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

Abstract

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.

Comparison of distributed memory algorithms for X-ray wave propagation in inhomogeneous media

“…As noted in Sec. 1, this finite difference method has been shown to outperform the full-array Fresnel multislice algorithm when comparing compute time for the same degree of accuracy on single node computers [23,24]. The general Helmholtz equation problem is known to be challenging to solve using finite difference methods [50].…”

“…Note that while a more recent formulation of an equivalent to Eq. (22) exists [24], the expression of Eq. (22) is sufficiently accurate for our purposes given the fact that we work at the hard X-ray energy regime.…”

“…To manage the distributed memory grid, we used PETSc's data-management distributed-array (DMDA) object [28] which is designed for optimal performance when using logically rectangular grids (it re-orders the memory mapping to suit typical differential equation solver operations). As mentioned earlier, previous implementations [23,24] of parabolic wave equation solvers relied on algorithms that were not easily scalable to distributed parallel compute nodes. PETSc enables using a wide variety of preconditioners and Krylov solvers which can be tuned to the problem at hand, thus allowing us to design an algorithm with superior performance and scaling characteristics for parallel computing.…”

“…2.3 is to use the Helmholtz equation as a starting point, and solve for the exit wave using finite difference (FD) methods for solving partial differential equations (PDEs). In calculations for wave propagation in X-ray waveguides, this FD approach has been shown to offer speed and accuracy advantages [23,24].…”

Three dimensions, two microscopes, one code: Automatic differentiation for x-ray nanotomography beyond the depth of focus limit

“…This will be especially important for fully exploiting the dramatic increases in coherent x-ray flux that the next generation of synchrotron light sources will provide (7). One approach to simulate wave propagation in a complex object is the finite-difference method (8). However, due to the need to solve a series of partial differential equations, the efficiency of this method relies on the availability of distributed differential equation solvers, which are usually sophisticated to implement.…”