Abstract

A method of calculating the effects of interfacial roughness on the grazing-incidence reflectivity and transmissivity of a synthetic multilayer x-ray reflector (SMR) is developed; this method makes it possible to investigate the dependence of an SMR’s performance on the spatial-frequency content of the roughness profile. The roughness is modeled by a superposition of volume diffraction gratings that are described mathematically by using a coupled-wave approach. The coupled-wave equations for both the electric and magnetic fields are derived without invoking the usual slowly varying envelope approximation, and exact analytical solutions are obtained for unslanted gratings. The effects of individual gratings with different spatial periods are investigated for 1.5-keV x rays, and it is found that the performance of the SMR is affected most strongly by short-period gratings, whereas gratings with longer periods have relatively little effect. At higher energies, guided wave resonances occur for gratings with short periods, giving rise to singular behavior of the equations describing the system; this behavior is discussed in some detail. The theory of random functions provides methods for constructing the superposition of many gratings, each with a different spatial frequency, so that a randomly rough surface is modeled. The reflectances and transmittances of typical SMR’s designed for x rays of 1.5, 8, and 17 keV are calculated for rough interfaces with different power spectra, and the results are compared with those of similar calculations for an ideal SMR. It is shown that the resonance effects are inconsequential for power spectra that characterize the interfacial roughness that is expected to occur in the fabrication of a typical SMR. The effects on SMR performance of interpenetration of the multilayer constituents at the layer interfaces are also calculated by using a simple averaging model, and it is shown that for these energies such effects are dominant in comparison with those of boundary roughness.

© 1992 Optical Society of America

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