X-ray reflectivity by rough multilayer gratings is treated in the framework of kinematical and dynamical theories. The kinematical scattering integral is calculated without the restrictions of the Fraunhofer approximation. The dynamical theory is presented by the matrix modal eigenvalue approach. In both theories we generalize the Fresnel reflection and transmission coefficients for the case of grating diffraction. We obtain one unique formalism which permits us to compare the results of both theories directly. Furthermore, interface and sidewall roughnesses are taken into account. The dynamical approach allows us to explain the experimental results obtained from a partially etched GaAs/InP periodic multilayer grating.

• Received 22 December 1997

DOI:https://doi.org/10.1103/PhysRevB.59.7632