Abstract

An analysis of the Strehl ratio and the optical transfer function as imaging quality parameters of optical elements with enhanced focal length is carried out by employing the Wigner distribution function. To this end, we use four different pupil functions: a full circular aperture, a hyper-Gaussian aperture, a quartic phase plate, and a logarithmic phase mask. A comparison is performed between the quality parameters and test images formed by these pupil functions at different defocus distances.

© 2000 Optical Society of America

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References

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  1. 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.
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  14. Lord Raleigh, “Investigations in optics with special reference to the spectroscopy,” Philos. Mag. 8, 261 (1879).
    [CrossRef]
  15. A. Maréchal, Ph.D. thesis (University of Paris, Paris, 1948).
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    [CrossRef]
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    [CrossRef]
  18. M. J. Bastiaans, “The Wigner distribution function applied to optical signal and systems,” Opt. Commun. 25, 26–30 (1978).
    [CrossRef]
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    [CrossRef]
  20. A. Papoulis, “Ambiguity function in Fourier optics,” J. Opt. Soc. Am. 64, 779–788 (1974).
    [CrossRef]
  21. K. H. Brenner, A. W. Lohmann, J. Ojeda-Castañeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323–326 (1983).
    [CrossRef]
  22. H. Bartelt, J. Ojeda-Castañeda, E. E. Sicre, “Misfocus tolerance seen by simple inspection of the ambiguity function,” Appl. Opt. 23, 2693–2696 (1984).
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  24. E. R. Dowsky, W. Thomas Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. 34, 1859–1866 (1995).
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  25. 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]
  26. G. Saavedra, W. D. Furlan, E. Silvestre, 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]
  27. D. Zalvidea, E. E. Sicre, “Phase pupil functions for focal depth enhancement derived from a Wigner distribution function,” Appl. Opt. 37, 3623–3627 (1998).
    [CrossRef]
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    [CrossRef]

1998 (2)

1997 (1)

G. Saavedra, W. D. Furlan, E. Silvestre, 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 (2)

E. R. Dowsky, W. Thomas Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. 34, 1859–1866 (1995).
[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]

1992 (1)

1991 (1)

1990 (1)

1988 (1)

1987 (1)

1986 (1)

1984 (1)

1983 (1)

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

1979 (2)

M. J. Yzuel, F. Calvo, “A study of the possibility of image optimization by apodization filters in optical systems of residual aberrations,” Opt. Acta 26, 1397–1406 (1979).
[CrossRef]

M. J. Bastiaans, “The Wigner distribution function and Hamilton characteristics of a geometric-optical system,” Opt. Commun. 30, 321–326 (1979).
[CrossRef]

1978 (1)

M. J. Bastiaans, “The Wigner distribution function applied to optical signal and systems,” Opt. Commun. 25, 26–30 (1978).
[CrossRef]

1974 (1)

1972 (1)

G. Häusler, “A method to increase the depth of focus by two steps image processing,” Opt. Commun. 6, 38–42 (1972).
[CrossRef]

1971 (2)

1966 (1)

H. H. Hopkins, “The use of diffraction-based criteria 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]

1879 (1)

Lord Raleigh, “Investigations in optics with special reference to the spectroscopy,” Philos. Mag. 8, 261 (1879).
[CrossRef]

Andrés, P.

Bará, S.

Bartelt, H.

Bastiaans, M. J.

M. J. Bastiaans, “The Wigner distribution function and Hamilton characteristics of a geometric-optical system,” Opt. Commun. 30, 321–326 (1979).
[CrossRef]

M. J. Bastiaans, “The Wigner distribution function applied to optical signal and systems,” Opt. Commun. 25, 26–30 (1978).
[CrossRef]

Berriel-Valdós, L. R.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1989), pp. 460–464.

Brenner, K. H.

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

Calvo, F.

M. J. Yzuel, F. Calvo, “A study of the possibility of image optimization by apodization filters in optical systems of residual aberrations,” Opt. Acta 26, 1397–1406 (1979).
[CrossRef]

Cathey, W. Thomas

Davidson, N.

Di´az, A.

Dowsky, E. R.

Duffieux, P. M.

P. M. Duffieux, L’intégrale de Fourier et ses Applications à l’Optique (Imprimeries Oberthur, Rennes, France, 1946).

Friesem, A. A.

Furlan, W. D.

G. Saavedra, W. D. Furlan, E. Silvestre, 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]

Granieri, S.

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]

Hasman, E.

Häusler, G.

G. Häusler, “A method to increase the depth of focus by two steps image processing,” Opt. Commun. 6, 38–42 (1972).
[CrossRef]

Hopkins, H. H.

H. H. Hopkins, “The use of diffraction-based criteria 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]

Jaroszewicz, Z.

Kolodziejczyck, A.

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]

Lohmann, A. W.

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

Maréchal, A.

A. Maréchal, Ph.D. thesis (University of Paris, Paris, 1948).

McCrickerd, J. T.

Mino, M.

Montes, E.

Morales, J.

Ojeda-Castañeda, J.

Okano, Y.

Papoulis, A.

Pons, A.

Raleigh, Lord

Lord Raleigh, “Investigations in optics with special reference to the spectroscopy,” Philos. Mag. 8, 261 (1879).
[CrossRef]

Saavedra, G.

G. Saavedra, W. D. Furlan, E. Silvestre, 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]

Sicre, E. E.

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

G. Saavedra, W. D. Furlan, E. Silvestre, 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, E. E. Sicre, “Analysis of the Strehl ratio using the Wigner distribution function,” Opt. Commun. 118, 207–214 (1995).
[CrossRef]

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]

Silvestre, E.

G. Saavedra, W. D. Furlan, E. Silvestre, 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.

Tepichin, E.

Welford, W. T.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1989), pp. 460–464.

Yzuel, M. J.

M. J. Yzuel, F. Calvo, “A study of the possibility of image optimization by apodization filters in optical systems of residual aberrations,” Opt. Acta 26, 1397–1406 (1979).
[CrossRef]

Zalvidea, D.

D. Zalvidea, E. E. Sicre, “Phase pupil functions 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. (7)

J. Opt. Soc. Am. (2)

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

Opt. Acta (2)

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

M. J. Yzuel, F. Calvo, “A study of the possibility of image optimization by apodization filters in optical systems of residual aberrations,” Opt. Acta 26, 1397–1406 (1979).
[CrossRef]

Opt. Commun. (6)

G. Häusler, “A method to increase the depth of focus by two steps image processing,” Opt. Commun. 6, 38–42 (1972).
[CrossRef]

M. J. Bastiaans, “The Wigner distribution function applied to optical signal and systems,” Opt. Commun. 25, 26–30 (1978).
[CrossRef]

M. J. Bastiaans, “The Wigner distribution function and Hamilton characteristics of a geometric-optical system,” Opt. Commun. 30, 321–326 (1979).
[CrossRef]

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

G. Saavedra, W. D. Furlan, E. Silvestre, 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. Lett. (3)

Philos. Mag. (1)

Lord Raleigh, “Investigations in optics with special reference to the spectroscopy,” Philos. Mag. 8, 261 (1879).
[CrossRef]

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]

Other (4)

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.

A. Maréchal, Ph.D. thesis (University of Paris, Paris, 1948).

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1989), pp. 460–464.

P. M. Duffieux, L’intégrale de Fourier et ses Applications à l’Optique (Imprimeries Oberthur, Rennes, France, 1946).

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Figures (4)

Fig. 1
Fig. 1

Normalized intensity of the hyper-Gaussian pupil aperture, the logarithmic phase mask, the circular pupil function, and the quartic pupil function for extended depth of focus, Δz=60 mm.

Fig. 2
Fig. 2

Optical transfer function of (a) circular pupil function, (b) logarithmic pupil function, (c) quartic pupil function, (d) hyper-Gaussian pupil function for extended depth of focus, Δz=60 mm.

Fig. 3
Fig. 3

Cross sections of the OTF of Fig. 2 for z=0 mm and z=24.119 mm. (a) Circular pupil function; some departure from the expected linear decrease (for z=0 mm) due to the computational limitations in data sampling can be observed. (b) Logarithmic pupil function. (c) Quartic pupil function. (d) Hyper-Gaussian pupil function.

Fig. 4
Fig. 4

Images of a test chart with (a) quartic pupil function for z=10 mm, (b) hyper-Gaussian pupil function for z=0 mm, (c) quartic pupil function for z=24.119 mm, (d) hyper-Gaussian pupil function for z=24.119 mm.

Equations (27)

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Wg(x, y; ν, μ)=-+gx+x2, y+y2×g*x-x2, y-y2×exp[-2πi(xν+yμ)]dxdy=-+G˜ν+ν2, μ+μ2×G˜*ν-ν2, μ-μ2×exp[2πi(xν+yμ)]dνdμ,
Wz(x, y; ν, μ)=W0(x-λzν, y-λzμ; ν, μ),
I(x, y; z)=-Wz(x, y; ν, μ)dνdμ,
t(x, y)=1t*(0, 0)-Wtx2, y2; ν, μ×exp[2πi(νx+μy)]dνdμ.
p(x, y; z=0)=Aλfexpi πλf(x2+y2)×-t(ξ,η)exp-2πiλf(xξ+yη)×dξdη,
Wi(x, y; ν, μ)=(λf)2Wtx-λfν, y-λfμ; xλf, yλf,
I(x, y; z)=-Wi(x-λzν, y-λzμ; ν, μ)dνdμ=(λf)2-dνdμWtx-λ(f+z)ν, y-λ(f+z)μ; xλf-zfν, yλf-zfμ.
S(z)=I(0, 0; z)I(0, 0; 0)=K-dζ1dζ2×Wt(x-λ(f+z)ζ1, y-λ(f+z)ζ2; ζ1, ζ2),
I(x, y; z)=|p(x, y; z|2.
I(0, 0; z)=02π0t(ρ)expiπzλf2ρ2ρdρdϕ2.
ρρ02=ζ+12.
I(0, 0; z)=-qζ+ζ2q*ζ-ζ2×exp2πi zρ022λf2ζdζdζ.
S(z)=-Wqρ02z2λf2, ζdζ.
Wqzρ022λf2, ζ=Wq0zρ022λf2-αζ, ζ.
t(ρ)=circρρ0exp-iπαρρ04-ρρ02+14.
Wqzρ022λf2; ζ=q0ζ+ζ2q0*ζ-ζ2×exp(-iπ2αζζ)expiπ zρ02λf2ζdζ.
zρ022λf2-αζ=0.
Wqzρ022λf2; ζ=zρ022λαf2=rectζ+ζ2rectζ-ζ2dζ=-2ζ+1ζ02ζ+1ζ0.
zmax=λf2αρ02.
H(ν; z)=0|p(r; z)|2J0(2πνr)rdr0|p(r; z=0)|2rdr,
|p(r; z)|2=0t(ρ)expπiλzf2ρ2J02πρrλfρdρ2=00t(ρ)t*(ρ)expπiλzf2[ρ2-(ρ)2]×J02πρrλfJ02πρrλfρρdρdρ.
H(ν; z)=10|p(r; 0)|2rdr0--q(ξ)q*(ξ)×J02πρ0λfξ+12 r×J02πρ0λfξ+12 r×expπiλρ02zf2(ξ-ξ)dξdξ×J0(2πνr)rdr,
H(ν; z)=10|p(r)|2rdr0--qζ+ζ2×q*ζ-ζ2×J02πρ0λfζ+ζ+12 r×J02πρ0λfζ-ζ-12 r×expπiλρ02zf2ζdζdζJ0(2πνr)rdr,
H(ν; z)=0-Wqˆ(ζ;r)ρ02z2 f2λ; ζdζJ0(2πνr)rdr0-Wqˆ(ζ,r)ρ02z2 f2λ; ζdζrdr,
qˆ(ζ; r)=q(ζ)J02πρ0λfζr.
t(ρ)=circρρ0expiπρ02λδzlnf0+δzρ2ρ02,
t(ρ)=circρρ0exp-2πγρρ02-122,

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