1. Introduction

Topological insulators are materials that are insulating in the bulk but possess conducting states on their surfaces [1,2]. They are a new phase of matter characterized by integer quantities known as topological invariants, which remain constant under arbitrary continuous deformations of the system. A well understood form of the topological phase is the Quantum Hall (QH) state [3], a two-dimensional electron gas in a static magnetic field, which belongs to a topological class that breaks the time-reversal (TR) symmetry. A different topological class that preserves the TR symmetry is the Quantum spin Hall (QSH) state [4,5], in which the spin-orbit interaction is responsible for the topological character. Theoretical concepts developed in the QSH states are generalized to three dimensions, leading to the more general class of topological insulators [6].

The most intriguing property of a topological insulator is the emergence of a pair of helical edge states in the band gap, which are protected by TR symmetry [7]. The two states with opposite spin counterpropagate at a given edge without backscattering even in the presence of disorder. The existence of edge states is determined by the topological structure of the bulk states, characterized by the $Z_2$ topological invariant [8] or spin Chern number [9], a feature known as the bulk-edge correspondence [1,2]. The robust transport of conducting states on a topological insulator may find potential applications in antenna arrays [10] and on-chip communications [11].

The novel concepts of topological phases were extended to photonic systems [12–14], and a variety of photonic topological insulators have been explored [15–26]. In photonic systems, the spin-orbit interaction can be emulated by the magnetoelectric couplings in bianisotropic media, which is manifest on the entanglement between the phase relationship in waves (spin state) and the polarization of dipole moment (orbital state) [19]. In photonic metamaterials [27–31], the bianisotropic response can further be engineered in artificial metamolecules modelled by metallic helices [32] or split rings [33].

In a topological insulator, the helical edge states form a so-called Kramers pair, which are doubly degenerate and TR partners to each other [34]. In the presence of spin-orbit interaction, the degeneracy of the Kramers pair is lifted and the phase becomes topologically nontrivial. The Kramers degeneracy theorem, however, is usually valid for a TR invariant system with spin 1/2 [7] and cannot readily apply to the photonic system with spin 1, unless additional symmetry has been imposed. In photonic systems, the ‘spin’-degenerate condition is introduced to form two pseudospin states by the linear combinations of the electric and magnetic fields [19], so that the system can be described by an effective Hamiltonian consisting of two subsystems for the pseudospin states [19,21,23]. This concept has also been employed in photonic metamaterials [31,35,36] to investigate the topological properties.

In the present study, we analyze the photonic topological phases in bianisotropic metamaterials characterized by a lossless and reciprocal magnetoelectric tensor. By introducing the pseudospin basis, the underlying medium can be described by a pair of spin-orbit Hamiltonians with spin 1 [31,35,37–39], which result in nonzero spin Chern numbers that determine the topological properties. In the presence of the chirality parameter, a nontrivial band gap is opened between the bulk modes, in which surface modes may exist. According to Maxwell’s boundary conditions, the surface modes at the interface between two bianisotropic media with opposite chirality are analytically formulated as elliptic or hyperbolic equations. In particular, two branches of hyperbolic surfaces in the band gap are degenerate at the frequency where the chiral nihility occurs, which depict the helical nature of edge states between two distinct topological phases. For illustration, the surface modes at an irregular boundary between two bianisotropic media are excited by a circularly polarized source. The surface waves propagate toward opposite directions for different handednesses of circular polarization, which are able to bend around sharp corners without backscattering.

2. Basic equations

2.1 Bulk modes

Consider a bianisotropic medium characterized by the constitutive relations:

The existence of a nontrivial solution of ${\mathbf E}$ and ${\mathbf H}$ requires that the determinant of the 6 $\times$ 6 matrix in Eq. (3) be zero, which gives the characteristic equation of the bulk modes as

Note that the features of bulk modes may change with the frequency for a dispersive medium (which is usually the case of metamaterials), depending on the choice of frequency range. In the neighborhood of a reference frequency $\omega _\text {ref}$, ${\varepsilon _n}$ ($n=t,z$) can be approximated as ${\varepsilon _n} \approx {\varepsilon _{n0}} + {\left. {\frac {{d{\varepsilon _n}}}{{d\omega }}} \right |_{\omega = {\omega _\text {ref}}}}\left ( {\omega - {\omega _\text {ref}}} \right ) \equiv {\varepsilon _{n0}} + {\tilde \varepsilon _n}\delta \omega /{\omega _\text {ref}}$, where ${\tilde \varepsilon }_n$ is positive definite [37]. A similar relation is valid for $\mu _n$ ($n=t,z$). In addition, we assume that the chirality parameter $\gamma$ vary smoothly around $\omega _{\rm ref}$ and can be treated as a constant for the first-order approximation [19,31].

2.2 Spin-orbit Hamiltonians

The electromagnetic duality of Maxwell’s equations dictates that the matrix in Eq. (3) holds a symmetric pattern when the degenerate condition $\underline \varepsilon =\underline \mu$ is satisfied. Here, we assume that the chiral nihility [43,44] occurs at the reference frequency $\omega _{\rm ref}$, that is, $\underline {\varepsilon }(\omega _{\rm ref})=\underline {\mu }(\omega _{\rm ref})=0$. This enables us to rewrite the wave equations as

Note that Eq. (8) is formulated as an eigensystem with $\delta \omega$ being the eigenvalue. The Hamiltonian ${\cal H}_\pm$ in Eq. (9) represents a modified form of the spin-orbit interaction ${\mathbf {k}}\cdot {\mathbf {S}}$ with spin 1, which is mathematically equivalent to the Hamiltonian of a magnetic dipole moment in the magnetic field [35,37].

2.3 Topological invariants

The topological properties of the spin-orbit Hamiltonians ${\cal H}_\pm$ can be characterized by the topological invariants obtained from the eigenfields. For this purpose, we calculate the Berry flux for the Hamiltonian over a closed surface in the wave vector space. The eigensystem for the spin-orbit Hamiltonian

The nonzero $C_\sigma$ ($\sigma =\pm 1$) characterize the topological properties of the system, where $\sigma$ refers to the helicity (or handedness) of the pseudospin states. In particular, the helical edge states are topologically protected, which means that their existence is guaranteed by the difference in band topology on two sides of the interface. In this system, the total Chern number $C=\sum\limits _\sigma {{C_\sigma }}=0$ and the spin Chern number $C_{\rm spin}=\sum\limits _\sigma {{\sigma C_\sigma }}=4$, which are consistent with the quantum spin Hall effect of light [45]. The topological invariants remain unchanged under arbitrary continuous deformations of the system. The topological properties of the isotropic medium will be retained when a certain anisotropy is included. For a more general anisotropic medium, the exact calculation of topological invariants can be obtained by the numerical integration of Berry curvatures [30].

The Hamiltonian ${\cal H}_\pm$ in Eq. (10) has degenerate eigenvalues: $\lambda ^{\sigma }_+=\lambda ^{\sigma }_-\equiv \lambda _\sigma$, and the combined eigensystem is written as

In Eq. (12), the combined system consists of two copies of the spin-orbit Hamiltonian, which is characteristic of the Bernevig-Hughes-Zhang (BHZ) model for the QSH system [5]. In particular, the combined Hamiltonian ${\cal H}_c$ formed by $\cal H_\pm$ is TR invariant under $T_p$ (see Appendix C):

2.4 Surface modes

Let the $xz$ plane be an interface between two bianisotropic media with opposite chirality, characterized by $\varepsilon _n$, $\mu _n$ ($n=t,z$), and $\pm \gamma$, at which the surface modes may exist. According to Maxwell’s boundary conditions: the continuity of tangential electric and magnetic field components at the interface, the characteristic equation of surface modes is given by (see Appendix D)

3. Results and discussion

3.1 Bulk modes

Let the frequency dependence of the bianisotropic medium be characterized by the Lorentz-type dispersive model: ${\varepsilon }_n = {\varepsilon _{n\infty }} - \omega _{p}^{2}/\left ( {{\omega ^{2}} - \omega _0^{2}} \right )$, ${\mu _n} = {\mu _{n\infty }} - \Omega _{{n}}\omega ^{2}/\left ( {{\omega ^{2}} - \omega _0^{2}} \right )$ ($n=t,z$) [29], and ${\gamma } = {\Omega _{xy}} \omega {\omega _p}/\left ( {{\omega ^{2}} - \omega _0^{2}} \right )$ [19,46,47], which is usually adopted in the study of metamaterials. Here, $\omega _p$ is the plasma frequency and $\omega _0$ is the resonance frequency. This model guarantees that the energy density in the dispersive medium is positive definite (see Appendix E).

In the present problem, we assume that the chiral nihility occurs at the reference frequency $\omega _{\rm ref}$, that is, $\varepsilon _n(\omega _{\rm ref})=\mu _n(\omega _{\rm ref})=0$ ($n=t,z$). It follows that $\varepsilon _t\varepsilon _z> 0$ and $\mu _t\mu _z> 0$ for either $\omega >\omega _{\rm ref}$ or $\omega <\omega _{\rm ref}$. It can be seen from Eq. (5) that the coefficients of $k_t^{2}$ and $k_z^{2}$ have the same sign, and the term $\varepsilon _t\mu _t-\gamma ^{2}$ plays a crucial role in determining the features of bulk modes. For a nonzero $\gamma$, there exists two frequencies $\omega _1$ and $\omega _2$, at which $\varepsilon _t\mu _t-\gamma ^{2}=0$ (see Appendix F). In terms of the frequency range, the bulk modes of the bianisotropic medium can be classified into three phases:

 

Fig. 1. Bulk modes of the bianisotropic metamaterial with $\varepsilon _{t\infty }=\varepsilon _{z\infty }=2.75$, $\mu _{t\infty }=1.75$, $\mu _{z\infty }=1.156$, ${\Omega _{ t}} = 0.7875$, ${\Omega _{ z}} = 0.5203$, $\omega _p/\omega _0=1.5$, and (a) $\Omega _{xy}=0$ and (b) $\Omega _{xy}=1.2$. Orange plane is the longitudinal mode. Greed dot is the Dirac point.

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For a nonzero chirality as shown in Fig. 1(b), a band gap belonging to phase (II) is opened between the upper and lower elliptic surfaces. The divergent lower surfaces in phase (I) that occur near the resonance frequency $\omega _0$ [cf. Figure 1(a)] do not appear since this frequency is now located inside the band gap. Note that the band gap is topologically nontrivial as a consequence of the nonzero topological invariants for the Hamiltonian of the photonic system (cf. Sec. 2.3). It will be shown later that surface modes exist in the nontrivial band gap, which is characteristic of the helical edge states and topological phase transitions in the photonic system.

3.2 Surface modes

Compared to the bulk modes, surface modes at the interface between two bianisotropic media with opposite chirality has a more complex expression [cf. Equation (15)]. Nevertheless, the surface modes for the isotropic case ($\varepsilon _t=\varepsilon _z\equiv \varepsilon$ and $\mu _t=\mu _z\equiv \mu$) can be rearranged as a standard quadratic equation [cf. Equation (16)]. In this situation, the features of surface modes are determined by the terms $\varepsilon \mu$ and $\varepsilon \mu -\gamma ^{2}$. Because of the occurrence of chiral nihility at $\omega _{\rm ref}$, $\varepsilon \mu >0$ for either $\omega >\omega _{\rm ref}$ or $\omega <\omega _{\rm ref}$. The other term $\varepsilon \mu -\gamma ^{2}$ thus plays a decisive role in determining the features of surface modes. It is noted that Eq. (15) has the same type of dispersion as Eq. (16) (see Appendix D). Similar to the bulk modes, the surface modes of the bianisotropic medium can be classified into three phases:

 

Fig. 2. (a) Surface modes at the interface between two bianisotropic metamaterials with opposite chirality ($\Omega _{xy}=\pm 1.2$), the rest material parameters being the same as in Fig. 1(b). Bulk modes are outlined in mesh for comparison. (b) Projection of surface modes (and bulk modes) on the $\omega -k_x$ plane at $k_z c/\omega _p=0.35$. Green and gray curves are surface and bulk modes, respectively. Green dot is the intersecting point of two surface curves. Light blue region is the band gap.

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For clarity, the projection of surface modes on the $\omega -k_x$ plane at a fixed $k_z$ is shown in Fig. 2(b). In phases (I) and (III), the surface modes are located next to the bulk modes, with similar elliptic dispersions. Notice that the surface and bulk modes are degenerate at $\omega _1$ and $\omega _2$, as a result of the common factor $\varepsilon _t\mu _t-\gamma ^{2}$ that appears in Eqs. (5) and (15). In phase (II), the two branches of hyperbolic surface meet at the reference frequency $\omega _{\rm ref}$, where the chiral nihility occurs. It is interesting to note that the intersecting point coincides with the Dirac point in the bulk modes for a zero chirality [cf. Figure 1(a)]. The two surface modes, which are degenerate at $\omega _{\rm ref}$, have the characteristic of helical edge states that occur between two distinct topological phases, their existence being consistent with the bulk-edge correspondence [1,2]. In this situation, the bianisotropic medium is regarded as a photonic analogue of the QSH system. The helical nature of the edge states is in accordance with the opposite helicity of the eigenstates for the spin-orbit Hamiltonian (cf. Sec. 2.2). The combined Hamiltonian respects the pseudo TR symmetry, leading to the topological protection of helical edge states in the photonic system (cf. Sec. 2.3).

Finally, the topological features of the bianisotropic metamaterial are illustrated with the propagation of surface modes at an irregular boundary [27,30,31,35,36]. For this purpose, a dipole source is placed at the interface between two bianisotropic metamaterials with opposite chirality to excite the surface modes in the their common band gap (where $\varepsilon \mu -\gamma ^{2}<0$), so that the waves are evanescent away from the interface on both sides. In Fig. 3, a pair of surface modes are excited with right- or left-handed circular polarizations (see Appendix G), which correspond to the opposite helicity of helical edge states. The surface waves propagate unidirectionally to the right or left along an irregular boundary with sharp corners. In particular, the surface waves counterpropagate at the boundary for different handednesses of circular polarization, showing the helical nature of edge states. The surface waves are able to bend around sharp corners without backscattering, which demonstrates that the edge states are topologically protected. Recall that the topological phase is nontrivial owing to the nonzero topological invariants (cf. Sec. 2.3). The topological edge states exist as a consequence of the bulk-edge correspondence.

 

Fig. 3. Surface wave propagation at the interface between two bianisotropic metamaterials with opposite chirality, where $\varepsilon _n=\mu _n=0.3$ ($n=t,z$), $\gamma =\pm 1$, and $k_z/k_0=0.7$ for (a) right-handed and (b) left-handed circular polarization. Green dot is the position of dipole source. Circular arrow denotes the handedness of circular polarization. Red and blue colors correspond to positive and negative values of Re[$E_z$], respectively.

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4. Concluding remarks

In conclusion, we have analyzed the photonic topological phases in lossless and reciprocal bianisotropic metamaterials. The photonic system is described by a pair of spin-orbit Hamiltonians with spin 1 based on the pseudospin basis, and the topological properties are characterized by the nonzero spin Chern numbers. A nontrivial band gap is opened between the bulk modes in the presence of the chirality parameter, in which the surface modes exist at the interface between two bianisotropic media with opposite chirality, and depict the typical features of helical edge states between two distinct topological phases. The underlying bianisotropic medium is regarded as a photonic analogue of the QSH system that respects that pseudo TR symmetry. The topological features of bianisotropic metamaterials are illustrated with the robust transport of surface modes at an irregular boundary.

Appendix

A. Spin-orbit Hamiltonians

The wave equation for ${\mathbf E}$ and ${\mathbf {H'}}$ in Eq. (7) can be rewritten as

(17)$${\tilde {\mathcal{H}_ \pm} }{{\tilde {\psi}} _ \pm } = {\tilde {\cal D } }{\tilde {\psi _ \pm} },$$

where

(18)$${\tilde {\mathcal{H}_ \pm} } = c\left( {\begin{array}{ccc} {\pm{k_z}} & {\frac{{\pm{k_x} - i{k_y}}} {{\sqrt 2 }}} & 0 \\ {\frac{{\pm{k_x} + i{k_y}}} {{\sqrt 2 }}} & 0 & {\frac{{\pm{k_x} - i{k_y}}} {{\sqrt 2 }}} \\ 0 & {\frac{{\pm{k_x} + i{k_y}}} {{\sqrt 2 }}} & { \mp {k_z}} \end{array} } \right),$$

(19)$${\tilde {\cal D } } = \omega \left( {\begin{array}{ccc} {{\varepsilon _t} - i{\gamma}} & 0 & 0 \\ 0 & {{\varepsilon _z}} & 0 \\ 0 & 0 & {{\varepsilon _t} + i{\gamma}} \\ \end{array}} \right),$$

and ${\tilde \psi _ + } = {\left (- {\frac {{ {E_x} - i{E_y}}}{{\sqrt 2 }},{E_z},\frac {{{E_x} + i{E_y}}}{{\sqrt 2 }}} \right )^{T}}$ and ${\tilde \psi _ - } = {\left ( -{\frac {{ {H_x'} + i{H_y'}}} {{\sqrt 2 }},{H_z'},\frac {{{H_x'} - i{H_y'}}}{{\sqrt 2 }}} \right )^{ T}}$ are the basis of the pseudospin states that include a $\pi /2$ phase difference between the transverse field components (with respect to the optical axis of the medium) [37]. In the neighborhood of the reference frequency $\omega _\text {ref}$, ${\varepsilon _n}$ ($n=t,z$) can be approximated as ${\varepsilon _n} \approx {\varepsilon _{n0}} + {\left. {\frac {{d{\varepsilon _n}}}{{d\omega }}} \right |_{\omega = {\omega _\text {ref}}}}\left ( {\omega - {\omega _\text {ref}}} \right ) \equiv {\varepsilon _{n0}} + {\tilde \varepsilon _n}\delta \omega /{\omega _\text {ref}}$, where ${\tilde \varepsilon }_n$ is positive definite [37]. Equation (17) is rearranged as

(20)$${\mathcal{H}_ \pm' }{\psi _ \pm } - {\cal D' }{\psi _ \pm } = \delta \omega {\psi _ \pm },$$

where

(21)$${{\cal{H}}_ \pm' } = \frac{c} {{\sqrt {{{\tilde \varepsilon }_t}{{\tilde \varepsilon }_z}} }}\left( {\begin{array}{ccc} {\sqrt {\frac{{{{\tilde \varepsilon }_z}}} {{{{\tilde \varepsilon }_t}}}} \left( { \pm {k_z} + \frac{{i{\omega _{{\text{ref}}}}\gamma }} {c}} \right)} & {\frac{{ \pm {k_x} - i{k_y}}} {{\sqrt 2 }}} & 0 \\ {\frac{{ \pm {k_x} + i{k_y}}} {{\sqrt 2 }}} & 0 & {\frac{{ \pm {k_x} - i{k_y}}} {{\sqrt 2 }}} \\ 0 & {\frac{{ \pm {k_x} + i{k_y}}} {{\sqrt 2 }}} & {\sqrt {\frac{{{{\tilde \varepsilon }_z}}} {{{{\tilde \varepsilon }_t}}}} \left( { \mp {k_z} - \frac{{i{\omega _{{\text{ref}}}}\gamma }} {c}} \right)} \\ \end{array} } \right),$$

(22)$${\cal D' } = {\omega _{{\text{ref}}}}\left( {\begin{array}{ccc} {\frac{{{\varepsilon _{t0}}}} {{{{\tilde \varepsilon }_t}}}} & 0 & 0 \\ 0 & {\frac{{{\varepsilon _{z0}}}} {{{{\tilde \varepsilon }_z}}}} & 0 \\ 0 & 0 & {\frac{{{\varepsilon _{t0}}}} {{{{\tilde \varepsilon }_t}}}} \\ \end{array}} \right),$$

and ${\psi _ \pm } = {U^{ - 1}}{\tilde \psi _ \pm }$ with $U = {\rm {diag}}\left ( {\sqrt {{{\tilde \varepsilon }_z}/{{\tilde \varepsilon }_t}} ,1,\sqrt {{{\tilde \varepsilon }_z}/{{\tilde \varepsilon }_t}} } \right )$. In the isotropic case, where ${\varepsilon _{t0}} = {\varepsilon _{z0}} \equiv \varepsilon$, ${\tilde \varepsilon _t} = {\tilde \varepsilon _z} \equiv \tilde \varepsilon$, Eq. (20) is simplified to

(23)$${\mathcal{H}_ \pm }{\psi _ \pm } - {\cal D }{\psi _ \pm } = \delta \omega {\psi _ \pm },$$

where ${\cal D } = {\omega _{{\text {ref}}}}\left ( {{\varepsilon / {\tilde \varepsilon }}} \right )$ and

(24)$${\cal{H}}_ +{=} \alpha \left( {{\mathbf{k}} \cdot {\mathbf{S}} + \frac{{i{\omega _{{\text{ref}}}}\gamma }} {c}{S_z}} \right){\text{ }},{\text{ }}{{\cal{H}}_ - } ={-} \alpha {\left( {{\mathbf{k}} \cdot {\mathbf{S}} + \frac{{i{\omega _{{\text{ref}}}}\gamma }} {c}{S_z}} \right)^{*}},$$

with $\alpha =c/{{\tilde \varepsilon }}$, ${\mathbf k}=k_x\hat x+k_y\hat y+k_z\hat z$, ${\mathbf {S}} = {S_x}\hat x + {S_y}\hat y + {S_z}\hat z$, and

(25)$${S_x} = \frac{1}{{\sqrt 2 }}\left( {\begin{array}{ccc} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \\ \end{array}} \right),\quad {S_y} = \frac{1}{{\sqrt 2 }}\left( {\begin{array}{ccc} 0 & { - i} & 0 \\ i & 0 & { - i} \\ 0 & i & 0 \\ \end{array}} \right),\quad {S_z} = \left( {\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & { - 1} \\ \end{array}} \right)$$

being the spin matrices for spin 1.

B. Topological invariants

In terms of the spherical coordinates, the Hamiltonian ${\cal H}_\pm$ [cf. Equation (24)] is rewritten as

(26)$${{\cal{H}}_ \pm } ={\pm} \frac{{\alpha {|{\mathbf k}|}}} {{\sqrt 2 }}\left( {\begin{array}{ccc} {\sqrt 2 \cos \theta } & {{e^{ {\mp} i\phi }}\sin \theta } & 0 \\ {{e^{ {\pm} i\phi }}\sin \theta } & 0 & {{e^{ {\mp} i\phi }}\sin \theta } \\ 0 & {{e^{ {\pm} i\phi }}\sin \theta } & { - \sqrt 2 \cos \theta } \\ \end{array}} \right),$$

where $k_x= |{\mathbf k}|\sin \theta \cos \phi$, $k_y= |{\mathbf k}|\sin \theta \sin \phi$, and ${{k_z} \pm \frac {{i{\omega _{{\rm {ref}}}}\gamma }}{c}}= |{\mathbf k}|\cos \theta$. Here, $\theta$ and $\phi$ are the polar and azimuthal angles, respectively, on the closed surface $S$: $k_x^{2} + k_y^{2} + {\left ( {{k_z} \pm \frac {{i{\omega _{{\rm {ref}}}}\gamma }}{c}} \right )^{2}}= |{\mathbf k}|^{2}$. The eigensystem for the Hamiltonian ${\cal H}_\pm$:

(27)$${\mathcal{H}_ \pm }\psi _ \pm ^{\sigma} =\lambda _ \pm ^{\sigma} \psi _ \pm ^{\sigma}$$

is solved to give the eigenvalues $\lambda _ \pm ^{\sigma } = \alpha |{\mathbf k}|\sigma$ ($\sigma =\pm 1, 0$) and the normalized eigenvectors as

(28)$$ \psi _ \pm ^{\sigma} = \frac{1} {2}\left( {\begin{array}{c} {\pm\sigma {e^{ {\mp} 2i\phi }}\left( {\pm\sigma + \cos \theta } \right)} \\ {\pm\sigma \sqrt 2 {e^{ {\mp} i\phi }}\sin \theta } \\ {1 \mp \sigma \cos \theta } \\ \end{array} } \right)\quad(\sigma={\pm} 1),$$

(29)$$ \psi _ \pm ^{\sigma} = \frac{1} {{\sqrt 2 }}\left( {\begin{array}{c} { - {e^{ {\mp} 2i\phi }}\sin \theta } \\ {\sqrt 2 {e^{ {\mp} i\phi }}\cos \theta } \\ {\sin \theta } \\ \end{array} } \right)\quad(\sigma=0).$$

Note here that the eigenvalue $\lambda _ \pm ^{\sigma }$ is related to $\delta \omega$ in Eq. (23) as $\lambda _ \pm ^{\sigma } = {\cal D} + \delta \omega$. Based on Eqs. (28) and (29), the Berry connections ${\mathbf A}_\pm ^{\sigma }=-i\left \langle {{\psi _\pm ^ \sigma }} \right.\left | {\nabla {\psi _\pm ^ \sigma }} \right \rangle$ are obtained as

(30)$${\mathbf A}_\pm^{\sigma}={\mp}\frac{1}{r}\left(\cot \frac{\theta }{2}\right)^{{\pm} \sigma}\hat \phi\quad(\sigma={\pm} 1),$$

(31)$${\mathbf A}_\pm^{\sigma}={\mp}\frac{1}{r}\csc \theta \hat \phi\quad(\sigma=0).$$

The Berry curvatures ${\mathbf F}_\sigma =\nabla \times {\mathbf A}_\pm ^{\sigma }$ are then given by

(32)$${\mathbf F}_\sigma=\sigma\frac{{\hat r}}{{{r^{2}}}}\quad(\sigma={\pm} 1, 0).$$

Integrating over the closed sphere $S$, the Chern numbers ${C_\sigma } = \frac {1}{2\pi }\int _S {{\mathbf F}_\sigma \cdot d{\mathbf {s}}}$ are calculated to give

(33)$${C_\sigma } = 2\sigma\quad(\sigma={\pm} 1, 0).$$

C. Pseudo time-reversal symmetry

The Hamiltonian for Maxwell’s equations [cf. Equation (3)] in a lossless and reciprocal medium is time-reversal (TR) invariant under $T_b$, that is,

(34)$$\left({T_b \otimes I}\right){{\cal H}_m \left( {\mathbf k} \right)}\left({T_b \otimes I}\right)^{ - 1} = {{\cal H}_m\left(-{\mathbf k} \right)},$$

where

(35)$${{\cal H}_m \left( {\mathbf k} \right)} = \left( {\begin{array}{cc} {{\omega}\underline{\varepsilon}} & {c{\mathbf{k}} \times \underline{I} + {{\omega}\underline{\xi}}} \\ {-c{\mathbf{k}} \times \underline{I} + {{\omega}\underline{\zeta}}} & { {\omega}\underline{\mu}} \\ \end{array}} \right),$$

${T_b} = {\sigma _z}K$ (with $T_b^{2}=1$) is the bosonic TR operator for photons, and $K$ is the complex conjugation [12]. The Hamiltonian ${\cal H}_m$, however, is not TR invariant under $T_f$, that is, $\left ({T_f \otimes I}\right ){{\cal H}_m \left ( {\mathbf k} \right )}\left ({T_f \otimes I}\right )^{ - 1} \ne {{\cal H}_m\left (-{\mathbf k} \right )}$, where ${T_f} = {i\sigma _y}K$ (with $T_f^{2}=-1$) is the fermionic TR operator for electrons [12].

The combined Hamiltonian formed by two spin-orbit Hamiltonians $\cal H_\pm$ [cf. Equation (24)], nevertheless, is TR invariant under $T_p$, that is,

(36)$$\left({T_p \otimes I}\right){{\cal H}_c \left( {\mathbf k} \right)}\left({T_p \otimes I}\right)^{ - 1} = {{\cal H}_c \left( -{\mathbf k} \right)},$$

where

(37)$${{\mathcal{H}}_c}\left( {\mathbf{k}} \right) = \left( {\begin{array}{cc} {\alpha \left( {{\mathbf{k}}\cdot{\mathbf{S}} + \frac{{i{\omega _{{\text{ref}}}}\gamma }} {c}{S_z}} \right)} & {\mathbf{0}} \\ {\mathbf{0}} & { - \alpha {{\left( {{\mathbf{k}}\cdot{\mathbf{S}} + \frac{{i{\omega _{{\text{ref}}}}\gamma }} {c}{S_z}} \right)}^{*}}} \\ \end{array} } \right),$$

and ${T_p}$ is the fermionic-like pseudo TR operator having the same form of $T_f$. The pseudo TR operator $T_p$ is inspired by noticing that ${\mathbf {E}} \leftrightarrow {\mathbf {H'}}$ during the TR operation. The pseudo TR operator is thus defined as ${T_p} = {T_b}{\sigma _x} = {\sigma _z}K{\sigma _x} = i{\sigma _y}K$ with $T_p^{2}=-1$ [23]. Here, $\sigma _x=\left (0,1;1,0\right )$, $\sigma _y=\left (0,-i;i,0\right )$, and $\sigma _z={\rm diag}\left (1,-1\right )$ are the Pauli matrices.

D. Surface wave equation

According to Maxwell’s equations, the eigenfields on either side of the interface ($y=0$) are given by the nontrivial solutions of ${\mathbf E}$ and ${\mathbf H}$ [cf. Equation (3)] or the null space of ${\cal H}_m$ [cf. Equation (35)]. On one side of the interface (say, $y>0$), we have

(38)$$ {{\mathbf{E}}^{\left( 1 \right)}} = \frac{1} {{k_0^{2}}}\left( {{\alpha _ - }{k_x},{\alpha _ - }k_y^{\left( 1 \right)},{\alpha _ + }{\alpha _ - } - {\varepsilon _t}{\mu _t}k_0^{2}} \right),{{\mathbf{H}}^{\left( 1 \right)}} = \frac{\varepsilon_t} {{{\eta _0}{k_0}}}\left(- {k_y^{\left( 1 \right)},{k_x},0} \right),$$

(39)$$ {{\mathbf{E}}^{\left( 2 \right)}} = \frac{\mu_t} {{{k_0}}}\left( {k_y^{\left( 2 \right)}, - {k_x},0} \right),{{\mathbf{H}}^{\left( 2 \right)}} = \frac{1} {\eta_0{k_0^{2}}}\left( {{\alpha _ + }{k_x},{\alpha _ + }k_y^{\left( 2 \right)},{\alpha _ + }{\alpha _ - } - {\varepsilon _t}{\mu _t}k_0^{2}} \right),$$

where $k_y^{\left ( 1 \right )} = \sqrt {\frac {{{\varepsilon _z}}}{{{\varepsilon _t}}}\left [ {\left ( {{\varepsilon _t}{\mu _t} - \gamma ^{2}} \right )k_0^{2} - k_z^{2}} \right ] - k_x^{2}}$ and $k_y^{\left ( 2 \right )} = \sqrt {\frac {{{\mu _z}}}{{{\mu _t}}}\left [ {\left ( {{\varepsilon _t}{\mu _t} - \gamma ^{2}} \right )k_0^{2} - k_z^{2}} \right ] - k_x^{2}}$ are the normal (to interface) wave vector components, ${\alpha _ \pm } = {k_z} \pm i{\gamma }{k_0}$, and the superscripts (1) and (2) refer to two independent polarizations. On the other side of the interface ($y<0$), the eigenfields are given by

(40)$$ {{\mathbf{E}}^{\left( 3 \right)}} = \frac{1} {{k_0^{2}}}\left( {{\alpha _ + }{k_x},{\alpha _ + }k_y^{\left( 3 \right)},{\alpha _ + }{\alpha _ - } - {\varepsilon _t}{\mu _t}k_0^{2}} \right),{{\mathbf{H}}^{\left( 3 \right)}} = \frac{\varepsilon_t} {{{\eta _0}{k_0}}}\left( -{k_y^{\left( 3 \right)}, {k_x},0} \right),$$

(41)$$ {{\mathbf{E}}^{\left( 4 \right)}} = \frac{\mu_t} {{{k_0}}}\left( {k_y^{\left( 4 \right)}, - {k_x},0} \right),{{\mathbf{H}}^{\left( 4 \right)}} = \frac{1} {{{\eta _0}k_0^{2}}}\left( {{\alpha _ - }{k_x},{\alpha _ - }k_y^{\left( 4 \right)},{\alpha _ + }{\alpha _ - } - {\varepsilon _t}{\mu _t}k_0^{2}} \right),$$

where $k_y^{\left ( 3 \right )} =-k_y^{\left ( 1 \right )}$ and $k_y^{\left ( 4 \right )} =-k_y^{\left ( 2 \right )}$ are the normal wave vector components, and the superscripts (3) and (4) refer to two independent polarizations. Note that the eigenfields in Eqs. (38)–(41) share the common tangential wave vector components $k_x$ and $k_z$ across the interface, as a direct consequence of the phase matching of electromagnetic fields. For the surface waves to exist on one side where $y>0$, $k_y^{\left ( 1 \right )}$ and $k_y^{\left ( 2 \right )}$ should be purely imaginary with a positive value, so that the waves decay exponentially away from the interface. On the other side where $y<0$, $k_y^{\left ( 3 \right )}$ and $k_y^{\left ( 4 \right )}$ should be purely imaginary with a negative value for a similar reason.

The tangential electric and magnetic field components are continuous at the interface:

(42)$$ C_{1}E_{x,z}^{(1)} + C_2E_{x,z}^{(2)} = C_3E_{x,z}^{(3)} + C_4E_{x,z}^{(4)},$$

(43)$$ C_{1}H_{x,z}^{(1)} + C_2H_{x,z}^{(2)} = C_3H_{x,z}^{(3)} + C_4H_{x,z}^{(4)},$$

where $C_1$, $C_2$, $C_3$, and $C_4$ are constants. The existence of a nontrivial solution of these constants requires that the determinant of the 4 $\times$ 4 matrix obtained from Eqs. (42) and (43) be zero, which gives the characteristic equation of the surface mode as

(44)$${\gamma ^{2}}k_x^{2} + \sqrt {{\varepsilon _t}\left[ {{\varepsilon _t}k_x^{2} + {\varepsilon _z}k_z^{2} - {\varepsilon _z}\left( {{\varepsilon _t}{\mu _t} - {\gamma ^{2}}} \right)k_0^{2}} \right]} \sqrt {{\mu _t}\left[ {{\mu _t}k_x^{2} + {\mu _z}k_z^{2} - {\mu _z}\left( {{\varepsilon _t}{\mu _t} - {\gamma ^{2}}} \right)k_0^{2}} \right]} = 0.$$

After some algebraic operations, Eq. (44) can be rearranged as a standard quadratic equation in the following form:

(45)$$\frac{{k_x^{2}}}\alpha + \frac{{k_z^{2}}}\beta = {{k_0^{2}}},$$

where

(46)$$\begin{aligned} \alpha = & \frac{1}{{2\left( {{\varepsilon _t}{\mu _t} + {\gamma ^{2}}} \right)}}\cdot\\ & \left[{\varepsilon _t}{\mu _t}\left( {{\varepsilon _z}{\mu _t} + {\varepsilon _t}{\mu _z}} \right) + \sqrt {{\varepsilon _t}{\mu _t}\left[ {{\varepsilon _t}{\mu _t}\left( {\varepsilon _z^{2}\mu _t^{2} + \varepsilon _t^{2}\mu _z^{2}} \right) - 2{\varepsilon _z}{\mu _z}\left( {\varepsilon _t^{2}\mu _t^{2} - 2{\gamma ^{4}}} \right)} \right]}\right], \end{aligned}$$

(47)$$\beta={{ {\varepsilon _t\mu _t - {\gamma ^{2}}} }}.$$

the Lorentz model used in the present medium (cf. Sec. 3.1), the coefficients $\alpha$ and $\beta$ in Eq. (45) are plotted in Fig. 4. Notice that $\alpha >0$ in the whole frequency range (except that $\alpha =0$ at $\omega _{\rm ref}$), while $\beta >0$ in the ranges $0<\omega <\omega _1$ and $\omega >\omega _2$, and $\beta <0$ in the range $\omega _1<\omega <\omega _2$. This indicates that the surface mode is represented by an elliptic equation in phases (I) and (III), or a hyperbolic equation in phase (II). The divergent behavior of $\alpha$ and $\beta$ in phase (II) is caused by the drastic change of material parameters near the resonance frequency $\omega _0$, which does not change the type of dispersion in this phase.

 

Fig. 4. Coefficients $\alpha$ and $\beta$ in the surface wave equation [Eq. (45)] for the bianisotropic metamaterial with the same material parameters as in Fig. 1(b). Light blue region is the band gap.

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The dispersion features stated above become more explicit for the isotropic case when $\varepsilon _t=\varepsilon _z\equiv \varepsilon$ and $\mu _t=\mu _z\equiv \mu$. In this situation, Eq. (45) is simplified to

(48)$${\frac{{k_x^{2}}}{{\varepsilon \mu }} + \frac{{k_z^{2}}}{{\varepsilon \mu - {\gamma ^{2}}}} = k_0^{2}}.$$

In the present problem, $\varepsilon \mu >0$ $(\omega \ne \omega _{\rm ref})$ because of the chiral nihility that occurs at $\omega _{\rm ref}$. The sign of $\varepsilon \mu - {\gamma ^{2}}$ thus determines the feature of surface mode as an elliptic type ($\varepsilon \mu >\gamma ^{2}$) or a hyperbolic type ($\varepsilon \mu <\gamma ^{2}$).

E. Electromagnetic energy density

The time-averaged energy density in a lossless bianisotropic medium is given by [46]

(49)$$\left\langle W \right\rangle =\frac{ 1}{4}{V^{\dagger} }{M}V,$$

where

(50)$${M} = \left( {\begin{array}{cccccc} {\frac{{\partial (\omega {\varepsilon _t})}}{{\partial \omega }}} & 0 & 0 & 0 & {i\frac{{\partial (\omega {\gamma})}}{{\partial \omega }}} & 0 \\ 0 & {\frac{{\partial (\omega {\varepsilon _t})}}{{\partial \omega }}} & 0 & { - i\frac{{\partial (\omega {\gamma})}}{{\partial \omega }}} & 0 & 0 \\ 0 & 0 & {\frac{{\partial (\omega {\varepsilon _z})}}{{\partial \omega }}} & 0 & 0 & 0 \\ 0 & {i\frac{{\partial (\omega {\gamma})}}{{\partial \omega }}} & 0 & {\frac{{\partial (\omega {\mu _t})}}{{\partial \omega }}} & 0 & 0 \\ { - i\frac{{\partial (\omega {\gamma})}}{{\partial \omega }}} & 0 & 0 & 0 & {\frac{{\partial (\omega {\mu _t})}}{{\partial \omega }}} & 0 \\ 0 & 0 & 0 & 0 & 0 & {\frac{{\partial (\omega {\mu _z})}}{{\partial \omega }}} \\ \end{array}} \right),V = \left( {\begin{array}{c} {\sqrt {{\varepsilon _0}} {E_x}} \\ {\sqrt {{\varepsilon _0}} {E_y}} \\ {\sqrt {{\varepsilon _0}} {E_z}} \\ {\sqrt {{\mu _0}} {H_x}} \\ {\sqrt {{\mu _0}} {H_y}} \\ {\sqrt {{\mu _0}} {H_z}} \\ \end{array} } \right),$$

with ${V^{\dagger} }$ being the Hermitian conjugate of $V$. The energy density must be positive definite, which implies that both the trace and determinant of ${M}$ are positive:

(51)$${\rm Tr}\left({M}\right)>0,\quad{\rm Det}\left({M}\right)>0.$$

Based on the Lorentz model used in the present medium (cf. Sec. 3.1), these quantities become

(52)$$\begin{aligned} & {\rm{Tr}}\left( {{M}} \right) =\frac{1}{\left( {{\omega ^{2}} - \omega _0^{2}} \right)^{2}}\cdot \\ & \left[ {3\omega _p^{2}\left( {{\omega ^{2}} + \omega _0^{2}} \right) + \left( {2{\varepsilon _{t\infty} } +\varepsilon _{z\infty}+ 2{\mu _{t\infty }} + {\mu _{z\infty }}} \right){{\left( {{\omega ^{2}} - \omega _0^{2}} \right)}^{2}}}{ - \left( {2{\Omega _{ t}} + {\Omega _{ z}}} \right){\omega ^{2}}\left( {{\omega ^{2}} - 3\omega _0^{2}} \right)}\right] \end{aligned}$$

and

(53)$$\begin{aligned} & {\rm{Det}}\left( {{M}} \right) = \frac{1}{{{{\left( {{\omega ^{2}} - \omega _0^{2}} \right)}^{12}}}}\cdot\\ & \left[ {{\varepsilon _{z\infty} }{{\left({\omega ^{2}} - \omega _0^{2}\right)}^{2}} + \omega _p^{2}\left({\omega ^{2}} + \omega _0^{2}\right)} \right]\left[ {{\mu _{z\infty }}{{\left({\omega ^{2}} - \omega _0^{2}\right)}^{2}} - {\Omega _{ z}}{\omega ^{2}}\left({\omega ^{2}} - 3\omega _0^{2}\right)} \right]\cdot \\ & \left\{{\left[ {{\varepsilon _{t\infty} }{{\left( {{\omega ^{2}} - \omega _0^{2}} \right)}^{2}} + \omega _p^{2}\left( {{\omega ^{2}} + \omega _0^{2}} \right)} \right]\left[ {{\mu _{t\infty }}{{\left( {{\omega ^{2}} - \omega _0^{2}} \right)}^{2}} - {\Omega _{ t}}{\omega ^{2}}\left( {{\omega ^{2}} - 3\omega _0^{2}} \right)} \right]}{ - 4\Omega _{xy}^{2}\omega _0^{4}\omega _p^{2}{\omega ^{2}}}\right\}^{2}, \end{aligned}$$

both of which are positive in the present study.

F. Band edge frequencies

Based on the Lorentz model used in the present medium (cf. Sec. 3.1), the reference frequency $\omega _{\rm ref}$ can be determined from ${\varepsilon _n(\omega _{\rm ref})}=0$ ($n=t,z$), which gives $\omega _{\rm ref} = \sqrt {\omega _0^{2} + \omega _p^{2}/\varepsilon _{\infty }}$. Here, we assume that $\varepsilon _{t\infty }=\varepsilon _{z\infty }\equiv \varepsilon _{\infty }$ and hence $\omega _{\rm ref}$ is uniquely defined. Having obtained $\omega _{\rm ref}$, the parameter ${\Omega _{ n}}$ is determined from $\mu _n(\omega _{\rm ref})=0$ to give ${\Omega _{ n}} = \mu _{n\infty }\left (1-\omega _0^{2}/\omega _{\rm ref}^{2}\right )$ ($n=t,z$). This condition is used in ${{\varepsilon _t}{\mu _t}-\gamma ^{2}}=0$ to solve the solutions of $\omega$ as

(54)$${\omega _1} = {\omega _{{\rm{ref}}}}\sqrt {1 + \frac{{{\omega _p}{\left|\Omega _{xy}\right|}\left( {{\omega _p}{\left|\Omega _{xy}\right|} - \sqrt {4{\varepsilon _\infty }{\mu _{t\infty }}\omega _0^{2} + \omega _p^{2}\Omega _{xy}^{2}} } \right)}}{{2{\varepsilon _\infty }{\mu _{t\infty }}\omega _0^{2}}}},$$

(55)$${\omega _2} = {\omega _{{\rm{ref}}}}\sqrt {1 + \frac{{{\omega _p}{\left|\Omega _{xy}\right|}\left( {{\omega _p}{\left|\Omega _{xy}\right|} + \sqrt {4{\varepsilon _\infty }{\mu _{t\infty }}\omega _0^{2} + \omega _p^{2}\Omega _{xy}^{2}} } \right)}}{{2{\varepsilon _\infty }{\mu _{t\infty }}\omega _0^{2}}}}.$$

Note that $\omega _1=\omega _2=\omega _{\rm ref}$ for $\Omega _{xy}=0$, and $\omega _1<\omega _{\rm ref}<\omega _2$ for $\Omega _{xy}\ne 0$.

G. Simulation

Let the $xy$ plane be the simulation domain with $k_z$ being the out-of-plane wave vector component, which is kept fixed in the simulation so that the eigenwaves possess the same $k_z$ [27]. The surface wave is excited at a certain point on the boundary between two bianisotropic media with opposite chirality, which can be implemented experimentally by a dipole antenna [16,20]. For the dipole to serve as the source of circularly polarized waves, two in-plane components with $\pm \pi /2$ phase difference are included to excite the right-handed or left-handed wave [51].

Funding

Ministry of Science and Technology, Taiwan (MOST 108-2221-E-002-155-MY3).

Acknowledgments

This work was supported in part by Ministry of Science and Technology, Taiwan (MOST 108-2221-E-002-155-MY3).

Disclosures

The authors declare no conflicts of interest.

Data availability

No data were generated or analyzed in the presented research.

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