Pulsed optical modulations with photon-counting receivers have been shown to be capable of very high photon efficiencies. With channel capacity as the relevant metric, we review the mathematical models of free-space receivers with photon-counting detectors in varying amounts of background noise. Although historically, such direct detection receivers have been designed to cover relatively wide fields of view because of turbulence, modern systems may be able to achieve performance with many fewer spatial modes, as well as near-matched filtering. We investigate the channel capacity in those cases and show the relative benefits of reducing both spatial and temporal modes of noise. In addition, because the exact capacity formulas are calculation-intensive and not intuitively helpful, we propose a particularly simple yet accurate mathematical approximation that greatly simplifies the analysis when the Poisson model is relevant, and is useful in link budgeting exercises.
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