# Reconstructing Polarization Multiplexing Terahertz Holographic Images with Transmissive Metasurface
## 3 Structure Design  
As shown in Figure 2a, the meta-atom is composed of a Si micro-pillar on a Si substrate, and the permittivity of Si is 11.9 [28]. \(P\), \(H\), \(L\), and \(W\) represent the period, height, length, and width of the meta-atom, respectively. The meta-atom is optimized to behave as a half-wave plate (HWP), which can convert the incident LCP(RCP) light into the RCP(LCP) light with an additional phase that is twice the rotation angle \(\theta\). Based on geometric phase or the Pancharatnam-Berry (PB) phase, the desired phase of the cross-polarized light can be achieved by varying the rotation angle \(\theta\) (0-180 degree). The mechanism of geometric phase can be easily revealed by the Jones matrix [12]. The incident LCP(RCP) light can be expressed with Jones' vector \(\frac{1}{\sqrt{2}} \begin{pmatrix}1\\ \pm i\end{pmatrix} \), where \"+\" and \"\(-\)\" correspond to the incident LCP and RCP lights, respectively. For the designed Si meta-atom with a rotation angle of \(\theta\), the transmitted light under the illumination of LCP(RCP) light with can be written as Formula (4):  
\[t=\frac{1}{2\sqrt{2}}\bigg{(}\Big{(}t_{xx}e^{i\varphi_{xx}}+t_{yy}e^{i\varphi_ {yy}}\Big{)} \begin{pmatrix}1\\ \pm i\end{pmatrix} +\Big{(}t_{xx}e^{i\varphi_{xx}}-t_{yy}e^{i\varphi_{yy}} \Big{)}e^{\pm i2\theta} \begin{pmatrix}1\\ \mp i\end{pmatrix} \bigg{)}, \tag{4}\]  
where \(t_{xx}\) and \(t_{yy}\) represent the transmission coefficients under x-linearly polarized (XLP) and y-linearly polarized (YLP) lights illumination, and \(\varphi_{xx}\) and \(\varphi_{yy}\) represent the corresponding phases. Moreover, the second term indicates the cross-polarized component with the additional phase of \(\pm 2\theta\). In order to make the meta-atom work as an HWP, the  
Figure 1: Schematic of the transmissive all-dielectric metasurface for the polarization multiplexing THz holograms. The hologram patterns with “EL” and “17” are reconstructed under the incident LCP and RCP light, respectively. The retrieved partial phase profiles \(\varphi_{\text{L}}\) and \(\varphi_{\text{R}}\) are also shown. transmission is near unity, and the phase difference between the \(x\)- and \(y\)-directions should be \(\pi\). The geometric optimization is carried out by using a commercial simulation software from FDTD Solutions based on the finite-difference time-domain (FDTD) method. Periodic boundary conditions are adopted in the \(x\)-\(y\) plane and a perfectly matched layer boundary condition is utilized in the \(z\)-direction. To ensure the convergence and accuracy of the results, the mesh cell size is set to 5 \(\upmu\)m in the \(x\)- and \(y\)-directions, and 10 \(\upmu\)m in the \(z\)-directions, which are smaller than \(\lambda/10\). Normal-incident circularly polarized light is established by two polarized lights along the \(x\)- and \(y\)-directions with the same amplitudes, and phase difference of \(\pi/2\). One 2D \"frequency-domain field and power\" monitor is placed in the transmitted field to collect the transmission data. By fixing \(P=80\)\(\upmu\)m, \(H=500\)\(\upmu\)m, and varying \(L\) and \(W\) of the meta-atom, the transmission coefficient and phase profile for the illumination of the XLP light at the frequency of 0.85 THz are shown in Figure 2b. Based on the mirror symmetry of the meta-atom corresponding to \(L=W\), the transmission coefficient and the phase profile under the YLP light illumination can be extracted from the results of \(t_{xx}\) and \(\varphi_{\text{xx}}\)[12]. At the geometric parameters of \(L=65\)\(\upmu\)m and \(W=15\)\(\upmu\)m, the transmission is near unity, and the phase difference in the two orthogonal directions is \(\pi\), consistent with the HWP-like operation. Therefore, the optimized structural parameters are \(L=65\)\(\upmu\)m, \(W=15\)\(\upmu\)m, \(H=500\)\(\upmu\)m, and \(P=80\)\(\upmu\)m, respectively. Meanwhile, we calculate the cross-polarized and co-polarized transmission spectra, which are plotted in Figure 2c.