Abstract

Since Woodward’s ambiguity function contains all the optical transfer functions with varying focus error, it can be applied to display visually the tolerance to focus error. We compare the tolerance to defocus of a transparent but finite extended pupil and a pupil with a central obscuration.

© 1984 Optical Society of America

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References

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  1. Lord Rayleigh, “Investigations in Optics with Special Reference to the Spectroscope,” Philos. Mag. 8, 261 (1879).
    [CrossRef]
  2. K. Strehl, Zeitfür Instrumkde 22, 213 (1902).
  3. A. Maréchal, Thesis, U. Paris (1948).
  4. H. H. Hopkins, “The Aberration Permissible in Optical Systems,” Proc. Phys. Soc. London Sect. B 70, 449 (1957).
    [CrossRef]
  5. H. H. Hopkins, “The Use of Diffraction-based Criteria of Image Quality in Automatic Optical Design,” Opt. Acta 13, 343 (1966).
    [CrossRef]
  6. P. M. Duffieux, L’integrale de Fourier et ses applications à l’Optique (Rennes, 1946).
  7. P. M. Woodward, Probability and Information Theory with Applications to Radar (Pergamon, New York, 1953).
  8. A. Papoulis, “Ambiguity Function in Fourier Optics,” J. Opt. Soc. Am. 64, 779 (1974).
    [CrossRef]
  9. K.-H. Brenner, A. W. Lohmann, J. Ojeda-Castaneda, “The Ambiguity Function as a Polar Display of the OTF,” Opt. Commun. 44, 323 (1983).
    [CrossRef]
  10. R. J. Marks, J. F. Walkup, T. F. Krile, “Ambiguity Function Display: an Improved Coherent Processor,” Appl. Opt. 16, 746 (1977).
    [CrossRef]
  11. K.-H. Brenner, A. W. Lohmann, “Wigner Distribution Function Display of Complex 1D Signals,” Opt. Commun. 42, 310 (1982).
    [CrossRef]
  12. Lord Rayleigh, “On the Diffraction of Object-Glasses,” Mon. Not. R. Astron. Soc. 33, 59 (1872).
  13. W. T. Welford, “Use of Annular Apertures to Increase Focal Depth,” J. Opt. Soc. Am. 50, 749 (1960).
    [CrossRef]
  14. J. T. McCrickerd, “Coherent Processing and Depth of Focus of Annular Aperture Imagery,” Appl. Opt. 10, 2226 (1971).
    [CrossRef] [PubMed]
  15. V. N. Mahajan, “Strehl Ratio for Primary Aberrations: Some Analytical Results for Circular and Annular Pupils,” J. Opt. Soc. Am. 72, 1258 (1982).
    [CrossRef]
  16. H. H. Hopkins, Wave Theory of Aberrations (Oxford U.P., London, 1959).

1983 (1)

K.-H. Brenner, A. W. Lohmann, J. Ojeda-Castaneda, “The Ambiguity Function as a Polar Display of the OTF,” Opt. Commun. 44, 323 (1983).
[CrossRef]

1982 (2)

K.-H. Brenner, A. W. Lohmann, “Wigner Distribution Function Display of Complex 1D Signals,” Opt. Commun. 42, 310 (1982).
[CrossRef]

V. N. Mahajan, “Strehl Ratio for Primary Aberrations: Some Analytical Results for Circular and Annular Pupils,” J. Opt. Soc. Am. 72, 1258 (1982).
[CrossRef]

1977 (1)

1974 (1)

1971 (1)

1966 (1)

H. H. Hopkins, “The Use of Diffraction-based Criteria of Image Quality in Automatic Optical Design,” Opt. Acta 13, 343 (1966).
[CrossRef]

1960 (1)

1957 (1)

H. H. Hopkins, “The Aberration Permissible in Optical Systems,” Proc. Phys. Soc. London Sect. B 70, 449 (1957).
[CrossRef]

1902 (1)

K. Strehl, Zeitfür Instrumkde 22, 213 (1902).

1879 (1)

Lord Rayleigh, “Investigations in Optics with Special Reference to the Spectroscope,” Philos. Mag. 8, 261 (1879).
[CrossRef]

1872 (1)

Lord Rayleigh, “On the Diffraction of Object-Glasses,” Mon. Not. R. Astron. Soc. 33, 59 (1872).

Brenner, K.-H.

K.-H. Brenner, A. W. Lohmann, J. Ojeda-Castaneda, “The Ambiguity Function as a Polar Display of the OTF,” Opt. Commun. 44, 323 (1983).
[CrossRef]

K.-H. Brenner, A. W. Lohmann, “Wigner Distribution Function Display of Complex 1D Signals,” Opt. Commun. 42, 310 (1982).
[CrossRef]

Duffieux, P. M.

P. M. Duffieux, L’integrale de Fourier et ses applications à l’Optique (Rennes, 1946).

Hopkins, H. H.

H. H. Hopkins, “The Use of Diffraction-based Criteria of Image Quality in Automatic Optical Design,” Opt. Acta 13, 343 (1966).
[CrossRef]

H. H. Hopkins, “The Aberration Permissible in Optical Systems,” Proc. Phys. Soc. London Sect. B 70, 449 (1957).
[CrossRef]

H. H. Hopkins, Wave Theory of Aberrations (Oxford U.P., London, 1959).

Krile, T. F.

Lohmann, A. W.

K.-H. Brenner, A. W. Lohmann, J. Ojeda-Castaneda, “The Ambiguity Function as a Polar Display of the OTF,” Opt. Commun. 44, 323 (1983).
[CrossRef]

K.-H. Brenner, A. W. Lohmann, “Wigner Distribution Function Display of Complex 1D Signals,” Opt. Commun. 42, 310 (1982).
[CrossRef]

Mahajan, V. N.

Maréchal, A.

A. Maréchal, Thesis, U. Paris (1948).

Marks, R. J.

McCrickerd, J. T.

Ojeda-Castaneda, J.

K.-H. Brenner, A. W. Lohmann, J. Ojeda-Castaneda, “The Ambiguity Function as a Polar Display of the OTF,” Opt. Commun. 44, 323 (1983).
[CrossRef]

Papoulis, A.

Rayleigh, Lord

Lord Rayleigh, “Investigations in Optics with Special Reference to the Spectroscope,” Philos. Mag. 8, 261 (1879).
[CrossRef]

Lord Rayleigh, “On the Diffraction of Object-Glasses,” Mon. Not. R. Astron. Soc. 33, 59 (1872).

Strehl, K.

K. Strehl, Zeitfür Instrumkde 22, 213 (1902).

Walkup, J. F.

Welford, W. T.

Woodward, P. M.

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

Appl. Opt. (2)

J. Opt. Soc. Am. (3)

Mon. Not. R. Astron. Soc. (1)

Lord Rayleigh, “On the Diffraction of Object-Glasses,” Mon. Not. R. Astron. Soc. 33, 59 (1872).

Opt. Acta (1)

H. H. Hopkins, “The Use of Diffraction-based Criteria of Image Quality in Automatic Optical Design,” Opt. Acta 13, 343 (1966).
[CrossRef]

Opt. Commun. (2)

K.-H. Brenner, A. W. Lohmann, J. Ojeda-Castaneda, “The Ambiguity Function as a Polar Display of the OTF,” Opt. Commun. 44, 323 (1983).
[CrossRef]

K.-H. Brenner, A. W. Lohmann, “Wigner Distribution Function Display of Complex 1D Signals,” Opt. Commun. 42, 310 (1982).
[CrossRef]

Philos. Mag. (1)

Lord Rayleigh, “Investigations in Optics with Special Reference to the Spectroscope,” 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 (1957).
[CrossRef]

Zeitfür Instrumkde (1)

K. Strehl, Zeitfür Instrumkde 22, 213 (1902).

Other (4)

A. Maréchal, Thesis, U. Paris (1948).

P. M. Duffieux, L’integrale de Fourier et ses applications à l’Optique (Rennes, 1946).

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

H. H. Hopkins, Wave Theory of Aberrations (Oxford U.P., London, 1959).

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Figures (5)

Fig. 1
Fig. 1

Loci of Hopkins’s tolerance of focus error for a uniform rectangular aperture. The maximum value tolerated for the defocus coefficient is proportional to the slope tanθ.

Fig. 2
Fig. 2

Computer-generated AF for a uniform rectangular aperture. Note the resemblance between the zero loci and the curve in Fig. 1.

Fig. 3
Fig. 3

In-focus OTFs for several rectangular apertures: - · - · - clear aperture; — central obscured aperture ɛ = ⅓; and - - - central obscured aperture ɛ = ⅔.

Fig. 4
Fig. 4

Loci of Hopkins’s tolerance to focus error for rectangular apertures. The notation is the same as in Fig. 3.

Fig. 5
Fig. 5

Computer-generated AF for rectangular aperture with a central obscuration. Note the resemblance between the zero loci and the curves in Fig. 4.

Equations (23)

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A ( μ , y ) = q ( ν + μ / 2 ) q * ( ν + μ / 2 ) exp ( i 2 π y ν ) d ν .
H ( ν , W 20 ) = A ( ν , y = 2 W 20 ν / λ ν 0 2 ) ,
M ( ν ; W 20 ) = H ( ν , W 20 ) / H ( ν , 0 ) = A ( ν , y ) / A ( ν , 0 ) 0.8.
q ( ν ) = 1 / 2 ν 0 if ν < ν 0 = 0 if ν > ν 0 .
A ( ν / ν 0 , y ) = ( 1 - ν / 2 ν 0 ) sinc 2 π y ν 0 ( 1 - ν 2 ν 0 ) ,
M ( ν ; W 20 ) = sinc 2 π y ν 0 ( 1 - ν / 2 ν 0 ) 0.8 ,
y ( 9 / 25 ν 0 ) ( 2 - ν / ν 0 ) - 1 .
ν / ν 0 = 1 ± 1 - 9 λ / ( 50 W 20 ) ,
W 20 9 λ / 50 λ / 6.
q ( ν ) = 1 / 2 ν 0 ( 1 - ɛ ) , if ɛ ν 0 < ν < ν 0 = 0 , elsewhere .
A ( ν , y ) = [ 1 - ν / ν 0 ( 1 - ɛ ) ] cos [ π ( 1 + ɛ ) ν 0 y ] × sinc π ( 1 - ɛ ) ν 0 y [ 1 - ν / ν 0 ( 1 - ɛ ) ] ,             if ν < ( 1 - ɛ ) ν 0 .
A ( ν , y ) = { [ ν - 2 ɛ ν 0 ] / 2 ( 1 - ɛ ) ν 0 } sinc π y ( ν - 2 ɛ ν 0 ) ,             if 2 ɛ ν 0 < ν < ( 1 + ɛ ) ν 0 .
A ( ν , y ) = { [ 2 ν 0 - ν ] / 2 ( 1 - ɛ ) ν 0 } sinc π y ( 2 ν 0 - ν ) ,             if ( 1 + ɛ ) ν 0 < ν < 2 ν 0 .
A ( ν , y ) = 0 ,             if ( 1 - ɛ ) ν 0 < ν < 2 ɛ ν 0 or ν > 2 ν 0.
M ( ν ; W 20 ) = cos [ π ( 1 + ɛ ) ν 0 y ] × sinc { π y ( 1 - ɛ ) ν 0 [ 1 - ν / ( 1 - ɛ ) ν 0 ] } 0.8 ,             if 0 < ν < ( 1 - ɛ ) ν 0 .
M ( ν ; W 20 ) = sinc 2 π ɛ ν 0 [ ( ν / 2 ɛ ν 0 ) - 1 ] 0.8 ,             if 2 ɛ ν 0 < ν < ( 1 + ɛ ) ν 0 .
M ( ν ; W 20 ) = sinc 2 π ν 0 y [ 1 - ν / 2 ν 0 ] 0.8 ,             if ( 1 + ɛ ) ν 0 < ν < 2 ν 0 .
y 0.2048 / ( 1 + ɛ ) ν 0 ,             if 0 < ν < ( 1 - ɛ 0 ) ν 0 ,
y ( 9 / 25 ν 0 ) ( ν / ν 0 - 2 ɛ ) - 1 ,             if 2 ɛ ν 0 < ν < ( 1 + ɛ ) ν 0 ,
y ( 9 / 25 ν 0 ) ( 2 - ν / ν 0 ) - 1 ,             if ν 0 ( 1 + ɛ ) < ν < 2 ν 0 .
W 20 ( 1 - 2 ɛ ) - 1 ( 9 λ / 50 ) ( 1 - 2 ɛ ) - 1 ( λ / 6 ) .
H ( ν = ν 0 ; W 20 = 0 ; ɛ = 0 ) = 0.5 ,
H ( ν = ν 0 ; W 20 = 0 ; ɛ = ) = 0.5 ( 1 - 2 ɛ ) / ( 1 - ɛ ) .

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