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

Calculation of dynamical diffraction patterns for X-ray topography and similar x-ray scattering-imaging techniques require the numerical integration of the Takagi-Taupin equations. This is usually done with a simple second order finite difference scheme on a sheared computational grid with two of the axes aligned with the wave vectors of the incident and scattered beams respectively. Here we present a finite difference scheme that carries out this integration 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 on similarly dense grids but is more computationally expensive due to the use of FFT operations.