A new interpretation of X-ray reflectivity in real space for low contrast multilayer systems I. Mathematical algorithm and numerical simulations

• A new X-ray reflectivity method for low contrast multilayer systems is proposed.
• We determine a probability to find an element of sort j at a depth z from the surface.
• We solve the ill posed problem for integral equation using the regularization method.
• We recover the phase of X-ray scattering using the logarithmic dispersion relation.
• Using the properties of the canonical functions we introduce the interface function.

It is shown that X-ray specular reflectivity may be described in terms of canonical distribution functions (CDFs) pj(z) which is a probability to find an element of sort j at a depth z from the sample surface. The problem reduces to determine K CDFs, where K is the number of elements in the multilayer sample. Using the properties of the canonical functions, we have introduced the interface function pint(z) and use it as one unknown function at solving the integral equation. The integral Fredholm equation of the first kind belongs to the class of ill-posed problems and for solving it needs special methods. We use the Tikhonov regularization method. This method is applied to study the low contrast multilayer systems.