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Abstract
In this paper, the one-dimensional photonic crystal Fano resonance heterostructure is used to achieve low-threshold and tunable graphene-based optical bistability of the transmitted and reflected light beam at optical communication band. The low-threshold of optical bistability (OB) originates from the local field enhancement owing to the Fano resonance excited by topological edge states mode and Fabry-Perot cavity mode. The study found that it is feasible to continuously adjust the hysteresis behavior and optical bistable thresholds by altering the Fermi energy of the left and right graphene respectively. Furthermore, the OB can also be controlled by changing the number of graphene layers or the angle of incident beam, which makes this structure a feasible object of experimental research at optical communication band in the future.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
As a representative nonlinear optical phenomenon, optical bistability (OB) describes the hysteresis switching phenomenon where a stable input light intensity could produce two different output light intensities [1]. OB has an important practical value in phototransistors [2], all-optical switches [3,4], optical storage [5,6], and all-optical logic devices [7] and so on. The production and regulation of micro/nano-structured OB were widely reported in metamaterials [8,9], photonic crystals [10,11], Fabry-Perot (F-P) cavities [12], nanostructures [13], etc. Meanwhile, graphene is ideal for OB devices because of its ultra-high nonlinear refractive index [14], ultra-high response time, modulation rate [15,16], broadband [17] and dynamically controllable conductivity [18]. Since the generation of OB requires a strong nonlinear effect, the current research on graphene-based OB are mostly focused on the Terahertz (THz) band. Graphene-based OB has attracted intensive research, for example, graphene-doped defective photonic multilayers [19], graphene-controlled fiber Bragg grating [20], graphene-wrapped dielectric composite [21], graphene-covered multilayer Otto configuration [22], etc. Since the nonlinear coefficient of graphene is inversely proportional to the third power of incident light frequency, the nonlinear effect of communication band with high frequency is relatively weak, but this frequency band has practical value in the actual operation process. Therefore, designing an optical bistable device that is easy to integrate, low power consumption, and suitable for optical communication is an extremely important job. Based on the above considerations, we propose that the one-dimensional photonic crystal heterostructure which can excite Fano resonance by coupling method to realize the low-threshold optical bistability of the transmitted light and reflected light beam in the optical communication band.
The OB threshold is closely related to the intensity of local electric field. Usually, micro/nano structures with strong local field characteristics are selected to combine with graphene to excite OB, such as low-threshold OB with graphene surface plasmons at THz frequency [23], low-threshold OB in graphene with one-dimensional gratings [24] and low-threshold OB in graphene with Tamm plasmon [25]. Recently, the emergence of topological edge state (TES) has attracted great attention due to its unique characteristics like topological protection, elimination of backscattering and local field enhancement [26]. The theory and experimental realization of TES have been proven applicable to photonic systems, such as chiral hyperbolic metamaterials [27], uniaxial metacrystal waveguide [28] and imaging in silicon photonics [29]. The TES based on one-dimensional photonic crystals [30,31] and systems [32] were also studied. The expected local field enhancement phenomenon appears at the interface of one-dimensional photonic crystal when TES exists [33]. It can be inferred that this feature plays a positive role in reducing the threshold of OB. The local field enhancement caused by the coupling of the TES mode and the F-P cavity mode creates conditions for the strong nonlinear optical effect of graphene at optical communication band. In addition, this approach provides a scheme and reference for the realization of the low-threshold OB phenomenon at optical communication band.
In this paper, we have theoretically analyzed the low-threshold and tunable graphene-based OB in one-dimensional photonic crystal (1D-PhC) Fano resonance heterostructure at optical communication band (1550 nm in wavelength). We built two parts of 1D-PhC to excite Fano resonance response: one part is to excite TES mode and the other is to form F-P cavity mode. At the same time, the transmission peaks of TES structure and F-P cavity appears near the wavelength of 1550 nm through parameter selection and optimization, which creates feasible conditions for the realization of low threshold OB in the optical communication band. We found that the OB threshold at 1550 nm can be greatly reduced in the proposed 1D-PhC Fano resonance heterostructure as a result of enhanced local electric field and the dynamic regulation of OB is achieved by altering the conductivity of graphene. The 1D-PhC Fano resonance heterostructure provides an alternative scheme for the study of graphene-based tunable and low-threshold OB at optical communication band and a feasible pathway for the design of optical bistable devices.
2. Theoretical model and method
We consider the heterostructure of photonic crystals composed of three photonic crystals, named PhC1, PhC2 and PhC3 from left to right. Meanwhile, two monolayers of graphene are embedded into the interface between PhC1 and PhC2, and the middle of the defect layer between PhC2 and PhC3, respectively, as shown in Fig. 1. For the convenience of description and calculation, the PhC is seen as a layered structure; the transmitted and reflected electromagnetic waves in each layer of the medium are marked as F and B; each layer of the medium (from left to right) is numbered from 1 to 25 and the background material is air, numbered 0 and 26. The refractive index is set to be n=1.46 for medium a (SiO2), n=2.82 for medium b (Si). Media a and b are alternately superimposed on each layer of the PhC and the thicknesses are ${d_{a1}} = 822\textrm{ nm}$, ${d_{b1}} = 690\textrm{ nm}$, ${d_{a2}} = 1300\textrm{ nm}$, ${d_{b2}} = 690\textrm{ nm}$, ${d_{a3}} = 822\textrm{ nm}$, and ${d_{b3}} = 690\textrm{ nm}$. The defect layer (D) is composed of medium a (SiO2) with a thickness of ${d_D} = 1070\textrm{ nm}$. The TES mode is excited in 1D-PhC heterostructure composed of PhC1, graphene (L) and PhC2. The F-P cavity consists of PhC2, defect layer, graphene (R) and PhC3. At present, micro/nano fabrication and measurement technology for preparing multi-layer dielectric structure is relatively mature. In addition, graphene can also be prepared by epitaxial growth, chemical vapor deposition and other methods. However, the structure fabrication in micro/nano size also needs to be seriously considered. Therefore, for the sake of simplicity, we do not consider the experimental verification in this paper, and the relevant results are obtained by theoretical calculation and numerical simulation. It is known that the graphene is a proven 2D material and has only one atomic layer. Therefore, we can use electrical conductivity to characterize the optical characteristics of graphene. Without considering the external magnetic field and under the random-phase approximation, the isotropic surface conductivity of graphene ${\sigma _0}$ is the sum of the interband ${\sigma _{inter}}$ and the intraband ${\sigma _{intra}}$, where [34,35]:
Fig. 1. Schematic diagram of the proposed Fano resonance heterostructure. Graphene (L) is placed between PhC1 and PhC2 to form the Topological Photonic Crystal; Graphene (R) is located in the defect layer between PhC2 and PhC3 to form the Fabry-Perot Cavity. A plane wave of amplitude ${E_i}$ is incident on the structure with incident angle $\theta$.
In this paper, the relationship among incidence, transmission and reflection waves is evaluated. With the electromagnetic field propagating along the z-axis that we assumed, graphene is parallel to the plane of the x and y axes. The incident electromagnetic field (${E_i},\textrm{ }{H_i}$) in TE polarization can be expressed as:
3. Results and discussions
We first discuss the relationship between the transmittance and wavelength of the left PhC heterostructure formed by PhC1 and PhC2, as shown in Fig. 2(a). It is found that PhC1 and PhC2 show obvious photonic band gap at near 1550 nm wavelength (near 193.5 THz in frequency). The wavelength range of the total reflection coincides with the band diagrams in Figs. 2(c) and 2(d) at 193.5 THz, where there are four bands in each figure. It is worth noting that the band gaps near 193.5 THz are wide and exhibit topological feature. Therefore, the TES mode is generated at the interface of PhC1 and PhC2 by calculating the sum of ZaK phases near 193.5 THz [30]. The solid line in Fig. 2(a) can be well explained by the characteristics of TES mode. The abnormal transmission phenomenon appears when PhC1 and PhC2 compose a heterostructure. A sharp transmission peak with 99.9% transmittance at the wavelength of 1554.32 nm is found on the transmission spectrum and a full width at half maxima Δλ1 of 2.04 nm. The TES mode in Fig. 2(b) is aroused at the interface of the PhC heterostructure, thus generating a local field enhancement [37]. The characteristics of abnormal transmission and local field enhancement of TES mode provide a good condition for enhancing the third-order nonlinear effect of graphene to generate OB.
Fig. 2. (a) The transmittance spectra of individual PhC1 (blue short-dashed line), PhC2 (red short-dot line) and the “PhC1+PhC2” heterostructure (black solid line); (b) the calculated electric field of the “PhC1+PhC2” heterostructure; the band structure of (c) PhC1 and (d) PhC2; the red band indicates a bandgap with the positive topological phase, and the blue indicates that with a negative topological phase.
The PhC heterostructure on the right side is taken as a whole to study the relationship between the transmittance and wavelength. The transmission spectrum of F-P cavity mode exhibits at Fig. 3(a) and the electric-field distribution of this structure exhibits at Fig. 3(b). It is found that transmission spectrum at Fig. 3(a) has a center wavelength 1556.64 nm and full width at half maxima Δλ2 of 2.952 nm. Besides, it is seen that the transmission peak is observed near the wavelength of 1550 nm in the shape of classic symmetric Lorentz lines on the transmission curve of the whole heterostructure as shown in Fig. 3(a), and the electric field shows an obvious local enhancement in the defect layer of the structure as shown in Fig. 3(b). It can be considered that the light of the stimulated radiation continues to oscillate in the defect layer of the structure, resulting in the increase of the optical energy density and the electric field strength. Consequently, this photonic crystal heterostructure is treated as an F-P cavity and the defect layer as a resonant cavity. At the same time, We can find that the line width of the mode generated by the PhC heterostructure is narrower than that in the F-P cavity. If we combine the PhC heterostructure and the F-P cavity together as shown in Fig. 1, their corresponding modes will couple with each other and the symmetry of the transmission spectrum will be broken.
Fig. 3. (a) The transmittance spectra and (b) the calculated electric field of F-P cavity in PhC heterostructure with a defect layer
Next, we calculate the transmittance and electric-field distribution in the whole PhC heterostructure. As shown in Fig. 4(a), an obvious transmission peak exists in a narrow frequency range around 1550 nm. On the interface of PhC1 and PhC2 heterostructure there is a topological edge mode. In the meantime, a Lorentz lines is generated in F-P cavity as shown in Fig. 3. The emergence of asymmetric resonance can be explained by the theory of approximate perturbation-a special state produced by the coupling of continuum states and discrete bound states [37]. The width of transmission line in this mode produced by TES PhC heterostructure (Δλ1=2.04 nm) is narrower than that in F-P cavity (Δλ2=2.952 nm). Therefore, it is reasonable to presume that the TES heterostructure generates clearly separate bound states and F-P cavity produces states of continua. Modes generated under TES and F-P cavity as shown in Fig. 1 can be coupled together to break the symmetry of the transmission spectrum due to the interruption of Fano resonance by the above two states [38]. The transmission of Fano resonance changes dramatically in a very narrow wavelength range under the destructive interference between the two states. Figure 4(b) shows the distribution of electric field in the whole Fano resonance heterostructure. Local field enhancement is easily observed at the interface of the 1D-PhC, being more concentrated and intense on the left side in terms of both range and strength.
Fig. 4. (a) The transmittance spectra and (b) the calculated electric field of the whole Fano resonance heterostructure
Taking into account the characteristics of Fano resonance, two monolayers of graphene are added to the field enhancement position, so that the low-threshold OB can better obtained. The role of Fano resonance is to create conditions for realizing low threshold OB in optical communication band. By adjusting the appropriate structural parameters, we can make the Fano resonance appear near 1550 nm. When the incident wavelength changes slowly around 1550nm, the electric field near the left and right graphene will be suppressed or enhanced. However, near a certain incident wavelength, the electric field is enhanced both in left and right graphene, coupling between two modes in this region is constructive, and the transmission appears as a sharp peak. This phenomenon plays a positive role in reducing the threshold of OB. As we know that the third-order nonlinear conductivity of graphene has a similar monotonically decreasing relationship with the incident light frequency, the third-order nonlinear conductivity of graphene is lower at high frequency. It seen from Eq. (3) that the third-order nonlinear conductivity is relatively weak due to the increase in frequency at the optical communication band of 1550 nm. According to the graphene conductivity equation $\sigma = {\sigma _0} + {\sigma _3}{|E |^2}$, the increase of the local electric field E caused by the Fano resonance at the position of the graphene has a positive effect on nonlinear characteristic of the graphene. Thus, an OB phenomenon with relatively low threshold is achieved in high-frequency communication band. In other words, the excitation of the Fano resonance in 1D-PhC heterostructure has played an active role in promoting the low-threshold OB phenomenon in the 1550 nm optical communication band.
Next, we discuss the regulation of graphene and structural parameters on OB. Firstly, we focus on the influence of the Fermi energy of graphene on OB, as shown in Fig. 5. It is found that there is an obvious OB phenomenon in the 1D-PhC Fano resonance heterostructure. As the Fermi energy increases, the upper and lower thresholds (i.e. ${|{{E_i}} |_{up}}$and ${|{{E_i}} |_{down}}$) rise, and so does the threshold width. For the convenience of analysis, we can assume that PhC1, PhC2 and PhC3 as three Bragg mirrors [20] and combine Eq. (1) with Eq. (2) to express the scattering matrix of the entire structure:
Fig. 5. Effects of incident electric field on transmitted electric field under different Fermi energy of (a) graphene (L) and (b) graphene (R). Here, $\lambda = 1550\;\textrm{nm},\textrm{N} = 1,\textrm{ }\theta = {0^o}.$
The intensity of the local field on the left side is relatively strong if the structure is taken as a whole. When comparing Fig. 5(a) with 5(b), it can be seen that the change of Fermi energy of the left graphene has a greater effect on the hysteresis curve than that of the right one. Consequently, the hysteresis width generated by regulating the left graphene would be:
The hysteresis width is $\Delta {E_i}_{\textrm{ }(gra,R)} \approx \textrm{0}\textrm{.954} \times \textrm{1}{\textrm{0}^\textrm{7}}\textrm{ V/m}$ when the Fermi energy is 0.40 eV, and $\Delta {E_i}_{\textrm{ }(gra,R)} \approx 1.208 \times \textrm{1}{\textrm{0}^\textrm{7}}\textrm{ V/m}$ when the Fermi energy is 0.6 eV. The hysteresis width increases by $0.954 \times \textrm{1}{\textrm{0}^\textrm{7}}\textrm{ V/m}$ with the increase of Fermi energy by 0.2 eV. According to $\sigma = {\sigma _0} + {\sigma _3}{|{{E_{(Graphene)}}} |^2}$, due to the obvious differences in the electric field enhancement between the sides of graphene (L) and graphene (R), there will be differences in the regulation of the graphene on the left and right sides, we will consider and deal with the graphene separately in the actual calculation process.
For graphene with fewer layers (n < 6), the conductivity can be approximately expressed as $\sigma \approx N{\sigma _0}$ in the case of non-contact monolayered graphene [39,40]. Therefore, it is easy to consider that OB is affected not only by Fermi energy, but also by the number of graphene layers. Next, the influence of the number of graphene layers on the OB phenomenon in detail will be discussed, as shown in Fig. 6. Increasing the number of graphene layers is equivalent to the linear increase of surface conductivity, thus increasing the real part of the reflection coefficient. And so that the optical bistable device with Fano resonance 1D-PhC heterostructure can be controlled by altering the number of graphene layers. Likewise, the difference between the number of graphene layers on the left and right side would result in different regulating effects on the hysteresis loop of OB due to different increments of local electric field.
Fig. 6. Effects of incident electric field on transmitted electric field under different number of graphene layer of (a) graphene (L) and (b) graphene (R). Here, $\lambda = 1550\;\textrm{nm},{ }{{\textrm E}_{\textrm F}}_{({{\textrm{Graphene}},{ }{\textrm L}} )} = 0.2\textrm{eV},\textrm{ }{{\textrm E}_{\textrm F}}_{({{\textrm {Graphene}},\textrm{ R}} )} = 0.4\;\textrm{eV},\textrm{ }\theta = {0^o}.$
In this section, we focused on the influence of the OB curve with the angle of incident light. The Fermi energy of the graphene is 0.2 eV on the left side and 0.4 eV on the right for simulation, other parameters are entirely consistent with those in Fig. 5. The effects of incident light intensity on the transmitted electric field and the transmittance in the left graphene are illustrated in Fig. 7 based on different incident angles. Similar to the effects of Fermi energy on hysteresis curves, the incident angle exerts its influence on the hysteresis loop by altering the threshold and hysteresis width. As shown in Fig. 7(a), the thresholds of OB and transmittance increase with the incident angle. The variations of transmittance and transmitted electric field, as shown in Fig. 7(b), are completely different from each other. The OB threshold width increases with the incidence angle ($\theta$), and the hysteresis width has a significant broadening effect also with the increase of incidence angle. For example, the OB threshold widths of the transmitted electric field are $1.208 \times \textrm{1}{\textrm{0}^\textrm{7}}\textrm{ V/m}$ at $\theta = {0^o}$and $19.28 \times \textrm{1}{\textrm{0}^\textrm{7}}\textrm{ V/m}$ at $\theta = {5^o}$. The pink dotted line in Fig. 7 shows the calculated results of COMSOL Multiphysics. The curve plotted at $\theta = {0^o}$ is taken as reference; the finite element method (FEM) is applied to check the linear and nonlinear response of the proposed structure; the rigorous coupled-wave theory is adopted to verify the accuracy of the numerical results. According to the FEM, the setting of structural parameters is consistent with the mathematical calculation. The figures have shown that the calculated results of COMSOL Multiphysics are in accord with the mathematical calculations, thus proving the accuracy of the mathematical calculations. It is obvious that the hysteresis width of OB on the left graphene changes significantly with the incident angle.
Fig. 7. Effects of incident electric field on (a) the transmitted electric field and (b) the transmittance according to the different angle of incident light of the left side graphene. The pink dotted line is the calculated results of COMSOL Multiphysics. Here, $\lambda = 1550\;\textrm{nm},{ }{{\textrm E}_{\textrm F}}_{({{\textrm {Graphene}},\textrm{L}} )} = 0.2\;\textrm{eV},\textrm{N} = 1.$
Finally, the effects of the incident electric field on the reflectance and reflected electric field according to the variation of Fermi energy in the left graphene are illustrated, as shown in Fig. 8. It can be seen that there is a continuous increase in the threshold of reflectance and reflected electric field and the width of hysteresis loop with the rise of Fermi energy, indicating a close dependence of the reflective OB on Fermi energy. Specifically, the pink dashed line in Fig. 8 reflects the accuracy of the resultant reflective OB by using the COMSOL multiphysics simulation software. The results of simulation are highly consistent with the numerical calculations. In general, the dynamic and adjustable nonlinear conductivity of graphene is the key to the generation of low-threshold reflective OB.
Fig. 8. Effects of incident electric field on (a) the reflected electric field and (b) the reflectance under different Fermi energy of Fermi energy in the graphene (L). Here, $\lambda = 1550\;\textrm{nm},{ }{{\textrm E}_{\textrm F}}_{({{\textrm {Graphene}},\textrm{L}} )} = 0.2\;\textrm{eV},{ }{{\textrm E}_{\textrm F}}_{({{\textrm {Graphene}},\textrm{R}} )} = 0.4\;\textrm{eV},\textrm{N} = 1,\textrm{ }\theta = {0^o}.$
4. Conclusion
In this study, the graphene-based low-threshold and tunable OB is analyzed in a 1D-PhC heterostructure on the basis of Fano resonance generation at optical communication band (near 1550 nm in wavelength). The simulation and calculation results show that the Fano resonance produced by the coupling of TES mode and F-P cavity mode plays a positive role in reducing the OB threshold. In the proposed structure, the excitation of Fano resonance can greatly increase the intensity of the local electric field and creating conditions for achieving a low-threshold OB through the enhancement of the nonlinear conductivity of the graphene. Meanwhile, the calculated results indicate that the conductivity of graphene could flexibly regulate the upper and lower thresholds and the hysteresis widths of OB, which are also closely related to the incident angle as well as the number of graphene layers. In addition, numerical simulations are conducted to verify the correctness of the mathematical calculations. This graphene-based tunable OB in one-dimensional Fano resonance photonic crystal heterostructure provides a feasible way to realize the integrated micro/nano-sized all-optical bistable device at optical communication band.
Funding
National Natural Science Foundation of China (11704119, 11774088, 11874269, 61875133, 11704259); Natural Science Foundation of Hunan Province (2018JJ3325, 2018JJ3557); Natural Science Foundation of Guangdong Province (2018A030313198); Science and Technology Planning Project of Shenzhen Municipality (JCYJ20180305125036005).
Disclosures
The authors declare no conflicts of interest.
References
1. H. M. Gibbs, Optical Bistability: Controlling Light with Light (Academic, 1985).
2. M. D. Tocci, M. J. Bloemer, M. Scalora, J. P. Dowling, and C. M. Bowden, “Thin-film nonlinear optical diode,” Appl. Phy. Lett. 66(18), 2324–2326 (1995). [CrossRef]
3. K. Nozaki, A. Lacraz, A. Shinya, S. Matsuo, T. Sato, K. Takeda, and M. Notomi, “All-optical switching for 10-Gb/s packet data by using an ultralow-power optical bistability of photonic-crystal nanocavities,” Opt. Express 23(23), 30379–30392 (2015). [CrossRef]
4. M. Notomi, A. Shinya, S. Mitsugi, G. Kira, and T. Tanabe, “Optical bistable switching action of si high-q photonic-crystal nanocavities,” Opt. Express 13(7), 2678–2687 (2005). [CrossRef]
5. T. Tanabe, M. Notomi, S. Mitsugi, A. Shinya, and E. Kuramochi, “Fast bistable all-optical switch and memory on a silicon photonic crystal on-chip,” Opt. Lett. 30(19), 2575–2577 (2005). [CrossRef]
6. V. R. Almeida and M. Lipson, “Optical bistability on a silicon chip,” Opt. Lett. 29(20), 2387–2389 (2004). [CrossRef]
7. P. Y. Wen, M. Sanchez, M. Gross, and S. Esener, “Vertical-cavity optical AND gate,” Opt. Commun. 219(1-6), 383–387 (2003). [CrossRef]
8. C. Argyropoulos, C. Ciracì, and D. R. Smith, “Enhanced optical bistability with film-coupled plasmonicnanocubes,” Appl. Phys. Lett. 104(6), 063108 (2014). [CrossRef]
9. Z. Huang, A. Baron, S. Larouche, C. Argyropoulos, and D. R. Smith, “Optical bistability with film-coupled metasurfaces,” Opt. Lett. 40(23), 5638–5641 (2015). [CrossRef]
10. S. Noda, M. Fujita, and T. Asano, “Spontaneous-emission control by photonic crystals and nanocavities,” Nat. Photonics 1(8), 449–458 (2007). [CrossRef]
11. A. G. Ardakani and F. B. Firoozi, “Highly tunable bistability using an external magnetic field in photonic crystals containing graphene and magnetooptical layers,” J. Appl. Phys. 121(2), 023105 (2017). [CrossRef]
12. H. Yuan, X. Jiang, F. Huang, and X. Sun, “Ultralow threshold optical bistability in metal/randomly layered media structure,” Opt. Lett. 41(4), 661–664 (2016). [CrossRef]
13. M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nat. Photonics 6(11), 737–748 (2012). [CrossRef]
14. H. Zhang, S. Virally, Q. Bao, L. K. Ping, S. Massar, N. Godbout, and P. Kockaert, “Z-scan measurement of the nonlinear refractive index of graphene,” Opt. Lett. 37(11), 1856–1858 (2012). [CrossRef]
15. J. H. Chen, C. Jang, S. Xiao, M. Ishigami, and M. S. Fuhrer, “Intrinsic and extrinsic performance limits of graphene devices on SiO 2,” Nat. Nanotechnol. 3(4), 206–209 (2008). [CrossRef]
16. J. J. Guo, J. Jiang, Z. M. Zheng, and B. C. Yang, “Enhanced performance of multilayer MoS2 transistor employing a polymer capping layer,” Org. Electron. 40, 75–78 (2017). [CrossRef]
17. F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4(9), 611–622 (2010). [CrossRef]
18. Z. Q. Li, E. A. Henriksen, Z. Jiang, Z. Hao, M. C. Martin, P. Kim, and D. N. Basov, “Dirac charge dynamics in graphene by infrared spectroscopy,” Nat. Phys. 4(7), 532–535 (2008). [CrossRef]
19. D. Zhao, Z. Q. Wang, H. Long, K. Wang, B. Wang, and P. X. Lu, “Optical bistability in defective photonic multilayers doped by graphene,” Opt. Quantum Electron. 49(4), 163 (2017). [CrossRef]
20. X. T. Gan, Y. D. Wang, F. L. Zhang, C. Y. Zhao, B. Q. Jiang, L. Fang, D. Y. Li, H. Wu, Z. Y. Ren, and J. L. Zhao, “Graphene-controlled fiber Bragg grating and enabled optical bistability,” Opt. Lett. 41(3), 603–606 (2016). [CrossRef]
21. Y. Huang, A. E. Miroshnichenko, and L. Gao, “Low-threshold optical bistability of graphene-wrapped dielectric composite,” Sci. Rep. 6(1), 23354 (2016). [CrossRef]
22. H. Wang, J. Wu, J. Guo, L. Jiang, Y. Xiang, and S. Wen, “Low-threshold optical bistability with multilayer graphene-covering Otto configuration,” J. Phys. D: Appl. Phys. 49(25), 255306 (2016). [CrossRef]
23. X. Y. Dai, L. Y. Jiang, and Y. J. Xiang, “Low threshold optical bistability at terahertz frequencies with graphene surface plasmons,” Sci. Rep. 5(1), 12271 (2015). [CrossRef]
24. J. Guo, L. Y. Jiang, Y. Jia, X. Y. Dai, Y. J. Xiang, and D. Y. Fan, “Low threshold optical bistability in one-dimensional gratings based on grapheneplasmonics,” Opt. Express 25(6), 5972–5981 (2017). [CrossRef]
25. L. Y. Jiang, J. Tang, J. Xu, Z. W. Zheng, J. Dong, J. Guo, S. Y. Qian, X. Y. Dai, and Y. J. Xiang, “Graphene Tamm plasmon-induced low-threshold optical bistability at terahertz frequencies,” Opt. Mater. Express 9(1), 139–150 (2019). [CrossRef]
26. C. Li, X. Hu, H. Yang, and Q. Gong, “Unidirectional transmission in 1D nonlinear photonic crystal based on topological phase reversal by optical nonlinearity,” AIP Adv. 7(2), 025203 (2017). [CrossRef]
27. W. L. Gao, M. Lawrence, B. Yang, F. Liu, F. Z. Fang, B. Béri, J. Li, and S. Zhang, “Topological photonic phase in chiral hyperbolic metamaterials,” Phys. Rev. Lett. 114(3), 037402 (2015). [CrossRef]
28. W. J. Chen, S. J. Jiang, X. D. Chen, B. Zhu, L. Zhou, J. W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5(1), 5782 (2014). [CrossRef]
29. M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photonics 7(12), 1001–1005 (2013). [CrossRef]
30. 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]
31. A. V. Poshakinskiy, A. N. Poddubny, L. Pilozzi, and E. L. Ivchenko, “Radiative topological states in resonant photonic crystals,” Phys. Rev. Lett. 112(10), 107403 (2014). [CrossRef]
32. C. W. Ling, M. Xiao, C. T. Chan, S. F. Yu, and K. H. Fung, “Topological edge plasmon modes between diatomic chains of plasmonic nanoparticles,” Opt. Express 23(3), 2021–2031 (2015). [CrossRef]
33. W. Gao, X. Y. Hu, C. Li, J. H. Yang, Z. Chai, J. Y. Xie, and Q. H. Gong, “Fano-resonance in one-dimensional topological photonic crystal heterostructure,” Opt. Express 26(7), 8634–8644 (2018). [CrossRef]
34. H. Cheng, S. Chen, P. Yu, J. X. Li, B. Y. Xie, Z. C. Li, and J. G. Tian, “Dynamically tunable broadband mid-infrared cross polarization converter based on graphene metamaterial,” Appl. Phys. Lett. 103(22), 223102 (2013). [CrossRef]
35. Y. J. Xiang, X. Dai, J. Guo, S. C. Wen, and D. Tang, “Tunable optical bistability at the graphene-covered nonlinear interface,” Appl. Phys. Lett. 104(5), 051108 (2014). [CrossRef]
36. S. A. Mikhailov and K. Ziegler, “Nonlinear electromagnetic response of graphene: frequency multiplication and the self-consistent-field effects,” J. Phys.: Condens. Matter 20(38), 384204 (2008). [CrossRef]
37. R. R. D. Meade and J. N. Winn, Photonic Crystals: Molding the flow of light (Princeton University Press, 1995).
38. B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9(9), 707–715 (2010). [CrossRef]
39. C. Casiraghi, A. Hartschuh, E. Lidorikis, H. Qian, H. Harutyunyan, and T. Gokus, “Rayleigh Imaging of Graphene and Graphene Layers,” Nano Lett. 7(9), 2711–2717 (2007). [CrossRef]
40. H. G. Yan, X. S. Li, B. Chandra, G. S. Tulevski, Y. Q. Wu, M. Freitag, and W. J. Zhu, “Tunable infrared plasmonic devices using graphene/insulator stacks,” Nat. Nanotechnol. 7(5), 330–334 (2012). [CrossRef]
