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X-ray Imaging Systems

In order for any optical system to form an image, it must satisfy the "Abbe sine condition", at least approximately. The Abbe sine condition states that an optical system will form an image of an infinitely distant object only if for each ray in the parallel beam emanating from the source:

h/sin(theta) = f

Diagram of the Abbe Sine Condition
Figure 1
where h is the (radial) distance of the ray from the optical axis, theta is the angle of the final path of the ray relative to its initial path (and thus the optical axis) and f is a constant for all rays. In other words, an image will be formed if the principal surface, defined as the locus of intersections of the initial and final paths of rays, is spherical. An optical system satisfying the Abbe sine condition thus acts as a simple spherical lens.

Paraboloid reflecting surface
Figure 2 - A simple parabolic mirror

A simple parabolic mirror was originally proposed in 1960 by Riccardo Giacconi and Bruno Rossi, the founders of extrasolar X-ray astronomy. This type of mirror is often used as the primary reflector in an optical telescope. However, it does not satisfy the Abbe sine condition, since the principal surface is the paraboloid itself (see Figure 2). Paraxial rays (rays impinging on the mirror parallel to the optical axis) will indeed focus to a point, but images of off-axis objects will be severely blurred. (An optical reflecting telescope gets around this by use of a secondary lens to correct this blurring.) The German physicist Hans Wolter showed in 1952 that the reflection off a combination of two elements, a paraboloid, followed by a confocal and coaxial hyperboloid, will allow the Abbe sine condition to be nearly satisfied. Wolter also showed that any odd number of coaxial conic sections will not form an image, but any even number can.

Nested pair of paraboloid and hyperboloid reflecting surfaces
Figure 3 - Two element reflection image system

Another system which forms real images consists of a set of two orthogonal parabolas of translation, off which incident X-rays reflect successively, as first proposed in 1948 by Kirkpatrick and Baez. A simplified Kirkpatrick-Baez design in shown in Figure 4a. In practical designs, the surface area is increased by using many approximately parallel mirrors (i.e., nesting) in each reflecting stage as in Figure 4b. The Kirkpatrick-Baez system offers inexpensive construction since the reflecting surfaces can be optically flat glass plates, bent to the proper curvature by mechanical stressing. On the other hand, the coalignment of many reflecting surfaces to form an optimum image is a difficult process. Since this was the first imaging telescope used for non-solar X-ray astronomy, the Kirkpatrick-Baez system is worthy of mention if only for historical reasons. The most commonly used X-ray mirrors are the cylindrically symmetric systems first described by Wolter.

The Kirkpatrick-Baez X-ray Telescope Design
Figure 4 - The Kirkpatrick-Baez X-ray Telescope Design

Wolter described three different imaging configurations, the Types I, II, and III, which are depicted in Figure 5. The design most commonly used by X-ray astronomers is the Type I since it has the simplest mechanical configuration (Figure 5a). In addition, the Type I design offers the possibility of nesting several telescopes inside one another, thereby increasing the useful reflecting area. This is an extremely important attribute, as virtually all X-ray sources are weak, and maximizing the light gathering power of a mirror system is critical. For comparable apertures and grazing angles, the primary advantage of Type II over Type I is that higher focal plane magnifications are attainable. This is so because the second reflection is off the outside of a surface, thus allowing longer focal lengths. However, since off-axis images suffer much more severely from blurring in Type II configurations, the Wolter Type II is useful only as a narrow-field imager or as the optic for a dispersive spectrometer. The Wolter Type III has never been employed for X-ray astronomy.

The Wolter X-ray Telescope Designs
Figure 5 - The Wolter X-ray Telescope Designs
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Contributed by Rob Petre of the Laboratory for High-Energy Astrophysics, GSFC

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