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Abstract
The phenomenon about optical activity has widespread applications in polarization optics, biosensing, and analytical chemistry. The optical activity in twisted graphene metasurface bilayers (TGMBs) is studied theoretically in this paper. It is found that the large circular dichroism (CD) value can be adjusted by various physical parameters of TGMBs such as separation distance, the voltage applied to metasurfaces, and twist angle. By adjusting the twist angle of TGMB, the shapes of the CD spectra, circular birefringence spectra, and ellipticity spectra can be manipulated in the broadband range. When the twisted bilayer metasurfaces are stacked with an ultrathin spacer, it is found that there might exist the strong optical activity responses near the rotated-$\sigma$-near-zero regime and topological transition $\sigma$-near-zero regime. The corresponding phenomena raise the prospect of tunable, compact, and on-chip terahertz devices with graphene metasurfaces based on optical activity.
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1. INTRODUCTION
As one of the most important optical effects in chiral materials, optical activity has received great attention in many areas such as analytical chemistry, spectroscopy, crystallography, and optics [1–7]. As the fundamental characteristic of the chiral materials, circular dichroism (CD) and circular birefringence (CB) can be used to behave the optical activity. Compared with three-dimensional (3D) chiral metamaterials [8], although the fabrication is relatively convenient, planar chiral metamaterials perform inferior in both maximum CD value and bandwidth due to isotropic response and limit thickness [9]. An efficient way is to stack planar chiral metamaterials to construct a quasi-type of three-dimensional chiral metamaterials. So far, several typical kinds of quasi-3D metamaterials stacked by planar chiral metamaterials have been reported. Type I is based on a tri-layer metal-dielectric-metal structure where the pattern of each layer is uniform [10–12]. Type III is based on an asymmetric multi-layer structure composed of different planar chiral metamaterials without mirror or rotational symmetry [13,14]. The second one (Type II) is a quasi-symmetric double-layer structure formed by two same planar metamaterials with mirror symmetry or a twist angle [14–25].
Metals are commonly used to fabricate 3D or quasi-3D chiral metamaterials [10–25]. Due to the restricted flexibility of the permittivity of metals, the fabrication of chiral metamaterials is fairly complicated, and the geometrical parameters of the corresponding structure are difficult to adjust. Appropriate materials that can dynamically and precisely tune the CD effect are highly desirable. Graphene is a two-dimension (2D) material composed of one monolayer of carbon atoms gathered in a honeycomb lattice. By changing the Fermi energy levels via chemical or electrostatic doping, graphene shows great tunability of its excellent electrical and optical properties via changing its surface conductivity. As the planar counterparts of metamaterials, metasurfaces have several intriguing properties such as compact size, lower loss, and higher efficiency [26]. Alternative metasurfaces based on graphene can be used to design tunable, broadband, and easier-to-implement ultrathin devices [27–30]. Recently, some literature reported that stacking hyperbolic metasurface bilayers with a twist angle between them can effectively tune the dispersion contours [31–37]. The corresponding reports have stimulated the new research area for twistronics about advanced low-dimensional nanodevices. The manipulation of the stacking in twisted graphene metasurface bilayers (TGMBs) provides a unique platform to investigate twist-induced photonic dispersion modifications which provide added flexibility for adjusting light–matter interactions. To the best of our knowledge, the optical activity in the TGMB has not yet been fully exploited, and it deserves further study.
Here we study theoretically the optical activity in the TGMB. It can be found that the large CD value can be manipulated by various physical parameters such as the separation distance, the voltage applied to metasurfaces, and twist angle. Through adjusting the twist angle of the TGMB, the CD spectra, CB spectra, and ellipticity spectra can be controlled in the broadband range. When the twisted bilayer metasurfaces are stacked with an ultrathin spacer, there might exist the strong optical activity responses near the rotated-$\sigma$-near-zero (RSNZ) regime and the transition $\sigma$-near-zero (TPSNZ) regime.
Fig. 1. (a) Schematic illustrating circular polarized beam launch onto the twisted graphene metasurface bilayer (TGMB). Geometry of the TGMB, consisting of two uniaxial metasurfaces with a dielectric spacer $h$ between them and a relative in-plane rotation $\Delta \theta = {\varphi _1} - {\varphi _2}$ (${\varphi _{1,2}}$ denote different rotation angles with respect to $x$). The background material $({{\varepsilon _1}})$ and spacer material $({{\varepsilon _2}})$ are assumed to be free space. (b) Schematic of the multiple reflection and transmission interference model in the TGMB.
2. MODEL AND FORMULAS
Figure 1 shows the schematic of circularly polarized beam launch onto the TGMB containing graphene nanoribbons, where $W$ denotes each graphene strip width and $G$ denotes the air gap separating neighboring strips. We assume deeply subwavelength periodicity $L = W + G \ll \lambda$, where $\lambda$ denotes the free-space wavelength. When this homogeneity condition is satisfied the metasurface response can be homogenized and the corresponding effective surface conductivity tensor can be analytically derived using effective medium theory as [27,38] $\hat \sigma = [{{\sigma _{\alpha \alpha}},0;\;0,{\sigma _{\beta \beta}}}]$, where ${\sigma _{\alpha \alpha}}= W{\sigma _G}/L$, ${\sigma _{\beta \beta}}= W{\sigma _G}{\sigma _c}/({L{\sigma _c} + {{W}}{\sigma _G}})$. Here ${\sigma _C} = - ({2i\omega {\varepsilon _0}L/\pi})\ln [{1/\sin ({\pi G/2L})}]$ denotes the effective strip conductivity, ${\varepsilon _0}$ denotes the free-space permittivity, $\omega = 2\pi f$ denotes the radial frequency, and ${\sigma _G}$ denotes the pristine graphene conductivity expressed by the Kubo formula [39]. Here $\beta$ and $\alpha$ denote the prime characteristic directions which are parallel and perpendicular to periodical directions, respectively. The TGMB can be formed by stacking two identical uniaxial metasurfaces [32,37], as shown in Fig. 1, separated by a dielectric gap with width $h$. The nature of the metasurface can be determined by the rotation angles ${\varphi _{1,2}}$ between the $x$ direction and the principal direction of each metasurface $\alpha$. If the incidence plane of a light is at a rotation angle ${\varphi _i}({i = 1,2})$ to the main axis of the metasurface, the rotated conductivity tensor can be expressed as [31]
A monochromatic Gaussian beam with wavelength $\lambda$ impinges onto the TGMB; it can be written as ${\tilde E_i} = ({{w_0}/\sqrt {2\pi}})\exp [{- w_0^2({k_{\textit{ix}}^2 + k_{\textit{iy}}^2})/4}],$ where ${w_0}$ is the beam waist, and ${k_{\textit{iy}}}$ and ${k_{\textit{ix}}}$ denote the wave vector components in the $y$ and $x$ directions, respectively. In the spin basis, the incident beam can be written as $\tilde E_i^V = i(1/\sqrt 2)({{{\tilde E}_{i -}} - {{\tilde E}_{i +}}})$ and $\tilde E_i^H = ({1/\sqrt 2})({{{\tilde E}_{i +}} + {{\tilde E}_{i -}}})$; here $V$ and $H$ present the vertical and horizontal polarization states, respectively. ${\tilde E_{i{ - }}}$ and ${\tilde E_{i{ + }}}$ are the right-handed circularly polarized (RCP) components and left-handed circularly polarized (LCP) components, respectively.
The light incident on the TGMB is not simply transmitted sequentially from the two metasurfaces but reflected and transmitted multiple times to form interference in the bilayers, which is similar to the Fabry–Perot (FP) cavity structure. Figure 1(b) shows the multiple transmission and reflection interference model that ignores the near-field coupling between metasurfaces. By matching the boundary condition, the transmitted light beams can be given as [40,41]
Based on the concept for extrinsic chirality [16], 3D chiral inclusions (as shown in Fig. 1) can be obtained by pairing two 2D graphene metasurfaces which twist each other. For the circularly polarized incident beam, the transmission coefficients can be obtained from [7,46]
3. RESULTS AND ANALYSIS
The calculation analysis and graphic simulation of this paper mainly use maple software. In Fig. 2(a) ${| {{t_{+ +}}} |^2}$ and ${| {{t_{{ - }\;{ - }}}} |^2}$ are used to represent the transmittance for the RCP and LCP waves, respectively. Figure 2(a) shows that the spectra of ${| {{t_{+ +}}} |^2}$ and ${| {{t_{{ - }\;{ - }}}} |^2}$ are obviously different, indicating the presence of 3D chirality. The CD, CB, and ellipticity of the TGMB can be calculated to further characterize the transmission difference. The larger the value for CD, the stronger the difference response of the LCP wave and RCP wave through the TGMB will be. As shown in Fig. 2, there exist the strong sign-changing resonance for CB and the maximum values for CD and ellipticity near the same resonant frequencies (about 10 THz and 15 THz).
Fig. 2. Chiral characteristics of the TGMB: (a) transmission spectra and CD spectra, (b) CB $\Phi$ spectra and ellipticity $\eta$ spectra. Some parameters are ${{L}} = {{50}}\;{\rm{nm}}$, ${{W}} = {{48}}\;{\rm{nm}}$, $\mu = 0.4\;{\rm eV}$, $h = 0.37\;\lambda$, ${\theta _i} = 0.0\;({\rm rad})$, $\Delta \theta = 1\;({\rm rad})$, and ${\varphi _1} = 1.4\;({\rm rad})$.
Figure 3(a) shows that there exists the periodic distribution of high CD in a wide range of frequency bands. As shown in Fig. 3(b), it can be seen that the transmission of the LCP wave has periodic peaks, which can be attributed to multiple transmission and reflection interference in the Fabry–Perot-like resonance system as shown in Fig. 1(b). In addition, it can be observed that there exists the periodic behavior for high CD. The corresponding phenomenon indicates that the extrinsic chirality can be enhanced by FP resonances in the TGMB. From Fig. 3(b), it can be seen that there exists the optimal separation distance at which the value of CD reaches the maximum, which is somewhat similar to that in twisted optical metamaterials composed of nanoinclusions [16].
Fig. 3. (a) Dependence of CD on the separation distance $h$ and frequency. (b) Dependence of CD and transmission on the separation distance $h$ for $f = 15\;{\rm THz}$. Some parameters are ${{L}} = {{50}}\;{\rm{nm}}$, ${{W}} = {{48}}\;{\rm{nm}}$, $\mu = 0.4\;{\rm eV}$, ${\theta _i} = 0.0\;({\rm rad})$, $\Delta \theta = 1\;({\rm rad})$, and ${\varphi _1} = 1.4\;({\rm rad})$.
As shown in Fig. 4(a), the CD values can be strongly affected by the chemical potential. It is worth noting that the corresponding phenomenon occurs in a large frequency band (about dozens of terahertz), which is very beneficial to the device application of the corresponding phenomenon, such as circular polarizers. It is worth noting that the chemical potential cannot be increased indefinitely but should be limited to a reasonable range ($0 \lt \mu\le 1\;{\rm eV}$), because an excessively large chemical potential makes the nonlocality caused by the intrinsic spatially dispersive response of the graphene strips more apparent, so that the formula for effective surface conductivity tensor mentioned above is no longer applicable and should be modified nonlocally [48]. Figure 4(b) shows the dependence of CD on the incident angle and frequency. It is clearly seen that in some frequency bands the CD property is robust against the incident angles changing over a wide range of incident angles about 1 (rad). Therefore, the proposed TGMBs will yield strong extrinsic chirality within a wide angle range and are more flexible for the polarization detection.
Fig. 4. (a) Dependence of CD on the chemical potential and frequency. (b) Dependence of CD on the incident angle ${\theta _i}$ and frequency. Some parameters are ${{L}} = {{50}}\;{\rm{nm}}$, ${{W}} = {{48}}\;{\rm{nm}}$, $h = 0.37\lambda$, ${\theta _i} = 0.0\;({\rm rad})$, $\Delta \theta = 1\;({\rm rad})$, and ${\varphi _1} = 1.4\;({\rm rad})$.
Figure 5(a) shows the twist-angle-dependent CD spectra of the TGMB. From Fig. 5(a) it can be found that the twist angle which is changed from 0 to $\pi$ strongly affects the corresponding resonant wavelength and the maximum value of CD. As one important physical feature for the corresponding structures, the change of the CD sign can be used to represent the different interaction between the left- and right-circularly polarized light and the corresponding structures [49]. From Fig. 5(b), it can be clearly seen that adjusting the twist angle of the TGMB can manipulate the CD sign and the shapes of the CD spectra, which is somewhat similar to the programming of CD spectra in twisted tri-layer graphene films with different symmetries [50]. Figure 5(c) shows that when the twist angle of the TGMB is an integer multiple of $\pi /2$, the transmittance of the LCP wave and RCP wave is symmetry along $n\pi /2$ ($n$ denotes integer), the period of transmittance is $\pi$, and the values of CD equal to zero. If we consider the monolayer graphene metasurface as a conductive surface with an elliptical conductivity magnitude in in-plane ($x$ and $y$ directions), it resembles a nanostructured planar metamaterial with dual rotational symmetry. When the twist angle of the TGMB is an integer multiple of $\pi /2$, the TGMB has symmetry in the out-of-plane ($z$ direction), and the mutual orientation of incident waves and the TGMB are the same from its mirror image. Then there exists this achiral phenomenon at this time. However, when the twist angle is not an integer multiple of $\pi /2$, the symmetry of the TGMB in the $z$ direction is broken, and the mutual orientation of incident waves and the TGMB are different from its mirror image, which leads to extrinsic chirality. From Fig. 5(c), it can be seen that there exist the maximum CD values near $\Delta \theta = ({2n + 1})\pi /4$ ($n$ denotes integer), corresponding to the largest extent of structure symmetry breaking.
Fig. 5. (a) Dependence of CD on the twist angle and frequency. (b) Dependence of CD on the frequency for different twist angle. Some parameters are ${{L}} = {{50}}\;{\rm{nm}}$, ${{W}} = {{48}}\;{\rm{nm}}$, $h = 0.37\lambda$, ${\theta _i} = 0.0\;({\rm rad})$, $\mu = 0.4\;{\rm eV}$, and ${\varphi _1} = 1.4\;({\rm rad})$.
Figure 6 shows the twist-angle-dependent CB and ellipticity spectra of the TGMB. Similar to the CD spectra in Fig. 5(a), the value of CB and ellipticity show nearly odd symmetrical distribution with respect to the twist angle of $\pi /2$ and $f \approx 12.8\;{\rm THz}$. From Fig. 6 it can be considered that through adjusting the twist angle of the TGMB, the shapes of the CB spectra and ellipticity spectra can be controlled. These interesting phenomena about optical chirality can be partly attributed to rotation-induced broken mirror symmetry [36]. The corresponding phenomenon here might be of importance to modulate terahertz and mid-infrared wave propagation, and can be used to develop novel highly efficient terahertz and mid-infrared polarization modulators.
Fig. 6. Dependence of (a) CB $\Phi$ and (b) ellipticity $\eta$ on the twist angle and frequency. Some parameters are ${{L}} = {{50}}\;{\rm{nm}}$, ${{W}} = {{48}}\;{\rm{nm}}$, $h = 0.37\lambda$, ${\theta _i} = 0.0\;({\rm rad})$, $\mu = 0.4\;{\rm eV}$, and ${\varphi _1} = 1.4\;({\rm rad})$.
It has been reported that when the spacer in the twisted metasurface bilayer is sufficiently thin ($h \ll \lambda$), the electromagnetic field resonances between the bilayer can be neglected and the twisted bilayer can be reduced to a monolayer [31]. Without accounting for the moiré superperiod dependence in the twisted bilayer [31], the effective conductivity tensor for the twisted bilayer can be written as a sum of the rotated conductivity tensors of the respectively two metasurfaces in Fig. 1:
Fig. 7. (a) Dependences of the effective conductivity ${\sigma _e} = ({4\pi /c}){\mathop{\rm Im}\nolimits} ({\tilde \sigma _{\alpha \alpha}})$ and CD on the frequency for different chemical potentials at $h = 0.01\lambda$, $\Delta \theta = 1.2\;({\rm rad})$, and ${\theta _i} = 0.0\;({\rm rad}).$ The red point denotes the topological transition $\sigma$-near-zero (TPSNZ) regime, and the black points denote the rotated-$\sigma$-near-zero (RSNZ) regimes. Solid lines denote CD and dashed lines denote ${\sigma _e}$. (b) Dependences of CD on the frequency for different thickness $h$ at $\mu = 1\;{\rm eV}$, $\Delta \theta = 1.2\;({\rm rad})$, and ${\theta _i} = 0.0\;({\rm rad}).$ (c) CD and ${| {{t_{\pm \pm}}} |^2}$changing with the twist angle for $h = 0.01\lambda$, $f = 12\;{\rm THz}$, $\mu = 1\;{\rm eV}$, and ${\theta _i} = 0.0\;({\rm rad}).$ (d) CD changing with the twist angle and incident angle ${\theta _i}$ for $h = 0.01\lambda$, $f = 12\;{\rm THz}$, and $\mu = 1\;{\rm eV}$. Some parameters are ${\rm{L}} = {{50}}\;{\rm{nm}}$, ${\rm{W}} = {{48}}\;{\rm{nm}}$, and ${\varphi _1} = 1.4\;({\rm rad})$.
Fig. 8. Dependences of (a) the effective conductivity, (b) the CB, and (c) ellipticity on the frequency for different chemical potentials at $\Delta \theta = 1.5\;({\rm rad})$. Some parameters are ${\rm{L}} = {{50}}\;{\rm{nm}}$, ${\rm{W}} = {{48}}\;{\rm{nm}}$, $h = 0.01\lambda$, ${\theta _i} = 0.0\;({\rm rad})$, and ${\varphi _1} = 1.4\;({\rm rad})$.
In Fig. 7(a), the red point denotes the TPSNZ regime where ${\mathop{\rm Im}\nolimits} ({{{\hat \sigma}_{\alpha \alpha (\beta \beta)}}})$ exhibits a sign-changing resonance. It has been reported that in the configurations with extrinsic chirality there might exist the additional $\sigma$-near-zero (SNZ) for the rotated effective conductivities given by effective surface conductivity tensor [28]. This scenario can be called the RSNZ regime in which ${\mathop{\rm Im}\nolimits} ({{{\hat \sigma}_{\alpha \alpha (\beta \beta)}}}) \approx 0$ [see the black points in Fig. 7(a)]. As the chemical potential increases, the positions of SNZ (both TPSNZ and RSNZ) move toward high frequency. Figure 7(a) shows that there exists one-to-one correspondence for the frequency values of RSNZ and the frequency values corresponding to the peaks of CD [see dotted lines in Fig. 7(a)]. Figure 7(b) shows that when $h \ll \lambda$, regardless of the spacer for the twisted metasurface bilayer change, the CD peak position is always at RSNZ. In the monolayer anisotropic 2DM system, to obtain CD one needs to use a substrate to break the corresponding symmetries [5,28,51]; however, here we do not need to use the substrate but use the twisted ultrathin bilayer to obtain CD whose maximum value can reach 0.13. Figure 7(c) shows that CD is zero when $\Delta \theta = n\pi /2$ ($n$ denotes integer), the transmittance of the LCP wave and RCP wave is symmetry along $n\pi /2$, and the period of transmittance is $\pi$. These features are similar to that for the CD and transmittance in monolayer black phosphorus (BP) [5], and the difference is that the periodic variable here is the twist angle for bilayer while it is the optical axis angle for monolayer BP in [5]. Figure 7(d) shows that the CD value decreases as the incident angle increases, and this phenomenon is the opposite of that in the monolayer BP where the CD value increases as the incident angle increases [5]. When the twist angle is large (except around $\pi /2$), the CD property has a certain degree of robustness to the incident angles changing [see Fig. 7(d)].
From Fig. 8, it can be found that there exist sign-changing resonance in $\Phi$ and deep troughs in $\eta$ at the frequency positions of TPSNZ which are affected by the chemical potential. This phenomenon is somewhat similar to the transmitted rotation resonance at the topological transition frequency of the monolayer metasurface based on graphene strips [28], but the resonance amplitude of the corresponding phenomenon here is not as large as that in [28]. It is worth noting that the corresponding phenomenon in Figs. 7 and 8 only exists in the case of $h \ll \lambda$. If the spacer in the twisted metasurface bilayer is not sufficiently thin, such as $h = 0.02\lambda$, the twisted bilayer cannot be reduced to a monolayer.
4. CONCLUSIONS AND DISCUSSION
In conclusion, we have studied theoretically the optical activity in the TGMB. It is found that the large CD value can be manipulated by various physical parameters of the TGMB (the separation distance, the chemical potential, and twist angle). By adjusting the twist angle of the TGMB, it is found that the CD spectra, CB spectra, and ellipticity spectra can be manipulated in the broadband range. When the twisted bilayer metasurfaces are stacked with an ultrathin spacer, there might exist the strong optical activity responses near the RSNZ regime and the TPSNZ regime. The corresponding phenomenon in this paper can be used to develop tunable chiral ultrathin devices which might find useful application in biomedicine, optical communication, and optoelectronics. Large CD can be generated because of the large light–matter interaction owing to the excitation of the plasmonic modes [26]. The giant optical activity can be obtained due to the electromagnetic coupling among the metal patterns [52–54]. The light–matter interaction is greatly enhanced by designing various metasurface pattern structures [55–58], utilizing plasmonic hybridization, synergistic effects of multiple resonance mechanisms, etc. Replacing the graphene nanoribbons in Fig. 1(a) with graphene metasurfaces with specific patterns, the enhanced electromagnetic field caused by the strong plasmonic behavior might further improve the CD value of the TGMB. In this paper, on the moderately wide bandwidth circular polarization selectivity can be supported in the TGMB after properly rotating one of the bilayers. It is conceivable that by stacking a large number of closely spaced graphene metasurfaces with proper sequential rotation and period, the bandwidth of the corresponding system might dramatically increase, similar to the bandwidth increase in a stack of rotated metamaterial metasurfaces [16]. Then the period and twist angle of the stack of graphene metasurfaces can be optimized to design a circular polarizer with the optimal operation. In addition, if the graphene metasurface in the stacking structure has the specific pattern structure, the strong resonance anisotropy of each metasurface might be transformed into magneto-electric coupling in the case of subwavelength stacking spacing, so as to broaden the bandwidth and meanwhile improve the CD value of the corresponding structure.
Funding
Natural Science Foundation of Fujian Province (2015J01219, 2017J01559, 2017J01560, 2018J01421); National Natural Science Foundation of China (11547254, 51277091, 61210016, 61575043).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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