  We provide an exact solution of the scattering problem for the potentials of
the form $v(x,y)=\chi_a(x)[v_0(x)+ v_1(x)e^{i\alpha y}]$, where $\chi_a(x):=1$
for $x\in[0,a]$, $\chi_a(x):=0$ for $x\notin[0,a]$, $v_j(x)$ are real or
complex-valued functions, $\chi_a(x)v_0(x)$ is an exactly solvable scattering
potential in one dimension, and $\alpha$ is a positive real parameter.If
$\alpha$ exceeds the wavenumber $k$ of the incident wave, the scattered wave
does not depend on the choice of $v_1(x)$. In particular, $v(x,y)$ is invisible
if $v_0(x)=0$ and $k<\alpha$. For $k>\alpha$ and $v_1(x)\neq 0$, the scattered
wave consists of a finite number of coherent plane-wave pairs $\psi_n^\pm$ with
wavevector: $\mathbf{k}_n=(\pm\sqrt{k^2-(n\alpha)^2},n\alpha)$, where
$n=0,1,2,\cdots<k/\alpha$. This generalizes to the scattering of wavepackets
and suggests means for generating quantum states with a quantized component of
momentum and pairs of states with an entangled momentum. We examine a
realization of these potentials in terms of certain optical slabs. If
$k=N\alpha$ for some positive integer $N$, $\psi_N^\pm$ coalesce and their
amplitude diverge. If $k$ exceeds $N\alpha$ slightly, $\psi_N^\pm$ have a much
larger amplitude than $\psi_n^\pm$ with $n<N$. This marks a resonance effect
that arises for the scattered waves whose wavevector makes a small angle with
the faces of the slab.
