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 ε. We consider the wideband, high-photon-efficiency extreme where ε approaches zero, and where the channel's “dark current” approaches zero proportionally with ε. Improving over a previously obtained first-order capacity approximation, we derive a refined approximation which also includes the second-order term. We also show that pulse-position modulation is optimal on this channel up to the second-order term in capacity.
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