Off-specular reflection from flat interfaces

The reflection of x ray or neutron from a flat interface with density or compositional fluctuations was studied. An integral equation relating the reflection intensities to the interfacial fluctuations was derived in the Fraunhofer limit. At the specular condition this integral equation, although derived for the off-specular intensities, has an expression identical to a well-established one for the specular component. This integral equation was applied to a flat surface onto which gold spheres of uniform size were randomly adhered. The out-of-plane components for both the real wave and the time reversed wave were approximated by the solution from the specular component. Based on the calculated off-specular intensities near the specular region, an approximation scheme was proposed to estimate the in-plane correlation length.