# Free Space Optic Receiver with Strongly Overlapped Photodetectors' Field of View
## 1 Introduction
### Conditions Preserving Parallel Amplifier Advantages
\begin{array}{l}P(A)=A-\big{(}A^{T}\overline{SZ}\big{)}\overline{SZ}\\ P(B)=B-\big{(}B^{T}\overline{SZ}\big{)}\overline{SZ}\\ P(C)=C-\big{(}C^{T}\overline{SZ}\big{)}\overline{SZ}\\ P(D)=D-\big{(}D^{T}\overline{SZ}\big{)}\overline{SZ}\end{array} \ . \tag{13}\]  
The final task is calculating surface \(S_{p}\) from the \(P\) (\(A\)), \(P\) (\(B\)), \(P\) (\(C\)), and \(P\) (\(D\)) corners:  
\[S_{p}=\frac{1}{2}\|\vec{e}\|\cdot\|\vec{f}\|\cdot\sin\vartheta. \tag{14}\]The diagonals \(\overline{e}\) and \(\overline{f}\) and the \(\vartheta\) angle are defined as:  
\[\overline{e}=P(D)-P(B),\ \ \ \overline{f}=P(C)-\ P(A),\ \ \ \cos\vartheta=\frac{ \overline{e}\cdot\overline{f}}{\|\overline{e}\|\cdot\|\overline{f}\|} \tag{15}\]  
The procedure above may be simplified for truncated pyramid holder dimensions much smaller than the distance between this holder and the point of illumination. First, there is no obstacle to having the points \(A\), \(B\), \(C\), and \(D\) stand symmetrically around the initial position [0 0 0]. Second, the operations \(rotx\) and \(rotz\) are sufficient to properly position the square; therefore, the operations _posy_ and _posz_ are omitted. As such, the ABCD square midpoint \(S\) holds the initial position over all rotations, and the vector \(\overline{SZ}\) is reduced to the point \(Z\) coordinates. The second step is to find all six photodetector surface screenings and their change over the \(\varphi_{z}\) angle (Figure 4). The illumination point \(Z\) Cartesian coordinates are calculated separately corresponding to the run along the circle, as shown in Figure 4, and they are not listed here. The only variable listed is the \(\varphi_{z}\) angle, which is the illumination point \(Z\) Polar coordinate (we assumed that midpoint S holds the initial position). Let there be a truncated pyramid holder positioned according to Figure 4a, where the pyramid edge points toward the illumination source. The illumination source runs along the circle from the \(x\)-axis toward the \(z\)-axis. The change in the photodetector surface screening for segments SA, SB, and SC is depicted in Figure 4a. Similarly, let the truncated pyramid holder be positioned according to Figure 4b, where the pyramid wall points toward the illumination source. Again, the illumination source runs along the circle. The change in the photodetector surface screening for segments SA, SB, SC, and SD is depicted in Figure 4b. The total surface screening from all segments can be found by their sum independent of orientation. For case (a), the total surface screening is the sum of the SA, SB, and SC segments multiplied by 2. For case (b), the total surface screening is the priority sum of the SB and SC segments multiplied by 2 and the additional sum of the SA and SD segments. The surface  
Figure 3: Shifts and rotations of square ABCD. (**a**) Rotating the square on the x-axis, (**b**) shifting the square along the \(y\)-axis, (**c**) shifting the square along the z-axis, (**d**) rotating the square on the z-axis. screening of the shadowed photodetectors is zero. The total surface screening vs. angle \(\varphi_{z}\) is shown in Figure 5. All calculations were performed in Excel software (Microsoft, Redmond, WA, USA, 2007). The minimum angle between the pyramid base and the direction of falling rays can be found as follows. Let each photodetector be described by the unit useful signal current and the unit noise current. It follows that \(SNR_{out}=1\). The parallel amplifier sums the useful signal currents and the noise currents of all photodetectors. The total useful signal current will be a six-multiple of one photodetector's useful signal current. The total noise current will be a \(\sqrt{6}\)-multiple of one photodetector's noise current. Figure 4: Photodetector surface screening for segments SA, SB, SC, and SD over the \(\varphi_{z}\) angle for (**a**) the pyramid edge oriented toward the illumination source and (**b**) the pyramid wall oriented toward the illumination source.