Abstract

One-dimensional topological photonic crystals (TPCs) with graphene sheet have been proposed to enhance the Faraday rotation (FR). Because of the strong localized field of the topological interface state, the enhanced FR angle with high transmittance has been confirmed. The effects of external magnetic field, unit cell number and multiple interface states of multilayers on FR angle and transmittance are studied. As a result, the FR is raised, which shows a field enhancement constraint at the interface between the TPCs with graphene. The FR angle can reach 16.2° with the high transmission (70%). By constructing multiple interface states, multiple transmission peaks and FR angles are further achieved. Our result would give a fresh idea, which could be applied in nonreciprocal photonic device or optical communication systems.

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

1. Introduction

The interaction between light and magnetic materials can cause various magneto-optical (MO) phenomena, such as Faraday and Kerr rotations, and these phenomena have potential application prospects in optical communication systems [1]. Faraday rotation is one of the well-known MO effects, in which the spin of a photon changes after passing through a material under an external magnetic field.

In the recent, Graphene is demonstrated to be a good MO material by theory and experiment [27]. Graphene is distinct from traditional bulk MO materials in that its FR angle is direct ratio to the ray propagation length. According to reports, The FR angle of the graphene experiment is about 0.15°, which cannot meet the needs of application. And then, it has been found that graphene embedded in the cavity can effectively enhance FR effect [8,9]. But, due to the trade-off between FR and transmittances, the transmittance of these graphene applications introduced is low, that is what limits its application in practical terms. Therefore, breakthrough the limit of graphene applications is challenging.

Topological insulator is a remarkable special material with non-trivial topological bounded state, and have attracted much attention owing to its nature of unidirectional Spin polarization transmission [10]. In recent years, topology has attracted increasing interest. An outstanding case is the Su-Schrieffer-Heeger structure of polyacetylene [1113]. And in the periodic system, generally refers to one-dimensional (1D) photonic crystals (PhCs). The results show that the interface state will appear when the Zak phase is not equal at both ends of the chain, and it could be obtained by gap conversion [14,15]. The interface state can produce a transmittance peak and a highly localized electric field, which can effectively enhance the magneto-optical effect [16,17]. Here, we will overcome the trade-off between transmission and FR angle through topological interface states.

In this paper, we designed a 1D topological photonic structure embedded with graphene sheet to achieve huge FR and high transmission. The method how to generate periodical topological edge modes was introduced by Choi [18]. The 1D PhC consists of two inversion centers. Changing the inversion center will change the Zak phase in the passband from π to 0 (or from 0 to π). Topological interface state patterns exist at the unit interfaces of different Zak phases. In the proposed structure, we put a single graphene layer as magnetic medium into the interface [19,20]. Through the topological interface state, we broke the trade-off between transmission and FR angle. By properly controlling the parameters of the two topological photonic crystals (TPCs), the amplitude (transmission) of the FR angle can reach 16.2° (0.70) at the selected operating frequency. Finally, we constructed the multiple interface states to achieve multi-channel transmission peaks and FR angles [21].

2. Theoretical model and method

Graphene is represented by its semiclassical conductivity tensor [22], the permittivity tensor can be written as [23]

$$\varepsilon = 1 + \frac{i}{{\omega {d_g}{\varepsilon _0}}}\left( {\begin{array}{ccc} {{\sigma_{xx}}}&{{\sigma_{xy}}}&0\\ { - {\sigma_{xy}}}&{{\sigma_{yy}}}&0\\ 0&0&{{\sigma_z}} \end{array}} \right).$$
And the dg is 0.34 nm, σz is related to the magnetic field. Semi-classical expression for the conductivity tensor is [8]:
$${\sigma _{xx}} = {\sigma _{yy}} = \frac{D}{\pi }\frac{{i({\omega + i/\tau } )}}{{{{(\omega + i/\tau )}^2} - \omega _c^2}},$$
$${\sigma _{xy}} ={-} {\sigma _{yx}} = \frac{D}{\pi }\frac{{{\omega _c}}}{{{{(\omega + i/\tau )}^2} - \omega _c^2}}.$$
where $D = {e^2}|{{E_F}} |/{\hbar ^2}$ is the Drude weight, ${\omega _c} = eBv_F^2/{E_F}$ is the cyclotron frequency, and τ is the relaxation time. e, $\hbar$, vF=106 m/s, and EF are electron charge, reduced Planck’s constant, Fermi velocity and energy. And cm2/Vs in $\tau = \mu {\textrm{E}_F}/eV_F^2$ is the carrier mobility. Therefore, the dielectric constant of graphene is isotropic when the magnetic field is zero.

The structure diagram is illustrated in Fig. 1. The graphene monolayer is located on the xy plane, sandwiched by two TPCs. Letter A and B means two isotropic dielectric layers, m means times of repetition, and the external magnetic field B is applied along z axis. Using TiO2 as the dielectric material of component A, and SiO2 is the component of B. The refractive indexes of TiO2 and SiO2 in the analyzed frequency range are 2.068 and 1.435, respectively. The thicknesses of the two PhC components are the quarter of the optical thickness, dA0/4nA, dB0/4nB with λ=59 um as the center wavelength.

 figure: Fig. 1.

Fig. 1. (a)Structure diagram, where graphene is caught between PhC X and PhC Y. An external magnetic field is parallel to the incident light;(b) Schematic diagram of the 1D TPC model. Both X and Y contain n unit cells. The slab A in x is sandwiched by half layer B, and slab B in Y is sandwiched by half layer A. Where the slab A is TiO2, slab B is SiO2.

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3. Results and discussion

Next, we construct topological photonic modes by shifting the inversion centers in the one-dimensional photonic crystal. We can set one of the inversion centers as the origin, then the Zak phase is 0 or π. As shown in Fig. 1(b), the origin of PhC X is A, and the origin of PhC Y is B. The 1D PhC structure composes of PhC X and PhC Y. A graphene sheet is caught between the X and Y, which have N unit cells.

The band structures are shown in Figs. 2(a) and (b), and can be obtain from the following dispersion relationship [24]:

$$\cos (Q\Lambda ) = \cos ({k_A}{d_A})\cos ({k_B}{d_B}) + (\frac{{{Z_A}}}{{{Z_B}}} + \frac{{{Z_B}}}{{{Z_A}}})\sin ({k_A}{d_A})\sin ({k_B}{d_B}).$$
Here, Q is the Bloch wavevector, ${Z_i} = \sqrt {{\mu _i}/{\varepsilon _i}}$ is the impedance (i = A or B), ${n_i} = \sqrt {{\mu _i}{\varepsilon _i}}$ is the fractive-index, di is the thickness of slab, and $\Lambda = {d_A} + {d_B}$ are the width of the unit cell, respectively.

 figure: Fig. 2.

Fig. 2. (a), (b) The band structures of PhC X and PhC Y, respectively. The Zak phase is marked by the green font in the center of the passband. The band gap is represented by blue and purple, and the value in the bracket on the right indicates the superposition of Zak below the band gap; (c) Transmission of PhC X + PhC Y, and the repetitions m is 6. Topological interface states exist on the red arrow surface.

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Each passband has a corresponding Zak phase (0 or π), which can be obtained by [20]

$$\theta _m^{Zak} = \int_{ - \frac{\pi }{\Lambda }}^{\frac{\pi }{\Lambda }} {[i\int\limits_{unit\textrm{ }cell} {\varepsilon (z)u_{m,K}^\ast (z){\partial _K}{u_{m,K}}(z)dz} ]dK} ,$$
where $\varepsilon (z )$ is the space permittivity, ${u_{m,K}}(z )$ indicate the Bloch eigenfunction of electric field at m-th photonic passband. The origin of the topological phase transition shown in Fig. 2 is associated with a special set of frequencies $\varpi $ given by $\sin ({n_B}{d_B}\varpi /c) = 0$.

When the $\varpi $ falls in the pass band, the Zak phase of this band is π. On the contrary, it is zero. However, the zeroth band is special and is determined by this function $exp (i\theta _0^{Zak}) = {\mathop{\rm sgn}} [{1 - {\varepsilon_A}{\mu_B}/({{\varepsilon_B}{\mu_A}} )} ]$. In Fig. 2(a), $\varpi $ is within the band 2, whereas for Fig. 2(b) $\varpi $ is within the band 1 and 3. Figures 2(a) and (b) displays the calculated structure of the belt. The Zak phase is marked in green.

We could search the characteristic state of each band gap by following formula to understand the position of topological interface mode. The sum of Zak phases of all the isolated bands below the nth gap. The sign of ${\varsigma ^{(n)}}$ has the following simple expression [20]:

$${\mathop{\rm sgn}} [{{\varsigma^{(n)}}} ]= {({ - 1} )^n}{({ - 1} )^l}\exp \left( {i\sum\limits_{m = 0}^{n - 1} {\theta_m^{Zak}} } \right),$$
Here, l means the intersection number under the nth gap. We can obtain a characteristic state about each band gap by Eq. (5). We mark it by purple when $\varsigma > 0$ and blue when $\varsigma < 0$. When the band gaps of the two PhCs have a good overlap, the topological interface state will appear at the opposite of ${\mathop{\rm sgn}} [{{\varsigma^{(n)}}} ]$, and it can produce peak transmittance, which means there is local field enhancement there. As we can see from Fig. 2(a) and (b), two PhCs have different characteristic state,such as band gap 1 and band gap 3 drawn by different colors [18]. We can see from the corresponding position in Fig. 2(c) that the superposition of band gaps of different colors will have a transmittance peaks (indicated by red arrows).

Both the FR angle and the light transmittance were computed using the (TMM). Figures 3(a) and (b) are the FR angle and the transmittance at B=3 T of graphene TPC structure (G-TPC) under normal incidence. For comparison, the results of the single graphene model, PhC X and PhC Y are also shown. Here, due to the reciprocity of the components, the FR angles of PhC X and PhC Y are both zero, and θ is significantly enhanced at a wavelength of 63.04 um in the graphene TPC structure. Figure 3(a) show a 0.76 transmittance peak at λ=63.04 μm. The FR angle in the figure is 7 times higher than that of single graphene. As shown in illustration of Fig. 3, the maximum FR angle θF=5.56° is obtained at a position close to the wavelength of the transmittance peak at λ=64.04 μm. Therefore, the important feature here is to obtain both FR effect and high transmission enhancement at the same time, which shows that the traditional trade-offs are solved. In Fig. 3(b), the calculation of the electric field distribution in the G-TPC structure shows that these field enhancements are confined between TPCs. Therefore, this is the reason why the FR effect is enhanced.

 figure: Fig. 3.

Fig. 3. (a) Transmittance and the FR angle versus wavelength of single graphene and G-TPC, where m=3; (b) Electric field of the structure.

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Figure 4(a) and (b) show how we explore how the repetition of the TPC cycle affects the MO characteristics. Other parameters are the same as before. The figure can see the relationship between FR angle and transmission and m, as m increases, the two behave oppositely. For example, when m=4, the transmittance and FR angles are 55% and 10.1°. The FR angle of m=4 is approximately 12 times that of single graphene. For m = 6, when λ=63.04 μm, the two are 0.16 (22.57°). This phenomenon is attributed to the increase in the Q factor of the cavity as m increases. As the quality factor increases, the electromagnetic loss of the resonator decreases, and a stronger local electric field effectively enhances Faraday rotation. Therefore, the number of FR angles can be adjusted as the number of repetitions of two TPCs changes.

 figure: Fig. 4.

Fig. 4. (a) Transmittance of 1D G-TPC with repetitions m=2, 4, and 6; (b) Relationship between FR angle and wavelength;

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Dirac properties of carriers in graphene can generate cyclotron resonance, and the square root correlation increases with the increase of magnetic field [7]. This offers new methods for optimizing MO effect. Figure 5(c) and (d) show the FR Angle and transmittance at B=3, 4, 5 T. In order to compare our result with reference [9], we choose the periodic number m=5. In reference [9], they get 21 times enhancement of FR Angle with T=28%. In Fig. 5(c), we get a 25 times enhancement, but also we have the method to get great transmittance.

 figure: Fig. 5.

Fig. 5. The real part of permittivity components (a) $ {\varepsilon _{xx}}$ and (b) ${\varepsilon _{xy}}$ of graphene in magnetic field B=3,4,5 T. (c) Transmittance and (d) FR angle fo 1D G-TPCs with m=5 under B=3, 4 and 5 T.

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In addition, as the magnetic field increases, the change in transmittance is related to Re(${\varepsilon _{xy}}$). The pink dotted line at 63.04 μm in Fig. 5(b) reveals this relationship. The larger the value of Re(${\varepsilon _{xy}}$), the lower the transmittance is. However, the change in FR Angle is related to Re(${\varepsilon _{xx}}$). As can be seen from Fig. 5(d), as the external magnetic field increases, the MO effect rise first and stay the same. We can analyze form Fig. 5(a). At 63.04 μm, the value of the red line and black line are equal and greater than the value of blue line. The FR Angle enhancement was due to the quantification of graphene Landau energy levels. By combining the strong electric field limitation of graphene with the local mode and Re(${\varepsilon _{xx}}$) will lead to further enhancement of FR angle at high magnetic. Regarding the transmission spectrum, the peak value depends on the designed TPC structure and Re(${\varepsilon _{xy}}$), so we can design the structure according to these characteristics. This characteristic is quite different from the single-layer model. For single-layer structure, its position of the FR angle changes with the change of the magnetic field [25,26]. The reason is that the enhancement of MO effect of G-TPC is primarily due to the spectrum position of topological interface state that comes from model/dielectric instead of cyclotron resonance. We can construct the MO device through this characteristic, the transmission is robust to the wavelength, and increasing the magnetic field can enhance the FR angle of the corresponding wavelength.

Finally, we achieved multiple transmittance peaks and multiple FR angles by constructing multiple interface states. In Fig. 6(a) and (b), we add PhC Y to the left of the structure PhC X-G-PhC Y to constitute the PhC Y-PhC X-G-PhC Y structure, which can generate two interface states. The two transmittance peaks in the figure are 81% and 91%, and FR angles are 5.6° and 1.9°, respectively, which make a relatively large improvement and have potential applications. The structures of (c) and (d) are PhC Y-PhC X-G-PhC Y-PhC X, and there are three interface states. The transmittance peaks are 0.52, 0.97, 0.60, and FR angles are 28°, 0°, and 20.6°, respectively. The reason for the 0° may be that the local fields generated by the interface states on both sides are superimposed in the middle, and the excessively high field causes a FR angle of 0 degree.

 figure: Fig. 6.

Fig. 6. (a) Transmission and (b) FR angle of structure PhC Y-PhC X-G-PhC Y; (c) Transmittance and (d) FR angle of structure PhC Y-PhC X-G-PhC Y-PhC X.

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4. Conclusion

In the article, we use the TMM method to theoretically prove the huge magneto-optical effect in topological photonic crystals with a single layer of graphene, achieving high transmittance and FR angle. In magneto-optical devices, the balance of these two physical quantities are pivotal in property. This result is due to the topological interface state formed between the two photonic crystals, which have a highly localized electric field. The position of the transmittance peak depends on the center wavelength of the interface state, and can be easier to tunable. Finally, by constructing multiple interface states, multiple transmittance peaks and FR angles exist simultaneously, which provides a novel idea for tunable optoelectronic devices such as optical switches and displays.

Funding

Science and Technology Planning Project of Shenzhen Municipality (JCYJ20180305124842330, JCYJ20180305125036005, JCYJ20180508152903208, JCYJ20190808143801672, JCYJ20190808150803580); Natural Science Foundation of Guangdong Province (2018A030313198).

Disclosures

The authors declare no conflicts of interest.

References

1. M. Mansuripur, The Physical Principles of Magneto-Optical Recording (Cambridge University Press, 1998).

2. Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photonics 5(7), 411–415 (2011). [CrossRef]  

3. F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4(9), 611–622 (2010). [CrossRef]  

4. J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, and P. Godignon, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487(7405), 77–81 (2012). [CrossRef]  

5. M. C. Lemme, T. J. Echtermeyer, M. Baus, and H. Kurz, “A graphene field-effect device,” IEEE Electron Device Lett. 28(4), 282–284 (2007). [CrossRef]  

6. N. Stander, B. Huard, and D. Goldhaber-Gordon, “Evidence for Klein tunneling in graphene p− n junctions,” Phys. Rev. Lett. 102(2), 026807 (2009). [CrossRef]  

7. Y. Zhang, Y.-W. Tan, H. L. Stormer, and P. Kim, “Experimental observation of the quantum Hall effect and Berry’s phase in graphene,” Nature 438(7065), 201–204 (2005). [CrossRef]  

8. A. Ferreira, J. Viana-Gomes, Y. V. Bludov, V. Pereira, N. M. R. Peres, and A. C. Neto, “Faraday effect in graphene enclosed in an optical cavity and the equation of motion method for the study of magneto-optical transport in solids,” Phys. Rev. B 84(23), 235410 (2011). [CrossRef]  

9. H. Da and G. Liang, “Enhanced Faraday rotation in magnetophotonic crystal infiltrated with graphene,” Appl. Phys. Lett. 98(26), 261915 (2011). [CrossRef]  

10. B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, “Quantum spin Hall effect and topological phase transition in HgTe quantum wells,” Science 314(5806), 1757–1761 (2006). [CrossRef]  

11. W. Su, J. R. Schrieffer, and A. J. Heeger, “Solitons in polyacetylene,” Phys. Rev. Lett. 42(25), 1698–1701 (1979). [CrossRef]  

12. W.-P. Su, J. R. Schrieffer, and A. J. Heeger, “Soliton excitations in polyacetylene,” Phys. Rev. B 22(4), 2099–2111 (1980). [CrossRef]  

13. A. J. Heeger, S. Kivelson, J. R. Schrieffer, and W.-P. Su, “Solitons in conducting polymers,” Rev. Mod. Phys. 60(3), 781–850 (1988). [CrossRef]  

14. O. A. Pankratov, S. V. Pakhomov, and B. A. Volkov, “Supersymmetry in heterojunctions: Band-inverting contact on the basis of Pb1-xSnxTe and Hg1-xCdxTe,” Solid State Commun. 61(2), 93–96 (1987). [CrossRef]  

15. M. Z. Hasan, S.-Y. Xu, and M. Neupane, “Topological insulators, topological crystalline insulators, and topological Kondo insulators,” Rev. Mod. Phys. 82(4), 3045–3067 (2010). [CrossRef]  

16. L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8(11), 821–829 (2014). [CrossRef]  

17. W. Gao, X. Hu, C. Li, J. Yang, Z. Chai, J. Xie, and Q. Gong, “Fano-resonance in one-dimensional topological photonic crystal heterostructure,” Opt. Express 26(7), 8634–8644 (2018). [CrossRef]  

18. K. H. Choi, C. W. Ling, K. F. Lee, Y. H. Tsang, and K. H. Fung, “Simultaneous multi-frequency topological edge modes between one-dimensional photonic crystals,” Opt. Lett. 41(7), 1644–1647 (2016). [CrossRef]  

19. J. Zak, “Berry’s phase for energy bands in solids,” Phys. Rev. Lett. 62(23), 2747–2750 (1989). [CrossRef]  

20. M. Xiao, Z. Q. Zhang, and C. T. Chan, “Surface impedance and bulk band geometric phases in one-dimensional systems,” Phys. Rev. X 4(2), 021017 (2014). [CrossRef]  

21. Š. Višňovský, K. Postava, and T. Yamaguchi, “Magneto-optic polar Kerr and Faraday effects in magnetic superlattices,” Czech. J. Phys. 51(9), 917–949 (2001). [CrossRef]  

22. T. Tang, C. Li, L. Luo, Y. Zhang, and Q. Yuan, “Thermo-optic Imbert-Fedorov effect in a prism-waveguide coupling system with silicon-on-insulator,” Opt. Commun. 370, 49–54 (2016). [CrossRef]  

23. W. Wang, S. P. Apell, and J. M. Kinaret, “Edge magnetoplasmons and the optical excitations in graphene disks,” Phys. Rev. B 86(12), 125450 (2012). [CrossRef]  

24. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley New York, 1984), 5.

25. I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. Van Der Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single-and multilayer graphene,” Nat. Phys. 7(1), 48–51 (2011). [CrossRef]  

26. I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic terahertz plasmons and magnetoplasmons in large scale monolayer graphene,” Nano Lett. 12(5), 2470–2474 (2012). [CrossRef]  

References

  • View by:

  1. M. Mansuripur, The Physical Principles of Magneto-Optical Recording (Cambridge University Press, 1998).
  2. Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photonics 5(7), 411–415 (2011).
    [Crossref]
  3. F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4(9), 611–622 (2010).
    [Crossref]
  4. J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, and P. Godignon, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487(7405), 77–81 (2012).
    [Crossref]
  5. M. C. Lemme, T. J. Echtermeyer, M. Baus, and H. Kurz, “A graphene field-effect device,” IEEE Electron Device Lett. 28(4), 282–284 (2007).
    [Crossref]
  6. N. Stander, B. Huard, and D. Goldhaber-Gordon, “Evidence for Klein tunneling in graphene p− n junctions,” Phys. Rev. Lett. 102(2), 026807 (2009).
    [Crossref]
  7. Y. Zhang, Y.-W. Tan, H. L. Stormer, and P. Kim, “Experimental observation of the quantum Hall effect and Berry’s phase in graphene,” Nature 438(7065), 201–204 (2005).
    [Crossref]
  8. A. Ferreira, J. Viana-Gomes, Y. V. Bludov, V. Pereira, N. M. R. Peres, and A. C. Neto, “Faraday effect in graphene enclosed in an optical cavity and the equation of motion method for the study of magneto-optical transport in solids,” Phys. Rev. B 84(23), 235410 (2011).
    [Crossref]
  9. H. Da and G. Liang, “Enhanced Faraday rotation in magnetophotonic crystal infiltrated with graphene,” Appl. Phys. Lett. 98(26), 261915 (2011).
    [Crossref]
  10. B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, “Quantum spin Hall effect and topological phase transition in HgTe quantum wells,” Science 314(5806), 1757–1761 (2006).
    [Crossref]
  11. W. Su, J. R. Schrieffer, and A. J. Heeger, “Solitons in polyacetylene,” Phys. Rev. Lett. 42(25), 1698–1701 (1979).
    [Crossref]
  12. W.-P. Su, J. R. Schrieffer, and A. J. Heeger, “Soliton excitations in polyacetylene,” Phys. Rev. B 22(4), 2099–2111 (1980).
    [Crossref]
  13. A. J. Heeger, S. Kivelson, J. R. Schrieffer, and W.-P. Su, “Solitons in conducting polymers,” Rev. Mod. Phys. 60(3), 781–850 (1988).
    [Crossref]
  14. O. A. Pankratov, S. V. Pakhomov, and B. A. Volkov, “Supersymmetry in heterojunctions: Band-inverting contact on the basis of Pb1-xSnxTe and Hg1-xCdxTe,” Solid State Commun. 61(2), 93–96 (1987).
    [Crossref]
  15. M. Z. Hasan, S.-Y. Xu, and M. Neupane, “Topological insulators, topological crystalline insulators, and topological Kondo insulators,” Rev. Mod. Phys. 82(4), 3045–3067 (2010).
    [Crossref]
  16. L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8(11), 821–829 (2014).
    [Crossref]
  17. W. Gao, X. Hu, C. Li, J. Yang, Z. Chai, J. Xie, and Q. Gong, “Fano-resonance in one-dimensional topological photonic crystal heterostructure,” Opt. Express 26(7), 8634–8644 (2018).
    [Crossref]
  18. K. H. Choi, C. W. Ling, K. F. Lee, Y. H. Tsang, and K. H. Fung, “Simultaneous multi-frequency topological edge modes between one-dimensional photonic crystals,” Opt. Lett. 41(7), 1644–1647 (2016).
    [Crossref]
  19. J. Zak, “Berry’s phase for energy bands in solids,” Phys. Rev. Lett. 62(23), 2747–2750 (1989).
    [Crossref]
  20. M. Xiao, Z. Q. Zhang, and C. T. Chan, “Surface impedance and bulk band geometric phases in one-dimensional systems,” Phys. Rev. X 4(2), 021017 (2014).
    [Crossref]
  21. Š. Višňovský, K. Postava, and T. Yamaguchi, “Magneto-optic polar Kerr and Faraday effects in magnetic superlattices,” Czech. J. Phys. 51(9), 917–949 (2001).
    [Crossref]
  22. T. Tang, C. Li, L. Luo, Y. Zhang, and Q. Yuan, “Thermo-optic Imbert-Fedorov effect in a prism-waveguide coupling system with silicon-on-insulator,” Opt. Commun. 370, 49–54 (2016).
    [Crossref]
  23. W. Wang, S. P. Apell, and J. M. Kinaret, “Edge magnetoplasmons and the optical excitations in graphene disks,” Phys. Rev. B 86(12), 125450 (2012).
    [Crossref]
  24. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley New York, 1984), 5.
  25. I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. Van Der Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single-and multilayer graphene,” Nat. Phys. 7(1), 48–51 (2011).
    [Crossref]
  26. I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic terahertz plasmons and magnetoplasmons in large scale monolayer graphene,” Nano Lett. 12(5), 2470–2474 (2012).
    [Crossref]

2018 (1)

2016 (2)

K. H. Choi, C. W. Ling, K. F. Lee, Y. H. Tsang, and K. H. Fung, “Simultaneous multi-frequency topological edge modes between one-dimensional photonic crystals,” Opt. Lett. 41(7), 1644–1647 (2016).
[Crossref]

T. Tang, C. Li, L. Luo, Y. Zhang, and Q. Yuan, “Thermo-optic Imbert-Fedorov effect in a prism-waveguide coupling system with silicon-on-insulator,” Opt. Commun. 370, 49–54 (2016).
[Crossref]

2014 (2)

M. Xiao, Z. Q. Zhang, and C. T. Chan, “Surface impedance and bulk band geometric phases in one-dimensional systems,” Phys. Rev. X 4(2), 021017 (2014).
[Crossref]

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8(11), 821–829 (2014).
[Crossref]

2012 (3)

I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic terahertz plasmons and magnetoplasmons in large scale monolayer graphene,” Nano Lett. 12(5), 2470–2474 (2012).
[Crossref]

W. Wang, S. P. Apell, and J. M. Kinaret, “Edge magnetoplasmons and the optical excitations in graphene disks,” Phys. Rev. B 86(12), 125450 (2012).
[Crossref]

J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, and P. Godignon, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487(7405), 77–81 (2012).
[Crossref]

2011 (4)

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photonics 5(7), 411–415 (2011).
[Crossref]

A. Ferreira, J. Viana-Gomes, Y. V. Bludov, V. Pereira, N. M. R. Peres, and A. C. Neto, “Faraday effect in graphene enclosed in an optical cavity and the equation of motion method for the study of magneto-optical transport in solids,” Phys. Rev. B 84(23), 235410 (2011).
[Crossref]

H. Da and G. Liang, “Enhanced Faraday rotation in magnetophotonic crystal infiltrated with graphene,” Appl. Phys. Lett. 98(26), 261915 (2011).
[Crossref]

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. Van Der Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single-and multilayer graphene,” Nat. Phys. 7(1), 48–51 (2011).
[Crossref]

2010 (2)

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4(9), 611–622 (2010).
[Crossref]

M. Z. Hasan, S.-Y. Xu, and M. Neupane, “Topological insulators, topological crystalline insulators, and topological Kondo insulators,” Rev. Mod. Phys. 82(4), 3045–3067 (2010).
[Crossref]

2009 (1)

N. Stander, B. Huard, and D. Goldhaber-Gordon, “Evidence for Klein tunneling in graphene p− n junctions,” Phys. Rev. Lett. 102(2), 026807 (2009).
[Crossref]

2007 (1)

M. C. Lemme, T. J. Echtermeyer, M. Baus, and H. Kurz, “A graphene field-effect device,” IEEE Electron Device Lett. 28(4), 282–284 (2007).
[Crossref]

2006 (1)

B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, “Quantum spin Hall effect and topological phase transition in HgTe quantum wells,” Science 314(5806), 1757–1761 (2006).
[Crossref]

2005 (1)

Y. Zhang, Y.-W. Tan, H. L. Stormer, and P. Kim, “Experimental observation of the quantum Hall effect and Berry’s phase in graphene,” Nature 438(7065), 201–204 (2005).
[Crossref]

2001 (1)

Š. Višňovský, K. Postava, and T. Yamaguchi, “Magneto-optic polar Kerr and Faraday effects in magnetic superlattices,” Czech. J. Phys. 51(9), 917–949 (2001).
[Crossref]

1989 (1)

J. Zak, “Berry’s phase for energy bands in solids,” Phys. Rev. Lett. 62(23), 2747–2750 (1989).
[Crossref]

1988 (1)

A. J. Heeger, S. Kivelson, J. R. Schrieffer, and W.-P. Su, “Solitons in conducting polymers,” Rev. Mod. Phys. 60(3), 781–850 (1988).
[Crossref]

1987 (1)

O. A. Pankratov, S. V. Pakhomov, and B. A. Volkov, “Supersymmetry in heterojunctions: Band-inverting contact on the basis of Pb1-xSnxTe and Hg1-xCdxTe,” Solid State Commun. 61(2), 93–96 (1987).
[Crossref]

1980 (1)

W.-P. Su, J. R. Schrieffer, and A. J. Heeger, “Soliton excitations in polyacetylene,” Phys. Rev. B 22(4), 2099–2111 (1980).
[Crossref]

1979 (1)

W. Su, J. R. Schrieffer, and A. J. Heeger, “Solitons in polyacetylene,” Phys. Rev. Lett. 42(25), 1698–1701 (1979).
[Crossref]

Alonso-González, P.

J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, and P. Godignon, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487(7405), 77–81 (2012).
[Crossref]

Apell, S. P.

W. Wang, S. P. Apell, and J. M. Kinaret, “Edge magnetoplasmons and the optical excitations in graphene disks,” Phys. Rev. B 86(12), 125450 (2012).
[Crossref]

Badioli, M.

J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, and P. Godignon, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487(7405), 77–81 (2012).
[Crossref]

Bao, Q.

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photonics 5(7), 411–415 (2011).
[Crossref]

Baus, M.

M. C. Lemme, T. J. Echtermeyer, M. Baus, and H. Kurz, “A graphene field-effect device,” IEEE Electron Device Lett. 28(4), 282–284 (2007).
[Crossref]

Bernevig, B. A.

B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, “Quantum spin Hall effect and topological phase transition in HgTe quantum wells,” Science 314(5806), 1757–1761 (2006).
[Crossref]

Bludov, Y. V.

A. Ferreira, J. Viana-Gomes, Y. V. Bludov, V. Pereira, N. M. R. Peres, and A. C. Neto, “Faraday effect in graphene enclosed in an optical cavity and the equation of motion method for the study of magneto-optical transport in solids,” Phys. Rev. B 84(23), 235410 (2011).
[Crossref]

Bonaccorso, F.

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4(9), 611–622 (2010).
[Crossref]

Bostwick, A.

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. Van Der Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single-and multilayer graphene,” Nat. Phys. 7(1), 48–51 (2011).
[Crossref]

Centeno, A.

J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, and P. Godignon, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487(7405), 77–81 (2012).
[Crossref]

Chai, Z.

Chan, C. T.

M. Xiao, Z. Q. Zhang, and C. T. Chan, “Surface impedance and bulk band geometric phases in one-dimensional systems,” Phys. Rev. X 4(2), 021017 (2014).
[Crossref]

Chen, J.

I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic terahertz plasmons and magnetoplasmons in large scale monolayer graphene,” Nano Lett. 12(5), 2470–2474 (2012).
[Crossref]

J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, and P. Godignon, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487(7405), 77–81 (2012).
[Crossref]

Choi, K. H.

Crassee, I.

I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic terahertz plasmons and magnetoplasmons in large scale monolayer graphene,” Nano Lett. 12(5), 2470–2474 (2012).
[Crossref]

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. Van Der Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single-and multilayer graphene,” Nat. Phys. 7(1), 48–51 (2011).
[Crossref]

Da, H.

H. Da and G. Liang, “Enhanced Faraday rotation in magnetophotonic crystal infiltrated with graphene,” Appl. Phys. Lett. 98(26), 261915 (2011).
[Crossref]

Echtermeyer, T. J.

M. C. Lemme, T. J. Echtermeyer, M. Baus, and H. Kurz, “A graphene field-effect device,” IEEE Electron Device Lett. 28(4), 282–284 (2007).
[Crossref]

Ferrari, A. C.

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4(9), 611–622 (2010).
[Crossref]

Ferreira, A.

A. Ferreira, J. Viana-Gomes, Y. V. Bludov, V. Pereira, N. M. R. Peres, and A. C. Neto, “Faraday effect in graphene enclosed in an optical cavity and the equation of motion method for the study of magneto-optical transport in solids,” Phys. Rev. B 84(23), 235410 (2011).
[Crossref]

Fung, K. H.

Gao, W.

Gaponenko, I.

I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic terahertz plasmons and magnetoplasmons in large scale monolayer graphene,” Nano Lett. 12(5), 2470–2474 (2012).
[Crossref]

Godignon, P.

J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, and P. Godignon, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487(7405), 77–81 (2012).
[Crossref]

Goldhaber-Gordon, D.

N. Stander, B. Huard, and D. Goldhaber-Gordon, “Evidence for Klein tunneling in graphene p− n junctions,” Phys. Rev. Lett. 102(2), 026807 (2009).
[Crossref]

Gong, Q.

Hasan, M. Z.

M. Z. Hasan, S.-Y. Xu, and M. Neupane, “Topological insulators, topological crystalline insulators, and topological Kondo insulators,” Rev. Mod. Phys. 82(4), 3045–3067 (2010).
[Crossref]

Hasan, T.

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4(9), 611–622 (2010).
[Crossref]

Heeger, A. J.

A. J. Heeger, S. Kivelson, J. R. Schrieffer, and W.-P. Su, “Solitons in conducting polymers,” Rev. Mod. Phys. 60(3), 781–850 (1988).
[Crossref]

W.-P. Su, J. R. Schrieffer, and A. J. Heeger, “Soliton excitations in polyacetylene,” Phys. Rev. B 22(4), 2099–2111 (1980).
[Crossref]

W. Su, J. R. Schrieffer, and A. J. Heeger, “Solitons in polyacetylene,” Phys. Rev. Lett. 42(25), 1698–1701 (1979).
[Crossref]

Hu, X.

Huard, B.

N. Stander, B. Huard, and D. Goldhaber-Gordon, “Evidence for Klein tunneling in graphene p− n junctions,” Phys. Rev. Lett. 102(2), 026807 (2009).
[Crossref]

Hughes, T. L.

B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, “Quantum spin Hall effect and topological phase transition in HgTe quantum wells,” Science 314(5806), 1757–1761 (2006).
[Crossref]

Huth, F.

J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, and P. Godignon, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487(7405), 77–81 (2012).
[Crossref]

Joannopoulos, J. D.

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8(11), 821–829 (2014).
[Crossref]

Kim, P.

Y. Zhang, Y.-W. Tan, H. L. Stormer, and P. Kim, “Experimental observation of the quantum Hall effect and Berry’s phase in graphene,” Nature 438(7065), 201–204 (2005).
[Crossref]

Kinaret, J. M.

W. Wang, S. P. Apell, and J. M. Kinaret, “Edge magnetoplasmons and the optical excitations in graphene disks,” Phys. Rev. B 86(12), 125450 (2012).
[Crossref]

Kivelson, S.

A. J. Heeger, S. Kivelson, J. R. Schrieffer, and W.-P. Su, “Solitons in conducting polymers,” Rev. Mod. Phys. 60(3), 781–850 (1988).
[Crossref]

Kurz, H.

M. C. Lemme, T. J. Echtermeyer, M. Baus, and H. Kurz, “A graphene field-effect device,” IEEE Electron Device Lett. 28(4), 282–284 (2007).
[Crossref]

Kuzmenko, A. B.

I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic terahertz plasmons and magnetoplasmons in large scale monolayer graphene,” Nano Lett. 12(5), 2470–2474 (2012).
[Crossref]

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. Van Der Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single-and multilayer graphene,” Nat. Phys. 7(1), 48–51 (2011).
[Crossref]

Lee, K. F.

Lemme, M. C.

M. C. Lemme, T. J. Echtermeyer, M. Baus, and H. Kurz, “A graphene field-effect device,” IEEE Electron Device Lett. 28(4), 282–284 (2007).
[Crossref]

Levallois, J.

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. Van Der Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single-and multilayer graphene,” Nat. Phys. 7(1), 48–51 (2011).
[Crossref]

Li, C.

W. Gao, X. Hu, C. Li, J. Yang, Z. Chai, J. Xie, and Q. Gong, “Fano-resonance in one-dimensional topological photonic crystal heterostructure,” Opt. Express 26(7), 8634–8644 (2018).
[Crossref]

T. Tang, C. Li, L. Luo, Y. Zhang, and Q. Yuan, “Thermo-optic Imbert-Fedorov effect in a prism-waveguide coupling system with silicon-on-insulator,” Opt. Commun. 370, 49–54 (2016).
[Crossref]

Liang, G.

H. Da and G. Liang, “Enhanced Faraday rotation in magnetophotonic crystal infiltrated with graphene,” Appl. Phys. Lett. 98(26), 261915 (2011).
[Crossref]

Lim, C. H. Y. X.

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photonics 5(7), 411–415 (2011).
[Crossref]

Ling, C. W.

Loh, K. P.

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photonics 5(7), 411–415 (2011).
[Crossref]

Lu, L.

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8(11), 821–829 (2014).
[Crossref]

Luo, L.

T. Tang, C. Li, L. Luo, Y. Zhang, and Q. Yuan, “Thermo-optic Imbert-Fedorov effect in a prism-waveguide coupling system with silicon-on-insulator,” Opt. Commun. 370, 49–54 (2016).
[Crossref]

Mansuripur, M.

M. Mansuripur, The Physical Principles of Magneto-Optical Recording (Cambridge University Press, 1998).

Neto, A. C.

A. Ferreira, J. Viana-Gomes, Y. V. Bludov, V. Pereira, N. M. R. Peres, and A. C. Neto, “Faraday effect in graphene enclosed in an optical cavity and the equation of motion method for the study of magneto-optical transport in solids,” Phys. Rev. B 84(23), 235410 (2011).
[Crossref]

Neupane, M.

M. Z. Hasan, S.-Y. Xu, and M. Neupane, “Topological insulators, topological crystalline insulators, and topological Kondo insulators,” Rev. Mod. Phys. 82(4), 3045–3067 (2010).
[Crossref]

Ni, Z.

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photonics 5(7), 411–415 (2011).
[Crossref]

Orlita, M.

I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic terahertz plasmons and magnetoplasmons in large scale monolayer graphene,” Nano Lett. 12(5), 2470–2474 (2012).
[Crossref]

Osmond, J.

J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, and P. Godignon, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487(7405), 77–81 (2012).
[Crossref]

Ostler, M.

I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic terahertz plasmons and magnetoplasmons in large scale monolayer graphene,” Nano Lett. 12(5), 2470–2474 (2012).
[Crossref]

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. Van Der Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single-and multilayer graphene,” Nat. Phys. 7(1), 48–51 (2011).
[Crossref]

Pakhomov, S. V.

O. A. Pankratov, S. V. Pakhomov, and B. A. Volkov, “Supersymmetry in heterojunctions: Band-inverting contact on the basis of Pb1-xSnxTe and Hg1-xCdxTe,” Solid State Commun. 61(2), 93–96 (1987).
[Crossref]

Pankratov, O. A.

O. A. Pankratov, S. V. Pakhomov, and B. A. Volkov, “Supersymmetry in heterojunctions: Band-inverting contact on the basis of Pb1-xSnxTe and Hg1-xCdxTe,” Solid State Commun. 61(2), 93–96 (1987).
[Crossref]

Pereira, V.

A. Ferreira, J. Viana-Gomes, Y. V. Bludov, V. Pereira, N. M. R. Peres, and A. C. Neto, “Faraday effect in graphene enclosed in an optical cavity and the equation of motion method for the study of magneto-optical transport in solids,” Phys. Rev. B 84(23), 235410 (2011).
[Crossref]

Peres, N. M. R.

A. Ferreira, J. Viana-Gomes, Y. V. Bludov, V. Pereira, N. M. R. Peres, and A. C. Neto, “Faraday effect in graphene enclosed in an optical cavity and the equation of motion method for the study of magneto-optical transport in solids,” Phys. Rev. B 84(23), 235410 (2011).
[Crossref]

Pesquera, A.

J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, and P. Godignon, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487(7405), 77–81 (2012).
[Crossref]

Postava, K.

Š. Višňovský, K. Postava, and T. Yamaguchi, “Magneto-optic polar Kerr and Faraday effects in magnetic superlattices,” Czech. J. Phys. 51(9), 917–949 (2001).
[Crossref]

Potemski, M.

I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic terahertz plasmons and magnetoplasmons in large scale monolayer graphene,” Nano Lett. 12(5), 2470–2474 (2012).
[Crossref]

Rotenberg, E.

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. Van Der Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single-and multilayer graphene,” Nat. Phys. 7(1), 48–51 (2011).
[Crossref]

Schrieffer, J. R.

A. J. Heeger, S. Kivelson, J. R. Schrieffer, and W.-P. Su, “Solitons in conducting polymers,” Rev. Mod. Phys. 60(3), 781–850 (1988).
[Crossref]

W.-P. Su, J. R. Schrieffer, and A. J. Heeger, “Soliton excitations in polyacetylene,” Phys. Rev. B 22(4), 2099–2111 (1980).
[Crossref]

W. Su, J. R. Schrieffer, and A. J. Heeger, “Solitons in polyacetylene,” Phys. Rev. Lett. 42(25), 1698–1701 (1979).
[Crossref]

Seyller, T.

I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic terahertz plasmons and magnetoplasmons in large scale monolayer graphene,” Nano Lett. 12(5), 2470–2474 (2012).
[Crossref]

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. Van Der Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single-and multilayer graphene,” Nat. Phys. 7(1), 48–51 (2011).
[Crossref]

Soljacic, M.

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8(11), 821–829 (2014).
[Crossref]

Spasenovic, M.

J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, and P. Godignon, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487(7405), 77–81 (2012).
[Crossref]

Stander, N.

N. Stander, B. Huard, and D. Goldhaber-Gordon, “Evidence for Klein tunneling in graphene p− n junctions,” Phys. Rev. Lett. 102(2), 026807 (2009).
[Crossref]

Stormer, H. L.

Y. Zhang, Y.-W. Tan, H. L. Stormer, and P. Kim, “Experimental observation of the quantum Hall effect and Berry’s phase in graphene,” Nature 438(7065), 201–204 (2005).
[Crossref]

Su, W.

W. Su, J. R. Schrieffer, and A. J. Heeger, “Solitons in polyacetylene,” Phys. Rev. Lett. 42(25), 1698–1701 (1979).
[Crossref]

Su, W.-P.

A. J. Heeger, S. Kivelson, J. R. Schrieffer, and W.-P. Su, “Solitons in conducting polymers,” Rev. Mod. Phys. 60(3), 781–850 (1988).
[Crossref]

W.-P. Su, J. R. Schrieffer, and A. J. Heeger, “Soliton excitations in polyacetylene,” Phys. Rev. B 22(4), 2099–2111 (1980).
[Crossref]

Sun, Z.

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4(9), 611–622 (2010).
[Crossref]

Tan, Y.-W.

Y. Zhang, Y.-W. Tan, H. L. Stormer, and P. Kim, “Experimental observation of the quantum Hall effect and Berry’s phase in graphene,” Nature 438(7065), 201–204 (2005).
[Crossref]

Tang, D. Y.

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photonics 5(7), 411–415 (2011).
[Crossref]

Tang, T.

T. Tang, C. Li, L. Luo, Y. Zhang, and Q. Yuan, “Thermo-optic Imbert-Fedorov effect in a prism-waveguide coupling system with silicon-on-insulator,” Opt. Commun. 370, 49–54 (2016).
[Crossref]

Thongrattanasiri, S.

J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, and P. Godignon, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487(7405), 77–81 (2012).
[Crossref]

Tsang, Y. H.

Van Der Marel, D.

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. Van Der Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single-and multilayer graphene,” Nat. Phys. 7(1), 48–51 (2011).
[Crossref]

Viana-Gomes, J.

A. Ferreira, J. Viana-Gomes, Y. V. Bludov, V. Pereira, N. M. R. Peres, and A. C. Neto, “Faraday effect in graphene enclosed in an optical cavity and the equation of motion method for the study of magneto-optical transport in solids,” Phys. Rev. B 84(23), 235410 (2011).
[Crossref]

Višnovský, Š.

Š. Višňovský, K. Postava, and T. Yamaguchi, “Magneto-optic polar Kerr and Faraday effects in magnetic superlattices,” Czech. J. Phys. 51(9), 917–949 (2001).
[Crossref]

Volkov, B. A.

O. A. Pankratov, S. V. Pakhomov, and B. A. Volkov, “Supersymmetry in heterojunctions: Band-inverting contact on the basis of Pb1-xSnxTe and Hg1-xCdxTe,” Solid State Commun. 61(2), 93–96 (1987).
[Crossref]

Walter, A. L.

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Figures (6)

Fig. 1.
Fig. 1. (a)Structure diagram, where graphene is caught between PhC X and PhC Y. An external magnetic field is parallel to the incident light;(b) Schematic diagram of the 1D TPC model. Both X and Y contain n unit cells. The slab A in x is sandwiched by half layer B, and slab B in Y is sandwiched by half layer A. Where the slab A is TiO2, slab B is SiO2.
Fig. 2.
Fig. 2. (a), (b) The band structures of PhC X and PhC Y, respectively. The Zak phase is marked by the green font in the center of the passband. The band gap is represented by blue and purple, and the value in the bracket on the right indicates the superposition of Zak below the band gap; (c) Transmission of PhC X + PhC Y, and the repetitions m is 6. Topological interface states exist on the red arrow surface.
Fig. 3.
Fig. 3. (a) Transmittance and the FR angle versus wavelength of single graphene and G-TPC, where m=3; (b) Electric field of the structure.
Fig. 4.
Fig. 4. (a) Transmittance of 1D G-TPC with repetitions m=2, 4, and 6; (b) Relationship between FR angle and wavelength;
Fig. 5.
Fig. 5. The real part of permittivity components (a) $ {\varepsilon _{xx}}$ and (b) ${\varepsilon _{xy}}$ of graphene in magnetic field B=3,4,5 T. (c) Transmittance and (d) FR angle fo 1D G-TPCs with m=5 under B=3, 4 and 5 T.
Fig. 6.
Fig. 6. (a) Transmission and (b) FR angle of structure PhC Y-PhC X-G-PhC Y; (c) Transmittance and (d) FR angle of structure PhC Y-PhC X-G-PhC Y-PhC X.

Equations (6)

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ε = 1 + i ω d g ε 0 ( σ x x σ x y 0 σ x y σ y y 0 0 0 σ z ) .
σ x x = σ y y = D π i ( ω + i / τ ) ( ω + i / τ ) 2 ω c 2 ,
σ x y = σ y x = D π ω c ( ω + i / τ ) 2 ω c 2 .
cos ( Q Λ ) = cos ( k A d A ) cos ( k B d B ) + ( Z A Z B + Z B Z A ) sin ( k A d A ) sin ( k B d B ) .
θ m Z a k = π Λ π Λ [ i u n i t   c e l l ε ( z ) u m , K ( z ) K u m , K ( z ) d z ] d K ,
sgn [ ς ( n ) ] = ( 1 ) n ( 1 ) l exp ( i m = 0 n 1 θ m Z a k ) ,

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