  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.
