1. Introduction

2. Materials and Methods

Figure 1 shows the schematic diagram for the dual mode reconfigurable multifunctional metasurface, which consists of three layers: indium telluride pillar unit layer (top), F4B substrate (middle), and graphene layer (bottom). Figure 1 shows the reconfigurable holographic imaging obtained by switching the incidence polarization in reflection and transmission modes, respectively. Figure 1c is the phase switching sketch map: metal when E_f = 0.9eV, insulator when E_f = 0.1eV. The proposed metasurface works in reflection and transmission modes when E_f =0.9eV and 0.1eV, respectively. The reconfiguration among the Chinese characters ‘’, ‘’ and ‘’are achieved by switching LCP, RCP and LP incidence, respectively. The operation frequency range is 1.15THz – 1.35THz (16%) for reflection mode, and 1.32 THz - 1.6 THz (19%) for transmission mode. The holographic imaging is co- and cross- polarization for reflection and transmission modes, respectively.

2.3. Gerchberg-Saxton (GS) Algorithm

The Gerchberg-Saxton (GS) algorithm, which was originally proposed by D. Gerchberg and W. Saxton in 1972 [22], is an iterative algorithm commonly used in phase retrieval for holographic imaging based on Fourier transform and inverse Fourier transform. The iteration starts from the measured amplitude information of the input image, and ends until the error threshold $(\left|E_i\right|-\left|E_0\right|)<\epsilon$is met. Here GS algorithm is used to obtain the phase distribution map of the phase gradient metasurface, and the basic steps and method are shown in Figure 2.

The required images ‘’, ‘’ in z axis direction are established by Computer holography technology based on MATLAB. First, painting software is used to draw the required images (including the black background and the white images). Second, the amplitude matrix A₀ (m×n) in the image area is obtained by MATLAB function ‘imread’ and ‘binary’. In A₀ (m×n), the intensities are “1” and “0” for points (x, y, 0) in white and black, respectively. Third, the MATLAB function ‘random’ generates a random phase matrix Φ_GS0 (m×n). Apply Fourier transform to the matrices A₀ and Φ_GS0, and obtain amplitude and phase matrices in frequency domain A₁ and Φ_GS1. Then, update all the amplitude matrix values in A₁ (m×n) into 1, that is, A^′₁(x, y) = 1. Perform inverse Fourier transform to A^′₁ and Φ_GS1, and obtain amplitude and phase matrices in time domain A₂ and Φ_GS2. Repeat iterative operations of step 2 and step 3, and update the i-th iteration amplitude A_i (x, y) to A^′_I (x, y) = 1 until the error threshold $(\left|E_i\right|-\left|E_0\right|)<\epsilon$, where $E=A\exp (i{\varphi }_{GS})$. Then the phase distribution map for the Metasurface is obtained.

2.4. The Total Compensated Phase Calculation Based on Transmission Mode

The metasurface is designed in transmission mode at f₁ when E_f = 0.1 eV. Under LCP incidence, a holographic imaging RCP ‘’ is generated in direction (θ₁, 0°). Under RCP incidence, a holographic imaging LCP ‘’ is generated in direction (θ₂, 0°). The phase compensation for the metasurface obtained by GS algorithm is Φ_GS, which is for the images in normal direction. An additional compensated phase is added for the desired imaging direction (θ, 0°), and the compensated phase for RCP ‘’ or LCP ‘’ for the metasurface unit located at (x, y, 0) is as follows:

$$\left\{\begin{array}{c}{\varphi }_{cross-RCP华}={\varphi }_{华}+{\displaystyle \frac{2\pi f_1}{c}}\times (\sin {\theta }_1)\times x\\ {\varphi }_{cross-LCP工}={\varphi }_{工}+{\displaystyle \frac{2\pi f_1}{c}}\times (\sin {\theta }_2)\times x\end{array}\right.$$

(8)

Then the compensated phase parameters (ϕ_x, ϕ_y, β) of the metasurface unit for reconfigurable multifunctional holographic imaging for ‘’, ‘’ and ‘’ is calculated by Equation (5) and Equation (8).

2.5. The Imaging Direction Deduction for Reflection Mode

By switching the bottom graphene layer from insulator to metal states, the metasurface designed in transmission mode can be made to work in reflection mode with the bottom graphene layer acting as a ground plane. According to Equation (6), the holographic imaging of (RCP, ‘’, θ₁, 0°) and (LCP, ‘’, θ₂, 0°) designed in transmission mode has changed into co-polarized imaging of (LCP, ‘’, θ₃, 0°) and (RCP, ‘’, θ₄, 0°) in reflection mode as shown in Figure 3. Though the phase distribution map of the metasurface is calculated based on transmission mode, (LCP, ‘’, θ₃, 0°) and (RCP, ‘’, θ₄, 0°) are generated under LCP and RCP incidences in reflection mode, respectively. Because the graphene permittivity is a function of its state, the operation frequency f₂ and direction θ₃₍₄₎ for reflection mode are different from f₁ and θ₁₍₂₎ in transmission mode. The direction θ₃₍₄₎ is calculated as follows:

$$\left\{\begin{array}{c}{\varphi }_{cross-RCP华}={\varphi }_{co-RCP华}={\varphi }_{华}+{\displaystyle \frac{2\pi f_2}{c}}\times (\sin {\theta }_4)\times x\begin{array}{c}\begin{array}{c}\end{array}(under\begin{array}{c}LCP\begin{array}{c}incidence)\end{array}\end{array}\end{array}\\ {\varphi }_{cross-LCP工}={\varphi }_{co-LCP工}={\varphi }_{工}+{\displaystyle \frac{2\pi f_2}{c}}\times (\sin {\theta }_3)\times x\begin{array}{c}\begin{array}{c}\end{array}(under\begin{array}{c}RCP\begin{array}{c}incidence)\end{array}\end{array}\end{array}\end{array}\right.$$

(9)

From Equations (8) and (9), θ₃ and θ₄ are deduced as follows.

$$\left\{\begin{array}{c}{\theta }_3=\arcsin \left({\displaystyle \frac{f_1}{f_2}}\sin \left({\theta }_2\right)\right)\\ {\theta }_4=\arcsin \left({\displaystyle \frac{f_1}{f_2}}\sin \left({\theta }_1\right)\right)\end{array}\right.$$

(10)

In summary, based on the above analysis, the proposed metasurface can be switched between transmission and reflection modes, and the holographic imaging can be reconfigured by switching the incident polarization when operating in transmission or reflection mode. Six holographic imaging channels are achieved, as shown in Table 1.

2.6. Unit Cell Design

The proposed dual-mode metasurface is excited by normal plane wave as in Figure 1 and Figure 3. Open boundaries were set as boundary conditions. As example, dual-mode reconfigurable multifunctional holographic imaging are designed in order to validate the above design method: (1) Reconfigurable Chinese characters with cross-polarized in transmission mode are achieved when E_f = 0.1 eV: (a) (RCP, ‘’, -14°, 0°), (b) (LCP, ‘’, 17.5°, 0°), and (c) both (RCP, ‘’, -14°, 0°) and (LCP, ‘’, 17.5°, 0°) by switching LCP, RCP and LP incidence, respectively. (2) Reconfigurable multifunctional holographic imaging (Chinese characters) with co-polarized are achieved in reflection mode when E_f = 0.9 eV: (a) (RCP, ‘’, -16°, 0°), (b) (LCP, ‘’, 20°, 0°), (c) both (RCP, ‘’, -16°, 0°) and (LCP, ‘’, 20°, 0°) by switching the RCP, LCP and LP incidence, respectively. All the curves and field patterns are simulated by CST Microwave Studio software.

To realize the designed dual-mode reconfigurable multifunctional holographic imaging, a unit cell is proposed to implement the phase (ϕ_x, ϕ_y, β) at (x, y, 0) calculated by Equation (5), which is shown in Figure 4. Figure 4a–c show the simulation Floquet model, top and side views for the unit, respectively, which consists of a star-shaped indium telluride pillar (ε_r =19, tan δ = 0.001) in the top and a 2D bottom graphene layer separated by an F4B substrate with a dielectric constant 2.2, tan δ = 0.003. All geometry parameters are marked in Figure 4, and the optimized parameters are as follows: p = 123 μm, w = 11 μm, h₁ = 150 μm, h₂ = 0.5 μm and h₃ = 0.05 μm. The propagation phases ϕ_x and ϕ_y are controlled by l_x and l_y, respectively, and the geometry phase is controlled by rotating the unit cell counter clockwise β degree.

Under normal x- (y-) polarized incident excitation: (1) For reflection mode at 1.31THz, 1.325THz and 1.336THz, Figure 5a shows the simulated S₁₁ and phase ϕ_x(y) as a function of l_x(y), and Figure 5b shows the simulated S₁₁ and phase ϕ_y(x) as a function of l_x(_y). (2) For transmission mode at 1.455THz, 1.465THz, and 1.473 THz: Figure 5c shows the simulated amplitude (S₂₁) and phase ϕ_x(y) curves as function of l_x(y), and Figure 5d is the simulated amplitude (S₂₁) and phase ϕ_y(x) curves as function of l_x(y). For both transmission and reflection modes when l_x(y) varies from 15 μm to 113 μm: (1) The phase of the co-polarization ϕ_x(y) increases from -450° to -90°, covering a 360^o phase range. (2) The phase shift of the cross-polarization ϕ_y(x) remains basically unchanged, indicating that ϕ_x and ϕ_y can be independently adjusted by l_x and l_y, respectively. S₁₁ > 0.99 for reflection mode, and S₂₁ > 0.85 for transmission mode. The phase ϕ_x(y) curves versus l_x(_y) for different frequency points (at 1.31THz, 1.325THz for reflection mode or at 1.455 THz, 1.465 THz, and 1.473 THz for transmission mode) are parallel, indicating that the metasurface can operate within a certain bandwidth. According to Figure 5a,c, a polynomial formula for the relationship between ϕ_x(_y) and l_x(_y) can be fitted as follows:

$$\begin{array}{c}{\varphi }_{x(y)}=392.7\times \sin (0.027\times l_{x(y)}+0.2954)+264.3\times \sin (0.075\times l_{x(y)}+0.1413)\\ +209.3\times \sin (0.08\times l_{x(y)}+2.699)\end{array}$$

(11)

When l_x =15 μm, Figure 6 shows the curves of the geometric phase as a function of the rotation angle β for reflection mode when l_x=15um (at 1.31THz, 1.325THz, and 1.336 THz) and transmission mode (at 1.455THz, 1.465THz, and 1.473THz). When β ranges from 0° to 180°, the geometric phase shift covers 360°, and the geometric phase vs β curves at different frequency points (1.31THz, 1.325THz and 1.336THz for reflection mode or 1.455THz, 1.465THz, and 1.473 THz for transmission mode) are parallel.

2.7. Metasurface Design

The transmission-reflection reconfigurable metasurface shown in Figure 3 is formed by the proposed 91×91 unit cells with size 11.193mm×11.193mm×0.15mm (50.4λ₀×50.4λ₀×0.68λ₀) for dual-mode reconfigurable multifunctional holographic imaging: (1) Reconfigurable cross-polarized holographic imaging in transmission mode when E_f = 0.1eV: (a) (RCP, ‘’, -14°, 0°), (b) (LCP, ‘’, 17.5°, 0°), and (c) both (RCP, ‘’, -14°, 0°) and (LCP, ‘’, 17.5°, 0°) by switching LCP, RCP and LP incidence, respectively. (2) Reconfigurable multifunctional co-polarized holographic imaging in reflection mode when E_f = 0.9eV: (a) (RCP, ‘’, -16°, 0°), (b) (LCP, ‘’, 20°, 0°), (c) both (RCP, ‘’, -16°, 0°) and (LCP, ‘’, 20°, 0°) by switching the RCP, LCP and LP incidence, respectively.

The compensated phase distributions for ϕ_x, ϕ_y and 2β according to Equation (5) are shown in Figure 7, and then the corresponding geometry length distributions for l_x, l_y in the metasurface are obtained according to Equation (11).

3. Results

The far-field radiation patterns at 1.325 THz or 1.465 THz under LCP, RCP and LP incidence are shown in Figure 8. Figure 8a,b show the three dimension (3D) and 2D far-field radiation patterns for reflection and transmission modes, respectively. Reconfigurable cross-polarized holographic imaging is achieved at 1.465 THz in transmission mode among (‘’, RCP, θ = -14°, φ = 0°), (‘’, LCP, θ = 17.5°, φ = 0°) and both (‘’, RCP, θ = -14°, φ = 0°) and (‘’, LCP, θ = 17.5°, φ = 0°) by switching LCP, RCP and LP incidence, respectively. Reconfigurable multifunctional co-polarized holographic imaging is achieved at 1.325 THz in reflection mode among (‘’, RCP, θ = -16°, φ = 0°), ‘’, LCP, θ = 20°, φ = 0°), and both (‘’, RCP, θ = -16°, φ = 0°) and (‘’, LCP, θ = 20°, φ = 0°) by switching the RCP, LCP and LP incidence, respectively. The simulated and calculated holographic imaging are in good agreement. Because LP wave can be decomposed into two equal LCP and RCP waves, both the LCP and RCP excitation are done simultaneously, and the holographic imaging for LP incidence are the holographic imaging sum of LCP and RCP incidences.

4. Discussion

5. Conclusions

This paper develops a multifunctional and reconfigurable reflection-transmission dual-mode THz dielectric holographic metasurface formed by propagation - geometry composite units. The dual-mode is achieved by controlling the Fermi energy level of graphene integrated into the metasurface unit, and the proposed metasurface works in reflection and transmission modes when E_f = 0.9 eV, 0.1eV, respectively. The reconfigurability is realized by switching incidence polarization. The metasurface is designed based on transmission mode, and the physical model switching to the reflection mode is established. For the first time, dynamic modulation reflection-transmission holographic imaging metasurface based on graphene is developed. As example, a multifunctional and reconfigurable metasurface has been developed: (1) Reconfigurable cross-polarized three-channel holographic imaging in transmission mode from 1.32THz to 1.6THz: (‘’, RCP, θ = -14°, φ = 0°), (‘’, LCP, θ = 17.5°, φ = 0°) and (‘’, RCP, θ = -14°, φ = 0° and ‘’, LCP, θ = 17.5°, φ = 0°) by switching LCP, RCP and LP incidence, respectively. (2) Reconfigurable co-polarized three-channel holographic imaging in reflection mode from 1.15THz to 1.35THz: (‘’, RCP, θ = -16°, φ=0°), (‘’, LCP, θ = 20°, φ = 0°), and (‘’, RCP, θ = -16°, φ = 0° and ‘’, LCP, θ = 20°, φ = 0°) by switching the RCP, LCP and LP incidences, respectively. Compare with published holographic imaging, ours has more channel numbers (six holographic imaging channels) and higher holographic efficiency (42.5% to 49%). These characteristics make the proposed metasurface has potential applications in information encryption transmission, multi-channel imaging, and other related fields.

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