Abstract

A new formulation of the sampling theorem for circular apertures is proposed that uses a Dini series approach instead of a Fourier–Bessel one. The results obtained are especially convenient for the fast evaluation of point-spread functions: even the case of thin-ring aperture is not troublesome.

© 1981 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), Chaps. 8 and 9.
  2. R. Barakat, “Application of the sampling theorem to optical diffraction theory,” J. Opt. Soc. Am. 54, 920–930 (1964).
    [Crossref]
  3. R. Barakat, “The calculation of integrals encountered in optical diffraction theory,” in The Computer in Optical Research, B. R. Frieden, ed. (Springer-Verlag, New York, 1980), pp. 35–80.
    [Crossref]
  4. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), Chap. 5.
  5. A. Boivin, Théorie et Calcul des Figures de Diffraction de Révolution (Laval U. Press, Quebec, 1964), pp. 164, 197–199.
  6. H. Bateman, A. Erdélyi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. II, Sec. 7.10.4.
  7. S. Szapiel, “Maréchal intensity criteria modified for circular apertures with nonuniform intensity transmission: Dini series approach,” Opt. Lett. 2, 124–126 (1978).
    [Crossref] [PubMed]
  8. F. W. J. Olver, ed., Royal Society Mathematics Tables, Vol. VII, Bessel Functions, Part III: Zeros and Associated Values (Cambridge U. Press, Cambridge, 1960), Tables I and VI.

1978 (1)

1964 (1)

Barakat, R.

R. Barakat, “Application of the sampling theorem to optical diffraction theory,” J. Opt. Soc. Am. 54, 920–930 (1964).
[Crossref]

R. Barakat, “The calculation of integrals encountered in optical diffraction theory,” in The Computer in Optical Research, B. R. Frieden, ed. (Springer-Verlag, New York, 1980), pp. 35–80.
[Crossref]

Bateman, H.

H. Bateman, A. Erdélyi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. II, Sec. 7.10.4.

Boivin, A.

A. Boivin, Théorie et Calcul des Figures de Diffraction de Révolution (Laval U. Press, Quebec, 1964), pp. 164, 197–199.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), Chaps. 8 and 9.

Erdélyi, A.

H. Bateman, A. Erdélyi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. II, Sec. 7.10.4.

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), Chap. 5.

Szapiel, S.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), Chaps. 8 and 9.

J. Opt. Soc. Am. (1)

Opt. Lett. (1)

Other (6)

F. W. J. Olver, ed., Royal Society Mathematics Tables, Vol. VII, Bessel Functions, Part III: Zeros and Associated Values (Cambridge U. Press, Cambridge, 1960), Tables I and VI.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), Chaps. 8 and 9.

R. Barakat, “The calculation of integrals encountered in optical diffraction theory,” in The Computer in Optical Research, B. R. Frieden, ed. (Springer-Verlag, New York, 1980), pp. 35–80.
[Crossref]

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), Chap. 5.

A. Boivin, Théorie et Calcul des Figures de Diffraction de Révolution (Laval U. Press, Quebec, 1964), pp. 164, 197–199.

H. Bateman, A. Erdélyi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. II, Sec. 7.10.4.

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Tables (2)

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Table 1 Successive Zeros λL, zL and Related Absolute Valuesa

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Table 2 Comparison of Intensity Distribution Generated by Thin-Ring Aperture Computed Exactly and by the Sampling Method Using 20 Sampling Points

Equations (16)

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G ( z ) = 2 0 1 T ( r ) J 0 ( z r ) r d r ,
G ( z ) = 4 J 0 ( z ) L = 1 G ( λ L ) [ 2 J 1 ( λ L ) λ L ] 1 ( λ L 2 z 2 ) 1 ,
2 J 1 ( z ) / z
T ( r ) = j = 1 G ( z j ) J 0 2 ( z j ) J 0 ( z j r ) ,
G ( z j ) = 2 0 1 T ( r ) J 0 ( z j r ) r d r
z J 1 ( z ) = 0 .
T ( r ) = T ( r ) + L = 1 G ( z L ) J 0 2 ( z L ) J 0 ( z L r ) ,
T ( r ) = G ( 0 ) = 2 0 1 T ( r ) r d r
T ( r ) = 1 .
T ( r ) = 1 + L = 1 G ( z L ) J 0 2 ( z L ) J 0 ( z L r ) .
G ( z ) = 2 J 1 ( z ) z + 2 L = 1 G ( z L ) J 0 2 ( z L ) × 0 1 J 0 ( z L r ) J 0 ( z r ) r d r .
0 1 J 0 ( a r ) J 0 ( b r ) r d r = a J 1 ( a ) J 0 ( b ) b J 1 ( b ) J 0 ( a ) a 2 b 2 , ( a b )
G ( z ) = 2 J 1 ( z ) z [ 1 + S ( z ) ] ,
S ( z ) = L = 1 G ( z L ) J 0 ( z L ) [ 1 ( z L z ) 2 ] 1 , z z L , z 0
G ( z ) = J 0 ( z ) ,
J 0 ( z ) = 2 J 1 ( z ) z { 1 + L [ 1 ( z L z ) 2 ] 1 } , z z L , z 0 .

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