Class of exactly solvable scattering potentials in two dimensions, entangled-state pair generation, and a grazing-angle resonance effect

Abstract

We provide an exact solution of the scattering problem for the potentials of the form $v(x,y)={\ensuremath{\chi}}_{a}(x)[{v}_{0}(x)+{v}_{1}(x){e}^{i\ensuremath{\alpha}y}]$, where ${\ensuremath{\chi}}_{a}(x):=1$ for $x\ensuremath{\in}[0,a]$, ${\ensuremath{\chi}}_{a}(x):=0$ for $x\ensuremath{\notin}[0,a]$, ${v}_{j}(x)$ are real or complex-valued functions, ${\ensuremath{\chi}}_{a}(x){v}_{0}(x)$ is an exactly solvable scattering potential in one dimension, and $\ensuremath{\alpha}$ is a positive real parameter. If $\ensuremath{\alpha}$ exceeds the wave number $k$ 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 $k<\ensuremath{\alpha}$. For $k>\ensuremath{\alpha}$ and ${v}_{1}(x)\ensuremath{\ne}0$, the scattered wave consists of a finite number of coherent plane-wave pairs ${\ensuremath{\psi}}_{n}^{\ifmmode\pm\else\textpm\fi{}}$ with wave vector: ${\mathbf{k}}_{n}=(\ifmmode\pm\else\textpm\fi{}\sqrt{{k}^{2}\ensuremath{-}{[n\ensuremath{\alpha}]}^{2}},n\ensuremath{\alpha})$, where $n=0,1,2,...<k/\ensuremath{\alpha}$. This generalizes to the scattering of wave packets 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 $k=N\ensuremath{\alpha}$ for some positive integer $N$, ${\ensuremath{\psi}}_{N}^{\ifmmode\pm\else\textpm\fi{}}$ coalesce and their amplitude diverge. If $k$ exceeds $N\ensuremath{\alpha}$ slightly, ${\ensuremath{\psi}}_{N}^{\ifmmode\pm\else\textpm\fi{}}$ have a much larger amplitude than ${\ensuremath{\psi}}_{n}^{\ifmmode\pm\else\textpm\fi{}}$ with $n<N$. This marks a resonance effect that arises for the scattered waves whose wave vector makes a small angle with the faces of the slab.