Abstract

Fresnel zone plates are used as pupil apertures to extend the depth of field of an incoherent imaging system. The defocused optical transfer function is evaluated approximately, and the results are confirmed by experiments and by subjective evaluations of the images of a test chart. With a pupil having four zones, for example, the resolution limit at four depths of field out of focus is improved by a factor of 10.

© 1984 Optical Society of America

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References

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  1. W. T. Welford, “Use of Annular Apertures to Increase Focal Depth,” J. Opt. Soc. Am. 50, 749 (1960); J. T. McCrickerd, “Coherent Processing and Depth of Focus of Annular Aperture Imagery,” Appl. Opt. 10, 2226 (1971).
    [CrossRef] [PubMed]
  2. H. H. Hopkin, Wave Theory of Aberrations (Oxford U. P., London, 1950).
  3. A. W. Lohman, Optical Information Processing (Physikalisches Institut der Universitat, 8520 Erlangen, EDR, 1978).
  4. C. J. C. Thompson, W. L. Wolfe, “Three-Dimensional Intensity Distribution in the Region of Focus of Two Superposed Converging Spherical Waves,” J. Opt. Soc. Am. A 1, 356 (1984).
    [CrossRef]
  5. M. Mino, Y. Okano, “Improvement in the OTF of a Defocused Optical System Through the Use of Shaded Apertures,” Appl. Opt. 10, 2219 (1971).
    [CrossRef] [PubMed]

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1971 (1)

1960 (1)

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Figures (4)

Fig. 1
Fig. 1

Approximate defocused OTF for an FZP pupil having four zones compared with the OTF of a clear aperture of the same size (a) in focus (z = 0); (b) two depth of field out of focus (z = 2); (c) z = 4; (d) (z = 6).

Fig. 2
Fig. 2

Evaluation of the quality of defocused images obtained with FZP apertures having different numbers N of zones. The measured parameter is the spatial frequency of a test chart corresponding to the limit of visual detection: (a) A, N = 0 (clear circular aperture), B, N = 4(+) (four zones, clear center), C, N = 4(−) (four zones, dark center); (b) A, N = 3(+) (three zones, clear center), B, N = 6(−) (six zones, dark center), C, superposition of three zones, clear center and six zones, dark center.

Fig. 3
Fig. 3

(A) Clear circular aperture (N = 0) (a) in focus, z = 0, (b) two depth of field defocus, z = 2, (c) z = 4, (d) z = 6. (B) Pupil with four zones, clear center (N = 4), (a)–(d) as in (A). (C) Pupil with six zones, clear center (N = 6); (a)–(d) as in (A).

Fig. 4
Fig. 4

(A) Clear circular aperture; (B) FZP pupil with five zones, clear center; (C) superposition of FZP N = 3(+) and 6(−); (D) thin annular aperture of ~0.1 width.

Equations (6)

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P ( ρ , z ) = n A n exp [ i 4 π ( z + n N ) ρ 2 ] circ ρ ,
T ( ρ ) = n A n exp ( i 4 π n N ρ 2 ) ,
H ( u , o , z ) = ϕ ( u , u ) exp ( - i 8 π z u u ) d u ,
ϕ ( u , u ) = P ( u + u / 2 v ) P * ( u - u / 2 , v ) d v ,
P ( + ) P * ( - ) = m , n A m A n exp { i 4 π N [ ( m - n ) ( u 2 + v 2 + u 2 / 4 ) + ( m + n ) u u ] } rect ( v ) rect [ u / ( 1 - u ) ] .
H ( u , o , z ) ~ A n 2 δ ( z - n N ) * sin [ 4 π z u ( 1 - u ) ] / ( 4 π z u ) .

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