Abstract

Results published previously [ J. Opt. Soc. Am. 57, 610 ( 1967)] concerning the focusing properties of Fresnel zone plates were obtained by implicitly invoking the Fraunhofer approximation. A comparison of these results with those derived from Fresnel diffraction theory shows that the focusing properties of Fresnel zone plates predicted by Fraunhofer diffraction theory are incorrect except for resolution defined by the Rayleigh criterion.

© 1991 Optical Society of America

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References

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  1. D. J. Stigliani, R. Mittra, R. G. Semonin, “Resolving power of a zone plate,”J. Opt. Soc. Am. 57, 610–613 (1967).
    [CrossRef]
  2. G. S. Waldman, “Variations on the Fresnel zone plate,”J. Opt. Soc. Am. 56, 215–218 (1966).
    [CrossRef]
  3. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964), Chap. 8.
  4. J. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 57–62.
  5. M. H. Horman, “Efficiencies of zone plates and phase zones plates,” Appl. Opt. 6, 2011–2013 (1967).
    [CrossRef] [PubMed]
  6. M. Bottema, “Fresnel zone-plate diffraction patterns,”J. Opt. Soc. Am. 59, 1632–1638 (1969).
    [CrossRef]
  7. A. Sommerfeld, Optics (Academic, New York, 1954), Chap. 5.

1969 (1)

1967 (2)

1966 (1)

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Figures (5)

Fig. 1
Fig. 1

Coordinate system involved in Eq. (1).

Fig. 2
Fig. 2

/(λf) versus the number of zones for positive and negative Fresnel zone plates (P-FZP and N-FZP, respectively).

Fig. 3
Fig. 3

Maximum value of the first sidelobe of the normalized intensity distribution versus the number of zones for positive Fresnel zone plates (P-FZP).

Fig. 4
Fig. 4

Maximum value of the first sidelobe of the normalized intensity distribution versus the number of zones for negative Fresnel zone plates (N-FZP).

Fig. 5
Fig. 5

Normalized intensity distributions at the focal plane: (a) positive Fresnel zone plate with N = 3, (b) negative Fresnel zone plate with N = 2. The solid curves show the results computed by using Eq. (6) of this paper, the dashed curves by using Eqs. (6) and (7) of Ref. 1.

Equations (6)

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u c = 2 ρ 1 ρ 2 - π / 2 π / 2 u 0 exp ( - j 2 π λ Δ L ) r d r d θ ,
- Δ L = r 0 2 2 F - r 0 r F sin θ .
u ( r 0 ) = exp ( j 2 π F / λ ) j λ F 0 0 2 π u ( r ) × exp [ j 2 π λ ( r 0 2 2 F - r 0 r F sin θ ) ] r d r d θ ,
u ( r 0 ) = 2 π j λ F exp ( j 2 π λ F ) exp ( j π λ F r 0 2 ) × 0 u ( r ) r J 0 ( 2 π r 0 λ F r ) exp ( j π λ F r 2 ) d r .
u ( 0 ) = { [ π U 0 ( N + 1 ) / ( 2 j ) ] exp ( j 2 π F / λ ) for positive zones [ π U 0 N / ( 2 j ) ] exp ( j 2 π F / λ ) for negative zones
u ( 0 ) = { ( N + 1 ) U 0 exp ( j 2 π F / λ ) for positive zones - N U 0 exp ( j 2 π F / λ ) for negative zones

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