X-ray Focusing Elements

Most applications of X-rays in medicine, dentistry, routine industrial testing and crystallography do not use any optical elements. Simple lenses and mirrors do not work, because the index of refraction for X-rays is very close to 1.0. Specialized optical elements to focus X-rays have become available only recently. Modern X-ray optics is required for high resolution X-ray microscopes and microprobes.

Zone Plates

Zone plates are circular diffraction gratings with radially increasing line density. The spacing is arranged so that the first order diffracted waves all meet at the primary focus. The zone boundaries are given by
r_n² = nfλ where n is the zone number, f is the focal length and λ is the wavelength.

As with all gratings, zone plates have multiple orders, and hence only a fraction of the incident X-rays are brought to the primary focus. That fraction depends on the thickness of the zones, and can reach a maximum of 40% for the ideal geometry.

Theoretical efficiency of zone plate lenses with different gold thickness

The focal length is inversely proportional to the X-ray wavelength, hence zone plates are highly chromatic. The ideal thickness also depends on the wavelength. However the resolution, or the size of the finest focal spot that can be formed depends only on the zone plate geometry, and is very close to the width of the outermost (and narrowest) zone, δR_n. The focal length can also be written in terms of the outer diameter of the zone plate, OD, as
f = OD δR_n/λ

For work with soft X-rays zone plates with 30 nm resolution are available. For hard X-ray applications the required thickness for reasonable efficiency is on the order of a micron, hence even for 50 nm resolution the thickness needs to be about 20 times the finest zone spacing. The fabrication of such tall and narrow structures poses difficult challenges.

A typical zone plate is no larger than a speck of dust, or smoke. It is supported on a transparent substrate for easy handling.

Scanning electron micrograph of the gold zones of a Frensnel zone plate objective.

Compound Refractive Lenses

It is not strictly true that lenses don't work. Because the real part of the index of refraction of materials is very close to, but slightly less than 1.0, or 1.0 – δ, where δ is a number of order 10^-5 the focal lengths of even the most highly curved lenses is measured in miles. Furthermore they work backwards: Concave lenses focus, while convex ones defocus. While single lenses are pretty useless, with a string of dozens of lenses in a row one can achieve a combined focusing effect with reasonable focal lengths. A major limitation is absorption of the X-rays in the material. One arranges the string of lenses in such a way that they constitute a series of holes that are nearly in contact, thereby minimizing the absorption, at least near the optic axis. The advantage of these compound refractive lenses is that they are much more robust than zone plates, and are suitable for focusing very high energy beams.

Kirkpatrick-Baez Mirrors

It is also not strictly true that mirrors do not work, but they only work at grazing incidence (or if prepared with a complex multilayer coating). In grazing incidence the reflectivity is close to 100% if the grazing angle is within the range of total external reflection, given by √(2δ). Grazing incidence mirrors can be curved slightly to image in one dimension, so as to image a point source to a line.

Taking two such mirrors, oriented at right angles, the combined effect can be arranged to create a point focus. Extremely smooth surfaces, precisely shaped figure, and careful alignment is required to achieve good focusing.

Wolter Mirrors

While Kirkpatrick-Baez mirrors image a point source on the optic axis to a point image, they do not form an image of an extended object. Wolter mirrors are required to perform that task. These are cylindrically symmetric grazing incidence mirrors made up of two segments with quadratic surfaces. These are particularly difficult to fabricate, and have found application mostly in X-ray astronomy.

Polycapillary Optics

Glass capillaries with smooth internal surfaces can guide X-rays by multiple grazing incidence reflection. By using a carefully shaped polycapillary array, as suggested by M. A. Kumakhov, X-rays from a laboratory source can be guided to a target for fluorescence analysis, for example. Such polycapillaries are now commercially available.

Monocapillary Optics

By carefully shaping the inside surface of the glass capillary to a paraboloidal or ellipsoidal shape, a collimated X-ray beam can be focused using just a single, grazing incidence reflection. Such optics have been particularly useful as condensers in X-ray microscopes, microprobes, and similar applications.