Abstract

A method for obtaining phase-retardation functions, which give rise to an increase of the image focal depth, is proposed. To this end, the Wigner distribution function corresponding to a specific aperture that has an associated small depth of focus in image space is conveniently sheared in the phase-space domain to generate a new Wigner distribution function. From this new function a more uniform on-axis image irradiance can be accomplished. This approach is illustrated by comparison of the imaging performance of both the derived phase function and a previously reported logarithmic phase distribution.

© 1998 Optical Society of America

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References

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    [Crossref]
  13. W. D. Furlan, G. Saavedra, and J. Lancis, “Phase-space representations as a tool for the evaluation of the polychromatic OTF,” Opt. Commun. 96, 208–213 (1993).
    [Crossref]
  14. E. R. Dowski, Jr., and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. 34, 1859–1866 (1995).
    [Crossref] [PubMed]
  15. D. Zalvidea, M. Lehman, S. Granieri, and E. E. Sicre, “Analysis of the Strehl ratio using the Wigner distribution function,” Opt. Commun. 118, 207–214 (1995).
    [Crossref]
  16. G. Saavedra, W. D. Furlan, E. Silvestre, and E. E. Sicre, “Analysis of the irradiance along different paths in the image space using the Wigner distribution function,” Opt. Commun. 139, 11–16 (1997).
    [Crossref]
  17. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), Chap. 9.
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    [Crossref]
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1997 (1)

G. Saavedra, W. D. Furlan, E. Silvestre, and E. E. Sicre, “Analysis of the irradiance along different paths in the image space using the Wigner distribution function,” Opt. Commun. 139, 11–16 (1997).
[Crossref]

1995 (3)

1994 (1)

Q. Gong and S. S. Hsu, “Aberration measurement using axial intensity,” Opt. Eng. 33, 1176–1186 (1994).
[Crossref]

1993 (1)

W. D. Furlan, G. Saavedra, and J. Lancis, “Phase-space representations as a tool for the evaluation of the polychromatic OTF,” Opt. Commun. 96, 208–213 (1993).
[Crossref]

1992 (1)

1991 (2)

L. V. Bourimborde, W. D. Furlan, and E. E. Sicre, “Off-axis analysis of the Strehl ratio using the Wigner distribution function,” J. Mod. Opt. 38, 1685–1689 (1991).
[Crossref]

N. Davidson, A. A. Friesem, and E. Hasman, “Holographic axilens: high resolution and long focal depth,” Opt. Lett. 16, 523–525 (1991).
[Crossref] [PubMed]

1988 (1)

1987 (1)

1985 (1)

1984 (1)

1979 (1)

M. J. Yzuel and 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]

1971 (2)

1960 (1)

Andrés, P.

Bará, S.

Bartelt, H.

Berriel-Valdos, L. R.

Born, M.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), Chap. 9.

Bourimborde, L. V.

L. V. Bourimborde, W. D. Furlan, and E. E. Sicre, “Off-axis analysis of the Strehl ratio using the Wigner distribution function,” J. Mod. Opt. 38, 1685–1689 (1991).
[Crossref]

Calvo, F.

M. J. Yzuel and 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]

Cathey, W. T.

Davidson, N.

Dowski, Jr., E. R.

Friesem, A. A.

Furlan, W. D.

G. Saavedra, W. D. Furlan, E. Silvestre, and E. E. Sicre, “Analysis of the irradiance along different paths in the image space using the Wigner distribution function,” Opt. Commun. 139, 11–16 (1997).
[Crossref]

W. D. Furlan, G. Saavedra, and J. Lancis, “Phase-space representations as a tool for the evaluation of the polychromatic OTF,” Opt. Commun. 96, 208–213 (1993).
[Crossref]

L. V. Bourimborde, W. D. Furlan, and E. E. Sicre, “Off-axis analysis of the Strehl ratio using the Wigner distribution function,” J. Mod. Opt. 38, 1685–1689 (1991).
[Crossref]

Gong, Q.

Q. Gong and S. S. Hsu, “Aberration measurement using axial intensity,” Opt. Eng. 33, 1176–1186 (1994).
[Crossref]

Granieri, S.

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

Hasman, E.

Hsu, S. S.

Q. Gong and S. S. Hsu, “Aberration measurement using axial intensity,” Opt. Eng. 33, 1176–1186 (1994).
[Crossref]

Jaroszewicz, Z.

Kolodziejczyk, A.

Lancis, J.

W. D. Furlan, G. Saavedra, and J. Lancis, “Phase-space representations as a tool for the evaluation of the polychromatic OTF,” Opt. Commun. 96, 208–213 (1993).
[Crossref]

Lehman, M.

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

McCrickerd, J. T.

Mino, M.

Montes, E.

Montes, E. L.

Ojeda-Castañeda, J.

Okano, Y.

Rosen, J.

Saavedra, G.

G. Saavedra, W. D. Furlan, E. Silvestre, and E. E. Sicre, “Analysis of the irradiance along different paths in the image space using the Wigner distribution function,” Opt. Commun. 139, 11–16 (1997).
[Crossref]

W. D. Furlan, G. Saavedra, and J. Lancis, “Phase-space representations as a tool for the evaluation of the polychromatic OTF,” Opt. Commun. 96, 208–213 (1993).
[Crossref]

Salik, B.

Sicre, E. E.

G. Saavedra, W. D. Furlan, E. Silvestre, and E. E. Sicre, “Analysis of the irradiance along different paths in the image space using the Wigner distribution function,” Opt. Commun. 139, 11–16 (1997).
[Crossref]

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

L. V. Bourimborde, W. D. Furlan, and E. E. Sicre, “Off-axis analysis of the Strehl ratio using the Wigner distribution function,” J. Mod. Opt. 38, 1685–1689 (1991).
[Crossref]

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

Silvestre, E.

G. Saavedra, W. D. Furlan, E. Silvestre, and E. E. Sicre, “Analysis of the irradiance along different paths in the image space using the Wigner distribution function,” Opt. Commun. 139, 11–16 (1997).
[Crossref]

Sochacki, J.

Soroko, L. M.

L. M. Soroko, “Axicons and meso-optical imaging devices,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1989), Vol. 27, pp. 109–160.
[Crossref]

Welford, W. T.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), Chap. 9.

Yariv, A.

Yzuel, M. J.

M. J. Yzuel and 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]

Zalvidea, D.

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

Appl. Opt. (4)

J. Mod. Opt. (1)

L. V. Bourimborde, W. D. Furlan, and E. E. Sicre, “Off-axis analysis of the Strehl ratio using the Wigner distribution function,” J. Mod. Opt. 38, 1685–1689 (1991).
[Crossref]

J. Opt. Soc. Am. (1)

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

Opt. Acta (1)

M. J. Yzuel and 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. Commun. (3)

W. D. Furlan, G. Saavedra, and J. Lancis, “Phase-space representations as a tool for the evaluation of the polychromatic OTF,” Opt. Commun. 96, 208–213 (1993).
[Crossref]

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

G. Saavedra, W. D. Furlan, E. Silvestre, and E. E. Sicre, “Analysis of the irradiance along different paths in the image space using the Wigner distribution function,” Opt. Commun. 139, 11–16 (1997).
[Crossref]

Opt. Eng. (1)

Q. Gong and S. S. Hsu, “Aberration measurement using axial intensity,” Opt. Eng. 33, 1176–1186 (1994).
[Crossref]

Opt. Lett. (4)

Other (2)

L. M. Soroko, “Axicons and meso-optical imaging devices,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1989), Vol. 27, pp. 109–160.
[Crossref]

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), Chap. 9.

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Figures (5)

Fig. 1
Fig. 1

(a) Gray-level display of the WDF of q 0(ζ) = rect(ζ). (b) Gray-level display of the WDF shown in (a) but sheared in the x-axis: W q 0 (x - αζ, ζ).

Fig. 2
Fig. 2

Schematic diagram of the procedure for obtaining the complex amplitude transmittance t(ρ) with an extended depth of focus.

Fig. 3
Fig. 3

Phase distribution Φ(ρ/ρ0) associated with the pupil aperture t(ρ) with an enhanced depth of focus.

Fig. 4
Fig. 4

WDF of the logarithmic phase distribution for (a) δz = 29 mm and (b) δz = 60 mm. The remaining parameters are f 0 = 1220 mm and λ = 633 nm.

Fig. 5
Fig. 5

Axial irradiance I(0; z) generated from the phase distribution given by Eq. (10) (solid curve) and the logarithmic phase (dashed curve) for δz = 60 mm. The values of the parameters are f 0 = 1220 mm and λ = 633 nm.

Equations (13)

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p r ;   z = 2 π   exp i   π r 2 λ f 0   t ρ exp i   π z ρ 2 λ f 2 × J 0 2 π r ρ / λ f ρ d ρ ,
t ρ = t 0 ρ exp i Φ ρ ,
I 0 ;   z = | p r = 0 ;   z | 2 = 4 π 2 0   t 0 ρ exp i Φ ρ exp i   π z λ f 2   ρ 2 ρ d ρ 2 .
ζ = ρ / ρ 0 2 - 1 / 2 ,
t 0 ρ = q 0 ζ .
I 0 ;   z / I 0 = -   W q 0 ρ 0 2 z 2 λ f 2 ,   ζ d ζ ,
W q 0 x ,   ζ = -   q 0 ζ + ζ 2 q 0 * ζ - ζ 2 exp 2 π ix ζ d ζ .
q ζ = 1 q * 0     W q x ,   ζ / 2 exp - 2 π ix ζ d x ,
W q x ,   ζ = W q 0 x - α ζ ,   ζ ,
q ˜ x = -   q ˜ 0 x exp i   π α x - x 2 d x ,
q ζ = exp - i π α ζ 2 rect ζ ,
t ρ = exp - i π α ρ / ρ 0 4 - ρ / ρ 0 2 + 1 / 4 circ ρ / ρ 0 .
Φ ρ = π ρ 0 2 λ δ z ln f 0 + δ z ρ 2 ρ 0 2 ,

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