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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 firstorder capacity approximation, we derive a refined approximation, which includes the exact characterization of the second-order term, as welldoi:10.1109/tit.2014.2320718 fatcat:2atzk74drzb3bb2i6s5q7awkvm