## 2 Time-Gated AQ-SPAD Receiver Modeling
### Single Time-Gated SPAD Receiver  
To describe the performance of time-gated AQ-SPAD, it is necessary to establish a photon counting model to characterize the SPAD. In Figure 1b, \(T_{L}\) is the photon arrival time of the previous symbol. \(T_{E}\) is the end time of the avalanche, \(T_{st}\) is the start time of the symbol, and \(T_{C}\) is the symbol duration. Considering ISI distortion, SPAD may have a blocking time after the beginning of the symbol. The blocking time \(T_{Bg}\) can be expressed as:  
\[T_{Bg}=\max\{T_{d}-(T_{st}-T_{L}),0\} \tag{1}\]  
where, \(T_{d}\) is dead time. Considering ISI distortion, the actual detection time of the SPAD is \(T_{C}-T_{B}\). It is worth noting that the blocking time \(T_{Bg}\) is determined by the gate-ON time, dead time, and symbol time.  
If the dead time is longer than the gate-OFF time, ISI distortion may occur in time-gated SPAD, and blocking time \(T_{Bg}\) meets \(T_{Bg}\in[0,T_{g}]\). If the dead time \(T_{d}\) meets \(T_{d}>T_{c}+T_{g}\)  
Figure 1: The output characteristics of free-running mode and time-gated mode. (**a**) Free-running AQ-SPAD and time-gated AQ-SPAD. (**b**) The output of time-gated SPAD. (If the gate-OFF time is less than dead time, ISI distortion may occur in AQ-SPAD).  
SPAD will not detect photons at the next gated-ON time. Therefore, the relationship between blocking time \(T_{Bg}\) and dead time \(T_{d}\) of the gated mode can be expressed as:  
\[T_{Bg}=\left\{\begin{array}{l}0,\hskip 14.226378 ptT_{\rm C}-T_{g}>T_{d}\\ \left[0,T_{g}\right]\hskip 28.452756 ptT_{\rm C}-T_{g}<T_{d}<T_{C}+T_{g}\\ T_{g},\hskip 14.226378 ptT_{\rm C}+T_{g}<T_{d}\end{array}\right. \tag{2}\]  
For the time-gated scheme, the maximum photon count of the SPAD receiver can be calculated as:  
\[K^{\prime}_{MAX}=\left\lfloor\frac{\left(T_{g}-T_{Bg}\right)}{T_{d}}\right \rfloor+1. \tag{3}\]  
The number of effective incident photons can be calculated:  
\[\lambda_{k^{\prime}}=\lambda\left(T_{g}-T_{Bg}-k^{\prime}T_{d}\right) \tag{4}\]  
where, \(k^{\prime}\) represents the actual photons count, \(\lambda\) is the photon incidence rate and the unit is photons/s. Then, the probability density distribution function of the time-gated SPAD photon count can be calculated [21; 22]:  
\[p_{K}(k^{\prime})=\left\{\begin{array}{l}\sum\limits_{i=0}^{k^{\prime}}u(i, \lambda_{k^{\prime}})-\sum\limits_{i=0}^{k^{\prime}-1}u(i,\lambda_{k^{\prime} }),\,k^{\prime}<K^{\prime}{}_{MAX}\\ 1-\sum\limits_{i=0}^{k^{\prime}-1}u(i,\lambda_{k^{\prime}}),\hskip 56.905512 ptk^{ \prime}=K^{\prime}{}_{MAX}\\ 0.\hskip 142.26378 ptk^{\prime}>K^{\prime}{}_{MAX}\end{array}\right. \tag{5}\]  
where, \(u(i,\lambda)=\frac{\lambda^{i}}{i^{\prime}}e^{-\lambda}\). Considering ISI distortion, SPAD counting means and counting variance can be calculated [23]:  
\[\mu_{K}=K^{\prime}_{\rm max}-\sum\limits_{k=0}^{K^{\prime}_{\rm max}-1}\sum \limits_{i=0}^{k^{\prime}}u(i,\lambda_{k^{\prime}}), \tag{6}\]  
\[\sigma_{k^{\prime}}^{2}=\sum\limits_{k^{\prime}=0}^{K^{\prime}_{\rm max}-1} \sum\limits_{i=0}^{k^{\prime}}\left(2k^{\prime}{}_{\rm max}-2k^{\prime}-1 \right)u(i,\lambda_{k^{\prime}})-\left(\sum\limits_{k^{\prime}=0}^{K^{\prime} _{\rm max}-1}\sum\limits_{i=0}^{k^{\prime}}u(i,\lambda_{k^{\prime}})\right)^{2} \tag{7}\]