Open Access
Add to BibSonomy
Abstract
A mono-layer metamaterial comprising four graphene-strips and one graphene-square-ring is proposed herein to realize triple plasmon-induced transparency (PIT). Theoretical results based on the coupled mode theory (CMT) are in agreement with the simulation results obtained using the finite-difference time-domain (FDTD). An optical switch is investigated based on the characteristics of graphene dynamic modulation, with modulation degrees of the amplitude of 90.1%, 80.1%, 94.5%, and 84.7% corresponding to 1.905 THz, 2.455 THz, 3.131 THz, and 4.923 THz, respectively. Moreover, the proposed metamaterial is insensitive to the change in the angle of polarized light, for which the triple-PIT is equivalent in the cases of both x- and y-polarized light. The optical switch based on the proposed structure is effective not only for the linearly polarized light in different directions but also for left circularly polarized and right circularly polarized light. As such, this work provides insight into the design of optoelectronic devices based on the polarization characteristics of the incident light field on the optical switch and PIT.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Surface plasmon polaritons (SPPs) [1], which can be regarded as the collective resonance of material surface electrons when they interact with electromagnetic fields, have been discovered and investigated on the surface of metals or graphene [2–6]. Plasmon-induced transparency (PIT) due to SPPs has also generated significant research interest [7–9]. The induced transparency can be realized on materials with metal or graphene structures. Different from Electromagnetically induced transparency (EIT) generated by atomic coherence, experimental conditions required by the PIT are relatively low [10]. In addition, graphene metamaterials have the advantage of dynamic modulation when compared to the static regulation of metal structures [11,12]. Therefore, most studies concerning single-PIT have been conducted on graphene metamaterials using a single source of linearly polarized incident light. And given the fact that multi-band EIT has been reported by researchers [13–17], thus, the multi-band PIT is also important.
Metamaterials based on graphene possess excellent property in the terms of not only stimulating dynamically tunable SPPs but also causing low loss [18–20]. Therefore, PIT, particularly that produced by graphene, has noteworthy research value and application prospects. PIT-based devices have been designed and proposed, such as optical switches, modulators, absorbers, and slow light devices [21–25]. Among the applications explored in numerous reports in literature, optical switches based on the dynamic modulation of graphene are the most common and classic applications. However, these devices are affected and limited by the polarization angle of the incident light because the PIT is based on SPPs [26,27]. Most studies have shown that SPPs can be excited by linearly polarized light without the introduction of other complex influences [28–30]. Based on the stimulated regular pattern of SPPs, when the polarization direction of the incident light is perpendicular to the long side of the rectangular hole, the SPPs can be effectively excited. This implies that the stimulation of SPPs is sensitive to the polarization angle of the incident light. However, in practical applications, the light source does not emit only linearly polarized light [31]. Irrespective of whether we consider single- and multi-frequency or synchronous and asynchronous optical switches, the effect of the PIT is noteworthy for linearly polarized light in one direction, but not for polarized light in other directions [32,33]. This limits the development of optical switches based on PIT. Thus, it is important to determine the reason behind the graphene structure being affected by linearly polarized light in a single direction, and to realize an optical switch that is insensitive to the polarization angle of incident light.
In this study, a metamaterial composed of four graphene-strips and one graphene-square-ring is proposed to realize tunable triple-PIT. The simulation results based on the finite-difference time-domain (FDTD) [34,35] are in agreement with the theoretical results using coupled mode theory [36]. It should be noted that this structure is insensitive to the angle of polarized light, in which case the triple-PIT is invariant for both x- and y-polarized light. Thus, the optical switch based on the proposed structure is effective not only for linearly polarized light with different directions, but also to left circularly polarized (LCP) and right circularly polarized (RCP) light. This work represents the first investigation of the design of optoelectronic devices based on the polarization characteristics of the incident light field on the optical switch and triple-PIT.
2. Structure and method
Figure 1(a) shows the periodic and monolayer structure of the graphene metamaterial consisting of a series of graphene arrays. The substrates of the upper and lower layers are dielectric silicon with a thickness of d = 0.2 µm, shown in Fig. 1(b). In addition, the rectangular sheet highlighted in blue is the electrode used to modulate the Fermi levels of graphene via tuning of the voltage. The design is provided with the biasing source and lines which are necessary for tuning the Fermi level of the graphene layer. In practice, the additional biasing line would also introduce some side effects to the original design. However, the dynamic modulation of Fermi level is based on simulation calculation not biasing source in our design. Thus, the side effects by biasing line in practice is ignored by idealization. The top view of the structural unit is shown in Fig. 1(c), which is joined by four graphene-strips and a graphene-square-ring. The parameters are as follows: l = 6 µm, l1 = l3 = 2.8 µm, l2 = l4 = 0.6 µm, l5 = 3.8 µm, l6 = 2.6 µm, s1 = s2 = 0.3 µm. The schematic of the preparation process in Fig. 1(d) can provide insight to clarify the proposed structure. When a polarized light beam in the x-direction is incident perpendicular to the structure, the different graphene structures that exist can produce a different transmission, as shown in Fig. 2(a). Here, the nonlinear effect of graphene is ignored owing to the weak intensity of the incident plane wave.
Fig. 1. (a) Schematic of the proposed periodic metamaterial in 3D. (b) A 3D structural unit of the periodic metamaterial. (c) 2D top view of the structural unit. (The graphene mobility is: µ=1.5m2/Vs) (d) Schematic of the process of preparation of the proposed structure metamaterials.
Fig. 2. (a) Transmission spectra of the metamaterial in the case that a polarized light beam in the x-direction is incident. (b-d) Electric field distribution of the separate structures at dip1, dip2, and dip3. (e-g) Electric field distribution of the entire structure at dip1, dip2, and dip3.
In the dark mode, when the vertical double graphene strips exist alone, they cannot directly interact with the incident light in the case of x-polarization. Thus, the curve of the transmission spectra is unchanged over the entire frequency band, as shown by the black curve in Fig. 2(a). In contrast, the horizontal double graphene strips can interact directly with the incident light and generate a green Lorenz line, which is known as the bright mode. Based on the electric field distribution in Fig. 2(c), it is evident that dip2 of the green Lorenz line originates from the energy localization of the corresponding structure. Similarly, two Lorenz lines are generated by the graphene-square-ring owing to the energy localization of different parts within the same structure, as shown in Figs. 2(b) and 2(d). According to CMT, the bright mode is corresponding to the resonant dip of Lorenz line. Thus, the graphene-square-ring can be seen as two bright modes. If the dark mode is nonexistent, the energy localization of the three bright modes exist and are mutually independent. When the entire structure is illuminated using the x-polarized light, three transparent windows appear at the frequency points dip1, dip2, and dip3, which are originally opaque, and the energy localization becomes weaker, as shown in Figs. 2(e)–2(g). Moreover, the dark mode is excited by the light field generated based on the interaction between the bright mode and the incident light, and the phenomenon of triple-PIT is observed. Thus, it is important to emphasize the role of the dark mode and the appearance of a transparent window at the resonant point of the bright mode for the evaluation of multi-PIT. In addition, the resonances excited by the graphene structure are complex, which means other resonant dips can appear in different frequencies (above 5 THz). However, these resonant modes are ignored because they are not satisfied with the fact that induced transparency window appears at the resonant frequency of the bright mode.
CMT [37–39] is employed to explain the triple-PIT, the schematic of which is plotted in Fig. 3(a), where each structure corresponds to a mode. The graphene-square-ring corresponds to resonators A and C, and it can be seen that the different parts of the graphene-square-ring are excited at different resonant frequencies f1 and f3. The electric field distribution around the horizontal double graphene strips is strong at f2. This implies that the horizontal double graphene strips correspond to resonator B. The vertical double graphene strips and the graphene-square-ring both illuminate at f4, which implies that the vertical double graphene strips are associated with the dark mode corresponding to resonator D. Their relationships can be expressed as follows [40]:
According to the principle of energy conservation, the transmission of energy can be expressed as:
Based on this, the transmission coefficients can be expressed as:
Thus, the transmission can be obtained as T=│t│2.
3. Results and discussion
The Fermi levels of graphene can be modulated as follows:[41]
where, Vg is the gate voltage. The transmission spectra of the simulation based on the finite-difference time-domain (FDTD) are illustrated in Fig. 4(a), where a blue shift of the blue curve is observed with the increase in the Fermi level of graphene from 0.8 eV to 1.1 eV in steps of 0.1 eV. The green dotted curves of the theoretical result obtained using CMT are in agreement with the blue curves. To observe the blue shift, the evolutionary processes are plotted for the increase of the Fermi levels in Fig. 4(b). As shown in Fig. 4(b), the opaque bands in blue, which consist of four resonant dips that change continuously, are oblique. This indicates that some dips can be coincident with the transparent windows at certain frequencies by means of tuning the Fermi levels of graphene, where an optical switch is realized. To highlight the effect of the optical switch, the contrast diagrams before and after the Fermi levels are changed are plotted in Figs. 4(c) and 4(d), respectively. In this case, the lower limit of the Fermi levels is set to 0.6 eV, and the Fermi levels of Ef = 1.0 eV are defined as the benchmark before modulation. As shown in Fig. 4(c), before modulation, the optical switch is set to “off” at 1.905 THz, which is the “on” state at 2.455 THz. The state change to “on” and “off” after modulation, respectively. Therefore, the effects of the switch are asynchronous. In addition, a synchronous optical switch is realized as shown in Fig. 4(d), and the states of the switch are both “off” before modulation and “on” after modulation. The modulation degree of the amplitude (MDA), which is used to describe the effect of the switch, can be expressed as follows: where, Aon and Aoff are the transmissions of the “on” and “off” states. Thus, the MDA values are 90.1%, 80.1%, 94.5%, and 84.7% corresponding to 1.905 THz, 2.455 THz, 3.131 THz, and 4.923 THz, respectively. The respective insertion losses of 14.6%, 11.2%,16.4%, and 7.4% are thus not negligible. Here it is defined as impermeable rate of light when the switch is on state, which is different from the insertion losses in ‘dB’. Additionally, the dephasing time is important with respect to the modulator for defining the performance of the switch, which is obtained by using T = 2ħ/FWHM [42–44]. Thus, the dephasing times are 5.5 ps, 4.8 ps, 4.2 ps, and 7.2 ps, respectively. As such, the optical switch is realized in the case of the x-polarized light. In addition, there will be an extra peak with increasing the THz bandwidth (above 5 THz) as shown in Figs. 4(c) and 4(d), which is not induced transparency based on the bright and dark mode theory. Thus, we do not furtherly analyze this complex situation in this paper.Fig. 4. (a) Transmission spectra obtained by FDTD simulation and CMT calculation at different Fermi levels. (b) Three-dimensional evolution of the triple-PIT at different Fermi levels. (c-d) Spectra contrast between the different Fermi levels.
In determining the response of the different graphene structures to the angle of polarized light, we note that the structures produce different transmission spectra for mutually perpendicular polarized light, as in Figs. 5(a)–5(c). The blue solid curves represent the condition in which the structures are illuminated using x-polarized light, and the red dotted curves represent y-polarized illumination. As previously discussed, when the x-polarized light is incident on the structure, the vertical double graphene strips are associated with the dark mode, and the horizontal double graphene strips with the role of the bright mode. However, the situation is reversed with a change in the polarized angle [45]. In the case of y-polarization, the dark mode corresponds the horizontal double graphene strips, as represented by the red dotted curves in Fig. 5(b). The process of mode conversion for a change of the angle of polarization is illustrated in Figs. 5(d)–5(e). The original unchanged curve generated by the dark mode gradually changes into the Lorentz line, which can be seen from the decreasing transmission at 3.743 THz in Fig. 5(d). Moreover, the minimum value (dip of Lorenz line in Fig. 5(b)) of the transmission spectra produced by the bright mode increases and disappears gradually with the change in the angle of polarization from x to y in Fig. 5(e). Thus, as the angle of polarization changes, the bright and dark modes are not significantly affected owing to mode synchronization-conversion. Therefore, this kind of synchronization-conversion contributes to the unchanged triple-PIT. However, the transmission spectra of the graphene-square-ring do not change due to centrosymmetry of the structure. As observed in Fig. 5(f), the two minimum values of the transmission spectra remain unchanged. As such, the spectral evolution of the triple-PIT generated by the entire structure is plotted in Fig. 5(g) with the change in the angle of polarization. The opaque bands in blue are vertical, and the widths are unchanged. This implies that the triple-PIT is not affected by the angle of polarization of the linearly polarized incident light. Thus, the triple-PIT of the x-polarization is equivalent to that of the y-polarization. As such, the effect of the optical switch based on the triple-PIT is desensitized to the polarization direction of the light source. In addition to optical switches, the properties of synchronization-conversion and centrosymmetry are highly suitable for the design of absorbers or other SPP-based devices, for which polarization-desensitization is required.
Fig. 5. (a-c) Transmission spectra for x-polarization and y-polarization. (d-f) Transmission minimum as a function of θ. (g) Three-dimensional evolution of the triple-PIT with the change of the polarization direction.
For further analysis of the more complex light sources, the transmission spectra of RCP and LCP light consisting of vertically polarized light with a phase difference of π/2 and -π/2 were used as the light source to illuminate the structure, as shown in Fig. 6(a). In this Fig, the two curves are coincident, meaning that the optical switch based on the proposed structure can be realized in the case of circularly polarized light. This result is significant for the design of an optical switch based on PIT. Indeed, the transmission spectra of the triple-PIT in Fig. 2(a) and Fig. 6(a) are coincident owing to the fact that the triple-PIT phenomenon is the same to both linearly and circularly polarized light. The triple-PIT is not changed with being shined by light of any polarized angle, which results in insensitive effect to LCP and RCP. In addition, the electric field distributions at four resonant frequencies in the case of RCP and LCP are plotted in Figs. 6(b)–6(i) to observe their effect on the structure. In this case, the red “x” represents the propagation direction of the incident light. It is evident that the excitation of the SPPs for circularly polarized light is more sufficient and well-distributed compared to that in the case of linearly polarized light. Interestingly, although the transmission spectra of RCP and LCP are similar, the electron field distributions are slightly different. Particularly for the first resonant frequency, the field of RCP is symmetrical to the field of LCP for incident light vertical to the plane of this page. This can be attributed to the interesting physical nature of RCP and LCP, which means that the first resonant dip of the triple-PIT can distinguish between these two polarization states based on the field distribution.
Fig. 6. (a) Transmission spectra for right-circularly polarized and left-circularly polarized light. (b-i) Electric field distribution at four resonant frequencies.
4. Conclusions
A metamaterial composed of four graphene-strips and one graphene-square-ring was proposed in this study to realize a triple-PIT. CMT was employed to explain the triple-PIT. The results are observed to be in good agreement with the simulation data for FDTD. The optical switch was studied by modulating the Fermi levels of graphene, for which the modulation degree of amplitude, insertion loss, and dephasing time were calculated. As the angle of the polarized light was changed, the centrosymmetric property of the graphene-square-ring resulted in an unchanged bright mode. Moreover, resonant mode synchronization-conversions among the four graphene strips caused a constant PIT. Thus, the triple-PIT was equivalent for both x-polarized and y-polarized light. This implies that the optical switch based on the proposed structure is effective not only for linearly polarized light in different directions, but also for LCP and RCP light. Overall, this work represents the first investigation of an optical switch that is insensitive to polarized light based on the triple-PIT.
Funding
Graduate Innovative Special Fund Projects of Jiangxi Province (YC2020-S312); Department of Science and Technology of Sichuan Province (2020YJ0137); National Natural Science Foundation of China (11804093, 11847026, 61764005); Natural Science Foundation of Jiangxi Province (20192BAB212003, 20202ACBL212005, 20202BABL201019).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408(3-4), 131–314 (2005). [CrossRef]
2. J. Wang, C. Song, J. Hang, Z. D. Hu, and F. Zhang, “Tunable Fano resonance based on grating-coupled and graphene-based Otto configuration,” Opt. Express 25(20), 23880–23892 (2017). [CrossRef]
3. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric Field Effect in Atomically Thin Carbon Films,” Science 306(5696), 666–669 (2004). [CrossRef]
4. T. Wang, H. Zhang, Y. Zhang, Y. Zhang, and M. Cao, “Tunable bifunctional terahertz metamaterial device based on Dirac semimetals and vanadium dioxide,” Opt. Express 28(12), 17434–17448 (2020). [CrossRef]
5. Y. Zhang, Z. Yi, X. Wang, P. Chu, W. Yao, Z. Zhou, S. Cheng, Z. Liu, P. Wu, and M. Pan, “Dual band visible metamaterial absorbers based on four identical ring patches,” Phys. E 127, 114526 (2021). [CrossRef]
6. Y. Yi, D. Xu, Z. Yi, Z. Li, X. Chen, H. Jile, J. Zhang, L. Zeng, and G. Li, “Terahertz tunable three band narrowband perfect absorber based on Dirac semimetal,” Physica E: Low-dimensional Systems and Nanostructures, 114750 (2021).
7. H. Xu, H. Li, Z. He, Z. Chen, M. Zheng, and M. Zhao, “Dual tunable plasmon-induced transparency based on silicon-air grating coupled graphene structure in terahertz metamaterial,” Opt. Express 25(17), 20780–20790 (2017). [CrossRef]
8. E. Gao, Z. Liu, H. Li, H. Xu, Z. Zhang, X. Luo, C. Xiong, C. Liu, B. Zhang, and F. Zhou, “Dynamically tunable dual plasmon-induced transparency and absorption based on a single-layer patterned graphene metamaterial,” Opt. Express 27(10), 13884–13894 (2019). [CrossRef]
9. S. X. Xia, X. Zhai, L. L. Wang, and S. C. Wen, “Plasmonically induced transparency in double-layered graphene nanoribbons,” Photonics Res. 6(7), 692–702 (2018). [CrossRef]
10. Q. Lin, X. Zhai, L. Wang, B. Wang, G. Liu, and S. Xia, “Combined theoretical analysis for plasmon-induced transparency in integrated graphene waveguides with direct and indirect couplings,” EPL 111(3), 34004 (2015). [CrossRef]
11. S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101(4), 047401 (2008). [CrossRef]
12. H. Xu, M. Zhao, M. Zheng, C. Xiong, B. Zhang, Y. Peng, and H. Li, “Dual plasmon-induced transparency and slow light effect in monolayer graphene structure with rectangular defects,” J. Phys. D: Appl. Phys. 52(2), 025104 (2019). [CrossRef]
13. R. Ning, Z. Jiao, and J. Bao, ““Multi-band and wide-band electromagnetically induced transparency in graphene metasurface of composite structure,” IET Microwaves, Antennas & Propagation 12, 380–384 (2018). [CrossRef]
14. M. H. Nam, V. T. H. Hanh, N. B. Tuong, B. S. Tung, B. X. Khuyen, V. D. Lam, L. Y. Chen, and Y. P. Lee, “Multi-Band Electromagnetically-Induced-Transparency Metamaterial Based on the Near-Field Coupling of Asymmetric Split-Ring and Cut-Wire Resonators in the GHz Regime,” Crystals 11(2), 164 (2021). [CrossRef]
15. Y. Ma, D. Li, Z. Chen, H. Qian, and R. Ning, “Multiband reciprocal polarization insensitivity electromagnetically induced transparency in metasurfaces,” J. Opt. 22(5), 055101 (2020). [CrossRef]
16. J. Ge, C. You, H. Feng, X. Li, M. Wang, L. Dong, G. Veronis, and M. Yun, “Tunable dual plasmon-induced transparency based on a monolayer graphene metamaterial and its terahertz sensing performance,” Opt. Express 28(21), 31781–31795 (2020). [CrossRef]
17. K. Zhang, C. Wang, L. Qin, R. W. Peng, D. H. Xu, X. Xiong, and M. Wang, “Dual-mode electromagnetically induced transparency and slow light in a terahertz metamaterial,” Opt. Lett. 39(12), 3539–3542 (2014). [CrossRef]
18. H. S. Skulason, H. V. Nguyen, A. Guermoune, V. Sridharan, M. Siaj, C. Caloz, and T. Szkopek, “110 GHz measurement of large-area graphene integrated in low-loss microwave structures,” Appl. Phys. Lett. 99(15), 153504 (2011). [CrossRef]
19. C. Qin, B. Wang, H. Huang, H. Long, K. Wang, and P. Lu, “Low-loss plasmonic supermodes in graphene multilayers,” Opt. Express 22(21), 25324–25332 (2014). [CrossRef]
20. Z. Wang, N. M. Han, Y. Wu, X. Liu, X. Shen, Q. Zheng, and J. K. Kim, “Ultrahigh dielectric constant and low loss of highly-aligned graphene aerogel/poly(vinyl alcohol) composites with insulating barriers,” Carbon 123, 385–394 (2017). [CrossRef]
21. X. Zhang, Z. Liu, Z. Zhang, E. Gao, F. Zhou, X. Luo, J. Wang, and Y. Wang, “Photoelectric switch and triple-mode frequency modulator based on dual-PIT in the multilayer patterned graphene metamaterial,” J. Opt. Soc. Am. A 37(6), 1002–1007 (2020). [CrossRef]
22. Z. Liu, E. Gao, Z. Zhang, H. Li, H. Xu, X. Zhang, X. Luo, and F. Zhou, “Dual-Mode On-to-Off Modulation of Plasmon-Induced Transparency and Coupling Effect in Patterned Graphene-Based Terahertz Metasurface,” Nanoscale Res Lett 15(1), 1 (2020). [CrossRef]
23. B. Zhang, H. Li, H. Xu, M. Zhao, C. Xiong, C. Liu, and K. Wu, “Absorption and slow-light analysis based on tunable plasmon-induced transparency in patterned graphene metamaterial,” Opt. Express 27(3), 3598–3608 (2019). [CrossRef]
24. Z. Liu, E. Gao, X. Zhang, H. Li, H. Xu, Z. Zhang, X. Luo, and F. Zhou, “Terahertz electro-optical multi-functional modulator and its coupling mechanisms based on upper-layer double graphene ribbons and lower-layer a graphene strip,” New J. Phys. 22(5), 053039 (2020). [CrossRef]
25. Z. Chen, H. Chen, H. Jile, D. Xu, Z. Yi, Y. Lei, X. Chen, Z. Zhou, S. Cai, and G. Li, “Mult-band multi-tunable perfect plasmon absorber based on L-shaped and double-elliptical graphene stacks,” Diamond Relat. Mater. 115, 108374 (2021). [CrossRef]
26. X. Zhang, Z. Liu, Z. Zhang, E. Gao, X. Luo, F. Zhou, H. Li, and Z. Yi, “Polarization-sensitive triple plasmon-induced transparency with synchronous and asynchronous switching based on monolayer graphene metamaterials,” Optics Express 28 (2020).
27. A. Farmani, M. Miri, and M. H. Sheikhi, “Design of a High Extinction Ratio Tunable Graphene on White Graphene Polarizer,” IEEE Photonics Technol. Lett. 30(2), 153–156 (2018). [CrossRef]
28. 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]
29. J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325(5947), 1513–1515 (2009). [CrossRef]
30. L. Ren, C. L. Pint, L. G. Booshehri, W. D. Rice, X. Wang, D. J. Hilton, K. Takeya, I. Kawayama, M. Tonouchi, and R. H. Hauge, “Carbon nanotube terahertz polarizer,” Nano Lett. 9(7), 2610–2613 (2009). [CrossRef]
31. M. Liu, Z. Tian, X. Zhang, J. Gu, C. Ouyang, J. Han, and W. Zhang, “Tailoring the plasmon-induced transparency resonances in terahertz metamaterials,” Opt. Express 25(17), 19844–19855 (2017). [CrossRef]
32. Z. Liu, E. Gao, H. Li, H. Xu, Z. Zhang, X. Zhang, X. Luo, and F. Zhou, “Investigation of plasmon-induced transparency and reflection in patterned graphene metamaterial,” J. Appl. Phys. 126(12), 123101 (2019). [CrossRef]
33. H. Xu, M. Zhao, Z. Chen, M. Zheng, C. Xiong, B. Zhang, and H. Li, “Sensing analysis based on tunable Fano resonance in terahertz graphene-layered metamaterials,” J. Appl. Phys. 123(20), 203103 (2018). [CrossRef]
34. K. Yee, “Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media,” IEEE Trans. Antennas Propag. 14(3), 302–307 (1966). [CrossRef]
35. L. Yang, J. Tian, K. Z. Rajab, and Y. Hao, “FDTD Modeling of Nonlinear Phenomena in Wave Transmission Through Graphene,” Antennas Wirel. Propag. Lett. 17(1), 126–129 (2018). [CrossRef]
36. K. Tsakmakidis, L. Shen, S. Schulz, X. Zheng, J. Upham, X. Deng, H. Altug, A. F. Vakakis, and R. Boyd, “Breaking Lorentz reciprocity to overcome the time-bandwidth limit in physics and engineering,” Science 356(6344), 1260–1264 (2017). [CrossRef]
37. A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. 9(9), 919–933 (1973). [CrossRef]
38. H. A. Haus and W. Huang, “Coupled-mode theory,” Proc. IEEE 79(10), 1505–1518 (1991). [CrossRef]
39. H. Xu, H. Li, Z. Chen, M. Zheng, M. Zhao, C. Xiong, and B. Zhang, “Novel tunable terahertz graphene metamaterial with an ultrahigh group index over a broad bandwidth,” Appl. Phys. Express 11(4), 042003 (2018). [CrossRef]
40. E. Gao, Z. Liu, H. Li, H. Xu, Z. Zhang, X. Zhang, X. Luo, and F. Zhou, “Dual plasmonically induced transparency and ultra-slow light effect in m-shaped graphene-based terahertz metasurfaces,” Appl. Phys. Express 12(12), 126001 (2019). [CrossRef]
41. Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. Castro Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487(7405), 82–85 (2012). [CrossRef]
42. I. Karakurt, D. Herman, H. Mathur, and A. J. Dahm, “Dephasing times in a nondegenerate two-dimensional electron Gas,” Phys. Rev. Lett. 85(5), 1072–1075 (2000). [CrossRef]
43. T. Klar, M. Perner, S. Grosse, G. Von Plessen, W. Spirkl, and J. Feldmann, “Surface-plasmon resonances in single metallic nanoparticles,” Phys. Rev. Lett. 80(19), 4249–4252 (1998). [CrossRef]
44. A. Ahmadivand, R. Sinha, B. Gerislioglu, M. Karabiyik, N. Pala, and M. Shur, “Transition from capacitive coupling to direct charge transfer in asymmetric terahertz plasmonic assemblies,” Opt. Lett. 41(22), 5333–5336 (2016). [CrossRef]
45. S. X. Xia, X. Zhai, L. L. Wang, and S. C. Wen, “Polarization-independent plasmonic absorption in stacked anisotropic 2D material nanostructures,” Opt. Lett. 45(1), 93–96 (2020). [CrossRef]
