Optical Communications, 2nd Edition

3.6 PHOTOCOUNTING WITH RANDOM PHOTOMUL TIPLICATION

When a photomultiplier is used to enhance the electron flow, the randomness of the photomultiplication mechanism will influence the statistics of the output count. If we let k₁ be the electron count over a volume based on primary electron flow (as computed in Section 3.1) then the output count k₂ (number of total output electrons that flow after photomultiplication) will have a probability

Here P(k₁) is the primary count probability, and P(k₂|k₁) is the conditional output count probability conditioned on a k₁ primary count. This conditional count depends on the type of photomultiplier used and the mechanism that generates the secondary emissions.

With a PMT having mean gain , the conditional probability is generally modeled as

where C is a normalizing factor. Equation (3.6.2) is a Gaussian-shaped count envelope (recall k₂ only takes on integer values) centered at , with a deviation spread of , with ξ the PMT spreading factor. The resulting ...