# Temporal Encoding to Reject Background Signals
## 2 Results
### Data Rates  
The achievable data rate of this system is determined by the number of pulse repetitions required to distinguish the signal, and hence the received power and the time separation between pulses. The SPAD response imposes a lower limit on this separation, due to the dead time \(\tau_{d}\) and pulse width, \(\tau_{pulse}\), giving an achievable data rate of:  
\[R_{data}=\frac{1}{N_{rep}(t_{pulse}+\tau_{d})}. \tag{6}\]  
where \(N_{rep}\) is the number of pulse repetitions required to see a distinguishable peak in the correlation histogram. Use of a SPAD array could lift the restrictions imposed by dead time through pulse combining techniques [23]. To demonstrate the system performance as a function of received power and data rate, bit error ratio (BER) measurements were taken. A target BER of \(1\times 10^{-3}\) was used, as FEC codes can be used to reduce this to effectively error free levels at a small overhead on data throughput [24]. A pseudo-random bit sequence (PRBS) of \(10^{4}\) bits was transmitted, limited by the data processing capabilities of the oscilloscope and PC components in the measurement setup. The ND filter wheel allowed control of received power, or equivalently, photon detection probability. Figure 4a shows BER curves for varying data transmission rates, taken with minimal background light. At 8.25 pW of average power, corresponding to an average of 0.34 incident photons per pulse, a data rate of 100 kb/s was possible with a BER of less than \(10^{-3}\). Received optical power can be reduced at the expense of data rate. A data rate of 10 kb/s can be achieved at the same BER with 2 pW, corresponding to 0.08 photons per pulse. The power measurements quoted here and used in Figures 4 and 5 are the incident optical power on the active area of the SPAD, calculated through numerical methods from the average detector count rate. Detector count rate is the parameter which governs BER performance, however the incident optical power will be influenced by the performance of the SPAD. Most importantly, the photon detection probability (PDP) at 405 nm is 18%, so the incident photon flux is significantly higher than the detector count rate. More efficient photon detection would improve BER performance in terms of required power. The system performance can also be described in terms of the number of received photons per bit. Figure 4b shows detected photons per bit for each data rate, at the level required for a BER of less than \(10^{-3}\). The fitted curve is calculated from the relationship between correlation counts, received power and data rate. The number of signal correlations depends on the square of received power and is inversely proportional to the data rate \(R_{data}\). To reach a given target BER, a certain constant number of signal correlation counts must be reached, meaning \((ph/s)^{2}\propto R_{data}\). As photons per bit is the required photons per second divided by data rate, the data follows a \(y=x^{-\frac{1}{2}}\) relationship. The 100 kb/s link is transmitting each bit with an average of 27 detected photons. This is relatively close to the standard quantum limit (SQL), set by Poissonian photon statistics [25]. For a BER of \(10^{-3}\), a minimum of seven photons is required to detect a \"1\". Therefore, an average of 3.5 photons per bit is required, assuming the probability of transmitting \"0\" or \"1\" is equal. The implemented scheme will be unable to reach the SQL, due to the correlation approach. Two photons are required for a single correlation detection, which itself has a Poissonian distribution that must be distinguished from noise. After correcting for detector efficiency, 27 detected photons equate to \(7.37\times 10^{-17}\) J incident on the detector per bit. This exceptionally low energy demonstrates the suitability of the transmission scheme in low power or high loss systems. As mentioned above, more efficient photon detection would allow further reductions in energy received per bit.