Abstract

We show that for any rotationally symmetric apodizer on the full aperture, there is a family of apodizers on the annular aperture with the same functional Strehl ratio vs defocus (W20) and vs spherical aberration (W40). However, in the latter case, the coefficients W20 and W40 are reduced by the factors (1 − ɛ2) and (1 − ɛ2)2, respectively, where ɛ is the central obscuration ratio. We indicate that the best focal plane is shifted from W20 = −W40 to W20 = −(1 + ɛ2)W40. These general results allow us to design and to compare novel apodizers on annular apertures which reduce the influence of W20 and W40. The Strehl ratios of a novel family of apodizers are discussed to illustrate our general results.

© 1988 Optical Society of America

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References

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  1. P. Jaquinot, B. Roizen-Dossier, “Apodization,” Prog. Opt. 3, 29 (1964).
  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. W. T. Welford, “Use of Annular Apertures to Increase Focal Depth,” J. Opt. Soc. Am. 50, 749 (1960).
    [Crossref]
  4. J. T. McCrickerd, “Coherent Processing and Depth of Focus of Annular Aperture Imagery,” Appl. Opt. 10, 2226 (1971).
    [Crossref] [PubMed]
  5. G. Indebetouw, H. X. Bai, “Imaging with Fresnel Zone Pupil Masks: Extended Depth of Field,” Appl. Opt. 23, 4299 (1984).
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  6. G. W. Sutton, M. M. Weiner, S. A. Mani, “Fraunhofer Diffraction Patterns from Uniformly Illuminated Square Output Apertures with Noncentered Square Obscurations,” Appl. Opt. 15, 2228 (1976).
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  7. V. N. Mahajan, “Imaging with Obscured Pupils,” Opt. Lett. 1, 128 (1977).
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  8. 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]
  9. S. C. Biswas, A. Boivin, “Influence of Spherical Aberration on the Performance of Optimum Apodizers,” Opt. Acta 23, 569 (1976).
    [Crossref]
  10. M. J. Yzuel, F. Calvo, “A Study of the Possibility of Image Optimization by Apodization Filters in Optical Systems with Residual Aberrations,” Opt. Acta 26, 1397 (1979).
    [Crossref]
  11. J. Ojeda-Castaneda, L. R. Berriel-Valdos, E. L. Montes, “Line-Spread Function Relatively Insensitive to Defocus,” Opt. Lett. 8, 458 (1983).
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  12. M. Ya Mints, E. D. Prilepskii, “Apodization of a Passive Optical System with Aberrations,” Opt. Spectrosc. 52, 538 (1982).
  13. J. Ojeda-Castaneda, L. R. Berriel-Valdos, E. L. Montes, “Spatial Filter for Increasing the Depth of Focus,” Opt. Lett. 10, 520 (1985).
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  14. J. P. Mills, B. J. Thompson, “Effect of Aberrations and Apodization on the Performance of Coherent Optical Systems. I. The Amplitude Impulse Response,” J. Opt. Soc. Am. A 3, 694 (1986).
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  15. J. Ojeda-Castaneda, L. R. Berriel-Valdos, “Arbitrarily High Focal Depth with Finite Apertures,” Opt. Lett. 13, 183 (1988).
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  16. B. M. Oliver, in Technical Digest of Topical Meeting on Imaging in Astronomy (Optical Society of America, Washington, DC, 1975), paper WB7.
  17. H. F. A. Tschunko, “Imaging Performance of Annular Apertures 3: Apodization and Modulation Transfer Functions,” Appl. Opt. 18, 3770 (1979).
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  18. J. Ojeda-Castaneda, P. Andrés, A. Díaz, “Annular Apodizers for Low Sensitivity to Defocus and to Spherical Aberration,” Opt. Lett. 11, 487 (1986).
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1988 (1)

1986 (2)

1985 (1)

1984 (1)

1983 (1)

1982 (1)

M. Ya Mints, E. D. Prilepskii, “Apodization of a Passive Optical System with Aberrations,” Opt. Spectrosc. 52, 538 (1982).

1979 (2)

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

H. F. A. Tschunko, “Imaging Performance of Annular Apertures 3: Apodization and Modulation Transfer Functions,” Appl. Opt. 18, 3770 (1979).
[PubMed]

1977 (1)

1976 (2)

1971 (2)

1965 (1)

1964 (1)

P. Jaquinot, B. Roizen-Dossier, “Apodization,” Prog. Opt. 3, 29 (1964).

1960 (1)

Andrés, P.

Bai, H. X.

Barakat, R.

Berriel-Valdos, L. R.

Biswas, S. C.

S. C. Biswas, A. Boivin, “Influence of Spherical Aberration on the Performance of Optimum Apodizers,” Opt. Acta 23, 569 (1976).
[Crossref]

Boivin, A.

S. C. Biswas, A. Boivin, “Influence of Spherical Aberration on the Performance of Optimum Apodizers,” Opt. Acta 23, 569 (1976).
[Crossref]

Calvo, F.

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

Díaz, A.

Houston, A.

Indebetouw, G.

Jaquinot, P.

P. Jaquinot, B. Roizen-Dossier, “Apodization,” Prog. Opt. 3, 29 (1964).

Mahajan, V. N.

Mani, S. A.

McCrickerd, J. T.

Mills, J. P.

Mino, M.

Montes, E. L.

Ojeda-Castaneda, J.

Okano, Y.

Oliver, B. M.

B. M. Oliver, in Technical Digest of Topical Meeting on Imaging in Astronomy (Optical Society of America, Washington, DC, 1975), paper WB7.

Prilepskii, E. D.

M. Ya Mints, E. D. Prilepskii, “Apodization of a Passive Optical System with Aberrations,” Opt. Spectrosc. 52, 538 (1982).

Roizen-Dossier, B.

P. Jaquinot, B. Roizen-Dossier, “Apodization,” Prog. Opt. 3, 29 (1964).

Sutton, G. W.

Thompson, B. J.

Tschunko, H. F. A.

Weiner, M. M.

Welford, W. T.

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).

Ya Mints, M.

M. Ya Mints, E. D. Prilepskii, “Apodization of a Passive Optical System with Aberrations,” Opt. Spectrosc. 52, 538 (1982).

Yzuel, M. J.

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

Appl. Opt. (5)

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (1)

Opt. Acta (2)

S. C. Biswas, A. Boivin, “Influence of Spherical Aberration on the Performance of Optimum Apodizers,” Opt. Acta 23, 569 (1976).
[Crossref]

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

Opt. Lett. (5)

Opt. Spectrosc. (1)

M. Ya Mints, E. D. Prilepskii, “Apodization of a Passive Optical System with Aberrations,” Opt. Spectrosc. 52, 538 (1982).

Prog. Opt. (1)

P. Jaquinot, B. Roizen-Dossier, “Apodization,” Prog. Opt. 3, 29 (1964).

Other (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).

B. M. Oliver, in Technical Digest of Topical Meeting on Imaging in Astronomy (Optical Society of America, Washington, DC, 1975), paper WB7.

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Figures (13)

Fig. 1
Fig. 1

Schematic representation of the procedure for obtaining 2-D apodizers on annular apertures.

Fig. 2
Fig. 2

Geometric mapping of the 1-D apodizer q ˜ (ζ) to generate 2-D apodizers either on the full aperture or on the annular aperture.

Fig. 3
Fig. 3

(a) Reduction of the defocus coefficient W20 and the spherical aberration coefficient W40 vs the central obscuration ratio ɛ. (b) Shift of the best focal plane as a function of ɛ.

Fig. 4
Fig. 4

Amplitude transmittance of the hyper-Gaussian annular apodizers on the full aperture. The parameter is specified in Eq. (21).

Fig. 5
Fig. 5

Same as Fig. 4 but for the annular aperture with central obscuration ratio ɛ = 0.5.

Fig. 6
Fig. 6

Same as Fig. 4 but for ɛ = 0.8.

Fig. 7
Fig. 7

Gray level pictures of the amplitude transmittances in Figs. 4, 5, and 6. For column (A) ɛ = 0, for (B) ɛ = 0.5, and for (C) ɛ = 0.8. For line (a) α = 0, clear aperture, while for lines (b), (c), (d), and (e) α = 1, 2, 3, and 4, respectively.

Fig. 8
Fig. 8

Light throughput as a function of the parameter α for the apertures represented by ɛ = 0, 0.5, and 0.8.

Fig. 9
Fig. 9

Strehl ratio vs defocus W20 for zero spherical aberration W40 = 0 for hyper-Gaussian apodizers on the full aperture.

Fig. 10
Fig. 10

Same as Fig. 9 but for W40 = 1.

Fig. 11
Fig. 11

Same as Fig. 9 but for W40 = 2.

Fig. 12
Fig. 12

Same as Fig. 9 but for W40 = 3.

Fig. 13
Fig. 13

Same as Fig. 9 but for W40 = 4.

Equations (27)

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p ( r , W 20 , W 40 ) = 2 π 0 Ω p ˜ ( ρ ) exp { i 2 π [ W 20 ( ρ / Ω ) 2 + W 40 ( ρ / Ω ) 4 ] } × J 0 ( 2 π r ρ ) ρ d ρ .
S ( W 20 , W 40 ) = p ( r = 0 , W 20 , W 40 ) 2 / p ( r = 0 , W 20 = 0 , W 40 = 0 ) 2 .
q ( W 20 + W 40 , W 40 ) 2 = p ( r = 0 , W 20 , W 40 ) 2 = π Ω 2 0 Ω p ˜ ( ρ ) exp { i 2 π { W 20 ( ρ / Ω ) 2 + W 40 ( ρ / Ω ) 4 ] } d [ ( ρ / Ω ) 2 ] 2 .
ζ = ( ρ / Ω ) 2 - 0.5 ,             q ˜ ( ζ ) = p ˜ ( ρ ) ,
q ( W 20 + W 40 , W 40 ) 2 = π Ω 2 - 0.5 0.5 q ˜ ( ζ ) × exp { i 2 π [ W 40 ζ 2 + ( W 40 + W 20 ) ζ ] } d ζ 2 .
ρ = Ω [ ( 1 - ɛ 2 ) ζ + 0.5 ( 1 + ɛ 2 ) ] 1 / 2 ,             A ˜ ( ρ , ɛ ) = q ˜ ( ζ ) .
A ( r , ɛ ) = 2 π ɛ Ω Ω A ˜ ( ρ , ɛ ) exp { i 2 π [ W 20 ( ρ / Ω ) 2 + W 40 ( ρ / Ω ) 4 ] } × J 0 ( 2 π r ρ ) ρ d ρ .
q A ( W 20 + W 40 , W 40 ) 2 = π Ω 2 Ω Ω A ˜ ( ρ , ɛ ) × exp { i 2 π [ W 20 ( ρ / Ω ) 2 + W 40 ( ρ / Ω ) 4 ] } d [ ( ρ / Ω ) 2 ] 2 .
q A ( W 20 + W 40 , W 40 ) 2 = π Ω 2 ( 1 - ɛ 2 ) - 0.5 0.5 q ˜ ( ζ ) × exp [ i 2 π ( 1 - ɛ 2 ) 2 W 40 ζ 2 ] × exp { i 2 π ( 1 - ɛ 2 ) × [ W 20 + W 40 ( 1 + ɛ 2 ) ] ζ } d ζ 2 .
q A ( W 20 + W 40 , W 40 ) 2 = ( 1 - ɛ 2 ) 2 q { ( 1 - ɛ 2 ) × [ W 20 + ( 1 + ɛ 2 ) W 40 ] , × ( 1 - ɛ 2 ) 2 W 40 } 2 .
S A ( W 20 , W 40 ) 2 = q A ( W 20 + W 40 , W 40 ) 2 / q A ( 0 , 0 ) 2 = q { ( 1 - ɛ 2 ) [ W 20 + ( 1 + ɛ 2 ) W 40 ] , × ( 1 - ɛ 2 ) 2 W 40 } 2 / q ( 0 , 0 ) 2 .
S A ( W 20 ) = S [ ( 1 - ɛ 2 ) W 20 ] .
W 20 = - ( 1 + ɛ 2 ) W 40 .
S A ( W 20 , W 40 ) = S [ ( 1 - ɛ 2 ) W 20 ,             ( 1 - ɛ 2 ) 2 W 40 ] .
T = π 0 Ω p ˜ ( ρ ) 2 ρ d ρ / π Ω 2 = - 0.5 0.5 q ˜ ( ζ ) 2 d ζ ;
T A = π ɛ Ω Ω A ( ρ , ɛ ) 2 ρ d ρ / π Ω 2 = ( 1 - ɛ 2 ) - 0.5 0.5 q ˜ ( ζ ) 2 d ζ .
T A = ( 1 - ɛ 2 ) T .
q ˜ ( ζ ) = exp ( - 2 π α ζ 2 ) .
A ˜ ( ρ , ɛ ) = exp ( - π α / 2 ) · exp ( - 2 π α [ ( ρ / Ω ) 2 - ɛ 2 ] ( 1 - ɛ 2 ) - 1 × { [ ( ρ / Ω ) 2 - ɛ 2 ] ( 1 - ɛ 2 ) - 1 - 1 } ) .
p ˜ ( ρ ) = exp ( - π α / 2 ) exp { - 2 π α ( ρ / Ω ) 2 [ ( ρ / Ω ) 2 - 1 ] } .
ρ = Ω 1 + ɛ 2 / 2 .
W 20 = - { [ ( 1 - ɛ 8 ) / 4 - ( 1 - ɛ 4 ) ( 1 - ɛ 6 ) / 6 ( 1 - ɛ 2 ) ] / [ ( 1 - ɛ 6 ) / 3 - ( 1 - ɛ 4 ) 2 / 4 ( 1 - ɛ 2 ) ] } W 40 .
1 - ɛ 8 = ( 1 + ɛ 4 ) ( 1 - ɛ 4 ) ,
1 - ɛ 6 = ( 1 - ɛ 2 ) ( 1 + ɛ 2 + ɛ 4 ) = ( 1 - ɛ 4 ) ( 1 + ɛ 2 + ɛ 4 ) / ( 1 + ɛ 2 ) ,
W 20 = - { [ 3 ( 1 - ɛ 4 ) ( 1 + ɛ 4 ) - 2 ( 1 - ɛ 4 ) ( 1 + ɛ 2 + ɛ 4 ) ] / [ 4 ( 1 - ɛ 4 ) ( 1 + ɛ 2 + ɛ 4 ) / ( 1 + ɛ 2 ) - 3 ( 1 - ɛ 4 ) ( 1 + ɛ 2 ) ] } W 40 ,
W 20 = - { [ 3 ( 1 + ɛ 4 ) - 2 ( 1 + ɛ 2 + ɛ 4 ) ] / [ 4 ( 1 + ɛ 2 + ɛ 4 ) - 3 ( 1 + ɛ 2 ) 2 ] } ( 1 + ɛ 2 ) W 40 .
W 20 = - ( 1 + ɛ 2 ) W 40 .

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