Asymptotic form of the reciprocity theorem with applications in x-ray scattering

The emission of electromagnetic waves from a source within or near a nontrivial medium (with or without boundaries, crystalline or amorphous, with inhomogeneities, absorption, and so on) is sometimes studied using the reciprocity principle which is a variation of the method of Green's functions. If one is only interested in the asymptotic radiation fields the generality of these methods may actually be a shortcoming: obtaining expressions valid for the uninteresting near fields is not just a wasted effort but may be prohibitively difficult. In this work we obtain a modified form of the reciprocity principle which gives the asymptotic radiation field directly. The method may also be used to study scattering problems. We give a few pedagogical examples and then, as more challenging applications, we calculate the specular reflection of x rays by a rough surface and by a smoothly graded surface taking polarization effects into account. In conventional treatments of reflection, x rays are treated as scalar waves; polarization effects are neglected. This is a good approximation at grazing incidence but becomes increasingly questionable for soft x rays and UV at higher incidence angles