Abstract

A quartic phase retardation function is described that reduces the variation of the intensity of the focal point of incoherent imaging systems suffering from primary third-order aberrations limited to coma and astigmatism. Corresponding modulation transfer functions are shown to remain practically invariant for moderate amounts of coma and astigmatism.

© 2006 Optical Society of America

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References

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  1. J. Ojeda-Castañeda, P. Andrés, and A. Díaz, "Annular apodizers for low sensitivity to defocus and spherical aberration," Opt. Lett. 11, 487-489 (1986).
    [Crossref] [PubMed]
  2. J. Ojeda-Castañeda, L. R. Berriel Valdós, and E. L. Montes, "Spatial filter for increasing depth of focus," Opt. Lett. 10, 520-522 (1985).
    [Crossref] [PubMed]
  3. E. R. Dowski and W. T. Cathey, "Extended depth of field through wave-front coding," Appl. Opt. 34, 1859-1866 (1995).
    [Crossref] [PubMed]
  4. S. Mezouari and A. R. Harvey, "Combined amplitude and phase filters for increased tolerance to spherical aberration," J. Mod. Opt. 50, 2213-2220 (2003).
  5. S. Mezouari and A. R. Harvey, "Phase functions for the reduction of defocus and spherical aberration," Opt. Lett. 28, 771-773 (2003).
    [Crossref] [PubMed]
  6. X. Liu, X. Cai, S. Chang, and C. P. Grover, "Optical system having a large focal depth for distant object tracking," Opt. Express 11, 3242-3247 (2003).
    [Crossref] [PubMed]
  7. G. W. Ritchey and H. Chrétien "Présentation du premier modèle de télescope aplanétique," Compt. Rend. 185, 266-268 (1927).
  8. V. A. Borovikov, Uniform Stationary Phase Method (Institution of Electrical Engineers, 1994).
  9. A. Papoulis, Signal Analysis (McGraw-Hill, 1977), p. 271.
  10. D. Zalvidea, C. Colatti, and E. E. Sicre, "Quality parameter analysis of optical imaging systems with enhanced focal depth using the Wigner distribution function," J. Opt. Soc. Am. A 17, 867-873 (2000).
    [Crossref]

2003 (3)

2000 (1)

1995 (1)

1994 (1)

V. A. Borovikov, Uniform Stationary Phase Method (Institution of Electrical Engineers, 1994).

1986 (1)

1985 (1)

1977 (1)

A. Papoulis, Signal Analysis (McGraw-Hill, 1977), p. 271.

1927 (1)

G. W. Ritchey and H. Chrétien "Présentation du premier modèle de télescope aplanétique," Compt. Rend. 185, 266-268 (1927).

Andrés, P.

Borovikov, V. A.

V. A. Borovikov, Uniform Stationary Phase Method (Institution of Electrical Engineers, 1994).

Cai, X.

Cathey, W. T.

Chang, S.

Chrétien, H.

G. W. Ritchey and H. Chrétien "Présentation du premier modèle de télescope aplanétique," Compt. Rend. 185, 266-268 (1927).

Colatti, C.

Díaz, A.

Dowski, E. R.

Grover, C. P.

Harvey, A. R.

S. Mezouari and A. R. Harvey, "Phase functions for the reduction of defocus and spherical aberration," Opt. Lett. 28, 771-773 (2003).
[Crossref] [PubMed]

S. Mezouari and A. R. Harvey, "Combined amplitude and phase filters for increased tolerance to spherical aberration," J. Mod. Opt. 50, 2213-2220 (2003).

Liu, X.

Mezouari, S.

S. Mezouari and A. R. Harvey, "Phase functions for the reduction of defocus and spherical aberration," Opt. Lett. 28, 771-773 (2003).
[Crossref] [PubMed]

S. Mezouari and A. R. Harvey, "Combined amplitude and phase filters for increased tolerance to spherical aberration," J. Mod. Opt. 50, 2213-2220 (2003).

Montes, E. L.

Ojeda-Castañeda, J.

Papoulis, A.

A. Papoulis, Signal Analysis (McGraw-Hill, 1977), p. 271.

Ritchey, G. W.

G. W. Ritchey and H. Chrétien "Présentation du premier modèle de télescope aplanétique," Compt. Rend. 185, 266-268 (1927).

Sicre, E. E.

Valdós, L. R.

Zalvidea, D.

Appl. Opt. (1)

Compt. Rend. (1)

G. W. Ritchey and H. Chrétien "Présentation du premier modèle de télescope aplanétique," Compt. Rend. 185, 266-268 (1927).

J. Mod. Opt. (1)

S. Mezouari and A. R. Harvey, "Combined amplitude and phase filters for increased tolerance to spherical aberration," J. Mod. Opt. 50, 2213-2220 (2003).

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

Opt. Express (1)

Opt. Lett. (3)

Other (2)

V. A. Borovikov, Uniform Stationary Phase Method (Institution of Electrical Engineers, 1994).

A. Papoulis, Signal Analysis (McGraw-Hill, 1977), p. 271.

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Figures (4)

Fig. 1
Fig. 1

Variation of the central intensity as a function of coma W 31 with a QF with α = 15 and α 0 = 9.75 (dashed curve) and without a QF (solid curve). The normalization coefficient for the QF intensity is 0.045.

Fig. 2
Fig. 2

Computed MTFs for imaging systems suffering from coma W 31 and astigmatism W 22 for clear aperture (left), reduced aperture (center), and a full aperture with a QF with α = 1 5 and α 0 = 9.75 (right). Aberration W is in units of wavelength.

Fig. 3
Fig. 3

Computed point-spread functions with and without QF ( α = 15 and α 0 = 9.75 ) in the presence of coma W 31 (right) and astigmatism W 22 (left). Aberration W is in units of wavelength.

Fig. 4
Fig. 4

Variation of the central intensity as a function of astigmatism W 22 with a QF with α = 15 and α 0 = 9.75 (dashed curve) and without a QF (solid curve). The normalization coefficient for the QF intensity is 0.046.

Equations (22)

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I = C 0 2 π 0 1 P ( r ) exp [ i k W ( r , θ ) ] r d r d θ 2 ,
I i = C 0 2 π 0 1 exp { i k [ W ( r , θ ) + Φ ( r ) ] } r d r d θ 2 .
W ( r , θ ) = W 31 r 3 cos θ + W 22 r 2 cos 2 θ .
r 2 = ξ + 1 2 .
W ̃ ( ξ , θ ) = W 31 ( ξ + 1 2 ) 3 2 cos θ + W 22 ( ξ + 1 2 ) cos 2 θ .
I = C 0 2 π 1 2 1 2 exp { i k [ W ̃ ( ξ , θ ) + Φ ̃ ( ξ , θ ) ] } d ξ d θ 2 ,
I ( W 31 ) = 0 2 π 1 2 1 2 exp { i k [ Φ ̃ ( ξ ) + W 31 ( ξ + 1 2 ) 3 2 cos θ ] } d ξ d θ 2 ,
I ( W 31 ) = 1 2 1 2 J 0 ( k W 31 ( ξ + 1 2 ) 3 2 ) exp [ i k Φ ̃ ( ξ ) ] d ξ 2 ,
I ( W 31 ) A [ J 0 ( k W 31 ( ξ s + 1 2 ) 3 2 ) ] 2 Φ ̃ ( ξ s ) ,
Φ ̃ ( ξ ) ξ s = 0 .
3 2 k W 31 ξ + 1 2 J 1 ( k W 31 ( ξ + 1 2 ) 3 2 ) J 0 ( k W 31 ( ξ + 1 2 ) 3 2 ) < k Φ ̃ ( ξ s ) 4 π for ξ s ± 1 2 > 4 π k Φ ̃ ( ξ s ) ,
I ( W 31 ) A [ J 0 ( k W 31 ( ξ s + 1 2 ) 3 2 ) ] 2 Φ ̃ ( ξ s ) A 1 Φ ̃ ( ξ s ) .
Φ ̃ ( ξ ) = α ξ 2 + α 0 ξ ,
Φ ( r ) = α ( r 2 1 2 ) 2 + α 0 ( r 2 1 2 ) .
ξ s = α 0 2 α ,
I ( W 31 ) A 2 α [ J 0 ( k W 31 ( α 0 2 α + 1 2 ) 3 2 ) ] 2 .
I ( W 22 ) = 0 2 π 1 2 1 2 exp { i k [ Φ ̃ ( ξ ) + W 22 ( ξ + 1 2 ) cos 2 θ ] } d ξ d θ 2 .
I ( W 22 ) = 1 2 1 2 J 0 ( k W 22 ( ξ + 1 2 ) 2 ) × exp { i k [ Φ ̃ ( ξ ) + ( W 22 2 ) ξ ] } d ξ 2 ,
I ( W 22 ) B [ J 0 ( 1 2 k W 22 ( ξ s + 1 2 ) ) ] 2 Φ ̃ ( ξ s ) ,
Φ ̃ ( ξ ) ξ s + W 22 2 = 0 .
1 2 k W 22 J 1 ( 1 2 k W 22 ( ξ + 1 2 ) ) J 0 ( 1 2 k W 22 ( ξ s + 1 2 ) ) < k Φ ̃ 4 π for ξ s ± 1 2 > 4 π k Φ ̃ .
I ( W 22 ) B 2 α [ J 0 ( 1 2 k W 22 ( W 22 4 α α 0 2 α + 1 2 ) ) ] 2 .

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