Open Access
Add to BibSonomy
Abstract
Topological one-way surface states allow light to pass through sharp corners without reflection. In order to enhance the capability of surface routing devices, multiple one-way surface modes are usually required. Different from previously reported multiple surface modes achieved with large Chern number photonic media, we realize multiple surface waves on a continuous medium with small Chern number, i.e., |C| = 1. The new topological phase is found when the hyperbolic and double semi-ellipsoid-like cone bands are simultaneously gapped by vacuum state. We also find the degeneracy of multiple one-way surface waves in the double semi-ellipsoid-like metamaterials. The propagation direction of the waves is determined by their own ellipticities. Our results may help to construct surface state devices with multiplexing capability and higher coupling efficiency.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Topological concept of matter states is a new paradigm for description and classification of condensed matters [1]. Topology, describing invariant properties under continuous deformation, which implies that this property is robust and global [2–6]. In recent years, the research on topologically nontrivial system has led to the exciting findings of topological insulators, Weyl semimetals, and nodal lines [2]. Topological insulator has the property of insulating in bulk, but it conducts electricity on its surface and can resist scattering and stable propagation [5]. In particular, the $Z_2$ invariants and Chern numbers are used to describe the global topological invariants of quantum spin Hall and quantum Hall system, respectively. In addition, when a topological insulator is paired with another material with different topology, one-way surface waves appear in the common band gap regions [7–14]. It is known as bulk-edge correspondence [15].
In recent years, lots of studies on topological photonics are focused on the excitation of one-way surface states for unidirectional transmission [16–24]. In condensed matter physics, one of the most attractive characteristics of the surface waves in quantum Hall system is the robust propagation [25]. Then, the one-way electromagnetic surface waves which are analog to the quantum Hall effect have attracted much attention [26–28]. In particular, Raghu and Haldane recently proposed the one-way electromagnetic modes similar to the surface state of quantum Hall effect [21], which shares the same topological root originated from that of electronic system. The remarkable advantages open a path for the research of new topological photonic phases of edge or surface optics. Very recently, the concept of topological multiple one-way surface states was proposed, and remarkable achievements have been made. Some feasible approaches have been established in the photonic crystals and metamaterials system for the realization of multiple one-way surface states [29–35]. To best of our knowledge, the two-dimensional honeycomb lattice structure is a general topic in the study of multiple one-way surface states phenomena. For instance, the multiple one-way surface states have been successfully found in the gyromagnetic photonic crystals [29–33]. Due to the nonzero external magnetic field, the chiral surface states exist in the band gap regions with broken time-reversal symmetry. Similarly, the multiple one-way surface states in reciprocal electromagnetic continuous media have also been proposed recently [35]. However, for the multiple one-way surface states, in the known photonic systems, all of them are realized under the condition of large Chern numbers, i.e., $|C|\ge 2$ [29–35]. In order to achieve this kind of large Chern numbers, it is necessary to tune multiple Dirac or quadratic degenerate points into a small region and make them work at the high frequency regime [35]. A natural question is whether or not the multiple one-way surface states can have a small gap Chern number $|C|=1?$
In the present work, we demonstrate the existence of topologically protected surface waves on the gyroelectric metamaterials. The surface waves are found in the hyperbolic and double semi-ellipsoid-like gyroelectric metamaterials, respectively. Contrary to previous results achieved with large Chern number photonic media [29–35], we realized multiple one-way surface states in the band gap regions with Chern number value of $|C|=1$. The two surface states form a degenerate point in the band diagram and can be distinguished by their ellipticities. The topological properties of the surface waves are further verified by the backscatter immune transmission around defects.
2. Band structures and their Chern numbers of gyroelectric metamaterials
Using the effective medium theory, the topological properties in electromagnetic continuous photonic materials have been widely studied recently [36–46]. In this paper, we study the topological photonic phase in gyroelectric metamaterials using the topological band theory. The corresponding electromagnetic constitutive relation is expressed as
The band structures of the gyroelectric metamaterial can be obtained by solving the Maxwell’s equations. We transform the Maxwell’s equations into an eigenfrequency form $\hat {H} \mathbf {\left | \Psi \rangle \right .}=\omega \mathbf {\left | \Psi \rangle \right .}$ [46], with eigenfrequency $\omega$ , operator $\hat {H}$, and eigenvector $\mathbf {\left | \Psi \rangle \right .}$:
We now calculate some typical dispersion relations using Eq. (1) and show how to transform an electromagnetic duality medium into a topological nontrivial system made of gyroelectric metamaterial. The evolution of the band structures is shown in Fig. 1. In Fig. 1(a), we start to analyze the band structure from the electromagnetic duality material. The corresponding electromagnetic parameters are $\epsilon _{t}=\mu _{t}=2, \epsilon _{z}=\mu _{z}=1, g=0$, and $k_z=0$, respectively. We choose the electromagnetic duality material as the starting point because it is topologically equivalent with vacuum. The duality medium system can be transformed into a hyperbolic metamaterial by varying $\mu _z$ from 1 to $-$1 and $k_z$ from 0 to 1, respectively. In this case, two complete degenerate bulk states are decoupled, and only one quadratic degeneracy point exists in the frequency space [as shown in Fig. 1(b)]. Owing to the existence of degenerate point in the band of the metamaterials, the system is topologically trivial. In Fig. 1(c), the gyrotropic effect is introduced by making $g$ nonzero $(g=1.5)$. The system becomes a hyperbolic gyroelectric metamaterial. The degenerate point is lifted and a stable band gap is formed. The Chern numbers of the bulk states are nonzero, which further demonstrates the nontrivial topological property of the hyperbolic gyroelectric system.
Fig. 1. The band structures and their band Chern numbers owing to the change of the material parameters. (a) Electromagnetic duality medium with $\epsilon _{z}=\mu _{z}=1, g=0$, and $k_z=0$. (b) Hyperbolic non-gyrotropic medium with $\epsilon _{z}=1, \mu _{z}=-1, g=0$, and $k_z=1$. (c) Hyperbolic gyroelectric medium with $\epsilon _{z}=1, \mu _{z}=-1, g=1.5$, and $k_z=1$. (d) Anisotropic gyroelectric medium with $\epsilon _{z}=\mu _{z}=1, g=0.5$, and $k_z=0$. (e) Hyperbolic gyroelectric medium with $\epsilon _{z}=-1, \mu _{z}=1, g=0.5$, and $k_z=0$. (f) Double semi-ellipsoid-like gyroelectric medium with $\epsilon _{z}=\mu _{z}=-1, g=0.5$, and $k_z=1$. The green cone band represents the vacuum state. Here, $\epsilon _{t}=\mu _{t}=2$ is set for all plots.
On the other hand, we can make another evolution path to realize the topological nontrivial double semi-ellipsoid-like gyroelectric model. We also start from the electromagnetic duality material as shown in Fig. 1(a). From Fig. 1(d), we add the gyrotropic coefficient $(g)$, which causes complete decoupling of the degenerate band. But there is still no complete gap between the bands. In Fig. 1(e), when the parameter $\epsilon _z$ changes to negative, the orange band bulk state extends downward to form a conical structure, while the blue one becomes flat. Moreover, there is a degenerate point between the two bands, so it is also topologically trivial. From Fig. 1(e) to Fig.1(f), when the hyperbolic gyroelectric medium system is transformed to a double semi-ellipsoid-like metamaterial by varying $\mu _z$ from 1 to $-$1 and $k_z$ from 0 to 1 ($\epsilon _t=\mu _t=2, \epsilon _z=-1$, and $g=0.5$), respectively. It is obvious that the double ellipsoid-like band bulk states in Fig. 1(f) can form a stable band gap with vacuum [see the green cone band in Fig. 1(f)]. It is an essential condition to realize nonreciprocal scatter-immune surface states. The double semi-ellipsoid-like band modes have nonzero Chern numbers, which demonstrates the topology of the material system. Until now, we have known that the band bulk states in Figs. 1(c) and 1(f) have nontrivial topological properties. In the following work, we will study and discuss the topological properties of the hyperbolic and the double semi-ellipsoid-like gyroelectric metamaterials system separately.
3. Nonreciprocal surface waves on gyroelectric metamaterials
Between topologically inequivalent media, the photonic surface states can exist according to the so-called bulk-edge correspondence. Now we are going to study how these topological features show up on the boundary of gyroelectric metamaterials. In what follows, we investigate systems with continuous translational invariance in the $z$-direction, thereby conserving $k_z$ . The surface wave exponentially decays from the interface along both directions, i.e., the half space $x>0$ is occupied by vacuum and the half space $x<0$ is occupied by gyroelectric metamaterials. We use the method proposed by Dyakonov [47] to calculate the nontrivial surface waves.
In the vacuum, the two orthogonal eigen modes (TE and TM) can be expressed as
Two independent eigen states of gyroelectric metamaterials can be expressed as
4. Convergence of Berry curvature and simulation of topologically protected surface waves on hyperbolic gyroelectric metamaterials
After solving Eq. (8), we obtain the dispersion curves of the surface waves for hyperbolic gyroelectric metamaterials with the constitutive parameters $\epsilon _{t}=\mu _{t}=2, \epsilon _{z}=1, \mu _{z}=-1, g=1.5$, and $k_z=1$, respectively, as shown in Fig. 2(a). The result shows that only one propagation mode is supported in the common band gap regions between the hyperbolic metamaterial bulk states and vacuum. The surface state starts from the metamaterial and ends in the vacuum. Different from the electronic materials, vacuum is not the insulator of photons, and there exist photonic bands in vacuum, i.e., the continuum of free space. Therefore, according to the bulk-edge correspondence principle, only in the common band gap region of the metamaterial and vacuum state can the surface state be topological, as shown in the shaded area in Fig. 2(a). Through the analysis of boundary conditions, we calculated the ellipticities of the surface state in the shadow part of the band gap region in Fig. 2(a).
Fig. 2. Polarization analysis of the surface states and Berry curvature distribution. (a) Bulk and surface states distribution of the 2D band structure. The gray shadow area represents the common band gap region between the hyperbolic gyroelectric metamaterial and vacuum bands. The orange (blue) solid lines and green dotted line represent bulk band structures of the metamaterial and vacuum, respectively. LEP and REP stand for left and right elliptical polarization, respectively. (b) The ellipticities of the surface states in vacuum (black line) and in the metamaterial (red line) sides. (c) The distribution of the Berry curvature of hyperbolic band (orange line) in (a). (d) The distribution of the Berry curvature of semi-ellipsoid-like band (blue line) in (a). The parameters of the metamaterial are $\epsilon _{t}=\mu _{t}=2, \epsilon _{z}=1, \mu _{z}=-1, g=1.5$, and $k_z=1$, respectively.
The electric field on the vacuum and gyroelectric metamaterial side can be obtained by summing the two independent eigenfields in Eqs. (4) - (7)
The ellipticities of the surface waves are different on the vacuum and metamaterial side [as shown in Fig. 2(b)]. But it can be found that the ellipticity of the surface state is always positive, which demonstrates the chiral property of the surface wave. Notably, the metamaterials have the gyrotropic effect (nonzero parameter $g$ in Eq. (2)) in the permittivity tensor, so the symmetry of electromagnetic duality is broken ($\overline {\boldsymbol {\epsilon }} \neq \overline {\boldsymbol {\mu }}$). In this case, the surface waves with elliptical polarizations arise on the interface.
The topological properties of the surface waves can be investigated by calculating the Berry phase of gyroelectric metamaterials as [14]
where ${\mathbf {\Omega (\mathbf {k})}}=-i{\bigtriangledown }_{\mathbf {k}}\times {\langle }U(\mathbf {k}){\vert }{\bigtriangledown }_{\mathbf {k}}{\vert }U(\mathbf {k}){\rangle }$ is the Berry curvature and $U(\mathbf {k})=[\mathbf {E},\mathbf {H}]^{T}$ represents the eigen-polarization states. The Berry curvature has the following symmetry: when the symmetry of time-reversal is preserved, ${\mathbf {\Omega (\mathbf {k})}}=-{\mathbf {\Omega (-\mathbf {k})}}$. When the symmetry of spatial inversion is preserved, ${\mathbf {\Omega (\mathbf {k})}}={\mathbf {\Omega (-\mathbf {k})}}$. Therefore, if the time-reversal symmetry and space-inversion symmetry are kept at the same time, the Berry curvature of the material is zero, that is to say, the topological invariants (Chern number) is also equal to zero. The corresponding material system is topologically trivial. We calculate the Berry curvature at $(k_x,k_y)$ on the $k_x-k_y-\omega$ surface, as shown in Figs. 2(c) and 2(d) at $k_z=1$. The Berry curvatures are mainly concentrated near the center of the $k$-space for both hyperbolic and double semi-ellipsoid-like gyroelectric metamaterials. When the wave vector increasees to infinity, our calculation shows that the Berry curvatures become negligible. Integrating the Berry curvatures in Eq. (9) on the bulk state of the hyperbolic gyroelectric metamaterials, one can obtain a value of $\pm 2\pi$, which corresponds to quantized Chern numbers of $\pm 1$ [see Fig. 1(c)].An unusual feature of the nonreciprocal surface states is the complete suppression of backscattering, which is an optical analog of lossless transmission of the chiral edge electrons in quantum Hall systems. We numerically investigated the transport properties of the one-way surface waves by full wave simulation software COMSOL Multiphysics. The line electric sources with a $z$-dependent phase gradient are used to excite surface modes. Here, we choose three frequencies $\omega _I$=0.35, 0.7, and 1.2, corresponding to the three points “A” , “B”, and “C” in Fig. 2(a), respectively. The simulation results are shown in Fig. 3. In Fig. 3(a), the frequency [“A” in Fig. 2(a)] is located in the metamaterial bulk state, which leads to the leakage of the surface wave mode into the metamaterial, and the transmission is not stable. When the frequency move to the common band gap region [“B” in Fig. 2(a)], the corresponding chiral surface wave can stably bypass the shape defect and propagate with topological protection [see Fig. 3(b)]. If $\omega _I$ lies in the common band of vacuum and the metamaterial [“C” in Fig. 2(a)], the surface wave diffuses into the entire space, as shown in Fig. 3(c). When the applied magnetic field acting on the $z$-direction of the metamaterial is reversed, the surface waves protected by the topology can also propagate but in reversed direction. This is clearly shown in Figs. 3(b) and 3(d) where the gyrotrogic coefficient is positive $(g=1.5)$ and negative $(g=-1.5)$, respectively. The topological properties of the surface modes are clearly illustrated by the robust transmission of electromagnetic waves at the interfaces between vacuum and the hyperbolic gyroelectric metamaterial.
Fig. 3. Full-wave simulation of the surface waves with topologically protected transmission. The color of simulation fields represents the distribution of electric field intensity. The electromagnetic parameters of the metamaterial are the same as in Fig. 2(a). The source (green arrow) frequency $\omega _I$ is set as (a) $\omega _I$=0.35, (b) $\omega _I$=0.7, and (c) $\omega _I$=1.2, corresponding to the surface state of points “A” , “B”, and “C” in Fig. 2(a). (d) Same excitation frequency as (b) except that the sign of the gyrotropic efficient is opposite $(g=-1.5)$.
5. Multiple one-way photonic surface waves on double semi-ellipsoid-like gyroelectric metamaterials
Figure 4(a) shows the dispersion curves of the surface waves for the double semi-ellipsoid-like gyroelectric metamaterials with $\epsilon _{t}=\mu _{t}=2, \epsilon _{z}=\mu _{z}=-1, g=0.5$, and $k_z=1$, respectively. There are two different forms of surface waves in the same gyrotropic direction $(g=0.5)$. The Chern number of the two bulk state of the matematrial is $+1$ and $-1$ , respectively. The surface states in the band gap have topologically protected transmission, otherwise it will leak into the vacuum or metamaterial bulk states. The two surfaces states have different group velocity direction, as shown in Fig. 4. The surface waves with positive/negative group velocities correspond to the right/left elliptical polarization states, respectively. Moreover, the polarization-dependent photonic edge wave is analogous to the spin-dependent edge wave in electronic systems [20]. Thus, the right/left polarization surface state in the bandgap region (Fig. 4) is similar to the spin-up/spin-down mode in the quantum-spin-Hall effect, respectively. Moreover, unlike the surface states in Fig. 4, the topological one-way mode can also be realized in a two-dimensional synthetic space by using an effective gauge field [48]. The topological feature of the surface waves in Fig. 4(a) can be demonstrated by full-wave simulation. We use an electric dipole to excite the stable surface waves, as shown in Fig. 4(b), where $\omega _I=0.9$ in the common band gap region. Through the simulation, we can see that the two surface modes in the common band gap region have properties of topologically protected transmission. One surface wave propagates in the positive $y$-direction (point “D”) and the other in the negative $y$-direction (point “E”), which directly verifies the unidirectional nature of the surface waves. The topological nontrivial of the system is intuitively verified by the stable transmission of the surface waves around square defects in both cases, as illustrated in Fig. 4(b).
Fig. 4. (a) Bulk and surface states of gyroelectric metamaterial with double semi-ellipsoid-like dispersion. The orange (blue) solid lines and green dotted line represent bulk state band of the metamaterial and vacuum, respectively. LEP and REP stand for left and right elliptical polarization, respectively. (b) Simulation of the propagation of the surface waves with $\omega _I=0.9$ at points “D” and “E”, respectively in (a). (c) The ellipticities of the surface states in the metamaterial side. (d) Simulations of surface waves with $\omega _I=0.72$ at the intersection point “F”. The parameters of the metamaterials are $\epsilon _t=\mu _t=2, \epsilon _z=\mu _z=-1, g=0.5$, and $k_z=1$, respectively.
As shown in the band structure in Fig. 4(a), the two surface states cross with each other and form a degenerate point. By calculating the ellipticities of two surface states in the shadow region of Fig. 4(a), we know that the ellipticity of the two surface states is opposite to each other, corresponding to the right and left elliptical polarized surface states, respectively. For the degenerate point “F”, it can be viewed as the superposition of two orthogonal states. These two orthogonal surface states can be distinguished by their ellipticities. The ellipticity change suddenly near the degenerate point as shown by the green shaded region in Fig. 4(c). Moreover, the surface waves at point “F” can well bypass the defect, and their robustness is determined by the topological nontrivial properties of the gyroelectric metamaterial. To excite stable unidirectional surface waves at point “F”, we use a pair of mutually orthogonal electric dipoles to excite left and right elliptical polarized surface waves, respectively. The surface waves can stably bypass the defects and propagate in their own directions.
6. Realization of the proposed gyroelectric metamaterials
Here, we propose a periodic layered structure as shown in Fig. 5 to show how to realize such gyroelectric metamaterials. We consider a bilayer superlattice composed of a gyrotropic slab (green layer) with relative permeability $\mu _1$, and a plasma slab (orange layer) with relative permeability $\mu _2=1-\omega _m^2/\omega ^2$ [49], where $\omega _m$ is plasma frequency. Their thicknesses are $d_1$ and $d_2$, respectively. According to the effective medium theory ($EMT$) [50,51], the effective relative permeability tensor of gyroelectric metamaterials can be written as
Fig. 5. Illustration of the proposed gyroelectric metamaterials as a supperlattice composed of a gyrotropic slab and a plasma slab. The layers are infinite in $xy$ plane.
For the plasma layer, the relative permittivity is $\epsilon _2$. For the gyrotropic layer, the relative permittivity tensor is
In Figs. 6(a) and 6(b), we calculate the band structures, Chern numbers, and surface state of the dispersive gyroelectric metamaterials using the above effective parameters [Eqs. (14) and (15)], respectively. In Fig. 6(a), the bandgap width decreases a lot owing to the effect of dispersion. In particular, many factors can affect the band structures in the dispersive gyroelectric metamaterials, such as resonance frequency $\omega _0$, resonance strength $\omega _e$, plasma frequency $\omega _m$ , and so on. We choose a parameter set of $\omega _e=0.5\omega _0$ and $\omega _m=1.02\omega _0$ because it can form a stable common bandgap between the dispersive gyroelectric metamaterials and the vacuum state. To show the surface state more clearly, the band structure in Fig. 6(b) corresponds to the part in the black frame of Fig. 6(a). Compared with the non-dispersion case [Fig. 2(a)], it can be seen that more bands are created in the dispersive gyroelectric metamaterials [Fig. 6(a)]. Moreover, in the dispersive band structure, the eigenfrequencies of the second band (blue) and the first band (black) are equal when $k_y$ approaches infinity. Therefore, there is also only one complete bandgap (gray region) in Fig. 6(a). It lies between the third band (orange) and the second band (blue). It can be seen that these two bands are quite similar to the bands in the non-dispersion case in a limited range of frequency.
Fig. 6. Band structures of the dispersive metamaterials and simulations of the surface waves. (a), (b) 2D band structures, Chern numbers, and surface state of the bilayer superlattice shown in Fig. 5, respectively. (c), (d) Simulation of the surface waves under the condition of $\omega _I=0.95/1.01$ corresponding to the points “G”/ “H” in (b), respectivly. The green arrows indicate the position of the line sources. The field patterns represent the $|E|$ distributions.
We use the efficient numerical algorithm reported in Ref. [54] to calculate the Chern numbers of dispersion bands [Fig. 6(a)]. Different from the computational scheme for the Chern number in Ref. [54] in the first Brillouin zone, the Chern number in Fig. 6(a) needs to be calculated in a large wave vector space. Notably, the Chern numbers on both sides of the bandgap (orange and blue bands) after considering dispersion [Fig. 6(a)] are consistent with the Chern numbers of non-dispersion in Fig. 2(a). It can verify that the band topology has not been altered after adding dispersion. Moreover, the topological nontrivial property of the dispersive gyroelectric metamaterials is also verified by the robust transmission of the surface wave around square defects, as shown in Fig. 6(c). Otherwise, it will leak to the metamaterials and vacuum regions [see Fig. 6(d)].
7. Conclusion
In conclusion, we have analyzed the topological photonic phase in gyroelectric metamaterials by topological band theory. The topological phase is characterized by the nonzero Chern numbers of the eigenstates. In the common band gap regions formed by the metamaterials bulk states and vacuum, there are unidirectional surface waves. In contrast to the previous results of large Chern number photonic media, we have realized multiple one-way photonic surface waves in the band gap region under condition of the Chern numbers are $\pm 1$. We find the degeneracy of multiple one-way surface waves on the double semi-ellipsoid-like metamaterials. Finally, through the calculation of specific effective electromagnetic parameters, we enrich the realization of topological photonic phase in electromagnetic continuum system. We believe our theoretical research may be helpful to construct surface state devices with multiplexing capability and higher coupling efficiency.
Funding
National Natural Science Foundation of China (11874132, 12074087, 61575055).
Disclosures
The authors declare no conflict of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonic,” Nat. Photonics 8(11), 821–829 (2014). [CrossRef]
2. M. Kim, Z. Jacob, and J. Rho, “Recent advances in 2D, 3D and higher-order topological photonics,” Light: Sci. Appl. 9(1), 130 (2020). [CrossRef]
3. Y. Z. Liu, X. B. Chen, and Y. Xu, “Topological phononics: from fundamental models to real materials,” Adv. Funct. Mater. 30(8), 1904784 (2020). [CrossRef]
4. Y. H. Yang, Y. Yamagami, X. B. Xu, P. Pitchappa, and B. L. Zhang, “Terahertz topological photonics for on-chip communication,” Nat. Photonics 14(7), 446–451 (2020). [CrossRef]
5. B. Y. Xie, H. F. Wang, X. Y. Zhu, M. H. Lu, Z. D. Wang, and Y. F. Chen, “Photonics meets topology,” Opt. Express 26(19), 24531 (2018). [CrossRef]
6. W. Gao, M. Lawrence, B. Yang, F. Liu, F. Fang, B. Béri, J Li, and S. Zhang, “Topological photonic phase in chiral hyperbolic metamaterials,” Phys. Rev. Lett. 114(3), 037402 (2015). [CrossRef]
7. Z. D. Song, L. Elcoro, and B. A. Bernevig, “Twisted bulk-boundary correspondence of fragile topology,” Science 367(6479), 794–797 (2020). [CrossRef]
8. N. Han, J. L. Liu, Y. Gao, K. Y. Zhou, and S. T. Liu, “Topologically protected and highly localized surface waves in gyro-electromagnetic metamaterials,” Ann. Phys. Berlin 532(8), 2000022 (2020). [CrossRef]
9. B. Yang, Q. Guo, B. Tremain, R. Liu, L. E. Barr, Q. Yan, W. Gao, H. Liu, Y. Xiang, J. Chen, C. Fang, A. Hibbins, L. Lu, and S. Zhang, “Ideal Weyl points and helicoid surface states in artificial photonic crystal structures,” Science 359(6379), 1013–1016 (2018). [CrossRef]
10. D. F. Jin, L. Lu, Z. Wang, C. Fang, J. D. Joannopoulos, M. Soljačić, L. Fu, and N. X. Fang, “Topological magnetoplasmon,” Nat. Commun. 7(1), 13486 (2016). [CrossRef]
11. M. Kim, W. Gao, D. Lee, T. Ha, T. T. Kim, S. Zhang, and J. Rho, “Extremely broadband topological surface states in a photonic topological metamaterial,” Adv. Opt. Mater. 7(20), 1900900 (2019). [CrossRef]
12. J. L. Liu, W. M. Ye, and S. Zhang, “Pseudospin-induced chirality with staggered optical graphene,” Light Sci. Appl. 5(8), e16094 (2016). [CrossRef]
13. F. D. M. Haldane, “Model for a quantum Hall effect without Landau levels: condensed-matter realization of the "parity anomaly",” Phys. Rev. Lett. 61(18), 2015–2018 (1988). [CrossRef]
14. M. Z. Hasan and C. L. Kane, “Colloquium: Topological insulators,” Rev. Mod. Phys. 82(4), 3045–3067 (2010). [CrossRef]
15. M. G. Silveirinha, “Bulk-edge correspondence for topological photonic continua,” Phys. Rev. B 94(20), 205105 (2016). [CrossRef]
16. D. Xiao, M. C. Chang, and Q. Niu, “Berry phase effects on electronic properties,” Rev. Mod. Phys. 82(3), 1959–2007 (2010). [CrossRef]
17. X. L. Qi and S. C. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys. 83(4), 1057–1110 (2011). [CrossRef]
18. L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological states in photonic systems,” Nat. Phys. 12(7), 626–629 (2016). [CrossRef]
19. S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A 78(3), 033834 (2008). [CrossRef]
20. A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2013). [CrossRef]
21. F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100(1), 013904 (2008). [CrossRef]
22. Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100(1), 013905 (2008). [CrossRef]
23. Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461(7265), 772–775 (2009). [CrossRef]
24. S. Barik, A. Karasahin, C. Flower, T. Cai, H. Miyake, W. DeGottardi, M. Hafezi, and E. Waks, “A topological quantum optics interface,” Science 359(6376), 666–668 (2018). [CrossRef]
25. T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilbergerg, and L. Carusotto, “Colloquium: topological photonics,” Rev. Mod. Phys 91(1), 015006 (2019). [CrossRef]
26. M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. Lond. Ser. A 392(1802), 45–57 (1984). [CrossRef]
27. A. B. Khanikaev and G. Shvets, “Two-dimensional topological photonics,” Nat. Photonics 11(12), 763–773 (2017). [CrossRef]
28. M. Kim, D. Lee, D. Lee, and J. Rho, “Topologically nontrivial photonic nodal surface in a photonic metamaterial,” Phys. Rev. B 99(23), 235423 (2019). [CrossRef]
29. S. A. Skirlo, L. Lu, and M. Soljačić, “Multimode one-way waveguides of large Chern numbers,” Phys. Rev. Lett. 113(11), 113904 (2014). [CrossRef]
30. S. A. Skirlo, L. Lu, Y. Igarashi, Q. Yan, J. D. Joannopoulos, and M. Soljačić, “Experimental observation of large Chern numbers in photonic crystals,” Phys. Rev. Lett. 115(25), 253901 (2015). [CrossRef]
31. H. C. Chan and G. Y. Guo, “Tuning topological phase transitions in hexagonal photonic lattices made of triangular rods,” Phys. Rev. B 97(4), 045422 (2018). [CrossRef]
32. B. Yang, H. Zhang, T. Wu, R. Dong, X. Yan, and X. Zhang, “Topological states in amorphous magnetic photonic lattices,” Phys. Rev. B 99(4), 045307 (2019). [CrossRef]
33. B. Yang, T. Wu, and X. Zhang, “Engineering topological edge states in two dimensional magnetic photonic crystal,” Appl. Phys. Lett. 110(2), 021109 (2017). [CrossRef]
34. X. Xi, K. P. Ye, and R. X. Wu, “Topological photonic crystal of large valley Chern numbers,” Photonics Rev. 8(9), B1 (2020). [CrossRef]
35. J. Y. Xu, Y. Liu, K. S. Chan, and J. Li, “Multiple one-way edge states from reciprocal continuous media,” Phys. Rev. Appl. 12(6), 064014 (2019). [CrossRef]
36. W. Gao, B. Yang, M. Lawrence, F. Fang, B. Béri, and S. Zhang, “Photonic Weyl degeneracies in magnetized plasma,” Nat. Commun. 7(1), 12435 (2016). [CrossRef]
37. S. A. Hassani Gangaraj and F. Monticone, “Physical Violations of the Bulk-Edge Correspondence in Topological Electromagnetics,” Phys. Rev. Lett. 124(15), 153901 (2020). [CrossRef]
38. Q. Guo, B. Yang, L. Xia, W. Gao, H. Liu, J. Chen, Y. Xiang, and S. Zhang, “Three dimensional photonic dirac points in metamaterials,” Phys. Rev. Lett. 119(21), 213901 (2017). [CrossRef]
39. M. G. Silveirinha, “Proof of the bulk-edge correspondence through a link between topological photonics and fluctuation-electrodynamics,” Phys. Rev. X 9, 011037 (2019). [CrossRef]
40. J. W. You, Z. Lan, and N. C. Panoiu, “Four-wave mixing of topological edge plasmons in graphene metasurfaces,” Sci. Adv. 6(13), eaaz3910 (2020). [CrossRef]
41. Z. Yu, G. Veronis, Z. Wang, and S. Fan, “One-way electromagnetic waveguide formed at the interface between a plasmonic metal under a static magnetic field and a photonic crystal,” Phys. Rev. Lett. 100(2), 023902 (2008). [CrossRef]
42. W. Gao, F. Fang, Y. Liu, and S. Zhang, “Chiral surface waves supported by biaxial hyperbolic metamaterials,” Light: Sci. Appl. 4(9), e328 (2015). [CrossRef]
43. J. Chen, W. Liang, and Z. Li, “Revealing photonic Lorentz force as the microscopic origin of topological photonic states,” Nanophotonics 9(10), 3217–3226 (2020). [CrossRef]
44. R. C. Shiu, H. C. Chan, H. X. Wang, and G. Y. Guo, “Photonic Chern insulators made of gyromagnetic hyperbolic metamaterials,” Phys. Rev. Mater. 4(6), 065202 (2020). [CrossRef]
45. X. D. Chen, W. M. Deng, F. L. Zhao, and J. W. Dong, “Accidental double Dirac cones and robust edge states in topological anisotropic photonic crystals,” Laser Photonics Rev. 12(11), 1800073 (2018). [CrossRef]
46. J. Hou, Z. Li, X. Luo, Q. Gu, and C. Zhang, “Topological Bands and Triply-Degenerate Points in Non-Hermitian Hyperbolic Metamaterials,” Phys. Rev. Lett. 124(7), 073603 (2020). [CrossRef]
47. M. I. Dyakonov, “New type of electromagnetic wave propagating at the interface,” Sov. Phys. JETP 67(4), 119–123 (1988).
48. Q. Shan, D. Yu, G. Li, L. Yuan, and X. Chen, “One-way topological states along vague boundaries in synthetic frequency dimensions including group velocity dispersion,” Prog. Electromagn. Res. 169, 33–43 (2020). [CrossRef]
49. J. B. Pendry, “Negative reflection makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000). [CrossRef]
50. D. P. Tsai, B. Wood, and J. B. Pendry, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B 74(11), 115116 (2006). [CrossRef]
51. V. I. Fesenko and V. R. Tuz, “Lossless and loss-induced topological transitions of isofrequency surfaces in a biaxial gyroelectromagnetic medium,” Phys. Rev. B 99(9), 094404 (2019). [CrossRef]
52. M. G. Silveirinha, “Chern invariants for continuous media,” Phys. Rev. B 92(12), 125153 (2015). [CrossRef]
53. A. Raman and S. Fan, “Photonic band structure of dispersive metamaterials formulated as a Hermitian eigenvalue problem,” Phys. Rev. Lett. 104(8), 087401 (2010). [CrossRef]
54. T. Fukui, H. Suzuki, and Y. Hatsugai, “Chern numbers in a discretized Brillouin zone: Efficient method to compute (spin) Hall conductances,” J. Phys. Soc. Jpn. 74(6), 1674–1677 (2005). [CrossRef]
