# Free Space Optic Receiver with Strongly Overlapped Photodetectors' Field of View
## 1 Introduction
### Conditions Preserving Parallel Amplifier Advantages  
Generally, an FSO multiple photodetector receiver is useful mainly in multipath light propagation environments, making all photodetectors useful in signal receiving. Situations may arise where the useful optical signal reaches a lower photodetector count than the FSO receiver. This reduces the total useful signal photocurrent, though the total photodiode noise current stays the same. As a result, the signal to noise ratio \(SNR_{out}\) in Equation (7) also becomes smaller because the shadowed photodetectors are only delivering noise. Therefore, the aim was to find some marginal conditions that ensure that the \(SNR_{out}\) of all photodetector assemblies (PDAs) are equal to the \(SNR_{out}\) of only one PDA  
Figure 2: Parallel amplifier block diagram. in use. These conditions might be expressed, for example, by a minimum angle between the pyramid base and the direction of falling rays. Unfortunately, there is no universal minimum angle, although many minimum angles are available that vary according to the pyramid's shape and the number of walls. The minimum angle may also be calculated for any particular case. Analytic geometry offers many powerful tools, some of which are further described. For more details, see [PERSON], [PERSON] et al. [28], [PERSON] [29]. The first step is calculation of all six photodetectors coordinates. The procedure listed below is related to the calculation of the coordinates of only one photodetector. Therefore, for all six photodetectors, the calculation must be performed six times because each photodetector has a different place on the truncated pyramid. Inputs are rotx and rotz angles, eventually posy and posz distances. Let ABCD be a square positioned initially. Its points are expressed by the vectors \(A=[\text{a}_{1}\ \text{a}_{2}\ \text{a}_{3}]^{\text{T}}\), \(B=[\text{b}_{1}\ \text{b}_{2}\ \text{b}_{3}]^{\text{T}}\), \(C=[\text{c}_{1}\ \text{c}_{2}\ \text{c}_{3}]^{\text{T}}\), and \(D=[\text{d}_{1}\ \text{d}_{2}\ \text{d}_{3}]^{\text{T}}\). These might represent the photodiode active area corners. To rotate the square on the x-axis or z-axis, the vectors must be multiplied by the matrix \(Arotx\) or \(Arotz\), respectively, as follows:  
\[Arotx=\left[ \begin{array}{ccc}1&0&0\\ 0&\cos(rotx)&-\sin(rotx)\\ 0&\sin(rotx)&\cos(rotx)\end{array} \right],\ Arotz=\left[ \begin{array}{ccc} \cos(rotz)&-\sin(rotz)&0\\ \sin(rotz)&\cos(rotz)&0\\ 0&0&1\end{array} \right]. \tag{8}\]  
The operations are depicted in Figure 3a,d. Written mathematically:  
\[A^{\prime}=Arot\cdot A,\ B^{\prime}=Arot\cdot B,\ C^{\prime}=Arot\cdot C,\ D^{ \prime}=Arot\cdot D. \tag{9}\]  
The shift along the \(y\)-axis or z-axis enables the matrix _posy_ or _posz_ as follows, respectively:  
\[posy=\left[ \begin{array}{ccc}0&vy&0\end{array} \right]^{T},\ posz=\left[ \begin{array}{ccc}0&0&vz\end{array} \right]^{T}. \tag{10}\]  
The operations are depicted in Figure 3b,c. Written mathematically:  
\[A^{\prime}=A+pos,\ B^{\prime}=B+pos,\ C^{\prime}=C+pos,\ D^{\prime}=D+pos. \tag{11}\]  
The operations above lead to the \(A^{\prime}\), \(B^{\prime}\), \(C^{\prime}\), \(D^{\prime}\) point coordinates introducing the square's final position. For simplicity, they are renamed as points \(A\), \(B\), \(C\), and \(D\), respectively. Knowing the final coordinates of square ABCD, the midpoint S of the square may be calculated. Now, it is rotated to calculate the vector \(\overline{SZ}\), where \(S\) and the \(Z\) are the square's midpoint and illumination point, respectively:  
\[\overline{SZ}=Z-S=\bigg{(}z_{1}-\frac{a_{1}+b_{1}+c_{1}+d_{1}}{4},z_{2}-\frac {a_{2}+b_{2}+c_{2}+d_{2}}{4},z_{3}-\frac{a_{3}+b_{3}+c_{3}+d_{3}}{4}\bigg{)}. \tag{12}\]  
The screening P of the square ABCD to the plane perpendicular to the direction of falling rays, i.e., vector \(\overline{SZ}\), can be found as individual points \(A\), \(B\), \(C\), \(D\) screening:  
\[