Estimation and detection of weak optical signals

In this paper we consider the problem of estimation and detection of Weak optical signals formulated as a problem in point process theory. Indeed at very low light intensity the only information available from an intensity detector is the random distribution of time instants <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\{t_i \}</tex> at which photons of the field are absorbed and photoelectrons emitted. There is also a noise due to thermoelectrons or to a background optical field. Starting from the statistical properties of the point process <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\{t_i\}</tex> , we formulate for various kinds of optical fields the problem of estimation of the light intensity of a modulated beam. We show that in some cases the number of photoelectrons is a sufficient statistic. In general, theoretical results are complex and we formulate the problem in the case of linear estimation, which is solved by means of the resolvent of an integral equation using the covariance function of the field. Some detection problems are also considered.