Share with Facebook
Tweet This
Post on reddit
Share with LinkedIn
Add to CiteULike
Add to Mendeley
Add to BibSonomy
Abstract
We present a semi-analytical model of the resonance phenomena occurring in a hybrid system made of a 1D array of periodic subwavelength slits deposited on an insulator/graphene layer. We show that the spectral response of this hybrid system can be fully explained by a simple semi-analytical model based on weak and strong couplings between two elementary sub-systems. The first elementary sub-system consists of a 1D array of periodic subwavelength slits viewed as a homogeneous medium. In this medium lives a metal-insulator-metal lattice mode interacting with surface and cavity plasmon modes. A weak coupling with surface plasmon modes on both faces of the perforated metal film leads to a broadband spectrum while a strong coupling between this first sub-system and a second one made of a graphene-insulator-metal gap leads to a narrow band spectrum. We provide a semi-analytical model based on these two interactions thus allowing efficient access of the full spectrum of the hybrid system.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Extraordinary optical transmission (EOT) [1] through an opaque metallic film perforated with subwavelength slits has received great interest over the past decade because of its numerous applications in optoelectronics such as mid-infrared spatial light modulators, linear signal processing or biosensing. Many theoretical and experimental works were carried out in order to understand and predict EOT and, especially, to highlight the role of surface waves [2–5]. More recently we provided, in [6], a simple and versatile model, for this phenomenon, involving a specific mode living in an equivalent homogeneous medium and a phase correction to account for surface waves. The proposed semi-analytical model is valid from the visible to the infrared frequencies ranges. On the other hand, significant efforts have been made to create active or tunable plasmonic devices operating from THz to mid-infrared frequencies. Thanks to its extraordinary electronic and optical properties, graphene, a single layer of arranged carbon atoms has attracted much attention in the last years. This material can support both TE an TM surface plasmons and can exhibit some remarkable properties such as flexible wide band tunability that can be exploited to build new plasmonic devices. The main challenge when designing a graphene-plasmon-based device is how to efficiently excite graphene surface plasmons with a free space electromagnetic wave since there is a huge momentum mismatch between the two electromagnetic modes. Generally two strategies are used. The first one consists in patterning the graphene sheet into nano-resonators [7–19]. In this case a surface plasmon of the obtained structure which is very similar to the graphene surface plasmon is excited and an absorption rate close to $100\%$ can be reached. In particular in [19], the authors presented an electrically tunable hybrid graphene-gold Fano resonator which consists of a square graphene patch and a square gold frame. They showed that the destructive interference between the narrow- and broadband dipolar surface plasmons, which are induced respectively on the surfaces of the graphene patch and the gold frame, leads to the plasmonic equivalent of electromagnetically induced transparency (EIT). However patterning a graphene sheet requires sophisticated processing techniques and deteriorates its extraordinary mobility. The second strategy consists in using a continuous graphene sheet instead of undesirable patterned graphene structure [20–25]. In this approach, the graphene sheet is coupled with nano-scatterers such as nano-particles, or nano-gratings. Gao et al. proposed [23] to use diffractive gratings to create a guided-wave resonance in the graphene film that can be directly observed from the normal incidence transmission spectra. In [22] Zhao et al. studied a tunable plasmon-induced-transparency effect in a grating-coupled double-layer graphene hybrid system at far-infrared frequencies. They used a diffractive grating to couple a normal incident wave and plasmonic modes living in a system of two graphene-films separated by a spacer. Zhang et al. [25] investigated optical field enhancements, in a wide mid-infrared band, originating from the excitation of graphene plasmons, by introducing a dielectric grating underneath a graphene monolayer. Usually, the optical response of all the grating-graphene based structures listed above is performed thanks to the finite difference time domain method (FDTD) or to the finite element method (FEM). However the features of these hybrid graphene-resonators devices is often linked to a plasmon resonance phenomenon. Therefore a modal method allowing for a full modal analysis of the couplings occurring in these plasmonic systems seems more suitable.
In this paper, we investigate an optical tunable plasmonic system involving two fundamental phenomena: an EOT phenomenon and a metal-insulator-graphene cavity plasmon mode excitation. We propose a semi-analytical model allowing to fully describe the spectrum behaviour of an hybrid plasmonic structure, made of a 1D periodic subwavelength slits array deposited on an insulator/graphene layers. The spectrum of the proposed hybrid system exhibits Lorentz and Fano-like resonances and also other broadband and narrow band resonances that are efficiently captured by our simplified model. In order to explain the origin of this particular behaviour, we first split the hybrid system into a couple of sub-systems. Second, thanks to a modal analysis through the polynomial modal method (PMM: one of the most efficient methods for modeling the electromagnetic properties of periodic structures) [26–29], we demonstrate that the scattering parameters of each sub-system can be computed through a concept of weak and strong couplings. Finally we provide analytical expressions of the reflection and transmission coefficients of the structure and describe the mechanisms leading to Lorentz and Fano resonances occurring in it.
2. Physical system
The hybrid structure under study is presented in Fig. 1. It consists of two sub-systems. The first sub-system earlier studied in [6] is a sub-wavelength periodic array of nano-slits with height $h_1=800nm$, period $d=165 nm << \lambda$ and slits-width $s=15nm$. The relative permittivity of the material filling the slits is denoted by $\varepsilon ^{(s)}$ while the dispersive relative permittivity of the metal (gold) is denoted by $\varepsilon ^{(m)}$ and described by the Drude-Lorentz model [31,32]. See Ref. [6] for the numerical parameters used for $\varepsilon ^{(m)}$ description. This first sub-structure is deposited on a dielectric spacer (with relative permittivity $\varepsilon ^{(2)}=1.54^2$ and hight $h_2=10nm$) itself deposited on a continuous graphene sheet. The monolayer graphene optical properties are modeled with an equivalent layer with thickness $\Delta$ and permittivity $\varepsilon (\omega )$ [30] :
where the optical conductivity of graphene includes both the interband and intraband contributions. The first term of Eq. (2) i.e. the interband contribution $\sigma _{inter}$ has the form $\sigma _{inter}=\sigma _{inter}^{\prime }+i\sigma _{inter}^{\prime \prime }$, where
Fig. 1. Sketch of hybrid structure made of a dispersive metal film perforated with a subwavelength periodic array of 1D nano-slits deposited on a dielectric spacer ended by a continuous graphene sheet.
Fig. 2. Reflection, transmission and absorption spectra of the hybrid system for $\mu _c=1eV$ (Fig. 2(a)) and $\mu _c=1.5eV$ (Fig. 2(b)). The hybrid structure exhibits both broadband and tunable narrow band resonances with respect to the chemical potential. Parameters: $\varepsilon ^{1}=\varepsilon ^{3}=\varepsilon ^{slit}=1$, incidence angle= $0^{o}$, $h=800nm$, $d=165nm$, $a=15nm$.
Fig. 3. Real part of the magnetic field $H_x(x,z)$ at $\lambda =4.17 \mu m$ (Fig. 3(a)) and at $\lambda =7.30 \mu m$ (Fig. 3(b)). Parameters: $\varepsilon ^{1}=\varepsilon ^{3}=\varepsilon ^{slit}=1$, incidence angle= $0^{o}$, $h=800nm$, $d=165nm$, $a=15nm$.
3. Modal analysis of the system
The sketch of vertical-to-horizontal cavity modes coupling outlined in the previous section is presented in Fig. 4, where $\gamma _0^{(1)}$ denotes effective index of the periodic slits array lattice mode in the $z$-direction while $\alpha _0^{(3)}$ denotes that of the metal/insulator/graphene cavity mode in the $x$-direction. The effective indices $\alpha _0^{(1)}$ and $\alpha _0^{(2)}$ will be introduced later. Modal methods are very suitable to deal with the current problem since it is related to mode resonances. Thus, all required effective indices are computed as eigenvalues of the generic operator $\mathcal {L}^{(k)}$:
Fig. 4. Sketch of the mechanism of the coupling between cavity lattice modes of the periodic array of nano-slits and the metal/insulator/graphene gap plasmon modes. Strong and weak couplings between three modes are responsible of the resonance phenomena of the hybrid structure.
Fig. 5. Configurations used for the computation of the required effective indices (eigenvalues of Eq. (5). $config.1$ is used for the computation of the modes of periodic arrays of nano-slits in general and in particular for the computation of the cavity lattice mode effective index $\gamma _0^{(1)}$. $config.2$ is used for the computation of the effective index $\alpha _0^{(2)}$ of the plasmon mode. The gap plasmon mode effective index $\alpha _0^{(3)}$ is computed thanks to $config.3$.
Fig. 6. Sketch of the weak coupling sub-system consisting of a periodic array of nano-slits encapsulated between $\varepsilon ^{(0)}$ and $\varepsilon ^{(3)}$ media. The lattice mode $\gamma _0^{(1)}$ is assumed to live in an $\sqrt {\varepsilon ^{(1)}}$ effective homogeneous medium. Two plasmon modes $\alpha _{sp}^{(0)}$ and $\alpha _0^{(2)}$ ensure the phase matching with the plane waves in media $\varepsilon ^{(0)}$ and $\varepsilon ^{(3)}$.
3.1 Weakly coupled sub-system
A semi-analytical model for the weakly coupled system has been already described in [6]. This system consists of a periodic array of subwavelength nano-slits encapsulated between media with relative permittivities $\varepsilon ^{(0)}$ and $\varepsilon ^{(3)}$. As pointed out in [6], the electromagnetic response of the system to an incident plane wave excitation, in the static limit ($d << \lambda$), is equivalent to that of a slab with equivalent permittivity $\varepsilon ^{(1)}=\langle 1/\varepsilon ^{(m,s)}(x) \rangle ^{-1}$ and height $h_1$. Its reflection and transmission coefficients $R_{12}$ and $T_{12}$ are then given by :
and where $r_{1}$, $t_{1}$ and $r_{2}$, $t_{2}$ are the Fresnel coefficients at the interfaces $\varepsilon ^{(0)}/\varepsilon ^{(1)}$ and $\varepsilon ^{(1)}/\varepsilon ^{(3)}$ under TM polarization:Fig. 8. Comparison between the reflection spectrum of the hybrid structure and the responses of the weakly coupled sub-system (a) and the strongly coupled sub-system (b). As expected, the weakly coupled sub-system reflection spectrum $|R_{12}(\lambda )|^2$ perfectly matches the broadband resonances of the hybrid structure. On the other hand, the strongly coupled sub-system spectrum characteristic function $|S_{11}(\lambda )+S_{12}(\lambda )|^2$ perfectly matches the narrow band resonances of the hybrid structure. Parameters: $\lambda \in [2,10]\mu m$, $\varepsilon ^{(1)}=\varepsilon ^{(3)}=\varepsilon ^{(s)}=1$, $\varepsilon ^{(2)}=1.54^2$, incidence angle= $0^{o}$, $\mu _c=1eV$.
3.2 Strongly coupled system
Consider now the strongly coupled system sketched in Figs. 9 and 10. Since the transverse geometrical parameters of the grating are smaller than the incident field wavelength $\lambda$ ($d << \lambda$), we can introduce for the lattice mode an effective index $\alpha _0^{(1)}$ along the $x$-axis as follows:
where $\alpha _0^{(1)}$ has a positive real part and a negative imaginary part. The $S$-parameters of the equivalent two ports network of Fig. 9 are then given by :Fig. 9. Sketch showing the strong coupling between the gap plasmon mode $\alpha _0^{(3)}$ living in an $\sqrt {\varepsilon ^{(2)}}$ homogeneous medium and $\alpha _0^{(1)}$ lattice mode in an $\sqrt {\varepsilon ^{(1)}}$ effective homogeneous medium.
Fig. 11. Comparison between the spectra of the hybrid-structure with the reflection and transmission curves obtained from the PMM for two values of the chemical potential $\mu _c$. The chemical potential is set to $\mu _c=1eV$, in Figs. 11(a) and (b), while $\mu _c=1.5eV$, in Figs. 11(c) and (d). All these results fit very well with the rigorous numerical simulations obtained with the PMM. Our model captures very well all resonances occurring in the hybrid system namely Lorentz and Fano ones. Parameters: $\lambda \in [2,10]\mu m$, $\varepsilon ^{(1)}=\varepsilon ^{(3)}=\varepsilon ^{(s)}=1$, $\varepsilon ^{(2)}=1.54^2$, incidence angle= $0^{o}$.
4. Analysis of the Lorentz and Fano resonances of the system
Analysing the reflection $|R|^2$, transmission $|T|^2$ from Eqs. (20), (21), it is possible to provide justifications for the curves shapes in Figs. 2(a) and (b). From these figures, we remark that:
- 1. In the frequency range close to the resonance frequencies of the weak sub-system, the reflection and transmission spectra generally exhibit asymmetric Fano-like shapes while the absorption presents Lorentz-like shapes (left inserts of Figs. 2(a) and (b)).
- 2. When the resonance frequency of both strongly and weakly coupled systems match each other, it results in an exaltation of the reflection and annihilation of both transmission and absorption. This can be seen as a sort of induced reflection.
- 3. In the frequency range far from the resonance frequencies of the weakly coupled sub-system, a Lorentz-like absorption enhancement can be observed (right inserts of Figs. 2(a) and (b)). The scattering efficiency vanishes and the absorption takes its maximum value close to unity.
For the second point, let us recall the resonance condition of the first sub-system. It is obtained from the zeros of the reflection coefficient $R_{12}$, in Eq. (6), as soon as:
There is an extinction of the reflection without any annihilation of the transmission. Knowing that the resonance condition of the strongly coupled system is given by when the latter resonance condition Eq. (23) meets the former Eq. (22), it results which leads to an exaltation of reflection, and an annihilation of the transmission (see Eq. (21)) and the absorption. The spectral responses of the structure are shown to be highly tunable by changing a gate voltage applied to the graphene sheet. The height $h_2$ of the horizontal cavity influences the system through the effective index $\alpha _0^{(3)}$. The dispersion curves of the effective index $\alpha _0^{(3)}$ are plotted in Fig. 12(a) for different values of $h_2$ while $\mu _c$ is kept constant and equal to $1eV$. It can be seen that increasing $h_2$ leads to a decrease of the real part of $\alpha _0^{(3)}$. Since the $x$ dependance of the electromagnetic field in the cavity may be approximated by $H_y(x)=A^+ exp(ik \alpha _0^{(3)}x)+A^- exp(-ik \alpha _0^{(3)}x)$, ($k=2\pi /\lambda$), for a given $d$-length cavity, the resonance wavelengths can be approximately obtained through a phase condition on the term $A^{\pm } sin(2\pi d \alpha _0^{(3)}/\lambda _r$). When $\alpha _0^{(3)}$ decreases, the resonance wavelength $\lambda _r$ brought by the strongly coupled sub-system also decreases. Consequently increasing spacer height pushes the resonance wavelengths resulting from the strongly coupling sub-system towards the visible wavelengths range. The same behavior can be observed when the height $h_2$ is kept constant while increasing the chemical potential $\mu _c$ (see Fig. 12(b)). This time it is $\mu _c$ that influences the system through the effective index of the horizontal cavity. Increasing $\mu _c$ decreases $\alpha _0^{(3)}$ and thereby leads to a decrease of the resonance wavelengths. By tuning the potential $\mu _c$, one can realize the condition of Eq. (24) leading to an induced reflection phenomenon. For the last point raised, the Lorentz resonance shape of the absorption is provided by the poles of the scattering parameters of the system i.e. when $1+r_1\phi _1 r_{13} r_2 \phi _2 \simeq 0$ leading to weak values of both reflection and transmission. Besides, the exaltation of the absorption always occurs around frequencies where both reflection and transmission are weak and equal and these frequencies are different from the zeros of the coefficient $r_{13}$. Therefore in the frequency range far from the resonance frequency of the EOT sub-system, the hybrid structure behaves as a tunable perfect absorber.Finally, it is worth noticing that the present model works very well for normal incidence and reasonably well for angles of incidence up to twenty degrees. For large angles of incidence, some new resonances appear in the spectra and are not captured by our model.
Fig. 12. Dispersion curves of the effective index $\alpha _0^{(3)}$ for different values of $h_2$, ($\mu _c=1eV$) (Fig. 12(a)) and for different values of $\mu _c$ ($h_2=10 nm$) (Fig. 12(b)). Increasing the chemical potential $\mu _c$ or the spacer width $h_2$, the real part of $\alpha _0^{(3)}$ decreases. Parameters: $\varepsilon ^{2}=1.54^2$.
5. Conclusion
In conclusion, we have proposed a simple model, allowing to deepen the comprehension of the resonance phenomena involving the EOT phenomenon and a metal/insulator/graphene gap plasmon excitation. We consider a hybrid structure that consists of a 1D array of periodic subwavelength slits ended by a metal/insulator/graphene gap. For our analysis, this hybrid structure is split into two sub-systems. Each sub-system is driven by eigenmodes operating in an appropriate coupling regime. The study of the first sub-system, characterised by modes operating in a weak coupling regime, allows to understand the broadband resonance of the hybrid system. We provided an analytical expression of the reflection and transmission coefficients of this first sub-system. The behavior of the second sub-system, characterized by modes acting in a regime of strong coupling allows to understand the narrow-band nature of the hybrid system. Here, the resonance frequencies directly depend on the metal-insulator-graphene horizontal Perot Fabry cavity effective index. Since the real part of this effective index decreases by increasing the graphene sheet chemical potential, the resonance wavelengths of the system become perfectly tunable ; better yet an induced reflection phenomenon or perfect absorption can be achieved with suited values of the graphene sheet Fermi level. We proposed a spectral function allowing not only to characterize the resonance frequencies of this second sub-system, but also showed that introducing this spectral function into the reflection and transmission coefficients of the first sub-system, we obtain an analytical expression of the reflection and transmission coefficients of the global hybrid system which are successfully compared with those obtained with rigorous numerical simulations (through the PMM approach). Finally, armed with these analytical expressions, we provided a full description of the resonance phenomena occurring in the system. Our analysis in terms of simple modes couplings can be extended to study the coupling of the lattice modes with a substrate made by a non-reciprocal photonic topological materials, of particular interest for energy management and transport [33] and for atomic manipulation [34]. The analysis of such complex hybrid configurations involving diffraction gratings coupled to hybrid graphene multilayer structures could also be applied to study and to estimate more complicated phenomena, like the Casimir effect [35] and the radiative heat transfer [36].
Funding
Agence Nationale de la Recherche (ANR) “Investissements d’Avenir,” IDEX-ISITE initiative (16-IDEX-0001) (CAP 20-25).
References
1. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelenght hole arrays,” Nature 391(6668), 667–669 (1998). [CrossRef]
2. L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pel- lerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of Extraordinary Optical Transmission through Subwavelength Hole Arrays,” Phys. Rev. Lett. 86(6), 1114–1117 (2001). [CrossRef]
3. L. Aigouy, P. Lalanne, J. P. Hugonin, G. Julie, V. Mathet, and M. Mortier, “Near field analysis of surface waves launched at nano-slit apertures,” Phys. Rev. Lett. 98(15), 153902 (2007). [CrossRef]
4. H. T. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature 452(7188), 728–731 (2008). [CrossRef]
5. A. Y. Nikitin, S. G. Rodrigo, F. J. Garcia-Vidal, and L. Martin-Moreno, “In the diffraction shadow: Norton waves versus surface plasmon polaritons in the optical region,” New J. Phys. 11(12), 123020 (2009). [CrossRef]
6. K. Edee, “Single mode approach with versatile surface wave phase correction for the extraordinary optical transmission comprehension of 1D period nano-slits arrays,” OSA Continuum 1(2), 613–624 (2018). [CrossRef]
7. 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]
8. S. Thongrattanasiri and F. J. García de Agajo, “Optical field enhancement by strong plasmon interaction in graphene nanostructures,” Phys. Rev. Lett. 110(18), 187401 (2013). [CrossRef]
9. H. G. Yan, T. Low, W. J. Zhu, Y. Q. Wu, and M. Freitag et al., “Damping pathways of mid-infrared plasmons in graphene nanostructures,” Nat. Photonics 7(5), 394–399 (2013). [CrossRef]
10. D. Rodrigo, A. Tittl, O. Limaj, F. J. García de Abajo, V. Pruneri, and H. Altug, “Double-layer graphene for enhanced tunable infrared plasmonics,” Light: Sci. Appl. 6(6), e16277 (2017). [CrossRef]
11. V. W. Brar, M. S. Jang, M. Sherrott, J. J. Lopez, and H. A. Atwater, “Highly Confined Tunable Mid-Infrared Plasmonics in Graphene Nanoresonators,” Nano Lett. 13(6), 2541–2547 (2013). [CrossRef]
12. J. H. Strait, P. Nene, W.-M. Chan, C. Manolatou, S. Tiwari, F. Rana, J. W. Kevek, and P. L. McEuen, “Confined Plasmons in Graphene Microstructures: Experiments and Theory,” Phys. Rev. B: Condens. Matter Mater. Phys. 87(24), 241410 (2013). [CrossRef]
13. S. Yi, M. Zhou, X. Shi, Q. Gan, J. Zi, and Z. Yu, “A Multiple- Resonator Approach for Broadband Light Absorption in a Single Layer of Nanostructured Graphene,” Opt. Express 23(8), 10081–10090 (2015). [CrossRef]
14. F. H. L. Koppens, D. E. Chang, and F. J. García de Agajo, “Graphene plasmonics: a platform for strong light-matter interactions,” Nano Lett. 11(8), 3370–3377 (2011). [CrossRef]
15. H. Yan, X. Li, B. Chandra, G. Tulevski, Y. Wu, M. Freitag, W. Zhu, P. Avouris, and F. Xia, “Tunable Infrared Plasmonic Devices Using Graphene/Insulator Stacks,” Nat. Nanotechnol. 7(5), 330–334 (2012). [CrossRef]
16. Z. Fang, Y. Wang, A. E. Schlather, Z. Liu, P. M. Ajayan, F. J. García de Abajo, P. Nordlander, X. Zhu, and N. J. Halas, “Active Tunable Absorption Enhancement with Graphene Nanodisk Arrays,” Nano Lett. 14(1), 299–304 (2014). [CrossRef]
17. A. Fallahi and J. Perruisseau-Carrier, “Design of Tunable Biperiodic Graphene Metasurfaces,” Phys. Rev. B: Condens. Matter Mater. Phys. 86(19), 195408 (2012). [CrossRef]
18. B. Zhao and Z. M. Zhang, “Strong Plasmonic Coupling between Graphene Ribbon Array and Metal Gratings,” ACS Photonics 2(11), 1611–1618 (2015). [CrossRef]
19. M. Amin, M. Farhat, and H. Bagci, “A dynamically reconfigurable Fano metamaterial through graphene tuning for switching and sensing applications,” Sci. Rep. 3(1), 2105 (2013). [CrossRef]
20. L. Tang, H. Shi, J. Yang, C. Du, F. Gao, J. Zhu, and J. Dua, “Complete optical absorption in graphene by using nano-gratings to excite graphene surface plasmons,” Microelectron. Eng. 145, 58–61 (2015). [CrossRef]
21. Z. Liu and K. Aydin, “Enhanced infrared transmission through gold nanoslit arrays via surface plasmons in continuous graphene,” Opt. Express 24(24), 27882 (2016). [CrossRef]
22. X. Zhao, L. Zhu, C. Yuan, and J. Yao, “Tunable plasmon-induced transparency in a grating-coupled double-layer graphene hybrid system at far-infrared frequencies,” Opt. Lett. 41(23), 5470–5473 (2016). [CrossRef]
23. W. Gao, J. Shu, C. Qiu, and Q. Xu, “Excitation of Plasmonic Waves in Graphene by Guided-Mode Resonances,” ACS Nano 6(9), 7806–7813 (2012). [CrossRef]
24. S.-X. Xia, X. Zhai, L.-L. Wang, B. Sun, J.-Q. Liu, and S.-C. Wen, “Dynamically tunable plasmonically induced transparency in sinusoidally curved and planar graphene layers,” Opt. Express 24(16), 17886 (2016). [CrossRef]
25. T. Zhang, L. Chen, B. Wang, and X. Li, “Tunable broadband plasmonic field enhancement on a graphene surface using a normal-incidence plane wave at mid-infrared frequencies,” Sci. Rep. 5(1), 11195 (2015). [CrossRef]
26. K. Edee, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings,” J. Opt. Soc. Am. A 28(10), 2006–2013 (2011). [CrossRef]
27. K. Edee, I. Fenniche, G. Granet, and B. Guizal, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings: weighting function, convergence and stability,” Prog. Electromagn. Res. 133, 17–35 (2013). [CrossRef]
28. K. Edee and J.-P. Plumey, “Numerical scheme for the modal method based on subsectional Gegenbauer polynomial expansion: application to biperiodic binary grating,” J. Opt. Soc. Am. A 32(3), 402–410 (2015). [CrossRef]
29. K. Edee, J.-P. Plumey, A. Moreau, and B. Guizal, “Matched coordinates in the framework of polynomial modal methods for complex metasurfaces modeling,” J. Opt. Soc. Am. A 35(4), 608–615 (2018). [CrossRef]
30. A. Vakil and N. Engheta, “One-Atom-Thick IR Metamaterials and Transformation Optics Using Graphene,” arXiv:1101.3585v1 (2011).
31. R. Brendel and D. Bormann, “An infrared dielectric function model for amorphous solids,” J. Appl. Phys. 71(1), 1–6 (1992). [CrossRef]
32. A. D. Rakiá, A. B. Djuriš ić, J. M. Elazar, and M. L. Majewski, “Optical properties of metallic films for vertical-cavity optoelectronic devices,” Appl. Opt. 37(22), 5271–5283 (1998). [CrossRef]
33. P. Doyeux, S. A. H. Gangaraj, G. W. Hanson, and M. Antezza, “Giant interatomic energy-transport amplification with nonreciprocal photonic topological insulators,” Phys. Rev. Lett. 119(17), 173901 (2017). [CrossRef]
34. M. G. Silveirinha, S. Ali Hassani Gangaraj, George W. Hanson, and Mauro Antezza, “Fluctuation-induced forces on an atom near a photonic topological material,” Phys. Rev. A 97(2), 022509 (2018). [CrossRef]
35. R. Messina, P. A. M. Neto, B. Guizal, and M. Antezza, “Casimir Interaction between a sphere and a grating,” Phys. Rev. A 92(6), 062504 (2015). [CrossRef]
36. B. Zhao, B. Guizal, Z. M. Zhang, S. Fan, and M. Antezza, “Near-field heat transfer between graphene/hBN multilayers,” Phys. Rev. B 95(24), 245437 (2017). [CrossRef]