Finite-difference propagation for the imulation of x-ray multilayer optics

Recent progress in nanofabrication, namely of multilayer optics, and the constructionof coherent hard x-ray sources has enabled high resolution x-ray microscopy with large numericalaperture optics for small focal spot sizes. Sub-10 nm and even sub-5 nm focal spot sizes havealready been achieved using multilayer optics such as multilayer Laue lenses and multilayerzone plates. However these optics can not be described by the kinematic theory given theirextreme aspect-ratio between the depth (thickness) and the layer width. Moreover, the numericalsimulation of these optics is challenging, and the absence of an accessible numerical frameworkinhibits further progress in their design and utilization. Here, we simulate the propagation of x-raywavefields within and behind optical multilayer elements using a finite-difference propagationmethod. We show that the method offers high accuracy at reasonable computational cost. Weinvestigate how small focal spot sizes and highest diffraction efficiency of multilayer opticscan be achieved, considering volume diffraction effects such as waveguiding and Pendellösung.Finally, we show the simulation of a novel imaging scheme, allowing for a detailed study ofimage formation and the development of customized phase retrieval schemes.