Onefold photoelectron-counting statistics of mixtures of thermal and coherent radiation

We consider onefold photoelectron-counting statistics, for arbitrary counting times, when the wave field on the detector is a mixture of thermal light of a specified spectral profile (incoherent component) and a harmonic signal (coherent component). The probability density function of the time-integrated intensity W(Ω) is obtained from the inverse Laplace transform of the moment-generating function by the Dubner–Abate algorithm. The corresponding probability distribution of the detected photoelectrons, P(m), is then evaluated directly by numerical integration of W(Ω). Numerical results for W(Ω) and P(m) have been obtained when the thermal line shape is Lorentzian for both the homodyne situation (coherent line coincides with the mean frequency of the thermal spectral profile) and the heterodyne situation (coherent line is now off center with respect to the mean frequency of the thermal spectral profile).