Abstract

Starting from the Culgoora paradox, the number of degrees of freedom of an image from a pupil of point-like elements has been determined. The influence of the pupil geometry and of the coherence characteristics has been examined. The results show that striking differences can exist between coherent and incoherent illumination.

© 1971 Optical Society of America

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References

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  1. B. Y. Mills and A. G. Little, Australian J. Phys. 6, 272 (1953).
    [Crossref]
  2. J. P. Wild, Proc. Roy. Soc. (London) A286, 499 (1965).
  3. D. Gabor, Rept. Progr. Phys. 32, 395 (1969).
    [Crossref]
  4. D. J. Cronin and G. O. Reynolds, J. Opt. Soc. Am. 59, 501A (1969).
  5. G. O. Reynolds and D. J. Cronin, J. Opt. Soc. Am. 60, 634 (1970).
    [Crossref]
  6. D. Yansen, G. O. Reynolds, and D. J. Cronin, J. Opt. Soc. Am. 60, 737A (1970).
  7. G. Toraldo di Francia, J. Opt. Soc. Am. 59, 799 (1969).
    [Crossref] [PubMed]
  8. A. Walther, J. Opt. Soc. Am. 60, 141 (1970).
    [Crossref]
  9. L. Mertz, Opt. Acta 16, 809 (1969).
    [Crossref]
  10. F. Gori and G. Guattari, Phys. Letters 31A, 131 (1970).
  11. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1969).
  12. As one of the referees noted, the use of the arrays in the radio-astronomy case presents some differences from the optical case. For instance, the interference between any two points of the array can be observed independent from the other points of the array. This is not the case in the optical analog.
  13. A. T. Moffet, IEEE Trans. AP-16, 172 (1968).
    [Crossref]
  14. F. Gori and G. Guattari, Phys. Letters 32A, 38 (1970).
  15. For the difference between the number of degrees of freedom for coherent or incoherent fields, see H. Gamo, in Progress in Optics III, edited by E. Wolf, (North-Holland, Amsterdam, 1964).

1970 (5)

G. O. Reynolds and D. J. Cronin, J. Opt. Soc. Am. 60, 634 (1970).
[Crossref]

D. Yansen, G. O. Reynolds, and D. J. Cronin, J. Opt. Soc. Am. 60, 737A (1970).

A. Walther, J. Opt. Soc. Am. 60, 141 (1970).
[Crossref]

F. Gori and G. Guattari, Phys. Letters 31A, 131 (1970).

F. Gori and G. Guattari, Phys. Letters 32A, 38 (1970).

1969 (4)

L. Mertz, Opt. Acta 16, 809 (1969).
[Crossref]

G. Toraldo di Francia, J. Opt. Soc. Am. 59, 799 (1969).
[Crossref] [PubMed]

D. Gabor, Rept. Progr. Phys. 32, 395 (1969).
[Crossref]

D. J. Cronin and G. O. Reynolds, J. Opt. Soc. Am. 59, 501A (1969).

1968 (1)

A. T. Moffet, IEEE Trans. AP-16, 172 (1968).
[Crossref]

1965 (1)

J. P. Wild, Proc. Roy. Soc. (London) A286, 499 (1965).

1953 (1)

B. Y. Mills and A. G. Little, Australian J. Phys. 6, 272 (1953).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1969).

Cronin, D. J.

D. Yansen, G. O. Reynolds, and D. J. Cronin, J. Opt. Soc. Am. 60, 737A (1970).

G. O. Reynolds and D. J. Cronin, J. Opt. Soc. Am. 60, 634 (1970).
[Crossref]

D. J. Cronin and G. O. Reynolds, J. Opt. Soc. Am. 59, 501A (1969).

Gabor, D.

D. Gabor, Rept. Progr. Phys. 32, 395 (1969).
[Crossref]

Gamo, H.

For the difference between the number of degrees of freedom for coherent or incoherent fields, see H. Gamo, in Progress in Optics III, edited by E. Wolf, (North-Holland, Amsterdam, 1964).

Gori, F.

F. Gori and G. Guattari, Phys. Letters 32A, 38 (1970).

F. Gori and G. Guattari, Phys. Letters 31A, 131 (1970).

Guattari, G.

F. Gori and G. Guattari, Phys. Letters 31A, 131 (1970).

F. Gori and G. Guattari, Phys. Letters 32A, 38 (1970).

Little, A. G.

B. Y. Mills and A. G. Little, Australian J. Phys. 6, 272 (1953).
[Crossref]

Mertz, L.

L. Mertz, Opt. Acta 16, 809 (1969).
[Crossref]

Mills, B. Y.

B. Y. Mills and A. G. Little, Australian J. Phys. 6, 272 (1953).
[Crossref]

Moffet, A. T.

A. T. Moffet, IEEE Trans. AP-16, 172 (1968).
[Crossref]

Reynolds, G. O.

D. Yansen, G. O. Reynolds, and D. J. Cronin, J. Opt. Soc. Am. 60, 737A (1970).

G. O. Reynolds and D. J. Cronin, J. Opt. Soc. Am. 60, 634 (1970).
[Crossref]

D. J. Cronin and G. O. Reynolds, J. Opt. Soc. Am. 59, 501A (1969).

Toraldo di Francia, G.

Walther, A.

Wild, J. P.

J. P. Wild, Proc. Roy. Soc. (London) A286, 499 (1965).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1969).

Yansen, D.

D. Yansen, G. O. Reynolds, and D. J. Cronin, J. Opt. Soc. Am. 60, 737A (1970).

Australian J. Phys. (1)

B. Y. Mills and A. G. Little, Australian J. Phys. 6, 272 (1953).
[Crossref]

IEEE Trans. (1)

A. T. Moffet, IEEE Trans. AP-16, 172 (1968).
[Crossref]

J. Opt. Soc. Am. (5)

D. J. Cronin and G. O. Reynolds, J. Opt. Soc. Am. 59, 501A (1969).

G. O. Reynolds and D. J. Cronin, J. Opt. Soc. Am. 60, 634 (1970).
[Crossref]

D. Yansen, G. O. Reynolds, and D. J. Cronin, J. Opt. Soc. Am. 60, 737A (1970).

G. Toraldo di Francia, J. Opt. Soc. Am. 59, 799 (1969).
[Crossref] [PubMed]

A. Walther, J. Opt. Soc. Am. 60, 141 (1970).
[Crossref]

Opt. Acta (1)

L. Mertz, Opt. Acta 16, 809 (1969).
[Crossref]

Phys. Letters (2)

F. Gori and G. Guattari, Phys. Letters 31A, 131 (1970).

F. Gori and G. Guattari, Phys. Letters 32A, 38 (1970).

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

J. P. Wild, Proc. Roy. Soc. (London) A286, 499 (1965).

Rept. Progr. Phys. (1)

D. Gabor, Rept. Progr. Phys. 32, 395 (1969).
[Crossref]

Other (3)

For the difference between the number of degrees of freedom for coherent or incoherent fields, see H. Gamo, in Progress in Optics III, edited by E. Wolf, (North-Holland, Amsterdam, 1964).

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1969).

As one of the referees noted, the use of the arrays in the radio-astronomy case presents some differences from the optical case. For instance, the interference between any two points of the array can be observed independent from the other points of the array. This is not the case in the optical analog.

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Figures (1)

F. 1
F. 1

Coordinate planes of a generalized optical system.

Equations (16)

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I ( x ) = + Γ ( ξ 1 , ξ 2 ) K ( ξ 1 ) K * ( ξ 2 ) × exp [ 2 π i ( ξ 1 ξ 2 ) · x / λ f ] d ξ 1 d ξ 2 ,
Γ ( ξ 1 , ξ 2 ) = + Γ ( y 1 , y 2 ) t ( y 1 ) t * ( y 2 ) × exp [ 2 π i λ f ( y 1 · ξ 1 y 2 · ξ 2 ) ] d y 1 d y 2 .
Γ ( y 1 , y 2 ) = + I ( σ ) exp [ 2 π i λ f ( y 1 y 2 ) · σ ] d σ .
I ( x ) = + K ( ξ 1 ) K * ( ξ 2 ) exp [ 2 π i λ f ( ξ 1 ξ 2 ) · x ] × [ + I ( σ ) t ( σ + ξ 1 ) t * ( σ + ξ 2 ) d σ ] d ξ 1 d ξ 2 .
K ( ξ ) = α = 1 n K α δ ( ξ ξ α ) .
I ( x ) = α = 1 n β = 1 n exp [ 2 π i λ f ( ξ α ξ β ) · x ] K α K β * × + I ( σ ) t ( ξ α + σ ) t * ( ξ β + σ ) d σ = I 0 + 2 α > β 1 , n I α , β cos [ 2 π λ f ( ξ α ξ β ) · x + φ α , β ] ,
I 0 = α = 1 n | K α | 2 + I ( σ ) | t ( σ + ξ α ) | 2 d σ [ I α , β exp ( i φ α , β ) ] α β = K α K β * + I ( σ ) t ( σ + ξ α ) t * ( σ + ξ β ) d σ .
N L = n ( n 1 ) + 1 .
N max = 1 + 2 ( n a 1 ) / 2 = n a .
η = N max / N L .
η = n 2 / 2 + 1 n ( n 1 ) + 1 ,
η = ( 2 n 1 ) / [ n ( n 1 ) + 1 ] ,
I 0 = α = 1 n | K α t ( ξ α ) | 2 [ α , β exp ( i φ α , β ) ] α β = K α t ( ξ α ) K β * t * ( ξ β ) = | K α t ( ξ α ) | | K β t ( ξ β ) | × exp { i [ arg K α t ( ξ α ) arg K β t ( ξ β ) ] } .
I 0 = α = 1 n | K α | 2 I + | t ( σ ) | 2 d σ [ I α , β exp ( i φ α , β ) ] α β = K α K β * I + t ( σ ) t * ( σ + ξ α ξ β ) d σ ,
I ( σ ) = γ = 1 m I γ δ ( σ σ γ ) ,
I 0 = α = 1 n | K α | 2 γ = 1 m I γ | t ( σ γ + ξ α ) | 2 [ I α , β exp ( i φ α , β ) ] α β = K α K β * γ = 1 m I γ t ( σ γ + ξ α ) t * ( σ γ + ξ β ) = K α K β * γ = 1 m I γ | t ( σ γ + ξ α ) | | t ( σ γ + ξ β ) | × exp { i [ arg t ( σ γ + ξ α ) arg t ( σ γ + ξ β ) ] } .