Angular spectrum simulation of X-ray focusing by Fresnel zone plates

Abstract

A computing simulation routine to model any type of circularly symmetric diffractive X-ray element has been implemented. The wavefield transmitted beyond the diffractive structures is numerically computed by the angular spectrum propagation method to an arbitrary propagation distance. Cylindrical symmetry is exploited to reduce the computation and memory requirements while preserving the accuracy of the numerical calculation through a quasi-discrete Hankel transform algorithm, an approach described by Guizar-Sicairos & Gutierrez-Vega [J. Opt. Soc. Am. A, (2004), 21, 53-58]. In particular, the code has been used to investigate the requirements for the stacking of two high-resolution Fresnel zone plates with an outermost zone width of 20 nm.

Keywords: Fresnel zone plate stacking; X-ray wavefield modeling; angular spectrum method; diffractive X-ray optics.

Figure 1

Scheme of the angular spectrum propagation method for a circularly symmetric optical wavefield, . The angular spectrum of the initial wavefield, , is calculated by a Hankel transform. Then, the propagated wavefield at a distance z, , is obtained by multiplying by the free-space propagator and applying a second Hankel transform.

Figure 2

Intensity of the propagated wavefield created by four different types of FZPs with a diameter of = 50 µm. (a) Ordinary, (b) zone-doubled, (c) zone-filled and (d) four-level staircase FZP geometries are considered. A photon energy of 6.2 keV, i.e. wavelength of 0.2 nm, is assumed.

Figure 3

Calculated diffraction efficiencies for four iridium FZP types considering an X-ray energy range from 6 to 12 keV. The diffraction efficiencies of the zone-doubled and zone-filled FZPs are about 35% lower than those from ordinary FZPs. The four-step staircase FZP displays a remarkable diffraction efficiency increase. The iridium L absorption edge generates an abrupt decrease of the diffraction efficiency at an energy of 11.2 keV.

Figure 4

Stacking of two FZPs. (a) Two identical FZPs separated by a very short distance can be stacked in the near-field to obtain an equivalent thicker structure and to achieve a substantial increase in diffraction efficiency. (b) The close proximity requirement on the two stacked FZPs can be relaxed by adjusting the diameter, , of the second diffractive optical element.

Figure 5

Calculated diffraction efficiency of two identical stacked gold FZPs with an outermost zone width of = 100 nm, a zone height of = 500 nm each and a photon energy of 6.2 keV. As the separation distance increases, the diffraction efficiency of the two stacked FZPs decreases. The diffraction efficiency of a single ordinary FZP with the same parameters is shown for comparison.

Figure 6

Simulated wavefield intensity in the vicinity of the focal spots created by two stacked zone-doubled FZPs with an outermost zone width of = 25 nm. (a) When two FZPs of identical diameter, = 45 µm, are stacked, the separation distance required for an acceptable focal spot shape is below = 10 µm. (b) The separation distance can be relaxed by adjusting the diameter, , of the second FZP. For a separation distance of = 25 µm an optimal focus profile is recovered for a diameter = 44.8 µm.

Figure 7

Normalized intensity of the focal spots created by two stacked zone-doubled FZPs with an outermost zone width of = 25 nm. (a) With two FZPs of identical diameter, = 45 µm, the focal spot is distorted by increasing their separation distance, . (b) For a chosen separation distance of = 25 µm, the optimal focal spot is recovered when the diameter of the second FZP is adjusted to = 44.8 µm.

Figure 8

Calculated diffraction efficiency for the two stacked high-resolution ordinary gold FZPs with an outermost zone width of = 20 nm and photon energy of 6.2 keV. (a) The diffraction efficiency decreases rapidly as a function of the separation distance when the two stacked FZPs have identical diameters. (b) The diffraction efficiency is kept constant when the diameter of the second FZP is optimized for every separation distance. (c) The diffractive efficiency for an experimentally realisable system in which the second FZP has a diameter adjusted for a separation distance of = 25 µm.