Finite difference methods for stationary and time-dependent X-ray propagation

We have generalized finite-difference (FD) simulations for time-dependent field propagation problems, in particular in view of ultra-short x-ray pulse propagation and dispersion. To this end, we first derive the stationary paraxial (parabolic) wave equation for the scalar field envelope in a more general manner than typically found in the literature. We then present an efficient FD implementation of propagators for different dimensionality for stationary field propagation, before we treat time-dependent problems by spectral decomposition, and suitable numerical sampling. We prove the validity of the numerical approach by comparison to analytical theory, using simple tractable propagation problems. Finally, we apply the framework to the problem of modal dispersion in X-ray waveguide. We show that X-ray waveguides can be considered as non-dispersive optical elements down to sub-femtosecond pulse width. Only when considering resonant absorption close to an X-ray absorption edge, we observe pronounced dispersion effects for experimentally achievable pulse widths. All code used for the work is made available as supplemental material.