# Reconstructing Polarization Multiplexing Terahertz Holographic Images with Transmissive Metasurface
## 2 Theoretical Analysis  
The design of the metalens is of great significance for reconstructing holographic images. The phase profile for a typical metalens that leads to a distinct focal point following the [PERSON]'s principle can be described by Formula (1) [41; 42]:  
\[\varphi(x,y)=-\frac{2\pi f_{0}}{c}\bigg{(}\sqrt{\left(x-x_{0}\right)^{2}+\left( y-y_{0}\right)^{2}+f^{2}}-f_{d}\bigg{)}, \tag{1}\]  
where \(f_{0}\) is the operating frequency, \((x_{0},y_{0})\) are the coordinates of the focal point on its focal plane, \(f\) is the focal length, \(f_{d}=\sqrt{x_{0}^{2}+y_{0}^{2}+f^{2}}\) is the distance between the center of the focal point and the center of the metalens, and \(c\) is the speed of light in vacuum. When a beam passes through the metalens with the phase profile in Formula (1), one focal point is produced on the focal plane of the metalens, which means a single bright spot can be reconstructed on the holographic image. To further generate a patterned holographic image, the consecutive foci are arranged in an \(m\times n\) array, and the phase profile of metalens can be written as Formula (2):  
\[\phi(x,y)=\arg\left\{\sum\limits_{i=1}^{m}\sum\limits_{j=1}^{n}e^{i\varphi_{ij}(x,y)}\right\}\!, \tag{2}\]  
where \(\varphi_{i,j}(x,y)=-\frac{2\pi f_{0}}{c}\Big{(}\sqrt{\left(x-x_{i}\right)^{2} +\left(y-y_{j}\right)^{2}+f^{2}}-f_{dij}\Big{)}\). Each pixel in the designed holographic image corresponds to the \(i\)th column and the \(j\)th row focal point with coordinates \((x_{i},y_{j})\) on the focal plane, where \(x_{i}=x_{1}+(i-1)d_{0}\), \(y_{j}=y_{1}+(j-1)d_{0}\), and \(d_{0}\) is the distance between two adjacent foci. The evaluation and optimization for the quality of the focal point is vital to achieve high-resolution and multiplexing holographic images. [PERSON] principle qualitatively points out that each point of the wavefront can serve as the secondary point source of the emitted waves [33]. When the incident light propagates from wave surface \(S_{1}\) to \(S_{2}\), according to [PERSON] principle, the complex amplitude of the wave emitted by point \(P_{0}\) in surface \(S_{1}\) at the point \(Q\) in surface \(S_{2}\) is: \(U(r_{0})\propto\frac{U_{0}e^{i\varphi_{0}}}{r_{0}}\), where \(U_{0}\) is the complex amplitude of the wave located at the \(z=0\) plane with surface area \(S\), \(r_{0}\) is the distance between points \(P_{0}\) and \(Q\), \(e^{i\varphi_{0}}\) is the phase factor, and \(k=2\pi/\lambda\) is the wavevector. It is worth noting that the magnitude of the amplitude is inversely proportional to the moving distance, and the phase change is \(k_{0}\) times the moving distance. By using Huygens principle and the principle of electromagnetic wave superposition, the complex amplitude at another point \(P\) can be obtained. According to the Huygens-Fresnel principle, the complex amplitude can be written as: \(U(P)=\frac{1}{i\lambda}\iint\limits_{S}U(r_{0})\frac{e^{i\varphi}}{r}K(\theta )\text{d}S\), where \(r\) is the distance between points \(Q\) and \(P\), and \(K(\theta)\) is an additional tilt factor reflecting the relationship between amplitude and direction. Based on the above [PERSON]'s mathematical expression, the Fresnel hypothesis and Huygens-Fresnel principle can be approximately simplified according to Kirchhoff diffraction theory, and the undetermined tilt factor \(K(\theta)\) in Fresnel theory can be given as: \(K(\theta)=\frac{\cos\left(\overrightarrow{n},\mathbf{r}\right)-\cos\left( \overrightarrow{n},\mathbf{r}_{0}\right)}{2}\), where normal direction \(\overrightarrow{n}\), \(\mathbf{r}_{0}\) is the vector between the source point and a point in the \(S_{1}\) plane, and \(\mathbf{r}\) is the vector between the point in the \(S_{1}\) plane and a point in the propagation distance \(z=f\).