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)=χ_a(x)\[v₀(x)+ v₁(x)e^iαy\], where χ_a(x):=1 for xin[0,a], χ_a(x):=0 for xnotin[0,a], v_j(x) are real or complex-valued functions, χ_a(x)v₀(x) is an exactly solvable scattering potential in one dimension, and α is a positive real parameter.If α exceeds the wavenumber k of the incident wave, the scattered wave does not depend on the choice of v₁(x). In particular, v(x,y) is invisible if v₀(x)=0 and k<α. For k>α and v₁(x)neq 0, the scattered wave consists of a finite number of coherent plane-wave pairs ψ_n^pm with wavevector: mathbf{k}_n=\(pmsqrt{k²-(nα\)²},nα), where n=0,1,2,cdots<k/α. 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=Nα for some positive integer N, ψ_N^pm coalesce and their amplitude diverge. If k exceeds Nα slightly, ψ_N^pm have a much larger amplitude than ψ_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.