Fast Simulation of Wide-Angle Coherent Diffractive Imaging

TL;DR

This work presents the propagation multi-slice Fourier-transform (pMSFT) as an accurate and efficient method for simulating wide-angle scattering in single-shot CDI at XUV and soft X-ray wavelengths. Derived from the scalar Helmholtz equation with a slice-by-slice approach, pMSFT separates vacuum propagation from a material correction in a split-step framework and uses a small-angle approximation to maintain tractable computations. Through a unified derivation, the paper shows how pMSFT encompasses and clarifies the approximations behind Hare's split-step, MSFT, Born, and SAXS, and provides a comprehensive benchmark against FDTD and Mie theory, demonstrating pMSFT’s superior accuracy across wide angles and a broad refractive-index range. The findings position pMSFT as the de facto standard for wide-angle CDI simulations, with SAXS remaining advantageous only in near-unity refractive-index and small-angle regimes, thus guiding practitioners in method selection for forward modeling and sample-property extraction.

Abstract

Single-shot coherent diffractive imaging (CDI) using intense XUV and soft X-ray pulses holds the promise to deliver information on the three dimensional shape as well as the optical properties of nano-scale objects in a single diffraction image. This advantage over conventional X-ray diffraction methods comes at the cost of a much more complex description of the underlying scattering process due to the importance of wide-angle scattering and propagation effects. The commonly employed reconstruction of the sample properties via iterative forward fitting of diffraction patterns requires an accurate and fast method to simulate the scattering process. This work introduces the propagation multi-slice Fourier transform method (pMSFT) and demonstrates its superior performance and accuracy against existing methods for wide-angle scattering. A derivation from first principles, a unified physical picture of the approximations underlying pMSFT and the existing methods, as well as a systematic benchmark that provides qualified guidance for the selection of the appropriate scattering method is presented.

Figures (4)

• Figure 1: Comparison of the differential scattering cross section of a truncated silver octahedron (150 nm diameter) for illumination with 90 eV (13.5 nm) soft X-ray radiation Barke2015 as predicted from different scattering simulation methods (as indicated). The result from an FDTD simulation (a) using optical properties of bulk-silver Henke1993 serves as a reference to benchmark the remaining approaches (c-g), see text for details. The results from pMSFT, Hare's method, MSFT, Born, and SAXS reflect the impact of the increasing level of approximation underlying the specific methods. Horizontal cuts through the respective two-dimensional k-space representations for $k_y=0$ enable the detailed comparison of quantitative signal strength and feature reproduction (b). Note that the Born and SAXS results are down-scaled by a factor of $0.03$ for convenience in all plots and that the data shown in (c-g) includes the polarization correction, see Section \ref{['sec:ff_detector']}.
• Figure 2: Schematic overview of different diffraction methods. (a) Propagation multislice Fourier transform (pMSFT) method with a series of alternating sub-steps representing material correction in real space (red) and vacuum propagation in k-space (blue) for each slice. (b) Conventional MSFT method with Beer-Lambert-type material projection (lilac) in real space on the path to the slice. After application of the scattering strength (grey) in the slice the resulting field is vacuum-propagated (material neglected) to the exit plane in k-space. (c) Born method with vacuum propagation to the slice in k-space, application of the scattering strength in the slice in real space followed by vacuum propagation in k-space to the exit plane. (d) SAXS approach with pure summation of the scattering strength over all slices resulting in a single projected scattering strength evaluated in the exit plane. For all cases the resulting exit field can be propagated to the detector using a far field transformation.
• Figure 3: Frequency dependent refractive index for various materials from the extreme ultraviolet (XUV) to the X-ray regime. The evolution of real and imaginary parts of the refractive index for selected materials (as indicated) are shown in panels (a) and (b), respectively. The corresponding evolutions in the complex plane are displayed in (c) together with additional data for all materials listed in the CXRO database Henke1993 (gray traces).
• Figure 4: Systematic benchmark of the different simulation methods for the scattering off a homogeneous sphere (diameter $D=10\lambda$) as a function of the real and imaginary part of the refractive index. False-color representations in (a-d) and (e-h) display the feature error $R(n)$ and the relative signal strength $Q(n)$ when compared to a Mie reference calculation, respectively (see text). As the signal strength in Born's approximation (e) greatly exceeds the range of the shared colorbar essentially everywhere, we have added grey contour lines for selected error levels. Representative angular profiles for three typical experimental scenarios, i.e. for the high energy limit with $n_1=1.000001$ (i), for silver at 90 eV with $n_2=0.89 + 0.09$i (j), and for helium at 23.5 eV with $n_3=1.03 + 0.03$i (k) illustrate the angle-dependent deviation underlying the aggregated accuracy measures (a-h). The data in (i-k) reflects the normalized scattered fraction in the plane with normal vector parallel to the polarization direction ($\varphi=0$).