Abstract

For band-limited optical systems, necessary and sufficient conditions have been obtained that enable us to characterize the mutual coherence functions of all partially coherent quasi-monochromatic sources that generate the same intensity at the output. Throughout this work, two different cases have been considered: In the first one, the general equation that links the output intensity with the mutual coherence function at the input plane has been assumed to be valid over the entire output plane, whereas in the second case its validity has been restricted to a finite region of this plane. The common characteristics of the sources may be directly calculated from the output-intensity data. Comparison with other previously reported results has been done. To elucidate whether a function really represents a true mutual coherence function, a general criterion has been established. Finally, from any given intensity distribution at the output plane, we have determined the explicit analytical expression of the mutual coherence function of a source that can reproduce the same output-intensity data.

© 1985 Optical Society of America

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References

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  1. J. Perina, Coherence of Light (Van Nostrand, London, 1971).
  2. In order to simplify the expressions, in this work we shall consider unidimensional variables. However, the results that we shall obtain can be extended in straightforward way to the bidimensional case.
  3. E. Wolf, “The radiant intensity from planar sources of any state of coherence,” J. Opt. Soc. Am. 68, 1597–1605 (1978).
    [CrossRef]
  4. E. Collet, E. Wolf, “New equivalence theorems for planar sources that generate the same distributions of radiant intensity,” J. Opt. Soc. Am. 69, 942–950 (1979).
    [CrossRef]
  5. R. Martínez-Herrero, P. M. Mejías, “Relation between planar sources that generate the same intensity distribution at the output plane,” J. Opt. Soc. Am. 72, 131–135 (1982).
    [CrossRef]
  6. R. Martínez-Herrero, P. M. Mejías, “Relation among planar sources that generate the same radiant intensity at the output of a general optical system,” J. Opt. Soc. Am. 72, 765–769 (1982).
    [CrossRef]
  7. A system is said to be band limited if the Fourier transform of its integral kernel possess bounded support. Many common optical systems satisfy this condition (see, for example, Refs. 8 and 9).
  8. R. Martínez-Herrero, M. Arenas, “Obtaining any given intensity at discrete close points at the output of band-limited optical systems,” Opt. Acta 31, 467–470 (1984).
    [CrossRef]
  9. R. Martínez-Herrero, P. M. Mejías, “Uniqueness in the inverse problem for homogeneous sources. Application to some optical systems,” Opt. Acta 31, 917–922 (1984).
    [CrossRef]
  10. S. Goldman, Information Theory (Dover, New York, 1968).
  11. H. Gamo, “Matrix treatment of partial coherence,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Vol. III, pp. 187–332.
    [CrossRef]
  12. It has been taken into account that the sinc functions appearing in Eq. (3) are orthogonal in ℝ.
  13. To prove the proposition, no explicit insertion of these coefficients is required.
  14. A. C. Zaanen, Linear Analysis (North-Holland, Amsterdam, 1960).
  15. This set of functions is also called the null space of the operator and is commonly denoted by ker{ }.
  16. R. Martínez-Herrero, “Expansion of complex degree of coherence,” Nuovo Cimento 54, 205–210 (1979).
  17. In the general case, a procedure similar to that shown in the present paper may be also followed: In fact, it will suffice to select from the global set of functions {K(n/4B, x), n an integer} the complete collection of linearly independent functions and apply to them an orthonormalization method. The resultant functions can be used in the same way as the family {φn(x)} in Eqs. (36)–(38). Note that, in some particular cases, all the functions K(n/4B, x) are linearly independent: This can easily be shown, for example, for any isoplanatic optical system [i.e., a system whose kernel is of the form K(y− x)] when Ω extends from −∞ to +∞.

1984 (2)

R. Martínez-Herrero, M. Arenas, “Obtaining any given intensity at discrete close points at the output of band-limited optical systems,” Opt. Acta 31, 467–470 (1984).
[CrossRef]

R. Martínez-Herrero, P. M. Mejías, “Uniqueness in the inverse problem for homogeneous sources. Application to some optical systems,” Opt. Acta 31, 917–922 (1984).
[CrossRef]

1982 (2)

1979 (2)

1978 (1)

Arenas, M.

R. Martínez-Herrero, M. Arenas, “Obtaining any given intensity at discrete close points at the output of band-limited optical systems,” Opt. Acta 31, 467–470 (1984).
[CrossRef]

Collet, E.

Gamo, H.

H. Gamo, “Matrix treatment of partial coherence,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Vol. III, pp. 187–332.
[CrossRef]

Goldman, S.

S. Goldman, Information Theory (Dover, New York, 1968).

Martínez-Herrero, R.

R. Martínez-Herrero, M. Arenas, “Obtaining any given intensity at discrete close points at the output of band-limited optical systems,” Opt. Acta 31, 467–470 (1984).
[CrossRef]

R. Martínez-Herrero, P. M. Mejías, “Uniqueness in the inverse problem for homogeneous sources. Application to some optical systems,” Opt. Acta 31, 917–922 (1984).
[CrossRef]

R. Martínez-Herrero, P. M. Mejías, “Relation between planar sources that generate the same intensity distribution at the output plane,” J. Opt. Soc. Am. 72, 131–135 (1982).
[CrossRef]

R. Martínez-Herrero, P. M. Mejías, “Relation among planar sources that generate the same radiant intensity at the output of a general optical system,” J. Opt. Soc. Am. 72, 765–769 (1982).
[CrossRef]

R. Martínez-Herrero, “Expansion of complex degree of coherence,” Nuovo Cimento 54, 205–210 (1979).

Mejías, P. M.

Perina, J.

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

Wolf, E.

Zaanen, A. C.

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

J. Opt. Soc. Am. (4)

Nuovo Cimento (1)

R. Martínez-Herrero, “Expansion of complex degree of coherence,” Nuovo Cimento 54, 205–210 (1979).

Opt. Acta (2)

R. Martínez-Herrero, M. Arenas, “Obtaining any given intensity at discrete close points at the output of band-limited optical systems,” Opt. Acta 31, 467–470 (1984).
[CrossRef]

R. Martínez-Herrero, P. M. Mejías, “Uniqueness in the inverse problem for homogeneous sources. Application to some optical systems,” Opt. Acta 31, 917–922 (1984).
[CrossRef]

Other (10)

S. Goldman, Information Theory (Dover, New York, 1968).

H. Gamo, “Matrix treatment of partial coherence,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Vol. III, pp. 187–332.
[CrossRef]

It has been taken into account that the sinc functions appearing in Eq. (3) are orthogonal in ℝ.

To prove the proposition, no explicit insertion of these coefficients is required.

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

This set of functions is also called the null space of the operator and is commonly denoted by ker{ }.

In the general case, a procedure similar to that shown in the present paper may be also followed: In fact, it will suffice to select from the global set of functions {K(n/4B, x), n an integer} the complete collection of linearly independent functions and apply to them an orthonormalization method. The resultant functions can be used in the same way as the family {φn(x)} in Eqs. (36)–(38). Note that, in some particular cases, all the functions K(n/4B, x) are linearly independent: This can easily be shown, for example, for any isoplanatic optical system [i.e., a system whose kernel is of the form K(y− x)] when Ω extends from −∞ to +∞.

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

In order to simplify the expressions, in this work we shall consider unidimensional variables. However, the results that we shall obtain can be extended in straightforward way to the bidimensional case.

A system is said to be band limited if the Fourier transform of its integral kernel possess bounded support. Many common optical systems satisfy this condition (see, for example, Refs. 8 and 9).

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Equations (46)

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I ( y ) = Ω Ω K * ( y , x 1 ) K ( y , x 2 ) Γ ( x 1 , x 2 ) d x 1 d x 2 ,
K * ( y , x 1 ) K ( y , x 2 ) = n sinc [ 4 π B ( y n 4 B ) ] × K * ( n 4 B , x 1 ) K ( n 4 B , x 2 ) ,
I ( y ) = n Γ n n sinc [ 4 π B ( y n 4 B ) ] ,
Γ n n = Ω Ω K * ( n 4 B , x 1 ) K ( n 4 B , x 2 ) Γ ( x 1 , x 2 ) d x 1 d x 2 .
I ( n 4 B ) = Ω Ω K * ( n 4 B , x 1 ) K ( n 4 B , x 2 ) Γ ( x 1 , x 2 ) d x 1 d x 2
Γ n n = I ( n 4 B ) .
u n ( y ) = a C a ( n ) sinc [ 4 π B ( y a 4 B ) ] ,
sinc [ 4 π B ( y k 4 B ) ] = b D b ( k ) u b ( y ) ,
b D b ( k ) C a ( b ) = δ k a ,
K * ( y , x 1 ) K ( y , x 2 ) = n , b D b ( n ) u b ( y ) K * ( n 4 B , x 1 ) K ( n 4 B , x 2 ) ,
I ( y ) = n , b Γ n n D b ( n ) u b ( y ) ,
k C k ( b ) A A I ( y ) u b ( y ) d y = n , k C k ( b ) D b ( n ) Γ n n = Γ k k ,
( L ̂ f ) ( x 1 , x 2 ) = K ( y , x 1 ) K * ( y , x 2 ) K * ( y , z 1 ) × K ( y , z 2 ) f ( z 1 , z 2 ) d z 1 d z 2 d y .
f n ( x 1 , x 2 ) = m f n ( m ) K ( m 4 B , x 1 ) K * ( m 4 B , x 2 ) ,
f 0 ( 1 ) ( x 1 , x 2 ) = n [ Γ ( 1 ) , f n ] f n ( x 1 , x 2 ) ,
f 0 ( 2 ) ( x 1 , x 2 ) = n [ Γ ( 2 ) , f n ] f n ( x 1 , x 2 ) ,
[ Γ ( i ) , f n ] Ω Ω Γ ( i ) ( x 1 , x 2 ) f n * ( x 1 , x 2 ) d x 1 d x 2 , i = 1 , 2 .
[ Γ ( i ) , f n ] = m [ f n ( m ) ] * Γ m m ( i ) , i = 1 , 2 ,
f 0 ( 1 ) ( x 1 , x 2 ) = f 0 ( 2 ) ( x 1 , x 2 ) .
[ Γ ( 1 ) , f n ] = [ Γ ( 2 ) , f n ] , n .
K * ( m 4 B , x 1 ) K ( m 4 B , x 2 ) = n g n ( m ) f n ( x 1 , x 2 ) ,
Γ m m ( i ) = n g n ( m ) [ Γ ( i ) , f n ] , i = 1 , 2 ,
Γ m m ( 1 ) = Γ m m ( 2 ) . Q . E . D .
Γ * ( x 1 , x 2 ) = Γ ( x 2 , x 1 ) ( hermiticity ) ,
i , j Γ ( x i , x j ) a i * a j 0 , a i , a j ( nonnegative definiteness ) .
Γ ( x 1 , x 2 ) = A H * ( x 1 , σ ) H ( x 2 , σ ) d σ .
Γ ( x 1 , x 2 ) = n λ n 2 φ n * ( x 1 ) φ n ( x 2 ) ,
( Γ ̂ f ) ( x ) = A Γ ( σ , x ) f ( σ ) d σ .
H ( x , σ ) = n λ n φ n ( x ) φ n * ( σ ) ,
Γ ( x 1 , x 2 ) = A H * ( x 1 , σ ) H ( x 2 , σ ) d σ ,
Γ * ( x 1 , x 2 ) = Γ ( x 2 , x 1 ) ( trivial )
i , j Γ ( x i , x j ) a i * a j = i , j A H * ( x i , σ ) a i * H ( x j , σ ) a j d σ = A [ i H ( x i , σ ) a i ] * [ j H ( x j , σ ) a j ] d σ 0 ,
φ n ( x ) = a A a ( n ) K ( a 4 B , x ) ,
K ( a 4 B , x ) = k B k ( a ) φ k ( x ) ,
k A b ( k ) B k ( a ) = δ a b ,
H 0 ( x , σ ) = a , p Y ( p 4 B ) A p ( a ) φ a ( x ) φ p * ( σ ) ,
| Y ( p / 4 B ) | 2 = I ( p / 4 B ) .
Γ 0 ( x 1 , x 2 ) = Ω H 0 * ( x 1 , σ ) H 0 ( x 2 , σ ) d σ = a , b , p I ( p / 4 B ) [ A p ( a ) ] * A p ( b ) φ a * ( x 1 ) φ b ( x 2 )
I o ( y ) Ω Ω K * ( y , x 1 ) K ( y , x 2 ) Γ 0 ( x 1 , x 2 ) d x 1 d x 2 = n , i , j , a , b , p sin [ 4 π B ( y n 4 B ) ] [ B i ( n ) ] * B j ( n ) × I ( p / 4 B ) [ A p ( a ) ] * A p ( b ) δ a i δ b j = n , a , b , p sinc [ 4 π B ( y n 4 B ) ] [ B a ( n ) ] * B b ( n ) × [ A p ( a ) ] * A p ( b ) I ( p / 4 B ) .
I 0 ( y ) = n I ( n 4 B ) sinc [ 4 π B ( y n 4 B ) ] = I ( y ) ,
H 0 ( x , σ ) = a , p α p A p ( a ) φ a ( x ) φ p * ( σ ) ,
| α p | 2 = k F k C p ( k )
F k = A + A I ( y ) u k ( y ) d y ,
Γ 0 ( x 1 , x 2 ) = Ω H 0 * ( x 1 , σ ) H 0 ( x 2 , σ ) d σ = a , p , c | α p | 2 [ A p ( a ) ] * A p ( c ) φ a * ( x 1 ) φ c ( x 2 )
I 0 ( y ) Ω Ω K * ( y , x 1 ) K ( y , x 2 ) Γ 0 ( x 1 , x 2 ) d x 1 d x 2 = a , b , c , n , p D b ( n ) | α p | 2 [ A p ( a ) ] * A p ( c ) [ B a ( n ) ] * B c ( n ) u b ( y ) = b , n , p D b ( n ) | α p | 2 δ n p u p ( y ) = b , n D b ( n ) | α n | 2 u b ( y ) ,
I 0 ( y ) = b , n , k D b ( n ) C n ( k ) F k u b ( y ) = b , k F k δ b k u b ( y ) = b F b u b ( y ) = I ( y ) . Q . E . D .

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