# Spatial Multiplexing Holography for Multi-User Visible Light Communication
## 2 Design Principle of Spatial Multiplexing Hologram  
### Spatial Multiplexing Hologram for Multi-User Power Allocation  
In VLC systems, it is essential to accommodate the diverse optical power demands of users who utilize different communication devices and engage in varied applications. To optimize resource utilization and minimize optical power wastage, the VLC system should be capable of dynamically allocating power among users. Conceptually, this allocation process can be likened to projecting an image of users onto different imaging plates, each situated at unique locations. For example, as shown in Figure 2, the little bunny object has been partitioned into multiple layers to create multiplexing holograms. By employing power allocation holography, we can enhance the heart region of the projected image. The creation of SMH for effective power allocation is depicted in Figure 3. A significant challenge in this process is the beam's diffraction transmission, which results in the expansion of the light plane and potential drift of the target position as the transmission distance increases [19; 20]. To mitigate this issue during the preprocessing stage, it is imperative to precisely sample the transmission space as shown in Figure 4. The source plate is defined as an SLM plate, characterized by a bilateral pixel count of \(N_{x}\times N_{y}\) and an initial pixel pitch of \(\Delta x_{0}\). The pixel intervals at a distance \(z\) from the source, as determined by Fresnel diffrac  
Figure 2: Application of power allocation holography in projection. tion, are \(\Delta x=z\lambda/(N_{x}\ \Delta x_{0})\) and \(\Delta y=z\lambda/\left(N_{y}\ \Delta x_{0}\right)\) for horizontal and vertical directions, respectively. Consequently, the light field's horizontal and vertical magnification factors, \(\alpha_{x}=\Delta x/\Delta x_{0}\) and \(\alpha_{y}=\Delta y/\Delta x_{0}\), can be computed. For a given user located at position \((x,y,z)\), these magnification coefficients are initially calculated based on the distance \(z\). Thus, the target's position within the source field can be accurately described as \(x_{0}=x/\alpha_{x}\) and \(y_{0}=y/\alpha_{y}\). Figure 4: Schematic diagram of precise transmission space sampling. (**a**) Transmission process without pre-control; (**b**) Transmission process with pre-control. Figure 3: (**a**) Process of generating SMH for power allocation; (**b**) Process of generating SMH for power coverage; (**c**) Process of generating SMH for OAM secure communication. A user's information can be represented as an electrical field \(E(x,y,z)\), where \(x\), \(y\), and \(z\) denote spatial coordinates. For multiple users, we define a vectorial representation as follows:  
\[\mathbf{E}=\left|E_{1}\quad E_{2}\quad\ldots\quad E_{n}\right| \tag{1}\]  
In reducing inter-hologram crosstalk, we employ Fourier holography complemented by a random phase vector \(\mathbf{\phi}\). The expression for the resulting hologram vector is [14; 21; 22; 23]:  
\[\mathbf{H}=\iint\mathbf{E}\text{exp}(i\mathbf{\phi})\text{exp}[i(\xi x+\eta y)]dxdy \tag{2}\]  
where \(\xi\) and \(\eta\) respectively represent the spatial electrical field distribution on SLM. Assuming the electrical fields \(E_{1}\) and \(E_{2}\) of two users are added as different random phase vectors \(\phi_{1}\) and \(\phi_{2}\), the resulting holograms can be expressed as  
\[ \begin{cases}H_{1}=\iint E_{1}\text{exp}(i\phi_{1})\text{exp}\left[i(\xi x+ \eta y)\right]dxdy\\ H_{2}=\iint E_{2}\text{exp}(i\phi_{2})\text{exp}\left[i(\xi x+\eta y)\right]dxdy \end{cases}  \tag{3}\]  
The orthogonality of \(H_{1}\) and \(H_{2}\) can be verified by a conjugate multiplication integral:  
\[ \begin{split}\iint H_{1}H_{2}^{*}d\xi d\eta=\iint E_{1}E_{2} \text{exp}[i(\phi_{1}-\phi_{2})]d\xi d\eta\\ =\left\{\iint E_{1}E_{2}dxdy\quad\phi_{1}=\phi_{2}\right.\\ 0\qquad\qquad\qquad\phi_{1}\
eq\phi_{2}\end{split}  \tag{4}\]  
As the difference between \(\phi_{1}\) and \(\phi_{2}\) increases, the degree of orthogonality between two holograms also increases [14].