# A Photon Imaging Detector Model with High Resolution and High Counting Rate
## 3 Model Description  
The arriving interval of photon events conforms to the Poisson distribution [22; 23]; the probability of N electron clouds emanating from MCP stack in time t is derived:  
\[P(N,t)=\frac{\left(Rt\right)^{N}\exp(-Rt)}{N!} \tag{1}\]  
where \(R\) is the average photon-counting rate, taking into account quantum efficiency. Depending on the bias voltage of the MCP, the total charge of the electron cloud conforms to Poisson or Gaussian distribution [24; 25]. The broadening of the electron cloud has to be optimized to fit the size of the anode, which is related to the amount of charge, the distance, and the voltage between the MCP and the XS anode. A too-narrow charge footprint results in an under-sampled charge distribution. At the same time, a too-wide charge footprint leads to charge division between a large number of anode electrodes and, consequently, to the reduction in the signal-to-noise ratio. Generally, when the electron cloud covers 5-7 strips, the most accurate centroid calculation results can be obtained [13]. The anode strip output signal is a short-duration current pulse. The CSA converts this current pulse to a voltage pulse whose amplitude is proportional to the input charge. The CSA output pulse can be approximately mathematically represented as:  
\[V_{\text{CSA}}(t)=A\cdot e^{-\frac{t}{\tau}} \tag{2}\]  
where \(\tau\) is the decay time constant of the CSA output pulse. \(A\) is the amplitude of the CSA output pulse, which is proportional to the corresponding anode strip charge. The decay time depends on the combination of the feedback resistance and the integral capacitance, as well as the parameters of the pole/zero cancellation circuit. At a high count rate, the next pulse will be superimposed on the tail of the previous pulse, making the accurate extraction of pulse amplitude more difficult. To prevent the pulse pile-up, a CSA with a short decay time (80 ns) is selected. Then, the CSA output pulse is digitized by a 12 bit ADC with a 62.5 MSPS sampling rate. Subsequently, a voltage variation corresponding to the noise of the front end of the detector is added/subtracted to the pulse in each channel. The noise at the front end of the detector mainly includes charge amplifier noise, charge division noise, capacitance noise, and ADC quantization noise, etc. The total noise can be generated according to the Gaussian distribution for a given RMS value by a random number generator. The digitized CSA pulse is input into FPGA to be shaped and filtered into a trapezoidal pulse. The structure of the digital trapezoidal shaping filter is shown in Figure 4. It is composed of four modules (DS1, DS2, HPD, and ACC2) [26; 27]. M is the time constant of the input pulse. For two superimposed pulses with an interval of 20 samples (320 ns), the filtered waveform is shown in Figure 5, where t\({}_{\text{p}}\) is the rise time and t\({}_{\text{ft}}\) is the flat-top time of the trapezoid-shaped pulse. The total duration of trapezoidal waveform is defined as t\({}_{\text{s}}\) = 2 \(\times\) t\({}_{\text{p}}\) + t\({}_{\text{ft}}\). The output waveform of different rise time t\({}_{\text{p}}\) and flat-top time t\({}_{\text{ft}}\) can be obtained by changing the delay parameters l and k (in Figure 4). Figure 4: Block diagram of the digital trapezoidal shaper. The elements are: DELAYn—a delay pipeline, \(\sum\)—an adder/ subtract, ACCn—an accumulator, and Xn—a multiplier. The point spread function (PSF) is used to characterize the overall resolution of the XS detector. The flow chart of the resolution measurement simulation is shown in Figure 6. At a single point with a fixed coordinate, a continuous electron cloud is generated randomly according to the distribution of time interval and amplitude. All the electron clouds are divided and integrated by the anode strips. Then, the charge on the anode is converted into digital voltage pulse by the CSA and the ADC. Subsequently, the signal is shaped and filtered by the trapezoidal shaping filter in FPGA and the peak value is extracted.