A computationally efficient method to solve the Takagi-Taupin equations for a large deformed crystal

A treatise is presented on solving the Takagi-Taupin equations in the case of a strain field with an additional, spatially slowly varying component (owing to, for example, heat expansion or angular compression). It is shown that such a component typically has a negligible effect on the shape of the reflectivity curve when considering the reflectivity of a microscopic surface area of the crystal. However, it makes the centroid of that curve shift in terms of the wavelength (or the incidence angle) as a function of the position of the mentioned area, which alters the shape of the overall reflectivity curve integrated over the crystal's macroscopic surface. The validity of the method is demonstrated by comparing computed reflectivity curves with experimental ones for bent silicon wafers. A good agreement is observed.