  The effect of the thickness of the dielectric boundary layer that connects a
material of refractive index $n_1$ to another of index $n_2$ is considered for
the propagation of an electromagnetic pulse. For very thin boundary layer the
scattering properties of the pulse mimics that found from the Fresnel jump
conditions for a plane wave - except that the transmission to incident
amplitudes are augmented by a factor of $\sqrt{n_2/n_1}$. As the boundary layer
becomes thicker one finds deviations away from the Fresnel conditions and
eventually one approaches WKB propagation. However there is found a small but
unusual dip in part of the transmitted pulse that persists in time. The quantum
lattice algorithm (QLA) used recovers the Maxwell equations to second order in
a small parameter -- but QLA still recovers Maxwell equations when this
parameter is unity. The expansion parameter is the speed of the pulse in medium
$n_1$.
