Abstract

We propose a method for designing apodizers that increase the depth of focus and reduce the influence of spherical aberration. These two properties are explicitly manifested in computer-generated pictures and values of the Strehl ratio for variable spherical aberration.

© 1986 Optical Society of America

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References

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  1. W. T. Welford, J. Opt. Soc. Am. 50, 749 (1960).
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  2. R. Barakat, A. Houston, J. Opt. Soc. Am. 55, 538 (1965).
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  3. C. J. R. Sheppard, T. Wilson, Opt. Lett. 3, 115 (1978).
    [CrossRef] [PubMed]
  4. T. Wilson, C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, London, 1984).
  5. H. H. Hopkins, Proc. Phys. Soc. London Ser. B 62, 22 (1949).
    [CrossRef]
  6. J. E. Villeneuve, A. Boivin, S. C. Biswas, Can. J. Phys. 63, 287 (1985), and references cited therein.
    [CrossRef]
  7. G. Indebetouw, H. Bai, Appl. Opt. 23, 4299 (1984).
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  8. C. W. McCutchen, J. Opt. Soc. Am. 54, 240 (1964).
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  9. J. Ojeda-Castaneda, L. R. Berriel-Valdos, E. Montes, Opt. Lett. 10, 520 (1985).
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  10. J. Ojeda-Castan̂eda, P. Andrés, A. Díaz (submitted to J. Opt. Soc. Am. A).
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    [CrossRef] [PubMed]
  13. J. Ojeda-Castaneda, L. R. Berriel-Valdos, E. Montes, Opt. Lett. 8, 458 (1983).
    [CrossRef] [PubMed]
  14. C. Varamit, G. Indebetouw, J. Opt. Soc. Am. A 2, 799 (1985).
    [CrossRef]
  15. M. J. Yzuel, F. Calvo, Opt. Acta 26, 1397 (1979).
    [CrossRef]
  16. J. W. Cooley, J. W. Tukey, Math. Comp. 19, 297 (1965).
    [CrossRef]

1985 (3)

1984 (1)

1983 (1)

1979 (1)

M. J. Yzuel, F. Calvo, Opt. Acta 26, 1397 (1979).
[CrossRef]

1978 (1)

1971 (1)

1965 (2)

R. Barakat, A. Houston, J. Opt. Soc. Am. 55, 538 (1965).
[CrossRef]

J. W. Cooley, J. W. Tukey, Math. Comp. 19, 297 (1965).
[CrossRef]

1964 (1)

1960 (1)

1949 (1)

H. H. Hopkins, Proc. Phys. Soc. London Ser. B 62, 22 (1949).
[CrossRef]

Andrés, P.

J. Ojeda-Castan̂eda, P. Andrés, A. Díaz (submitted to J. Opt. Soc. Am. A).

Bai, H.

Barakat, R.

Berriel-Valdos, L. R.

Biswas, S. C.

J. E. Villeneuve, A. Boivin, S. C. Biswas, Can. J. Phys. 63, 287 (1985), and references cited therein.
[CrossRef]

Boivin, A.

J. E. Villeneuve, A. Boivin, S. C. Biswas, Can. J. Phys. 63, 287 (1985), and references cited therein.
[CrossRef]

Calvo, F.

M. J. Yzuel, F. Calvo, Opt. Acta 26, 1397 (1979).
[CrossRef]

Cooley, J. W.

J. W. Cooley, J. W. Tukey, Math. Comp. 19, 297 (1965).
[CrossRef]

Díaz, A.

J. Ojeda-Castan̂eda, P. Andrés, A. Díaz (submitted to J. Opt. Soc. Am. A).

Hopkins, H. H.

H. H. Hopkins, Proc. Phys. Soc. London Ser. B 62, 22 (1949).
[CrossRef]

Houston, A.

Indebetouw, G.

Jacquinot, P.

P. Jacquinot, B. Roizen-Dossier, in Progress in Optics III, E. Wolf, ed. (North-Holland, Amsterdam, 1964), p. 62.

McCutchen, C. W.

Mino, M.

Montes, E.

Ojeda-Castan^eda, J.

J. Ojeda-Castan̂eda, P. Andrés, A. Díaz (submitted to J. Opt. Soc. Am. A).

Ojeda-Castaneda, J.

Okano, Y.

Roizen-Dossier, B.

P. Jacquinot, B. Roizen-Dossier, in Progress in Optics III, E. Wolf, ed. (North-Holland, Amsterdam, 1964), p. 62.

Sheppard, C. J. R.

C. J. R. Sheppard, T. Wilson, Opt. Lett. 3, 115 (1978).
[CrossRef] [PubMed]

T. Wilson, C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, London, 1984).

Tukey, J. W.

J. W. Cooley, J. W. Tukey, Math. Comp. 19, 297 (1965).
[CrossRef]

Varamit, C.

Villeneuve, J. E.

J. E. Villeneuve, A. Boivin, S. C. Biswas, Can. J. Phys. 63, 287 (1985), and references cited therein.
[CrossRef]

Welford, W. T.

Wilson, T.

C. J. R. Sheppard, T. Wilson, Opt. Lett. 3, 115 (1978).
[CrossRef] [PubMed]

T. Wilson, C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, London, 1984).

Yzuel, M. J.

M. J. Yzuel, F. Calvo, Opt. Acta 26, 1397 (1979).
[CrossRef]

Appl. Opt. (2)

Can. J. Phys. (1)

J. E. Villeneuve, A. Boivin, S. C. Biswas, Can. J. Phys. 63, 287 (1985), and references cited therein.
[CrossRef]

J. Opt. Soc. Am. (3)

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

Math. Comp. (1)

J. W. Cooley, J. W. Tukey, Math. Comp. 19, 297 (1965).
[CrossRef]

Opt. Acta (1)

M. J. Yzuel, F. Calvo, Opt. Acta 26, 1397 (1979).
[CrossRef]

Opt. Lett. (3)

Proc. Phys. Soc. London Ser. B (1)

H. H. Hopkins, Proc. Phys. Soc. London Ser. B 62, 22 (1949).
[CrossRef]

Other (3)

J. Ojeda-Castan̂eda, P. Andrés, A. Díaz (submitted to J. Opt. Soc. Am. A).

P. Jacquinot, B. Roizen-Dossier, in Progress in Optics III, E. Wolf, ed. (North-Holland, Amsterdam, 1964), p. 62.

T. Wilson, C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, London, 1984).

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Figures (4)

Fig. 1
Fig. 1

Coordinate transformation for mapping a 1-D pupil into a 2-D pupil.

Fig. 2
Fig. 2

Amplitude transmittance of the annular apodizer in Eq. (12).

Fig. 3
Fig. 3

(a) In-focus images and (b) out-of-focus images (W20 = 2λ) obtained with (A) a clear aperture and (B) the annular apodizer in Fig. 2.

Fig. 4
Fig. 4

Strehl-ratio, or normalized, on-axis irradiance at the best focal plane for variable spherical aberration: (dotted curve) clear aperture, (dashed curve) the best apodizer in Ref. 15, (solid curve) the annular apodizer in Fig. 2.

Equations (12)

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I ( W 20 , W 40 ) = | 2 π 0 ρ 0 p ˜ ( ρ ) exp { i 2 π [ W 20 ( ρ / ρ 0 ) 2 + W 40 ( ρ / ρ 0 ) 4 ] } ρ d ρ | 2 ,
I ( W 20 , W 40 ) = | π ρ 0 2 - 1 / 2 1 / 2 q ˜ ( ξ ) exp { i 2 π [ W 40 ξ 2 + ( W 40 + W 20 ) ξ ] } d ξ | 2 ,
ξ = ( ρ / ρ 0 ) 2 - 0.5 ;             q ˜ ( ξ ) = p ˜ ( ρ ) .
q ( α ) = - 1 / 2 1 / 2 q ˜ ( ξ ) exp ( i 2 π α ξ ) d ξ ,
I ( W 20 , W 40 = 0 ) = | π ρ 0 2 - 1 / 2 + 1 / 2 q ( ξ ) exp ( i 2 π W 20 ξ ) d ξ | 2 = π ρ 0 2 q ( W 20 ) 2 .
I ( W 20 = - W 40 , W 40 ) = | π ρ 0 2 - 1 / 2 1 / 2 q ˜ ( ξ ) exp ( i 2 π W 40 ξ 2 ) d ξ | 2
p ( W 20 ) = - 1 1 p ˜ ( ν ) exp ( i 2 π W 20 ν 2 ) d ν
q ˜ ( ξ ) = ( 1 - 4 ξ 2 ) n - 1 rect ( ξ ) ,
p ˜ ( ρ ) = ( 2 ρ / ρ 0 ) 2 n - 1 [ 1 - ( ρ / ρ 0 ) 2 ] n - 1 / 2 if ρ ρ 0 = 0 elsewhere .
p ˜ ( ρ ) = 3 sinc ( ρ / ρ 0 ) - cos ( π ρ / ρ 0 ) ( π ρ / ρ 0 ) 2 circ ( ρ / ρ 0 ) ,
q ˜ ( ξ ) = 3 sinc ( 2 ξ ) - cos ( 2 π ξ ) ( 2 π ξ ) 2 rect ( ξ ) .
p ˜ ( ρ ) = 3 sinc [ 2 ( ρ / ρ 0 ) 2 - 0.5 ] - cos { 2 π [ ( ρ / ρ 0 ) 2 - 0.5 ] } { 2 π [ ( ρ / ρ 0 ) 2 - 0.5 ] } 2 × circ ( ρ / ρ 0 ) ,

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