# Spatial Multiplexing Holography for Multi-User Visible Light Communication
## 2 Design Principle of Spatial Multiplexing Hologram
### Spatial Multiplexing Hologram Using OAM for Multi-User Secure Communication  
Ensuring secure communication is of utmost importance in VLC systems. One novel approach to enhance security involves manipulating the OAM of light beams during the generation of SMH. This method leverages the unique property of OAM, wherein only specific OAM modes can reconstruct the optical power intended for specific users effectively [18; 26]. The core concept is that incorporating an OAM phase into the beam's structure acts as a form of encryption--akin to a key. Incorrect or false OAM keys cause the resultant optical field to assume a toroidal (doughnut-shaped) configuration, leading to the dispersion of optical power. This characteristic ensures that only receivers equipped with the correct OAM decoding capabilities can successfully focus the light into a usable signal, thereby enhancing the overall security of the communication within the VLC system. SLM-based OAM multi-user communication has traditionally been classified into two main categories: mode-division multiplexing (MDM) and space-division multiplexing (SDM). In MDM, OAM modes are transmitted coaxially, with different users using different OAM keys at similar locations to decode the signals. In SDM, OAM modes are modulated in different directions through a single SLM, allowing different users to receive signals from different forward directions but at similar depths. However, a significant drawback of these methods is the inability to control the forming depth of OAM, requiring users to be positioned on the same plane, which is uncommon in realistic scenarios. Additionally, the enlargement of OAM rings over distance results in substantial power dispersion. By utilizing SMH, these challenges can be addressed effectively. In addition to the previously outlined steps, the OAM phase within the complex amplitude domain is represented as  
\[P=\exp(il\theta) \tag{11}\]  
Here, \(l\) denotes the mode number of OAM, while \(\theta\) refers to the angular component of the holograms, which are generated by Equation (2) in polar coordinates. Accordingly, the array of OAM modes corresponding to different users can be articulated as \(P=\left|P_{1}\quad P_{2}\quad\ldots\quad P_{n}\right|\). Thus, we can construct the OAM-enriched SMH through the following summation:  
\[H_{max}=\sum_{i=1}^{n}H_{i}F_{i}P_{i} \tag{12}\]  
In the reconstructed light field, users typically observe ring-shaped light spots, which may lead to diminished received optical power. To counteract this issue and enhance power reception, it is essential to incorporate the correct inverse OAM modes. By applying these inverse modes before reception, the ring-shaped illumination can transform into concentrated, solid light, thereby significantly increasing the received optical power. To further explain the theory of this method, we take a two-user scenario for example. Assuming that an SMH multiplexes two users with OAM keys \(l_{1}\) and \(l_{2}\), respectively, the OAM part of the optical field can be expressed as  
\[E=E_{0}[\exp(il_{1}\theta)+\exp(il_{2}\theta)] \tag{13}\]  
At the location of User 1, we use the inverse OAM phase to decode. The OAM of the optical field at the receiver plane can be expressed as  
\[ \begin{array}{l}E_{1}=E_{0}[exp(il_{1}\theta)+exp(il_{2}\theta)]exp(-il_{1} \theta)\\ =E_{0}+E_{0}exp[i(12-1l)\theta]\end{array}  \tag{14}\]  
The optical intensity is given by  
\[ \begin{array}{l}I_{1}=\iint E_{1}E_{1}^{*}dxdy\\ =\iint\left|E_{0}\right|^{2}+2\left|E_{0}\right|^{2}\exp[i(l2-l1)\theta]+\left| E_{0}\right|^{2}\exp[i2(l2-l1)\theta]dxdy\\ =\left|E_{0}\right|^{2}+\delta_{1,2}\end{array}  \tag{15}\]  
It is noteworthy that the part \(E_{0}\)exp\([i(l2-l1)\theta]\) is eliminated when calculating intensity due to the orthogonality of OAM. Here, \(\delta_{1,2}\) represents the crosstalk between the two users, which is minimal.