# Free Space Optic Receiver with Strongly Overlapped Photodetectors' Field of View
## 1 Introduction
### Parallel Amplifier Topology  
Before proceeding to the problem of parallel amplifying and receiving with several photodetectors, the receiving using only one photodetector was analyzed. It is important to separate the optical and electrical parts of signal detection [26] and to determine in which part noise might prevail. Written mathematically, the signal-to-noise ratio \(SNR_{in}\) in the photodetector's optical part is given by the total captured useful signal power \(p_{s}\) to the total captured noise power \(p_{n}\) in an optical channel:  
\[SNR_{in}=\frac{p_{s}}{p_{n}}\ [/\cdot]. \tag{1}\]The electrical part of the optical photodetector has a larger number of factors. Noise from the photodetector itself and the follow-up amplifier must be considered. Let this noise be expressed by the quantity \(p_{nA}\). The signal-to-noise ratio (SNR) in the photodetector's electrical part is given by:  
\[SNR_{out}=\frac{p_{s}K_{21}^{2}}{p_{n}K_{21}^{2}+p_{nA}}\ [/]. \tag{2}\]  
The \(p_{s}/p_{n}\) ratio is given by the photodetector's optical part and there is no possibility of increasing the ratio in the photodetector's electrical part. However, if sufficient power amplification \(K_{21}{}^{2}\) is not available, the final signal to noise ratio \(SNR_{out}\) is much lower due to \(p_{nA}\). The SC technique can produce the highest \(SNR_{in}\) since the spatial angle, along with unwanted spurious signals, is reduced. However, complexity and low reliability come as trade-offs. Note, no easy way of recognizing the strongest signal photodetector exists. Many interference optical signals are due to, for example, artificial illumination, sunlight or lightning, which might be stronger than the optical signal used and can confuse even the strongest signal photodetector selection circuit. The parallel amplifier accommodates \(s\) photodetectors and their follow-up amplifiers, which work continuously in a common resistive load (Figure 2). Let all photodetectors have uniform signal and noise coverage. The useful optical signal power \(p_{s}\), together with optical noise power \(p_{n}\), enter the receiver in the same way. The total useful signal and noise currents must be \(s\) times higher than those for one photodetector, and similarly, the total useful signal and noise powers must be \(s^{2}\) higher than those for one photodetector. Each photodetector and its follow-up amplifier contribute their own noise power \(p_{nA}\). Since all photodetectors are standalone components, their noise powers are uncorrelated and expressed as \(p_{nA1}\), \(p_{nA2}\),..., \(p_{nAs}\). To prove the merits of a parallel amplifier, its uncorrelated noise powers must be summed. The procedure is as follows [27]:  
Let the random and uncorrelated sequences \(X_{n}\), \(n\) = 1,2,..., \(N\) and \(Y_{m}\), \(m\) = 1,2,..., \(N\) have the same number of realizations \(N\) such that each member of the first sequence \(X_{n}\) makes \(N\) possible sums with the members of the second sequence \(Y_{m}\). This is mathematically expressed as \(X_{n}+Y_{m}\), where \(m\) = 1,2,..., \(N\). Since the sequences are the same size, \(N^{2}\) sums with different indexes \(nm\) exist as a result. Figure 1: The truncated pyramid holder used for fixing six photodetectors \(\mathrm{PD_{1}}\) to \(\mathrm{PD_{6}}\). Points A, B, C, and D, and S1, S2, and S mark the boundary and midpoints of one segment. The vector \(\overline{n}\) means the normal vector of \(\mathrm{PD_{1}}\) photodetector active area. PD: photodetectors. The mean power value of the sum of the single \(X_{n}\) sequence member and the all \(Y_{m}\) sequence members can be written as:  
\[\frac{1}{N}\sum_{m=1}^{N}(X_{n}+Y_{m})^{2}=\frac{1}{N}\sum_{m=1}^{N}X_{n}^{2}+Y_{ m}^{2}+2X_{n}Y_{m}=X_{n}^{2}+\frac{1}{N}\sum_{m=1}^{N}Y_{m}^{2}+2X_{n}\frac{1}{N} \sum_{m=1}^{N}Y_{m}. \tag{3}\]  
If the \(Y_{m}\) sequence mean value is zero, then the last part of Equation (3) must also be zero.