Abstract

A general explicit form of the correlation functions of all the partially coherent quasi-monochromatic sources that generate identical intensity distributions at the far (Fraunhofer) zone is given. The common characteristic part of all of these correlation functions is pointed out. Also, the possibility is shown for reconstructing (in unique way), from intensity data at the far zone, any source whose correlation function at some region Ω depends on the coordinate difference only.

© 1981 Optical Society of America

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References

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  1. E. Wolf, J. Opt. Soc. Am. 68, 1597–1605 (1978).
    [Crossref]
  2. E. Collet, E. Wolf, Opt. Lett. 2, 27–29 (1978).
    [Crossref]
  3. E. Collet, E. Wolf, J. Opt. Soc. Am. 69, 943–950 (1979).
    [Crossref]
  4. sΓi (r1, r2) denotes the mutual coherence function at τ = 0.
  5. J. Perina, V. Perinova, Opt. Acta 16, 309–320 (1969).
    [Crossref]
  6. B. J. Thompson, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1969), Vol. VII, p. 171.
  7. A. C. Zaanen, Linear Analysis (North-Holland, Amsterdam, 1960).
  8. Since the eigenfunctions of Ĥ′Ĥ generate the subspace orthogonal to the kernel of Ĥ′Ĥ, any function Pi(r1, r2) may be written as sum (finite or infinite) of functions orthogonal to all such eigenfunctions.
  9. R. P. Kanwal, Linear Integral Equations (Academic, New York, 1971).
  10. J. Perina, Coherence of Light (Van Nostrand, London, 1971).
  11. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980).
  12. H. Gamo, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Vol. III, p. 189.
  13. B. E. A. Saleh, Photoelectron Statistics (Springer-Verlag, Berlin, 1978).
  14. R. Marímez-Herrero, Nuovo Cimento 54, 205–210 (1979).
  15. M. J. Beran, G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, N.J., 1974).

1979 (2)

E. Collet, E. Wolf, J. Opt. Soc. Am. 69, 943–950 (1979).
[Crossref]

R. Marímez-Herrero, Nuovo Cimento 54, 205–210 (1979).

1978 (2)

1969 (1)

J. Perina, V. Perinova, Opt. Acta 16, 309–320 (1969).
[Crossref]

Beran, M. J.

M. J. Beran, G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, N.J., 1974).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980).

Collet, E.

E. Collet, E. Wolf, J. Opt. Soc. Am. 69, 943–950 (1979).
[Crossref]

E. Collet, E. Wolf, Opt. Lett. 2, 27–29 (1978).
[Crossref]

Gamo, H.

H. Gamo, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Vol. III, p. 189.

Kanwal, R. P.

R. P. Kanwal, Linear Integral Equations (Academic, New York, 1971).

Marímez-Herrero, R.

R. Marímez-Herrero, Nuovo Cimento 54, 205–210 (1979).

Parrent, G. B.

M. J. Beran, G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, N.J., 1974).

Perina, J.

J. Perina, V. Perinova, Opt. Acta 16, 309–320 (1969).
[Crossref]

J. Perina, Coherence of Light (Van Nostrand, London, 1971).

Perinova, V.

J. Perina, V. Perinova, Opt. Acta 16, 309–320 (1969).
[Crossref]

Saleh, B. E. A.

B. E. A. Saleh, Photoelectron Statistics (Springer-Verlag, Berlin, 1978).

Thompson, B. J.

B. J. Thompson, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1969), Vol. VII, p. 171.

Wolf, E.

E. Collet, E. Wolf, J. Opt. Soc. Am. 69, 943–950 (1979).
[Crossref]

E. Collet, E. Wolf, Opt. Lett. 2, 27–29 (1978).
[Crossref]

E. Wolf, J. Opt. Soc. Am. 68, 1597–1605 (1978).
[Crossref]

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980).

Zaanen, A. C.

A. C. Zaanen, Linear Analysis (North-Holland, Amsterdam, 1960).

J. Opt. Soc. Am. (2)

E. Wolf, J. Opt. Soc. Am. 68, 1597–1605 (1978).
[Crossref]

E. Collet, E. Wolf, J. Opt. Soc. Am. 69, 943–950 (1979).
[Crossref]

Nuovo Cimento (1)

R. Marímez-Herrero, Nuovo Cimento 54, 205–210 (1979).

Opt. Acta (1)

J. Perina, V. Perinova, Opt. Acta 16, 309–320 (1969).
[Crossref]

Opt. Lett. (1)

Other (10)

sΓi (r1, r2) denotes the mutual coherence function at τ = 0.

M. J. Beran, G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, N.J., 1974).

B. J. Thompson, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1969), Vol. VII, p. 171.

A. C. Zaanen, Linear Analysis (North-Holland, Amsterdam, 1960).

Since the eigenfunctions of Ĥ′Ĥ generate the subspace orthogonal to the kernel of Ĥ′Ĥ, any function Pi(r1, r2) may be written as sum (finite or infinite) of functions orthogonal to all such eigenfunctions.

R. P. Kanwal, Linear Integral Equations (Academic, New York, 1971).

J. Perina, Coherence of Light (Van Nostrand, London, 1971).

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980).

H. Gamo, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Vol. III, p. 189.

B. E. A. Saleh, Photoelectron Statistics (Springer-Verlag, Berlin, 1978).

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Figures (1)

Fig. 1
Fig. 1

Illustration of the notation used in this work.

Equations (33)

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I ( r ) = A Ω Ω Γ s i ( r 1 , r 2 ) × exp [ i k r D ( r 2 r 1 ) ] d 2 r 2 d 2 r 1 , i = 1 , 2
A = k 2 cos 2 θ / D 2 ,
( Ĥ a ) ( r ) = A Ω Ω exp [ i k r D ( r 2 r 1 ) ] a ( r 1 , r 2 ) d 2 r 1 d 2 r 2
( Ĥ b ) ( r 1 , r 2 ) = A Ω Ω exp [ i k r D ( r 2 r 1 ) ] b ( r ) d 2 r .
Ĥ Ĥ f n = λ n 2 f n ; Ĥ Ĥ g n = λ n 2 g n , Ĥ f n = λ n g n ; Ĥ g n = λ n f n .
Γ i S ( r 1 , r 2 ) = φ ( r 1 , r 2 ) + P i ( r 1 , r 2 ) , i = 1 , 2 , 3 , ( r 1 , r 2 ) Ω × Ω ,
Ω Ω exp [ i k r D ( r 2 r 1 ) ] P i ( r 1 , r 2 ) d 2 r 1 d 2 r 2 = 0
Ω Ω φ ( r 1 , r 2 ) P i ( r 1 , r 2 ) * d 2 r 1 d 2 r 2 = 0 ,
φ ( r 1 , r 2 ) = A Ω ( r ) exp [ i k r D ( r 2 r 1 ) ] d 2 r ,
I ( r ) = ( Ĥ Ĥ ) ( r ) .
Γ i s ( r 1 , r 2 ) * = Γ i s ( r 2 , r 1 ) ,
P i ( r 1 , r 2 ) * = P i ( r 2 , r 1 ) .
( i P ̂ a ) ( r 1 ) = Ω P i ( r 1 , r 2 ) a ( r 2 ) d 2 r 1 .
P i ( r 1 , r 2 ) = n S n i a n i ( r 1 ) a n i ( r 2 ) * .
n S n i | b n i ( r ) | 2 = 0 , r ,
b n i ( r ) = Ω exp ( i k r r 1 D ) a n i ( r 1 ) d 2 r 1 .
Γ i s ( r 1 , r 2 ) = Ω ( r ) exp { i k r D ( r 2 r 1 ) ] d 2 r + n S n i a n i ( r 1 ) a n i ( r 2 ) * ,
r 2 r 1 = r , r 2 + r 1 = r .
exp ( i k r r D ) g ( r ) P ( r ) d 2 r = 0 ,
g ( r ) = { ( 2 a x ) ( 2 a y ) , 2 a < x < 0 , 2 a < y < 0 ( 2 a x ) ( 2 a + y ) , 2 a < x < 0 , 0 > y > 2 a ( 2 a + x ) ( 2 a y ) , 0 > x > 2 a , 2 a < y < 0 ( 2 a + x ) ( 2 a + y ) , 0 > x > 2 a , 0 > y > 2 a 0 otherwise
g ( r ) P ( r ) = 0 ,
P ( r 2 r 1 ) = 0 ,
Ĥ ( Γ 1 s Γ 2 s ) = 0 .
P ( r 1 , r 2 ) = Γ i s ( r 1 , r 2 ) Γ 2 s ( r 1 , r 2 ) ,
Γ i = p ˜ Γ i s + Q ˜ Γ i s , i = 1 , 2 P ˜ P ( r 1 , r 2 ) = 0 s .
p ˜ Γ 1 s P ˜ Γ 2 s = 0 ,
φ ( r 1 , r 2 ) = ( p ˜ Γ i s ) ( r 1 , r 2 ) = n ( Γ i s , f n ) f n ( r 1 , r 2 )
f n ( r 1 , r 2 ) = A λ n Ω g n ( r ) exp [ i k r D ( r 2 r 1 ) ] d 2 r ,
φ ( r 1 , r 2 ) = A Ω ( r ) exp [ i k r D ( r 2 r 1 ) ] d 2 r ,
( r ) = n ( s Γ i , f n ) λ n g n ( r ) ,
Ĥ φ + Ĥ P 1 = Ĥ φ + Ĥ P 2 .
Ĥ Ĥ = I . Q . E . D .
Ĥ Ĥ υ = 0 υ = 0 ,

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