Evolution of the statistical properties of photons passed through atraveling-wave laser amplifier

The authors determine the evolution of the photon statistics of a light beam as it passes through a traveling-wave laser amplifier, modeled as a birth-death immigration (BDI) medium. The relationship between the input and output probability distributions and probability generating functions with given (but possibly varying) birth, death, and immigration rates for arbitrary input statistics is obtained. The case of constant birth, death, and immigration rates is considered in particular detail. The photon statistics at the output of a general BDI traveling-wave amplifier are always broader than those at the input, and they can take many forms. The most general solution can be applied when the input distribution to the amplifier takes the form of a negative-binomial transform