  Contrary to praxis, we provide an analytical expression, for a physical
locally periodic structure, of the average $\langle S\rangle$ of the scattering
matrix, called optical $S$ matrix in the nuclear physics jargon, and
fundamentally present in all scattering processes. This is done with the help
of a strictly analogous nonlinear dynamical mapping where iteration time is the
number $N$ of scatterers. The ergodic property of chaotic attractors implies
the existence and analyticity of $\langle S\rangle$. We find that the optical
$S$ matrix depends only on the transport properties of a single cell, and that
the Poisson kernel is the distribution of the scattering matrix $S_N$ in the
large size limit $N\rightarrow \infty$. The theoretical distribution shows
perfect agreement with numerical results for a chain of delta potentials. A
consequence of our findings is the a priori knowledge of $\langle S\rangle$
without resort to experimental data.
