High-resolution bandpass x-ray imaging with crystal reflectors: overcoming geometric aberrations

Abstract

The imaging problem of a specular reflector is revisited. Retaining terms through second order in the reflector surface expansion, we derive the form of the aberration-limiting aperture for arbitrary magnification assuming no bandwidth limitations. A permissible relative aperture size of the reflector is limited by a set relative aberration tolerance and scales with the tangent of the central glancing angle of incidence. These limiting aberrations become practically insignificant near backscattering. The results extend to x-ray diffracting crystals in symmetric Bragg geometry shaped as an ellipsoid of revolution. This geometry permits polychromatic imaging for hard x-rays over a bandwidth defined by the accepted range of Bragg angles, thereby suppressing aberrations of higher orders. We assess ellipsoidal crystal imagers using ray tracing simulations for two high-magnification designs with Bragg angles far from and close to backscattering. In both cases the ellipsoidal crystals produce images of higher quality compared to those formed by equivalent toroidal crystal imagers.

Figures (10)

• Figure 1: DuMond diagram of a crystal reflection for the energy-angle ($E$-$\theta$) entrance phase space. For practical values of accepted angular divergence $\Delta \psi$ the energy bandwidth selected by the crystal reflection is dominated by the Bragg's law dispersion $\Delta E = E \Delta \psi /\tan{\theta}$ for the case of intermediate Bragg angles ($\Delta E_1$) as well as for near-backscattering case ($\Delta E_2$). The intrinsic width of the reflection region is exaggerated for clarity.
• Figure 2: Ellipsoidal reflector for arbitrary magnification: an object to be imaged and its Gaussian image are centered at $F_1$ and $F_2$, respectively. An arbitrary point on the reflector $P$ is defined in the local Cartesian system of coordinates [$u$,$\omega$,$l$] with origin O at the center of the reflector. The object and the image are assumed to be small portions of cylindrical surfaces centered at O and oriented normal to the central rays $F_1O$ and $OF_2$. The coordinates of the object point $A_1$ and its image $A_2$ are defined in the local system using their lateral $\epsilon$ and vertical $\eta$ angular distances. The limiting form of the reflector's aperture is obtained by a requirement that the lateral and vertical components of angular aberrations $\Delta \epsilon$ and $\Delta \eta$ combined in quadrature do not exceed a fixed value $\Delta \rho$.
• Figure 3: Limiting apertures of aspherical reflectors for aberration tolerance $\Delta \rho/\rho = 0.1$ and angles of incidence $\alpha~=~10^o, 22.5^o, 45^o$ by Eqs. \ref{['eq:ell0']} (black). For the case $\alpha = 10^o$ several ellipses are plotted using more precise Eqs. \ref{['eq:V123']} assuming an angular field of view with $\rho = 0.1$. These ellipses (colored) correspond to the outermost points along the diagonals across a circular field of view $\epsilon = \pm \rho/\sqrt{2}, \eta = \pm \rho/\sqrt{2}$.
• Figure 4: Mask used in the numerical simulations: a square grid with a period of 20$\mu$m; the size of transparent sections (squares) is 10$\mu$m. The image is obtained in a simulation where the detector is placed next to the mask with zero offset. The size of the detector pixels is 1$\times$1 $\mu$m$^2$.
• Figure 5: (a) Layout of simulated ellipsoidal geometry using Si 331 reflecting crystal in the meridional midplane (not to scale). The x-ray source plane is located at $F_1$ normal to incident rays (blue); the detector plane is placed at $F_2$ normal to reflected rays (red). (b) Ray-traced images using ellipsoidal reflector with aberration-limiting apertures and photon energy range set to aberration tolerance $\Delta \rho/\rho = 0.05$ (parameters are shown in Table \ref{['tab:param']}). (c) Ray-traced images using ellipsoidal reflector with aberration-limiting apertures and photon energy range reduced by factor of 2 and thus set to $\Delta \rho/\rho = 0.025$. (d) Ray-traced image using an equivalent toroidal reflector with the reduced aberration-limiting apertures and photon energy range as in (c). Images in (b-c) are shown in the source coordinates $x,y$ (i.e., coordinates normalized by the nominal magnification). For each case the main figure (top) corresponds to a simulation using a uniform x-ray source with a mask (Fig. \ref{['fig:mask']}), and the supplemental figure (bottom) corresponds to a simulation using a $5~\times~5~\mu m^2$ (FWHM) Gaussian source placed in the center of the object plane. The sources have uniform x-ray divergences and a uniform distribution of photon energies within the specified bandwidths.