## 4 The Geometries of the Proposed Transmitters
### Honeycomb Hexagonal Array  
As shown in Figure 14, a honeycomb-shaped hexagonal array transmitter is proposed with a separation \(s\) between each LED. Each element has a half FOV of \(\phi_{\frac{1}{2}}^{\prime}\). Again, each LED \(e_{i,j}\) is oriented along a vector \(\mathbf{O}_{i,j}\) crossing the location of \(e_{i,j}\) on the \(y=0\) plane and the location of \(e_{i,j}^{\prime}\) on the \(y=d_{0}\) plane. Consistently, the \(y=d_{0}\) plane is defined such that the two honeycomb hexagonal arrays formed by the \(e_{i,j}^{\prime}\) points from the two transmitters become tangent on this plane (illustrated in Figure 15). To meet this condition, the separation between the \(e_{i,j}^{\prime}\) points should be \(D/5\). With the naming convention of Figure 14, the locations of the elements \(e_{i,j}\) and \(e_{i,j}^{\prime}\) relative to the center element \(e_{0,0}\) are determined with  
\[\mathrm{P}(y,0,s\frac{\sqrt{3}}{2}i),\quad i=\text{even}; \tag{36}\]  
Figure 14: Honeycomb hexagonal-shaped array geometry. Figure 15: Honeycomb hexagonal arrays formed by the \(e_{i,j}^{\prime}\) points on the \(y=d_{0}\) plane. \[\mathrm{P}(s\,\mathrm{sign}(j)(|j|-0.5),d_{0},s\frac{\sqrt{3}}{2}i),\quad i= \mathrm{odd}. \tag{37}\]  
and  
\[\mathrm{P}(\frac{D}{N}j,d_{0},\frac{D}{N}\frac{\sqrt{3}}{2}i),\quad i=\mathrm{ even}; \tag{38}\]  
\[\mathrm{P}(\frac{D}{N}\,\mathrm{sign}(j)(|j|-0.5),d_{0},\frac{D}{N}\frac{ \sqrt{3}}{2}i),\quad i=\mathrm{odd}. \tag{39}\]  
respectively. Thus, the required orientation vectors \(\mathbf{O}_{i,j}\) can be calculated by  
\[\langle(\frac{D}{N}-s)j,d_{0},(\frac{D}{N}-s)\frac{\sqrt{3}}{2}i\rangle,\quad i =\mathrm{even}; \tag{40}\]  
\[\langle(\frac{D}{N}-s)\,\mathrm{sign}(j)(|j|-0.5)j,d_{0},(\frac{D}{N}-s)\frac{ \sqrt{3}}{2}i\rangle,\quad i=\mathrm{odd}. \tag{41}\]  
Similar to the considerations made for the other arrays, the power of each LED in the honeycomb hexagonal array transmitter is determined to be \(P_{0}/25\) since it includes a total of 25 LEDs. Proportional to the power level divided by the squared tangent of the half FOV, the intensity \(I_{5}\) of each LED of the array is set as follows:  
\[I_{5}=\frac{1}{25}\frac{\tan^{2}\phi_{\frac{1}{2}}}{\tan^{2}\phi_{\frac{1}{2}} ^{\prime}}I_{0} \tag{42}\]  
The initial values of the design parameters for the honeycomb hexagonal array are listed in Table 2. Using Equations (2) and (3), and the values given in Table 2, the irradiance distribution on the plane \(y=1\) m is plotted using MATLAB software. The result is given in Figure 6f. Established as a reference, we take the irradiation distribution of the single-LED transmitters at \(y=1\) m, which is shown in Figure 6a. An area with an irradiance of at least 0.5 is considered illuminated. The height of the illuminated area is given as \(H\), and the width as \(W\), as shown in Figure 6a. We aim to keep the values of \(H\) and \(W\) equal for all the configurations so that the illuminated areas remain approximately the same. As seen in Figure 6, all the array transmitters manage to illuminate the area that is illuminated by the reference configuration of single-LED transmitters. The \(H\) and \(W\) values for the different configurations are given in Figure 14:0 The array transmitters illuminate larger areas compared to the single LEDs with a current value of \(d_{0}\). Therefore, we can adjust the value of \(d_{0}\) to enhance the system performance as long as the \(H\) and \(W\) values do not become smaller than those of the single-LED transmitters. points \(e^{\prime}_{i,j}\), which are the points where the orientation vectors of the two array transmitters intersect the plane \(y=d_{0}\), from the two transmitters are tangent to each other. We expect a higher SNR with a decreasing half FOV \(\phi^{\prime}_{\frac{1}{2}}\) since the DC channel gain is proportional to the Lambertian coefficient \(m\) (refer to Equation (7)), whereas \(m\) is inversely proportional to the half FOV \(\phi^{\prime}_{\frac{1}{2}}\) (refer to Equation (3)).