Abstract

The ambiguity function was employed as a merit function to design an optical system with a high depth of focus. The ambiguity function with the desired enlarged-depth-of-focus characteristics was obtained by using a properly designed joint filter to modify the ambiguity function of the original pupil in the phase-space domain. From the viewpoint of the filter theory, we roughly propose that the constraints of the spatial filters that are used to enlarge the focal depth must be satisfied. These constraints coincide with those that appeared in the previous literature on this topic. Following our design procedure, several sets of apodizers were synthesized, and their performances in the defocused imagery were compared with each other and with other previous designs.

© 2005 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. W. T. Welford, “Use of annular aperture to increase focal depth,” J. Opt. Soc. Am. 50, 749–753 (1960).
    [CrossRef]
  2. M. Mino, Y. Okano, “Improvement in the optical transfer function of a defocused optical system through the use of shaded aperture,” Appl. Opt. 10, 2219–2225 (1971).
    [CrossRef] [PubMed]
  3. J. Ojeda-Castañeda, L. R. Berriel-Valdos, E. L. Montes, “Spatial filter for increasing the depth of focus,” Opt. Lett. 10, 520–522 (1985).
    [CrossRef] [PubMed]
  4. 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]
  5. J. Ojeda-Castañeda, E. Tepichin, A. Diaz, “Arbitrary high focal depth with finite aperture,” Opt. Lett. 13, 183–185 (1988).
    [CrossRef]
  6. J. Ojeda-Castañeda, R. Ramos, A. Noyola-Isgleas, “High focal depth by apodization and digital restoration,” Appl. Opt. 27, 2583–2586 (1988).
    [CrossRef] [PubMed]
  7. J. Ojeda-Castañeda, L. R. Berriel-Valdos, “Zone plate for arbitrarily high focal depth,” Appl. Opt. 29, 994–997 (1990).
    [CrossRef] [PubMed]
  8. D. Zalvidea, G. Colautti, E. E. Sicre, “Quality parameters analysis of optical imaging systems with enhanced focal depth using the Wigner distribution function,” J. Opt. Soc. Am. A 17, 867–873 (2000).
    [CrossRef]
  9. H. H. Hopkins, “The aberration permissible in optical systems,” Proc. Phys. Soc. London Sect. B 70, 449–470 (1957).
    [CrossRef]
  10. H. H. Hopkins, “The use of diffraction-based criterion of image quality in automatic optical design,” Opt. Acta 13, 343–369 (1966).
    [CrossRef]
  11. H. Bartelt, J. Ojeda-Castañeda, E. E. Sicre, “Misfocus tolerance seen by simple inspection of the ambiguity function,” Appl. Opt. 23, 2693–2696 (1984).
    [CrossRef] [PubMed]
  12. J. Ojeda-Castañeda, P. Andrés, E. Montes, “Phase-space representation of the Strehl ratio: ambiguity function,” J. Opt. Soc. Am. A 4, 313–317 (1987).
    [CrossRef]
  13. D. Zalvidea, M. Lehman, S. Granieri, E. E. Sicre, “Analysis of the Strehl ratio using the Wigner distribution function,” Opt. Commun. 118, 207–214 (1995).
    [CrossRef]
  14. D. Zalvidea, E. E. Sicre, “Phase pupil function for focal-depth enhancement derived from a Wigner distribution function,” Appl. Opt. 37, 3623–3627 (1998).
    [CrossRef]
  15. W. D. Furlan, G. Saavedra, E. Silvestre, P. André, M. J. Yzuel, “Polychromatic axial behavior of aberrated optical systems: Wigner distribution function approach,” Appl. Opt. 36, 9146–9151 (1997).
    [CrossRef]
  16. D. Zalvidea, S. Granieri, E. E. Sicre, “Space and spectral behaviour of the optical systems under broadband illumination by using a Wigner distribution function approach,” Opt. Commun. 204, 99–106 (2002).
    [CrossRef]
  17. Q. Yang, L. Liu, H. Lang, “Computation of the ambiguity function for circularly symmetric pupils,” J. Opt. A 7, 431–437 (2005).
    [CrossRef]
  18. Q. Yang, L. Liu, H. Lang, “Defocus transfer function for circularly symmetric pupils under polychromatic illumination,” in Proc. SPIE 5896, 166–173 (2005).
  19. K. Brenner, A. Lohmann, J. Ojeda-Castañeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323–326 (1983).
    [CrossRef]
  20. A. R. FitzGerrell, E. R. Dowski, W. T. Cathey, “Defocus transfer function for circularly symmetric pupils,” Appl. Opt. 36, 5796–5804 (1997).
    [CrossRef] [PubMed]
  21. J. Ojeda-Castañeda, L. R. Berriel-Valdos, E. Montes, “Ambiguity function as a design tool for high focal depth,” Appl. Opt. 27, 790–795 (1988).
    [CrossRef] [PubMed]
  22. E. R. Dowski, W. T. Cathey, “Extended depth of field through wavefront coding,” Appl. Opt. 34, 1859–1866 (1995).
    [CrossRef] [PubMed]
  23. S. S. Sherif, W. T. Cathey, E. R. Dowski, “Phase plate to extend the depth of field of incoherent hybrid imaging systems,” Appl. Opt. 43, 2709–2721 (2004).
    [CrossRef] [PubMed]
  24. A. Castro, J. Ojeda-Castañeda, “Asymmetric phase masks for extended depth of field,” Appl. Opt. 43, 3474–3479 (2004).
    [CrossRef] [PubMed]

2005 (2)

Q. Yang, L. Liu, H. Lang, “Computation of the ambiguity function for circularly symmetric pupils,” J. Opt. A 7, 431–437 (2005).
[CrossRef]

Q. Yang, L. Liu, H. Lang, “Defocus transfer function for circularly symmetric pupils under polychromatic illumination,” in Proc. SPIE 5896, 166–173 (2005).

2004 (2)

2002 (1)

D. Zalvidea, S. Granieri, E. E. Sicre, “Space and spectral behaviour of the optical systems under broadband illumination by using a Wigner distribution function approach,” Opt. Commun. 204, 99–106 (2002).
[CrossRef]

2000 (1)

1998 (1)

1997 (2)

1995 (2)

E. R. Dowski, W. T. Cathey, “Extended depth of field through wavefront coding,” Appl. Opt. 34, 1859–1866 (1995).
[CrossRef] [PubMed]

D. Zalvidea, M. Lehman, S. Granieri, E. E. Sicre, “Analysis of the Strehl ratio using the Wigner distribution function,” Opt. Commun. 118, 207–214 (1995).
[CrossRef]

1990 (1)

1988 (3)

1987 (1)

1986 (1)

1985 (1)

1984 (1)

1983 (1)

K. Brenner, A. Lohmann, J. Ojeda-Castañeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323–326 (1983).
[CrossRef]

1971 (1)

1966 (1)

H. H. Hopkins, “The use of diffraction-based criterion of image quality in automatic optical design,” Opt. Acta 13, 343–369 (1966).
[CrossRef]

1960 (1)

1957 (1)

H. H. Hopkins, “The aberration permissible in optical systems,” Proc. Phys. Soc. London Sect. B 70, 449–470 (1957).
[CrossRef]

André, P.

Andres, P.

Andrés, P.

Bartelt, H.

Berriel-Valdos, L. R.

Brenner, K.

K. Brenner, A. Lohmann, J. Ojeda-Castañeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323–326 (1983).
[CrossRef]

Castro, A.

Cathey, W. T.

Colautti, G.

Diaz, A.

Dowski, E. R.

FitzGerrell, A. R.

Furlan, W. D.

Granieri, S.

D. Zalvidea, S. Granieri, E. E. Sicre, “Space and spectral behaviour of the optical systems under broadband illumination by using a Wigner distribution function approach,” Opt. Commun. 204, 99–106 (2002).
[CrossRef]

D. Zalvidea, M. Lehman, S. Granieri, E. E. Sicre, “Analysis of the Strehl ratio using the Wigner distribution function,” Opt. Commun. 118, 207–214 (1995).
[CrossRef]

Hopkins, H. H.

H. H. Hopkins, “The use of diffraction-based criterion of image quality in automatic optical design,” Opt. Acta 13, 343–369 (1966).
[CrossRef]

H. H. Hopkins, “The aberration permissible in optical systems,” Proc. Phys. Soc. London Sect. B 70, 449–470 (1957).
[CrossRef]

Lang, H.

Q. Yang, L. Liu, H. Lang, “Computation of the ambiguity function for circularly symmetric pupils,” J. Opt. A 7, 431–437 (2005).
[CrossRef]

Q. Yang, L. Liu, H. Lang, “Defocus transfer function for circularly symmetric pupils under polychromatic illumination,” in Proc. SPIE 5896, 166–173 (2005).

Lehman, M.

D. Zalvidea, M. Lehman, S. Granieri, E. E. Sicre, “Analysis of the Strehl ratio using the Wigner distribution function,” Opt. Commun. 118, 207–214 (1995).
[CrossRef]

Liu, L.

Q. Yang, L. Liu, H. Lang, “Defocus transfer function for circularly symmetric pupils under polychromatic illumination,” in Proc. SPIE 5896, 166–173 (2005).

Q. Yang, L. Liu, H. Lang, “Computation of the ambiguity function for circularly symmetric pupils,” J. Opt. A 7, 431–437 (2005).
[CrossRef]

Lohmann, A.

K. Brenner, A. Lohmann, J. Ojeda-Castañeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323–326 (1983).
[CrossRef]

Mino, M.

Montes, E.

Montes, E. L.

Noyola-Isgleas, A.

Ojeda-Castañeda, J.

A. Castro, J. Ojeda-Castañeda, “Asymmetric phase masks for extended depth of field,” Appl. Opt. 43, 3474–3479 (2004).
[CrossRef] [PubMed]

J. Ojeda-Castañeda, L. R. Berriel-Valdos, “Zone plate for arbitrarily high focal depth,” Appl. Opt. 29, 994–997 (1990).
[CrossRef] [PubMed]

J. Ojeda-Castañeda, L. R. Berriel-Valdos, E. Montes, “Ambiguity function as a design tool for high focal depth,” Appl. Opt. 27, 790–795 (1988).
[CrossRef] [PubMed]

J. Ojeda-Castañeda, R. Ramos, A. Noyola-Isgleas, “High focal depth by apodization and digital restoration,” Appl. Opt. 27, 2583–2586 (1988).
[CrossRef] [PubMed]

J. Ojeda-Castañeda, E. Tepichin, A. Diaz, “Arbitrary high focal depth with finite aperture,” Opt. Lett. 13, 183–185 (1988).
[CrossRef]

J. Ojeda-Castañeda, P. Andrés, E. Montes, “Phase-space representation of the Strehl ratio: ambiguity function,” J. Opt. Soc. Am. A 4, 313–317 (1987).
[CrossRef]

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]

J. Ojeda-Castañeda, L. R. Berriel-Valdos, E. L. Montes, “Spatial filter for increasing the depth of focus,” Opt. Lett. 10, 520–522 (1985).
[CrossRef] [PubMed]

H. Bartelt, J. Ojeda-Castañeda, E. E. Sicre, “Misfocus tolerance seen by simple inspection of the ambiguity function,” Appl. Opt. 23, 2693–2696 (1984).
[CrossRef] [PubMed]

K. Brenner, A. Lohmann, J. Ojeda-Castañeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323–326 (1983).
[CrossRef]

Okano, Y.

Ramos, R.

Saavedra, G.

Sherif, S. S.

Sicre, E. E.

Silvestre, E.

Tepichin, E.

Welford, W. T.

Yang, Q.

Q. Yang, L. Liu, H. Lang, “Computation of the ambiguity function for circularly symmetric pupils,” J. Opt. A 7, 431–437 (2005).
[CrossRef]

Q. Yang, L. Liu, H. Lang, “Defocus transfer function for circularly symmetric pupils under polychromatic illumination,” in Proc. SPIE 5896, 166–173 (2005).

Yzuel, M. J.

Zalvidea, D.

D. Zalvidea, S. Granieri, E. E. Sicre, “Space and spectral behaviour of the optical systems under broadband illumination by using a Wigner distribution function approach,” Opt. Commun. 204, 99–106 (2002).
[CrossRef]

D. Zalvidea, G. Colautti, E. E. Sicre, “Quality parameters analysis of optical imaging systems with enhanced focal depth using the Wigner distribution function,” J. Opt. Soc. Am. A 17, 867–873 (2000).
[CrossRef]

D. Zalvidea, E. E. Sicre, “Phase pupil function for focal-depth enhancement derived from a Wigner distribution function,” Appl. Opt. 37, 3623–3627 (1998).
[CrossRef]

D. Zalvidea, M. Lehman, S. Granieri, E. E. Sicre, “Analysis of the Strehl ratio using the Wigner distribution function,” Opt. Commun. 118, 207–214 (1995).
[CrossRef]

Appl. Opt. (11)

J. Ojeda-Castañeda, R. Ramos, A. Noyola-Isgleas, “High focal depth by apodization and digital restoration,” Appl. Opt. 27, 2583–2586 (1988).
[CrossRef] [PubMed]

A. R. FitzGerrell, E. R. Dowski, W. T. Cathey, “Defocus transfer function for circularly symmetric pupils,” Appl. Opt. 36, 5796–5804 (1997).
[CrossRef] [PubMed]

W. D. Furlan, G. Saavedra, E. Silvestre, P. André, M. J. Yzuel, “Polychromatic axial behavior of aberrated optical systems: Wigner distribution function approach,” Appl. Opt. 36, 9146–9151 (1997).
[CrossRef]

D. Zalvidea, E. E. Sicre, “Phase pupil function for focal-depth enhancement derived from a Wigner distribution function,” Appl. Opt. 37, 3623–3627 (1998).
[CrossRef]

E. R. Dowski, W. T. Cathey, “Extended depth of field through wavefront coding,” Appl. Opt. 34, 1859–1866 (1995).
[CrossRef] [PubMed]

J. Ojeda-Castañeda, L. R. Berriel-Valdos, “Zone plate for arbitrarily high focal depth,” Appl. Opt. 29, 994–997 (1990).
[CrossRef] [PubMed]

H. Bartelt, J. Ojeda-Castañeda, E. E. Sicre, “Misfocus tolerance seen by simple inspection of the ambiguity function,” Appl. Opt. 23, 2693–2696 (1984).
[CrossRef] [PubMed]

J. Ojeda-Castañeda, L. R. Berriel-Valdos, E. Montes, “Ambiguity function as a design tool for high focal depth,” Appl. Opt. 27, 790–795 (1988).
[CrossRef] [PubMed]

M. Mino, Y. Okano, “Improvement in the optical transfer function of a defocused optical system through the use of shaded aperture,” Appl. Opt. 10, 2219–2225 (1971).
[CrossRef] [PubMed]

S. S. Sherif, W. T. Cathey, E. R. Dowski, “Phase plate to extend the depth of field of incoherent hybrid imaging systems,” Appl. Opt. 43, 2709–2721 (2004).
[CrossRef] [PubMed]

A. Castro, J. Ojeda-Castañeda, “Asymmetric phase masks for extended depth of field,” Appl. Opt. 43, 3474–3479 (2004).
[CrossRef] [PubMed]

J. Opt. A (1)

Q. Yang, L. Liu, H. Lang, “Computation of the ambiguity function for circularly symmetric pupils,” J. Opt. A 7, 431–437 (2005).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Opt. Acta (1)

H. H. Hopkins, “The use of diffraction-based criterion of image quality in automatic optical design,” Opt. Acta 13, 343–369 (1966).
[CrossRef]

Opt. Commun. (3)

D. Zalvidea, M. Lehman, S. Granieri, E. E. Sicre, “Analysis of the Strehl ratio using the Wigner distribution function,” Opt. Commun. 118, 207–214 (1995).
[CrossRef]

D. Zalvidea, S. Granieri, E. E. Sicre, “Space and spectral behaviour of the optical systems under broadband illumination by using a Wigner distribution function approach,” Opt. Commun. 204, 99–106 (2002).
[CrossRef]

K. Brenner, A. Lohmann, J. Ojeda-Castañeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323–326 (1983).
[CrossRef]

Opt. Lett. (3)

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

H. H. Hopkins, “The aberration permissible in optical systems,” Proc. Phys. Soc. London Sect. B 70, 449–470 (1957).
[CrossRef]

Proc. SPIE (1)

Q. Yang, L. Liu, H. Lang, “Defocus transfer function for circularly symmetric pupils under polychromatic illumination,” in Proc. SPIE 5896, 166–173 (2005).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

(A) Plots of the magnitude of the AF for a clear rectangular aperture; (B) gray-scale picture of (A) (for the enhancement of visibility, the image’s gray scale has been adjusted).

Fig. 2
Fig. 2

Plots of transmittance profiles for pupil functions a–g.

Fig. 3
Fig. 3

Gray-scale picture of the ambiguity function for pupil functions b–g.

Fig. 4
Fig. 4

Plots of the OTFs for a defocused optical system with apodizers a–e. The defocus coefficients are (A) W20 = 0, (B) W20 = 1, (C) W20 = 2, and (D) W20 = 5 in units of wavelength.

Fig. 5
Fig. 5

Simulated photographs of a spoke target from an optical system with apodizers a, b, c, d, and e in the presence of defocus. From the left to the right column, the defocus coefficients are (A) W20 = 0, (B) W20 = 1, (C) W20 = 2, and (D) W20 = 3 in units of wavelength.

Fig. 6
Fig. 6

Plots of transmittance profiles for pupil functions h–k.

Fig. 7
Fig. 7

Gray-scale picture of the ambiguity function for pupil functions h–k.

Equations (22)

Equations on this page are rendered with MathJax. Learn more.

A P ( ρ , x ) = P ( u + ρ / 2 ) P * ( u ρ / 2 ) exp ( j 2 π xu ) d u ,
P ( ρ ) = 1 P * ( 0 ) A P ( ρ , x ) exp ( j π x ρ ) d x .
H ( ρ ) = A P ( ρ , 2 W 20 ρ ) ,
A Q ( x , ρ ) = A P ( x , ρ ) * ( x ) h ( x , ρ ) ,
h ( x , ρ ) = f ( x ) G ( ρ ) .
Q ( ρ ) = P * ( 0 ) Q * ( 0 ) P ( ρ ) G ( ρ ) F ( ρ 2 ) ,
( a ) P ( ρ ) = rect ( ρ ) ,
( b ) Q ( ρ ) = sinc ( 2 ρ ) rect ( ρ ) ,
( c ) Q ( ρ ) = sinc ( 2 ρ ) ( 1 4 ρ 2 ) rect ( ρ ) ,
( d ) Q ( ρ ) = sinc ( 2 ρ ) { 1 exp [ 2 π ( 0 0.1 ) 2 ] + exp [ 2 π ( 4 ρ 2 0.1 ) 2 ] } rect ( ρ ) ,
( e ) Q ( ρ ) = sinc ( 2 ρ ) { 1 + exp [ 2 π ( 0 0.1 ) 2 ] exp [ 2 π ( 4 ρ 2 0.1 ) 2 ] } rect ( ρ ) ,
( f ) Q ( ρ ) = sinc 2 ( 4 ρ ) rect ( ρ ) ,
( g ) Q ( ρ ) = sinc 2 ( 4 ρ ) ( 1 + 4 ρ 2 ) rect ( ρ ) ,
( h ) Q ( ρ ) = 1 1 + ( 4 π ρ ) 2 rect ( ρ ) ,
( i ) Q ( ρ ) = 1 1 + ( 4 π ρ ) 2 ( 1 4 ρ 2 ) rect ( ρ ) ,
( j ) Q ( ρ ) = sech ( 4 π ρ ) rect ( ρ ) ,
( k ) Q ( ρ ) = sech ( 4 π ρ ) ( 1 4 ρ 2 ) rect ( ρ ) ,
A Q ( x , ρ ) = A P ( α , ρ ) h ( x α , ρ ) d α .
A Q ( x , ρ ) = G ( ρ ) A P ( α , ρ ) f ( x α ) d α .
Q ( ρ ) = G ( ρ ) Q * ( 0 ) A P ( α , ρ ) f ( x α ) × exp ( j π x ρ ) d α d x .
Q ( ρ ) = G ( ρ ) Q * ( 0 ) A P ( α , ρ ) exp ( j π α ρ ) d α × f ( β ) exp ( j π β ρ ) d β = P * ( 0 ) Q * ( 0 ) P ( ρ ) G ( ρ ) F ( ρ 2 ) .
F ( ρ ) = f ( x ) exp ( j 2 π ρ x ) d x .

Metrics