  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
E. We consider the wideband, high-photon-efficiency extreme where E approaches
zero, and where the channel's "dark current" approaches zero proportionally
with 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.
