Multiplied-Poisson noise in pulse, particle, and photon detection

Abstract

Multiplication effects in point processes are important in a number of areas of electrical engineering and physics. We examine the properties and applications of a point process that arises when each event of a primary Poisson process generates a random number of subsidiary events, with a given time course. The multiplication factor is assumed to obey the Poisson probability law, and the dynamics of the time delay are associated with a linear filter of arbitrary impulse response function; special attention is devoted to the rectangular and exponential cases. The process turns out to be a doubly stochastic Poisson point process whose stochastic rate is shot noise; it has application in pulse, particle, and photon detection. Explicit results are obtained for the single and multifold counting statistics (distribution of the number of events registered in a fixed counting time), the time statistics (forward recurrence time and interevent probability densities), and the power spectrum (noise properties). These statistics can provide substantial insight into the underlying physical mechanisms generating the process. An example of the applicability of the model is provided by cathodoluminescence (the phenomenon responsible for the television image) where a beam of electrons (the primary process) impinges on a phosphor, generating a shower of visible photons (the secondary process). Each electron produces a random number of photons whose emission times are determined by the (possibly random) lifetime of the phosphor, so that multiplication effects and time delay both come into play. We use our formulation to obtain the forward-recurrence-time probability density for cathodoluminescence in YVO4:Eu³⁺, the excess cathodoluminescence noise in ZnS:Ag, and the counting distribution for radioiuminescence photons produced in a glass photomultiplier tube. Agreement with experimental data is very good in all cases. A variety of other applications and extensions of the model are considered.