Unique Determination of Periodic Polyhedral Structures by Scattered Electromagnetic Fields Ii: the Resonance Case

This paper is concerned with the unique determination of a threedimensional polyhedral bi-periodic diffraction grating by the scattered electromagnetic fields measured above the grating. It is shown that the uniqueness by any given incident field fails for seven simple classes of regular polyhedral gratings. Moreover, if a regular bi-periodic polyhedral grating is not uniquely identifiable by a given incident field, then it belongs to a non-empty class of the seven classes whose elements generate the same total field as the original grating when impinged upon by the same incident field. The new theory provides a complete answer to the unique determination of regular bi-periodic polyhedral gratings without any restrictions on Rayleigh frequencies, thus extending our early results (2011) which work under the assumption of no Rayleigh frequencies.