Capacity of the low-photon-rate direct-detection optical pulse-position-modulation channel in the presence of noise photons

We derive expressions for the capacity of the pulse-position-modulated (PPM) direct-detection photon-counting channel in the presence of noise photons in addition to the signal-dependent shot noise that is normally considered in studying photon counting at low photon rates. We note that even a small mean number of noise photons per PPM count bin significantly decreases the capacity of the channel. These results are useful for comparisons of performance that are obtained by use of real coding and synchronization algorithms with photon-counting PPM schemes that are currently being considered for deep-space optical communications.