## 4 Uncoded BER of APD OOK Receivers
In one extreme, shot noise is negligible vs. thermal noise, and the RFE operates purely in the thermal-noise limit (\(TNL\)):  
\[Q_{TNL}=\frac{R\ \cdot\langle P_{Rx}\rangle}{\sigma_{t}}=\frac{R\ \cdot\langle P_{Rx}\rangle}{i_{t}\cdot\sqrt{B}}=\frac{\sqrt{2r}\cdot R\ \cdot E_{Ph}}{i_{t}}\cdot\langle N\rangle \tag{20}\]  
Thus, \(Q\) is linear to \(<\)\(N\)\(>\), and to achieve a constant \(Q\), the bitwise sensitivity will decrease by \(\sqrt{r}\), i.e., larger bandwidth shall be beneficial in terms of sensitivity. This relation however holds only as long as the receiver's amplifier is not adopted by \(B\). But an adoption might be required to achieve a certain data rate, which at the same time meanschoosing the highest possible \(R_{T\!I}\). Then again, the data rate advantage of \(T\!NL\) practically will be compensated, compare Figure 10.  
In the other ideal extreme of shot-noise limit (\(SNL\)), multiplication \(M\) cancels, excess noise \(F_{A}\) becomes one, and thermal noise \(i_{t}\) is negligible vs the shot-noise:  
\[Q_{SNL}=\frac{M\ R\ \cdot\!2\langle P_{Rx}\rangle}{\sigma_{s,1}}=\sqrt{\frac{2R \ \cdot\langle N\rangle\cdot E_{pb}}{q}}=\sqrt{2\eta\langle N\rangle} \tag{21}\]  
So in \(SNL\), the sensitivity in \(Ppb\) is independent from the data rate, and the run of \(Q\) now corresponds linear to \(\sqrt{\langle N\rangle}\) ([45] ch. 4.5.2). The results here always assume Gaussian noise distributions and the presence of thermal noise during a zero bit. This theoretically is not true for an ideal photon-counting receiver without BGL or dark current, and becomes obvious when the received number of photons is low. There will be no noise during zero, thus, the decision threshold will be at nearly zero, and furthermore, the photon arrival probability will be Poisson-distributed, not Gaussian anymore [53]. The sensitivity numbers calculated with (20) and (21), therefore, deviate from ideal statistics when the numbers of photon per bit becomes very low. In our regime of dozens of photons per one bit, however, the preceding formulas provide a sufficiently accurate approximation. The above findings can be compared on logarithmic scales for the received power, as shown in Figures 8-10, employing real-world APD parameters from Table 11. In Figures 8 and 9, the data rate is varied while the TIAs noise density stays constant. Thus, only the reception filter is adopted by the data rate. We find that when \(M=25\), this is an advantageous fixed value for \(1\) Gbps, when no \(I_{d}\) or \(P_{BGL}\) are present (compare Figure 11), which is, however, only an idealized case. The sensitivity runs of the different RFE types become steeper, regarding SNL via APD to TNL, and typically a 10 dB sensitivity step is seen for the same BER in between these technologies (Figure 8, comparing the plots of the same line style). However, going from PIN to APD is only a moderate technology step, while building and operating a SNL (photon counting) receiver at 15 xx nm wavelength requires much higher complexity and expenditure [54]. Figure 9 plots the BER sensitivity in photons per bit for a fixed M = 25 and a TIA covering rates up to 10 Gbps. Since the TI amplifier's noise density is assumed  
Figure 8: RFE sensitivity for the three RFE cases where \(M=25\). The RFE here has a constant TIA noise density but an adoptive Rx lowpass. constant here (2.1 pA/sqrt(Hz) for all data rates, and only the bandwidth of the separate reception filter is adopted by the data rate, then the sensitivity is rate dependent. This assumption, however, is impractical for many systems since the selection of the TIA component would then depend on the highest employed bandwidth (here 10 Gbps), which would mean unnecessary high noise density values for the lower rates (1 Gbps or 100 Mbps). In other words, the TIAs noise density in a data rate-optimized RFE setup must follow its bandwidth.