Diffraction phenomena inside thick Fresnel zone plates

The parabolic wave equation method is used to describe the complex wave field inside the body of a thick zone plate used to focus X ray radiation. Two analytical approaches are applied: (1) Diffraction of a plane wave incident onto a separate interface between opaque and open zones is considered. We construct an approximate analytical solution to the classical problem of diffraction by a dielectric wedge in terms of the Fresnel integral and a new special function. (2) Coupled wave theory is used to describe collective effects of diffraction by many interfaces. The zone plate is considered as thick diffraction grating with slowly varying period. An analytical solution is found for three interacting modes. A possibility to optimize the zone plate performance is shown. Taken together, these results describe all the main features observed in the output field of realistic X ray zone plates.