A Refined Analysis of the Poisson Channel in the High-Photon-Efficiency Regime

We study the discrete-time Poisson channel under the constraint that its average input power (in photons per channel use) must not exceed some constant
\( \mathcal {E}\)
. We consider the wideband, high-photon-efficiency extreme where
\( \mathcal {E}\)
approaches zero, and where the channel's dark current approaches zero proportionally with
\( \mathcal {E}\)
. Improving over a previously obtained first-order capacity approximation, we derive a refined approximation, which includes the exact characterization of the second-order term, as well as an asymptotic characterization of the third-order term with respect to the dark current. We also show that pulse-position modulation is nearly optimal in this regime.