# Free Space Optic Receiver with Strongly Overlapped Photodetectors' Field of View
## 2 Free Space Optic (FSO) Receiver Design
### Selecting a Suitable Follow-Up Amplifier
Based on the recommendations above, a careful designer looks for the lowest voltage noise \(e_{n}\) and the lowest input capacitance \(C_{a}\) in a discrete FET or the input FET operational amplifier. However, it has been proven that FET voltage noise \(e_{n}\) and FET input capacitance \(C_{a}\) are reciprocal [1]: the lower the voltage noise \(e_{n}\), the higher the input capacitance \(C_{a}\) and vice versa. In other words, capacitive matching is needed:  
\[C_{a}=C_{p}. \tag{30}\]  
For completeness, stability is required. Figure 7 is helpful with this explanation. \[\frac{1}{Z_{in}}=\frac{1}{R_{in}}+j\omega C_{in}=\left(\frac{1}{R_{p}}+\frac{ 1}{R_{a}}+\frac{1}{R_{fb}}\right)+j\omega\Big{(}C_{p}+C_{a}+C_{fb}\Big{)}\quad \text{[S]}. \tag{23}\]0 Stability investigation of (**a**) uncompensated, (**b**) low frequency; and (**c**) high frequency PDA using Bode plots. The blue line is the noise gain \(NG\) defined, for example, by [PERSON] [32]; the red line is the follow-up amplifier \(K_{21}\) high frequency roll-off. System stability is determined by the intersection angle between the blue and red lines. If the angle is equal to the red line and the horizontal axis angle, the system is stable; if it is greater, the system is unstable. The pool frequency \(f_{p}\) of the amplifier passive network is defined as:  
\[\frac{1}{Z_{in}}=\frac{1}{R_{in}}+j\omega C_{in}=\left(\frac{1}{R_{p}}+\frac{ 1}{R_{a}}+\frac{1}{R_{fb}}\right)+j\omega\Big{(}C_{p}+C_{a}+C_{fb}\Big{)}\quad \text{[S]}. \tag{23}\]1  
The zero frequency is determined only by feedback impedance:  
\[\frac{1}{Z_{in}}=\frac{1}{R_{in}}+j\omega C_{in}=\left(\frac{1}{R_{p}}+\frac{ 1}{R_{a}}+\frac{1}{R_{fb}}\right)+j\omega\Big{(}C_{p}+C_{a}+C_{fb}\Big{)}\quad \text{[S]}. \tag{23}\]2  
The transit frequency \(f_{T}\) is determined by the amplifier roll-off and the 0 dB level intersection, which is also known as the gain-bandwidth product of an operational amplifier (GBW). Finally, the amplifier roll-off and noise gain also intersect, whose frequency \(f_{x}\) is:  
\[\frac{1}{Z_{in}}=\frac{1}{R_{in}}+j\omega C_{in}=\left(\frac{1}{R_{p}}+\frac{ 1}{R_{a}}+\frac{1}{R_{fb}}\right)+j\omega\Big{(}C_{p}+C_{a}+C_{fb}\Big{)}\quad \text{[S]}. \tag{23}\]3  
The frequency \(f_{x}\) should equal \(f_{h}\), which is the high cut-off frequency of the PDA. For the FSO receiver discussed, this is \(f_{h}=f_{x}=1.5\) MHz. Based on the theoretical analysis, it is now possible to proceed with the calculation of the specific components. The suitable low input voltage noise \(e_{n}\) and capacitance \(C_{a}\) operational amplifiers are listed in \[\frac{1}{Z_{in}}=\frac{1}{R_{in}}+j\omega C_{in}=\left(\frac{1}{R_{p}}+\frac{ 1}{R_{a}}+\frac{1}{R_{fb}}\right)+j\omega\Big{(}C_{p}+C_{a}+C_{fb}\Big{)}\quad \text{[S]}. \tag{23}\]4  
In Table 2, input capacitance \(C_{a}\), input voltage noise \(e_{n}\), and transit frequency \(f_{T}\) are typical values seen in operational amplifier datasheets. Total input capacitance \(C_{in}\) is the sum of the BP104F photodiode typical capacitance \(C_{p}=16\) pF (10 V reverse biased), stray capacitance \(C_{s}=3\) pF and the operational amplifier input capacitance \(C_{a}\). The frequency \(f_{x}\) was matched to the PDA's high cut-off frequency \(f_{h}\) (\(f_{x}=f_{h}=1.5\) MHz). Due to the high frequency PDA, high cut-off frequency \(f_{h}\) is the same as the zero frequency \(f_{z}\), shown in Figure 7\(c,f_{z}=f_{h}=1.5\) MHz. The amplifier output noise current density \(e_{n}{}^{2}\omega_{x}{}^{2}C_{in}{}^{2}\) was determined by calculation. The \(R_{in}\) and \(I_{a}\) conditions are derived from Equations (27) and (28), respectively. Similarly, pool frequency \(f_{p}\), feedback resistance \(R_{fb}\) (which determines \(R_{in}\)), and feedback capacitance \(C_{fb}\) are derived from Equations (33), (31) and (32), respectively.