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Abstract
We present a controlled Fano resonance in the mid-infrared (MIR) region from a kind of plasmonic metasurface consisting of a single-atom-thick surface layer with a periodic pattern of graphene nanostrip pairs on a dielectric substrate. Both the numerical and theoretical results indicate that the Fano resonance spectrum can be flexibly tailored through adjusting the geometrical parameters, such as the asymmetric distance and coupling gap between each pair of graphene nanostrips. Particularly, we achieve the dynamic tunability of the plasmonic Fano resonance spectrum by controlling the polarization of incident light and the Fermi level of graphene. The theoretical calculations agree well with the numerical simulations. These results could find significant applications in nanoscale light control and functional devices operating in the MIR region.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
As a typical two-dimensional (2D) atomic crystal, graphene has attracted particular attentions in the fields of photonics and optoelectronics due to its fantastic electronic and optical properties, containing the ultrahigh charge carrier mobility, ultrabroad operating frequency range, dynamic tunability, and compatibility with conventional optical elements [1, 2]. Until now, a large amount of photonic devices based on graphene have been demonstrated experimentally and investigated numerically, such as photodetectors [3], modulators [4], nanospasers [5], polarizers [6], fiber lasers [7], biosensors [8], and so on. Especially, the propagation of surface plasmon polaritons (SPPs) in graphene was predicted theoretically and verified experimentally in the terahertz (THz) and mid-infrared (MIR) regions [9–12]. Since then, SPPs in graphene have become a hot topic and received the rapid development, promoting a new emerging field named graphene plasmonics (GPs) [12–16]. SPPs derived from the collective oscillations of electrons on the surface of conductors can effectively break the diffraction limit of light and concentrate light on the deep-subwavelength scale, which provide a promising platform for the realization of novel optical phenomena and highly integrated devices [17–27]. GPs has presented the favorable features including ultrahigh confinement of light, relatively low dissipative loss, and flexible tunability, making graphene alternative to the conventional metal-based plasmonics [12]. Recently, a lot of significant photonic effects were reported in graphene-based plasmonic systems [25, 28–46]. Yu et al. investigated the coupling effects between the localized and delocalized SPPs in the graphene nanostructures [28]. Chen et al. theoretically realized the dual-band asymmetric transmission in graphene planar chiral metasurfaces [29]. Luo et al. found the tailorable plasmonic reflection in the defect-engineered graphene sheet [30]. Bao et al. experimentally demonstrated the plasmonic effects of graphene edges in graphene nanowaveguides [32]. Gonçalves et al. proposed the universal description of channel plasmonic wave in graphene and the hybridization of plasmonics in related 2D materials [40, 44]. Amin et al. investigated the graphene-controlled switching and sensing from plasmonic metamaterials in the THz region [43]. AbdollahRamezani et al. numerically realized the analog computation and beam manipulation in graphene-based metalines and metasurfaces [25, 46]. In addition, graphene-supported plasmonic nanofocusing [33], slow-light effect [34], plasmonic absorption [35], band-stop filtering [36], optical gradient force [37], and modulation [38] were proposed for the realization of ultracompact functional devices. Even so, exploring novel photonic phenomena in various graphene-based plasmonic architectures is still crucial for broadening the applications of graphene in photonics and optoelectronics.
Fano resonance discovered by Ugo Fano in atomic systems derives from the quantum interference effect and possesses an asymmetric spectral lineshape, which is different from the symmetric lineshape of the Lorentzian resonance [47]. Recently, the physical phenomena analogous to Fano resonance were observed in the classical optical systems, especially plasmonic nanostructures [48–51]. This kind of Fano resonances has found significant applications in novel optical functionalities such as sensing, second harmonic generation, and so on [51–53]. Here, we investigate the generation and tunability of Fano resonance in a plasmonic metasurface with a periodic array of single-layer graphene nanostrip pairs on the dielectric substrate. The achieved results show that the Fano spectral profile can be effectively tailored by the asymmetric distance and coupling gap between each pair of graphene nanostrips. The theoretical results are in good agreement with the numerical simulations. Particularly, we find that the Fano resonance spectrum can be flexibly tuned by controlling the polarization of incident light and Fermi level of graphene. The results will open a new avenue for nanoscale light manipulation and functional devices in the MIR region.
2. Structure and model
As shown in Fig. 1, the proposed plasmonic metasurface consists of a single-layer graphene array with a nanostrip pair in each unit cell on the dielectric substrate. P1 and P2 stand for the pitches of the graphene array in the x- and y-axis directions, respectively. θ is the polarization angle of incident light. g and a are the coupling gap and asymmetric distance between the graphene nanostrip pair in the unit cell of the metasurface, respectively. W1 and L1 (W2 and L2) are the width and length of the graphene nanostrip 1 (2), respectively. The relative permittivity of dielectric substrate (εs) is set as 3.9. The single-layer graphene can be characterized by the surface conductivity σg, which is generally described using the Kubo formula [54,55]. The surface conductivity can be expressed as σg = σintra + σinter, where σintra and σinter correspond to the intraband and interband electron-photon scattering in graphene, respectively. σintra can be written as
When ħω>>kBT and |Ef|>>kBT, σinter can be written aswhere e, kB, T, Ef, ħ, and τ represent the electron charge, Boltzmann’s constant, temperature, Fermi level of graphene, reduced Planck's constant, and relaxation time of charge carrier scattering, respectively. In graphene, τ is related to μ, which can be described as τ = μEf/(evf2). ω = 2πc/λ is the angular frequency of incident light, and λ is the incident wavelength of light in vacuum. Here, the Fermi velocity vf is set as 106 m/s [54]. The carrier mobility of graphene on the silica substrate could reach 4 × 104 cm2V−1s−1 at room temperature (T = 300 K) [56]. In our calculations, a moderate carrier mobility of 1 × 104 cm2V−1s−1 is set to ensure the credibility of results. Ef is assumed to be 0.5 eV, and thus τ is obtained as 0.5 ps [54]. The carrier mobility is reasonable for the graphene even with a high doping level [56]. The thickness of single-layer graphene can be set as Δ = 1 nm [9]. Thus, the equivalent relative permittivity of graphene can be expressed as εg = 1 + iσg/(ωΔε0), where ε0 is the permittivity of vacuum [9]. The optical response in the graphene structures can be numerically investigated by the finite-difference time-domain (FDTD) simulations. In our simulations, the perfectly matched layer absorbing boundary conditions are set at the top and bottom of computational space, and the periodic boundary conditions are set on the other four planes of the unit cell [57]. In the x- and y-axis directions, the mesh size is set as 0.5 nm. In the z-axis direction, the non-uniform mesh is employed. The maximum mesh sizes for the graphene and other layers are 0.2 and 0.5 nm, respectively. When the MIR light is normally incident on the metasurface, the plasmonic resonance will be excited on the graphene nanostrips. Thus, the graphene nanostrips can be regarded as plasmonic resonators [58]. Once the plasmonic resonators gather together, the strong coupling effect may be generated and induce the appearance of special phenomena, for example Fano resonance [59–61].
Fig. 1 Schematic diagram of the plasmonic metasurface composed of a single-layer graphene array with a nanostrip pair in each cell unit on a dielectric substrate. The inset shows the graphene nanostrip pair in each unit cell. Here, P1: the pitch of graphene array in the x-axis direction, P2: the pitch of graphene array in the y-axis direction, g: the coupling gap between the graphene nanostrip pair, a: the asymmetric distance between the centers (dashed lines) of the graphene nanostrip pair, θ: the polarization angle of incident light, W1 (W2): the width of the graphene nanostrip 1 (2), L1 (L2): the length of the graphene nanostrip 1 (2).
3. Results and analysis
We numerically investigate the plasmonic response of the graphene metasurface by using the FDTD simulations. When the graphene nanostrip pairs are symmetric along the x axis (i. e. a = 0), as depicted in Fig. 2(a), there are obvious reflection spectral peaks at the wavelengths of 6.40 and 6.36 μm for the x- (θ = 0°) and y-polarized (θ = 90°) incident light, respectively. The reflection spectra possess the typical symmetric Lorentzian profiles. Here, the geometrical parameters are assumed as P1 = 70 nm, P2 = 40 nm, W1 = 20 nm, L1 = 21 nm, W2 = 20 nm, L2 = 16 nm, and g = 6 nm. Figures 2(b) and 2(c) depict the electric field distributions in the plane of 1 nm above the graphene nanostrip pairs at the reflection spectral peaks for the x- and y-polarized incident light. From Fig. 2(b), we can see that the x-polarized light can excite the plasmonic resonance in the graphene nanostrip 2 at λ = 6.4 μm. However, the resonance is not distinctly excited in the graphene nanostrip 1 at this wavelength because L1 is not equal to L2. The plasmonic resonance in the graphene nanostrip 1 is excited at a short wavelength (5.7 μm), as shown in the inset of Fig. 2(a). For the y-polarized light, the plasmonic resonance in the graphene nanostrip 1 is excited at λ = 6.36 μm. At this wavelength, the resonance in the graphene nanostrip 2 is not obvious, as shown in Fig. 2(c). The resonant electric fields mainly locate at the corners of graphene nanostrips, forming localized surface plasmon resonance. Obviously, the plasmonic coupling between the graphene nanostrip pair for both x- and y-polarized incident light is particularly weak when a = 0 nm. Subsequently, we break the symmetry of the graphene nanostrip pairs and explore the spectral features of the plasmonic metasurface. Figure 3(a) depicts the reflection spectra of the metasurface with different asymmetric distances a for x-polarized incident light. It shows that the symmetry of original spectrum is broken, generating the asymmetric spectral lineshape. Moreover, the asymmetric spectral profile is strongly dependent on a. With the increase of a, the dip of reflection spectrum becomes specially distinct when a approaches 11 nm. The spectral peak at the right side of the dip disappears gradually with increasing a, while the left-side peak grows obviously. Different from the x-polarized incident light, the reflection peaks for the y-polarized incident light possess the reverse evolution, as shown in Fig. 3(b). Here, the Wood's anomalies are not observed in the MIR wavelengths of interest for normal incidence.
Fig. 2 (a) Reflection spectra of the plasmonic metasurface with the symmetric graphene nanostrip pairs (a = 0) for the x- and y-polarized incident light. (b)-(c) Normalized electric field distributions |E|2 in the plane at the distance of 1 nm above the graphene nanostrip pairs at the reflection spectral peaks for x- and y-polarized incident light. The red arrows in (b) and (c) denote the x- and y-polarized light, respectively. The inset in (a) shows the normalized field distribution |E|2 at the spectral peak (λ = 5.7 μm) for x-polarized incident light. The geometrical parameters are set as P1 = 70 nm, P2 = 40 nm, W1 = 20 nm, L1 = 21 nm, W2 = 20 nm, L2 = 16 nm, and g = 6 nm.
Fig. 3 (a)-(b) Reflection spectra of the plasmonic metasurface with different asymmetric distances a for the x- and y-polarized incident light. Here, P1 = 70 nm, P2 = 40 nm, W1 = 20 nm, L1 = 21 nm, W2 = 20 nm, L2 = 16 nm, and g = 6 nm.
The asymmetric spectral response of graphene metasurface can be considered as a classical analogue of Fano resonance [48–52]. In the metasurface, the x-polarized light can only directly excite the plasmonic resonance in the graphene nanostrip 2 at the wavelengths around 6.4 μm. The coupling between the graphene nanostrips will induce the resonant mode (with the resonant wavelength of 6.36 μm) in the graphene nanostrip 1. The plasmonic coupling and interference between the graphene nanostrips result in the formation of Fano resonance spectrum in Fig. 4(a) and the disappearance of electric field in the graphene nanostrip 2 at the resonance dip (λ = 6.36 μm), as depicted in Fig. 4(b). To further clarify the mechanism of asymmetric spectral profile, we theoretically investigate the spectral response of the metasurface by fitting the simulation results. According to the temporal coupled-mode theory (CMT) [26, 51, 62], the reflection spectrum of the metasurface with the graphene nanostrip pairs can be derived as
where ω1 represents the resonance frequency of the graphene nanostrip supporting the directly excited mode, whose decay rate is denoted by γ1. δ = ω1-ω2 is the detuning resonance frequency between the graphene nanostrip pair, γ2 is the decay rate of the resonance mode supported by another graphene nanostrip, κ is the coupling coefficient (coupling strength) between the graphene nanostrip pair, C1 is a reflection coefficient, and C2 is the reflection attributed to the substrate. By using the Eqs. (3) and (4), we can fit the reflection spectrum of the metasurface. It is worth noting that the theoretical fitting results agree well the numerical simulations. For the spectrum in Fig. 4(a), the fitting parameters are achieved as C1 = 0.8644, κ = 2.574 × 1012 rad/s, ω1 = 2.954 × 1014 rad/s, γ1 = 1.1 × 1012 rad/s, δ = −0.898 × 1012 rad/s, γ2 = 2.444 × 1012 rad/s, and C2 = 0.1065. C2 agrees well with the theoretical reflectivity (0.1073) of the substrate. From Eq. (4), we can see that the spectral profile is mainly dependent on the multiplication of the term I (containing κ) and term II (relying on ω1 and γ1). The results of the terms I and II with the above parameters are depicted in the inset of Fig. 4(a). It is shown that the Fano resonance spectrum can be considered as an asymmetric spectrum (from the term I) modulated by a Lorentzian resonance (from the term II) with the centre frequency ω1 [63]. The right-side peak of the asymmetric spectrum is closer to the Lorentzian resonance and higher than the left-side peak, which contributes to the formation of the Fano spectral profile in Fig. 4(a). According to the Fano equation in [63], the asymmetry parameter can be fitted as about 8.2. From Fig. 4(a), we can see that the spectral profile of Fano resonance possesses larger slope than that of Lorentzian resonance in Fig. 2(a). This feature contributes to improving the performance of sensors and switches [52, 64]. As depicted in Fig. 5(a), the resonance frequency ω1 ascends with increasing a for both x- and y-polarized incident light. For the x-polarized incident light, the higher peak of asymmetric spectrum gradually moves from the left to right side with increasing a, and the wavelength of Lorentzian resonance possesses a blue-shift between the two peaks of asymmetric spectrum. As shown in the inset of Fig. 5(a), the detuning resonance frequency δ increases from −1.18 × 1012 to 1.3 × 1012 rad/s when a changes from 3 to 17 nm. It corresponds to the shift of the dip for asymmetric spectrum from the left to right side of Lorentzian resonance, as depicted in Fig. 5(b). Thus, it is not difficult to understand the evolution process of Fano spectral profile in Fig. 3(a). For the y-polarized incident light, δ exhibits an opposite alternation with that of the x-polarized incident light, contributing to the evolution feature of Fano spectral profile in Fig. 3(b). The Fano resonance spectrum also depends on the pitches of graphene array [45]. For example, the reflection peak possesses a blue-shift with the increase of P1 (not shown here).Fig. 4 (a) Numerical (FDTD simulation) and theoretical (CMT fitting) results of reflection spectrum for the x-polarized light impinging on the metasurface with a = 5 nm. The inset shows the results of the term I and II in Eq. (4). (b) Normalized electric field distribution |E|2 at the dip (λ = 6.36 μm) of reflection spectrum in (a). The inset shows the corresponding field distribution of Ez component. Here, P1 = 70 nm, P2 = 40 nm, W1 = 20 nm, L1 = 21 nm, W2 = 20 nm, L2 = 16 nm, and g = 6 nm.
Fig. 5 (a) Resonance frequency ω1 of the metasurface with different asymmetric distances a for the x- and y-polarized incident light. These results are obtained by fitting the simulation results in Fig. 3. The inset shows the detuning resonance frequency δ as a function of a. (b) Spectral profile of the term I in Eq. (4) for the x-polarized incident light with different a. The arrows denote the positions of Lorentzian resonance from the term II in Eq. (4). Here, P1 = 70 nm, P2 = 40 nm, W1 = 20 nm, L1 = 21 nm, W2 = 20 nm, L2 = 16 nm, and g = 6 nm.
Subsequently, we investigate the dependence of Fano resonance spectrum on the coupling distance g between the graphene nanostrip pair. As shown in Fig. 6(a), the spectral width of the Fano resonance dip gradually becomes sharper and shallower with increasing g for both x- and y-polarized incident light. When g approaches 10 nm, the resonance dip will disappear. It is found that the wavelength of Fano resonance dip has a blue-shift with the increase of g. The spectral peaks around the resonance dip present different evolution processes with the change of g. For the x-polarized incident light, the left-side peak ascends with increasing g from 4 to 10 nm, while the right-side peak gradually drops. The case is reverse for the y-polarized incident light, as depicted in Fig. 6(b). To understand these behaviors of Fano resonance, we fit the simulation results using the CMT theoretical formula and analyze the spectral response. The theoretical results are fully consistent with the numerical simulations. Figure 7(a) shows that the resonance frequency ω1 of the metasurface rises up as g increases for both x- and y-polarized incident light. When g increases from 4 to 10 nm, δ ascends from −0.39 × 1012 to 1.36 × 1012 rad/s for the x-polarized incident light, and descends from 0.18 × 1012 to −1.28 × 1012 rad/s for the y-polarized incident light, as shown in the inset of Fig. 7(b). The opposite trend of δ contributes to the reverse evolution of spectral peaks for the x- and y-polarized incident light in Fig. 6. As depicted in Fig. 7(b), the coupling strength between the graphene nanostrip pair increases with the decrease of g. From Eq. (4), we can see that the Fano resonance spectrum is strongly dependent on the coupling strength κ. When κ approaches zero, the Fano resonance evolves into Lorentzian resonance. Thus, it can be seen in Fig. 6 that the spectral profile becomes relatively symmetric when g reaches 10 nm. When the coupling strength increases, the modulation of Lorentzian resonance is much stronger, and thus the Fano resonance becomes broader. There exists a slight difference between the coupling strengths for the x- and y-polarized incident light, which decreases with increasing g and nearly approaches zero when g>9nm.
Fig. 6 (a)-(b) Reflection spectra of the metasurface with different coupling distance g for the x- and y-polarized incident light. The solid circles and curves stand for the FDTD simulation and CMT fitting results, respectively. The parameters are set as P1 = 70 nm, P2 = 40 nm, W1 = 20 nm, L1 = 21 nm, W2 = 20 nm, L2 = 16 nm, and a = 12 nm.
Fig. 7 Resonance frequency ω1 of the metasurface with different coupling distances g for the x- and y-polarized incident light. These results are achieved by the CMT fitting in Fig. 6. The inset shows the detuning resonance frequency δ as a function of g. (b) Coupling coefficient κ between the graphene nanostrip pair with different g. Here, P1 = 70 nm, P2 = 40 nm, W1 = 20 nm, L1 = 21 nm, W2 = 20 nm, L2 = 16 nm, and a = 12 nm.
We also study the evolution of reflection spectrum from the graphene metasurface with the polarization angle θ of incident light. The results in Fig. 8(a) show that the Fano resonance spectrum is strongly dependent on θ. The resonance dip is relatively distinct when θ reaches 0° or 90°. Moreover, the long-wavelength peak around the resonance dip dominates when θ changes from 0° to 90°, while the short-wavelength peak turns stronger when θ alters from 90° to 180°. The long- and short-wavelength peaks reach the highest values when θ reaches 55° and 135°, respectively. As shown in Fig. 8(b), the theoretical fitting results are in accordance with the FDTD simulations. From Eq. (4), we can see that the spectral asymmetry is dependent on the coefficient κ. Here, we plot the fitting results of κ at different θ in the inset of Fig. 8(c). It is found that κ decreases when θ changes from 0° to 55°, increases when θ alters from 55° to 90°, declines when θ changes from 90° to 135°, and then rises when θ alters from 135° to 180°. There exists the strongest asymmetry for the Fano spectrum when θ = 0° and 90°. These theoretical results agree well with the simulations in Fig. 8(a). The frequency ω1 reaches the minimum (maximum) value when θ = 55° (135°), which corresponds to the position of long (short)-wavelength peak. If the incident light is circularly polarized, the Fano spectral response can also be observed. This polarization-dependent spectral response is particularly significant for the polarization selection and spectral shaping in the MIR region. SPPs in graphene can be effectively tuned by controlling the Fermi level of graphene via the chemical doping or external gating voltage, which facilitates the active tunability of plasmonic response in graphene systems [9, 65–67]. Here, we investigate the dependence of Fano spectral profile on the Fermi level Ef of graphene. As shown in Fig. 8(d), the Fano resonance dip possesses a linear blue-shift with the increase of Ef for the x-polarized incident light. Although the spectral peaks have a slight rise as Ef increases, the Fano spectral profile almost keeps unchanged. The shift of Fano resonance spectrum can be qualitatively explained by the resonance condition of graphene nanostrips, which can be described as 2Re{kGPs}L + 2ϕ = 2mπ [58]. Here, kGPs≈iε0(εs + 1)ω/δg is the wavevector of plasmonic mode in graphene resonators (nanostrips) [68], L is the resonance length (W2 for the graphene nanostrip 2 and L1 for the graphene nanostrip 1), ϕ is the phase shift of graphene resonator edges, and m is the order of resonant mode. Here, the fundamental mode (m = 1) is excited in graphene nanostrips, as can be seen in Fig. 4(b). In the MIR region, the surface conductivity of graphene can be approximately written as: δg≈ie2Ef/[πħ2(ω + iτ−1)] [68]. Thus, Re(kGPs)≈πħ2ε0(εs + 1)ω2/(e2Ef) is inversely proportional to Ef. When Ef increases, ω will ascend to satisfy the resonance condition, which induces the blue-shift of resonance frequencies of graphene nanostrips [58]. Therefore, we can observe the blue-shift of Fano resonance spectrum with increasing Ef. In practice, the ion gel layer can be deposited on the graphene metasurface to exert the gating voltage for the adjustment of Fermi level in graphene [38].The tunability of Fano resonance will provide an effective way for the realization of active plasmonic devices, especially MIR modulators.
Fig. 8 (a) Evolution of reflection spectrum from the metasurface with the polarization angle θ of incident light when a = 12 nm. (b) Reflection spectra with different θ. The solid circles and curves denotes the FDTD simulation and CMT fitting results, respectively. (c) Fitting results of the resonance frequency ω1 with different θ. The inset shows the coefficient κ as a function of θ. (d) Reflection spectra with different Fermi levels Ef of graphene for the x-polarized incident light when a = 5 nm. The geometrical parameters are set as P1 = 70 nm, P2 = 40 nm, W1 = 20 nm, L1 = 21 nm, W2 = 20 nm, L2 = 16 nm, and g = 6 nm.
4. Summary
We have investigated the MIR spectral response of a novel plasmonic metasurface consisting of single-layer graphene nanostrip pairs on the dielectric substrate. The results indicate that a Fano resonance spectral profile can be generated in the metasurface due to the coupling between the plasmonic modes in the asymmetric graphene nanostrip pair. It is found that the Fano resonance spectrum is particularly dependent on the geometric parameters, such as the asymmetric distance and coupling gap between the graphene nanostrip pair. The formation and evolution mechanisms of the Fano resonance spectrum are reasonably analyzed by the CMT theoretical calculations. The FDTD numerical simulations agree well with the theoretical results. Especially, we have obtained the flexible tunability of Fano resonance spectrum by adjusting the polarization angle of incident light and the Fermi level of graphene. Our results will open a new way for light control and functionalities in the MIR region, especially the polarization selection, spectral shaping, sensing, and active modulation.
Funding
National Key R&D Program of China (2017YFA0303800); National Natural Science Foundation of China (11774290, 11634010, 61705186, 61475188, 61675170, and 61705257); Natural Science Basic Research Plan in Shaanxi Province of China (2017JQ1023); Technology Foundation for Selected Overseas Chinese Scholar of Shaanxi Province (2017007); and Fundamental Research Funds for the Central Universities (3102016OQD031).
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