Despite being an accepted model for a wide variety of optical channels, few general results on optimal signalling for discrete-time Poisson (DTP) channels are known. Among the most significant is that under simultaneous peak and average constraints, the capacity-achieving distributions are discrete with a finite number of mass points. In this paper, several fundamental properties of capacity-achieving distributions for DTP channels are established. In particular, we demonstrate that all capacity-achieving distributions of the DTP channel have zero as a mass point. In the case of only a peak constraint, it is further shown that the optimal distribution always has a mass point at the maximum amplitude. Finally, under solely an average power constraint, it is shown that a finite number of mass points are insufficient to achieve the capacity. In addition to these analytical results, a numerical algorithm based on deterministic annealing is presented which can efficiently compute both the channel capacity and the associated optimal input distribution under peak and average power constraints. Numerical lower bounds based on the envelope of information rates induced by the maxentropic distributions are also shown to be extremely close to the capacity, especially in the low power regime.

Discrete-time Poisson (DTP) channels exist in many scenarios including space laser communication systems which operate over long distances and which can be corrupted by reflected and scattered light. Through simulation, binary-input distributions have been observed to be optimal in many cases, however, little analytical work exists on conditions for optimality or the form of optimal signalling. In this second part, the general properties of Part I are extended to the case of DTP channels where binary-inputs are optimal. Necessary and sufficient conditions on the optimality of binary (i.e. two mass point) distributions are presented by leveraging the general properties of DTP capacity-achieving distributions. Closed-form expressions of the capacity-achieving distributions are derived in several important special cases including zero dark current and for high dark current. Numerical results are presented to elucidate the developed analytical work.