# Multiplexing Perfect Optical Vortex for Holographic Data Storage
## 2 Materials and Methods  
The scheme of the proposed method is illustrated in Figure 1, where Figure 1a-c show the recording process, the multiplexing process, and the reconstructed process, respectively. In the recording process (Figure 1a), the original data are a binary sequence, which can be mapped into four gray levels. In the Figure 1a, the original data are converted into data page, which means that the single cell in the data page represents 2-bit binary data. Figure 1: The scheme of the proposed method. (**a**) The recording process. (**b**) The multiplexing process. (**c**) The reconstruction process. To achieve the \(k\)th point-data page, the \(k\)th data page is multiplied by the sampling array, resulting in a representation known as  
\[f_{k}(x,y)=D_{k}(x,y)\times\delta(x,y), \tag{1}\]  
where \((x,y)\) represents the Cartesian coordinates, and \(f_{k}(x,y)\), \(D_{k}(x,y)\), and \(\delta(x,y)\) represent the \(k\)th point-data page, the \(k\)th data page, and the sampling array, respectively. The sampling array can be described as  
\[\delta(x,y)= \begin{cases}1&x=x_{p};y=y_{q}\\ 0&x\
eq x_{p};y\
eq y_{q}\end{cases} , \tag{2}\]  
where \((x_{p},y_{q})\) represents the sampling point position with the sampling distance \(d\) (\(x_{p}-x_{p-1}=d;y_{q}-y_{q-1}=d\)). \((p,q)\) represents the rows and columns of the cell in data page. Hence, the \(k\)th point-data page (\(f_{k}(x,y)\)) can be expressed as  
\[ \begin{split} f_{k}(x,y)=\sum_{\begin{subarray}{c}1\leq p\leq P \\ 1\leq q\leq Q\end{subarray}}I_{p,q}\delta(x-x_{p},y-y_{q})\\ I_{p,q}=[I_{1}\ I_{2}\ I_{3}\ I_{4}],\end{split}  \tag{3}\]  
where \(I_{p,q}\) represents the different intensity levels and \((P,Q)\) represents the total rows and columns of the point in the point-data page. The \(k\)th point-data page (\(f_{k}(x,y)\)) is discretely Fourier transformed into the computer-generated hologram (CGH, \(F_{k}(u,v)\)), which can be described as  
\[F_{k}(u,v)=\ \ \sum_{x=1}^{M}\sum_{y=1}^{N}f_{k}(x,y)exp[i(xu+yv)], \tag{4}\]  
where \((u,v)\) represents the orthogonal coordinates in the hologram. \(M\) and \(N\) are the boundary of the point-data page. In our experiment, for higher accuracy, the GSW algorithm [23] is used to implement the discrete Fourier transform. Then, to achieve the \(k\)th POV-recording hologram (\(H_{k}(u,v)\)), the \(k\)th CGH is multiplied by the optical vortex phase and axicon phase, which can be described as  
\[H_{k}(u,v)=F_{k}(u,v)exp(i\ell_{k}\varphi)exp(i\beta(d_{POV})), \tag{5}\]  
where \(\ell_{k}\) is the topological charge of \(k\)th data page, which can achieve the OV selectivity. \(\beta(k_{r})\) represents the axicon phase, which can realize the perfect optical vortex [24]. The axicon phase can be described as  
\[\beta(d_{POV})=\frac{2\pi r}{d_{POV}}, \tag{6}\]  
where \(d_{POV}\) and \(r\) are the axicon period and the radial coordinate, respectively. By inverse Fourier transform (IFT), the the \(k\)th POV-recording hologram (\(H_{k}(u,v)\)) can be transformed into the POV array (\(l_{k}(x,y)\)), and each POV carries a different intensity level, which can be described in Appendix A. The diameter of the single POV (\(r_{POV}\)) in the POV array is independent of its topological charge, and the minimum sampling distance of POV can be described as  
\[d_{min}=MIN(r_{POV})=\frac{\lambda f}{MAX(d_{POV})}, \tag{7}\]  
where \(MAX(\cdot)\) and \(MIN(\cdot)\) represent the maximum and minimum values. \(\lambda\) and \(f\) are the wavelength and the focal length of a spherical lens, respectively. On the other hand, the diameter of an OV (\(r_{OV}\)) is related to \(\ell\), and the minimum sampling distance of OV can be described as  
\[d_{min}=MIN(r_{OV})=2\sqrt{|\ell_{MAX}|+1}\omega_{0}, \tag{8}\]where \(\omega_{0}\) corresponds to the waist of the fundamental Gaussian beam. Hence, in the same storage space, the storage density of POV is better than that of OV.