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Optical and near-optical band communication systems are known to be intrinsically more secure than comparable RF channels, due to their narrow beamwidths and, in some cases, their high atmospheric absorption. The use of coding against wiretapping for such channels is investigated. For the degraded Poisson wiretap channel model, the secrecy capacity is determined exactly. Moreover, a complete characterization of the rate-equivocation region is presented. For achievability, an optimal code is constructed explicitly by using a code designed by Wyner for the Poisson channel. The converse is proved in two ways: the first method leverages the low-SNR nature of the channel and relies only on simple properties of conditional expectation and classical information inequalities. The second method uses a link recently established between minimum mean square error estimation and mutual information over Poisson channels.