# Multiplexing Perfect Optical Vortex for Holographic Data Storage
## 2 Materials and Methods
For instance, when considering the case of \(\ell=10\) and \((p=1,q=2)\), the comparison between perfect optical vortex (POV) and optical vortex (OV) is illustrated in Figure 2. The results are obtained under the same conditions (\(d=170\) pixels, \(\lambda=632.8\) nm, \(\omega_{0}=5\) um, and \(d_{POV}=0.36\)). In the multiplexing process (Figure 1b), the orthogonality of POV provides valuable selectivity to enhance the storage capacity through the multiplexing of the POV-recording holograms. The topological charge (\(\ell\)) is considered a distinctive label for the POV-recording hologram. The multiple POV-recording holograms are superimposed to create a POV-multiplexing hologram, which can be described as  
\[H_{MUX}(u,v)=\sum_{k=1}^{K}F_{k}(u,v)exp(i\ell_{k}\varphi)exp(i\beta(d_{POV})), \tag{9}\]  
where \(K\) is the total multiplexing number. The POV-multiplexing hologram can be imprinted and recorded by a spatial light modulator (SLM) in our experiment. Some other storage media (such as Metasurface [25; 26], Diffractive optical element (DOE) [27; 28], etc.) can be used to record the POV-multiplexing hologram. In the reconstruction process (Figure 1c), upon illuminating the POV-multiplexing hologram with the POV reading beam, characterized by an inverse topological charge (\(-\ell_{k}\)) and the axicon phase (\(-\beta(d_{POV})\)), the POV undergoes a conversion to Gaussian mode. As a result of this conversion, the \(k\)th point-data page can be successfully retrieved, yielding an output represented as  
\[H_{MUX}(u,v)\cdot[exp(-il_{k}\varphi)exp(-i\beta(d_{POV}))] = F_{1}(u,v)exp[i(1_{1}-l_{k})\varphi] \tag{10}\] \[\vdots\] \[+ F_{k}(u,v)exp(i0\varphi)\] \[H_{k}(u,v)=F_{k}(u,v)exp(i\ell_{k}\varphi)exp(i\beta(d_{POV})), \tag{5}\]0 \[H_{k}(u,v)=F_{k}(u,v)exp(i\ell_{k}\varphi)exp(i\beta(d_{POV})), \tag{5}\]1  
From Equation (10), the \(k\)th point-data page is retrieved in Gaussian mode, and the remaining point-data pages maintain their original OV mode. This observation validates the preservation of the OV mode in the majority of the retrieved data. The acquired results reveal a distinct pattern in the reconstructed data: the \(k\)th point-data page exhibits a solid-spot structure with varying intensity levels, while the other point-data pages retain their characteristic \"doughnut\" shape. This distinction enables the implementation of a  
\[H_{k}(u,v)=F_{k}(u,v)exp(i\ell_{k}\varphi)exp(i\beta(d_{POV})), \tag{5}\]2 (**a**) The experimental image of optical vortex (OV) array with \(d=170\) pixels. (**b**) The experimental image of perfect optical vortex (POV) array with \(d=170\) pixels. mode-selective aperture array, denoted \(\delta^{{}^{\prime}}(x,y)\), to effectively obtain the Gaussian modes. After passing through the Fourier lens and the mode-selective aperture array, the \(k\)th reconstructed point-data page (\(f^{{}^{\prime}}_{k}(x,y)\)) can be obtained, that is,  
\[H_{k}(u,v)=F_{k}(u,v)exp(i\ell_{k}\varphi)exp(i\beta(d_{POV})), \tag{5}\]3  
where \(IFT\{\cdot\}\) represents the inverse Fourier transform. In the decoding process, different intensity thresholds (\(T_{1}\),\(T_{2}\), and \(T_{3}\)) are set to recover 2-bit binary data. If the POV reading beam with \(\ell=0\) or an unmatched topological charge is used, no data can be obtained. This observation also highlights the inherent confidentiality of the proposed method. Hence, the matched POV reading beam can retrieve corresponding data and the unmatched POV reading beam can obtain null.