## 2 Gain and Noise in Avalanche Photodiodes  
In this section we consider the APDs sensitivity for an unmodulated or continuous wave (CW) optical signal illuminating the photosensitive detector area. The received optical power, \(P_{Rx}\), in the form of photons impinging onto the detection area of a signal receiver, is the exciting of photoelectrons from the semiconductor's valence-band, via the detector material's responsivity, \(R\). This responsivity depends on signal wavelength, \(\lambda\), and the semiconductor's conversion efficiency, \(\eta\). An additional multiplication gain, \(M\), is achieved in the APDs by an electric field inside the detection volume, which accelerates the primary photoelectrons. These again strike out further (secondary) electrons in the multiplication region. This way, the signal current is increased by an internal multiplication factor of \(M\):  
\[I_{Sig}=M\cdot P_{Rx}\cdot R=M\cdot P_{Rx}\cdot\left(\eta\cdot\frac{q\lambda}{ hc}\right) \tag{1}\]  
Figure 1: Practical InGaAs-APD receiver diagram for high-speed free-space optical (FSO) communications. \(c=2.998\cdot 10^{8}\) m/s (speed of light in vacuum)  
\(h=6.626\cdot 10^{-34}\) Ws\({}^{2}\)  
\(q=1.602\cdot 10^{-19}\) As  
\(R\), Responsivity [A/W], for \(\lambda=1550\) nm is \(R=\eta\cdot 1.250\) A/W  
\(\eta\), quantum efficiency of the diode's detection region (typ. \(\sim\)0.75 to 0.9)  
The reception of a static power level is subject to variations in the photon-excited photoelectrons due to stochastic conversion variations and the variations of the photon-arrival probability itself. In APDs (as well as in most electron multiplication devices), the multiplication effect itself is subject to variations, resulting in additional noise effects. Expressions for the accordant distributions have been derived by [PERSON] [38; 39], confirmed experimentally by [PERSON] [40], and reviewed and simplified by [PERSON] [41]. The signal-to-noise power ratio is derived for CW illumination as the ratio of the square of the signal current (from constant illumination, where we neglect the influence from level-offsets by background light and dark currents) to the sum of shot-noise and thermal-noise variances in the current (\({\sigma_{s}}^{2}\) and \({\sigma_{t}}^{2}\)), where shot-noise variance \({\sigma_{s}}^{2}\) in [\(A^{2}\)] is calculated from any flowing current \({I\sigma_{s}}^{2}=2q\cdot I\cdot B\), and thermal noise density \(i_{t}\) [\(A/\sqrt{Hz}\)] defines the thermal noise variance \({\sigma_{t}}^{2}={i_{t}}^{2}\cdot B\). \[ \begin{array}{c}\frac{S}{N}=\left(\frac{{I_{sig}}}{{I_{n}}}\right)^{2}=\frac {{I_{sig}}^{2}}{{\sigma_{s}}^{2}+{\sigma_{t}}^{2}}=\\ =\frac{({P_{R}}_{R}RM)^{2}}{{B\cdot\left\{2q\right\}}[{I_{th}}+({P_{Rx}}{F_{sm }}+({I_{dm}}+{R\cdot P_{RGL}}){F_{dm}})\cdot M^{2}]+{i_{t}}^{2}\right\}}\end{array}  \tag{2}\]  
With the APDs multiplication factor \(M\), the unmultiplied dark current component is \(I_{dt}\), multiplied dark current component is \(I_{dm}\), responsivity of the detector material is \(R\), excess noise factor for signal current is \(F_{sm}\), excess noise factor for the multiplied dark current is \(F_{dm}\), and observation bandwidth is \(B\). \(I_{dm}\) is given through the biasing of the detector material itself, but is constant around the operating bias point, \(U_{APD}\); it will however generate a dark-current component, which is variable by its multiplication with \(M\). The thermal noise current density of the succeeding amplifier stage \(i_{t}\) can practically be found in data sheets, or is stated with the load resistor, \(R_{L}\), noise figure, and other constants (compare Appendix A.2). False background light power, \(P_{BGL}\), can illuminate the detector and is added to the multiplied dark current. By making \(M\) large, the \(M^{2}\) term in the denominator of (2) becomes larger than the thermal noise of the amplifiers, improving the SNR towards shot noise limitation. However, the APDs excess noise factor, \(F_{A}\), will cause a local maximum here.