Abstract

Apodizers with relatively high transmittance over an annular region of the exit pupil can reduce the sensitivity to defocusing and to spherical aberration [ Opt. Lett. 11, 487 ( 1986)]. Here, we analyze the imaging properties (pupil functions, point spread functions, optical transfer functions, and Strehl ratios) of the Bessel type of annular apodizers. We also show some computer-simulated images, obtained with and without this kind of annular apodizer.

© 1987 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).
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  5. C. J. R. Sheppard, T. Wilson, “Imaging Properties of Annular Lenses,” Appl. Opt. 18, 3764 (1979).
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  7. 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]
  8. M. J. Yzuel, F. Calvo, “A Study of the Possibility of Image Optimization Filters in Optical Systems with Residual Aberrations,” Opt. Acta 26, 1397 (1979).
    [CrossRef]
  9. S. C. Biswas, A. Boivin, “Performance of Optimum Apodizers in Presence of Primary Coma,” Can. J. Phys. 57, 1388 (1979).
    [CrossRef]
  10. J. Ojeda-Castaneda, L. R. Berriel-Valdos, E. Montes, “Line-Spread Function Relatively Insensitive to Defocus,” Opt. Lett. 8, 458 (1983).
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  11. J. Ojeda-Castaneda, L. R. Berriel-Valdos, E. Montes, “Spatial Filter for Increasing the Depth of Focus,” Opt. Lett. 10, 520 (1985).
    [CrossRef] [PubMed]
  12. J. Ojeda-Castaneda, P. Andres, A. Diaz, “Annular Apodizers for Low Sensitivity to Defocus and to Spherical Aberration,” Opt. Lett. 11, 487 (1986).
    [CrossRef] [PubMed]
  13. P. Jacquinot, B. Roizen-Dossier, “Apodisation,” Prog. Opt. 3, 29 (1964).
    [CrossRef]

1986

1985

J. Ojeda-Castaneda, L. R. Berriel-Valdos, E. Montes, “Spatial Filter for Increasing the Depth of Focus,” Opt. Lett. 10, 520 (1985).
[CrossRef] [PubMed]

Z. S. Hegedous, “Annular Pupil Arrays Application to Confocal Scanning,” Opt. Acta 32, 815 (1985).

1983

1979

M. J. Yzuel, F. Calvo, “A Study of the Possibility of Image Optimization Filters in Optical Systems with Residual Aberrations,” Opt. Acta 26, 1397 (1979).
[CrossRef]

S. C. Biswas, A. Boivin, “Performance of Optimum Apodizers in Presence of Primary Coma,” Can. J. Phys. 57, 1388 (1979).
[CrossRef]

C. J. R. Sheppard, T. Wilson, “Imaging Properties of Annular Lenses,” Appl. Opt. 18, 3764 (1979).
[PubMed]

1974

1971

1965

1964

P. Jacquinot, B. Roizen-Dossier, “Apodisation,” Prog. Opt. 3, 29 (1964).
[CrossRef]

1960

Andres, P.

Barakat, R.

Berriel-Valdos, L. R.

Biswas, S. C.

S. C. Biswas, A. Boivin, “Performance of Optimum Apodizers in Presence of Primary Coma,” Can. J. Phys. 57, 1388 (1979).
[CrossRef]

Boivin, A.

S. C. Biswas, A. Boivin, “Performance of Optimum Apodizers in Presence of Primary Coma,” Can. J. Phys. 57, 1388 (1979).
[CrossRef]

Calvo, F.

M. J. Yzuel, F. Calvo, “A Study of the Possibility of Image Optimization Filters in Optical Systems with Residual Aberrations,” Opt. Acta 26, 1397 (1979).
[CrossRef]

Diaz, A.

Hegedous, Z. S.

Z. S. Hegedous, “Annular Pupil Arrays Application to Confocal Scanning,” Opt. Acta 32, 815 (1985).

Houston, A.

Jacquinot, P.

P. Jacquinot, B. Roizen-Dossier, “Apodisation,” Prog. Opt. 3, 29 (1964).
[CrossRef]

McCrickerd, J. T.

Mino, M.

Montes, E.

Ojeda-Castaneda, J.

Okano, Y.

Roizen-Dossier, B.

P. Jacquinot, B. Roizen-Dossier, “Apodisation,” Prog. Opt. 3, 29 (1964).
[CrossRef]

Sheppard, C. J. R.

Tschunko, H. F. A.

Welford, W. T.

Wilson, T.

Yzuel, M. J.

M. J. Yzuel, F. Calvo, “A Study of the Possibility of Image Optimization Filters in Optical Systems with Residual Aberrations,” Opt. Acta 26, 1397 (1979).
[CrossRef]

Appl. Opt.

Can. J. Phys.

S. C. Biswas, A. Boivin, “Performance of Optimum Apodizers in Presence of Primary Coma,” Can. J. Phys. 57, 1388 (1979).
[CrossRef]

J. Opt. Soc. Am.

Opt. Acta

Z. S. Hegedous, “Annular Pupil Arrays Application to Confocal Scanning,” Opt. Acta 32, 815 (1985).

M. J. Yzuel, F. Calvo, “A Study of the Possibility of Image Optimization Filters in Optical Systems with Residual Aberrations,” Opt. Acta 26, 1397 (1979).
[CrossRef]

Opt. Lett.

Prog. Opt.

P. Jacquinot, B. Roizen-Dossier, “Apodisation,” Prog. Opt. 3, 29 (1964).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Radial profile of the amplitude transmittance for the pupil functions in Eq. (6).

Fig. 2
Fig. 2

Logarithmic display of the irradiance point spread functions that are associated with the pupil functions in Fig. 1.

Fig. 3
Fig. 3

Logarithmic display of the Strehl ratio vs defocusing, or normalized on-axis irradiance, of the apodizers in Fig. 1.

Fig. 4
Fig. 4

In-focus optical transfer functions of the apodizers in Fig. 1.

Fig. 5
Fig. 5

Out-of-focus, W20 = λ, optical transfer functions.

Fig. 6
Fig. 6

A, In-focus and B, out-of-focus images of a spoke pattern obtained with the apodizers: (a) n = ½, (b) n = 1, (c) n = 3/2, (d) n = 2, and (e) n = 5/2.

Equations (8)

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p ( r . W 20 ) = 2 π 0 p ˜ ( ρ ) exp [ i 2 π W 20 ( ρ / Ω ) 2 ] J 0 ( 2 π r ρ ) ρ d ρ .
q ( W 20 ) 2 = p ( r = 0 , W 20 ) 2 = π Ω 2 0 p ˜ ( ρ ) exp [ i 2 π W 20 ( ρ / Ω ) 2 ] d [ ( ρ / Ω ) 2 ] 2 .
ζ = ( ρ / Ω ) 2 - 0.5 ,             q ˜ ( ζ ) = p ˜ ( ρ ) ,
q ( W 20 ) 2 = | π Ω 2 - 0.5 0.5 q ( ζ ) exp ( i 2 π W 20 ζ ) d ζ | 2 .
q ˜ ( ζ ) = ( 1 - 4 ζ 2 ) n - 1 / 2 rect ( ζ ) ,
p ˜ ( ρ ) = ( 2 ρ / Ω ) 2 n - 1 [ 1 - ( ρ / Ω ) 2 ] n - 1 / 2 if ρ Ω = 0 otherwise ,
Y = 1.0 + 0.25 log [ p ( r , W 20 = 0 ) 2 / p ( r = 0 , W 20 = 0 ) 2 ] .
Y = 1.0 + log [ q ( W 20 ) 2 / q ( W 20 = 0 ) 2 ] .

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