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Optics Express

Optics Express

  • Vol. 16, Iss. 16 — Aug. 4, 2008
  • pp: 12050–12059

An analytical approach to estimating aberrations in curved multilayer optics for hard x-rays: 1. Derivation of caustic shapes

J-P. Guigay, Ch. Morawe, V. Mocella, and C. Ferrero  »View Author Affiliations

Optics Express, Vol. 16, Issue 16, pp. 12050-12059 (2008)

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An analytical approach has been developed to derive aberration effects in parabolic and elliptic multilayer optics with weak interaction between photons and matter. The method is based on geometrical ray tracing including refraction effects up to the first order of the refractive index decrement δ. In the parabolic case, the derivation leads to simple parametric equations for the caustic shape. In the elliptic case, the analytical results more involved, but can be well approximated by the parabolic solution. Both geometries are compared with regard to the fundamental impact on their focusing properties.

© 2008 Optical Society of America

OCIS Codes
(080.1010) Geometric optics : Aberrations (global)
(080.2720) Geometric optics : Mathematical methods (general)
(340.6720) X-ray optics : Synchrotron radiation
(340.7470) X-ray optics : X-ray mirrors
(080.4228) Geometric optics : Nonspherical mirror surfaces

Original Manuscript: May 23, 2008
Revised Manuscript: July 9, 2008
Manuscript Accepted: July 13, 2008
Published: July 28, 2008

J.-P. Guigay, Ch. Morawe, V. Mocella, and C. Ferrero, "An analytical approach to estimating aberrations in curved multilayer optics for hard x-rays: 1. Derivation of caustic shapes," Opt. Express 16, 12050-12059 (2008)

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  1. W. Chao, B. D. Harteneck, J. A. Liddle, E. H. Anderson, and D. T. Attwood, "Soft X-ray microscopy at a spatial resolution better than 15 nm," Nature 435, 1210-1213 (2005). [CrossRef] [PubMed]
  2. C. G. Schroer, O. Kurapova, J. Patommel, P. Boye, J. Feldkamp, B. Lengeler, M. Burghammer, C. Riekel, L. Vincze, A. van der Hart, and M. Küchler, "Hard X-ray nanoprobe based on refractive X-ray lenses," Appl. Phys. Lett. 87, 124103 (2005). [CrossRef]
  3. A. Jarre, C. Fuhse, C. Ollinger, J. Seeger, R. Tucoulou, and T. Salditt, "Two-dimensional hard X-ray beam compression by combined focusing and waveguide optics," Phys. Rev. Lett. 94, 074801 (2005). [CrossRef] [PubMed]
  4. D. H. Bilderback, S. A. Hoffman, and D. J. Thiel, "Nanometer spatial resolution achieved in hard X-ray imaging and Laue diffraction experiments," Science 263, 201-203 (1994). [CrossRef] [PubMed]
  5. H. Mimura, S. Matsuyama, H. Yumoto, H. Hara, K. Yamamura, Y. Sano, M. Shibahara, K. Endo, Y. Mori, Y. Nishino, K. Tamasaku, M. Yabashi, T. Ishikawa, and K. Yamauchi, "Hard X-ray diffraction-limited nanofocusing with Kirkpatrick-Baez mirrors," J. J. Appl. Phys. 44, L539 - L542 (2005).
  6. W. Liu, G. E. Ice, J. Z. Tischler, A. Khounsary, C. Liu, L. Assoufid, and A.T. Macrander, "Short focal length Kirkpatrick-Baez mirrors for a hard X-ray nanoprobe," Rev. Sci. Instr. 76, 113701 (2005). [CrossRef]
  7. Ch. Morawe, O. Hignette, P. Cloetens, W. Ludwig, Ch. Borel, P. Bernard, and A. Rommeveaux, "Graded multilayers for focusing hard X rays below 50 nm," Proc. SPIE 6317, 63170F (2006). [CrossRef]
  8. F. Pfeiffer, C. David, J. F. van der Veen, and C. Bergemann, "Nanometer focusing properties of Fresnel zone plates described by dynamical diffraction theory," Phys. Rev. B 73, 245331 (2006). [CrossRef]
  9. C. G. Schroer and B. Lengeler, "Focusing hard X rays to nanometer dimensions by adiabatically focusing lenses," Phys. Rev. Lett. 94, 054802 (2005). [CrossRef] [PubMed]
  10. K. Yamauchi, K. Yamamura, H. Mimura, Y. Sano, A. Saito, A. Souvorov, M. Yabashi, K. Tamasaku, T. Ishikawa, Y. Mori, "Nearly diffraction-limited line focusing of hard X-ray beam with an elliptically figured mirror," J. Synchrotron Rad. 9, 313 - 316 (2002). [CrossRef]
  11. C. M. Kewish, L. Assoufid, A. T. Macrander, and J. Qian, "Wave-optical simulation of hard X-ray nanofocusing by precisely figured elliptical mirrors," Appl. Opt. 46, 2010 - 2021 (2007). [CrossRef] [PubMed]
  12. C. G. Schroer, "Focusing hard X rays to nanometer dimensions using Fresnel zone plates," Phys. Rev. B 74, 033405 (2006). [CrossRef]
  13. H. Yan, J. Maser, A.T. Macrander, Q. Shen, S. Vogt, G.B. Stephenson, H.C. Kang, "Takagi-Taupin description of X-ray dynamical diffraction from diffractive optics with large numerical aperture," Phys. Rev. B 76, 115438 (2007). [CrossRef]
  14. H. Mimura, S. Matsuyama, H. Yumoto, S. Handa, T. Kimura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, K. Yamauchi, "Reflective optics for sub-10nm hard X-ray focusing," Proc. SPIE 6705, 67050L (2007). [CrossRef]
  15. A. Authier, Dynamical Theory of X-ray Diffraction, IUCr monographs, (Oxford, University Press. 2001)

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