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Abstract
We demonstrate the enhanced polarization modulation of electromagnetic fields through hybrid skew-ring-resonator-graphene meta-surfaces that can dynamically transform the linearly polarized waves into its cross-linearly polarized counterparts or the circularly polarized waves. Such a meta-surface consists of a grounded skew-ring resonator array inserted with a monolayer graphene sheet that controls the electromagnetic interactions between the skew-ring resonators and the ground. Especially, the reconfigurable characteristic of graphene enables the reflections to be capable of converting from the cross-linearly polarized fields to the circularly polarized waves by setting different Fermi energies with the same original co-linearly polarized incidence. Finally, we demonstrate that the bandwidth of the cross-polarization conversion would be greatly expanded when the monolayer graphene sheet is integrated with skew-bar-resonator meta-surfaces.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Meta-surfaces have been verified as promising candidates to manipulate polarizations of electromagnetic fields [1–5] where a linearly polarized (LP) wave could be converted to its cross-LP counterpart or the circularly polarized (CP) wave through redistributing the energy in the spatial domain, or re-assigning the amplitudes of co- and cross-polarized components equally with a fixed phase difference of $\pm$90 degrees.
Graphene, on the other hand, is more agility to manipulate electromagnetic fields on the merits of the complex surface conductivities [6–8] and tunable field effects [9–12]. Especially, the meta-surfaces can fulfill the reconfigurable design of polarization conversions of electromagnetic fields when integrated with graphene, where the typical way is to replace the metallic meta-atoms by graphene patches and perform the polarization manipulations through prescribing specific Fermi energy for each graphene unit. These designs have demonstrated the valid conversions between the co- and cross-LP electromagnetic fields [13–17], as well as the perfect transformations between the LP and the CP travelling waves [18,19]. However, such a strategy is normally complex in the practical implementation as we have to build up a feeding network to manipulate every graphene patch of the meta-surfaces, which makes it difficult in both design and fabrication. An alternative proposal is to construct the complementary hollowed-out graphene meta-surfaces where a static electricity bias is applied to the entire graphene layer for the required electromagnetic parameters and such a design is also shown to be capable of dynamically tuning the polarization states of electromagnetic fields in a desired manner [20,21].
In the meanwhile, hybrid resonator-graphene meta-surfaces, integrating the periodic meta-resonators with a whole layer of graphene sheet, offers a much more efficient way to control of electromagnetic fields in the practical implementation. Such devices have been widely applied to the transmittance adjustments [22–27], reflectance adjustments [28–30] and the phase modulations [31–33] of electromagnetic fields. And it is also qualified to tune the polarization states of the electromagnetic fields [34–36]. The graphene layer in these hybrid structures are usually inserted between the meta-atoms and the dielectric base directly or laid over closely upon the meta-resonators, where a unified Fermi energy is imposed to the whole graphene sheet that can manipulate the overall performance of such hybrid resonator-graphene meta-surfaces. However, the respective functionalities of the graphene sheet and the periodic meta-surface resonators are still unclear when under the illumination of electromagnetic fields. Especially, the electromagnetic interactions between the graphene sheet and the periodic meta-surface resonators are lack of theoretical explorations to reveal the fundamental connections. In this paper, we demonstrate dynamically tuning polarizations of electromagnetic fields based on hybrid skew-resonator-graphene meta-surfaces, where by tuning the fermi energy imposed on the graphene sheet, the polarization conversion effects of the meta-surfaces would be adjusted continuously. More specifically, such meta-surfaces consist of a grounded skew-ring resonator array inserted with a monolayer graphene sheet in the middle of the dielectric layer, and the reconfigurable characteristics of such a graphene layer enable the reflections to be capable of converting from the cross-LP fields to CP waves by setting different Fermi energies with the same original co-LP incidence. Especially, we illustrate the generalized signal flows between the skew-ring arrays and the ground, and demonstrate that the graphene sheet would fulfill the control of the electromagnetic interactions within such hybrid structures and can thus determine the polarization states of reflections. Finally, we will show that the bandwidth of the cross polarization conversion would be greatly extended when the skew-ring is replaced by the skew-dual-bar resonator.
2. Hybrid skew-ring-resonator-graphene meta-surfaces
Figure 1 illustrates the configuration of the hybrid skew-ring-resonator-graphene meta-surfaces under the illumination of LP electromagnetic fields, where the polarization states of reflections are controllable, either appearing to be the cross LP fields or presenting as the CP waves. The meta-surfaces consist of periodical skew-rings and a ground separated by PTFE dielectric spacer ($\varepsilon _s = 2.65$) with a mono-layer graphene inserted in the middle of it. The embedded picture demonstrates detailed structural information of 45 $^\circ$-skew-ring resonator, and the geometrical parameters are chosen to be $h = 25\,\mu \textrm {m}$, $L = 65\,\mu \textrm {m}$, $a = 20\,\mu \textrm {m}$, $b = 58\,\mu \textrm {m}$, and $w = 4.5\,\mu \textrm {m}$ in this investigation.
Fig. 1. Schematic view of hybrid skew-ring-resonator-graphene meta-surfaces. The ring array on the top and the ground at the bottom are assumed as perfect electric conductors with zero thickness. The mono-layer of graphene is integrated in the middle of the dielectric spacer made of PTFE with $\varepsilon _s = 2.65$. The embedded picture refers to the geometric dimensions of the skew-ring structure of meta-surface.
We carry out the full-wave simulations (COMSL studio) to demonstrate polarization conversion from such hybrid skew-ring-resonator-graphene meta-surfaces with an incidence of a LP electromagnetic fields in $x$-direction. The reflection coefficients of the co- or the cross-LP waves can be defined as ${\tilde R_{xx/yx}} \equiv {R_{xx/yx}}\exp \left ( {j{\Phi _{xx/yx}}} \right ) = {{E_{x/y}^{ref}} \mathord {\left / {\vphantom {{E_{x/y}^{ref}} {E_x^{inc}}}} \right.} {E_x^{inc}}}$ , and we employ the normalized Stokes parameters to fully describe the polarization conversion state. More specifically, the polarization state of the LP reflections relate to ${S_1} = {{\left ( {{{\left | {{E_x}} \right |}^2} - {{\left | {{E_y}} \right |}^2}} \right )} \mathord {\left / {\vphantom {{\left ( {{{\left | {{E_x}} \right |}^2} - {{\left | {{E_y}} \right |}^2}} \right )} {\left ( {{{\left | {{E_x}} \right |}^2} + {{\left | {{E_y}} \right |}^2}} \right )}}} \right.} {\left ( {{{\left | {{E_x}} \right |}^2} + {{\left | {{E_y}} \right |}^2}} \right )}}$ where the values of 1 and −1 refer to the transformation into the co- and cross-LP electromagnetic fields respectively from the original $x$-LP incidence. In the meanwhile, the CP reflections can be defined by ${S_3} = {{\left ( {{{\left | {{E_r}} \right |}^2} - {{\left | {{E_l}} \right |}^2}} \right )} \mathord {\left / {\vphantom {{\left ( {{{\left | {{E_r}} \right |}^2} - {{\left | {{E_l}} \right |}^2}} \right )} {\left ( {{{\left | {{E_r}} \right |}^2} + {{\left | {{E_l}} \right |}^2}} \right )}}} \right.} {\left ( {{{\left | {{E_r}} \right |}^2} + {{\left | {{E_l}} \right |}^2}} \right )}}$ where the values of 1 and −1 thus refer to the conversions into the right CP (RCP) and left CP (LCP) fields from the original $x$-LP incidence. We can observe that the ${S_1}$ in Fig. 2(a) would decrease as the frequency goes up at 0 eV Fermi energy of graphene sheet, and then starts to experience some ascent when the frequency is beyond 2.74 THz. As a result, we can have a wideband cross polarization conversion performance with ${S_1}$ below −0.9 from 2.24 to 2.74 THz, and the ${S_1}$ would reach nearly −1 at 2.3 THz demonstrating very pure cross LP reflections. We can also observe that the bandwidth of the cross polarization conversion shrinks with increased minimum of $S_1$ as the Fermi energy increases. On the other hand, the ${S_3}$ in Fig. 2(b) presents opposite trends with expanded bandwidth of the RCP conversion as the Fermi energy increases. Especially, more than 90 % of the energy from the incident wave is converted to the RCP field over the frequency range from 2.06 to 2.60 THz when Fermi energy of graphene sheet is set as 1.0 eV, and can achieve almost perfect conversions from $x$-LP to RCP waves with 0.997 of $S_3$ at 2.3 THz. Figure 2(c) and Fig. 2(d) demonstrate the amplitude and phase difference $\Delta \Phi = \arg \left ( {{R_{xx}}} \right ) - \arg \left ( {{R_{yx}}} \right )$ of the reflection coefficients from the hybrid skew-ring-resonator-graphene meta-surfaces. We can observe that the amplitude of the reflected cross-LP field is much greater than the one of the reflected co-LP wave from 2.24 to 2.74 THz when the graphene sheet is imposed with 0.0 eV Fermi energy, and the co- and cross-LP component of the reflections would become identical at 2.3 THz with 90-degree phase difference forming the RCP fields when graphene sheet is having 1.0 eV Fermi energy. These results agree quite well with the ${S_1}$ and ${S_3}$ from the hybrid skew-ring-resonator-graphene meta-surfaces and demonstrate the great capacities of polarization conversion. There are two resonances for the hybrid skew-ring-resonator-graphene meta-surfaces at 2.3 THz and 2.68 THz respectively when imposed with the Fermi energy of 0.0 eV. The corresponding field distribution of the meta-surfaces and the flow directions are demonstrated in Figs. 2(e) and (f) respectively. Clearly, such resonances come from the corner part and the long edge of the ring array respectively, as the corresponding electric fields at 2.3 THz are strongest at such certain parts over the ring. On the other hand, such a device can provide a continuous tuning of the polarization states as illustrated in Fig. 3(a). We can observe that the value of ${S_1}$ goes up from −1 to 0 basically as the Fermi energy increases, while the ${S_3}$ turns out to have the trend running from 0 to 1 accordingly when the Fermi energy grows. In addition, the Stokes parameter of ${S_2} = {{\left ( {{{\left | {{E_{ + 45^\circ }}} \right |}^2} - {{\left | {{E_{ - 45^\circ }}} \right |}^2}} \right )} \mathord {\left / {\vphantom {{\left ( {{{\left | {{E_{ + 45^\circ }}} \right |}^2} - {{\left | {{E_{ - 45^\circ }}} \right |}^2}} \right )} {\left ( {{{\left | {{E_{ + 45^\circ }}} \right |}^2} + {{\left | {{E_{ - 45^\circ }}} \right |}^2}} \right )}}} \right.} {\left ( {{{\left | {{E_{ + 45^\circ }}} \right |}^2} + {{\left | {{E_{ - 45^\circ }}} \right |}^2}} \right )}}$ experiences very few changes as the Fermi energy varies, indicating there are no $45^\circ$-inclined LP waves induced from such a hybrid skew-ring-resonator-graphene meta-surfaces. The corresponding Stokes parameters are thus illustrated in Fig. 3(b) in the Poincare sphere. We can observe that the hybrid skew-ring-resonator-graphene meta-surfaces enable the dynamically tuning of polarizations of the electromagnetic fields by switching from a cross polarization conversion to a linear-to-circular polarization conversion with different Fermi energies.
Fig. 2. The polarization conversion performances from the hybrid skew-ring-resonator-graphene meta-surfaces. The Stokes parameters $S_1$ (a) and $S_3$ (b) versus frequency when the graphene sheet is imposed with different Fermi energies. The amplitude of the reflection coefficient $R_{xx/yx}$ and phase difference $\Delta \Phi$ versus the frequency when the graphene sheet is having the Fermi energy of (c) 0.0 eV and (d)1.0 eV. The $z$-component of the electic fields on the top layer of the hybrid skew-ring-resonator-graphene meta-surfaces at the resonance frequencies of (e) 2.3 THz and (g) 2.68 THz. The electric fields are normaliez by $2\times 10^6\ \rm V/m$. The streamlines refer to the corresponding flow directions of the electric fields.
Fig. 3. Dynamically tuning the polarizations of electromagnetic fields by the hybrid skew-ring-resonator-graphene meta-surfaces. (a) Normalized Stokes parameters versus the Fermi energy. (b) The demonstration of the corresponding polarization states of the reflections on the Poincare sphere.
Figure 4 illustrates the general signal flow of such a hybrid skew-ring-resonator-graphene meta-surfaces to reveal the polarization manipulation of electromagnetic fields. The whole meta-surfaces can be considered as a cascade multi-layered structure with each layer modelled to be a generalized two port network. Therefore, tuning of the sub-network characteristics of the graphene layer may well change the whole responses of the entire meta-surfaces system. More specifically, the layer-I consists of the skew-ring and the half thick PTFE dielectric spacer. The layer-II is composed of a mono-layer of graphene sheet controlling the interactions between the skew-ring and the ground, offering the reconfigurable designs of the polarization conversion. And the layer-III consists of another half thick PTFE dielectric substrate spacer and the ground. The total reflection matrix can thus be expressed as follows
Fig. 5. Electromagnetic responses of each layer of the hybrid skew-ring-resonator-graphene meta-surfaces over the frequency range from 2 to 3 THz. The amplitudes (a) and phase (b) of co-reflection/transmission coefficient $\tilde R_{co}^{I1}/\tilde T_{co}^{I1}$ and cross-reflection/transmission coefficient $\tilde R_{cr}^{I1}/\tilde T_{cr}^{I1}$. The amplitudes (c) and phase (d) of co-reflection/transmission coefficient $\tilde R_{co}^{I2}/\tilde T_{co}^{I2}$ and cross-reflection/transmission coefficient $\tilde R_{cr}^{I2}/\tilde T_{cr}^{I2}$. The amplitude and the phase of the reflection/transmission coefficient ${\tilde R^G}/{\tilde T^G}$ with (e) 0.0 eV and (f) 1.0 eV Fermi energy imposed on the graphene sheet. The analytical results of the reflectance ${R_{xx/yx}}$ and $\Delta \Phi$ with (g) 0.0 eV and (h)1.0 eV Fermi energy imposed on the graphene sheet.
The layer-II consists of the mono-layer of graphene in the PTFE dielectrics, where the graphene sheet could be modelled as an infinity thin layer characterized by the surface conductivity tensor $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over \sigma } \left ( {\omega ,{\mu _c},\Gamma ,\textrm {T}} \right ) = \hat x\hat x\sigma _{xx} + \hat x\hat y{\sigma _{xy}} + \hat y\hat x{\sigma _{yx}} + \hat y\hat y{\sigma _{yy}}$ . The conductivity should thus be in a scalar form [37], as we only employ the electrostatic bias to control the character of graphene layer
The scattering matrix of layer-II would be
As for the layer-III, the ground and the substrate form a co-LP reflector with the reflection coefficient of
3. Hybrid skew-bar-resonator-graphene meta-surfaces
We continue to demonstrate another polarization conversion meta-surfaces with skew-dual-bar resonators by removing two short sides (the red dashed rectangular region) of the corresponding ring resonators, as shown in Fig. 6(a). Such hybrid skew-bar-resonator-graphene meta-surfaces possess the merits of broadband cross polarization conversion. Figure 6(b) shows that the bandwidth of cross polarization conversion would reach 73.7 % with the $S_1$ below −0.9 from 1.31 to 2.84 THz when the graphene sheet is imposed with 0.2 eV Fermi energy. In the meanwhile, we can also observe in Fig. 6(c) that the ${S_3}$ reaches 0.992 at 1.8 THz when Fermi energy of the graphene sheet is set as 1.0 eV indicating a perfect conversion from LP wave to RCP field. These results are also verified by both theoretical results and the full wave simulations, where the varying trends of ${\tilde R_{xx}}$ and ${\tilde R_{yx}}$ as illustrated in Figs. 6(d) to (g) agree well with the ${S_1}$ and ${S_3}$. Such a broadband cross linear polarization conversion comes from the multiple resonant behaviors of the skew-dual-bar resonator array with three resonances at 1.41 THz, 2 THz and 2.75 THz, while the skew-ring resonator only has two resonances at 2.3 THz and 2.68 THz. Clearly, the resonant mode at 1.41 THz in Fig. 6(h) would firstly come from the dual-bar array for the interactions between the adjacent unit cells. In addition, the mutual coupling between the dual bars would bring in another resonant mode at 2 THz as shown in Fig. 6(i). Finally, the skew-bar itself would introduce the resonant mode at 2.75 THz as shown in Fig. 6(j). In the same way as the hybrid skew-ring-resonator-graphene meta-surfaces, it can also provide a continuous tuning of the polarization states as illustrated in Fig. 7(a). We can observe that the $S_1$ and $S_3$ will run from −1 to 0 and 0 to 1 respectively as the Fermi energy grows. The graphene sheet imposed with 0.74 eV Fermi energy will lead to the strongest energy loss from such hybrid skew-bar-resonator-graphene meta-surfaces and such an absorption will degrade both the cross linear polarization conversion and linear-to-circular polarization conversion as $S_1$ and $S_3$ are both small. In the meanwhile, $S_2$ reaches the peak when setting the Fermi energy to be 0.74 eV and the absolute values of $S_1$ and $S_3$ are almost identical. Hence, the reflections of the electromagnetic fields in such a scenario would be much more closed to the elliptical polarization. The Fig. 7(b) thus illustrates the corresponding Stokes parameters on the Poincare sphere where we can observe that such hybrid skew-bar-resonator-graphene meta-surfaces are capable of dynamically tuning the polarizations of electromagnetic fields from the cross linear polarization transformation to the right hand circular polarization conversion from the original LP incidence.
Fig. 6. Polarization conversions from the hybrid skew-bar-resonator-graphene meta-surfaces. (a) The configuration of the hybrid skew-bar-resonator-graphene meta-surfaces by removing the two short bars (the red dashed rectangular region) from the hybrid skew-ring-resonator-graphene meta-surfaces. The Stokes parameters $S_1$ (b) and $S_3$ (c) versus frequency when the graphene sheet is imposed with different Fermi energies. (d) and (e) The theoretical and numerical results of the reflectance and $\Delta \Phi$ versus the frequency at Fermi energy of 0.2 eV. (f) and (g) The theoretical and numerical results of the reflectance and $\Delta \Phi$ versus the frequency at Fermi energy of 1.0 eV. The $z$-component of the electric fields on the top layer of the hybrid skew-bar-resonator-graphene meta-surfaces at (h) 1.41 THz, (i) 2 THz, and (j) 2.75 THz when the grapheme sheet is imposed with 0.2 eV Fermi energy. The electric fields are normalized by $4.5\times 10^6\ \rm V/m$, $1\times 10^6\ \rm V/m$ and $2\times 10^6\ \rm V/m$ respectively. The streamlines refer to the corresponding flow directions of the electric fields.
Fig. 7. Dynamically tuning the polarizations of electromagnetic fields by the hybrid skew-bar-resonator-graphene meta-surfaces. (a) Normalized Stokes parameters versus the Fermi energy. (b) The demonstration of the corresponding polarization states of the reflections on the Poincare sphere.
4. Conclusions
In conclusion, we have demonstrated the dynamically tuning the polarizations of electromagnetic fields based on the hybrid skew-resonator-graphene meta-surfaces consisting of a grounded skew resonator array inserted with a monolayer graphene sheet. The reconfigurable characteristics of the graphene layer enable the meta-surfaces to switch from a cross polarization conversion to a linear-to-circular polarization conversion with different Fermi energies. Furthermore, we have also illustrated the generalized signal flows between the skew-ring arrays and the ground, and revealed the electromagnetic interactions within such hybrid structures. It is noted that such a dynamically tuning of polarizations using hybrid skew-resonator-graphene meta-surfaces are also valid for the general inputs of electromagnetic fields with different polarization states. We can observe that in Figs. 8(a) and (b), the reflections from the hybrid skew-ring-resonator-graphene meta-surfaces would experience a cross circular polarization conversion when the graphene sheet is imposed with 0.0 eV or a circular-to-linear polarization conversion when given the Fermi energy of 1.0 eV under the illumination of the LCP incidence at 2.3 THz. On the other hand, we can also observe that in Figs. 8(c) and (d), the reflections from the hybrid skew-bar-resonator-graphene meta-surfaces can fulfill a cross circular polarization conversion when the graphene sheet is imposed with 0.2 eV or a circular-to-linear polarization transformation with 1.0 eV Fermi energy under the illumination of the LCP incidence at 1.8 THz. In addition, the Stokes parameter $S_3$ in such a case is all above 0.9 from 1.31 THz to 2.84 THz when we impose the 0.2 eV Fermi energy on the graphene sheet and thus possesses the wideband cross circular polarization conversion with 73.7 % bandwidth, which is similar to the cases under the illumination of the $x$-LP electromagnetic fields for the hybrid skew-bar-resonator-graphene meta-surfaces. We expect our design of hybrid skew-resonator graphene meta-surfaces pave the way for the reconfigurable design of polarization conversions of electromagnetic fields, and offer an efficient way to characterize electromagnetic behaviors of such hybrid meta-surfaces with an exploration of the analytical model.
Fig. 8. The polarization conversion performances of the hybrid skew-resonator-graphene meta-surfaces under the illumination of LCP electromagnetic fields. (a) Normalized Stokes parameters versus the frequency of the hybrid skew-ring-resonator-graphene meta-surfaces. (b) The polarization states of the reflections from the hybrid skew-ring-resonator-graphene meta-surfaces on the Poincare sphere. (c) Normalized Stokes parameters versus the frequency of the hybrid skew-bar-resonator-graphene meta-surfaces. (d) The demonstration of the corresponding polarization states of the reflections of the hybrid skew-bar-resonator-graphene meta-surfaces on the Poincare sphere.
Funding
National Natural Science Foundation of China (61671344, 61301072).
Disclosures
The authors declare no conflicts of interest.
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