Aharonov-Bohm beam deflection: Shelankov's formula, exact solution, asymptotics and an optical analogue

Using a paraxial analysis, Shelankov (1998 Europhys. Lett. 43 623) has shown that charged particles in a beam of small angular width l/w, aimed at a magnetic flux line with quantum flux a, are deflected through an angle D proportional to sin(2 pi alpha)/w, vanishing in the classical limit and also vanishing if the incident beam has zero intensity at the flux line. These properties are confirmed by numerical calculations based on an exact solution of the Schrodinger equation, and the paraxial wavefunction is obtained as an asymptotic approximation for large iv. Paraxial theory suggests that the same deflection will occur for a light beam reflected by a mirror containing a step of height pi alpha. A theory of this optical phenomenon, based on an exact solution (for which D is not periodic in alpha), shows that convergence to the paraxial D is fast for alpha << 1, and slow for alpha near 1/2.