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) = chi(a)(x)[v(0)(x) + v(1)(x)e(i alpha y)], where chi(a)(x) := 1 for x is an element of [0,a], chi(a)(x) := 0 for x is an element of [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 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< alpha. For k > alpha and v(1)(x) = 0, the scattered wave consists of a finite number of coherent plane-wave pairs Psi(+/-)(n) with wave vector: k(n) = (+/-root k(2)- [n alpha](2),n alpha), where n = 0,1,2, . . .< k/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 alpha for some positive integer N, Psi(+/-)(N) coalesce and their amplitude diverge. If k exceeds N alpha slightly, Psi(+/-)(N) have a much larger amplitude than Psi(+/-)(n) 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.