A finite difference scheme for integrating the Takagi–Taupin equations on an arbitrary orthogonal grid

Abstract

Calculating dynamical diffraction patterns for X-ray diffraction imaging techniques requires numerical integration of the Takagi–Taupin equations. This is usually performed with a simple, second-order finite difference scheme on a sheared computational grid in which two of the axes are aligned with the wavevectors of the incident and scattered beams. This dictates, especially at low scattering angles, an oblique grid of uneven step sizes. Here a finite difference scheme is presented that carries out this integration in slab-shaped samples on an arbitrary orthogonal grid by implicitly utilizing Fourier interpolation. The scheme achieves the expected second-order convergence and a similar error to the traditional approach for similarly dense grids.