A Class of Exactly Solvable Scattering Potentials in Two Dimensions, Entangled State Pair Generation, and a Grazing Angle Resonance Effect

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.