# A Roadmap for Gigabit to Terabit Optical Wireless Communications Receivers
## 3 Results
### Impact of Non-Linearity on BER  
The count rate for a SiPM, such as a J30020, can be related to the irradiance of monochromatic light falling on the SiPM, \(L\), via Equation (4):  
\[C_{rate} =N_{cells}\alpha L/\left(1+\alpha\tau_{p}L\right) \tag{4}\]  
where \(N_{cells}\) is the number of microcells in the SiPM and \(\tau_{p}\) is a characteristic time. In addition, parameter \(\alpha\) is as follows:  
\[\alpha=\eta\left(V_{ov},\ \lambda\right)A_{\mu}/E_{p} \tag{5}\]  
where \(\eta\left(V_{ov},\ \lambda\right)\) is the PDE of the SiPM at a particular over-voltage, \(V_{ov}\), and wavelength, \(E_{p}\) is the energy of a photon at this wavelength, and \(A_{\mu}\) is the active area of a microcell. Equation (4) was derived assuming that a microcell cannot detect photons for a time, known as the dead time, whilst it recovers after detecting a photon [4]. This equation has previously been used to explain the relationship between the current needed to sustain the SiPM bias voltage and the monochromatic irradiance falling on the SiPM [11]. However, recent simulations have shown that the assumption that a microcell cannot detect a photon during its dead time is not correct [19]. Fortunately, if \(\tau_{p}\) is approximately 2.2 times the recovery time listed in the datasheets and Table 1, Equation (4) can be used to explain the current needed to sustain the SiPM bias voltage. In addition, it has been shown that this non-linear response occurs on the fast output used to create a receiver [19]. Previously, SiPMs' non-linear responses occurred because SiPMs were exposed to ambient light [19]; however, they will also occur at high transmitter irradiances. An important difference between ambient light and light from the transmitter is that the light from the transmitter is modulated. For a J30020 SiPM, at OOK data rates of 1 Gbps or higher \(\tau_{p}\) is greater than 33 times the bit time. Under these conditions the varying irradiance from the transmitter is expected to be indistinguishable from a constant ambient irradiance. It is therefore expected that the non-linearity observed at high ambient light irradiances will also  
Figure 3: The impact of the extinction ratio of the transmitter on the number of photons per bit needed to represent a one and achieve a BER of 3.8 \(\times\) 10\({}^{-3}\). be observed at higher OOK data rates. If this is the case, then the irradiance, \(L_{NL}\), needed to achieve a particular count rate, \(C_{rate}\), can be calculated by rearranging Equation (4):  
\[L_{NL}=C_{rate}/\left(\alpha\big{(}N_{cells}-\tau_{p}C_{rate}\big{)}\right) \tag{6}\]  
The usefulness of (6) has been investigated using Monte Carlo simulations. In these simulations the BERs achieved at different data rates when the SiPM is assumed to have a linear response were first calculated. Then, the BER was calculated when Equation (6) was used to compensate for a non-linearity. The results in Figure 4a show the simulated BER for a J30020 SiPM with an output pulse width of zero. These results show that, if the SiPM is assumed to have a linear response, then the target BER is only achieved for data rates less than 1 Gbps. In contrast, using Equation (6) to determine the irradiance needed to achieve the required count rate maintains the BER at less than the target BER up to 100 Gbps. The deviations in the BER from the target value arise because Equation (6) is not a perfect representation of the non-linearity, and the results in Figure 1 show that the BER is very sensitive to the number of detected photons per bit. Taking these factors into consideration, the results in Figure 4a confirm that the same non-linearity occurs when high count rates arise from either ambient light or high data rates. The irradiances used to obtain the results in Figure 4a are shown in Figure 4b. These results show that the non-linearity has very little effect on irradiances less than the maximum irradiance that would be available from a 405 nm transmitter in a typical office [12].