A theory of reflective X-ray multilayer structures with graded period and its applications

In this paper a theory of reflective X-ray multilayer structures with a graded (slowly varying) period based on the coupled waves method and quasi-classical asymptotic expansions is reported. 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. The developed theory is then 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. A number numerical experiments is conducted 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, an upper limit on the integral reflectivity achievable with a graded multilayer is estimated.