Optical elements for X-rays are based on the principle of grazing incidence reflection. Reflection, absorption, and transmission are expressed through the complex index of refraction, which can be written as n = 1 - 8 - ip where 8 describes the phase change and P accounts for the absorption. The optical constants 8 and P are functions of the wavelength or the photon energy (Fig. 6.1). For X-rays, the real part of n, 1 - 8, is slightly less than unity for matter whereas it is exactly unity in vacuum. X-rays propagating in vacuum, therefore, undergo total external reflection when incident below the critical grazing angle at with cos at = 1 - 8
according to Snell's law. Because of the nonvanishing value of P the reflection is actually not total for a < at, but is less than unity, and X-rays are reflected at angles even larger than the critical angle at . There are also characteristic absorption edges resulting from the specific atomic structure of each element.
Generally, 8 is proportional to the atomic number Z and proportional to the squared wavelength X. In the case of heavy elements, the critical angle for 8 C 1 can be estimated as at = 56 x ^Jp x X
for a material with density p and at « V25, with at in arcminutes, p in g/cm3, and X in nanometer . Converting the wavelength X into a photon energy E -which is more appropriate for X-rays - one gets at = 69 x ^p/E
with E in kiloelectronvolts.
Imaging X-ray optical systems were introduced by Hans Wolter when he published his paper (in German) on "Grazing Incidence Mirror Systems as Imaging Optics for X-rays" in 1952 . Originally meant for X-ray microscopy, they were
recognized as the appropriate optics for X-ray telescopes when the new branch of X-ray astronomy developed, after the detection of the first X-ray source in the sky in 1962 .
The type of optics proposed by Wolter - now called Wolter optics - makes use of the fact that X-rays are reflected by smooth surfaces under small angles of incidence. However, a single mirror optics like a paraboloid is not able to focus X-rays properly because it cannot satisfy Abbe's sine condition (Fig. 6.2).
The sine condition means that the principal surface has to be a sphere with radius f. For single mirrors, the principal surface is always identical to the mirror surface itself. Therefore, the sine condition is approximately satisfied only if the reflection is almost perpendicular to the mirror's surface. Unlike optical mirrors, grazing incidence X-ray mirrors obviously do not fulfil this condition. The solution for X-ray optics is - as Wolter elaborated - an optical system of at least two mirrors. Wolter could also show that mirror systems with even numbers of mirror elements can satisfy the sine condition whereas systems with odd numbers of mirror elements cannot (as long as reflections are less than 90° to the optical axis). In practice, two-mirror systems are strongly favored because any system with four and more mirror elements would increase the losses due to scattering and reflection significantly; furthermore, the alignment procedure for such systems would be complicated.
principal surface d opticalaxis
Fig. 6.2 Abbe's sine condition: The optic's principal surface has to be a sphere making the distance to the focus f the same for all paraxial rays
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