Photo-Counting Inversion in Presence of Dead Time Effects

Abstract

The problem of obtaining information about the statistical properties of light from the photocounting statistics is of considerable interest and is referred to as the photocounting inversion problem. It is well known[1] that if one assumes the different photocounts to be statistically independent then the counting distribution p(n,T) is the Poisson transform of the probability density P(W) of the integrated intensity W:

$${\rm{p}}\left( {{\rm{n,T}}} \right) = \mathop \smallint \limits_o^\infty {\rm{dW}}\,{\rm{P}}\left( {\rm{W}} \right){{\rm{e}}^{{\rm{ - \alpha W}}}}\frac{{{{\left( {{\rm{\alpha W}}} \right)}^{\rm{n}}}}}{{{\rm{n!}}}}{\rm{;W}} = \mathop \smallint \limits_o^T {\rm{I}}\left( {\rm{t}} \right){\rm{dt}}.$$

(1)