Reconfigurable coherent perfect absorption of different polarized electromagnetic fields through vanadium-dioxide based asymmetry transmissive meta-surfaces sandwiched with graphene meta-gratings

Shanfang Zhang; Rui Yang

doi:10.1364/OE.504084

1. Introduction

Graphene meta-surfaces with great capacities of coherent perfect absorptions (CPAs) of electromagnetic fields, have been widely applied to the development of optical sensing, optoelectronic devices, light-to-light control, and signal modulation [1–4]. Zhang et al. experimentally demonstrated electrically tunable broadband CPA based on a graphene-dielectric-graphene structure, and dynamically modulated the coherent absorptivity from 50${\% }$ to 100${\% }$ by controlling the Fermi level of graphene [5]. Meng et al. realized a tunable and ultrathin five-band CPA based on graphene and a high-contrast grating, achieving multi- and narrow-band CPA under large-angle incidence [6]. Li et al. illustrated the multi-band CPAs in the mid-infrared region through a vertically stacked metal-dielectric-graphene heterostructure, and the CPA wavelengths were dynamically tuned by changing the gate-dependent chemical potential of graphene [7]. These coherent absorbing meta-surfaces with tunable operating frequencies and controllable absorption efficiency have all achieved the desired wave extinction of incoming radiations [8–18]. However, the present CPAs are only applied to the same polarized control wave and the signal wave with identical amplitude and the 180-degree phase differences [19–29]. So far, the current theory is still limited to the cases where the signal and control waves have the same polarization. On the other hand, the interferences between different polarized signal waves and control waves will deal with both polarized reflections and transmissions at the same time. Especially, the scattering matrix should also include the polarization conversion components of the dual incidences in both reflection and transmission regimes. As a result, challenges will emerge when the design is required to achieve the CPAs for both the x-polarized and the y-polarized reflections and transmissions simultaneously. However, if a meta-surface can utilize just one polarized control wave to interfere with both co- and cross-linearly polarized (LP) waves, it should greatly facilitate the signal modulation and promote the integration of multipolar devices. In this way, a big progress in this research area should be achieved by employing just one polarized control wave to perfectly trap both the co-polarized and cross-polarized signal fields, but still, it is noteworthy that polarization conversion should be the very first step to perform before conducting the cross-polarized CPA.

Asymmetry transmissive (AT) meta-surfaces [30–34] capable of simultaneously regulating different polarized waves should provide an effective way to fulfill the multi-polarized CPA. However, it will be more challenging to achieve the CPAs of co-LP and cross-LP waves simultaneously in the same frequency band, as the control wave should always be updated when the signal changes in the reciprocal system. Therefore, it is necessary to introduce additional control variables to break the original reciprocal system with the constant control wave. VO$_{2}$ is a well-known phase change material and undergoes a reversible insulator-to-metal transition upon heating [35–37], where the insulator-to-metal transition in VO$_{2}$ occurs at a temperature of Tc$\approx$340 K. The transition from the conducting state to the insulating state of VO$_{2}$ can be used to change the structural properties of the meta-surface to achieve multi-polarization control, and the regulation of the Fermi level of graphene to control the absorption efficiency. Based on these considerations, we demonstrate the reconfigurable CPAs of different polarized electromagnetic fields through VO$_{2}$ based AT meta-surfaces sandwiched with graphene meta-gratings. We will show that either the x- or the y-polarized signal fields can be perfectly trapped when interacting with the x-polarized control wave by imposing specific Fermi level over the graphene together with conducting- or insulating-state VO$_{2}$ at 3 THz. On the other hand, the y-polarized control wave will also be capable of achieving the coherent cancellation with both LP signals at 3.65 THz with the assistance of tunable materials of VO$_{2}$ and graphene [38–45]. This paper is organized as follows: in Section 2, the design principles and simulation results of co-polarized and cross-polarized CPAs from VO$_{2}$-metal-graphene meta-surface are demonstrated. We start to present the principles and simulation results for co-polarized and cross-polarized CPAs. After that, the underlying principle and transmission coefficients for realizing different polarized CPAs with the proposed meta-surface are demonstrated. We analyze the dynamic control of the absorption rate for co-polarized and cross-polarized CPAs, and discuss the influence of structural parameters on the absorptivity. Conclusions are finally drawn in Section 3. Our approach, using AT meta-surfaces for more advanced coherent control of different LP electromagnetic fields with the assistance of tunable materials, should pave the way for building up multipolar and multifunctional absorbers.

2. Modeling and simulation results

Figure 1 demonstrates the reconfigurable CPAs of different polarized electromagnetic fields through the proposed meta-surfaces. At 3 THz, co-polarized CPA between the x-polarized control wave and x-polarized signal wave is attained with conducting-state VO${_2}$ and graphene Fermi level of 1 eV, as depicted in Fig. 1(a). On the other hand, cross-polarized CPA between the x-polarized control wave and y-polarized signal wave is also achieved as shown in Fig. 1(b), when the VO${_2}$ is tuned to be insulating-state by temperature control and the graphene Fermi level is fixed at 0.7 eV by applying a bias voltage. Additionally, at 3.65 THz, as shown in Fig. 1(c), the cross-polarized CPA between the y-polarized control wave and the x-polarized signal wave is accomplished when the VO${_2}$ is controlled to be insulating-state by temperature and the Fermi level of graphene is tuned to be 0.2 eV by applying a bias voltage. Meanwhile, co-polarized CPA is achieved between the y-polarized control and y-polarized signal waves, as depicted in Fig. 1(d), under the conditions of conducting-state VO${_2}$ and graphene Fermi level of 0.1 eV. Figure 1(e) demonstrates the detail information of the meta-cell, consisting of two outermost VO${_2}$-metal layers and symmetrical graphene meta-gratings, as well as an intermediate C-slit metallic layer. More specifically, we have VO${_2}$ partially filled in pound-key-slit array etched in the metallic outermost layer. The thermal controlled conducting- or insulating-state will determine the wave transmission. In the insulating state, VO${_2}$ has a dielectric constant of 9 and a conductivity of less than 200 S/m, while in the conducting state, its conductivity will reach 10${^5}$ S/m. The dielectric material used is polyimide, which has a dielectric constant of 2 + 0.025i.

Fig. 1. Schematic diagram of reconfigurable CPAs of different polarized electromagnetic fields through the proposed meta-surfaces. (a) CPA of x-polarized signal wave and x-polarized control wave at 3 THz. (b) CPA of y-polarized signal wave and x-polarized control wave at 3 THz. (c) CPA of x-polarized signal wave and y-polarized control wave at 3.65 THz. (d) CPA of y-polarized signal wave and y-polarized control wave at 3.65 THz. (e) Unit structure diagram. The structural parameters are p = 40 $\mu$m, t = 2 $\mu$m, d$_{1}$ = 0.5 $\mu$m, d$_{2}$ = 1.25 $\mu$m, L$_{1}$ = 35.5 $\mu$m, k = 4 $\mu$m, L$_{2}$ = 27 $\mu$m, D$_{1}$ = 6.5 $\mu$m, D$_{2}$ = 5 $\mu$m, $\alpha$ = $45^\circ$, $\beta$ = $45^\circ$. (f) Schematic diagram showing the principle of multi-polarized CPAs.

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The inner layer consists of periodic 45-degree skew graphene gratings, together with the intermediate C-slit metallic layer, performing the polarization conversion of the transmitting fields. Considering the potential implementation, the DC voltage of ${{\rm {V}}_{\rm {g}}}$ can be applied between a 40 nm thick polycrystalline silicon and the graphene gratings with a 10 nm thick ${\rm {A}}{{\rm {l}}_2}{{\rm {O}}_3}$ spacer in between [38], to effectively tune the Fermi level imposed over the graphene [39,40]. The polycrystalline silicon layer is deposited on the substrate, and the thin ${\rm {A}}{{\rm {l}}_2}{{\rm {O}}_3}$ is fabricated on it. Finally, the graphene layer can be transferred onto the insulating layer and patterned by electron-beam lithography [41]. The influences of the thin polycrystalline silicon layer and thin ${\rm {A}}{{\rm {l}}_2}{{\rm {O}}_3}$ layer on the device response can be neglected.

The surface conductivity of graphene $\sigma$ consists of two components, intraband conductivity ${\sigma _{{\mathop {\rm int}} ra}}$ and interband conductivity ${\sigma _{{\mathop {\rm int}} er}}$, according to the Kubo formula [42]:

(1)$$\sigma = {\sigma _{{\mathop{\rm int}} ra}} + {\sigma _{{\mathop{\rm int}} er}},$$

(2)$${\sigma _{{\mathop{\rm int}} ra}} ={-} j\frac{{{e^2}{k_B}T}}{{\pi {\hbar ^2}(\omega - j2\Gamma )}}\left( {\frac{{{\mu _c}}}{{{k_B}T}} + 2\ln ({e^{\frac{{ - {\mu _c}}}{{{k_B}T}}}} + 1)} \right),$$

(3)$${\sigma _{{\mathop{\rm int}} er}} = \frac{{ - je}}{{4\pi \hbar }}\ln \left( {\frac{{2\left| {{\mu _c}} \right| - (\omega - j2\Gamma )\hbar }}{{2\left| {{\mu _c}} \right| + (\omega - j2\Gamma )\hbar }}} \right)$$

where $e$ is the electron charge, ${k_B}$ is the Boltzmann constant, T = 300 K represents temperature, $\hbar$ is the reduced Planck constant, $\omega$ is the electromagnetic wave angular frequency, $\Gamma = \hbar /(2\tau )$ is the scattering rate, $\tau$=0.1ps is the relaxation time, ${\mu _c}$ is the Fermi level. The temperature variation exhibits a relatively very small influence on the surface conductivity properties of graphene. When$\left | {{\mu _c}} \right | \gg {k_B}T$ , the surface conductivity of graphene can be simplified to a Drude model [43]:

(4)$$\sigma \approx \frac{{{e^2}{\mu _c}}}{{\pi {{\hbar}^2}}}.\frac{j}{{\omega + j/\tau }}$$

The relationship between the graphene Fermi level and the applied voltage is [44]:

(5)$${\mu _c} \approx \hbar {v_f}\sqrt {\frac{{\pi {\varepsilon _0} \cdot {\varepsilon _r}{v_g}}}{{e} \cdot {t}}}$$

where ${v_f}$= 1.1*$10^6$ m/s is the Fermi velocity, ${v_g}$ is the applied bias voltage, ${\varepsilon _0}$ and ${\varepsilon _r}$ are the permittivity of vacuum and the dielectric layer, respectively, and $t$ is the thickness of the dielectric layer.

Figure 1(f) schematically demonstrates the principle of multi-polarized CPAs through the proposed meta-surfaces. For an arbitrarily polarized signal wave, it can be considered as a superposition of x-polarized and y-polarized incident components, denoted as ${{\text {I}}_{\text {P }\!\!\_\!\!\text { x}}}$ and ${{\text {I}}_{\text {P }\!\!\_\!\!\text { y}}}$ from the P direction. Similarly, for an arbitrarily polarized control wave, it can also be considered as a superposition of x-polarized and y-polarized incident components, denoted as ${{\text {I}}_{\text {N }\!\!\_\!\!\text { x}}}$ and ${{\text {I}}_{\text {N }\!\!\_\!\!\text { y}}}$ from the N direction. Likewise, ${{\text {O}}_{\text {P }\!\!\_\!\!\text { x}}}$,${{\text {O}}_{\text {P }\!\!\_\!\!\text { y}}}$, ${{\text {O}}_{\text {N }\!\!\_\!\!\text { x}}}$,${{\text {O}}_{\text {N }\!\!\_\!\!\text { y}}}$ refer to the x-polarized component in the P direction, y-polarized component in the P direction, x-polarized component in the N direction, and y-polarized component in the N direction of the output electromagnetic waves, respectively. The relationship between the incidences and the corresponding transmitted and reflected waves can be described by the following scattering matrix [45]:

(6)$${\left[ \begin{array}{c} O_{P\_x}\\ O_{P\_y}\\ O_{N\_x}\\ O_{N\_y}\\ \end{array} \right]}=\mathbf{\hat{S}}{\left[ \begin{array}{c} I_{P\_x}\\ I_{P\_y}\\ I_{N\_x}\\ I_{N\_y}\\ \end{array} \right]},\qquad \mathbf{\hat{S}}={\left[ \begin{array}{cccc} r_{P\_xx} & r_{P\_yx} & t_{P\_xx} & t_{P\_yx}\\ r_{P\_xy} & r_{P\_yy} & t_{P\_xy} & t_{P\_yy}\\ t_{N\_xx} & t_{N\_yx} & r_{N\_xx} & r_{N\_yx}\\ t_{N\_xy} & t_{N\_yy} & r_{N\_xy} & r_{N\_yy}\\ \end{array} \right]}$$

where $r_{P/N\_xx}$ represents the reflection coefficient for $x$-polarized incidence converted to the $x$-polarized reflection in the $P/N$ direction, $r_{P/N\_yx}$ represents the reflection coefficient for $y$-polarized incidence converted to the $x$-polarized reflection in the $P/N$ direction, $t_{P/N\_xx}$ represents the transmission coefficient for $x$-polarized incidence converted to the $x$-polarized transmission in the $P/N$ direction, $t_{P/N\_yx}$ represents the transmission coefficient for $y$-polarized incidence converted to the $x$-polarized transmission in the $P/N$ direction, $r_{P/N\_xy}$ represents the reflection coefficient for $x$-polarized incidence converted to the $y$-polarized reflection in the $P/N$ direction, $r_{P/N\_yy}$ represents the reflection coefficient for $y$-polarized incidence converted to the $y$-polarized reflection in the $P/N$ direction, $t_{P/N\_xy}$ represents the transmission coefficient for $x$-polarized incidence converted to the $y$-polarized transmission in the $P/N$ direction, and $t_{P/N\_yy}$ represents the transmission coefficient for $y$-polarized incidence converted to the $y$-polarized transmission in the $P/N$ direction.

Based on the scattering matrix, the coherent absorptivity $A$ can thus be written as:

(7)$$A=1-\frac{|O_{P\_x}|^{2}+|O_{P\_y}|^{2}+|O_{N\_x}|^{2}+|O_{N\_y}|^{2}}{|I_{P\_x}|^{2}+|I_{P\_y}|^{2}+|I_{N\_x}|^{2}+|I_{N\_y}|^{2}}$$

when the incident waves from the $P$ and $N$ directions are both $x$-polarized, the amplitudes of the $y$-polarized waves from the $N$ and $P$ directions are zero, that is, $|I_{P\_y}|=|I_{N\_y}|=0$. The amplitudes of the two incident waves are equal with $|I_{P\_x}|=|I_{N\_x}|=1$, and there is no phase difference between them as well. We can obtain the absorptivity of the co-polarized CPA as:

(8)$$\begin{aligned} A_{co}=&1-\frac{1}{2}\cdot[|r_{P\_xx}|^{2}+|t_{P\_xx}|^{2}+2\cdot|r_{P\_xx}|\cdot|t_{P\_xx}|\cdot\cos(\phi_{1})+|r_{P\_xy}|^{2}+|t_{P\_xy}|^{2} \\ &+2\cdot|r_{P\_xy}|\cdot|t_{P\_xy}|\cdot\cos(\phi_{2})+|r_{N\_xx}|^{2}+|t_{N\_xx}|^{2}+2\cdot|r_{N\_xx}|\cdot|t_{N\_xx}|\cdot\cos(\phi_{3}) \\ &+|r_{N\_xy}|^{2}+|t_{N\_xy}|^{2}+2\cdot|r_{N\_xy}|\cdot|t_{N\_xy}|\cdot\cos(\phi_{4})] \end{aligned}$$

where $\phi _{1}$ represents the phase difference between $r_{P\_xx}$ and $t_{P\_xx}$, $\phi _{2}$ represents the phase difference between $r_{P\_xy}$ and $t_{P\_xy}$, $\phi _{3}$ represents the phase difference between $r_{N\_xx}$ and $t_{N\_xx}$, $\phi _{4}$ represents the phase difference between $r_{N\_xy}$ and $t_{N\_xy}$. When the transmission and reflection coefficients are the same in the $N$ and $P$ directions with $|r_{P\_xx}|=|r_{N\_xx}|=|r_{xx}|$, $|t_{P\_xx}|=|t_{N\_xx}|=|t_{xx}|$, $\phi _{1}=\phi _{3}$, $|r_{P\_xy}|=|r_{N\_xy}|=|r_{xy}|$, $|t_{P\_xy}|=|t_{N\_xy}|=|t_{xy}|$, $\phi _{2}=\phi _{4}$, the absorptivity of the co-polarized CPA can be simplified as:

(9)$$A_{co}=1-[|r_{xx}|^{2}+|t_{xx}|^{2}+2\cdot|r_{xx}|\cdot|t_{xx}|\cdot\cos(\phi_{1})+|r_{xy}|^{2}+|t_{xy}|^{2}+2\cdot|r_{xy}|\cdot|t_{xy}|\cdot\cos(\phi_{2})]$$

On the other hand, when the incident wave is $x$-polarized from the $P$ direction and $y$-polarized from the $N$ direction, the amplitudes of the $y$-polarized incident wave from the $P$ direction and the $x$-polarized incident wave from the $N$ direction should be zero, that is, $|I_{P\_y}|=|I_{N\_x}|=0$. The amplitudes of the two incident waves are equal with no phase difference. We can obtain the absorptivity of the cross-polarized CPA as:

(10)$$\begin{aligned} A_{co}=&1-\frac{1}{2}\cdot[|r_{P\_xx}|^{2}+|t_{P\_yx}|^{2}+2\cdot|r_{P\_xx}|\cdot|t_{P\_yx}|\cdot\cos(\phi_{5})+|r_{P\_xy}|^{2}+|t_{P\_yy}|^{2} \\ &+2\cdot|r_{P\_xy}|\cdot|t_{P\_yy}|\cdot\cos(\phi_{6})+|r_{N\_yx}|^{2}+|t_{N\_xx}|^{2}+2\cdot|r_{N\_yx}|\cdot|t_{N\_xx}|\cdot\cos(\phi_{7}) \\ &+|r_{N\_yy}|^{2}+|t_{N\_xy}|^{2}+2\cdot|r_{N\_yy}|\cdot|t_{N\_xy}|\cdot\cos(\phi_{8})] \end{aligned}$$

where $\phi _{5}$ represents the phase difference between $r_{P\_xx}$ and $t_{P\_yx}$, $\phi _{6}$ represents the phase difference between $r_{P\_xy}$ and$t_{P\_yy}$, $\phi _{7}$ represents the phase difference between $r_{N\_yx}$ and $t_{N\_xx}$, $\phi _{8}$ represents the phase difference between $r_{N\_yy}$ and $t_{N\_xy}$. When the transmission and reflection coefficients are the same in the N and P directions with $|r_{P\_xx}|=|r_{N\_yy}|=|r_{xx}|$, $|t_{P\_yx}|=|t_{N\_xy}|=|t_{yx}|$, $\phi _{5}=\phi _{8}$, $|r_{P\_xy}|=|r_{N\_yx}|=|r_{xy}|$, $|t_{P\_yy}|=|t_{N\_xx}|=|t_{yy}|$, $\phi _{6}=\phi _{7}$, the absorptivity of the cross-polarized CPA can be simplified as:

(11)$$A_{cro}=1-[|r_{xx}|^{2}+|t_{yx}|^{2}+2\cdot|r_{xx}|\cdot|t_{yx}|\cdot\cos(\phi_{5})+|r_{xy}|^{2}+|t_{yy}|^{2}+2\cdot|r_{xy}|\cdot|t_{yy}|\cdot\cos(\phi_{6})]$$

Full-wave simulations (CST Microwave Studio) are carried out to verify the proposed design in Fig. 2. We are using the Floquet mode analysis with the boundary conditions virtually repeating the modeled structure periodically in x and y directions to mimic the interactions between the electromagnetic fields and the proposed meta-surfaces as shown in Fig. 2(a). Figure 2(b) demonstrates the absorptivity characteristics of the co- and cross-polarized CPAs, and we can observe that there are multiple resonant modes for both co-polarized and cross-polarized CPAs within the frequency band from 2 THz to 4.5 THz, and resonant absorbing peaks of both polarized CPAs occur at 3 THz. In addition, the co-polarized CPA also emerges at 4.21 THz with the corresponding absorptivity of 51.7%. The cross-polarized CPAs happen at 3.69 THz and 4.15 THz with corresponding absorptivities of 83.5%, and 69.1% respectively. However, the absorptivities at these resonant frequencies are low and the concurrent realization of co-polarized and cross-polarized CPA cannot be attained at the same frequency band. On the other hand, given the conducting-state VO$_{2}$ and the graphene with ${\mu _{\rm {c}}}$= 1 eV Fermi energy, the CPA will be achieved at 3 THz with 97.9% absorptivity for the x-polarized control wave and x-polarized signal wave, as shown in Fig. 2(c). Similarly, given the proposed meta-surface having insulating-state VO$_{2}$ and graphene with ${\mu _{\rm {c}}}$= 0.7 eV Fermi energy, an absorptivity of 95.3% can be realized for achieving CPA of the x-polarized control wave and y-polarized signal wave at the same frequency, as shown in Fig. 2(d). Figure 2(e-f) illustrate the corresponding distributions of the E-field intensity at the top and bottom layers of the five-layered cascaded VO$_{2}$-metal-graphene meta-surface for the cases of co-polarized and cross-polarized CPA, respectively. We can observe that the E-field distributions are parallel for the co-polarized CPA, while being perpendicular to each other for the cross-polarized CPA. The E-fields are normalized by 0.44 V/m and 0.18 V/m, respectively.

Fig. 2. CPAs through the proposed meta-surface with the x-polarized control wave. (a) Floquet mode analysis. (b) Both co- and cross-polarized CPA characteristics. (c) Co-polarized CPA absorptivity and (d) Cross-polarized CPA absorptivity varied with the frequency and graphene Fermi energy. (e)Normalized E-field intensity of co-polarized CPA and (f) Normalized E-field intensity of cross-polarized CPA at 3 THz.

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Similar results can be observed through the proposed meta-surface with the $y$-polarized control wave, as shown in Fig. 3. Figure 3(a) demonstrates the Floquet mode analysis of the proposed meta-surface with the $y$-polarized control wave. Figure 3(b) demonstrates the absorptivity characteristics of the co- and cross-polarized CPAs, and we can observe that there are multiple resonant modes for both co-polarized and cross-polarized CPAs within the frequency band from 2 THz to 4.5 THz, and resonant absorbing peaks of both polarized CPAs occur at 3.65 THz. In addition, the co-polarized CPAs also emerge at 3.23 THz and 4.04 THz with the corresponding absorptivities of 65.6%, and 80.5% respectively. The cross-polarized CPAs happen at 2.91 THz and 4.18 THz with the corresponding absorptivities of 87.9%, and 71.2% respectively. On the other hand, given the conducting-state VO$_{2}$ and the graphene with $\mu _{c}= 0.1$ eV Fermi energy, the CPA will be achieved at 3.65 THz with 99% absorptivity for the $y$-polarized control wave and $y$-polarized signal wave, as shown in Fig. 3(c). Similarly, given the proposed meta-surface having insulating-state VO$_{2}$ and graphene with $\mu _{c}= 0.2$ eV Fermi energy, an absorptivity of 99.7% can be realized for achieving CPA of the $y$-polarized control wave and $x$-polarized signal wave at the same frequency, as shown in Fig. 3(d). Figure 3(e-f) illustrate the corresponding distributions of the E-field intensity at the top and bottom layers of the five-layered cascaded VO$_{2}$-metal-graphene meta-surface for the cases of co-polarized and cross-polarized CPA, respectively. Similarly, we can observe that the E-field distributions are parallel for the co-polarized CPA, while being perpendicular to each other for the cross-polarized CPA. The E-fields are normalized by 0.72 V/m and 0.21 V/m, respectively.

Fig. 3. CPAs through the proposed meta-surface with the y-polarized control wave. (a) Floquet mode analysis. (b) Both co- and cross-polarized CPA characteristics. (c) Co-polarized CPA absorptivity and (d) Cross-polarized CPA absorptivity varied with the frequency and graphene Fermi energy. (e) Normalized E-field intensity of co-polarized CPA and (f) Normalized E-field intensity of cross-polarized CPA at 3.65 THz.

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Figure 4 continues to demonstrate the operating principle of the proposed meta-surface. As shown in Fig. 4(a), when VO$_{2}$ is in the conducting state, the front and back metals of the meta-surface together with VO$_{2}$ act as a conductor. In this case, the slot structures on the metal surfaces of the meta-surface are identical, realizing co-polarized CPA. Specifically, the incident $x$-polarized wave transmits as $x$-polarized wave after interacting with the meta-surface, whereas the incident $y$-polarized wave will also maintain the same polarization as the $y$-polarized wave. On the other hand, when VO$_{2}$ is in the insulating state, the slot structure on the back of the meta-surface is identical to the front slot structure rotated counterclockwise by $90^{\circ }$. In such a case, the $y$-polarized incident wave is converted to $x$-polarized transmission after interacting with the meta-surface. Figure 4(b) illustrates the transmission coefficients as well as the reflection coefficients at 3 THz for the co-polarized CPA. We can observe the transmissions and reflections have the relationship of $|r_{P\_xx}|=|r_{N\_xx}|=|r_{xx}|$, $|t_{P\_xx}|=|t_{N\_xx}|=|t_{xx}|$, $|r_{P\_xy}|=|r_{N\_xy}|=|r_{xy}|$, $|t_{P\_xy}|=|t_{N\_xy}|=|t_{xy}|$, $\phi _{r_{P\_xx}}=\phi _{r_{N\_xx}}$, $\phi _{t_{P\_xx}}=\phi _{t_{N\_xx}}$, $\phi _{r_{P\_xy}}=\phi _{r_{N\_xy}}$, $\phi _{t_{P\_xy}}=\phi _{t_{N\_xy}}$. At 3 THz, co-polarized CPA is achieved when the transmission coefficients $t_{xx}$ of the $x$-polarized signal wave equals the reflection coefficients $r_{xx}$ of the $x$-polarized control wave in magnitude with a phase difference of $180^{\circ }$. The absorptivity of the co-polarized CPA is 97.9% rather than 100% at 3 THz is due to the $r_{xy}$ and $t_{xy}$ components are not exactly identical with each other and thus degrade the overall coherent absorption. Figure 4(c) illustrates the transmission coefficients as well as the reflection coefficients at 3 THz for the cross-polarized CPA, where $|r_{P\_xx}|=|r_{N\_yy}|=|r_{xx}|$, $|t_{P\_yx}|=|t_{N\_xy}|=|t_{yx}|$, $|r_{P\_xy}|=|r_{N\_yx}|=|r_{xy}|$, $|t_{P\_yy}|=|t_{N\_xx}|=|t_{yy}|$, $\phi _{r_{P\_xx}}=\phi _{r_{N\_yy}}$, $\phi _{t_{P\_yx}}=\phi _{t_{N\_xy}}$, $\phi _{r_{P\_xy}}=\phi _{r_{N\_yx}}$, $\phi _{t_{P\_yy}}=\phi _{t_{N\_xx}}$. At 3 THz, cross-polarized CPA is realized when the transmission coefficients $t_{yx}$ of the $y$-polarized signal wave after polarization conversion matches the reflection coefficients $r_{xx}$ of the $x$-polarized control wave in magnitude with a $180^{\circ }$ phase difference. The absorptivity of the cross-polarized CPA is 95.3% rather than 100% is due to the in-phase mode of $r_{xy}$ and $t_{yy}$ components leading to the coherent enhancement, and thus degrade the overall coherent absorption. Similarly, Fig. 4(d) illustrates the corresponding transmission coefficients as well as the reflection coefficients at 3.65 THz for the co-polarized CPA. We can observe the transmissions and reflections have the relationship of $|r_{P\_yy}|=|r_{N\_yy}|=|r_{yy}|$, $|t_{P\_yy}|=|t_{N\_yy}|=|t_{yy}|$, $|r_{P\_yx}|=|r_{N\_yx}|=|r_{yx}|$, $|t_{P\_yx}|=|t_{N\_yx}|=|t_{yx}|$, $\phi _{r_{P\_yy}}=\phi _{r_{N\_yy}}$, $\phi _{t_{P\_yy}}=\phi _{t_{N\_yy}}$, $\phi _{r_{P\_yx}}=\phi _{r_{N\_yx}}$, $\phi _{t_{P\_yx}}=\phi _{t_{N\_yx}}$. At 3.65 THz, co-polarized CPA is achieved when the transmission coefficients $t_{yy}$ of the $y$-polarized signal wave equals the reflection coefficients $r_{yy}$ of the $y$-polarized control wave in magnitude with a phase difference of $180^{\circ }$. The absorptivity of the co-polarized CPA is 99%, as the very small amplitudes of $r_{yx}$ and $t_{yx}$ components haves little impact on the overall coherent absorption. Figure 4(e) illustrates the corresponding transmission coefficients as well as the reflection coefficients at 3.65 THz for the cross-polarized CPA, where $|r_{P\_xx}|=|r_{N\_yy}|=|r_{xx}|$, $|t_{P\_yx}|=|t_{N\_xy}|=|t_{yx}|$, $|r_{P\_xy}|=|r_{N\_yx}|=|r_{xy}|$, $|t_{P\_yy}|=|t_{N\_xx}|=|t_{yy}|$, $\phi _{r_{P\_xx}}=\phi _{r_{N\_yy}}$, $\phi _{t_{P\_yx}}=\phi _{t_{N\_xy}}$, $\phi _{r_{P\_xy}}=\phi _{r_{N\_yx}}$, $\phi _{t_{P\_xx}}=\phi _{t_{N\_yy}}$. At 3.65 THz, cross-polarized CPA is realized when the transmission coefficients $t_{yx}$ after polarization conversion of the $y$-polarized control wave matches the reflection coefficients $r_{xx}$ of the $x$-polarized signal wave in magnitude with a $180^{\circ }$ phase difference. The absorptivity of the cross-polarized CPA is 99.7%, and the very small amplitudes $r_{xy}$ of $t_{yy}$ and components affect little on the overall coherent absorption.

Fig. 4. The underlying principle and transmission coefficients for realizing different polarized CPAs with the proposed meta-surface. (a) The structural variation with different VO$_{2}$ states. The transmission and reflection characteristics (b) when the control wave and the signal wave are both x-polarized. (c) when the control wave is x-polarized and the signal wave is y-polarized. (d) when the control wave and the signal wave are both y-polarized. (e) when the control wave is y-polarized and the signal wave is x-polarized.

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Figure 5 illustrates the dynamically tuning of the absorptivity through the Fermi energy level imposed over the graphene and the relative phases of control wave. Given the x-polarized control wave, the absorption efficiencies will vary from 27.8${\% }$ to 97.9${\% }$ for the x-polarized signal fields at 3 THz, and range from 7.6${\% }$ to 95.3${\% }$ for the y-polarized signal fields. Especially, the absorptivity can always exceed 90${\% }$ as demonstrated by the red dashed region in Fig. 5(a) and Fig. 5(b), given the Fermi energy of graphene from 0.7 to 0.9 eV with $\pm 30^\circ$ relative phase modulation of the control wave. Similarly, for the y-polarized control waves, the absorptivity can be adjusted from 7.8${\% }$ to 99.7${\% }$ when the signal fields is x-polarized at 3.65 THz, and will vary from 16.5${\% }$ to 99${\% }$ for the y-polarized signal fields. By tuning the Fermi level of graphene from 0.1 to 0.2 eV, while restricting the relative phase of the control wave within $\pm 35^\circ$ of the optimal value, the CPA will achieve the absorptivity above 90${\% }$, as delineated by the portion of the plot enclosed by the red dashed boundary in Fig. 5(c) and Fig. 5(d).

Fig. 5. Dynamic modulation of the coherent absorptivity. The absorptivity at 3 THz varies with the graphene Fermi level and the phase difference change between the x-polarized control wave and (a) x-polarized, (b) y-polarized signal wave. The absorptivity at 3.65 THz varies with the graphene Fermi level and the phase difference change between the y-polarized control wave and (c) x-polarized, (d) y-polarized signal wave.

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Finally, we examine the absorptivities of the co-polarized and the cross-polarized CPAs varied with structural parameters of the proposed meta-surface, as shown in Fig. 6. We can observe that in Fig. 6(a-d), the absorbing peak of both co-polarized and cross-polarized CPAs will shift to lower frequencies with the increase of $L_2$ from 24 $\mu$m to 28 $\mu$m. Similarly, Fig. 6(e-f) exhibits that as $L_1$ increases from 33.5 $\mu$m to 37.5 $\mu$m, the operating frequencies for co-polarized and cross-polarized CPAs will move towards lower frequencies. In addition, we can also observe that the influence of the variations of $L_1$ has a greater influence on the absorbing frequency bands compared to $L_2$. As a result, the proposed design can be changed to obtain multi-polarized CPAs for other frequencies when we properly optimize the structural parameters.

Fig. 6. The absorptivities of the co-polarized and the cross-polarized CPAs varied with structural parameters. Absorptivity versus $L_2$ for (a) $x$-polarized control wave and $x$-polarized signal wave, (b) $x$-polarized control wave and $y$-polarized signal wave, (c) $y$-polarized control wave and $y$-polarized signal wave, and (d) y-polarized control wave and $x$-polarized signal wave. Absorptivity versus $L_1$ for (e) $x$-polarized control wave and $x$-polarized signal wave, (f) $x$-polarized control wave and $y$-polarized signal wave, (g) $y$-polarized control wave and $y$-polarized signal wave, and (h) $y$-polarized control wave and $x$-polarized signal wave.

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3. Conclusions

In conclusion, we have demonstrated reconfigurable CPAs of different polarized electromagnetic fields through VO$_{2}$ based AT meta-surfaces sandwiched with graphene meta-gratings. Different from the previous implementations employing co-polarized waves, the proposed design has achieved CPAs with different polarized signal- and control-waves. Specifically, with an x-polarized control wave at 3 THz and a y-polarized control wave at 3.65 THz, we have demonstrated both co- and cross-polarized between control wave and signal waves. We expect the proposed design using AT meta-surfaces for more advanced coherent control of different LP electromagnetic fields with the assistance of tunable materials, should pave the way for building up multipolar and multifunctional absorbers.

Funding

Hongque Innovation Center (HQ202202002); National Natural Science Foundation of China (61301072, 61671344).

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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Reconfigurable coherent perfect absorption of different polarized electromagnetic fields through vanadium-dioxide based asymmetry transmissive meta-surfaces sandwiched with graphene meta-gratings

Abstract

1. Introduction

2. Modeling and simulation results

3. Conclusions

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (6)

Equations (11)

Optics Express