Abstract

Diffraction of quasimonochromatic partially space-coherent light by annular apertures, which are of fundamental importance when reflecting components are used, has been examined theoretically. The work on slit and circular apertures by previous workers has been briefly reviewed. Use has been made of Schell’s theorem which gives the relationship between the far-field intensity and the Fourier transform of the product of the source autocorrelation function and the correlation function across the diffracting aperture. Exponential, besinc, and sinc forms of the degree of coherence have been assumed and the results are presented for three typical values of the central obstruction of the circular aperture.

© 1969 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1964), Ch. 10, p. 491.
  2. M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Inc., Englewood Cliffs, N. J., 1964).
  3. M. Françon, Optical Interferometry (Academic Press Inc., New York, 1966), Chs. 1, 8, pp. 1, 162.
  4. M. Françon, Diffraction; Coherence in Optics (Pergamon Press, Inc., New York, 1966).
  5. W. H. Steel, Interferometry (Cambridge University Press, 1967).
  6. H. H. Hopkins, in Advances in Optical Techniques, A. C. S. van Heel, Ed. (North-Holland Publ. Co., Amsterdam, 1967), Ch. 6, p. 190.
  7. M. Françon and S. Mallick, in Progress in Optics VI, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1967), Ch. 3, p. 73.
  8. E. Wolf, Proc. Roy. Soc. (London) A230, 246 (1955).
  9. A. C. Schell, Ph.D. thesis, Massachusetts Institute of Technology (1961).
  10. G. B. Parrent, J. Opt. Soc. Am. 49, 787 (1959); also Ref. 2, p. 40.
    [Crossref]
  11. J. Bakos and K. Kantor, Nuovo Cimento 22, 519 (1961).
    [Crossref]
  12. G. B. Parrent and T. J. Skinner, Opt. Acta. 8, 93 (1961).
    [Crossref]
  13. W. T. Cathey, J. Opt. Soc. Am. 55, 1035 (1965).
    [Crossref]
  14. K. D. Mielenz, J. Opt. Soc. Am. 57, 66 (1967).
    [Crossref]
  15. A. Röseler, Optik 27, 179 (1968).
  16. R. A. Shore, Air Force Cambridge Research Laboratories Unclassified Research Report 62-715, August, 1962. Also in Proceedings of Symposium on Electro-magnetic Theory and Antennas, Copenhagen, Denmark, June, 1962, E. C. Jordan, Ed. (Pergamon Press (Ltd.)London, 1963), p. 787.
  17. J. G. Meadors, Opt. Acta. 12, 379 (1965); and dissertation submitted to Ohio State University, Columbus, Ohio (1964).
    [Crossref]
  18. J. L. Poirier, W. Rotman, and R. A. Shore, Appl. Opt. 4, 1321 (1965).
    [Crossref]
  19. R. A. Shore, B. J. Thompson, and R. E. Whitney, J. Opt. Soc. Am. 56, 733 (1966).
    [Crossref]
  20. J. V. Cornacchio and R. P. Soni, Nuovo Cimento 38, 1169 (1965).
    [Crossref]
  21. J. V. Cornacchio and K. A. Farnham, Nuovo Cimento 42, 108 (1966).
    [Crossref]
  22. A. Consortini, L. Ronchi, A. M. Scheggi, and G. Toraldo di Francia, Alta Frequenza 32, 178E (1963).
  23. A. Consortini, L. Ronchi, A. M. Scheggi, and G. Toraldo di Francia, Radio Science 1, 523 (1966).
  24. P. Burlamacchi and A. Consortini, Opt. Acta. 14, 17 (1967).
    [Crossref]
  25. D. L. Fried, J. Opt. Soc. Am. 56, 1372 (1966).
    [Crossref]
  26. D. L. Fried, Proc. IEEE 55, 57 (1967).
    [Crossref]
  27. D. L. Fried, J. Opt. Soc. Am. 57, 169 (1967).
    [Crossref]
  28. W. H. Steel, Rev. Opt. 32, 4, 143, 269 (1953). Also thesis, University of Paris (1953).
  29. E. L. O’Neill, J. Opt. Soc. Am. 46, 285 (1956). Also see Introduction to Statistical Optics (Addison–Wesley Publ. Co., Reading, Mass., 1963), p. 99, and E. B. Brown, Modern Optics (Reinhold Publ. Corp., New York, 1965), p. 496.
    [Crossref]
  30. R. Barakat and T. Asakura, Oyo Buturi 30, 728 (1961).
  31. R. Barakat and A. Houston, J. Opt. Soc. Am. 55, 538 (1965).
    [Crossref]
  32. J. V. Cornacchio, J. Opt. Soc. Am. 57, 1325 (1967).
    [Crossref] [PubMed]
  33. E. H. Linfoot and E. Wolf, Proc. Phys. Soc. (London) B66, 145 (1953). See also E. H. Linfoot, Recent Advances in Optics (Oxford at the Clarendon Press, 1955), p. 41.
  34. J. R. Izatt, J. Opt. Soc. Am. 55, 106 (1965).
    [Crossref]
  35. C. A. Taylor and B. J. Thompson, J. Opt. Soc. Am. 48, 844 (1958).
    [Crossref]
  36. B. J. Thompson, J. Opt. Soc. Am. 55, 145 (1965).
    [Crossref]
  37. W. T. Welford, J. Opt. Soc. Am. 50, 749 (1960).
    [Crossref]
  38. G. Toraldo di Francia, Nuovo Cimento Suppl. 9, 426 (1952).
    [Crossref]
  39. G. Lansraux, Rev. Opt. 32, 475 (1953).
  40. B. Roizen–Dossier, Rev. Opt. 33, 57, 147, 267 (1954).

1968 (1)

A. Röseler, Optik 27, 179 (1968).

1967 (5)

1966 (4)

A. Consortini, L. Ronchi, A. M. Scheggi, and G. Toraldo di Francia, Radio Science 1, 523 (1966).

D. L. Fried, J. Opt. Soc. Am. 56, 1372 (1966).
[Crossref]

R. A. Shore, B. J. Thompson, and R. E. Whitney, J. Opt. Soc. Am. 56, 733 (1966).
[Crossref]

J. V. Cornacchio and K. A. Farnham, Nuovo Cimento 42, 108 (1966).
[Crossref]

1965 (7)

1963 (1)

A. Consortini, L. Ronchi, A. M. Scheggi, and G. Toraldo di Francia, Alta Frequenza 32, 178E (1963).

1961 (3)

J. Bakos and K. Kantor, Nuovo Cimento 22, 519 (1961).
[Crossref]

G. B. Parrent and T. J. Skinner, Opt. Acta. 8, 93 (1961).
[Crossref]

R. Barakat and T. Asakura, Oyo Buturi 30, 728 (1961).

1960 (1)

1959 (1)

1958 (1)

1956 (1)

1955 (1)

E. Wolf, Proc. Roy. Soc. (London) A230, 246 (1955).

1954 (1)

B. Roizen–Dossier, Rev. Opt. 33, 57, 147, 267 (1954).

1953 (3)

G. Lansraux, Rev. Opt. 32, 475 (1953).

E. H. Linfoot and E. Wolf, Proc. Phys. Soc. (London) B66, 145 (1953). See also E. H. Linfoot, Recent Advances in Optics (Oxford at the Clarendon Press, 1955), p. 41.

W. H. Steel, Rev. Opt. 32, 4, 143, 269 (1953). Also thesis, University of Paris (1953).

1952 (1)

G. Toraldo di Francia, Nuovo Cimento Suppl. 9, 426 (1952).
[Crossref]

Asakura, T.

R. Barakat and T. Asakura, Oyo Buturi 30, 728 (1961).

Bakos, J.

J. Bakos and K. Kantor, Nuovo Cimento 22, 519 (1961).
[Crossref]

Barakat, R.

R. Barakat and A. Houston, J. Opt. Soc. Am. 55, 538 (1965).
[Crossref]

R. Barakat and T. Asakura, Oyo Buturi 30, 728 (1961).

Beran, M. J.

M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Inc., Englewood Cliffs, N. J., 1964).

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1964), Ch. 10, p. 491.

Burlamacchi, P.

P. Burlamacchi and A. Consortini, Opt. Acta. 14, 17 (1967).
[Crossref]

Cathey, W. T.

Consortini, A.

P. Burlamacchi and A. Consortini, Opt. Acta. 14, 17 (1967).
[Crossref]

A. Consortini, L. Ronchi, A. M. Scheggi, and G. Toraldo di Francia, Radio Science 1, 523 (1966).

A. Consortini, L. Ronchi, A. M. Scheggi, and G. Toraldo di Francia, Alta Frequenza 32, 178E (1963).

Cornacchio, J. V.

J. V. Cornacchio, J. Opt. Soc. Am. 57, 1325 (1967).
[Crossref] [PubMed]

J. V. Cornacchio and K. A. Farnham, Nuovo Cimento 42, 108 (1966).
[Crossref]

J. V. Cornacchio and R. P. Soni, Nuovo Cimento 38, 1169 (1965).
[Crossref]

Farnham, K. A.

J. V. Cornacchio and K. A. Farnham, Nuovo Cimento 42, 108 (1966).
[Crossref]

Françon, M.

M. Françon, Optical Interferometry (Academic Press Inc., New York, 1966), Chs. 1, 8, pp. 1, 162.

M. Françon, Diffraction; Coherence in Optics (Pergamon Press, Inc., New York, 1966).

M. Françon and S. Mallick, in Progress in Optics VI, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1967), Ch. 3, p. 73.

Fried, D. L.

Hopkins, H. H.

H. H. Hopkins, in Advances in Optical Techniques, A. C. S. van Heel, Ed. (North-Holland Publ. Co., Amsterdam, 1967), Ch. 6, p. 190.

Houston, A.

Izatt, J. R.

Kantor, K.

J. Bakos and K. Kantor, Nuovo Cimento 22, 519 (1961).
[Crossref]

Lansraux, G.

G. Lansraux, Rev. Opt. 32, 475 (1953).

Linfoot, E. H.

E. H. Linfoot and E. Wolf, Proc. Phys. Soc. (London) B66, 145 (1953). See also E. H. Linfoot, Recent Advances in Optics (Oxford at the Clarendon Press, 1955), p. 41.

Mallick, S.

M. Françon and S. Mallick, in Progress in Optics VI, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1967), Ch. 3, p. 73.

Meadors, J. G.

J. G. Meadors, Opt. Acta. 12, 379 (1965); and dissertation submitted to Ohio State University, Columbus, Ohio (1964).
[Crossref]

Mielenz, K. D.

O’Neill, E. L.

Parrent, G. B.

G. B. Parrent and T. J. Skinner, Opt. Acta. 8, 93 (1961).
[Crossref]

G. B. Parrent, J. Opt. Soc. Am. 49, 787 (1959); also Ref. 2, p. 40.
[Crossref]

M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Inc., Englewood Cliffs, N. J., 1964).

Poirier, J. L.

Roizen–Dossier, B.

B. Roizen–Dossier, Rev. Opt. 33, 57, 147, 267 (1954).

Ronchi, L.

A. Consortini, L. Ronchi, A. M. Scheggi, and G. Toraldo di Francia, Radio Science 1, 523 (1966).

A. Consortini, L. Ronchi, A. M. Scheggi, and G. Toraldo di Francia, Alta Frequenza 32, 178E (1963).

Röseler, A.

A. Röseler, Optik 27, 179 (1968).

Rotman, W.

Scheggi, A. M.

A. Consortini, L. Ronchi, A. M. Scheggi, and G. Toraldo di Francia, Radio Science 1, 523 (1966).

A. Consortini, L. Ronchi, A. M. Scheggi, and G. Toraldo di Francia, Alta Frequenza 32, 178E (1963).

Schell, A. C.

A. C. Schell, Ph.D. thesis, Massachusetts Institute of Technology (1961).

Shore, R. A.

R. A. Shore, B. J. Thompson, and R. E. Whitney, J. Opt. Soc. Am. 56, 733 (1966).
[Crossref]

J. L. Poirier, W. Rotman, and R. A. Shore, Appl. Opt. 4, 1321 (1965).
[Crossref]

R. A. Shore, Air Force Cambridge Research Laboratories Unclassified Research Report 62-715, August, 1962. Also in Proceedings of Symposium on Electro-magnetic Theory and Antennas, Copenhagen, Denmark, June, 1962, E. C. Jordan, Ed. (Pergamon Press (Ltd.)London, 1963), p. 787.

Skinner, T. J.

G. B. Parrent and T. J. Skinner, Opt. Acta. 8, 93 (1961).
[Crossref]

Soni, R. P.

J. V. Cornacchio and R. P. Soni, Nuovo Cimento 38, 1169 (1965).
[Crossref]

Steel, W. H.

W. H. Steel, Rev. Opt. 32, 4, 143, 269 (1953). Also thesis, University of Paris (1953).

W. H. Steel, Interferometry (Cambridge University Press, 1967).

Taylor, C. A.

Thompson, B. J.

Toraldo di Francia, G.

A. Consortini, L. Ronchi, A. M. Scheggi, and G. Toraldo di Francia, Radio Science 1, 523 (1966).

A. Consortini, L. Ronchi, A. M. Scheggi, and G. Toraldo di Francia, Alta Frequenza 32, 178E (1963).

G. Toraldo di Francia, Nuovo Cimento Suppl. 9, 426 (1952).
[Crossref]

Welford, W. T.

Whitney, R. E.

Wolf, E.

E. Wolf, Proc. Roy. Soc. (London) A230, 246 (1955).

E. H. Linfoot and E. Wolf, Proc. Phys. Soc. (London) B66, 145 (1953). See also E. H. Linfoot, Recent Advances in Optics (Oxford at the Clarendon Press, 1955), p. 41.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1964), Ch. 10, p. 491.

Alta Frequenza (1)

A. Consortini, L. Ronchi, A. M. Scheggi, and G. Toraldo di Francia, Alta Frequenza 32, 178E (1963).

Appl. Opt. (1)

J. Opt. Soc. Am. (13)

Nuovo Cimento (3)

J. Bakos and K. Kantor, Nuovo Cimento 22, 519 (1961).
[Crossref]

J. V. Cornacchio and R. P. Soni, Nuovo Cimento 38, 1169 (1965).
[Crossref]

J. V. Cornacchio and K. A. Farnham, Nuovo Cimento 42, 108 (1966).
[Crossref]

Nuovo Cimento Suppl. (1)

G. Toraldo di Francia, Nuovo Cimento Suppl. 9, 426 (1952).
[Crossref]

Opt. Acta. (3)

J. G. Meadors, Opt. Acta. 12, 379 (1965); and dissertation submitted to Ohio State University, Columbus, Ohio (1964).
[Crossref]

P. Burlamacchi and A. Consortini, Opt. Acta. 14, 17 (1967).
[Crossref]

G. B. Parrent and T. J. Skinner, Opt. Acta. 8, 93 (1961).
[Crossref]

Optik (1)

A. Röseler, Optik 27, 179 (1968).

Oyo Buturi (1)

R. Barakat and T. Asakura, Oyo Buturi 30, 728 (1961).

Proc. IEEE (1)

D. L. Fried, Proc. IEEE 55, 57 (1967).
[Crossref]

Proc. Phys. Soc. (London) (1)

E. H. Linfoot and E. Wolf, Proc. Phys. Soc. (London) B66, 145 (1953). See also E. H. Linfoot, Recent Advances in Optics (Oxford at the Clarendon Press, 1955), p. 41.

Proc. Roy. Soc. (London) (1)

E. Wolf, Proc. Roy. Soc. (London) A230, 246 (1955).

Radio Science (1)

A. Consortini, L. Ronchi, A. M. Scheggi, and G. Toraldo di Francia, Radio Science 1, 523 (1966).

Rev. Opt. (3)

W. H. Steel, Rev. Opt. 32, 4, 143, 269 (1953). Also thesis, University of Paris (1953).

G. Lansraux, Rev. Opt. 32, 475 (1953).

B. Roizen–Dossier, Rev. Opt. 33, 57, 147, 267 (1954).

Other (9)

A. C. Schell, Ph.D. thesis, Massachusetts Institute of Technology (1961).

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1964), Ch. 10, p. 491.

M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Inc., Englewood Cliffs, N. J., 1964).

M. Françon, Optical Interferometry (Academic Press Inc., New York, 1966), Chs. 1, 8, pp. 1, 162.

M. Françon, Diffraction; Coherence in Optics (Pergamon Press, Inc., New York, 1966).

W. H. Steel, Interferometry (Cambridge University Press, 1967).

H. H. Hopkins, in Advances in Optical Techniques, A. C. S. van Heel, Ed. (North-Holland Publ. Co., Amsterdam, 1967), Ch. 6, p. 190.

M. Françon and S. Mallick, in Progress in Optics VI, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1967), Ch. 3, p. 73.

R. A. Shore, Air Force Cambridge Research Laboratories Unclassified Research Report 62-715, August, 1962. Also in Proceedings of Symposium on Electro-magnetic Theory and Antennas, Copenhagen, Denmark, June, 1962, E. C. Jordan, Ed. (Pergamon Press (Ltd.)London, 1963), p. 787.

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Figures (10)

Fig. 1
Fig. 1

Coordinate system and the illustration of symbols used in various formulae.

Fig. 2
Fig. 2

Calculated far-field irradiance distribution of an annular aperture. Exponential correlation. α = 0, 0.5, 1.0, 2.0, and 5.0; for = 0.25.

Fig. 3
Fig. 3

Calculated far-field irradiance distribution of an annular aperture. Exponential correlation. α = 0, 0.5, 1.0, 2.0, and 5.0; for = 0.50.

Fig. 4
Fig. 4

Calculated far-field irradiance distribution of an annular aperture. Exponential correlation. α = 0, 0.5, 1.0, 2.0, and 5.0; for = 0.75.

Fig. 5
Fig. 5

Calculated far-field irradiance distribution of an annular aperture. Besinc form of correlation. α = 0, 1.0, 2.0, and 5.0; for = 0.25.

Fig. 6
Fig. 6

Calculated far-field irradiance distribution of an annular aperture. Besinc form of correlation. α = 0, 1.0, 2.0, and 5.0; for = 0.50.

Fig. 7
Fig. 7

Calculated far-field irradiance distribution of an annular aperture. Besinc form of correlation. α = 0, 1.0, 2.0, and 5.0; for = 0.75.

Fig. 8
Fig. 8

Comparison between the irradiance distribution for exponential and besinc correlation; for = 0.25.

Fig. 9
Fig. 9

Comparison between the irradiance distribution for exponential and besinc correlation. For = 0.50, α = 0.25.

Fig. 10
Fig. 10

Comparison between the irradiance distribution for exponential and besinc correlation. For = 0.75, α = 0.25.

Equations (13)

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Γ ( P 1 , P 2 , τ ) = [ exp ( - 2 π i ν ¯ τ ) / ( 2 π ) 2 ] × σ σ ( 1 - i k ¯ r 1 ) ( 1 + i k ¯ r 2 ) cos θ 1 cos θ 2 r 1 2 r 2 2 × Γ ( S 1 , S 2 , 0 ) exp [ i k ¯ ( r 1 - r 2 ) ] d σ 1 d σ 2
I ( P ) = cos 2 θ / ( λ ¯ 2 R 2 ) σ σ Γ ( S 1 , S 2 , 0 ) × exp [ i k ¯ sin θ p ˆ ( S 2 - S 1 ) ] d σ 1 d σ 2 .
Γ ( S 1 , S 2 , 0 ) = I ( S 1 ) 1 2 I ( S 2 ) 1 2 γ ( S 2 - S 1 , 0 ) .
I ( P ) = A     cos 2 θ λ ¯ 2 R 2 σ s C ( S ) γ ( S ) exp ( i k ¯ sin θ p ˆ · S ) d S ,
C ( S ) = ( 1 / A ) σ ( s ) I ( S 1 ) 1 2 I ( S 1 + S ) 1 2 d σ ,
I ( θ ) = ( π a 2 / λ ¯ 2 ) 0 2 a d ρ 0 2 π d ϕ ρ γ ( ρ ) C ( ρ ) × exp [ i k ¯ θ ρ cos ϕ ] ,
I ( θ ) = ( 2 π 2 a 2 / λ ¯ 2 ) 0 2 ρ d ρ γ ( a ρ ) c ( ρ ) J 0 ( k a θ ρ ) .
C ( ρ ) = [ π ( 1 - 2 ) ] - 1 ( A + B + C ) ,
A = 2 [ arc cos ( ρ / 2 ) - ( ρ / 2 ) sin arc cos ( ρ / 2 ) ] 0 ρ 2 = 0 ρ 2 B = 2 2 [ arc cos ( ρ / 2 ) - ( ρ / 2 ) sin arc cos ( ρ / 2 ) ] 0 ρ 2 = 0 ρ 2
C = - 2 π 2 0 ρ 1 - = - 2 π 2 + 2 sin arc cos Ω + ( 1 + 2 ) × arc cos Ω - 2 ( 1 - 2 ) arc tan [ ( 1 + ) / ( 1 - ) tan arc cos Ω ] 1 - ρ 1 - = 0 , ρ 1 +
γ ( a ρ ) = exp ( - ρ / L ) = exp ( - α ρ )             ( exponential correlation ) γ ( a ρ ) = 2 J 1 ( ρ / L ) / ( ρ / L ) = 2 J 1 ( α ρ ) / ( α ρ )             ( besinc correlation ) γ ( a ρ ) = sin ( ρ / L ) / ( ρ / L ) = sinc ( α ρ ) / ( α ρ )             ( sinc correlation ) γ ( a ρ ) = exp ( - ρ 2 / L 2 ) = exp ( - α 2 ρ 2 ) ,
α = 2 π a r / ( λ f 1 )
I ( θ ) = I 0 ( 1 - 2 ) 2 [ 2 J 1 ( k a θ ) k a θ - 2 { 2 J 1 ( k a θ ) k a θ } ] 2 .