Effect of silicon anisotropy on mirror substrate design

Single-crystal silicon is one of the substrate materials often used for x-ray optical components, such as mirrors and monochromators. Silicon crystal is elastically anisotropic, mechanical properties are direction dependent. Because anisotropic analysis is complicated, isotropic approximation is commonly used in the design of optical substrates. This approximation is satisfactory in most cases. However, a full anisotropic analysis is required to precisely characterize the performance of optical substrates or determine the effect of anisotropy. In this paper, single-crystal-silicon anisotropy and its effects on deformation of bendable optics are discussed. The resulting anticlastic bending is described, and a complete numerical solution as well as approximate analytical formulations, is provided. Anisotropy can be used advantageously to accentuate anticlastic bending.