# Towards a Counting Point Detector for Nanosecond Coherent X-ray Science
## 4 Discussion: High-Dynamic Range Measurement  
Noise-free photon counting over a wide dynamic range is demonstrated by repeating the oscilloscope data acquisition and off-line peak-finding analysis for different combinations of aluminum filters placed before the detector. The uncertainty in the observed count rate \(N_{o}\) is calculated based on Poission statistics where the random error in the number of counts is \(\sqrt{N_{o}t_{obs}}\), where \(t_{obs}\) is the observation time, equal to one hundred round trips of the storage ring (100 us, or ten successive oscilloscope traces such as shown in Figure 2). At a relatively low flux, the detector count rate response is linear, as shown in Figure 4. However, at higher count rates, a nonlinearity occurs because there is no pulse-height analysis of the output signal, and therefore multiple photon hits in one pulse cannot be distinguished from a single photon [28; 29]. Figure 3: A binned probability distribution of photons per rf-bucket, which approximates the exponential distribution \(P=e^{-RT}\) of a continuous random variable where \(R\) is the detected photon rate (34 MHz in this example) and \(T\) is the time between pulses, or 2 ns. Figure 2: (a) Waveform traces from the APD recorded from 10 revolution clock cycles. Peaks with the amplitude exceeding the threshold of 2.5 mV are recorded as X-ray photons and marked by red circles. (b) Histogram of time between two successive pulses. The most important criterion for such pulse counting at a high photon flux is the minimum resolvable time separation between two adjacent events. In most high-speed applications, the limiting time arising from the combined effect of the detector response and the associated electronics is called the dead-time, which ultimately determines the temporal resolution of the detection scheme. For correctly recovering unbiased counting statistics from measured data, one needs to compensate for the electronics and detector dead-times. For several decades, many researchers have proposed models to correct for the detection dead-times [28, 30, 31, 32]. Here, we explore two different dead-time models. First, we consider the _isolated dead-time model_ that is applicable for a detector that is not limited by the pulse structure, i.e., it has a dead-time \(\tau_{iso}\) less than the pulse spacing (\(T\)), and the only difference between the true (\(N_{t}\)) and observed (\(N_{o}\)) count rates are a result of the probability of multiple photons arriving within a single pulse, which can not be distinguished from a single photon observation,  
\[N_{t}=-\frac{1}{\tau_{iso}}ln(1-N_{o}\tau_{iso}). \tag{1}\]  
In Figure 5, this model (dashed red line) is applied to APD counts measured over a wider range of upstream X-ray filter transmission. The saturation behavior yields a dead-time of 2.17 \(\pm\) 0.10 ns, which is slightly larger than the actual pulse separation time of \(T=2.00\) ns. The discrepancy between the model and the saturation data indicates that we are just beyond the applicable range of the isolated model, where the pulse repetition rate has no effect on detector performance. Next, we compare the saturation curve to the _non-extended dead-time model_, which allows for the detector to be rendered inoperative for a fixed time \(\tau_{hex}\) after each observed count,  
\[N_{t}=\frac{N_{o}}{1-\tau_{hex}N_{o}}. \tag{2}\]  
This model is shown in Figure 5 (dot-dashed green line) and shows a better agreement with the saturation curve data than the isolated model, indicating that the detector recovery time has a measurable but minor and correctable affect on the observed count rate at 500 MHz. The non-extended dead-time obtained from the nonlinear curve fit of the data is 1.90 \(\pm\) 0.04 ns, just below the pulse separation time. Here, to analyze the high count-rate data sets shown in Figure 5, we perform a nonlinear curve fitting using Equation (1) on data and obtained an error of 0.1 ns. We note that the error bars in Figure 5 are too small to be seen due to the high count rate.