Abstract

Focal depth is assessed by the average value of the square modulus of the slope associated with the complex amplitude along the optical axis. Then, the calculus of variations is used for identifying the optimum apodizer, characterized by a Strehl ratio vs defocus with high focal depth, for a specified light throughput. We show that a certain Lorentzian profile is a quasioptimum solution for the above requirements. This apodizer has real and positive transmittance, and it can be modified to achieve arbitrarily high focal depth. A closed formula relates focal depth to light throughput.

© 1989 Optical Society of America

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References

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  1. J. Ojeda-Castaneda, L. R. Berriel-Valdos, E. Montes, “Line-Spread Function Relatively Insensitive to Defocus,” Opt. Lett. 8, 458–460 (1983).
    [CrossRef] [PubMed]
  2. G. Indebetouw, H. Bai, “Imaging with Fresnel Zone Pupils Masks: Extended Depth of Field,” Appl. Opt. 23, 4299–4302 (1984).
    [CrossRef] [PubMed]
  3. C. Varamit, G. Indebetouw, “Imaging Properties of Defocused Partitioned Pupils,” J. Opt. Soc. Am A 2, 799–807 (1985).
    [CrossRef]
  4. 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]
  5. J. Ojeda-Castaneda, P. Andres, A. Diaz, “Annular Apodizers for Low Sensitivity to Defocus and to Spherical Aberration,” Opt. Lett. 11, 487–489 (1986).
    [CrossRef] [PubMed]
  6. J. Ojeda-Castaneda, L. R. Berriel-Valdos, E. Montes, “Bessel Annular Apodizers: Imaging Characteristics,” Appl. Opt. 26, 2770–2772 (1987).
    [CrossRef] [PubMed]
  7. J. Ojeda-Castaneda, L. R. Berriel-Valdos, E. Montes, “Ambiguity Function as a Design Tool for High Focal Depth,” Appl. Opt. 27, 790–794 (1988).
    [CrossRef] [PubMed]
  8. W. H. Steel, “Precision Focusing with Photoelectric Detection,” J. Opt. Soc. Am. 52, 1153–1155 (1962).
    [CrossRef]
  9. B. R. Frieden, “On Arbitrarily Perfect Imagery with a Finite Aperture,” Opt. Acta 16, 795–807 (1969).
    [CrossRef]
  10. J. Ojeda-Castaneda, L. R. Berriel-Valdos, “Arbitrarily High Focal Depth with Finite Apertures,” Opt. Lett. 13, 183–185 (1988).
    [CrossRef] [PubMed]
  11. R. K. Luneberg, Mathematical Theory of Optics (U. California Press, Berkeley, CA, 1964) 353.
  12. T. Asakura, T. Veno, “Apodization for Minimizing the Second Moment of the Intensity Distribution in the Fraunhofer Diffraction Pattern (I),” Nouv. Rev. d’Opt. 7, 199–203 (1976).

1988

1987

1986

1985

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]

C. Varamit, G. Indebetouw, “Imaging Properties of Defocused Partitioned Pupils,” J. Opt. Soc. Am A 2, 799–807 (1985).
[CrossRef]

1984

1983

1976

T. Asakura, T. Veno, “Apodization for Minimizing the Second Moment of the Intensity Distribution in the Fraunhofer Diffraction Pattern (I),” Nouv. Rev. d’Opt. 7, 199–203 (1976).

1969

B. R. Frieden, “On Arbitrarily Perfect Imagery with a Finite Aperture,” Opt. Acta 16, 795–807 (1969).
[CrossRef]

1962

Andres, P.

Asakura, T.

T. Asakura, T. Veno, “Apodization for Minimizing the Second Moment of the Intensity Distribution in the Fraunhofer Diffraction Pattern (I),” Nouv. Rev. d’Opt. 7, 199–203 (1976).

Bai, H.

Berriel-Valdos, L. R.

Diaz, A.

Frieden, B. R.

B. R. Frieden, “On Arbitrarily Perfect Imagery with a Finite Aperture,” Opt. Acta 16, 795–807 (1969).
[CrossRef]

Indebetouw, G.

C. Varamit, G. Indebetouw, “Imaging Properties of Defocused Partitioned Pupils,” J. Opt. Soc. Am A 2, 799–807 (1985).
[CrossRef]

G. Indebetouw, H. Bai, “Imaging with Fresnel Zone Pupils Masks: Extended Depth of Field,” Appl. Opt. 23, 4299–4302 (1984).
[CrossRef] [PubMed]

Luneberg, R. K.

R. K. Luneberg, Mathematical Theory of Optics (U. California Press, Berkeley, CA, 1964) 353.

Montes, E.

Ojeda-Castaneda, J.

Steel, W. H.

Varamit, C.

C. Varamit, G. Indebetouw, “Imaging Properties of Defocused Partitioned Pupils,” J. Opt. Soc. Am A 2, 799–807 (1985).
[CrossRef]

Veno, T.

T. Asakura, T. Veno, “Apodization for Minimizing the Second Moment of the Intensity Distribution in the Fraunhofer Diffraction Pattern (I),” Nouv. Rev. d’Opt. 7, 199–203 (1976).

Appl. Opt.

J. Opt. Soc. Am A

C. Varamit, G. Indebetouw, “Imaging Properties of Defocused Partitioned Pupils,” J. Opt. Soc. Am A 2, 799–807 (1985).
[CrossRef]

J. Opt. Soc. Am.

Nouv. Rev. d’Opt.

T. Asakura, T. Veno, “Apodization for Minimizing the Second Moment of the Intensity Distribution in the Fraunhofer Diffraction Pattern (I),” Nouv. Rev. d’Opt. 7, 199–203 (1976).

Opt. Acta

B. R. Frieden, “On Arbitrarily Perfect Imagery with a Finite Aperture,” Opt. Acta 16, 795–807 (1969).
[CrossRef]

Opt. Lett.

Other

R. K. Luneberg, Mathematical Theory of Optics (U. California Press, Berkeley, CA, 1964) 353.

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Figures (9)

Fig. 1
Fig. 1

Amplitude transmittance profile in the 1-D representation of the Lorentzian apodizer. The width of the curves reduces when T decreases.

Fig. 2
Fig. 2

Same as Fig. 1 but in a true 2-D representation.

Fig. 3
Fig. 3

In focus pictures, column A), of a spoke pattern when imaged by a) clear aperture, b) Lorentzian apodizer with T = 0.1, and c) with T = 0.02. In column B), the same pictures but out of focus with W20 = λ.

Fig. 4
Fig. 4

Strehl ratio vs defocus for the pupil apertures in Fig. 2.

Fig. 5
Fig. 5

Light throughput vs focal depth for a clear aperture and the Lorentzian apodizer.

Fig. 6
Fig. 6

In focus, irradiance point spread functions for the pupil apertures in Fig. 2.

Fig. 7
Fig. 7

Same as in Fig. 6 but in the out of focus plane W20 = λ.

Fig. 8
Fig. 8

Optical transfer functions for the irradiance PSFs in Fig. 6, with W20 = 0.

Fig. 9
Fig. 9

Same as Fig. 8, but for the PSFs in Fig. 7, with W20 = λ.

Equations (21)

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p ( W 20 ) = 2 π 0 p ˜ ( ρ ) exp [ i 2 π W 20 ( ρ / Ω ) 2 ] ρ d ρ .
ζ = ( ρ / Ω ) 2 0 . 5 , q ˜ ( ζ ) = p ˜ ( ρ ) .
p ( W 20 ) = π Ω 2 exp ( i π W 20 ) q ( W 20 ) ,
q ( W 20 ) = 0 . 5 0 . 5 q ˜ ( ζ ) exp [ i 2 π W 20 ζ ] d ζ .
A = | q ( W 20 ) | 2 d W 20 ,
T = ( π Ω 2 ) 1 2 π 0 | p ˜ ( ρ ) | 2 ρ d ρ = 1 / 2 1 / 2 | q ˜ ( ζ ) | 2 d ζ .
T = | q ( W 20 ) | 2 d W 20 .
Y = A + α T = F ( q , q , W 20 ) d W 20 ;
F ( q , q , W 20 ) = | q ( W 20 ) | 2 + α | q ( W 20 ) | 2 .
lim W 20 ± q ( W 20 ) = 0 ;
d d W 20 F q F q = 0 .
q ( W 20 ) α q ( W 20 ) = 0 .
q ( W 20 ) = B exp ( α | W 20 | ) ,
q ( W 20 ) q ( W 20 ) = α q ( W 20 ) δ ( W 20 ) .
B 2 α = A = T 1 ,
q ( W 20 ) = 2 T exp [ 4 T | W 20 | ] .
q ˜ ( ζ ) = 1 1 + ( π ζ / 2 T ) 2 ,
p ˜ ( ρ ) = 1 1 + ( π / 2 T ) 2 [ ( ρ / Ω ) 2 0 . 5 ] 2 .
S ( W 20 ) = | q ( W 20 ) | 2 / | q ( 0 ) | 2 = e 8 T | W 20 | .
W 20 [ ln ( 1 / 0 . 8 ) ] / 8 T ,
W 20 ( 36 T ) 1 .

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