Predicting the angular resolution of x-ray mirrors

Abstract

The angular resolution of a grazing incidence focusing mirror is an important parameter in x-ray optics. It is often quantified by either the point spread function (PSF)—the distribution of the focused rays as a function of the distance from the center of the focal spot—or the half-energy width, which is double the PSF median value. The focusing of x-rays (with wavelengths 1000 shorter than visible light) is very sensitive to both a mirror’s surface profile and roughness (see Figure 1), i.e., surface relief on a variety of scales down to a few angstroms in height. PSF calculations for a focusing mirror must therefore account for both these factors. The surface roughness typically causes an x-ray scattering interference effect, which generally increases with x-ray energy. There are, however, several problems with computing the PSF by considering profile and roughness separately. It has long been known that x-ray focusing degradation caused by long-period deformations—e.g., Figure 1(a)—and surface roughness of mirrors—see Figure 1(c)—can be analyzed separately by treating the x-rays as rays or waves, respectively. When treating the x-rays as rays, geometric optics are applied. A well-established x-ray scattering (perturbation) theory1 is used when they are treated as waves, but it can be applied only to very smooth surfaces and to short-period features. However, the boundary between these two regimes2 is difficult to define and not clear-cut. Degradations in optical behavior caused by surface features in an intermediate spatial frequency range—see Figure 1(b)—cannot always be treated with either of the two methods mentioned.3 In addition, neither the geometric optics nor the scattering approach account for the effect of aperture diffraction. It can also be difficult to combine the separately derived geometry and scattering PSF values to obtain the total PSF of the mirror. We have created a new and self-consistent method that does not require boundaries between mirror geometries, profile frequencies, or surface roughnesses to be set.3 Our method is based Figure 1. Behavior of x-rays interacting with non-smooth grazing focusing mirrors. (a) Geometric optics can be applied to mirror defects with long spatial periods. (b) At mid-frequencies (i.e., centimeter to millimeter wavelength range) x-ray behavior is more uncertain. (c) Surface diffraction from micro-roughness is described by perturbation theory. (d) All three defects, as in (a), (b), and (c), are included and are difficult to treat consistently.