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Abstract
We demonstrate that a topological edge state can enhance the Goos–Hänchen (GH) shift on the interface of a magnetic photonic crystal (MPC) fabricated by ferrite rings in a square lattice. The GH shift is nonreciprocal because of the time reversal symmetry breaking, and the shift is negative, which is associated with the incident angles and direction of bias DC magnetic field. In particular, the nonreciprocal GH shift presents at normal incidence, and is further verified by experiments. The nonreciprocal negative GH shifts provide a new way to control the flow of light, which could be applied to sensing and even the slow light waveguide to trap the light.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
When a light beam impinges on the interface of two media with different refractive indices, the reflected light beam will experience a lateral shift along the interface from the ideal position predicted by the geometric optics. This phenomenon is known as the GH effect that was first experimentally observed by Goos and Hänchen in total internal reflection [1]. Though it is known for decades, the GH shift still attracts much attention, especially after the observation of giant and negative GH shift [2–3]. The GH shift has been extensively studied in various material systems, such as metal-dielectric nanocomposite [4], grating [5], epsilon-near-zero material [6], atomic coherence cavities [7] and photonic crystal (PC) [8–9]. In certain circumstances such as on the metal surface, the GH shift is possible to get a significant value [10]. The shift is derived by phase variation of the reflection coefficient with the incident angle based on the stationary phase method [11]. Usually, the GH effect is shown at oblique incidence; however, recent studies show it is also possible at normal incidence [12–17]. Though the GH effect can be interpreted by several ways [11,18–19], the effect is understood by the energy flow along the interface, which enhances the lateral shift. The enhanced GH shift provides a wide range of applications such as in switching [20], transducers [21], optical sensing and detection [22,23].
In this paper, we explore the GH effect on the surface of MPC made of yttrium-iron-garnet (YIG) ferrite rings in square lattice. We reveal that the excitation of the topological edge states inside the light cone results in a significant negative GH shift for total reflection waves. Particularly, the enhanced GH shift is nonreciprocal because of time-reversal symmetry breaking of the MPC. Even at normal incidence, the GH shift is also significant and nonreciprocal, which is verified by experiments. The enhanced nonreciprocal negative GH shift provides a new platform to manipulate the wave propagation, which may apply to such as sensing, switching, slow light waveguide and even stop the light by the waveguide built from MPCs [24].
2. Nonreciprocal negative GH shifts excited by topological edge states of MPC slabs
We consider a two-dimensional MPC slab as shown in Fig. 1(a). The slab consists of YIG ferrite rings in a vacuum background. The outer radius of the rings is R = 10 mm and the inside radius r = 6 mm. The rings are arranged in the square lattice and the lattice constant is d = 22 mm. The axes of rings are parallel to the z-axis. The material parameters of YIG are, respectively, the relative permittivity ɛr = 15.26, and the saturation magnetization 4πMs = 1884Ga. Suppose the MPC is in the Voigt configuration; the direction of bias DC magnetic field is along the z-axis, parallel to the polarization of the incident wave. When the ferrite is fully magnetized, its permeability is a tensor written as [25]:
Fig. 1. (a) Schematics of geometric structure of MPC slab. The outer radius of ferrite rings is R = 10 mm, and inner radius r = 6 mm. The lattice constant of the MPC is d = 22 mm. An electromagnetic wave beam is incident at angle θ with respect to the normal of the MPC slab. (b) Projected band structure of the five-layer MPC slab and full MPC along the Г-X direction. The band of full MPC is shaded in blue. The dots represent the bands of the MPC slab. Two edge bands represented by red and blue dots in the bandgap of full MPC are the edge bands for upper and lower edges of the slab, respectively. The light cone is shaded in yellow. (c) Projected band structure of the MPC slab displayed in the inset. The domain wall in the inset is created by applying bias magnetic field which is in opposite direction in the two domains. (d) The electric field distribution of the topological edge state along the domain wall at the mid-gap frequency 0.432. The defect is displayed in white rectangle. The electric field is excited by a current line marked by the star.
We first calculated the projected bands of the MPC slab and full MPC along the x-direction by the supercell technique [26]. In the calculations, the supercell is terminated by the PML boundaries on the upper and lower boundaries, and the periodic boundary condition is used along the x-direction. The band structure was calculated by using commercial software COMSOL MULTIPHYSICS with RF module. The solution steps were first using an eigenvalue solver to obtain initial values, and then with nonlinear solver to get eigenfrequencies by sweeping the wave number from kx -0.5 to 0.5. Figure 1(b) plots the results under the bias magnetic fields H0=800Oe. In the figure, the dots represent the band structure of the slab and blue shaded region is the band structure of full MPC. The parameters k0 and kx is the wave number in free space and the wave number along the x direction, respectively. We see a wide photonic bandgap of MPC presents in the normalized frequencies 0.405 to 0.475, corresponding to the working frequencies from 5.5 GHz to 6.45 GHz. In this frequency range, MPC likes a perfect mirror reflecting the incident waves. For the MPC slab, we notice two new energy bands, the edge bands, appear in the bandgap of full MPC. The two edge bands correspond to the upper and lower edge of the slab, respectively. The edge bands span the bandgap forming a gapless edge state, which is a characteristic of topological edge state. Generally, the topological nature of a bandgap is determined by the Chern number of all the bands below the bandgap. However, it is difficult to determine the bandgap in this way in our case because of a great number of flat bands below the Mie resonance frequency [27]. A round way is to calculate the edge states in the bandgap according to the principle of the “bulk-surface” correspondence [28]. Since the gap Chern number is the same as the number of topologically edge states in the bandgap, the red edge band in Fig. 1(b) has positive slope, implying that the Chern number of the bandgap is one. As another demonstration, we calculated the band structure of the MPC slab with a domain wall created by applied bias magnetic field in opposite direction for the two domains, as shown by the inset in Fig. 1(c). The upper and lower the domains are terminated by the PML boundaries. We suppose the direction of the bias magnetic field in the lower part of the domain is along the z-axis, and the upper one is along -z-axis. In this case, two edge states with positive slop should appear in the bandgap if it is topological. Figure 1(c) plots the band structure in this case. Indeed, two red gapless edge states along the domain wall appear in the bandgap, and the slope of the edge bands indicate that the waves of two edge modes are locked in the same direction. Two blue edge states are coincident, corresponding to the upper and lower edge of the slab in the inset, respectively. As verification, we simulated the wave propagation on the domain wall. The results are shown in Fig. 1(d). In the simulation, we introduce one big defect, the metal plate, to block the wave channel in order to explore if the waves of edge modes are robust [29]. As shown in the figure, the wave excited by a current source is localized at the domain wall and decays rapidly away from the wall. The wave propagates only along right-hand and is stopped on the opposite side. Specifically, the propagating wave circumvents the defect, showing the robustness of edge mode waves. All the results above confirm the bandgap is topological, thus the corresponding edge states are topological edge states.
In the following, we concentrate on the edge band in red shown in Fig. 1(b). The edge band corresponds to the edge states on the upper surface of MPC slab. The band has a positive slope, implying the wave of the edge modes is locked in the x-direction. This direction is opposite to the direction of the incident wave along the interface in the regime kx<0. It is known negative GH shifts can originate from the contribution of the leaky guided wave or surface wave which propagates back to the direction of the incident wave along the interface [30]. Thus, the one-way edge states in the regime kx<0 maybe result in an enhanced negative GH shift. When phase-matching conditions are satisfied on the surface of the MPC slab, the incident radiation with kx<0 will couple into the slab first, forming a surface wave that propagates back along x-direction. The wave will reemit to the air because the edge band is within the light cone shaded in yellow. As a result, the GH shift will be negative and enhanced remarkably. On the contrary, in the regime kx>0, no edge states are shown in the bandgap under the same incident angle, and thus take place will be a conventional GH shift that is usually small.
We calculate the GH shift on the surface of MPC slab by stationary phase method. For linearly polarized reflected beam, the shift is a function of reflection phase written as
where λ is the incident wavelength, φr is the phase of reflection coefficients at incident angle θ. Equation (2) indicates that a large GH shift is expected at the phases with a big jump, which is usually associated with the resonance provided by exciting a leaky guided wave, or surface wave. Figure 2(a) shows the normalized GH shift D/λ as a function of incident angle and working frequency that covers the bandgap of MPC. The results are obtained by Eq. (2) through the simulations of the reflection coefficient at the MPC interface. In the simulations, the supercell technique was used. We took one column of rings from the MPC slab and applied the periodic boundary condition along x-direction to simulate an infinite slab. The wave ports were used in the y-direction to imitate the plane wave projecting on the rods. The figure shows for every frequency in the bandgap, there is an incident angle at which the GH shift is negative and enhanced. The enhanced GH shift forms a blue line in the figure when it varies with frequency and incident angle. Because kx continues at the interface of MPC and the air while kx in the air is k0sinθ, we redraw the edge band and the enhanced GH shift together as shown in Fig. 2(b). The figure shows the GH shifts fit the edge band, indicating the enhanced GH shift results from the topological edge states. Table 1 lists the specific values of the GH shifts displayed in blue dots in Fig. 2(b).From Fig. 2(a), we also see the GH shift is nonreciprocal. For example, at the frequency f = 5.67 GHz, the GH shift is negative and the shift is about 4.6λ at incident angle θ=-45° (where kx<0), but it is positive and its value is about 2.9λ at θ=+45° (where kx>0). The negative GH shift is remarkably enhanced for θ < 0. This phenomenon is different from the GH shift observed in the dielectric PCs where no nonreciprocal effect takes place [31], though all the shifts can be understood by the exciting leaky waves.
Fig. 2. (a) Normalized GH shifts D/λ of the TE waves reflected from the lossless MPC slab under bias magnetic field H0=+800Oe. (b) The variation of the enhanced GH shift and the edge band in the (θ, f) plane. The blue dots are the position where the enhanced GH shifts appear, and the red line is edge band. (c)-(d) The simulations of incident wave beam reflected from MPC. The arrows represent power flow. Panel (c) for the incident angle θ= -45° and panel (d) for θ=45°. The bias magnetic field is H0=800Oe. The incident wave is a Gaussian beam with beam waist radius w0=2λ, working at frequency f = 5.67 GHz.
As an illustration, we simulated the wave reflection on the surface of the slab. For a clearer illustration, we take the incident wave beam, the Gaussian beam, with beam waist radius w0=2λ, where λ is the incident wavelength. Figures 2(c) and 2(d), respectively, show the results at incident angles θ=±45°. The simulation is done at working frequency 5.67 GHz and bias magnetic field H0=800Oe. When the wave incidents from the upper right (θ=-45°) as displayed in Fig. 2(c), we see the incident wave excites the strong leaky surface wave. The power flow, which is displayed in red arrows, is in the opposite direction of the tangential wave vector kx of the incident wave. The wave propagates and reemits radiation into the air forming a reflected wave beam that has a clear shift along the interface. The GH shift is about 2.5λ, measured by the peak of the reflected beam to the peak of the geometric-optics reflected beam. The GH shift is different from the one calculated by Eq. (2) because of the narrow beam waist of the incident Gaussian beam [30–31]. Figure 3 shows the intensity profiles of the beams reflected from the MPC slab under the different incident beamwidth. The result shows that the shapes of the reflected beams are appreciably different from the Gaussian form. The distortion is more obvious for the beam waist radius w0=2λ. The wider the incident beam is, the lesser the reflected beam is distorted. As a result, the GH shift measured by the intensity profile is closer to the theoretical one. For example, the GH shift D4 is about 3.8λ for the wave beam with waist radius w0=5λ, which is much closer to the theoretical result of 4.6λ listed in Table 1 compared with the result of the beam waist 2λ. In contrast, when the wave beam comes from the upper left (θ=+45°), kx is positive and the MPC works in the bandgap, the MPC slab likes as a total reflector. In this case, Fig. 2(d) shows the incident wave does not excite surface, and the GH shift is small and hardly distinguished in the figure. In addition, the nonreciprocal GH shift depends on not only the incident angle but also the direction of the bias DC magnetic field. The reason is that the operation of reversing the direction of the magnetic field is equivalent to the mirror operation of mapping x→-x [32].
Fig. 3. Intensity profiles of the beams reflected from the MPC slab under the different incident beamwidth. The incident Gaussian beam works at the frequency f = 5.67 GHz. and the incident angle θ= -45°. The thick black dotted line displays the position where the peak of the geometric-optics reflected beam appears. The peaks of the reflected beam with the different incident beam width are marked by thin black dotted lines.
Table 1. The GH shift caused by topological edge state under different frequencies and incident angles.
We further consider the effect of magnetic loss on the GH shift. A typical value of the magnetic damping of the material YIG is α=2 × 10−3. Keeping the other parameters of the ferrite, we calculated the reflectivity and the normalized GH shifts of the MPC slab. Figure 4(a) shows the reflectivity as a function of incident angle and working frequency. We see a narrow dip in the contour map, which indicates the coupling of the incident wave and the MPC due to the edge sates. The magnetic damping does not change the edges state of the MPC. Figure 4(b) shows the normalized GH shift is the function of incident angles for different incident frequencies with and without magnetic damping. The results show the GH shift and its dependence on the incident angles are not affected by the magnetic damping. Therefore, the GH shift is insensitive to the absorption of materials.
Fig. 4. (a) The reflectivity of the TE waves reflected from the loss MPC slab under bias magnetic field H0=+800Oe. (b) The variation of the normalized GH shift is as the function of incident angles at different frequencies with magnetic damping α = 0 (solid lines) and α = 2 × 10−3 (circles).
3. Normal-incidence GH shift
One special in Fig. 2(a) is the enhanced GH shift at normal incidence at which the incident angle is θ=0. We notice at the frequency f = 6.4 GHz the GH shift is about 3.5λ, which is bigger than that of any other angles nearby. At this frequency, we are inferred from Fig. 1(b) that the group velocity of the edge mode is nonzero and positive. This means the normal incident radiation will couple into the edge mode of MPC, exciting a surface wave propagating along x-direction on the MPC interface. The wave will lead to a lateral shift of the reflected beam, i.e. the enhancing the GH shift at normal incidence. Further, the GH shift at normal incidence is also nonreciprocal. Reversing the direction of bias magnetic field, the GH shift changes its sign. As an intuitive illustration, Figs. 5(a)–5(c) plot the simulation results about the wave reflection at the MPC slab; Fig. 5(a) is the electric field distribution of the incident Gaussian beam, and Fig. 5(b) and Fig. 5(c) are the scattered field distributions when the bias magnetic field is H0=800Oe but in opposite directions, respectively. We see the lateral shifts of the two cases move in the opposite direction with respect to the incident wave beam, showing an obvious non-reciprocity. The lateral shift is about 2.3λ, measured by the central positions between the reflected beam and the incident beam. Though similar phenomena are reported on the interfaces of ferrite slab or arrays under the condition of total reflection, they are caused by magnetic surface plasmon resonance, and the shift is small. For example, the GH shift is bound by a maximum value of λ/π on the interface of homogenous ferrites [15], and it is about the wavelength on the interface of ferrite array [16].
Fig. 5. (a)-(c) are simulations for a wave beam scattering from the MPC. (a) The incident field, (b) the scattered field at H0=800Oe, and (c) the scattered field with H0=-800Oe. The wave beam is the Gaussian beam working at frequency 6.4 GHz. The white line in the panels (b) and (c) plots the reference position, the centerline of the incident wave beam to the reflected one. The simulation is done by software COMSOL.
Experimentally, we investigated GH effects caused by topological edge states at normal incidence. We fabricated ferrite rings by a set of thin YIG rods with radius 2 mm to decrease the demagnetization of the rings. Figure 6(a) displays the schematics of the ring fabricated by thin rods. The two types of rings have the same average radius of Ra = 8 mm. The bias magnetic field was provided by circular permanent magnets (NdFeB); they were placed at the bottom of each ring. The MPC sample contains only a chain of rings, which avoids the difficulty in sample making, caused by strong interaction between the magnets. We simulated the effects of ring fabrication and single chain slab on the GH effect. The result is shown in Fig. 6(b). It shows those changes do not affect the feature of the GH shift; the scattered field is almost the same as that of Fig. 5(b) and the surface wave is still concentrated at the outermost surface of the rings.
Fig. 6. (a) Schematics of the replacement of ring by a set of rods. (b) Scattered field distribution at H0=800Oe. The scattered field has a clear wave beam shift to the white line, the incident beam centerline. The incident Gaussian beam is working at frequency of 6.4 GHz. (c) The image of experimental setup. (d)-(g) are the electric field distributions by simulations and experiments. (d) and (f) are, respectively, the simulation and experimental results when bias field along positive z-axis. (e) and (g) are the results when bias field along negative z-axis. The working frequency is 6.56 GHz for experiments and 6.4 GHz for simulations.
Figure 6(c) shows the image of the experiment setup. The sample is sandwiched in two parallel metal plates with a gap of 10 mm and enclosed with absorbers around. The incident wave beam is formed in the waveguide, which was excited by a probe. The wave was polarized perpendicular to the parallel plates. The wave beam perpendicularly projected on the sample away from the waveguide port is about 2λ. Another probe was used to measure the total field distribution near the sample by field mapping technique. The average magnetization field was measured by a Gauss meter; it was about 810Oe in experiments. Under the conditions of the experimental setup, we simulated total field distribution near the sample. Figure 6(d) gives the results when the bias field is along + z-direction. We see the reflected wave beam shifts up. The intensity of the reflected wave becomes weaker because a small amount of the incident energy passes through the sample. In the figure, the white dashed frames denote the boundaries of the absorber. As expected, the wave beam shifts in the opposite direction when we reverse the direction of the bias magnetic field, as shown in Fig. 6(e).
Figure 6(f) and Fig. 6(g) plot, respectively, the measurement results under the opposite direction of the bias magnetic field. In two cases, the field excited in the MPC offsets to one side of the chain, showing the wave in the chain propagates and attenuate in one direction. The experimental results are in good agreement with the simulation ones shown in Fig. 6(d) and Fig. 6(e), and their beam shifts are almost the same. We note their working frequencies have a small shift to the simulation ones. The reason is that the magnetization state is different from the simulation one. Since the bias magnetic field is provided by permanent magnets in experiments, magnetization in ferrite rods along its axis is non-uniform. However, in the simulations the ferrite rods are assumed to be uniform and fully magnetized. Our experiments verify the nonreciprocal GH shifts are caused by the topological edge states of the MPC.
4. Conclusion
In summary, we have demonstrated that the topological edge states of MPC can result in an enhanced negative GH shift. We show that the enhanced negative GH shift is non-reciprocal, and the shift depends on the incident angle and bias DC magnetic field. The nonreciprocal GH shift presents even at normal incidence, which is verified by experiments. The enhanced nonreciprocal negative GH shifts provide a new way to control the flow of light, which can be used in integrated optics and the design of brand of new devices, such as waveguide to trap the light.
Funding
National Natural Science Foundation of China (61671232, 61771237).
Acknowledgments
The authors acknowledge support from Priority Academic Program Development of Jiangsu Higher Education Institutions, and Jiangsu Provincial Key Laboratory of Advanced Manipulating Technique of Electromagnetic Waves.
Disclosures
The authors declare no conflicts of interest.
References
1. F. Goos and H. Hänchen, “Ein neuer und fundamentaler versuch zur totalreflexion,” Ann. Phys. 436(7-8), 333–346 (1947). [CrossRef]
2. I. Shadrivov, A. Zharov, and Y. S. Kivshar, “Giant Goos-Hänchen effect at the reflection from left-handed metamaterials,” Appl. Phys. Lett. 83(13), 2713–2715 (2003). [CrossRef]
3. L. G. Wang, H. Chen, and S. Y. Zhu, “Large negative Goos–Hänchen shift from a weakly absorbing dielectric slab,” Opt. Lett. 30(21), 2936–2938 (2005). [CrossRef]
4. J. H. Huang and P. T. Leung, “Nonlocal optical effects on the Goos-Hänchen shift at an interface of a composite material of metallic nanoparticles,” J. Opt. Soc. Am. A 30(7), 1387–1393 (2013). [CrossRef]
5. R. Yang, W. Zhu, and J. Li, “Giant positive and negative Goos–Hänchen shift on dielectric gratings caused by guided mode resonance,” Opt. Express 22(2), 2043–2051 (2014). [CrossRef]
6. Y. D. Xu, C. T. Chan, and H. Y. Chen, “Goos-Hänchen effect in epsilon-near-zero metamaterials,” Sci. Rep. 5(1), 8681 (2015). [CrossRef]
7. L. G. Wang, M. Ikram, and M. S. Zubairy, “Control of the Goos-Hänchen shift of a light beam via a coherent driving field,” Phys. Rev. A 77(2), 023811 (2008). [CrossRef]
8. D. Felbacq, A. Moreau, and R. Smaâli, “Goos-Hänchen effect in the gaps of photonic crystals,” Opt. Lett. 28(18), 1633–1635 (2003). [CrossRef]
9. I. V. Soboleva, V. V. Moskalenko, and A. A. Fedyanin, “Giant Goos-Hänchen effect and Fano resonance at photonic crystal surfaces,” Phys. Rev. Lett. 108(12), 123901 (2012). [CrossRef]
10. X. B. Yin, L. Hesselink, Z. W. Liu, N. Fang, and X. Zhang, “Large positive and negative lateral optical beam displacements due to surface plasmon resonance,” Appl. Phys. Lett. 85(3), 372–374 (2004). [CrossRef]
11. K. Artmann, “Berechnung der Seitenversetzung des totalreflektierten Strahles,” Ann. Phys. 437(1-2), 87–102 (1948). [CrossRef]
12. F. Lima, T. Dumelow, J. A. P. da Costa, and E. L. Albuquerque, “Lateral shift of far infrared radiation on normal incidence reflection off an antiferromagnet,” EPL 83(1), 17003 (2008). [CrossRef]
13. A. Huamán and G. Usaj, “Anomalous Goos-Hänchen shift in the Floquet scattering of Dirac fermions,” Phys. Rev. A 100(3), 033409 (2019). [CrossRef]
14. R. Macêdo, R. L. Stamps, and T. Dumelow, “Spin canting induced nonreciprocal Goos-Hänchen shifts,” Opt. Express 22(23), 28467–28478 (2014). [CrossRef]
15. W. J. Yu, H. Sun, and L. Gao, “Magnetic control of Goos-Hanchen shifts in a yttrium-iron-garnet film,” Sci. Rep. 7(1), 45866 (2017). [CrossRef]
16. W. J. Yu, H. Sun, and L. Gao, “Enhanced normal-incidence Goos-Hänchen effects induced by magnetic surface plasmons in magneto-optical metamaterials,” Opt. Express 26(4), 3956–3973 (2018). [CrossRef]
17. H. B. Wu, Q. L. Luo, H. J. Chen, Y. Han, X. N. Yu, and S. Y. Liu, “Magnetically controllable nonreciprocal Goos-Hänchen shift supported by a magnetic plasmonic gradient metasurface,” Phys. Rev. A 99(3), 033820 (2019). [CrossRef]
18. R. H. Renard, “Total reflection: A new evaluation of the Goos–Hänchen shift,” J. Opt. Soc. Am. 54(10), 1190–1196 (1964). [CrossRef]
19. M. Mcguirk and C. K. Carniglia, “An angular spectrum representation approach to the Goos-Hänchen shift,” J. Opt. Soc. Am. 67(1), 103–109 (1977). [CrossRef]
20. T. Sakata, H. Togo, and F. Shimokawa, “Reflection-type 2×2 optical waveguide switch using the Goos–Hänchen shift effect,” Appl. Phys. Lett. 76(20), 2841–2843 (2000). [CrossRef]
21. J. J. Sun, X. P. Wang, C. Yin, P. P. Xiao, H. G. Li, and Z. Q. Cao, “Optical transduction of E. Coli O157:H7 concentration by using the enhanced Goos-Hänchen shift,” J. Appl. Phys. 112(8), 083104 (2012). [CrossRef]
22. X. B. Yin and L. Hesselink, “Goos–hänchen shift surface plasmon resonance sensor,” Appl. Phys. Lett. 89(26), 261108 (2006). [CrossRef]
23. Y. Y. Nie, Y. H. Li, Z. J. Wu, X. P. Wang, W. Yuan, and M. H. Sang, “Detection of chemical vapor with high sensitivity by using the symmetrical metal-cladding waveguide-enhanced Goos–hänchen shift,” Opt. Express 22(8), 8943–8948 (2014). [CrossRef]
24. K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “Trapped rainbow’ storage of light in metamaterials,” Nature 450(7168), 397–401 (2007). [CrossRef]
25. H. Ma and R. X. Wu, “Nonreciprocal normal-incidence lateral shift for transmitted wave beams through the magnetic photonic crystal slab,” Appl. Phys. Lett. 116(7), 071104 (2020). [CrossRef]
26. A. Mekis, S. H. Fan, and J. D. Joannopoulos, “Bound states in photonic crystal waveguides and waveguide bends,” Phys. Rev. B 58(8), 4809–4817 (1998). [CrossRef]
27. S. T. Chui, S. Y. Liu, and Z. F. Lin, “Multiple flat photonic bands with finite Chern numbers,” Phys. Rev. E 88(3), 031201 (2013). [CrossRef]
28. Y. F. Ren, Z. Qiao, and Q. Niu, “Topological Phases in Two-Dimensional Materials: A Brief Review,” Rep. Prog. Phys. 79(6), 066501 (2016). [CrossRef]
29. Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461(7265), 772–775 (2009). [CrossRef]
30. T. Tamir and H. L. Bertoni, “Lateral displacement of optical beams at multilayered and periodic structures,” J. Opt. Soc. Am. 61(10), 1397 (1971). [CrossRef]
31. M. Miri, A. Khavasi, F. Alishahi, K. Mehrany, and B. Rashidian, “Approximate expressions for resonant shifts in the reflection of Gaussian wave packets from two-dimensional photonic crystal waveguides,” J. Opt. Soc. Am. B 29(4), 683–690 (2012). [CrossRef]
32. Z. F. Yu, Z. Wang, and S. F. Fan, “One-way total reflection with one-dimensional magneto-optical photonic crystals,” Appl. Phys. Lett. 90(12), 121133 (2007). [CrossRef]
