# A Roadmap for Gigabit to Terabit Optical Wireless Communications Receivers
## 3 Results  
### Poisson Statistics and BER  
The dominant noise source in a SiPM receiver is expected to be Poisson noise. If this is the case and an OOK signal is transmitted, the BER can be calculated using the following equation [6]:  
\[BER=\frac{1}{2}\left[\sum_{k=0}^{n_{T}}\frac{\left(N_{Tx}+N_{b}\right)^{k}}{k! }.e^{-\left(N_{Tx}+N_{b}\right)}+\sum_{k=n_{T}}^{\infty}\frac{\left(N_{b} \right)^{k}}{k!}.e^{-N_{b}}\right] \tag{1}\]  
where \(N_{Tx}\) is the number of additional detected photons per bit when a one is received, and \(N_{b}\) is the average number of photons detected per bit time when a zero is received.  
In addition, \(n_{T}\) is the threshold used to differentiate a zero from a one. The value of \(n_{T}\) that is used is the value that minimizes the BER.  
One important consequence of Equation (1) is demonstrated in Figure 1. The results in this figure show that the BER is very sensitive to the number of detecting photons per bit. In addition, this figure shows a linear fit to the relationship between the BER and \(N_{Tx}\) obtained using the linear fitting tool in MATLAB. This fit means that, at least in this range of BERs, the relationship between BER, \(BER\), and \(N_{Tx}\) is as follows:  
\[N_{Tx}=-\left(log_{10}(BER)+0.33\right)/0.42 \tag{2}\]  
The relationship between \(N_{b}\) and the average value of \(N_{Tx}\), determined using the optimum thresholds, is shown in Figure 2. One important feature of Figure 2 is that \(N_{Tx}\) is independent of \(N_{b}\) for small values of \(N_{b}\). The other important feature is that \(N_{Tx}\) is proportional to the square root of \(N_{b}\) at high values of \(N_{b}\)[18]. Previously, these features have been used to predict that, when ISI is negligible, increasing the area of a SiPM by a factor, f, will reduce the irradiance needed to achieve a particular BER by a factor between f\({}^{1/2}\) and f [18]. Unfortunately, the results in Table 1 show that increasing the area of a SiPM also increases the width of the output pulses. This strategy may therefore only be successful at lower data rates.  
In the past, optical filters have been used to reduce the impact of ambient light; however, these filters will not protect a SiPM from one non-ideal characteristic of some transmitters. In particular, transmitters need a wide bandwidth to support data rates of several Gbps, and wide bandwidths are achieved by not turning off transmitters when they are transmitting a zero. The ratio between transmitters' output powers when transmitting a zero, \(P_{0}\), and when transmitting a one, \(P_{1}\), is characterized by the extinction ratio (\(EXR\)):  
\[EXR=P_{1}/P_{0} \tag{3}\]  
The results in Figure 3 show that the extinction ratio can have a significant impact on the number of photons per bit needed to achieve a particular BER. Previously, the extinction ratio of a L405P20 405 nm laser diode used to transmit OOK data rates of less than 2.4 Gbps  
Figure 1: The number of recieved photons required to represent a logic one, \(N_{Tx}\), to achieve a range of BERs when \(N_{b}=0\). The range of BERs shown is close to the value of 3.8 \(\times\) 10\({}^{-3}\) needed when forward error correction is used to significantly reduce the final BER. The results obtained with different integer numbers of additional photons are shown as black circles. The red line is then a linear fit to the points.  
Figure 2: The average number of additional detected photons per bit required to achieve a BER of 3.8 \(\times\) 10\({}^{-3}\) when varying numbers of photons per bit are detected when a zero is being received.  
was found to be 15. The impact of this extinction ratio had to be taken into account when predicting the performance of a system accurately [18]. However, at lower extinction ratios, for example, those less than five, the number of photons per bit increases very rapidly as the _EXR_ reduces. The _EXR_ of a transmitter could therefore play a key role in determining the performance of a system.