International
Tables for
Crystallography
Volume H
Powder diffraction
Edited by C. J. Gilmore, J. A. Kaduk and H. Schenk

International Tables for Crystallography (2018). Vol. H, ch. 2.2, p. 56

## Section 2.2.3.3. Compound refractive lens

A. Fitcha*

aESRF, 71 Avenue des Martyrs, CS40220, 38043 Grenoble Cedex 9, France
Correspondence e-mail: fitch@esrf.fr

#### 2.2.3.3. Compound refractive lens

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The refractive index n of a material for X-rays is given (Gullikson, 2001; Spiller, 2000) bywhere fn = f1 + if2 is the complex scattering factor for forward scattering for atom n and Nn is the number of atoms of type n per unit volume. δ and β are known as the refractive index decrement and the absorption index, respectively, and vary with photon energy depending on the proximity of an absorption edge. The real part of the refractive index is therefore slightly less than 1, with δ typically of the order 10−6–10−9 depending on the energy. Thus a hole drilled in a piece of metal can act like a conventional convex lens, as the hole has a higher refractive index than the surrounding metal. With such a small difference in n between hole and metal, the focusing power is very slight; however, a series of holes (Fig. 2.2.9) can be used to focus the X-ray beam over a reasonable distance (Snigirev et al., 1997, 1998). For a series of cylindrical lenses, the focal length, f, is given by f = r/2Nδ, where r is the radius of the hole and N is the number of holes.

 Figure 2.2.9 | top | pdf |Schematic diagram of a set of refractive lenses.

Note that further away from the axis of the device the X-ray beam must pass through increasing amounts of material which absorb the radiation. Hence, only relatively small holes and apertures are possible (a maximum of a few mm in diameter) and weakly absorbing metals such as Be and Al are preferred. With hard-energy photons, Ni lenses are possible, and indeed the construction of such a device is a compromise between refractive power, absorption, aperture and the desired focal length. Such devices can be placed in the monochromatic beam or in a polychromatic beam with cooling.

Many variants of the basic scheme exist, with lenses pressed from foil with a parabolic form to eliminate spherical aberrations, with axial symmetry to focus in both the horizontal and vertical simultaneously (Lengeler et al., 1999), etched via lithography from plastic or other material, or with a more complex profile to minimize the amount of redundant material attenuating the transmitted beam by absorption and so allowing a larger aperture. A `transfocator' can be constructed whereby series of lenses can be accurately inserted or removed from the beam path, thus allowing the focusing power to be adjusted depending on the desired focal distance and the wavelength of the experiment (Vaughan et al., 2011).

### References

Gullikson, E. M. (2001). Atomic scattering factors. X-ray Data Booklet, edited by A. C. Thompson & D. Vaughan. Lawrence Berkeley National Laboratory, USA. http://xdb.lbl.gov/Section1/Sec_1-7.pdf .Google Scholar
Lengeler, B., Schroer, C., Tümmler, J., Benner, B., Richwin, M., Snigirev, A., Snigireva, I. & Drakopoulos, M. (1999). Imaging by parabolic refractive lenses in the hard X-ray range. J. Synchrotron Rad. 6, 1153–1167.Google Scholar
Snigirev, A. A., Filseth, B., Elleaume, P., Klocke, Th., Kohn, V., Lengeler, B., Snigireva, I., Souvorov, A. & Tuemmler, J. (1997). Refractive lenses for high-energy X-ray focusing. Proc. SPIE, 3151, 164–170.Google Scholar
Snigirev, A., Kohn, V., Snigireva, I., Souvorov, A. & Lengeler, B. (1998). Focusing high-energy X rays by compound refractive lenses. Appl. Opt. 37, 653–662.Google Scholar