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 show n that charged particles in a beam of small angular width l/w, aimed at a m agnetic flux line with quantum flux a, are deflected through an angle D pro portional to sin(2 pi alpha)/w, vanishing in the classical limit and also v anishing if the incident beam has zero intensity at the flux line. These pr operties 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 containi ng 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 conve rgence to the paraxial D is fast for alpha << 1, and slow for alpha near 1/ 2.