Early X-ray microscopes of the reflection type were beset with geometrical aberrations which seriously limited their resolution. In addition, the problem was clouded by the role of surface defects to which an intolerable amount of non-symmetrical broadening of the image was attributed. Solution of the Kirchhoff diffraction integral shows that diffraction maxima are to be expected on the high angle side of the line image when spherical aberration is present and the magnification is greater than unity. A similar type of diffraction integral analysis leads to the discovery of the figure of an aspherical surface which is represented by a third order polynomial for which the spherical aberration is reduced to a negligible amount. The new figure increases the attainable resolution by an order of magnitude over that of spherical or cylindrical surfaces. A visible-light test of the aspherical figure at normal incidence is suggested by a solution of the diffraction integral for the case of reflection at normal incidence instead of grazing incidence as required for X-rays.


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  1. 1.
    Jentzsch, F.: Physik. Z. 30, 268 (1929).Google Scholar
  2. 2.
    Kirkpatrick, P., and A. V. Baez: J. Opt. Soc. Am. 38, 766 (1948).Google Scholar
  3. 3.
    Montel, M.: Rev. Optique 32, 585 (1953); — Opt. Acta 1, 117 (1954); — Compt. Rend. 239, 39 (1954).Google Scholar
  4. 4.
    Ehrenberg, W.: Nature 160, 330 (1947); — J. Opt. Soc. Am. 39, 741, 746 (1949).CrossRefGoogle Scholar
  5. 5.
    Hesser, D. R., and J. F. Mcgee: J. Opt. Soc. Am. 53, 525 (1963).Google Scholar
  6. 6.
    Corson, E. T.: Asymptotic expansions, p. 27, 107. London: Cambridge University Press 1965.Google Scholar
  7. 7.
    Chester, C., B. Friedman, and F. Ursell: Proc. Cambridge Phil. Soc. 53, 599–611 (1957).MathSciNetzbMATHGoogle Scholar
  8. 8.
    Mcgee, J. F., and J. W. Milton, in: X-ray microscopy and X-ray microanalysis, edit. by Eng-Strom et al. New York: Elsevier Publ. Co. 1960.Google Scholar
  9. 9.
    Baez, A. V., and M. Weissbluth: Phys. Rev. 93, 942 (1954).Google Scholar
  10. 10.
    Dyson, J.: Proc. Phys. Soc. (London) B 65, 580 (1952).ADSCrossRefGoogle Scholar
  11. 11.
    Baez, A. V., in: X-ray microscopy and micro-radiography, edit. by Cosslett, et al., p. 186. New York: Academic Press Inc. 1957.Google Scholar
  12. 12.
    Mcgee, J. F., and Ignacio Arrazola: Proceedings of the Fifth Internat. Congr. on X-Ray Optics and Microanalysis, Tübingen, Germany. Berlin-Heidelberg-New York: Springer 1969.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1969

Authors and Affiliations

  • J. F. McGee
    • 1
  • D. R. Hesser
    • 1
  • J. W. Milton
    • 1
  1. 1.Saint Louis UniversitySaint LouisUSA

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