Exact scattering theory for any straight reflectors in two dimensions

Abstract

The exact Green function for the scalar wave equation in a plane with any set of perfectly reflecting straight mirrors, which may be joined to form corners, is given as a diffraction scattering series. Instances would be slit diffraction in optics, or the Schrodinger equation inside (or outside) a general polygonal enclosure ('quantum polygon billiards'). The method is based on the seminal 1896 Riemann helicoid surface solution by Sommerfeld for optical diffraction by a single corner. It is generalised to account for multiple scatter by adapting the analysis of Stovicek for a closely related problem: a collection of magnetic flux lines (points) in a plane, the multi-flux Aharonov-Bohm effect. The short wavelength limit is shown to yield the 'geometrical theory of diffraction'. For slit diffraction the exact series is shown to coincide with that of Schwarzschild in 1902.