## 5 Optimum \(M\) with Background Light
### Optimum Multiplication Factor with BGL  
From the above analysis (19), we compare the BER with \(M\), without and with BGL. The optimum multiplication factor for the examples in Figure 11 varies from \(\sim\)10 to \(\sim\)40, and a wrong value can reduce the BER by orders of magnitude, e.g., with a 50 nW BGL and a 200 Ppb \(M_{opt}\) = 12, whereas, without BGL the \(M_{opt}\) = 28, and using the optimum \(M\) from before would increase the BER from \(10^{-8}\) (at \(M\) = 28) to \(10^{-6}\). Controlling \(M\) dependent on the varying <\(P_{Rx}\)> and BGL, therefore, is a must in optimized FSO-APD-receivers. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**W/(m\({}^{2}\) nm sr)** & \(\lambda\) **= 850 nm** & \(\lambda\) **= 1064 nm** & \(\lambda\) **= 1550 nm** \\ \hline on Sun-disk & \(20\times 10^{3}\) & \(10\times 10^{3}\) & \(2\times 10^{3}\) \\ blue sky zenith & \(2.0\times 10^{-3}\) & \(2.3\times 10^{-3}\) & \(1.2\times 10^{-3}\) \\ blue sky \(30^{\circ}\) el. & \(3.5\times 10^{-3}\) & \(4.0\times 10^{-3}\) & \(2.0\times 10^{-3}\) \\ blue sky horizon & \(30\times 10^{-3}\) & \(30\times 10^{-3}\) & \(25\times 10^{-3}\) \\ sunlit cloud & \(200\times 10^{-3}\) & \(80\times 10^{-3}\) & \(20\times 10^{-3}\) \\ overcast cloud & \(20\times 10^{-3}\) & \(8\times 10^{-3}\) & \(2\times 10^{-3}\) \\ on Moon-disk & \(400\times 10^{-3}\) & \(220\times 10^{-3}\) & \(20\times 10^{-3}\) \\ full Mars-disk & \(11\times 10^{-12}\) & \(8\times 10^{-12}\) & \(3\times 10^{-12}\) \\ \hline \end{tabular}
\end{table} 
Table 1: BGL spectral irradiance at different wavelengths. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Scene** & \(L_{e,\Omega\lambda}\)**in W/(m\({}^{2}\) nm sr)** & \(A\) **in m\({}^{2}\)** & \(\omega\) in \(\upmu\)rad** & \(\Delta\lambda\) **in nm** & \(P_{\text{BGL}}\) **in nW** \\ \hline towards & & & & & \\ horizon & & & & & \\ towards & & & & & \\ horizon & \(25\times 10^{-3}\) & 0.7 & 100 & 40 & 88 \\ to zenith & \(1.2\times 10^{-3}\) & 0.1 & 200 & 40 & 2.4 \\ to zenith & \(1.2\times 10^{-3}\) & 0.7 & 100 & 40 & 4.2 \\ \hline \end{tabular}
\end{table} 
Table 2: Typical values of \(P_{BGL}\) for a 1550 nm wavelength. As with (7), from (19) we can derive an optimum \(M\) for the maximum \(Q\) for APD data receivers, with the exponential approximation of \(F_{A}\) and with NRZ modulation. Now, again, regarding \(I_{dm}\) and BGL:  
\[ \begin{array}{l}\frac{dQ}{dM}=0\ \to d\left(\frac{M}{\sqrt{M^{2+x}A+B+ }\sqrt{M^{2+x}(A+C)+B}}\right)/dM=0\\ with\quad A=2q(RP_{BGL}+I_{dm})\ ;\ B=i_{t}^{2}\ ;\ C=2 qR\cdot 2\langle P_{Rx} \rangle\ ;\ x=k_{A}0^{.355}\end{array}  \tag{23}\]  
In contrast to the term with CW illumination, the derivation here results in:  
\[M_{opt}=\left(\frac{-b+\sqrt{b^{2}-4 ac}}{2a}\right)^{\frac{1}{2+x}} \tag{24}\]  
with  
\( \begin{array}{l}a=x^{2}AC(A+C)\\ b=x^{2}BC(2A+C)\\ c=-4(x+1)B^{2}C\end{array} \)  
As shown in Figure 12 for different \(k_{A}\), \(M_{opt}\) with data modulation differs significantly from CW illumination (compare Figure 3). The relationship in Chapter 4.5.2 of [45], and from [46], also describe the relation for an optimum \(M\); however, this requires numerical evaluation. For the following comparisons, we assume the values from Table 2 with a 1 Gbps data rate. The optimization of \(M\) is independent of the data rate (which only requires the fraction of on-time vs. off-time to be 50:50) but depends on the received signal power <\(P_{Rx}\)>. When regarding bit length, however, we can relate this received power to energy per bit. With the optimization of \(M\) from (23) and (24), we understand the importance of adjusting \(M\) to the received power, especially for the low-power/high-BER regime. Figure 13 shows the effect of the optimized vs. fixed \(M\) on the BER, with and without background light and dark current. Figure 12: Optimum multiplication factor \(M_{opt}\) with an NRZ-50:50 modulated signal for different ionization ratios, according to (24).