Theoretical study of multilayer x-ray mirrors with a wide spectral band of reflection

Our method is composed of three steps. First, we use an approximate analytical expression to describe the x-ray reflectivity spectral dependence of any arbitrary graded-multilayer structure. Such an approximation is obtained by assuming the multilayer d-spacing profile to be a monotone function of the depth. The d-spacing gradient is also assumed to be large enough, so that the reflection condition for each given energy is fulfilled in a depth zone that is small compared to the thickness of the multilayer stack. On this basis we could derive the differential equation that describes the change of period necessary to guarantee a given spectral reflectivity profile. Then, a computer code, written for solving the inverse problem, lead us to the desired multilayer period profile. Finally, the effective spectral dependence of reflectivity was accurately computed using a standard matrix method.

Recent progress in the manufacturing of X-ray multilayers has opened up new possibilities in the field of hard xray optics allowing to produce wide bandpass optical elements through the design of depth-graded multilayer coatings. However, the inverse problem consisting of inferring the composition profile of the multilayer has only been addressed in a semi-empirical way, which encouraged us to develop a new (theoretical) approach.