This paper addresses the discrete-time Poisson wiretap channel (DT-PWC) in an optical wireless communications system based on intensity modulation and direct detection. Subject to nonnegativity and peak intensity as well as bandwidth constraints imposed on the channel input, 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. 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 a finite support. In general, the number of mass point of the optimal distributions are greater than two. This is in contrast with the continuous-time PWC where the secrecy capacity and the entire boundary of the rate-equivocation region are achieved by binary distributions when the signaling bandwidth is not restricted. Additionally, we shed light on the asymptotic behavior of the secrecy capacity in the low intensity regime and observe that the secrecy capacity scales quadratically with the peak intensity constraint. Finally, Our numerical results indicate that there is a tradeoff between the secrecy capacity and the capacity in the sense that both may not be achieved simultaneously.