Advertisement

Diffractive Optics II Zone Plates

  • Alan G. Michette

Abstract

Zone plates may be considered to be circular diffraction gratings with radially increasing line densities. Their focusing properties have long been well known,(1) and they offer perhaps the best chance in the foreseeable future of diffraction-limited imaging using soft X rays, where they are used in transmission. Figure 8.1 shows the requirement for focusing, namely that radiation emitted from an object point P is focused to an image point P’. The complex disturbance at P’ due to a spherical wavefront emitted by P is
$$Y\left( {P'} \right) = A\exp \left[ {ik(R + R')} \right]/(R + R')$$
(8.1)
where A is the amplitude at unit radius from P, R is the perpendicular distance from P to the focusing device, R’ is the perpendicular distance from the focusing device to P’, and k is the wavenumber \(( = 2\pi /\lambda )\). In equation (8.1) the periodic time factor has been omitted since soft X-ray detectors only respond to time-averaged signals, and an inclination factor expressing the amplitude variation with distance has been ignored since (due to the small sizes of the optical components) only small angle diffraction need be considered. Using the Huygens-Fresnel wave propagation principle, each point on the wavefront emitted by P is considered to be a point source of spherical wavelets; all such points are in phase. The total effect at P’ is found by summing the effects from all points on the wavefront with relative phases at P’ determined solely by the optical distances from the wavefront to P’. If two signals on the wavefront have an optical path difference to P’ of an integral number of wavelengths, they will contribute in phase (i.e., interfere constructively), while if the optical path difference is an odd-integral number of half-wavelengths, they will contribute completely out of phase (i.e., interfere destructively). The wavefront may be imagined as being divided into a set of Fresnel (or half-period) zones by spheres differing in radius by \(\lambda /2\) and centered on P’ Adjacent zones will then transmit radiation with opposite signs of phase—the resultant disturbance from these zones gives the propagating spherical wave in free space. If alternate zones are blocked, then the disturbances at P’ will have the same sign of phase and the summation shows an increase proportional to the number of zones. Note that it is not necessary for the distance R + R’ between P and P’ to be an integral number of half-wavelengths; in Figure 8.1, δ is introduced to take account of this, i.e., δ corresponds to an overall arbitrary phase.

Keywords

Point Spread Function Modulation Transfer Function Diffraction Efficiency Zone Plate Focus Intensity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J. L. Soret, Concerning diffraction by circular gratings, Ann. Phys. Chem., 156, 99–106 (1875).CrossRefGoogle Scholar
  2. 2.
    M. Born and E. Wolf, Principles of Optics, 5th ed., pp. 378–386, Pergamon Press, Elmsford, N.Y. (1975).Google Scholar
  3. 3.
    A. Boivin, On the theory of diffraction by concentric arrays of ring-shaped apertures, J. Opt. Soc. Am., 42, 60–64 (1952).CrossRefGoogle Scholar
  4. 4.
    G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, London (1922).Google Scholar
  5. 5.
    K. Kamiya, Theory of Fresnel zone plate, Sci. Light (Tokyo), 12, 35–49 (1963).Google Scholar
  6. 6.
    M. Born and E. Wolf, Principles of Optics, 5th ed., p. 441, Pergamon Press, Elmsford, N.Y. (1975).Google Scholar
  7. 7.
    M. Born and E. Wolf, Principles of Optics, 5th ed., pp. 397–398, Pergamon Press, Elmsford, N.Y. (1975).Google Scholar
  8. 8.
    J. W. Goodman, Introduction to Fourier Optics, p. 13, McGraw-Hill, New York (1968).Google Scholar
  9. 9.
    M. Born and E. Wolf, Principles of Optics, 5th ed., pp. 484–490, Pergamon Press, Elmsford, N.Y. (1975).Google Scholar
  10. 10.
    M. J. Simpson, Design considerations of zone plate optics for a scanning transmission x-ray microscope, Ph.D. thesis, London University (1984).Google Scholar
  11. 11.
    M. J. Simpson and A. G. Michette, Imaging properties of modified Fresnel zone plates, Opt. Acta, 31, 403–413 (1984).Google Scholar
  12. 12.
    H. Rarback and J. Kirz, Optical performance of apodized zone plates, High Resolution Soft X-Ray Optics, Proc. SPIE 316, 120–125 (1981).Google Scholar
  13. 13.
    H. F. A. Tschunko, Imaging performance of annular apertures, Appl. Opt., 13, 1820–1823 (1974).PubMedCrossRefGoogle Scholar
  14. 14.
    W. T. Welford, Aberrations of the Symmetrical Optical System, p. 60, Academic Press, New York (1974).Google Scholar
  15. 15.
    H. E. Hart, J. B. Scrandis, R. Mark, and R. D. Hatcher, Diffraction characteristics of a linear zone plate, J. Opt. Soc. Am., 56, 1018–1023 (1966).CrossRefGoogle Scholar
  16. 16.
    C. Gomez-Reino, J. M. Cuadrado, and M. V. P. Martinez, Optically produced linear zone plates, Appl. Opt., 18, 3032–3034 (1979).PubMedCrossRefGoogle Scholar
  17. 17.
    L. C. Janicijevic, Diffraction characteristics of square zone plates, J. Opt. (Paris), 13, 199–206 (1982).Google Scholar
  18. 18.
    T. R. Welberry and R. P. Williams, On certain non-circular zone plates, Opt. Acta, 23, 237–244 (1976).Google Scholar
  19. 19.
    C. Gomez-Reino, J. M. Cuadrado, and M. V. Perez, Elliptical and hyperbolic zone plates, Appl. Opt., 19, 1541–1545 (1980).PubMedCrossRefGoogle Scholar
  20. 20.
    J. M. Cuadrado, R. C. Gomez, and M. V. P. Martinez, Zone Plates produced by cylindrical wavefronts: Recording and reconstruction, Opt. Acta, 29, 717–723 (1982).Google Scholar
  21. 21.
    Z. Knittl and D. Lostakova-Roupova, Fresnel-Soret zone plates with manipulated focus patterns, Opt. Acta, 230, 927–942 (1983).Google Scholar
  22. 22.
    K. K. Dey and P. Khastgir, Frequency dependence of a compounded microwave zone plate, Proc. Natl. Acad. Sci. India Sect. A, 51, 45–48 (1981).Google Scholar
  23. 23.
    G. S. Waldman, Variations on the Fresnel zone plate, J. Opt. Soc. Am., 56, 215–218 (1966).CrossRefGoogle Scholar
  24. 24.
    Lord Rayleigh, Wave theory, in: Encyclopaedia Brittanica, 9th ed., Vol. 24, pp. 429–451 (1888).Google Scholar
  25. 25.
    R. W. Wood, Phase-reversed zone plates and diffraction telescopes, Philos. Mag. Ser. 5, 45, 511–522 (1898).Google Scholar
  26. 26.
    J. Kirz, Phase zone plates for x-rays and the extreme uv, J. Opt. Soc. Am., 64, 301–309 (1974).CrossRefGoogle Scholar
  27. 27.
    R. O. Tatchyn, Optimum zone plate theory and design, in: X-Ray Microscopy (G. Schmahl and D. Rudolph, eds.), Springer Series in Optical Sciences, Vol. 43, pp. 40–50, Springer, Berlin (1984).Google Scholar

Copyright information

© Plenum Press, New York 1986

Authors and Affiliations

  • Alan G. Michette
    • 1
  1. 1.King’s CollegeLondonEngland

Personalised recommendations