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Abstract
In this paper, a graphene plasmonic waveguide consisting of Si graded gratings and a SiO2 separator has been designed in order to rainbow trap and release in the mid-infrared frequencies. Tunability of the light trapping and releasing in this proposed structure has been realized thanks to the adjustable chemical potential of the graphene. Using this structure, the light velocity has been decreased by a slowdown factor above 1270 with a trapping bandwidth of 3.5 µm. Due to the high tunability of this miniaturized structure, it can be used in a variety of applications including optical switches, buffers, and storages.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Slow light can be realized by propagating an optical pulse with very low group velocity. Its applications include optical memories, buffers, and switches, nonlinear enhancement, and optical signal processing [1]. Applying slow light for realizing a real-time adjustable buffer for optical pulses is very important. This device can be used for delaying an entire packet of optical information in order to substantially increase the efficiency of routers in optical telecommunication networks [2].
Optical pulses propagate in a medium with refractive index n by the group velocity of vg = c/(n + ωdn/dω) = c/S, where ω is the light angular frequency, c is the light velocity in the vacuum, and S is the slowdown factor [3]. Given this relation, S depends on two main factors [4]: refractive index and material dispersion of the medium. As we know, the refractive index of optical materials is limited; in addition, the dispersion of available optical materials is low [3]. So, naturally, the speed of light cannot be reduced significantly in a single optical material; therefore, dispersion engineering should be utilized.
Quantum interference effect and stimulated Brillouin scattering are two mechanisms for realization of dispersion engineering in the optical structures [5,6]. However, efficiency of these techniques in slow light is not satisfactory and the size of their corresponding devices is relatively large [5,6]. Photonic crystals, as the third solution, are efficient slow light structures, but their size is still large [7]. As a key solution to overcome this challenge, plasmonic graded grating waveguide structures based on noble metals have been proposed. These plasmonic waveguides have small sizes and high levels of slowdown factors; but their performance has been limited due to their inherent ohmic losses [8,9].
In recent years, a new material called graphene has been proposed with unique properties in various aspects of optics, chemistry, materials and etc. Graphene is a two-dimensional material consisting of carbon atoms in a honeycomb lattice [10,11]. Thanks to the unique properties of this material, such as tunability of its Fermi energy by chemical doping and electrical gating, high optical confinement, and low ohmic losses, it is preferred over noble metals in plasmonics [12]. So, graphene surface plasmon has been used in order to realize several optical components such graphene-loaded waveguides, tunable metamaterials, photodetectors, bio-sensors, plasmon-induced light absorbers, and multi-band perfect plasmonic absorbers [13,14]. Also, periodic graphene nanoribbons have been utilized with sinusoidally shaped boundaries to excite crest modes [15]. On the other hand, the plasmon-induced transparency and slow light effect have been proposed based on terahertz chip scale plasmonic semiconductor-insulator-semiconductor waveguide system [16]. Also, planarly bended periodic graphene gratings have been introduced to excite the localized graphene surface plasmon polaritons (SPPs) with an ultrahigh Q-factor [17].
In the field of slow SPP and light trapping, a tunable nanostructure consisting of a monolayer graphene on a Si-based graded grating structure with a SiO2 spacer has been proposed. According to the theoretical and numerical results, it has been shown that this structure exhibits an ultra-high slowdown factor above 450 for the propagation of SPPs excited in graphene in the mid-infrared wavelength region within a bandwidth of approximately 2.1 µm and on a length scale less than 1/6 of the operating wavelength [18]. Also, a tunable graphene-spacer-grating-based (GSG-based) device with graded periods of the Si/SiO2 grating has been proposed to achieve a slowdown factor over 350 at mid-infrared region with a trapping bandwidth of approximately 0.7 µm [19]. On the other hand, a mid-infrared band-tunable dual-parallel graphene nanoribbon array deposited on both sides of a SiO2 substrate has been proposed to realize a slowdown factor over 340. It has been illustrated that rainbow trapping can be achieved by gradiently distribution of the nanoribbon width [20]. At last, a tunable graphene-based slow light structure has been proposed and numerically investigated to realize a slowdown factor over 131 with the bandwidth of 0.85 THz [21].
In this paper, we design a graphene plasmonic waveguide including optimized Si graded gratings and a SiO2 separator to rainbow trapping and releasing in the mid-infrared spectrum. We show that the tunability of the light trapping and releasing in this structure can be realized using the adjustable chemical potential of the graphene. Using the potential of this proposed structure, slowdown factor above 1270 with trapping bandwidth of 3.5 µm can be achieved. Also, the location of the trapped wavelengths along the propagation direction can be tuned by applied voltage. Thanks to the high tunability of this miniaturized waveguide, it has a variety of applications including optical switches, buffers, and storages.
2. Electrical and optical properties of graphene
In the mid-infrared range, the graphene surface conductivity σg has a complex value calculated by the Kubo formula as a function of ω (angular frequency of photons), µc (chemical potential or Fermi energy), τ (carrier relaxation time) and T (ambient temperature) [22–25]. This quantity consists of intraband and interband components. The first part illustrates the electron-photon scattering within the band and is obtained from the following relation [19–23]:
In this equation, e is the electron charge, ℏ is the reduced Planck constant, and kB is the Boltzmann constant. The second part of the surface conductivity is due to the interband transmission, which considering ℏ ω >> kBT and |µc| >> kBT, we have [23–25]:
In all simulations of this paper, the ambient temperature has been assumed as T = 300 K. In graphene, τ is dependent on carrier mobility µ and can be calculated as τ = µµc/evf2. Here, vf represents the Fermi velocity that is considered equal to 106 m/s [26]. Values of the carrier mobility µ are in the range of 20,000 cm2V−1s−1 to 200,000 cm2V−1s−1 [27]; here, it is considered equal to 25,000 cm2V−1s−1. Also, the chemical potential is calculated by µc =ℏvf(πns)1/2 [26]. The graphene doping level ns can be computed by [22]:
At last, dielectric constant and refractive index of graphene can be calculated as ɛg = 1 + iσg/ɛ0ωtg and ng = ɛg1/2; where, tg indicates the effective thickness of graphene [28]. In this paper, tg is considered equal to its actual value, 0.33 nm.
As shown in Fig. 1, real part of ɛg is negative for the applied bias voltages 20 V to 100 V, in the frequency range of 10 THz to 45 THz; so, in these conditions graphene can be assumed as a conductor in plasmonic structures. In addition, the imaginary part of ng that indicates the optical loss in this material decreases for high frequencies.
Fig. 1. Investigation of the effect of frequency and voltage changes on (a) the graphene dielectric constant and (b) the graphene refractive index. Here, ɛd = 3.9 and h = 100 nm. The curves of imaginary part of the graphene dielectric constant have overlaps for the various bias voltages.
3. Simulation results
3.1 Graphene-based plasmonic slab waveguide
Figure 2 shows a graphene plasmonic slab waveguide. This structure consists of a Si substrate and a graphene layer that separated by an insulator. To determine the level of ns, an external bias voltage Vb is applied between Si and graphene. The dispersion relation of SPP modes in this waveguide is obtained as follows [29]:
Here, k0 = 2π ⁄λ is the wavenumber in the free space and λ is the operating wavelength. Also, ɛc and ɛd are dielectric constants of the air and the separator, respectively.
Fig. 2. Transverse cross-section of the plasmonic slab waveguide consists of a graphene layer and a Si substrate separated by an insulator layer with thickness h.
First, according to Fig. 3(a), we examine behavior of the effective refractive index neff for the dominant guided SPP mode of the slab waveguide considering 1 ≤ ɛd ≤ 4 in order to determine an appropriate insulator to achieve minimum optical loss. The values of neff have been calculated by numerically solving the dispersion relation (4). The real part of neff can be approximately calculated as follows [18]:
Fig. 3. Effective refractive index for the dominant guided plasmonic mode of the slab waveguide versus (a) 1 ≤ ɛd ≤ 4, and (b) operating wavelength. The dielectric constants ɛSiO2 = 3.9, ɛAl2O3 = 1.14, and ɛPMMA = 2.25 [30,31] have been marked as real candidates for the separation layer. Here, Vb = 60 V, h = 100 nm.
The real and imaginary parts of neff represent the confinement and loss of the guided plasmonic mode, respectively. As shown in Fig. 3(a), the highest mode confinement with the lowest optical loss has been obtained considering SiO2 as the separator. On the other hand, according to definition of the slowdown factor in section 1, SiO2 is a better candidate due to its key role in realization of higher values for the real part of neff. The dispersion curve of the plasmonic slab waveguide for its dominant guided (TM) mode together with its electric field profile have been shown in Fig. 3(b) that illustrate high optical confinement in the graphene layer.
Figure 4 indicates the real and imaginary parts of neff for the dominant plasmonic mode of the slab waveguide as a function of h, λ, and Vb considering SiO2 as the separator. According to these results, both of the real and imaginary parts of neff increase with increasing h and decrease with increasing Vb for the operating wavelengths 7 µm to 12 µm.
Fig. 4. The real and imaginary parts of the effective refractive index for the dominant plasmonic mode of the slab waveguide as a function of h, λ, and Vb considering SiO2 as the separator. In (a) and (b): Vb = 60 V. In (c) and (d): h = 100 nm.
3.2 Graphene-based plasmonic Bragg grating waveguide
Bragg grating waveguides are periodic structures that act as filters or mirrors [32,33]. Here, three Bragg grating waveguides including square, sawtooth, and triangular gratings constructed on Si substrate of the slab waveguide have been studied. These waveguides and their structural parameters have been shown in Fig. 5. The minimum thickness of the SiO2 layer t, the periodicity of the grating p, and the length of the structure in the x direction have been assumed respectively equal to 100 nm, 40 nm, and 1280 nm in all simulations of this paper.
Fig. 5. Graphene-based plasmonic Bragg grating waveguides with (a) square, (b) sawtooth, and (c) triangular gratings constructed on Si substrate of the slab waveguide. A red arrow shows the propagation direction of the incident light.
According to Bragg's condition, the wavelengths in which the SPP modes of the grating waveguide are reflected can be calculated as [18]:
where, m equals zero in the wavelength range of interest and neff,1 and neff,2 represent the effective refractive indices of the dominant SPP mode for the graphene slab waveguide considering respectively t and t + d as distances between graphene and Si substrate.Figure 6 illustrates reflectivity of the SPP wave in the waveguides shown in Fig. 5 as a function of grating depth and operating wavelength. These results have been obtained numerically by the COMSOL Multiphysics software. The reflectance spectrum of these Bragg grating structures is also dependent on the applied voltage Vb, because of the relation between graphene doping and this bias voltage [18]. As shown in Fig. 6, the reflected wavelengths red-shifts with increasing d. Also, according to Fig. 6(a), theoretical results obtained by Solving Eq. (6) indicate same treatment. As shown, a good agreement has been obtained between the simulation and theoretical results.
Fig. 6. Reflectivity of the SPP wave in graphene based plasmonic Bragg grating structures as a function of grating depth and operating wavelength for (a) square, (b) sawtooth, and (c) triangular gratings. The red areas and the white circles represent respectively the maximum reflection and the theoretical results. The length of 1280 nm, w1 = w2 = 20 nm and Vb = 60 V have been considered for three structures.
On the other hand, the reflective bandwidths of these structures are different. In fact, the square Bragg grating structure has the widest reflective bandwidth. However, each of these structures has its own application. For example, with a few modifications, the sawtooth Bragg grating structure can be used as a single wavelength plasmonic filter. In this paper, the main criterion for selecting the optimal structure is the wide reflective bandwidth; therefore, we choose the square Bragg grating structure.
The Bragg grating structure decreases the speed of light near its cut-off wavelength, which can be obtained by solving the dispersion Eq. (7) [9,18].
Here, K represents the Bloch wave number of the SPP waves in the x direction, k1 = k0neff,1 and k2 = k0neff2, are respectively wave numbers in the SiO2 regions with thicknesses of t (effective refractive index neff,1) and t + d (effective refractive index neff,2).
By numerically solution of the characteristic Eq. (7), dispersion and slowdown factor curves can be obtained according to Figs. 7(a), (b) for different values of grating depth d in the square Bragg grating structure. According to these results, the slowdown factor is strongly dependent on the depth of gratings. The group velocities are obtained by calculating the slope of the dispersion curves shown in Fig. 7(a) [3]. Dashed lines in Fig. 7(b) represent the cut-off frequencies corresponding to different values of d. It is obvious that the cut-off frequency decreases with increasing d. Also, the obtained values for S are over 470 at frequencies near cut-off. This factor is limited by the ohmic loss of graphene, so it can be increased by physical parameters such as carrier mobility [18].
Fig. 7. (a) Dispersion curves for different values of grating depth considering w2 = 20 nm, (b) Slowdown factor for different values of grating depth as a function of the operating frequency considering w2 = 20 nm, (c) Slowdown factor for different values of grating width as a function of the operating frequency considering d = 250 nm, and (d) Slowdown factor for different values of grating depth and width as a function of the operating frequency. All of these figures are for the structure shown in Fig. 5(a).
On the other hand, Figs. 7(c), (d) show the slowdown factor for different values of grating width and depth of the structure shown in Fig. 5(a) as a function of the operating frequency. As can be seen, the slowdown factor is strongly dependent on the geometrical parameters of the structure. Also, it can be increased to a high value of 1270 for d = 223 nm and w2 = 6 nm.
It is clear that the graphene plasmonic Bragg grating structure is capable to reduce the velocity of operating wavelength very near to the Bragg wavelength. So, in this case, a very limited bandwidth of the light can be slow down efficiently. To overcome this limitation challenge, graded grating graphene plasmonic waveguides are designed.
3.3 Graded grating graphene plasmonic waveguide
The Si graded grating graphene plasmonic waveguide can be realized in three cases including (a) graded changes in grating depth considering constant grating width [18], (b) graded variations in grating width with constant grating depth [19], and (c) combination of both (a) and (b), i.e., graded changes in grating depth and width.
Figure 8 shows the above proposed first and second cases, which the mentioned gradual changes of the gratings are in the propagation direction x. To realize gradual variations in grating depth d, a linear function (Linear) and two nonlinear quadratic functions, i.e., Nonlinear1 (concave up) and Nonlinear2 (concave down), have been derived for d with respect to x. The constant coefficients of these linear and nonlinear functions have been determined considering the variations of d from 100 nm to 400 nm in the x-interval [0, 1280 nm] using MATLAB curve fitting toolbox.
Fig. 8. Gradual increase of (a) the grating depth, and (b) the grating width in the propagation direction of the graphene-based plasmonic grating waveguide.
Figure 9 shows distribution of the normalized electric field intensity (|Ey|2) in the xy plane of the structure for an incident wavelength of 9 µm. Light trapping occurs earlier in the case of Nonlinear2 function. So, Nonlinear2 is better than Linear and Nonlinear1 for efficient light trapping. Our criterion for the location of light trapping in the longitudinal direction of the slow light structure is the location that the normalized electric field intensity is decayed to 1/e of its initial value [18].
Fig. 9. Profiles of the normalized |Ey|2 in the xy plane of the structure shown in Fig. 8(a) for an incident wavelength of 9 µm considering Vb = 60 V and quadratic equation d = Ax2 + Bx + C assuming A = 0, B ≈ 0.23, and C = 100 × 10−9 for Linear function; A ≈ 121252.20, B ≈ 0.08 and C = 100 × 10−9 for Nonlinear1 function; and A ≈ ‒110229.27, B ≈ 0.38 and C = 100 × 10−9 for Nonlinear2 function.
On the other hand, Fig. 10 shows distribution of the normalized electric field intensity (|Ey|2) in the xy plane of the structure considering Nonlinear2 and Linear functions for variations of d at three deferent wavelengths. According to these results, trapping bandwidth for these cases have been obtained respectively 1.8 µm and 1.6 µm which indicates dependence of the trapping bandwidth on the variations type of the grating depth. Also, these results show that the Nonlinear2 provides wider bandwidth for efficient light trapping. However, this structure is not a good candidate for rainbow trapping, because of its limited bandwidth.
Fig. 10. Profiles of the normalized |Ey|2 in the xy plane of the structure shown in Fig. 8(a) for three deferent wavelengths considering (a) Nonlinear2 function and (b) Linear function for d and Vb = 60 V.
On the other hand, in order to investigate light trapping in the structure shown in Fig. 8(b) by gradual variations in grating width w2, two linear functions and a nonlinear quadratic function have been derived for w2 with respect to x. These functions for w2 have been introduced in the caption of Fig. 11. The constant coefficients of the nonlinear function have been determined considering the variations of w2 from 5 nm to 36 nm in the x-interval [0, 1280 nm] using MATLAB curve fitting toolbox.
Fig. 11. Profiles of the normalized |Ey|2 in the xy plane of the structure shown in Fig. 8(b) for three deferent wavelengths considering for (a) w2 = [5: 1: 36] nm, (b) w2 = [5: 0.5: 20.5] nm, (c) w2 = Ax2 + Bx + C assuming A ≈ −1.8311× 10−5, B ≈ 0.047656, and C = 5 × 10−9; d = 250 nm, and Vb = 60 V.
Figure 11 shows distribution of the normalized electric field intensity (|Ey|2) in the xy plane of the structure considering linear and nonlinear functions for variations of w2 at three deferent wavelengths. As shown in this figure, the mentioned wavelengths have been trapped at different locations in the propagation direction. Also, according to these results, trapping bandwidth for these cases have been obtained respectively 1.3 µm, 1 µm, and 0.8 µm which indicates dependence of the trapping bandwidth on the variations type of the grating width. Also, these results show that the first linear function provides wider bandwidth for efficient light trapping. However, this structure is not a good candidate for rainbow trapping, because of its limited bandwidth same as the previous waveguide.
As said in the beginning of this section, the Si graded grating graphene plasmonic waveguide can be realized by graded changes both in grating depth and width. For this purpose, a graded grating graphene plasmonic waveguide has been proposed according to Fig. 12, that is a combination of the structures shown in Figs. 8(a) and (b).
Fig. 12. Gradual increase of the grating depth and the grating width in the propagation direction of the graphene-based plasmonic grating waveguide.
Figure 13 shows distribution of the normalized electric field intensity (|Ey|2) in the xy plane of the structure for incident wavelengths of 7.0, 9.0, and 10.0 µm considering Nonlinear2 function for variations of d while w2 has been changed linearly in the range of 5 nm to 36 nm with steps of 1 nm. According to these results, the wavelengths have been trapped in different locations of the waveguide. Also, trapping of light with a very wide bandwidth of 3.5 µm (7.0 µm to 10.5 µm) has been achieved, indicating a significant improvement over previous works. Therefore, this structure is a good candidate for rainbow trapping.
Fig. 13. Profiles of the normalized |Ey|2 in the xy plane of the structure shown in Fig. 12 for three deferent wavelengths considering Nonlinear2 function for variations of d, w2 = [5 nm: 1 nm: 36 nm], and Vb = 60 V.
It should be noted that the multi-color light trapping can be realized if the condition of adiabaticity is met [34]. This condition has been satisfied for all of the graded grating structures designed in this paper for their various geometrical parameters.
Another interesting property of this structure is the possibility of releasing the trapped wavelengths, which is done by increasing the applied bias voltage Vb. Thus, each of the wavelengths that have been trapped at unique locations of the longitudinal direction of the structure can be propagated longer distance by increasing Vb.
According to Fig. 14, the trapped wavelengths can be released and propagated along the whole length of the waveguide by applying their corresponding bias voltages. As can be seen, more voltages are required to release the wavelengths trapped at the shorter lengths of the structure. Also, operating bandwidth of the slow SPP waveguide for light trapping is 3.5 µm at all the considered initial bias voltages (VRef).
Fig. 14. Bias voltages required to release the trapped wavelengths in the structure shown in Fig. 12 considering Nonlinear2 function for variations of d and w2 = [5 nm: 1 nm: 36 nm] for initial bias voltages VRef = 40 V (a), VRef = 50 V (b), VRef = 60 V (c), and VRef = 70 V (d).
An interesting feature of this structure is the ability to blue-shift and red-shift its operating bandwidth and reducing its bias voltages required to release the trapped wavelengths by tuning its initial bias voltage. For example, a considerable red-shift in the operating bandwidth and a significant decrease in the required bias voltages to release the trapped wavelengths can be seen in Fig. 14(a) compared to other cases.
4. Conclusion
At first, a planar plasmonic waveguide was designed and analyzed numerically. Because of the importance of the optical propagation loss in the plasmonic structures, a low-loss material, i.e., SiO2, was selected as a spacer between graphene and Si. Then, square, sawtooth, and triangular Bragg grating waveguides was analyzed numerically in order to design a grating structure with a wider bandwidth in reflection spectrum. So, the square grating waveguide was selected with S factor over 1270. Although, its bandwidth was limited. Therefore, graded variations for depth or width of the gratings in the propagation direction were proposed to realize broad bandwidth. In the following, it was shown that applying graded variations simultaneously in the grating depth and width leads to more considerable improvements in the operating bandwidth (3.5 µm) with red and blue-shift capability. At last, the tunability of the trapped wavelengths locations along the propagation direction by applied bias voltage was shown as a remarkable property of the designed graphene plasmonic waveguide in rainbow trapping and releasing in the mid-infrared wavelengths. This miniaturized nano waveguide has a variety of applications including optical switches, buffers, and storages.
Acknowledgment
We thank members of the Electromagnetics and Photonics Research Group (EPRG) at Sahand University of Technology for useful discussions.
Disclosures
The authors declare no conflicts of interest.
References
1. T. F. Krauss, “Why do we need slow light?” Nat. Photonics 2(8), 448–450 (2008). [CrossRef]
2. J. B. Khurgin and R. S. Tucker, Slow light: Science and applications (CRC press, 2018).
3. R. Hao, X.-L. Peng, E.-P. Li, Y. Xu, J.-M. Jin, X.-M. Zhang, and H.-S. Chen, “Improved slow light capacity in graphene-based waveguide,” Sci. Rep. 5(1), 15335 (2015). [CrossRef]
4. S. He, Y. He, and Y. Jin, “Revealing the truth about ‘trapped rainbow’storage of light in metamaterials,” Sci. Rep. 2(1), 583 (2012). [CrossRef]
5. L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397(6720), 594–598 (1999). [CrossRef]
6. Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett. 94(15), 153902 (2005). [CrossRef]
7. Y. A. Vlasov, M. O’boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438(7064), 65–69 (2005). [CrossRef]
8. D. Janner, G. Galzerano, G. Della Valle, P. Laporta, S. Longhi, and M. Belmonte, “Slow light in periodic superstructure Bragg gratings,” Phys. Rev. E 72(5), 056605 (2005). [CrossRef]
9. L. Chen, G. P. Wang, Q. Gan, and F. J. Bartoli, “Trapping of surface-plasmon polaritons in a graded Bragg structure: Frequency-dependent spatially separated localization of the visible spectrum modes,” Phys. Rev. B 80(16), 161106 (2009). [CrossRef]
10. Q. Bao, H. Hoh, and Y. Zhang, Graphene Photonics, Optoelectronics, and Plasmonics (CRC Press, 2017).
11. S. Syama and P. Mohanan, “Comprehensive application of graphene: emphasis on biomedical concerns,” Nano-Micro Lett. 11(1), 6 (2019). [CrossRef]
12. P. Tassin, T. Koschny, M. Kafesaki, and C. M. Soukoulis, “A comparison of graphene, superconductors and metals as conductors for metamaterials and plasmonics,” Nat. Photonics 6(4), 259–264 (2012). [CrossRef]
13. Z. Zhang, J. Yang, Y. Han, X. He, J. Huang, and D. Chen, “Plasmon-induced light absorption in mid-infrared based on hexagonal-shape graphene,” Mater. Res. Express 6(12), 125602 (2019). [CrossRef]
14. S.-X. Xia, X. Zhai, Y. Huang, J.-Q. Liu, L.-L. Wang, and S.-C. Wen, “Multi-band perfect plasmonic absorptions using rectangular graphene gratings,” Opt. Lett. 42(15), 3052–3055 (2017). [CrossRef]
15. S.-X. Xia, X. Zhai, L.-L. Wang, Q. Lin, and S.-C. Wen, “Localized plasmonic field enhancement in shaped graphene nanoribbons,” Opt. Express 24(15), 16336–16348 (2016). [CrossRef]
16. Z. Zhang, J. Yang, X. He, Y. Han, J. Huang, and D. Chen, “Tunable plasmon-induced transparency and slow light in terahertz chipscale semiconductor plasmonic waveguides,” J. Phys. D: Appl. Phys. 53(31), 315101 (2020). [CrossRef]
17. S.-X. Xia, X. Zhai, L.-L. Wang, G.-D. Liu, and S.-C. Wen, “Excitation of surface plasmons in sinusoidally shaped graphene nanoribbons,” J. Opt. Soc. Am. B 33(10), 2129–2134 (2016). [CrossRef]
18. H. Lu, C. Zeng, Q. Zhang, X. Liu, M. M. Hossain, P. Reineck, and M. Gu, “Graphene-based active slow surface plasmon polaritons,” Sci. Rep. 5(1), 8443 (2015). [CrossRef]
19. S. Ye, B. Wei, C. Sun, C. Dong, and S. Jian, “Graphene-spacer-grating-based structure to realize tunable mid-infrared narrow-band plasmonically induced transparency and slow light effect,” Appl. Phys. A 123(10), 637 (2017). [CrossRef]
20. B. Wei and S. Jian, “Analog of midinfrared electromagnetically induced-transparency and slow rainbow trapping light based on graphene nanoribbon-coated silica substrate,” J. Nanophotonics 11(2), 026011 (2017). [CrossRef]
21. H. Zhuang, H. Xu, F. Kong, Y. Wang, M. Gao, and K. Li, “Tunable slow light based on detuned coupling between graphene nanoribbon and square ring splitting modes,” Opt. Quantum Electron. 50(12), 444 (2018). [CrossRef]
22. P.-Y. Chen and A. Alu, “Atomically thin surface cloak using graphene monolayers,” ACS Nano 5(7), 5855–5863 (2011). [CrossRef]
23. F. Jabbarzadeh, M. Heydari, and A. Habibzadeh-Sharif, “A comparative analysis of the accuracy of Kubo formulations for graphene plasmonics,” Mater. Res. Express 6(8), 086209 (2019). [CrossRef]
24. F. Jabbarzadeh and A. Habibzadeh-Sharif, “Double V-groove dielectric loaded plasmonic waveguide for sensing applications,” J. Opt. Soc. Am. B 36(3), 690–696 (2019). [CrossRef]
25. F. Jabbarzadeh and A. Habibzadeh-Sharif, “High performance dielectric loaded graphene plasmonic waveguide for refractive index sensing,” Opt. Commun. 479, 126419 (2021). [CrossRef]
26. E. V. Castro, K. Novoselov, S. Morozov, N. Peres, J. L. Dos Santos, J. Nilsson, F. Guinea, A. Geim, and A. C. Neto, “Biased bilayer graphene: semiconductor with a gap tunable by the electric field effect,” Phys. Rev. Lett. 99(21), 216802 (2007). [CrossRef]
27. K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. E. Hone, P. Kim, and H. Stormer, “Ultrahigh electron mobility in suspended graphene,” Solid State Commun. 146(9-10), 351–355 (2008). [CrossRef]
28. S. He, X. Zhang, and Y. He, “Graphene nano-ribbon waveguides of record-small mode area and ultra-high effective refractive indices for future VLSI,” Opt. Express 21(25), 30664–30673 (2013). [CrossRef]
29. X. Y. He and R. Li, “Comparison of graphene-based transverse magnetic and electric surface plasmon modes,” IEEE J. Sel. Top. Quantum Electron. 20(1), 62–67 (2013). [CrossRef]
30. G. Ghosh, Handbook of optical constants of solids: Handbook of thermo-optic coefficients of optical materials with applications (Academic Press, 1998).
31. L. Luo, K. Wang, C. Ge, K. Guo, F. Shen, Z. Yin, and Z. Guo, “Actively controllable terahertz switches with graphene-based nongroove gratings,” Photonics Res. 5(6), 604–611 (2017). [CrossRef]
32. M. Nevière, “Bragg–Fresnel multilayer gratings: electromagnetic theory,” J. Opt. Soc. Am. A 11(6), 1835–1845 (1994). [CrossRef]
33. R. Kashyap, Fiber bragg gratings (Academic press, 2009).
34. Q. Gan, Y. Gao, K. Wagner, D. Vezenov, Y. J. Ding, and F. J. Bartoli, “Experimental verification of the rainbow trapping effect in adiabatic plasmonic gratings,” Proc. Natl. Acad. Sci. 108(13), 5169–5173 (2011). [CrossRef]
