Abstract

Some general theorems are derived about spatial coherence properties of a two-dimensional statistically homogeneous source that gives rise to any prescribed angular distribution of radiant intensity. The results are applied to sources that radiate in accordance with Lambert’s law. It is found that lambertian sources cannot be spatially strictly incoherent and that they have, in fact, certain unique coherence properties. This result is illustrated by calculations for a blackbody source. The idealized case of a spatially completely incoherent source is discussed for comparison.

© 1975 Optical Society of America

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References

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  1. E. Wolf and W. H. Carter, Opt. Commun. 13, 205 (1975).
    [Crossref]
  2. M. Beran and G. Parrent, Nuovo Cimento 27, 1049 (1963).
    [Crossref]
  3. T. J. Skinner, Ph.D. thesis (Boston University, 1965), p. 46.
  4. A. Walther, J. Opt. Soc. Am. 58, 1256 (1968).
    [Crossref]
  5. E. W. Marchand and E. Wolf, Opt. Commun. 6, 305 (1972).
    [Crossref]
  6. In this paper, the assumption of statistical homogeneity of the source is to be understood as meaning the property expressed by Eq. (1). For a fuller discussion of this point see footnote 3 of Ref. 1. We implicitly assume that for a finite source, the function Fω(r), introduced in Eq. (1), can be defined for all values of r by analytic continuation; however, within the accuracy of the present theory (which applies only to sufficiently large sources), the exact form of Fω(r) is immaterial when r= |r| is large compared with the linear dimensions of the area of coherence on the source.
  7. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. P., Cambridge, 1922), p. 26, Eq. (5) (with an obvious substitution).
  8. I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1965), p. 682, formula 2 of 6.554.
  9. The term “large” is to be understood here in the sense explained in the paragraph that follows Eq. (2).
  10. Somewhat-less-precise versions of this result were obtained previously, in Refs. 2 and 4.
  11. C. L. Mehta and E. Wolf, Phys. Rev. 161, 1328 (1967), Eqs. (3.9)–(3.11).
    [Crossref]
  12. (a)In this connection, see the papers by B. Karczewski: Phys. Lett. 5, 191 (1963); (b)Nuovo Cimento 30, 906 (1963).
  13. E. W. Marchand and E. Wolf, J. Opt. Soc. Am. 64, 1219 (1974).
    [Crossref]
  14. M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1975), Sec. 8.5.2.

1975 (1)

E. Wolf and W. H. Carter, Opt. Commun. 13, 205 (1975).
[Crossref]

1974 (1)

1972 (1)

E. W. Marchand and E. Wolf, Opt. Commun. 6, 305 (1972).
[Crossref]

1968 (1)

1967 (1)

C. L. Mehta and E. Wolf, Phys. Rev. 161, 1328 (1967), Eqs. (3.9)–(3.11).
[Crossref]

1963 (2)

(a)In this connection, see the papers by B. Karczewski: Phys. Lett. 5, 191 (1963); (b)Nuovo Cimento 30, 906 (1963).

M. Beran and G. Parrent, Nuovo Cimento 27, 1049 (1963).
[Crossref]

Beran, M.

M. Beran and G. Parrent, Nuovo Cimento 27, 1049 (1963).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1975), Sec. 8.5.2.

Carter, W. H.

E. Wolf and W. H. Carter, Opt. Commun. 13, 205 (1975).
[Crossref]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1965), p. 682, formula 2 of 6.554.

Karczewski, B.

(a)In this connection, see the papers by B. Karczewski: Phys. Lett. 5, 191 (1963); (b)Nuovo Cimento 30, 906 (1963).

Marchand, E. W.

E. W. Marchand and E. Wolf, J. Opt. Soc. Am. 64, 1219 (1974).
[Crossref]

E. W. Marchand and E. Wolf, Opt. Commun. 6, 305 (1972).
[Crossref]

Mehta, C. L.

C. L. Mehta and E. Wolf, Phys. Rev. 161, 1328 (1967), Eqs. (3.9)–(3.11).
[Crossref]

Parrent, G.

M. Beran and G. Parrent, Nuovo Cimento 27, 1049 (1963).
[Crossref]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1965), p. 682, formula 2 of 6.554.

Skinner, T. J.

T. J. Skinner, Ph.D. thesis (Boston University, 1965), p. 46.

Walther, A.

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. P., Cambridge, 1922), p. 26, Eq. (5) (with an obvious substitution).

Wolf, E.

E. Wolf and W. H. Carter, Opt. Commun. 13, 205 (1975).
[Crossref]

E. W. Marchand and E. Wolf, J. Opt. Soc. Am. 64, 1219 (1974).
[Crossref]

E. W. Marchand and E. Wolf, Opt. Commun. 6, 305 (1972).
[Crossref]

C. L. Mehta and E. Wolf, Phys. Rev. 161, 1328 (1967), Eqs. (3.9)–(3.11).
[Crossref]

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1975), Sec. 8.5.2.

J. Opt. Soc. Am. (2)

Nuovo Cimento (1)

M. Beran and G. Parrent, Nuovo Cimento 27, 1049 (1963).
[Crossref]

Opt. Commun. (2)

E. Wolf and W. H. Carter, Opt. Commun. 13, 205 (1975).
[Crossref]

E. W. Marchand and E. Wolf, Opt. Commun. 6, 305 (1972).
[Crossref]

Phys. Lett. (1)

(a)In this connection, see the papers by B. Karczewski: Phys. Lett. 5, 191 (1963); (b)Nuovo Cimento 30, 906 (1963).

Phys. Rev. (1)

C. L. Mehta and E. Wolf, Phys. Rev. 161, 1328 (1967), Eqs. (3.9)–(3.11).
[Crossref]

Other (7)

T. J. Skinner, Ph.D. thesis (Boston University, 1965), p. 46.

In this paper, the assumption of statistical homogeneity of the source is to be understood as meaning the property expressed by Eq. (1). For a fuller discussion of this point see footnote 3 of Ref. 1. We implicitly assume that for a finite source, the function Fω(r), introduced in Eq. (1), can be defined for all values of r by analytic continuation; however, within the accuracy of the present theory (which applies only to sufficiently large sources), the exact form of Fω(r) is immaterial when r= |r| is large compared with the linear dimensions of the area of coherence on the source.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. P., Cambridge, 1922), p. 26, Eq. (5) (with an obvious substitution).

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1965), p. 682, formula 2 of 6.554.

The term “large” is to be understood here in the sense explained in the paragraph that follows Eq. (2).

Somewhat-less-precise versions of this result were obtained previously, in Refs. 2 and 4.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1975), Sec. 8.5.2.

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Figures (2)

FIG. 1
FIG. 1

Illustrating the notation.

FIG. 2
FIG. 2

The low-frequency part WLF(r1, r2, ω) of the cross-spectral density function, the low-frequency part of its spatial-frequency spectrum Fω(f), and the angular distribution of radiant intensity Jω(s) (a) for a large two-dimensional lambertian source (Ref. 9) and (b) for a large two-dimensional spatially incoherent source. All of the curves are normalized as indicated on the top of the vertical axes.

Equations (51)

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W ( r 1 , r 2 , ω ) = F ω ( r 1 - r 2 )             ( r 1 S ,             r 2 S )
γ ω ( r 1 - r 2 ) W ( r 1 , r 2 , ω ) [ W ( r 1 , r 1 , ω ) W ( r 2 , r 2 , ω ) ] 1 / 2 = F ω ( r 1 - r 2 ) F ω ( 0 )
J ω ( s ) = k 2 A F ˆ ω ( k s ) cos 2 θ .
F ˆ ω ( f ) = 1 ( 2 π ) 2 ( z = 0 ) F ω ( r ) e - i f · r d 2 r ,
k = ω / c .
F ˆ ω ( f ) = 1 ( 2 π ) 2 ( z = 0 ) W ( r 1 + r , r 1 ) e - i f · r d 2 r ,
s 2 = s x 2 + s y 2 = 1 - s z 2 = 1 - cos 2 θ 1 ,
f x 2 + f y 2 = k 2
F ω ( r ) = F ω LF ( r ) + F ω HF ( r ) ,
F ω LF ( r ) = f 2 k 2 F ˆ ω ( f ) e i f · r d 2 f ,
F ω HF ( r ) = f 2 > k 2 F ˆ ω ( f ) e i f · r d 2 f .
( z = 0 ) F ω LF ( r ) e - i f · r d 2 r = 0             when     f 2 > k 2 ,
( z = 0 ) F ω HF ( r ) e - i f · r d 2 r = 0             when f 2 k 2 .
W ( r 1 , r 2 , ω ) = W L F ( r 1 , r 2 , ω ) + W H F ( r 1 , r 2 , ω ) ,
W LF ( r 1 , r 2 , ω ) = F ω LF ( r 1 - r 2 ) ,
W HF ( r 1 , r 2 , ω ) = F ω HF ( r 1 - r 2 ) .
F ω LF ( r ) = k 2 s 2 1 F ˆ ω ( k s ) e i k s · r d 2 s .
F ˆ ω ( k s ) = 1 k 2 A J ω ( s ) 1 - s x 2 - s y 2 ,
F ω LF ( r ) = 1 A s x 2 + s y 2 1 J ω ( s ) 1 - s x 2 - s y 2 e i k ( s x x + s y y ) d s x d s y ,
W LF ( r 1 , r 2 , ω ) = 1 A s x 2 + s y 2 1 J ω ( s ) 1 - s x 2 - s y 2 × e i k [ s x ( x 1 - x 2 ) + s y ( y 1 - y 2 ) ] d s x d s y ,
( z = 0 ) W HF ( r 1 + r , r 1 ) e - i f · r d 2 r = 0
f = k s
f x = k s x ,             f y = k s y ,
F ˆ ω ( f ) = 1 A J ω ( f x / k , f y / k , + k 2 - f x 2 - f y 2 / k ) k 2 - f x 2 - f y 2
f 2 f x 2 + f y 2 k 2 .
J ω ( s ) = C cos θ ,
cos θ = + 1 - s x 2 - s y 2 ,
W LF ( r 1 , r 2 , ω ) = C A s x 2 + s y 2 1 1 1 - s x 2 - s y 2 × e i k [ s x ( x 1 - x 2 ) + s y ( y 1 - y 2 ) ] d s x d s y .
s x = ρ cos χ ,             s y = ρ sin χ ,
x 1 - x 2 = r cos ϕ ,             y 1 - y 2 = r sin ϕ ,
W LF ( r 1 , r 2 , ω ) = C A 0 2 π d χ 0 1 1 1 - ρ 2 e i k r ρ cos ( ϕ - χ ) ρ d ρ .
0 2 π e i k r ρ cos ( ϕ - χ ) d χ = 2 π J 0 ( k r ρ ) ,
W LF ( r 1 , r 2 , ω ) = 2 π C A 0 1 1 1 - ρ 2 J 0 ( k r ρ ) ρ d ρ .
W LF ( r 1 , r 2 , ω ) = 2 π C A [ sin ( k r 1 - r 2 ) k r 1 - r 2 ] .
J ω ( f x / k , f y / k , k 2 - f x 2 - f y 2 / k ) = C 1 - ( f x / k ) 2 - ( f y / k ) 2 ,
F ˆ ω ( f ) = C k A 1 k 2 - f 2 ,
W i j ( r 1 , r 2 , ω ) = π A { δ i j [ j 0 ( k r ) - 1 k r j 1 ( k r ) ] + r i r j r 2 j 2 ( k r ) } ,
A = 2 w 3 π c 3 1 e w / K T - 1 ,
j 0 ( k r ) = sin k r k r , j 1 ( k r ) = sin k r ( k r ) 2 - cos k r k r , j 2 ( k r ) = [ 3 ( k r ) 3 - 1 k r ] sin k r - 3 ( k r ) 2 cos k r .
Tr W i j ( r 1 , r 2 , ω ) = π A { 3 j 0 ( k r ) - 3 k r j 1 ( k r ) + j 2 ( k r ) }
Tr W i j ( r 1 , r 2 , ω ) = C ( sin k r k r ) ,
C = 4 ω 3 c 3 1 e ω / K T - 1 .
Tr W i j ( r 1 , r 2 , ω ) = Tr W i j LF ( r 1 , r 2 , ω ) + Tr W i j HF ( r 1 , r 2 , ω ) ,
Tr W i j LF ( r 1 , r 2 , ω ) = C sin k r 1 - r 2 k r 1 - r 2 ,
Tr W i j HF ( r 1 , r 2 , ω ) = 0 ,
W ( r 1 , r 2 , ω ) = C δ ( 2 ) ( r 1 - r 2 ) ,
F ˆ ( f ) = C ( 2 π ) 2 ( z = 0 ) δ ( 2 ) ( r ) e - i f · r d 2 r = C ( 2 π ) 2 ,
J ω ( s ) = ( k 2 π ) 2 A C cos 2 θ ,
W LF ( r 1 , r 2 , ω ) = ( k 2 π ) 2 C × s x 2 + s y 2 1 e i k [ s x ( x 1 - x 2 ) + s y ( y 1 - y 2 ) ] d s x d s y
W LF ( r 1 , r 2 , ω ) = k 2 C 2 π 0 1 J 0 ( k r ρ ) ρ d ρ .
W LF ( r 1 , r 2 , ω ) = k 2 C 4 π ( 2 J 1 ( k r 1 - r 2 ) k r 1 - r 2 ) ,