Theory of Reflective X-Ray Multilayer Structures with Graded Period and its Applications

We report a theory of reflective X-ray multilayer structures with a graded (slowly-varying) period based on the coupled-wave method and quasiclassical asymptotic expansions. A number of exact solutions of the coupled wave equations is obtained and analyzed demonstrating suitability of this method for the description of the reflective properties of the graded multilayers. Then the developed theory is used as a basis for the solution of the inverse problem, i.e., designing multilayer structures with a pre-specified reflectivity dependence on the wavelength or grazing angle. We conduct a number of numerical experiments to demonstrate the capabilities of the proposed method in designing reflective multilayer coatings with an arbitrary shape of the reflectivity curve. The problem of maximization of the integral reflectivity is considered and a second-order differential equation, which solutions correspond to multilayer structures with the maximal reflectivity, is derived. Finally, we estimate an upper limit on the integral reflectivity achievable with a graded multilayer.