Abstract

The influence that Seidel spherical aberration has on the irradiance distribution along the optical axis, or the Strehl ratio versus defocus, is expressed in terms of a differential operator. This permits us to describe separately the following two effects: a focal shift and a shape variation. We analyze two annular apodizers that reduce the shape variation of the Strehl ratio versus defocus.

© 1988 Optical Society of America

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References

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  1. M. Mino, Y. Okano, “Improvement in the OTF of a defocused optical system through the use of shade apertures,” Appl. Opt. 10, 2219–2225 (1971).
    [CrossRef] [PubMed]
  2. J. Ojeda-Castañeda, L. R. Berriel-Valdos, E. Montes, “Line-spread function relatively insensitive to defocus,” Opt. Lett. 8, 458–460 (1983).
    [CrossRef] [PubMed]
  3. G. Indebetouw, H. Bai, “Imaging with Fresnel-zone pupil: extended depth of field,” Appl. Opt. 23, 4299–4302 (1984).
    [CrossRef] [PubMed]
  4. C. Varamit, G. Indebetouw, “Imaging properties of defocused partitioned pupils,” J. Opt. Soc. Am. A 2, 799–802 (1985).
    [CrossRef]
  5. J. Ojeda-Castañeda, L. R. Berriel-Valdos, E. Montes, “Spatial filter for increasing the depth of focus,” Opt. Lett. 10, 520–522 (1985).
    [CrossRef] [PubMed]
  6. J. N. Maggo, K. N. Chopra, G. S. Bhatnagar, “A method for improving the optical transfer function of an astigmatic aperture,” Opt. Acta 21, 801–808 (1974).
    [CrossRef]
  7. 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, 1397–1406 (1979).
    [CrossRef]
  8. J. Ojeda-Castañeda, P. Andres, A. Diaz, “Annular apodizers for low sensitivity to defocus and to spherical aberration,” Opt. Lett. 11, 487–489 (1986).
    [CrossRef] [PubMed]
  9. H. H. Hopkins, Wave Theory of Aberrations (Oxford U. Press, Oxford, 1950), p. 51.
  10. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), p. 462.
  11. J. Ojeda-Castañeda, “Focus-error operator and related special functions,”J. Opt. Soc. Am. 73, 1042–1047 (1983).
    [CrossRef]
  12. J. Ojeda-Castañeda, A. Boivin, “The influence of wave aberrations: an operator approach,” Can. J. Phys. 63, 250–253 (1985).
    [CrossRef]
  13. See Ref. 10, p. 472.
  14. H. H. Hopkins, M. J. Yzuel, “The computation of diffraction patterns in the presence of aberrations,” Opt. Acta 47, 157–182 (1970); in particular, p. 174.
    [CrossRef]
  15. This formula predicts a maximum at W20= −W40only if q˜(ζ)is an even function with a maximum value at ζ= 0.

1986 (1)

1985 (3)

1984 (1)

1983 (2)

1979 (1)

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, 1397–1406 (1979).
[CrossRef]

1974 (1)

J. N. Maggo, K. N. Chopra, G. S. Bhatnagar, “A method for improving the optical transfer function of an astigmatic aperture,” Opt. Acta 21, 801–808 (1974).
[CrossRef]

1971 (1)

1970 (1)

H. H. Hopkins, M. J. Yzuel, “The computation of diffraction patterns in the presence of aberrations,” Opt. Acta 47, 157–182 (1970); in particular, p. 174.
[CrossRef]

Andres, P.

Bai, H.

Berriel-Valdos, L. R.

Bhatnagar, G. S.

J. N. Maggo, K. N. Chopra, G. S. Bhatnagar, “A method for improving the optical transfer function of an astigmatic aperture,” Opt. Acta 21, 801–808 (1974).
[CrossRef]

Boivin, A.

J. Ojeda-Castañeda, A. Boivin, “The influence of wave aberrations: an operator approach,” Can. J. Phys. 63, 250–253 (1985).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), p. 462.

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, 1397–1406 (1979).
[CrossRef]

Chopra, K. N.

J. N. Maggo, K. N. Chopra, G. S. Bhatnagar, “A method for improving the optical transfer function of an astigmatic aperture,” Opt. Acta 21, 801–808 (1974).
[CrossRef]

Diaz, A.

Hopkins, H. H.

H. H. Hopkins, M. J. Yzuel, “The computation of diffraction patterns in the presence of aberrations,” Opt. Acta 47, 157–182 (1970); in particular, p. 174.
[CrossRef]

H. H. Hopkins, Wave Theory of Aberrations (Oxford U. Press, Oxford, 1950), p. 51.

Indebetouw, G.

Maggo, J. N.

J. N. Maggo, K. N. Chopra, G. S. Bhatnagar, “A method for improving the optical transfer function of an astigmatic aperture,” Opt. Acta 21, 801–808 (1974).
[CrossRef]

Mino, M.

Montes, E.

Ojeda-Castañeda, J.

Okano, Y.

Varamit, C.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), p. 462.

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, 1397–1406 (1979).
[CrossRef]

H. H. Hopkins, M. J. Yzuel, “The computation of diffraction patterns in the presence of aberrations,” Opt. Acta 47, 157–182 (1970); in particular, p. 174.
[CrossRef]

Appl. Opt. (2)

Can. J. Phys. (1)

J. Ojeda-Castañeda, A. Boivin, “The influence of wave aberrations: an operator approach,” Can. J. Phys. 63, 250–253 (1985).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Acta (3)

H. H. Hopkins, M. J. Yzuel, “The computation of diffraction patterns in the presence of aberrations,” Opt. Acta 47, 157–182 (1970); in particular, p. 174.
[CrossRef]

J. N. Maggo, K. N. Chopra, G. S. Bhatnagar, “A method for improving the optical transfer function of an astigmatic aperture,” Opt. Acta 21, 801–808 (1974).
[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, 1397–1406 (1979).
[CrossRef]

Opt. Lett. (3)

Other (4)

H. H. Hopkins, Wave Theory of Aberrations (Oxford U. Press, Oxford, 1950), p. 51.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), p. 462.

This formula predicts a maximum at W20= −W40only if q˜(ζ)is an even function with a maximum value at ζ= 0.

See Ref. 10, p. 472.

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Figures (9)

Fig. 1
Fig. 1

Irradiance along the optical axis, or Strehl ratio versus defocus, for a clear aperture free from aberrations (dashed line) and for a clear aperture with a spherical aberration W40 = −2 (solid line).

Fig. 2
Fig. 2

Amplitude transmittances of the annular masks in (a) Eq. (14) and (b)Eq. (13), which are proposed to reduce the influence of spherical aberration.

Fig. 3
Fig. 3

Strehl ratio versus focus error for the apodizer in Eq. (13): dashed line, W40 = 0; solid line, W40 = −2.

Fig. 4
Fig. 4

Same as in Fig. 3 but for the apodizer in Eq. (14).

Fig. 5
Fig. 5

Strehl ratio at the best focus, W20 = −W40, for variable W40: dashed line, for a clear aperture; curve a, for the apodizer in Eq. (13); curve b, for the apodizer in Eq. (14).

Fig. 6
Fig. 6

Nonnormalized on-axis irradiance, |p(W40)|2, at the best focus for the same apodizers as in Fig. 5.

Fig. 7
Fig. 7

OTF’s for a clear aperture and for several values of defocus (W20) and primary spherical aberration (W40).

Fig. 8
Fig. 8

Same as in Fig. 7 but for the apodizer in Eq. (13).

Fig. 9
Fig. 9

Same as in Fig. 7 but for the apodizer in Eq. (14).

Equations (15)

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p ˜ ( ρ , W 20 , W 40 ) = p ˜ 0 ( ρ ) exp { i 2 π [ W 20 ( ρ / ρ 0 ) 2 + W 40 ( ρ / ρ 0 ) 4 ] } .
p ( r , W 20 , W 40 ) = 2 π 0 p ˜ ( ρ , W 20 , W 40 ) J 0 ( 2 π ρ r ) ρ d ρ ,
ζ = ( ρ / ρ 0 ) 2 0.5 , with q ˜ ( ζ ) = p ˜ 0 ( ρ ) .
p ( W 20 , W 40 ) = exp ( i ϕ ) q ˜ ( ζ ) exp ( i 2 π W 40 ζ 2 ) × exp [ i 2 π ( W 20 + W 40 ) ζ ] d ζ ,
ϕ = π ( W 20 + W 40 / 2 ) .
q ( W 20 ) = q ˜ ( ζ ) exp ( i 2 π W 20 ζ ) d ζ ,
q ( W 20 + W 40 ) = q ˜ ( ζ ) exp [ i 2 π ( W 20 + W 40 ) ζ ] d ζ .
D 2 W 20 q ( W 20 + W 40 ) = q ˜ ( ζ ) ( i 2 π ζ ) 2 × exp [ i 2 π ( W 20 + W 40 ) ζ ] d ζ .
exp ( D 2 W 20 ) q ( W 20 + W 40 ) = q ˜ ( ζ ) exp [ ( i 2 π ζ ) 2 ] × exp [ i 2 π ( W 20 + W 40 ) ζ ] d ζ .
exp [ i ( W 40 / 2 π ) D 2 ] q ( W 20 + W 40 ) = q ˜ ( ζ ) exp ( i 2 π W 40 ζ 2 ) exp [ i 2 π ( W 20 + W 40 ) ζ ] d ζ ;
p ( W 20 , W 40 ) = 2 π exp ( i ϕ ) × exp [ i ( W 20 / 2 π ) D 2 W 20 ] q ( W 20 + W 40 ) .
S ( W 20 , W 40 ) = | p ( W 20 0 , W 40 0 ) | 2 / | p ( W 20 = 0 , W 40 = 0 ) | 2 = | exp [ i ( W 40 / 2 π ] D 2 W 20 ) × q ( W 20 + W 40 ) | 2 / | q ( 0 ) | 2 .
S ( W 20 , W 40 = 0 ) = | q ( W 20 ) | 2 / | q ( 0 ) | 2 .
q ( ζ ) = ( sinc 4 ζ 2 ) rect ( ζ ) ,
q ( ζ ) = 3 ( cos 2 π ζ sinc 2 ζ ) ( 2 π ζ ) 2 rect ( ζ ) .

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