Numerical analysis of the efficiency of multilayer-coated gratings using integral method

An analysis is made of a rigorous and an approximate approach to the solution of the diffraction problem for a multilayer-coated X-ray grating by the integral equation formalism. Whereas a rigorous analysis involving the integral method requires a lot of computer resources, even for gratings with a small number of layers, the approximate approach based on a modification of the solution of the integral equation at the lower boundary with a finite conductivity is practically independent of the number of layers and is readily tractable with the use of a standard PC. The efficiencies of multilayer gratings measured at grazing angles with synchrotron soft X-ray radiation are compared with the values calculated using the integral approaches for ideal groove profiles.