Efficient Analytical Approaches to the Optics of Compound Refractive Lenses for Use with Synchrotron X-rays
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I.
INTRODUCTION
IN recent decades, synchrotron X-radiation imaging has become a versatile and indispensable tool in materials science. The highly collimated, high flux (i.e., high brilliance) X-ray beam from a modern synchrotron source allows for rapid non-destructive characterization of microstructural features in bulk samples, and has given new insight into a wide range of phenomena in materials science. As technology advances, better sources and X-ray detectors allow for resolution at increasingly small length and time scales. At the same time there is a rapid development in the use of X-ray optics. Many branches of X-ray optics exist, relying on different physical phenomena, such as diffraction, e.g., Fresnel zone plates for soft and medium energy X-rays (energies lower than 20 keV),[1,2] Bragg diffraction,[3] and total external reflection, e.g., Kirkpatrick-Baez mirrors,[4] Wolter mirrors,[5] ‘‘lobster-eye lenses,’’[6] and Kumakhov lenses.[7] Refractive optics, although tremendously successful in the visible light regime, were long thought impractical for X-rays due to the small deviations from unit index of refraction of materials for these photon energies. The complex refractive index for X-rays is conventionally written n ¼ 1 d þ ib;
½1
where d is the (real) refractive index decrement, and b is the (real) absorption index, which respectively govern the strength of refraction and attenuation, and the refractive index decrement takes on typical values of
STEFAN OTHMAR POULSEN, Post Doctoral Researcher, formerly with DTU Energy Conversion, Technical University of Denmark, Frederiksborgvej 399, 4000 Roskilde, Denmark, is now with the Department of Materials Science and Engineering, Northwestern University, Evanston, IL 60208. Contact e-mail: stefan.poulsen@ northwestern.edu HENNING FRIIS POULSEN, Professor, is with NEXMAP, Department of Physics, Technical University of Denmark, Anker Engelunds Vej 1, 2800 Kgs. Lyngby, Denmark. Manuscript submitted January 8, 2014. METALLURGICAL AND MATERIALS TRANSACTIONS A
d 104 to 107 . Thus, the contrast to vacuum is very low. This may be compared to e.g., visible light refracting in the common lens material crown glass, which has a refractive index of n 1:5. A conceptually simple solution to allow practical use of refractive optics for hard X-rays was first demonstrated by Snigirev et al. in 1996[8]; a number of individual lenses were arranged in a linear array as sketched in Figure 1, to create a so-called compound refractive lens (CRL), by which the optical power of the compound system was enhanced greatly above that of an individual lens. Since their introduction, the technology has matured, and CRLs have found widespread use at synchrotrons. Among the advantages of CRLs as compared to the types of optics listed above, is the relatively low cost and the high adaptability, meaning that a wide range of X-ray energies may be accommodated by controlling the number of individual lenses in the beam path. This adaptability has been exploited in the development of transf
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