  This paper addresses the degraded discrete-time Poisson wiretap channel
(DT--PWC) in an optical wireless communication system based on intensity
modulation and direct detection. Subject to nonnegativity, peak- and
average-intensity as well as bandwidth constraints, we study the
secrecy-capacity-achieving input distribution of this wiretap channel and prove
it to be unique and discrete with a finite number of mass points; one of them
located at the origin. Furthermore, we establish that every point on the
boundary of the rate-equivocation region of this wiretap channel is also
obtained by a unique and discrete input distribution with finitely many mass
points. In general, the number of mass points of the optimal distributions is
greater than two. This is in contrast with the degraded continuous-time PWC
when the signaling bandwidth is not restricted and where the secrecy capacity
and the entire boundary of the rate-equivocation region are achieved by binary
distributions. Furthermore, we extend our analysis to the case where only an
average-intensity constraint is active. For this case, we find that the secrecy
capacity and the entire boundary of the rate-equivocation region are attained
by discrete distributions with countably \textit{infinite} number of mass
points, but with finitely many mass points in any bounded interval.
