Abstract

We show that a zone plate with a prespecified number of foci, which are separated axially by Rayleigh’s limit of resolution, can create an arbitrarily high focal depth when used as an apodizer over an otherwise clear aperture. We discuss the resolution and light gathering power of this method.

© 1990 Optical Society of America

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References

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  1. P. Jaquinot, B. Roizen-Dossier, “Apodization,” in Progress in Optics, Vol. 3, E. Wolf, Ed. (North-Holland, Amsterdam, 1964), p. 29.
  2. W. B. Wetherell, “The Calculation of Image Quality,” in Applied Optics and Optical Engineering, Vol. 3, R. R. Shannon, J. C. Wyant, Eds. (Academic, New York, 1980).
  3. G. Indebetouw, H. X. Bai, “Imaging with Fresnel Zone Pupil Masks: Extended Depth of Field,” Appl. Opt. 23, 4299–4302 (1984).
    [CrossRef] [PubMed]
  4. J. Ojeda-Castaneda, L. R. Berriel-Valdos, E. Montes, “Line-Spread Function Relatively Insensitive to Defocus,” Opt. Lett. 8, 458–460 (1983).
    [CrossRef] [PubMed]
  5. J. Ojeda-Castaneda, L. R. Berriel-Valdos, E. Montes, “Spatial Filter for Increasing the Depth of Focus,” Opt. Lett. 10, 520–522 (1985).
    [CrossRef] [PubMed]
  6. J. Ojeda-Castaneda, L. R. Berriel-Valdos, “Arbitrarily High Focal Depth with Finite Apertures,” Opt. Lett. 13, 183–185 (1988).
    [CrossRef] [PubMed]
  7. J. Ojeda-Castaneda, A. Diaz, “High Focal Depth by Quasibifocus,” Appl. Opt. 27, 4163–4165 (1988).
    [CrossRef]
  8. R. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1978), p. 189.

1988 (2)

1985 (1)

1984 (1)

1983 (1)

Bai, H. X.

Berriel-Valdos, L. R.

Bracewell, R.

R. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1978), p. 189.

Diaz, A.

Indebetouw, G.

Jaquinot, P.

P. Jaquinot, B. Roizen-Dossier, “Apodization,” in Progress in Optics, Vol. 3, E. Wolf, Ed. (North-Holland, Amsterdam, 1964), p. 29.

Montes, E.

Ojeda-Castaneda, J.

Roizen-Dossier, B.

P. Jaquinot, B. Roizen-Dossier, “Apodization,” in Progress in Optics, Vol. 3, E. Wolf, Ed. (North-Holland, Amsterdam, 1964), p. 29.

Wetherell, W. B.

W. B. Wetherell, “The Calculation of Image Quality,” in Applied Optics and Optical Engineering, Vol. 3, R. R. Shannon, J. C. Wyant, Eds. (Academic, New York, 1980).

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Figures (7)

Fig. 1
Fig. 1

Schematic diagram that shows the superposition of multiple axial amplitude responses.

Fig. 2
Fig. 2

Strehl ratio vs defocus of the proposed apodizer for an increased number of terms.

Fig. 3
Fig. 3

Amplitude transmittance of the equivalent 1-D pupil function.

Fig. 4
Fig. 4

True amplitude transmittance of the 2-D circular symmetric apodizer.

Fig. 5
Fig. 5

Grey level picture of the point spread functions at various focal planes (columns) and various foci (lines) of the zone plate.

Fig. 6
Fig. 6

Light throughput vs number of foci.

Fig. 7
Fig. 7

Grey level picture of the OTFs associated with the point spread functions in Fig. 5.

Equations (10)

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p ( r ; W 20 ) = 2 π 0 Ω p ˜ ( ρ ) J 0 ( 2 π r ρ ) exp [ i 2 π ( ρ / Ω ) 2 W 20 ] ρ d ρ .
p ( r = 0 , W 20 ) = π Ω 2 exp ( i π W 20 ) q ( W 20 ) ,
q ( W 20 ) = 0 Ω p ˜ ( ρ ) × exp { i 2 π W 20 [ ( ρ / Ω ) 2 - 0.5 ] } d [ ( ρ / Ω ) 2 ] = - q ˜ ( ζ ) rect ( ζ ) exp ( i 2 π W 20 ζ ) d ζ .
ζ = ( ρ / Ω ) 2 - 0.5 , q ˜ ( ζ ) rect ( ζ ) = p ˜ ( ρ ) .
q ( W 20 ) = sinc ( W 20 ) ,
q ( W 20 ) = m = - M M δ ( W 20 - m ) * sinc ( W 20 ) ,
q ˜ ( ζ ) = ( 2 M + 1 ) - 1 [ 1 + m = 1 M cos ( 2 π m ζ ) ] rect ( ζ ) ,
p ˜ ( ρ ) = ( 2 M + 1 ) - 1 × [ 1 + m = 1 M ( - 1 ) m cos ( 2 π m ρ 2 / Ω 2 ) ] circ ( ρ / Ω ) .
T = ( π Ω 2 ) - 1 2 π 0 p ˜ ( ρ ) 2 ρ d ρ = - q ˜ ( ζ ) 2 d ζ .
T = ( 2 M + 1 ) - 2 [ 1 + 4 m = 1 M - 1 / 2 1 / 2 cos ( 2 π m ζ ) d ζ + 4 m = 1 M n = 1 M - 1 / 2 1 / 2 cos ( 2 π m ζ ) cos ( 2 π n ζ ) d ζ ] = ( 2 M + 1 ) - 1 .

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