mode, however, a separation of Te and Tm modes is impossible for $m\neq 0$.

To calculate the dispersion diagrams of higher-order modes, we should determine those values of $\beta$, for which the homogeneous system of Eqs. (22) and (23) has non-zero solutions at a given frequency and a given $m$. To that end, we consider the regions $\rho <\rho _0$ and $\rho >\rho _0$, separately. For a given $\beta$, (22) and (23) can be solved in the region $\rho >\rho _{0}$ if $e_{\phi },h_{\phi }$ are specified at $\rho =\rho _{0}$. (We again assume the outer bi-anisotropic layer to be infinitely thick for simplicity and assume the electromagnetic field to vanish as $\rho \rightarrow \infty$.) The calculation is performed using an in-house developed code based on the finite element method (FEM), and the result obtained is used to evaluate $e_{z},h_{z}$ at $\rho =\rho _{0}$. Since we are dealing with linear equations, it follows that

Figure 8 shows the propagation constants of higher-order modes $0<m\leq 10$ at $f=1\,\mathrm {THz}$ for $\rho _0=150\,\mu {m}$, $\nu =1.3036\sqrt {\mu _0\epsilon _0}$, $\nu _\phi =\sqrt {\mu _0\epsilon _0}$ and $\chi =2.1029\sqrt {\mu _0\epsilon _0}$ calculated using proposed numerical method. For the sake of comparison, COMSOL results are also included. Propagation constant is increased by increasing the mode number. This is unlike conventional waveguides, but quite common in gyrotropic surface waveguides [8].

The Te and Tm modes cannot be strictly separated for higher-order modes. Nonetheless, in those modes propagating in the $+z$-direction, the $e_\phi$ component has a much larger amplitude than the $h_{\phi }$ component. The opposite situation occurs for modes propagating in the $-z$-direction. Figure 9 shows the mode profile of the waves with $m=2$ as an example. It should be noted that the field profile of the modes propagating toward positive $z$-direction is exactly the same for modes propagating toward negative $z$-direction by exchanging $e_\phi$ and $h_\phi$ as expected from (22) and (23). Hence, although higher-order modes are Te-Tm hybrids, they practically behave as Te or Tm modes depending on the direction of propagation.

## 6. Practical realization of bianisotropic metamaterial

A possible method for implementing bi-anisotropic media is by means of $\Omega$ particles embedded in a dielectric background as shown in Fig. 10 schematically [13]. This arrangement consists of arrays of horizontal (brown colour) and vertical (black colour) $\Omega$ particles and is repeated in the $z$-direction with a period of $D_c$. The distance between $\Omega$ particles in the $\phi$-direction is $D_c$, too. But the distance between $\Omega$ particles in the $\rho$-direction for vertical and horizontal $\Omega$ particles are $D_c/{2\pi }$ and $D_c$, respectively. With such arrangement for $\Omega$ particles, a medium is obtained with effective permittivity, permeability and magnetoelectric coupling tensors given by [24]:

It should be noted that due to the material dispersion, the use of $\omega$ particles results in a narrow-band design because the condition in (4) can be satisfied only for the frequency

However, this limitation is due to use of $\Omega$-particles, not the operation principle of the waveguide.The $\Omega$ particles were designed for operation at $f=1\,\mathrm {THz}$ assuming that the background medium is air. We took $\epsilon _b=\epsilon _0$ and $\mu _b=\mu _0$, and assumed the particles to be built from Aluminum with a conductivity of $\sigma =36.398\times 10^6\,S/m$. The dimensions of $\Omega$ particles are $r_0=1\,\mu {m}$, $a=16.98\,\mu {m}$, and $l=16.98\,\mu {m}$ to satisfy the condition in (57). A volume concentration of $N=3.94\times 10^{13}\,1/m^3$ was assumed that corresponds to $D_c=56.94\,\mu {m}$. This design leads to $\epsilon _t=(1.3613-j0.0029)\epsilon _0$, $\mu _t=(1.3613-j0.0029)\mu _0$ and $\chi =(2.1029-j0.0166)\sqrt {\mu _0\epsilon _0}$. As a result, $\nu =(1.3613-j0.0029)\sqrt {\mu _0\epsilon _0}$, $\nu _\phi =\sqrt {\mu _0\epsilon _0}$, and $\nu _\perp =(-1.8873+j0.0418)\sqrt {\mu _0\epsilon _0}$. We used these values in previously shown simulation results throughout the paper. It is worth noting that by applying a $180^\circ$ twist to the $\Omega$ particle loops, the sign of the parameter $\chi$ is reversed.

## 7. Conclusion

We presented a theoretical study of surface wave propagation at the cylindrical interface between two bianisotropic media with anti-symmetric magnetoelectric coupling tensors. The analysis was carried out using a transformation which allows us to model the bianisotropic medium as an anisotropic medium with gyrotropic permittivity and permeability tensors that operate on pseudo-electromagnetic fields. It was shown that the rotationally symmetric modes are unidirectional with pseudo-electric and pseudo-magnetic polarizations propagating in opposite directions. The results were verified by performing full-wave simulations using a commercial EM solver. Moreover, using the latter, it was demonstrated that the unidirectional waves could pass through obstacles and discontinuities without any reflection. The higher-order modes of the waveguide were also investigated. Although these modes are hybrids, and not strictly unidirectional, they have similar behavior to Te or Tm modes depending on the direction of propagation.

## Funding

Iran National Science Foundation (97012482).

## Disclosures

The authors declare no conflicts of interest.

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