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New equations for paralyzable, non paralyzable and hybrid DT models, valid for any time dependent sources are presented. We show how such new equations include the equations already used for constant rate sources, and how it’s is possible to correct DT losses in the case of time dependent sources. Montecarlo simulations were performed to compare the equations behavior with the three DT models. Excellent accordance between equations predictions and Montecarlo simulation was found. We also obtain good results in the experimental validation of the new hybrid DT equation. Passive quenched SPAD device was chosen as a device affected by hybrid DT losses and active quenched SPAD with 50 ns DT was used as DT losses free device.

When using an electronic signal counter, the Dead Time (DT) must be considered, to be sure that the measurement is free from losses or that the losses have been correctly calculated. The DT is the time after the detection of an event, in which the acquisition system is not able to detect new events. On increasing the counting rate the DT effect on the total number of counts detected increases. The DT can be due or to the sensor device or to the electronics readout system. Theoretical studies of the DT behavior are often used to infer the real amount of count rate from the measured one [

There are two elementary idealized models for DT, named paralyzable (P-DT) and non paralyzable (NP-DT). In the P-DT case all events occurring during the DT are not registered, but are able to prolong the period during which the detector is not able to reveal further signals. In the NP-DT case, any event arriving during the DT is neither registered nor has any influence on the device. In the real equipments parallel or series combinations of these two kinds of DT can be found. One of the most important configuration is the hybrid model [

The Avalanche Photon Diode (APD) can be used over the breakdown voltage like a GM device [

Some studies have performed the analysis of short-lived time dependent source [_{0}^{−}^{λt}_{0}

With our theoretical approach it is possible to evaluate the DT losses only by using a data analysis process, apart from the time dependence of the source, and without the necessity of developing complex electronic systems. In this work we present three new equations that describe the time dependent effect of paralyzable, non paralyzable, and hybrid DTs, and the accordance of the equations predictions with Montecarlo simulations and experimental data.

The goal of our study is to extend the dynamic range of our imaging device developed by using SPAD technology [

We realized a modulated light source using a commercial Light Emitting Diode (LED) powered by a pulse generator, model PB-4. The shape of the light source obtained is defined by an exponential grow, with a selectable characteristic time between 0.05 μs and 10 μs, and an exponential fall down with a selectable time between 0.5 μs and 1000 μs. Both counter devices were contemporaneously enlighten by the same light source. The duration of the acquisition was 35 μs and, to increase the statistics, about 10E6 run have been performed. The readout system has been designed to obtain an histogram of the measured count rate

As said, the signals that arrive during the NP-DT interval after a detected signal are lost. In the case of NP-DT the total inactive time of the sensor can be calculated by multiplying the measured count rate (_{np}

_{i}_{i}_{np}_{np}_{np}_{np}_{np}_{np}

A deeper analysis of this equation allows to solve the problem of hybrid DT model. Inside the _{np}

The quantity is characteristic of the DT type (_{np}

Using the exponential distribution of time intervals between Poissonian random events occurring at a constant rate

In the case of time dependent count rate the two values of the true rate _{p}

where _{p}_{p}

As before, we can separate the lost rate due to the P-DT (_{p}

In the hybrid DT model, introduced by Lee and Gardner [

A detailed description and the derivation of this equation can be found in [

In the time dependent count rate, we must consider which previous intervals produce losses into the _{p}_{np}

Note that the

The non steady-state equation must be valid also in the steady-state, therefore imposing the steady-state condition in the Equations

In fact in steady-state conditions _{i}_{i}_{np}_{p}

Moreover in the steady-state condition the size of the time interval (Δ) can be chosen as short as possible, so using the time precision of the readout system the losses produced inside the

For the sake of clarity the mathematical steps that give Equation

A code, developed in Matlab, was performed to simulate the behavior of the three DT types. We found that, in this simulation approach, a sharp increase of the illumination from zero to a constant value, gives best information about the DT losses and the equation validity. The predictions of the three proposed equations (

We chose that the illumination value started with zero and remained zero for a time interval longer than DT. In this way Equations _{0}, _{0} + 1, _{0} + 2, _{i}_{i}

Montecarlo simulation uses the Poisson statistics to generate the source count distribution, and the DT laws to determine which counts are measured and which counts are lost. In the zero illumination zone no count is introduced. In the constant illumination region a Poisson distribution was used to extract the random values of photons that should be put into the illumination time window. The time position of these photons was randomly chosen inside the illumination time window. The histogram of the simulated photon source count rate is shown in

The agreement between the simulations and the data calculated with the Equations

In the simulation, the first 300Δ are defined without photons, after that the illumination source starts with a constant rate. In the following 105Δ, corresponding to the total DT duration, the shape of the three DT types are the same, because in this period, independently from the type of the occurring active DT, only one photon can be detected. Therefore this first part of the histogram shows the arriving time position of the first photons. This shape represents the probability to have no photon in illumination condition. This probability drastically decreases on increasing the time. After a time equal to the total DT the type of DT generates changes in the shape of the response. The P-DT that takes into account all lost counts shows a constant output rate, due to the constant lost incoming rate. The NP-DT shows an increase of counts due to the end of the DT of the first measured count not extended by the others photons. The hybrid model shows a behavior in the middle of the two fundamental DT types, according to the proportion of the DT component of hybrid model.

The P-DT reaches the steady-state value of Equation

To test the features of the proposed new hybrid DT Equation

Exposing the test device to a time dependent source we observed that the behavior changed with time. At the beginning the device was affected only by the negligible DT losses produced by the previous darkcount rate, 1 kcps, so the measured rate was very close to real one. Immediately after the measured rate fell down for the DT losses produced by the first detected rate. When the DT losses production reached the maximum value, the measured rate continued with a slope different from that produced by the reference counter.

The Equation

Assuming that during the dark state the count losses can be neglected, it is possible to evaluate the real true rate from the measured one using the Equation

The difference between predicted and measured rate are in the first part of the light pulse because, due to the rapid change in the excitation rate, the device doesn’t show exactly the hybrid DT behavior but something close. After a time double the DT duration the predicted and the measured rates show a very good agreement, being deviation less than 5%.

The presented new approach for evaluating DT losses is able to show how the two elementary DT types and their combination works with time dependent sources. The theoretical derivation, the completely accordance with the Montecarlo simulation, the inclusion of classic DT equation, and the experimental validation are a well-round presentation of this three new equations, that could be useful in all counting system devices and application. Worth to note that our DT compensation technique requires only a simple data analysis process, apart from the type of time dependent source and without the necessity of developing complex electronic systems.

Our theoretical approach can be also useful to deduce new equations for other series or parallel combinations of P-DT and NP-DT. The presented equation can be also used to predict the behavior of many devices in time dependent applications, so allowing to select the most appropriate device.

In particular we plan to use this technique in the development of a new time resolved single-photon imaging sensor (TRIS) [

This work has been supported by the Istituto Nazionale di Fisica Nucleare (INFN) and was done inside the activities of the projects SinPhos2 and TRIS. The authors wish to thank Antonio Causa of Mathematical Department of Catania University for the useful discussion.

Histograms of rate