Achievability of the Rate ${1/2}\log(1+\es)$ in the Discrete-Time   Poisson Channel
  A simple lower bound to the capacity of the discrete-time Poisson channel with average energy $\es$ is derived. The rate ${1/2}\log(1+\es)$ is shown to be the generalized mutual information of a modified minimum-distance decoder, when the input follows a gamma distribution of parameter 1/2 and mean $\es$. 
Alfonso Martinez