# Free Space Optic Receiver with Strongly Overlapped Photodetectors' Field of View
## 2 Free Space Optic (FSO) Receiver Design
### Selecting a Suitable Follow-Up Amplifier
Since all components are in parallel, the input impedance \(Z_{in}\) may be expressed as:  
\[\frac{1}{Z_{in}}=\frac{1}{R_{in}}+j\omega C_{in}=\left(\frac{1}{R_{p}}+\frac{ 1}{R_{a}}+\frac{1}{R_{fb}}\right)+j\omega\Big{(}C_{p}+C_{a}+C_{fb}\Big{)}\quad \text{[S]}. \tag{23}\]  
The whole PDA noise power current spectral density is given by the equation:  
\[\overline{J_{PDA}^{2}}=\frac{FN}{f}+2 qI_{0}+2 qI_{a}+\frac{4 kT}{R_{in}}+\frac {4 kT\psi}{G}\omega^{2}C_{in}^{2}\quad\Big{[}\text{A}^{2}\text{Hz}^{-1}\Big{]}. \tag{24}\]  
where \(FN/f\) is the flicker noise of the photodiode and amplifier; \(2 qI_{0}\) and \(2 qI_{a}\) are the shot noises of the photodiode and amplifier, respectively; \(4 kT/R_{in}\) is the thermal noise of the total input resistance \(R_{in}\) according to Equation (23) and \(4 kT\left(\Psi/G\right)\omega^{2}C_{in}{}^{2}\) is the amplifier output noise current density calculated for the amplifier input. This current density can also be expressed in shorter form as \(e_{n}{}^{2}\omega^{2}C_{in}{}^{2}\). Since the shot and thermal noises are independent of frequency, they are white noise and may be substituted by element \(2 qI_{n}=2 qI_{0}+2 qI_{a}+4 kT/R_{in}\). It is useful to unify Equation (24):  
\[\Xi(f)=\frac{\overline{I_{PDA}^{2}(f)}}{2 qI_{n}}=\frac{FN}{2 qI_{n}}\frac{1}{ f}+1+\frac{\left(2\pi C_{in}\right)^{2}\overline{e_{n}^{2}}}{2 qI_{n}}f^{2}= \frac{f_{lowN}}{f}+1+\frac{f^{2}}{f_{highN}^{2}}\quad[/]. \tag{25}\]  
Figure 6: (**a**) PDA direct topology; (**b**) PDA feedback topology; (**c**) PDA noise model including photodiode and amplifier intrinsic elements and discrete passive elements needed for operation. From Equation (25), the low cut-off and the high cut-off frequencies of white noise will be:  
\[f_{lowN}=\frac{FN}{2 qI_{n}}\quad[\mathrm{Hz}],\ \ \ f_{highN}=\frac{\sqrt{2 qI_{n}}}{2 \pi C_{in}\overline{e_{n}}}\quad[\mathrm{Hz}]. \tag{26}\]  
After preparation of the above, the low frequency PDA and the high frequency PDA can be classified and optimized. Because the described FSO receiver is a high frequency PDA (with high cut-off frequency \(f_{h}=1.5\) MHz), only the latter optimization is discussed. High frequency DPA noise optimization is performed with respect to the \(e_{n}{}^{2}\omega^{2}C_{in}{}^{2}\) element, which becomes dominant. An amplifier with the lowest input voltage noise \(e_{n}\) and the lowest input capacitance \(C_{a}\) (determining \(C_{in}\)) is necessary. The other conditions are as follows:  
\[\frac{4 kT}{R_{in}}\leq\overline{e_{n}^{2}}\omega_{h}^{2}C_{in}^{2}\quad \Rightarrow\quad R_{in}\geq\frac{4 kT}{\overline{e_{n}^{2}}\omega_{h}^{2}C_{in }^{2}} \tag{27}\]  
\[\frac{4 kT}{R_{in}}=2 qI_{a}\quad\Rightarrow\quad I_{a}=\frac{4 kT}{2 qR_{in}}= \frac{4U_{T}}{2R_{in}}=\frac{2U_{T}}{R_{in}} \tag{28}\]  
\[\frac{1}{2\pi R_{in}C_{in}}\geq f_{h} \tag{29}\]  
where \(R_{in}\) in Equation (28) is calculated from Equation (27). Equation (29) serves only to verify that the high cut-off frequency of PDA \(f_{h}\) is fulfilled. If not, high frequency amplification loss should be compensated by a separate corrective circuit. An operational amplifier whose gain is big enough can cancel this loss and maintain the stability of the amplification up to frequency \(f_{x}\) (see below). Even though much has been discussed about follow-up amplifier design, bipolar or unipolar transistor use should be defined. Generally, the sources of noise in unipolar transistors are mutually independent and may be optimized much more easily. This is not the case for bipolar transistors, in which the lowering of input noise is linked to the lowering of input current, which further leads to lower transistor gain and higher output noise effects. Because of negligible FET input currents, they are the best choice in most cases. [PERSON], M.A. [1] provides more details. The final point concerns the FET itself.