Abstract

The irradiance distribution along the optical axis, or the Strehl ratio versus defocus, of a rotationally symmetric system is described in terms of a phase-space representation: Woodward’s ambiguity function. We show that the Fourier spectrum of the Strehl ratio versus defocus for variable spherical aberration can be analyzed from a single picture. This result is applied to analyze the use of shade pupils for reducing the influence of spherical aberration.

© 1987 Optical Society of America

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References

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  1. A. Papoulis, “Ambiguity function in Fourier optics,”J. Opt. Soc. Am. 64, 779–788 (1974).
    [CrossRef]
  2. M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
    [CrossRef]
  3. K. Dutta, J. W. Goodman, “Reconstruction of images of partially coherent objects from samples of mutual intensity,”J. Opt. Soc. Am. 67, 796–803 (1977).
    [CrossRef]
  4. A. Walther, “Radiometry and coherence,”J. Opt. Soc. Am. 58, 1256–1259 (1968).
    [CrossRef]
  5. J.-P. Guigay, “The ambiguity function in diffraction and isoplanatic imaging by partially coherent beams,” Opt. Commun. 26, 136–138 (1978).
    [CrossRef]
  6. K.-H. Brenner, J. Ojeda-Castañeda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta 31, 213–223 (1984).
    [CrossRef]
  7. J. Ojeda-Castañeda, E. E. Sicre, “Bilinear optical systems: Wigner distribution function and ambiguity function representations,” Opt. Acta 31, 255–260 (1984).
    [CrossRef]
  8. J. Ojeda-Castañeda, E. E. Sicre, “Quasi ray-optical approach to longitudinal periodicities of free and bounded fields,” Opt. Acta 32, 17–26 (1985).
    [CrossRef]
  9. N. Bolognini, J. Ojeda-Castañeda, E. E. Sicre, “Interferometry based on the Lau effect: a quasi-ray description,” Opt. Acta 32, 409–422 (1985).
    [CrossRef]
  10. M. J. Yzuel, F. Calvo, “A study of the possibility of image optimization by apodizations,” Opt. Acta 26, 1397–1406 (1979).
    [CrossRef]
  11. J. Ojeda-Castañeda, P. Andrés, A. Díaz, “Strehl ratio with low sensitivity to spherical aberration,” J. Opt. Soc. Am. A (submitted).
  12. P. M. Woodward, Probability and Information Theory with Applications to Radar (McGraw-Hill, New York, 1953).
  13. C. W. McCutchen, “Generalized aperture and three dimensional diffraction image,”J. Opt. Soc. Am. 54, 240–244 (1964).
    [CrossRef]
  14. A. W. Lohmann, “Three-dimensional properties of wavefields,” Optik (Stuttgart) 51, 105–117 (1978).
  15. 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]
  16. L. R. Berriel-Valdos, E. Montes, J. Ojeda-Castañeda, “Ambiguity function applied to increase depth of focus,” Acta Polytech. Scand. Appl. Phys. Ser. 149, 260 (1985).
  17. L. R. Berriel-Valdos, J. Ojeda-Castañeda, E. Montes, “Ambiguity function as a design tool for high focal depth,” Appl. Opt. (submitted).

1985 (3)

J. Ojeda-Castañeda, E. E. Sicre, “Quasi ray-optical approach to longitudinal periodicities of free and bounded fields,” Opt. Acta 32, 17–26 (1985).
[CrossRef]

N. Bolognini, J. Ojeda-Castañeda, E. E. Sicre, “Interferometry based on the Lau effect: a quasi-ray description,” Opt. Acta 32, 409–422 (1985).
[CrossRef]

L. R. Berriel-Valdos, E. Montes, J. Ojeda-Castañeda, “Ambiguity function applied to increase depth of focus,” Acta Polytech. Scand. Appl. Phys. Ser. 149, 260 (1985).

1984 (2)

K.-H. Brenner, J. Ojeda-Castañeda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta 31, 213–223 (1984).
[CrossRef]

J. Ojeda-Castañeda, E. E. Sicre, “Bilinear optical systems: Wigner distribution function and ambiguity function representations,” Opt. Acta 31, 255–260 (1984).
[CrossRef]

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 (1)

M. J. Yzuel, F. Calvo, “A study of the possibility of image optimization by apodizations,” Opt. Acta 26, 1397–1406 (1979).
[CrossRef]

1978 (3)

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

J.-P. Guigay, “The ambiguity function in diffraction and isoplanatic imaging by partially coherent beams,” Opt. Commun. 26, 136–138 (1978).
[CrossRef]

A. W. Lohmann, “Three-dimensional properties of wavefields,” Optik (Stuttgart) 51, 105–117 (1978).

1977 (1)

1974 (1)

1968 (1)

1964 (1)

Andrés, P.

J. Ojeda-Castañeda, P. Andrés, A. Díaz, “Strehl ratio with low sensitivity to spherical aberration,” J. Opt. Soc. Am. A (submitted).

Bastiaans, M. J.

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

Berriel-Valdos, L. R.

L. R. Berriel-Valdos, E. Montes, J. Ojeda-Castañeda, “Ambiguity function applied to increase depth of focus,” Acta Polytech. Scand. Appl. Phys. Ser. 149, 260 (1985).

L. R. Berriel-Valdos, J. Ojeda-Castañeda, E. Montes, “Ambiguity function as a design tool for high focal depth,” Appl. Opt. (submitted).

Bolognini, N.

N. Bolognini, J. Ojeda-Castañeda, E. E. Sicre, “Interferometry based on the Lau effect: a quasi-ray description,” Opt. Acta 32, 409–422 (1985).
[CrossRef]

Brenner, K.-H.

K.-H. Brenner, J. Ojeda-Castañeda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta 31, 213–223 (1984).
[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]

Calvo, F.

M. J. Yzuel, F. Calvo, “A study of the possibility of image optimization by apodizations,” Opt. Acta 26, 1397–1406 (1979).
[CrossRef]

Díaz, A.

J. Ojeda-Castañeda, P. Andrés, A. Díaz, “Strehl ratio with low sensitivity to spherical aberration,” J. Opt. Soc. Am. A (submitted).

Dutta, K.

Goodman, J. W.

Guigay, J.-P.

J.-P. Guigay, “The ambiguity function in diffraction and isoplanatic imaging by partially coherent beams,” Opt. Commun. 26, 136–138 (1978).
[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]

A. W. Lohmann, “Three-dimensional properties of wavefields,” Optik (Stuttgart) 51, 105–117 (1978).

McCutchen, C. W.

Montes, E.

L. R. Berriel-Valdos, E. Montes, J. Ojeda-Castañeda, “Ambiguity function applied to increase depth of focus,” Acta Polytech. Scand. Appl. Phys. Ser. 149, 260 (1985).

L. R. Berriel-Valdos, J. Ojeda-Castañeda, E. Montes, “Ambiguity function as a design tool for high focal depth,” Appl. Opt. (submitted).

Ojeda-Castañeda, J.

L. R. Berriel-Valdos, E. Montes, J. Ojeda-Castañeda, “Ambiguity function applied to increase depth of focus,” Acta Polytech. Scand. Appl. Phys. Ser. 149, 260 (1985).

J. Ojeda-Castañeda, E. E. Sicre, “Quasi ray-optical approach to longitudinal periodicities of free and bounded fields,” Opt. Acta 32, 17–26 (1985).
[CrossRef]

N. Bolognini, J. Ojeda-Castañeda, E. E. Sicre, “Interferometry based on the Lau effect: a quasi-ray description,” Opt. Acta 32, 409–422 (1985).
[CrossRef]

K.-H. Brenner, J. Ojeda-Castañeda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta 31, 213–223 (1984).
[CrossRef]

J. Ojeda-Castañeda, E. E. Sicre, “Bilinear optical systems: Wigner distribution function and ambiguity function representations,” Opt. Acta 31, 255–260 (1984).
[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]

L. R. Berriel-Valdos, J. Ojeda-Castañeda, E. Montes, “Ambiguity function as a design tool for high focal depth,” Appl. Opt. (submitted).

J. Ojeda-Castañeda, P. Andrés, A. Díaz, “Strehl ratio with low sensitivity to spherical aberration,” J. Opt. Soc. Am. A (submitted).

Papoulis, A.

Sicre, E. E.

J. Ojeda-Castañeda, E. E. Sicre, “Quasi ray-optical approach to longitudinal periodicities of free and bounded fields,” Opt. Acta 32, 17–26 (1985).
[CrossRef]

N. Bolognini, J. Ojeda-Castañeda, E. E. Sicre, “Interferometry based on the Lau effect: a quasi-ray description,” Opt. Acta 32, 409–422 (1985).
[CrossRef]

J. Ojeda-Castañeda, E. E. Sicre, “Bilinear optical systems: Wigner distribution function and ambiguity function representations,” Opt. Acta 31, 255–260 (1984).
[CrossRef]

Walther, A.

Woodward, P. M.

P. M. Woodward, Probability and Information Theory with Applications to Radar (McGraw-Hill, New York, 1953).

Yzuel, M. J.

M. J. Yzuel, F. Calvo, “A study of the possibility of image optimization by apodizations,” Opt. Acta 26, 1397–1406 (1979).
[CrossRef]

Acta Polytech. Scand. Appl. Phys. Ser. (1)

L. R. Berriel-Valdos, E. Montes, J. Ojeda-Castañeda, “Ambiguity function applied to increase depth of focus,” Acta Polytech. Scand. Appl. Phys. Ser. 149, 260 (1985).

J. Opt. Soc. Am. (4)

Opt. Acta (5)

K.-H. Brenner, J. Ojeda-Castañeda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta 31, 213–223 (1984).
[CrossRef]

J. Ojeda-Castañeda, E. E. Sicre, “Bilinear optical systems: Wigner distribution function and ambiguity function representations,” Opt. Acta 31, 255–260 (1984).
[CrossRef]

J. Ojeda-Castañeda, E. E. Sicre, “Quasi ray-optical approach to longitudinal periodicities of free and bounded fields,” Opt. Acta 32, 17–26 (1985).
[CrossRef]

N. Bolognini, J. Ojeda-Castañeda, E. E. Sicre, “Interferometry based on the Lau effect: a quasi-ray description,” Opt. Acta 32, 409–422 (1985).
[CrossRef]

M. J. Yzuel, F. Calvo, “A study of the possibility of image optimization by apodizations,” Opt. Acta 26, 1397–1406 (1979).
[CrossRef]

Opt. Commun. (3)

J.-P. Guigay, “The ambiguity function in diffraction and isoplanatic imaging by partially coherent beams,” Opt. Commun. 26, 136–138 (1978).
[CrossRef]

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

Optik (Stuttgart) (1)

A. W. Lohmann, “Three-dimensional properties of wavefields,” Optik (Stuttgart) 51, 105–117 (1978).

Other (3)

L. R. Berriel-Valdos, J. Ojeda-Castañeda, E. Montes, “Ambiguity function as a design tool for high focal depth,” Appl. Opt. (submitted).

J. Ojeda-Castañeda, P. Andrés, A. Díaz, “Strehl ratio with low sensitivity to spherical aberration,” J. Opt. Soc. Am. A (submitted).

P. M. Woodward, Probability and Information Theory with Applications to Radar (McGraw-Hill, New York, 1953).

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Figures (7)

Fig. 1
Fig. 1

Schematic diagram of the image-formation process.

Fig. 2
Fig. 2

Geometrical representation of the change of variable for transforming a two-dimensional radial pupil into a one-dimensional rectangular pupil.

Fig. 3
Fig. 3

Geometrical-optics considerations for showing that there is geometrical isoplanatism along the optical axis.

Fig. 4
Fig. 4

Transfer of a input cosinusoidal irradiance variation along the optical axis into a cosinusoidal irradiance variation.

Fig. 5
Fig. 5

Use of phase space (η, N) to visualize the changes of the transfer function for a given spatial frequency η = η0 and variable W40.

Fig. 6
Fig. 6

Modulus of the ambiguity functions associated with (a) a clear aperture, (b) the annular apodizer in Eq. (18), and (c) the shade mask in Eq. (19).

Fig. 7
Fig. 7

Amplitude transmittance of three pupils functions: (a) clear aperture, (b) annular apodizer, (c) shade mask.

Equations (23)

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p ( r ; W 20 ; W 40 ) = 2 π 0 p ˜ ( ρ ) exp { i 2 π [ W 20 ( ρ / ρ 0 ) 2 + W 40 ( ρ / ρ 0 ) 4 ] } J 0 ( 2 π r ρ ) ρ d ρ .
S ( W 20 ; W 40 ) = p ( r = 0 ; W 20 ; W 40 ) 2 / p ( r = 0 ; W 20 = 0 ; W 40 = 0 ) 2 0.8.
ζ = ( ρ / ρ 0 ) 2 - 0.5
Q ˜ ( ζ ) = p ˜ ( ρ ) / [ 2 π 0 p ˜ ( ρ ) ρ d ρ ] .
S ( W 20 , W 40 ) = Q ( W 20 ; W 40 ) 2 0.8 ,
Q ( W 20 ; W 40 ) = exp ( i ϕ ) - Q ˜ ( ζ ) × exp { i 2 π [ W 40 ζ 2 + ( W 40 + W 20 ) ζ ] } d ζ .
S ˜ ( ζ ; W 40 ) = - S ( W 20 ; W 40 ) exp ( - i 2 π ζ W 20 ) d W 20 .
I i ( W 20 ) = 1 + M cos 2 π η W .
I 0 ( W 20 ; W 40 ) = - I i ( W 20 ) S ( W 20 - W 20 , W 40 ) d W 20
I 0 ( W 20 ; W 40 ) = - I i ( ζ ) S ˜ ( ζ , W 40 ) exp ( i 2 π W 20 ζ ) d ζ .
I ˜ i ( ζ ) = - I i ( W 20 ) exp ( - i 2 π ζ W 20 ) d W 20 = δ ( ζ ) + 0.5 M δ ( ζ - η ) + 0.5 M δ ( ζ + η ) .
I 0 ( W 20 ; W 40 ) = S ˜ ( 0 , W 40 ) + S ˜ ( η , W 40 ) M × cos [ 2 π η W 40 + Φ ( η , W 40 ) ] ,
S ˜ ( η , W 40 ) = S ˜ ( η , W 40 ) exp [ i Φ ( η , W 40 ) ] .
S ˜ ( η ; W 40 ) = - Q ( W 20 ; W 40 ) 2 exp ( - i 2 π η W 20 ) d W 20 = - Q ˜ ( ζ ; W 40 ) Q ˜ * ( ζ ; W 40 ) × exp [ i 2 π W 40 ( ζ 2 - ζ 2 ) ] × exp [ i 2 π ( W 40 + W 20 ) ( ζ - ζ ) ] × exp ( - i 2 π η W 20 ) d ζ d ζ d W 20
= - Q ˜ ( ζ ; W 40 ) Q ˜ * ) ζ ; W 40 ) × exp { i 2 π W 40 [ ζ 2 - ζ 2 + ζ - ζ ] } × δ ( η + ζ - ζ ) d ζ d ζ
= exp [ i 2 π W 40 ( η + η 2 ) ] × - Q ˜ ( η + ζ ; W 40 ) Q * ( ζ ; W 40 ) × exp [ i 2 π ( 2 W 40 η ) ζ ] d ζ
= exp ( i 2 π W 40 η ) × - Q ˜ ( ζ + η / 2 ; W 40 ) Q ˜ * ( ζ - η / 2 ; W 40 ) × exp [ i 2 π ( 2 W 40 η ) ζ ] d ζ .
A ( η , N ) = - Q ˜ ( ζ + η / 2 ) Q ˜ * ( ζ + η / 2 ) exp ( i 2 π N ζ ) d ζ .
S ˜ ( η ; W 40 ) = A ( η ; N = 2 W 40 η ) .
Q ˜ ( ζ ) = rect ( ζ ) .
A ( η , N ) = [ 1 - η ] rect ( η / 2 ) sinc ( [ 1 - η ] N ) ,
Q ˜ ( ζ ) = 3 [ cos 2 π ζ - sinc 2 ζ ] ( 2 π ζ ) 2 rect ( ζ ) .
Q ˜ ( ζ ) = ( 0.45 - 1.1 ζ ) rect ( ζ ) .

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