Optical switches based on dielectric nanostructures are highly desired at present. To enhance the wavelength-selective light absorption, and achieve an absorption-induced switching effect, here we propose a graphene-based metamaterial absorber that consists of a dielectric grating, a graphene monolayer, and a photonic crystal. Numerical results reveal that the dual-band absorption with an ultranarrow spectrum of the system is enhanced greatly due to the critical coupling, which is enabled by the combined effects of guided mode resonances and photonic band gap. The quality factor of the absorber can achieve a high value (>500), which is basically consistent with the coupled mode theory. Slow light emerges within the absorption window. In addition, electrostatic gating of graphene in the proposed structure provides dynamic control of the absorption due to the change of the chemical potential of the graphene, resulting in an optional multichannel switching effect. Unlike other one-dimensional devices, these effects can be applied to another polarization without changing the structure parameters, and the quality factor is significantly enhanced (>1000). The tunable light absorption offered by the simple structure with an all-dielectric configuration will provide potential applications for graphene-based optoelectronic devices in the near-infrared range, such as narrowband selective filters, detectors, optical switches, modulators, slow optical devices, etc.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Graphene, an one-dimensional (1D) material has attracted considerable attention due to its extraordinary electric and optical properties in nanoelectronic devices and optoelectronic devices [1–3]. A variety of graphene-based optoelectronic devices have been proposed in the frequency spectrum range from visible (Vis) to terahertz (THz) [4–16]. In the infrared (IR) and THz ranges, graphene can support graphene plasmons [17, 18], resulting in enhanced optical absorption by strong light-graphene interaction. However, it behaves mostly like an absorptive dielectric and cannot sustain plasmonic resonances at Vis and near-infrared (NIR) regions. It has been theoretically and experimentally demonstrated that the optical absorption of monolayer graphene reaches a value of 2.3% in the Vis and NIR regions, which limits its relevant applications. Therefore, the optical absorption enhancement of monolayer graphene has attracted significant attention.
Recently, various mechanisms have been presented to enhance the absorption [19–27]. More specifically, Piper and Fan theoretically utilized the critical coupling induced by guided mode resonances (GMR) of two-dimensional (2D) silicon photonic crystal slab to achieve perfect absorption in graphene monolayer successfully . This phenomenon of critical coupling can be explained by the coupled mode theory (CMT) . Afterwards, Huang et al. designed a total absorber with graphene strips covered on a three-period-silica/silicon grating . Grande et al. proposed a 1D dielectric grating to enhance absorption in a monolayer graphene based on GMR, which results in an absorption enhancement to about 60% . Long et al. numerically studied graphene perfect absorbers with ultranarrow bandwidth [32, 33]. Bian et al. numerically demonstrated the double-mode absorption can be achieved in a double-cavity photonic crystal with solely one graphene monolayer . Guo et al. experimentally demonstrated that the absorption over 99% can be obtained in a monolayer graphene coupled with subwavelength grating on top of a back gold mirror , etc. However, considering the advantages of its prominent optical properties and stability, the graphene-based design with an all-dielectric form is still in urgent need, especially in the NIR range. In addition, achieving high absorption efficiency with dynamic controllability in the communication band is still challenging. Fortunately, the optical properties of graphene can be controlled by applying a gate voltage [36–38].
In this paper, a dual-band perfect absorption in monolayer graphene is investigated in NIR region, with the maximum absorption efficiency reaching to nearly 100%. Slow light (i.e., it refers to a very low group velocity of light. If the dispersion relation of the permittivity changes rapidly over a small range of frequencies, then the group velocity might be very low, thousands or millions of times less than the speed of light) emerges within the absorption window. In our design, the perfect absorption of graphene is enhanced greatly due to the critical coupling, which can be understood from the two aspects of CMT and impedance matching. The finite element method (FEM) is employed to calculate the optical performance of the system. Then, the electromagnetic field distributions are calculated to provide a qualitative understanding of this enhanced light absorption effect based on critical coupling. Intriguingly, the system exhibits a unique absorption-induced multichannel switching effect due to the conductivity of graphene can be easily controlled by applying a gate voltage on purpose. As the chemical potential changes, the system can support on/on state, on/off state, and off/off state. Unlike other 1D devices, these effects (e.g., dual-band high absorption, slow light, and multichannel switching effect) also exist for another polarization without changing the structure parameters. We believe that the proposed structure may hold potential in engineering many graphene-based optoelectronic devices.
2. Structure and theory
The proposed configuration is schematically depicted in Fig. 1(a). The monolayer graphene is placed between the dielectric grating and the photonic crystal (PC). A SiO2 film with thickness of hs is set on the top of the PC. The PC is composed of Si and SiO2 alternately. The number of periods for PC is N. The refractive indices of Si and SiO2 are 3.47 and 1.44, respectively . The structure parameters of the designed absorber are: P = 1.21 μm, W = 1.11 μm, hd = 0.3 μm, hs = 0.32 μm, and N = 6, respectively. The thickness of Si and SiO2 are set to ha = λc/(4nSi) = 0.12 μm and hb = λc/(4nSiO2) = 0.28 μm, where the central wavelength λc is set to 1600 nm. The structure can work in both TE (electric field is parallel to y-axis) and TM (magnetic field is parallel to y-axis) polarizations. Here, we first simulate a plane wave of TE polarization with normal incidence impinging on the structure.
The graphene can be modeled as a ultrathin dielectric layer with permittivity ∊g = 1 + iσg/(ω∊0t), where σg is graphene surface conductivity, ω is the angular frequency of incident wave, ∊0 is vacuum permittivity, and t = 0.34 nm is the monolayer graphene thickness [34, 40]. The conductivity σg can be calculated from the Kubo formula :40, 42]. The permittivities of monolayer graphene at different incident wavelengths are shown in Fig. 1(b).Eq. (2), the 100% absorptivity can be achieved at resonance frequency (ω = ω0), when the external leakage and intrinsic loss rates are the same (γ = δ). This phenomenon is known as critical coupling.
3. Results and analysis
Figure 2(a) shows the calculated absorption spectrum at normal incidence for the proposed graphene absorber. Undoubtedly, the absorber we designed can significantly enhance the absorptivity of graphene monolayer in NIR region. Two near perfect absorption peaks reaching 98.3% at λ1 = 1558.5 nm (GMR1 mode) and 99.5% at λ0 = 1658.5 nm (GMR0 mode) are observed. Meanwhile, the absorption spectra from CMT give a comparison with the FEM. One can clearly find that there is a great agreement between FEM and CMT, indicating that CMT can provide a good interpretation for the single port system. Specifically, for the case of GMR0 mode, the full width at half maximum (FWHM) is only Δλ0 = 3.2 nm, which demonstrates a remarkably narrow linewidth for spectral selective absorption. The quality factor Q of the GMR0 mode can be defined as Q0 = λ0/Δλ0, reaching about 518.3. According to CMT, the quality factors Qδ = ω0/2δ and Qγ = ω0/2γ represent the intrinsic loss and external leakage of the GMR0 mode, respectively. Therefore, the quality factor in theory can be calculated by QCMT = (Qδ·Qγ)/(Qδ+Qγ). Particularly, Qδ and Qγ are equal when critical coupling occurs. So the formula of QCMT can be simplified to QCMT = Qδ/2 = ω0/4δ. The fitted δ0 = γ0 = 0.55 ×1012 Hz is obtained. Then, the value of QCMT0 is 516.6. The Q0 and QCMT0 are nearly identical, indicating that the total absorption of the configuration can be attributed to critical coupling. Analogously, for the case of GMR1 mode, the FWHM is Δλ1 = 8.4 nm, and Q1 = 185.5. Accordingly, the value of QCMT1 is 184.8 (i.e., δ1 = γ1 = 1.636×1012 Hz).
From the point of impedance matching, when the effective impedance of the system matches the impedance of free space, reflection can be effectively suppressed and total absorption can be achieved, as shown in Fig. 2(b). Taking GMR0 mode as an example, it is found that at the resonant wavelength of λ0 = 1658.5 nm, the impedance is Z0 = 1.141 − 0.0473i, which is substantially equal to the impedance of the free space. We additionally present the phase of reflection coefficient as a function of wavelength for the system, as shown in Fig. 2(c). One can clearly see that, the involved resonance mode experiences a distinctively large jump astride the resonance wavelength . Interestingly, the absorber we designed not only achieves dual-band perfect absorption, but also exhibits slow light characteristics, as depicted in Fig. 2(d). An obvious positive group time delay (i.e., tg = dΦ/dω) larger than 0.014 ps can be obtained within the absorption window. Many similar works have been proposed. For example, Qing et al. investigated a dual-band absorber based on graphene-SiC hybrid system, achieving a group time delay of tg > 0.045 ps . Li et al. proposed a plasmonic absorber with a group time delay of tg > 0.012 ps . This characteristic can be applied to slow light devices for controlling the absorption of optical signals.
To gain a qualitative understanding of this enhanced light absorption effect based on critical coupling, |E| and Ey field distributions in the configuration at two resonance wavelengths are shown in Figs. 3(a) and 3(b), respectively. The absorption peaks of 1558.5 nm and 1658.5 nm correspond to different orders of GMR: the first order GMR at 1558.5 nm (i.e., GMR1 mode), and zeroth order GMR at 1658.5 nm (i.e., GMR0 mode). The incident fields are trapped as the guided modes near the graphene sheet and further induce the effects of resonant absorption, that is, the electromagnetic fields locally near the graphene layer is more easily consumed by the graphene layer, thus greatly increasing the absorptivity of the graphene monolayer. Taking GMR0 mode as an example, the periodical mode forms a typical standing wave profile along the x direction which indicates the GMR in the system. In other words, when the phase matching between the incident wave and a leaky lateral waveguide mode is satisfied, GMR can be excited successfully .
Next, we investigate the influence of the chemical potential of the graphene on the absorption of the system, as shown in Fig. 4(a). The GMR1 mode is almost unaffected while varying μc until 0.398 eV. Then absorption starts to decrease. The behavior of the GMR0 mode is quite different, where the absorption significantly decreases beyond 0.375 eV. As stated in , we clearly note that the μc of the graphene is positively correlated with the gate voltage; hence the permittivity of graphene can be tuned by altering μc via bias voltage. Figure 4(b) illustrates the relationship between μc and the absorption of different GMR modes. For the case of GMR1 mode, the absorption efficiency sharply drops from 97.3% to 3.1% as μc increases from 0.35 eV to 0.425 eV. Similarly, for the case of GMR0 mode, the absorption efficiency decreases from 98.3% to 2.3%. Although the trends of the two modes are basically the same, the corresponding mutation positions are different, which is essentially because graphene has different responses to chemical potentials at different wavelengths, as shown in Fig. 1(b). By increasing μc, graphene can be transformed from a lossy material to a lossless material. It should be noted that only graphene is used as loss material in the system. Therefore, changing the state of graphene will directly affect the absorption of the system. As mentioned above, when γ = δ, the system can achieve perfect absorption. Once the state of graphene changes, the critical coupling condition is no longer satisfied, which results in a significant decrease in the absorption of the system. Due to this unique characteristic, an optional multichannel absorption-induced optical switching can be easily achieved by changing the chemical potential of graphene. Accordingly, there are three states in the system, namely, on/on state, on/off state, and off/off state, corresponding to two absorption peaks, one absorption peak and zero absorption peak, respectively. To distinguish these three states, we calculated the difference between the absorption of the two modes, that is, |ΔA| = |A1 − A0|, as depicted in Fig. 4(c). When μc < 0.375 eV, |ΔA| < 0.5, which means that both modes can be well excited and maintain a high absorption, thus corresponding to the on/on state of the system. In contrast, when μc > 0.398 eV, neither mode can be effectively excited, so there is no absorption peak, and the system is in the off/off state. Particularly, when 0.375 eV < μc < 0.398 eV, |ΔA| is above 0.5, which means that there is a significant difference in the absorptivity between the two modes, and the state of the system is on/off. Figure 4(d) depicts the resonance absorption of the system for different μc, i.e., μc = 0.3 eV, 0.386 eV, and 0.5 eV. Dual-band critical coupling of the GMR modes occurs at μc = 0.3 eV. As μc increases to 0.386 eV, GMR0 mode vanishes, leaving only GMR1 mode. When μc is 0.5 eV, the system can be considered as a total reflector. Unlike the optical switches based on metallic structures, our structure is made of dielectric materials and therefore more stable. This impressive absorption modulation provides an effective method for the design of graphene-based optional multichannel switches.
Figure 5 shows the absorption spectra as a function of wavelength and geometrical parameters. Except as indicated, the geometric parameters are fixed to the default values. Figure 5(a) illustrates the normal incidence absorption spectra of the absorber, with the period increased from 1.15 μm to 1.3 μm. As P increases, the resonance wavelength of GMR1 mode presents an obvious blueshift. At the same time, the resonance wavelength of GMR0 mode undergoes a slight redshift. Similarly, with the increase of W or hd, the resonance wavelengths of GMR1 mode and GMR0 mode move with the same redshift tendency, as shown in Figs. 5(b) and 5(c). Particularly, when hs increases from 0.25 μm to 0.35 μm, the resonance wavelength of GMR1 mode redshifts from 1537 nm to 1567 nm, while the GMR0 mode is hardly affected because the energy of the GMR0 mode is mainly localized in the grating layer, as shown in Figs. 5(d) and 3(b).
In addition, when P is larger than 1.24 μm, the multiorder GMR modes in the PC (i.e., PCGMR ) can be excited and hybridized with GMR1 mode, resulting in splitting of absorption peaks. To understand the absorption behaviors based on PCGMR, we further simulate the absorption spectra when P = 1.26 μm, as shown in Figs. 6(a) and 6(b). Five absorption peaks reaching 36.4% at 1506.3 nm, 57% at 1532 nm, 85.5% at 1549 nm, 89.5% at 1551.5 nm, and 87.6% at 1661 nm are observed. Obviously, the fifth absorption peak is the GMR0 mode, the first two absorption peaks are PCGMR2 mode and PCGMR1 mode, the third and fourth absorption peaks are hybrid modes caused by the hybridization of PCGMR0 mode and GMR1 mode. The mode profiles of PCGMR2 mode, PCGMR1 mode, and hybrid modes are plotted in Figs. 6(c) to 6(f).
Finally, we investigate the characteristics of the absorber under TM polarization. It is found that there are two absorption peaks at 1550.8 nm (95.6%) and 1664.2 nm (92.3%), as shown in Fig. 7(a). Although both absorption peaks decrease slightly compared with the case of TE polarization, the corresponding quality factors are greatly enhanced. The FWHM of the first absorption peak is 1.3 nm, so the quality factor is 1192.9; for the second absorption peak, the FWHM is 1.8 nm, and thus the quality factor is 924.6. As can be seen from Fig. 7(c), slow light with group time delay larger than 0.0135 ps appears in the absorption window. Similar to the case of TE polarization, the system still maintains multichannel switching effect, as depicted in Figs. 7(b) and 7(d). When μc < 0.374 eV, the system is in on/on state; when 0.374 eV < μc < 0.4 eV, the system is in on/off state; when μc is further increased, the system is in off/off state. These characteristics of the system (e.g., dual-band high absorption, slow light, and multichannel switching effect) coexist for TE and TM waves, indicating that the structure has excellent applicability, which is not available in most other 1D structures. To evaluate the performance of the designed structure, we compared the results with other similar structures, as listed in the Table 1. Obviously, our structure exhibits both dual-channel absorption and high quality factor simultaneously.
A graphene-based absorber is proposed and investigated to enhance light absorption in the NIR range. The mechanism of the dual-band perfect optical absorption in our absorber is based on critical coupling, which can be successfully proved by CMT and impedance matching. Slow light emerges within the absorption window. Moreover, the absorption-induced multichannel switching effect can be dynamically controlled by chemical potential of graphene. Simulation results reveal that the absorption characteristics can be significantly affected by tuning the structural parameters. Interestingly, these characteristics also exist in another polarized light. This actively tunable light absorption offered by the simple structure will hold potential in the design of graphene-based devices, including optical switch, modulator, high-performance optoelectronic detector, narrowband selective filter, etc.
National Key Research and Development Program of China (2017YFA0700201, 2017YFA0700202, 2017YFA0700203); National Natural Science Foundation of China (61522106, 61571117, 61501117, 61501112, 61631007, 61701108, 61831006); 111 Project (111-2-05); Natural Science Foundation of Jiangsu Province (BK20150020).
1. 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] [PubMed]
2. F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4(9), 611–622 (2010). [CrossRef]
3. S. D. Sarma, S. Adam, E. H. Hwang, and E. Rossi, “Electronic transport in two-dimensional graphene,” Rev. Mod. Phys. 83(2), 407–470 (2011). [CrossRef]
7. H. S. Chu and C. How Gan, “Active plasmonic switching at mid-infrared wavelengths with graphene ribbon arrays,” Appl. Phys. Lett. 102(23), 231107 (2013). [CrossRef]
8. H. J. Li, L. L. Wang, J. Q. Liu, Z. R. Huang, B. Sun, and X. Zhai, “Investigation of the graphene based planar plasmonic filters,” Appl. Phys. Lett. 103(21), 211104 (2013). [CrossRef]
9. H. J. Li, L. L. Wang, H. Zhang, Z. R. Huang, B. Sun, X. Zhai, and S. C. Wen, “Graphene-based mid-infrared, tunable, electrically controlled plasmonic filter,” Appl. Phys. Express 7(2), 024301 (2014). [CrossRef]
10. G. Wang, C. Chen, Z. Zhang, G. Ma, K. Zhang, and C. W. Qiu, “Dynamically tunable infrared grating based on graphene-enabled phase switching of a split ring resonator,” Opt. Mater. Express 9(1), 56–64 (2019). [CrossRef]
11. L. Peng, L. Zhang, J. Yuan, C. Chen, Q. Bao, C. W. Qiu, Z. Peng, and K. Zhang, “Gold nanoparticle mediated graphene plasmon for broadband enhanced infrared spectroscopy,” Nanotechnology 28(26), 264001 (2017). [CrossRef] [PubMed]
12. Q. Xu, T. Ma, M. Danesh, B. N. Shivananju, S. Gan, J. Song, C. W. Qiu, H. M. Cheng, W. Ren, and Q. Bao, “Effects of edge on graphene plasmons as revealed by infrared nanoimaging,” Light. Sci. Appl. 6, e16204 (2017). [CrossRef] [PubMed]
13. J. Song, L. Zhang, Y. Xue, Q. Y. S. Wu, F. Xia, C. Zhang, Y. L. Zhong, Y. Zhang, J. Teng, M. Premaratne, C. W. Qiu, and Q. Bao, “Efficient Excitation of Multiple Plasmonic Modes on Three-Dimensional Graphene: An Unexplored Dimension,” ACS Photonics 3(10), 1986–1992 (2016). [CrossRef]
14. S. Biabanifard, M. Biabanifard, S. Asgari, S. Asadi, and C. E. Mustapha, “Tunable ultra-wideband terahertz absorber based on graphene disks and ribbons,” Opt. Commun. 427, 418–425 (2018). [CrossRef]
15. S. Asgari, Z. G. Kashani, and N. Granpayeh, “Tunable nano-scale graphene-based devices in mid-infrared wavelengths composed of cylindrical resonators,” J. Opt. 20(4), 045001 (2018). [CrossRef]
16. S. Asgari and N. Granpayeh, “Applications of Tunable Nanoscale Midinfrared Graphene Based Slot Cavity in Nanophotonic Integrated Circuits,” IEEE Trans. Nanotechnol. 17(3), 533–542 (2018). [CrossRef]
17. A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photonics 6(11), 749–758 (2012). [CrossRef]
18. A. Y. Nikitin, F. Guinea, F. J. Garcia-Vidal, and L. Martin-Moreno, “Surface plasmon enhanced absorption and suppressed transmission in periodic arrays of graphene ribbons,” Phys. Rev. B 85(8), 081405 (2012). [CrossRef]
19. Y. Liu, A. Chadha, D. Zhao, J. R. Piper, Y. Jia, Y. Shuai, L. Menon, H. Yang, Z. Ma, S. Fan, F. Xia, and W. Zhou, “Approaching total absorption at near infrared in a large area monolayer graphene by critical coupling,” Appl. Phys. Lett. 105(18), 181105 (2014). [CrossRef]
20. W. Wang, A. Klots, Y. Yang, W. Li, I. I. Kravchenko, D. P. Briggs, K. I. Bolotin, and J. Valentine, “Enhanced absorption in two-dimensional materials via Fano-resonant photonic crystals,” Appl. Phys. Lett. 106(18), 181104 (2015). [CrossRef]
21. J. T. Liu, N. H. Liu, J. Li, X. Jing Li, and J. H. Huang, “Enhanced absorption of graphene with one-dimensional photonic crystal,” Appl. Phys. Lett. 101(5), 052104 (2012). [CrossRef]
23. T. J. Cui, “Microwave metamaterials - From passive to digital and programmable controls of electromagnetic waves,” J. Opt. 19, 084004 (2017). [CrossRef]
24. X. Wang, X. Jiang, Q. You, J. Guo, X. Dai, and Y. Xiang, “Tunable and multichannel terahertz perfect absorber due to Tamm surface plasmons with graphene,” Photonics Res. 5(6), 536–542 (2017). [CrossRef]
25. Y. Jiang, H. D. Zhang, J. Wang, C. N. Gao, J. Wang, and W. P. Cao, “Design and performance of a terahertz absorber based on patterned graphene,” Opt. Lett. 43(17), 4296–4299 (2018). [CrossRef] [PubMed]
26. T. J. Cui, “Microwave metamaterials,” Natl. Sci. Rev. 5(2), 134–136 (2018). [CrossRef]
27. J. Zhou, S. Yan, C. Li, J. Zhu, and Q. H. Liu, “Perfect ultraviolet absorption in graphene using the magnetic resonance of an all-dielectric nanostructure,” Opt. Express 26(14), 18155–18163 (2018). [CrossRef] [PubMed]
28. J. R. Piper and S. Fan, “Total absorption in a graphene monolayer in the optical regime by critical coupling with a photonic crystal guided resonance,” ACS Photonics 1(4), 347–353 (2014). [CrossRef]
29. S. Fan and W. Suh, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20(3), 569–572 (2003). [CrossRef]
30. J. H. Hu, Y. Q. Huang, X. F. Duan, Q. Wang, X. Zhang, J. Wang, and X. M. Ren, “Enhanced absorption of graphene strips with a multilayer subwavelength grating structure,” Appl. Phys. Lett. 105(22), 221113 (2014). [CrossRef]
31. M. Grande, M. A. Vincenti, T. Stomeo, G. V. Bianco, D. de Ceglia, N. Akozbek, V. Petruzzelli, G. Bruno, M. De Vittorio, M. Scalora, and A. D’Orazio, “Graphene-based absorber exploiting guided mode resonances in one-dimensional gratings,” Opt. Express 22(25), 31511–31519 (2014). [CrossRef]
32. Y. Long, L. Shen, H. Xu, H. Deng, and Y. Li, “Achieving ultranarrow graphene perfect absorbers by exciting guided-mode resonance of one-dimensional photonic crystals,” Sci. Rep. 6, 32312 (2016). [CrossRef] [PubMed]
33. Y. Long, Y. Li, L. Shen, W. Liang, H. Deng, and H. Xu, “Dually guided-mode-resonant graphene perfect absorbers with narrow bandwidth for sensors,” J. Phys. D: Appl. Phys. 49(32), 32LT01 (2016). [CrossRef]
34. L. A. Bian, L. Yang, P. Liu, Y. Chen, H. Liu, and Q. Zhou, “Controllable perfect absorption in a double-cavity photonic crystal with one graphene monolayer,” J. Phys. D: Appl. Phys. 51(2), 025106 (2018). [CrossRef]
35. C. C. Guo, Z. H. Zhu, X. D. Yuan, W. M. Ye, K. Liu, J. F. Zhang, W. Xu, and S. Q. Qin, “Experimental demonstration of total absorption over 99% in the near infrared for monolayer-graphene-based subwavelength structures,” Adv. Opt. Mater. 4(12), 1955–1960 (2016). [CrossRef]
39. E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1998).
40. H. Lu, X. Gan, D. Mao, and J. Zhao, “Graphene-supported manipulation of surface plasmon polaritons in metallic nanowaveguides,” Photonics Res. 5(3), 162–167 (2017). [CrossRef]
41. G. W. Hanson, “Dyadic Green’s functions and guided surface waves for a surface conductivity model of graphene,” J. Appl. Phys. 103(6), 064302 (2008). [CrossRef]
43. Y. M. Qing, H. F. Ma, S. Yu, and T. J. Cui, “Tunable dual-band perfect metamaterial absorber based on a graphene-SiC hybrid system by multiple resonance modes,” J. Phys. D: Appl. Phys. 52(1), 015104 (2019). [CrossRef]
44. D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E 71(3), 036617 (2005). [CrossRef]
45. Y. M. Qing, H. F. Ma, and T. J. Cui, “Tailoring anisotropic perfect absorption in monolayer black phosphorus by critical coupling at terahertz frequencies,” Opt. Express 26(25), 32442–32450 (2018). [CrossRef]
47. H. J. Li, Y. Z. Ren, M. Qin, and L. L. Wang, “Multispectral perfect absorbers using plasmonically induced interference,” J. Appl. Phys. 123(20), 203102 (2018). [CrossRef]
49. A. Mahigir and G. Veronis, “Nanostructure for near total light absorption in a monolayer of graphene in the visible,” J. Opt. Soc. Am. B 35(12), 3153–3158 (2018). [CrossRef]
50. Y. S. Fan, C. C. Guo, Z. H. Zhu, W. Xu, F. Wu, X. D. Yuan, and S. Q. Qin, “Monolayer-graphene-based perfect absorption structures in the near infrared,” Opt. Express 25(12), 13079–13086 (2017). [CrossRef] [PubMed]