## 2 Time-Gated AQ-SPAD Receiver Modeling
### Time-Gated SPAD Array Receiver  
According to the single time-gated AQ-SPAD photon counting model, we build the AQ-SPADs array photon counting model. We assume that each unit in the SPAD array has the same physical characteristics. For the AQ-SPAD array, blocking time vectors \(\mathbf{T_{Bg}}\) can be expressed as:  
\[\mathbf{T_{Bg}}=\{T_{Bg,1},T_{Bg,2},\cdots,T_{Bg,n}\}^{T}. \tag{8}\]  
We assume that the SPAD in the array is an independent count, and the output of the SPADs array can be expressed as:  
\[Y=\sum\limits_{i=1}^{N_{array}}k_{i}(\mathbf{T_{Bg,i}}). \tag{9}\]  
where \(k_{i}\) represents the actual output of each SPAD. The mean and variance of the AQ-SPAD array then can be represented as:  
\[\mu_{Y}=\sum\limits_{i=1}^{N_{array}}u_{k^{\prime}_{i}}(\lambda_{k^{\prime}},T _{Bg,i}), \tag{10}\]\[\sigma_{Y}^{2}=\sum\limits_{i=1}^{N_{array}}\sigma_{K_{i}}^{2}(\lambda_{k^{\prime} },T_{Bg,i}). \tag{11}\]  
According to the mean and variance, the counting model of the time-gated AQ-SPAD array can be approximately calculated as:  
\[P\big{(}y_{n},\mathbf{T_{Bg}}\big{)}=\frac{1}{\sqrt{2\pi\sum\limits_{i=1}^{N_{ array}}\sigma_{K_{i}}^{2}(\lambda_{k^{\prime}},T_{Bg,i})}}\exp\left(-\frac{\left(y_{n}- \sum\limits_{i=1}^{N_{array}}u_{K^{\prime}_{i}}(\lambda_{k^{\prime}_{i^{ \prime}}},T_{Bg,i})\right)^{2}}{2\sum\limits_{i=1}^{N_{array}}\sigma_{K_{i}}^{ 2}(\lambda_{k^{\prime}},T_{Bg,i})}\right) \tag{12}\]  
The probability mass function of the time-gated SPAD array is shown in Figure 2. The histogram of the photon counting distribution is fitted with the Gaussian distribution curve. The gate-ON times are defined as \(0.25T_{C}\), \(0.5T_{C}\), \(0.75T_{C}\), and \(T_{C}\). With the increase of gate-ON time, the count means and variance of \"1\" signal photon and \"0\" signal photon increase. When the gate-ON time is \(0.75T_{C}\) and \(T_{C}\), the output of the SPAD has ISI distortion. The random blocking time makes the fitting accurate, but basically reflects the photon count statistical characteristics of the time-gated SPAD.  
Figure 3 shows the effect of gate-ON time on the photon detection probability and variance of SPAD. We use photon detection probability and variance as the research objects. The ratio of photon detection probability and variance can be approximately regarded as the signal-to-noise ratio of the time-gated SPAD receiver. When the photon flux is weak, increasing the incident photon can improve the photon detection probability of the SPAD. This is because that low photon flux is the main reason affecting the performance of time-gated AQ-SPAD. Increasing the gate-ON time can effectively improve the photon flux. The detection performance of SPAD increases with the rises of gate-ON time. With the rise of incident photon flux, the photon detection probability of SPAD still benefits from the rise of gate-ON time, and the variance of detection probability tends to be stable. Increasing the gate-ON time is beneficial to the performance of time-gated SPAD. Further increasing photon flux and gate-ON time, the promotion of the SPAD receiver detection  
performance is no longer significant. SPAD only requires a shorter gate-ON time to meet the detection performance requirements, the variance is slight, and the statistical characteristics of photons are better. Therefore, when the photon flux is weak, the gate-ON time can be increased to enhance the detection performance of the SPAD. When the photon flux is intense, the time-gated SPAD can be optimized without loss of performance by selecting a reasonable gate-ON time.