Simulating and optimizing compound refractive lens-based X-ray microscopes

Abstract

A comprehensive ray-transfer matrix formalism for the use and design of compound refractive lens-based X-ray microscopes is presented, including closed analytical expressions for key optical parameters. These form the basis of an optimization, which reveals that thick-lens imaging geometries are ideal in terms of spatial resolution and achromaticity.

The fractional Fourier transform as a simulation tool for lens-based X-ray microscopy

“…sponding to an object to entry-of-lens distance d 1 = 0.302 m and an exit-of-lens to detector distance d 2 = 3.54 m [equations (18) and (19) of Simons et al (2017)]. Using equation (16) and performing the FrFT propagation we find the image plane to be 0.46 mm closer to the CRL.…”

“…where a is the RMS acceptance angle of the CRL, given by Simons et al (2017), and ' CRL = ðT=f Þ 1=2 [see the supporting information (SI) for the derivation]. As the pupil function is Gaussian, the PSF is Gaussian as well.…”

“…9(b) shows the PSFs from geometrical optics. The geometrical optics PSF from the chromatic aberration alone is calculated from equation (38) of Simons et al (2017), and then convoluted with the CRL resolution PSF to give an effective PSF. The energy distributions used to weight the sums of the fully coherent simulations are not normalized, and so the increase in PSF intensity reflects the increase in flux.…”

Translative lens-based full-field coherent X-ray imaging

“…The amplitude attenuation suffered by Ψ in is determined by the function A r 1 s n , which has Gaussian profile characterized by the variance σ . H r 1 s n represents the finite aperture size of the lens and serves to transmit only the spatial frequencies of the wavefield Ψ in within the radius of the physical aperture [25]. χ(r 1 , α pq ) is the aberration function of the lens.…”

Modelling phase imperfections in compound refractive lenses

“…Both implementations, although based on different schemes, ray tracing (Alianelli et al, 2007) and wave optics respectively, are based on an ideal model combining refraction and absorption for the stacked lenses. Since then, much has been done in terms of refining the modelling of ideal X-ray lenses (Umbach et al, 2008;Sanchez del Rio & Alianelli, 2012;Osterhoff et al, 2013;Simons et al, 2017;Pedersen et al, 2018) and, to a certain extent, the modelling of optical imperfections (Pantell et al, 2001;Andrejczuk et al, 2010;Gasilov et al, 2017;Osterhoff et al, 2017). With exception of the work of Roth et al (2014), investigating the inner structure of X-ray lenses, the present models consider only lens shape and departure from a perfect parabolic shape to either test the limits of figure errors and fabrication defects, or to improve lens shape and focusing quality.…”