## 3 Optimum Multiplication for CW Illumination  
Depending on \(P_{0}\), the optimum multiplication factor \(M_{opt,CW}\) can be found for the CW case by solving \(d(SNR)/dM=0\).  
With the exact relation or the linear approximation for \(F_{A}\), further simplifications would be necessary to solve for \(M_{opt,CW}\). However, with the exponential approximation (5), we can derive a closed-form solution for the optimum multiplication factor with CW-illumination:  
\[M_{opt,\ CW}=\left(\frac{2q\ I_{dm}+i_{t}{}^{2}}{x\cdot q\ (RP_{Rx}+RP_{BGL}+I_{ dm})}\right)^{\frac{1}{2+2}} \tag{7}\]  
with \(x=k_{A}{}^{0.355}\).  
\(M_{opt,CW}\) is a function of the received power and further RFE parameters, but is independent of the observation bandwidth \(B\).  
For typical InGaAs-APDs and TIAs (using for \(i_{t}\) a typical 2.1 pA Hz\({}^{-0.5}\)), we find \(10<M_{Opt}<30\) (see Figure 3).  
We assume here that the excess noise factors for signal electrons and multiplied dark current electrons are equal; this assumption causes negligible errors when the signal current is dominating the total avalanche current.  
Figure 2: Comparison of approximations (5) and (6) with exact solution (4).  
Dependency of M from Bias Voltage and Temperature  
The electric field induced through the reverse voltage, \(U_{R}\), causes the electron-hole avalanche, starting with M = 1 at a minimum operating voltage (typically one-third of the breakdown voltage, \(U_{BD}\)). Beyond \(\sim\)0.5-\(U_{BD}\), the multiplication increases according to (8) up to a pole at \(U_{BD}\) (see Figure 4). When no optical data power is applied, only the APDs dark current is multiplied, and the \(U_{BD}\) is then typically defined as the voltage, in which the reverse current exceeds 100 \(\upmu\)A.  
The relationship between bias voltage, \(U_{R}\), and gain, \(M\), is related by a simple rational term [42; 43]:  
\[M(U_{R})=\frac{1}{1-\left(\frac{U_{R}-U_{BD}\cdot R_{R}}{U_{BD}}\right)^{n}} \tag{8}\]  
Figure 4: Measured and calculated \(M\), according to (8), over the reverse voltage \(U_{R}\) for \(U_{BD}\) = 55 V and \(n\) = 1.15.  
Figure 3: Optimum multiplication factor \(M_{opt}\) with CW illumination, for different \(k_{A}\), without and with \(L_{int}\) = 2.5 nA and \(P_{BGL}\) = 50 nW.  
The term \(I_{APD}\cdot R_{S}\) in (8) indicates the voltage lost across the additional series resistance, \(R_{S}\), of the photodiode (resistance of contacts and nondepleted semiconductor region). This resistance is typically around 1 k\(\Omega\), and the voltage-term is thus negligible for data communication. But it needs to be regarded in a situation with high optical power, especially since \(I_{APD}\), again, is a function of the multiplication factor. Exponent \(n\) can be adopted to an improved fitting to measured behavior. Typically, its value is slightly above 1.  
Figure 4 compares the measured \(M\) of an InGaAs-APD-RFE (derived from RFE #7 of [35]) with the fit, according to (8). We understand that for this type of APD, the \(U_{R}\) is typically 4% to 8% below the breakthrough.  
\(U_{BD}\), as well as \(U_{R}\), reduces along with the temperature, and can be approximated by a linear fit using the temperature coefficient \(\rho_{T}=\Delta U_{R}/\Delta T\)[44]  
\[U_{R}=U_{R,ref}+\rho_{T}\cdot\left(T-T_{ref}\right) \tag{9}\]  
We find the required operating voltage, \(U_{R}\), for a certain \(M\) by solving (8) with (9) (neglecting the series resistance), as illustrated in Figure 5:  
\[U_{R}=\left[U_{BD,ref}+\rho_{T}\cdot\left(T-T_{ref}\right)\right]\cdot\left(1 -\frac{1}{M}\right)^{\frac{1}{n}} \tag{10}\]  
We now can use the optimum \(M\) from (7) to describe the complete APD voltage control loop for CW illumination, including the temperature compensation.