Discrete thin-film multilayer design for X-ray and neutron supermirrors

Multilayer structures, analogous to broadband optical filters, may be used to reflect X-rays or neutrons at angles larger than the total external reflection angle intrinsic to the bulk mirror material. These so-called supermirrors have generally been designed using continuum theories, which predict that the thickness of the jth layer in the multilayer should vary smoothly as j^-1/4. A new approach is proposed, based on considering the contribution of each bilayer to the extinction in a given stack of bilayers, and the discrete equations governing the choice of layer thicknesses are derived and solved. In the limit of zero layer thickness, the continuum result is recovered. The design produces essentially perfect reflectivity over the entire supermirror range. An optimal technique for trading reflectivity to gain angular range also emerges naturally from the physics of the design.