## 5 Geometric Implications for Optimizing PD Arrays for Received Power
### Parameters and Simulation Design  
Two arrays, one with three and the other with five PDs, are evaluated in both convex (C-type) and concave (D-type) configurations (Figure 4). In both configurations, the inclination of PD placement along an ellipse is varied by the distance of the vertex. The degree of convexity or concavity increases from a flat (F-type) array to depths of 10, 25, 40, and 55 mm. These arrays are labeled as \(C\chi\) or \(D\chi\), where \(\chi\) refers to the depth of convexity or concavity. For example, C25 refers to a convex array with a depth of 25 mm. The PDs are spaced equidistant along the curve. The co-vertex is 50 mm in each case. The concave and convex arrays have the PDs placed inside and outside of the transparent curved surface, respectively. The LED intensity distribution pattern and PD parameters used are from real-world devices: an Osram LZ4-00G108 LED [28] and an OSI Optoelectronics SD200-12-22-041 Silicon PD [29], respectively. Figure 4: Array configuration and naming convention. Array center axis is orthogonal to the tangent to curve at the middle PD, with coordinate \((x_{c},\,y_{c})\). C-type and D-type arrays observe the light from the outside and inside of the curve, respectively. Orange circles and purple squares are representative of the PD placements of the 3 PD and 5 PD arrays, respectively. Each PD is equidistantly placed along the curve starting from the array center. Figure 3: Relative intensity distribution over annuli concentric about the LED center axis. For each PD, the reference noise-equivalent power (NEP) is \(1.6\times 10^{-13}\) W/\(\surd\)Hz [29]. The cumulative typical noise power \(P_{\eta_{tot}}\) at the modulation frequency (BW) can be estimated as Equation (17), where \(I_{PD}\) is the total number of PDs in the array. The total received power \(P_{\tau_{tot}}\) by the array is given by Equation (18), where \(P_{\tau_{a_{i}}}\) is optical power received by the \(i^{th}\) PD. Thus, the SNR may also be evaluated as Equation (19). For this study, the optical transmit power was set at \(P_{T}=10\) W, and a \(BW=5\) MHz has been used throughout. Here, we consider the modulation frequency to be fixed at the upper limit of the modulation bandwidth. This frequency is well within the cut-off bandwidth of the selected photodiode. The dynamic relationship of shot noise with received power is not emulated in this case, which should not affect the model's accuracy because the contribution of shot noise where SNR \(\approx 1\) is infinitesimally small compared to those of thermal noise and dark current. The received power is estimated at the PD's planar surface area resolved for the AOI, not considering its spectral sensitivity. These decisions were taken to better isolate and observe the geometrical implications of the array design from the influences of PD's optical parameters, which are dependent on the manufacturer and choice of photodiode. \[P_{\eta_{tot}}=NEP\times\sqrt{BW}\times I_{PD} \tag{17}\]  
\[P_{\tau_{tot}}=\sum_{i=1}^{I_{PD}}P_{\tau_{a_{i}}} \tag{18}\]  
\[SNR=\frac{P_{\eta_{tot}}}{P_{\tau_{tot}}} \tag{19}\]  
The arrays were first analyzed under movement by horizontal and vertical displacements about the positive quadrant of an x-y cartesian plane, with the origin at the location of the LED emitter. One quadrant is deemed sufficient due to the symmetrical intensity profile about the center axis, collinear with the environment's x-axis. The incremental displacement resolution in both directions was 0.1 m. The results were generated for both three-PD and five-PD arrays for rotations from \(-90^{\circ}\) to \(+90^{\circ}\) for all arrays about the array's center coordinate in incremental intervals of \(30^{\circ}\), for wavelengths of 450 nm (blue), 500 nm (green), and 550 nm (line-green/yellow), to observe the received power while moving within the Jerlov seawater types. The corresponding attenuation parameters are in Table 3. Finally, a bit-error analysis was conducted for each of the previous wavelengths and Jerlov seawater types.