Abstract

The complex degree of spectral coherence for both the source distribution function and the field amplitude over the source plane are found for isotropic homogeneous planar sources that do not produce evanescent plane waves. This calculation is extended to also include similar sources that are quasi-homogeneous rather than homogeneous. All the calculated coherence functions are expressed in simple mathematical form and are tabulated for easy reference.

© 1984 Optical Society of America

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References

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  1. A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 58, 1256–1259 (1968).
    [CrossRef]
  2. E. Wolf, “Coherence and radiometry,” J. Opt. Soc. Am. 68, 6–17 (1978).
    [CrossRef]
  3. W. H. Carter, “Statistical radiometry,” Radio Sci. 18, 149–158 (1983).
    [CrossRef]
  4. W. H. Carter, E. Wolf, “Coherence properties of Lambertian and non-Lambertian sources,” J. Opt. Soc. Am. 65, 1067–1071 (1975).
    [CrossRef]
  5. H. P. Baltes, B. Steinle, G. Antes, “Radiometric and correlation properties of bounded planar sources,” in Proceedings of the Fourth Rochester Conference on Coherence and Quantum Optics, L. Mandel, E. Wolf, eds. (Plenum, New York, 1978).
  6. H. P. Baltes, B. Steinle, G. Antes, “Spectral coherence and the radiant intensity from statistically homogeneous and isotropic planar sources,” Opt. Commun. 18, 242–246 (1976).
    [CrossRef]
  7. W. H. Carter, E. Wolf, “Correlation theory of wavefields generated by fluctuating three-dimensional primary scalar sources. I. General theory,” Opt. Acta 28, 227–244 (1981).
    [CrossRef]
  8. E. Wolf, W. H. Carter, “Coherence and radiant intensity in scalar wave fields generated by fluctuating primary planar sources,” J. Opt. Soc. Am. 68, 953–963 (1978).
    [CrossRef]
  9. W. H. Carter, E. Wolf, “Coherence and radiometry with quasi-homogeneous planar sources,” J. Opt. Soc. Am. 67, 785–796 (1977).
    [CrossRef]
  10. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1966).
  11. W. H. Carter, E. Wolf, “Correlation theory of wavefields generated by fluctuating three-dimensional primary scalar sources. II. Radiation from isotropic model sources,” Opt. Acta 28, 245–259 (1981).
    [CrossRef]
  12. W. H. Carter, “Some difficulties in the derivation of Hopkins’s formula,” submitted to J. Opt. Soc. Am. A.
  13. W. H. Carter, “Radiant intensity from inhomogeneous sources and the concept of averaged cross-spectral density,” Opt. Commun. 26, 1–4 (1978).
    [CrossRef]
  14. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

1983 (1)

W. H. Carter, “Statistical radiometry,” Radio Sci. 18, 149–158 (1983).
[CrossRef]

1981 (2)

W. H. Carter, E. Wolf, “Correlation theory of wavefields generated by fluctuating three-dimensional primary scalar sources. I. General theory,” Opt. Acta 28, 227–244 (1981).
[CrossRef]

W. H. Carter, E. Wolf, “Correlation theory of wavefields generated by fluctuating three-dimensional primary scalar sources. II. Radiation from isotropic model sources,” Opt. Acta 28, 245–259 (1981).
[CrossRef]

1978 (3)

W. H. Carter, “Radiant intensity from inhomogeneous sources and the concept of averaged cross-spectral density,” Opt. Commun. 26, 1–4 (1978).
[CrossRef]

E. Wolf, W. H. Carter, “Coherence and radiant intensity in scalar wave fields generated by fluctuating primary planar sources,” J. Opt. Soc. Am. 68, 953–963 (1978).
[CrossRef]

E. Wolf, “Coherence and radiometry,” J. Opt. Soc. Am. 68, 6–17 (1978).
[CrossRef]

1977 (1)

1976 (1)

H. P. Baltes, B. Steinle, G. Antes, “Spectral coherence and the radiant intensity from statistically homogeneous and isotropic planar sources,” Opt. Commun. 18, 242–246 (1976).
[CrossRef]

1975 (1)

1968 (1)

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1966).

Antes, G.

H. P. Baltes, B. Steinle, G. Antes, “Spectral coherence and the radiant intensity from statistically homogeneous and isotropic planar sources,” Opt. Commun. 18, 242–246 (1976).
[CrossRef]

H. P. Baltes, B. Steinle, G. Antes, “Radiometric and correlation properties of bounded planar sources,” in Proceedings of the Fourth Rochester Conference on Coherence and Quantum Optics, L. Mandel, E. Wolf, eds. (Plenum, New York, 1978).

Baltes, H. P.

H. P. Baltes, B. Steinle, G. Antes, “Spectral coherence and the radiant intensity from statistically homogeneous and isotropic planar sources,” Opt. Commun. 18, 242–246 (1976).
[CrossRef]

H. P. Baltes, B. Steinle, G. Antes, “Radiometric and correlation properties of bounded planar sources,” in Proceedings of the Fourth Rochester Conference on Coherence and Quantum Optics, L. Mandel, E. Wolf, eds. (Plenum, New York, 1978).

Carter, W. H.

W. H. Carter, “Statistical radiometry,” Radio Sci. 18, 149–158 (1983).
[CrossRef]

W. H. Carter, E. Wolf, “Correlation theory of wavefields generated by fluctuating three-dimensional primary scalar sources. I. General theory,” Opt. Acta 28, 227–244 (1981).
[CrossRef]

W. H. Carter, E. Wolf, “Correlation theory of wavefields generated by fluctuating three-dimensional primary scalar sources. II. Radiation from isotropic model sources,” Opt. Acta 28, 245–259 (1981).
[CrossRef]

W. H. Carter, “Radiant intensity from inhomogeneous sources and the concept of averaged cross-spectral density,” Opt. Commun. 26, 1–4 (1978).
[CrossRef]

E. Wolf, W. H. Carter, “Coherence and radiant intensity in scalar wave fields generated by fluctuating primary planar sources,” J. Opt. Soc. Am. 68, 953–963 (1978).
[CrossRef]

W. H. Carter, E. Wolf, “Coherence and radiometry with quasi-homogeneous planar sources,” J. Opt. Soc. Am. 67, 785–796 (1977).
[CrossRef]

W. H. Carter, E. Wolf, “Coherence properties of Lambertian and non-Lambertian sources,” J. Opt. Soc. Am. 65, 1067–1071 (1975).
[CrossRef]

W. H. Carter, “Some difficulties in the derivation of Hopkins’s formula,” submitted to J. Opt. Soc. Am. A.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1966).

Steinle, B.

H. P. Baltes, B. Steinle, G. Antes, “Spectral coherence and the radiant intensity from statistically homogeneous and isotropic planar sources,” Opt. Commun. 18, 242–246 (1976).
[CrossRef]

H. P. Baltes, B. Steinle, G. Antes, “Radiometric and correlation properties of bounded planar sources,” in Proceedings of the Fourth Rochester Conference on Coherence and Quantum Optics, L. Mandel, E. Wolf, eds. (Plenum, New York, 1978).

Walther, A.

Wolf, E.

W. H. Carter, E. Wolf, “Correlation theory of wavefields generated by fluctuating three-dimensional primary scalar sources. I. General theory,” Opt. Acta 28, 227–244 (1981).
[CrossRef]

W. H. Carter, E. Wolf, “Correlation theory of wavefields generated by fluctuating three-dimensional primary scalar sources. II. Radiation from isotropic model sources,” Opt. Acta 28, 245–259 (1981).
[CrossRef]

E. Wolf, W. H. Carter, “Coherence and radiant intensity in scalar wave fields generated by fluctuating primary planar sources,” J. Opt. Soc. Am. 68, 953–963 (1978).
[CrossRef]

E. Wolf, “Coherence and radiometry,” J. Opt. Soc. Am. 68, 6–17 (1978).
[CrossRef]

W. H. Carter, E. Wolf, “Coherence and radiometry with quasi-homogeneous planar sources,” J. Opt. Soc. Am. 67, 785–796 (1977).
[CrossRef]

W. H. Carter, E. Wolf, “Coherence properties of Lambertian and non-Lambertian sources,” J. Opt. Soc. Am. 65, 1067–1071 (1975).
[CrossRef]

J. Opt. Soc. Am. (1)

E. Wolf, W. H. Carter, “Coherence and radiant intensity in scalar wave fields generated by fluctuating primary planar sources,” J. Opt. Soc. Am. 68, 953–963 (1978).
[CrossRef]

J. Opt. Soc. Am. (4)

Opt. Acta (1)

W. H. Carter, E. Wolf, “Correlation theory of wavefields generated by fluctuating three-dimensional primary scalar sources. I. General theory,” Opt. Acta 28, 227–244 (1981).
[CrossRef]

Opt. Commun. (1)

H. P. Baltes, B. Steinle, G. Antes, “Spectral coherence and the radiant intensity from statistically homogeneous and isotropic planar sources,” Opt. Commun. 18, 242–246 (1976).
[CrossRef]

Opt. Acta (1)

W. H. Carter, E. Wolf, “Correlation theory of wavefields generated by fluctuating three-dimensional primary scalar sources. II. Radiation from isotropic model sources,” Opt. Acta 28, 245–259 (1981).
[CrossRef]

Opt. Commun. (1)

W. H. Carter, “Radiant intensity from inhomogeneous sources and the concept of averaged cross-spectral density,” Opt. Commun. 26, 1–4 (1978).
[CrossRef]

Radio Sci. (1)

W. H. Carter, “Statistical radiometry,” Radio Sci. 18, 149–158 (1983).
[CrossRef]

Other (4)

H. P. Baltes, B. Steinle, G. Antes, “Radiometric and correlation properties of bounded planar sources,” in Proceedings of the Fourth Rochester Conference on Coherence and Quantum Optics, L. Mandel, E. Wolf, eds. (Plenum, New York, 1978).

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1966).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

W. H. Carter, “Some difficulties in the derivation of Hopkins’s formula,” submitted to J. Opt. Soc. Am. A.

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Figures (1)

Fig. 1
Fig. 1

Illustrating the notation.

Tables (3)

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Table 1 Relations for Homogeneous Planar Sources

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Table 2 Relations for Secondary Quasi-homogeneous Sources

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Table 3 Transform Pairs

Equations (67)

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J ω ( s ) = n = 0 a n cos n θ ,
( 2 + k 2 ) ψ ( x , ω ) = - 4 π ρ ( x , ω ) ,
δ ( ω - ω ) W ρ ( x 1 , x 2 ) = ρ ( x 1 , ω ) ρ * ( x 2 , ω ) ,
δ ( ω - ω ) W ( x 1 , x 2 ) = ψ ( x 1 , ω ) ψ * ( x 2 , ω ) ,
W ˜ ( 0 ) ( ξ 1 , ξ 2 ) = W ˜ ρ ( 0 ) ( ξ 1 , ξ 2 ) / ζ 1 ζ 2 * ,
W ˜ ( 0 ) ( ξ 1 , ξ 2 ) = - d 2 x 1 - d 2 x 2 × W ( 0 ) ( x 1 , x 2 ) exp [ - 2 π i ( ξ 1 · x 1 + ξ 2 · x 2 ) ] ,
W ˜ ρ ( 0 ) ( ξ 1 , ξ 2 ) = - d 2 x 1 - d 2 x 2 × W ρ ( 0 ) ( x 1 , x 2 ) exp [ - 2 π i ( ξ 1 · x 1 + ξ 2 · x 2 ) ]
ζ n = { ( 1 / λ 2 - ξ n 2 - η n 2 ) 1 / 2 if ξ n 2 + η n 2 1 / λ 2 i ( ξ n 2 + η n 2 - 1 / λ 2 ) 1 / 2 if ξ n 2 + η n 2 > 1 / λ 2 .
W ρ ( 0 ) ( x 1 , x 2 ) = I ρ ( 0 ) μ ρ ( 0 ) ( x 1 - x 2 )
μ ρ ( x 1 , x 2 ) = W ρ ( x 1 , x 2 ) [ I ρ ( x 1 ) ] 1 / 2 [ I ρ ( x 2 ) ] 1 / 2
I ρ ( x ) = W ρ ( x , x )
I ρ ( 0 ) ( x ) = I ρ ( 0 ) .
W ˜ ρ ( 0 ) ( ξ 1 , ξ 2 ) = I ρ ( 0 ) δ ( 2 ) ( ξ 1 + ξ 2 ) μ ˜ ρ ( 0 ) ( ξ 1 - ξ 2 2 ) ,
W ˜ ( 0 ) ( ξ 1 , ξ 2 ) = I ρ ( 0 ) δ ( 2 ) ( ξ 1 + ξ 2 ) μ ˜ ρ ( 0 ) [ ( ξ 1 - ξ 2 ) / 2 ] ζ 1 ζ 2 * ,
μ ˜ ρ ( 0 ) ( ξ ) = - μ ρ ( 0 ) ( x - ) exp ( - 2 π i ξ · x - ) d 2 x -
x - = x 1 - x 2
W ( 0 ) ( x 1 , x 2 ) = I ρ ( 0 ) g ( x 1 - x 2 ) ,
g ( x - ) = - μ ˜ ρ ( 0 ) ( ξ ) ζ 2 exp ( 2 π i ξ · x - ) d 2 ξ .
J ω ( s ) = A I ρ ( 0 ) μ ˜ ρ ( 0 ) ( s / λ ) ,
μ ˜ ρ ( 0 ) ( s / λ ) = 1 A I ρ ( 0 ) n = 1 a n [ ( 1 - s x 2 - s y 2 ) 1 / 2 ] n Π ( s ) ,
μ ρ ( 0 ) ( x - ) = 1 λ 2 A I ρ ( 0 ) n = 1 a n - [ ( 1 - s x 2 - s y 2 ) 1 / 2 ] n × Π ( s ) exp [ i k ( s x x - + s y y - ) ] d s x d s y .
μ ρ ( 0 ) ( x - ) = 2 π λ 2 A I ρ ( 0 ) n = 1 a n 2 n / 2 Γ ( n 2 + 1 ) J n / 2 + 1 ( k x - ) ( k x - ) n / 2 + 1 ,
g ( x - ) = 2 π A I ρ ( 0 ) n = 1 a n 2 n / 2 - 1 Γ ( n / 2 ) J n / 2 ( k x - ) ( k x - ) n / 2 .
μ ρ ( 0 ) ( 0 ) = 1 ,
J ν ( k r ) ( k r ) ν ~ 1 2 ν Γ ( ν + 1 )             as k r 0
n = 1 a n n + 2 = λ 2 A I ρ ( 0 ) 2 π .
a n = { 0 for all n m λ 2 A I ρ ( 0 ) 2 π ( m + 2 ) for n = m
J ω ( s ) = λ 2 A I ρ ( 0 ) 2 π ( m + 2 ) cos m θ ,
μ ρ ( 0 ) ( x - ) = 2 m / 2 ( m + 2 ) Γ [ ( m / 2 ) + 1 ] × J m / 2 + 1 ( k x - ) ( k x - ) m / 2 + 1 ,
g ( x - ) = λ 2 ( m + 2 ) 2 m / 2 - 1 Γ ( m / 2 ) J m / 2 ( k x - ) ( k x - ) m / 2 .
j n ( z ) = ( π 2 z ) 1 / 2 J n + 1 / 2 ( z )
W ρ ( 0 ) ( x 1 , x 2 ) = I ρ ( 0 ) ( x + ) μ ρ ( 0 ) ( x - ) .
W ( 0 ) ( x 1 , x 2 ) = I ( 0 ) ( x + ) μ ( 0 ) ( x - ) .
x + = x 1 + x 2 2 ,
x - = x 1 - x 2 .
J ω ( s ) = W ˜ ρ ( 0 ) ( s / λ , - s / λ ) .
J ω ( s ) = I ˜ ρ ( 0 ) ( 0 ) μ ˜ ρ ( s / λ ) ,
I ˜ ρ ( 0 ) ( ξ ) = - I ρ ( 0 ) ( x + ) exp ( - 2 π i ξ · x + ) d 2 x +
μ ρ ( 0 ) ( x - ) = 1 λ 2 I ρ ( 0 ) ( 0 ) n = 1 a n 2 n / 2 Γ [ ( n / 2 ) + 1 ] J n / 2 + 1 ( k x - ) ( k x - ) n / 2 + 1 ,
W ( 0 ) ( x 1 , x 2 ) = - d 2 ξ 2 - d 2 ξ 1 × I ˜ ρ ( 0 ) ( ξ 1 + ξ 2 ) μ ˜ ρ ( 0 ) [ ( ξ 1 - ξ 2 ) / 2 ] ζ 1 ζ 2 * exp [ 2 π i ( ξ 1 · x 1 + ξ 2 · x 2 ) ] .
I ρ ( 0 ) ( 0 ) = A I ρ ( 0 )
J ω ( s ) = I ˜ ( 0 ) ( 0 ) μ ˜ ( 0 ) ( s / λ ) cos 2 θ / λ 2 ,
I ˜ ( 0 ) ( ξ ) = - I ( 0 ) ( x + ) exp ( - 2 π i ξ · x + ) d 2 x +
μ ˜ ( 0 ) ( ξ ) = - μ ( 0 ) ( x - ) exp ( - 2 π ξ · x - ) d 2 x - .
μ ( 0 ) ( x - ) = λ 2 I ˜ ( 0 ) ( 0 ) n = 1 a n - [ ( 1 - s x 2 - s y 2 ) 1 / 2 ] n - 2 × Π ( s ) exp [ i k ( s x x - + s y - ) ] d s x d s y .
μ ( 0 ) ( x - ) = 2 π I ˜ ( 0 ) ( 0 ) n = 1 a n 2 ( n / 2 ) - 1 Γ ( n 2 ) J n / 2 ( k x - ) ( k x - ) n / 2 .
μ ( 0 ) ( 0 ) = 1.
n = 1 a n / n = I ˜ ( 0 ) ( 0 ) / 2 π .
a n = { 0 for all n m I ˜ ( 0 ) ( 0 ) 2 π for n = m
μ ( 0 ) ( x - ) = m 2 m / 2 - 1 Γ ( m / 2 ) J m / 2 ( k x - ) ( k x - ) m / 2 .
μ ( 0 ) ( x - ) = m λ 2 ( m + 2 ) g ( x - ) ,
W ρ ( 0 ) ( x 1 , x 2 ) = - d 2 ξ 1 - d 2 ξ 2 ( ζ 1 , ζ 2 * ) × I ˜ ( 0 ) ( ξ 1 + ξ 2 ) μ ˜ ( 0 ) ( ξ 1 - ξ 2 2 ) exp [ 2 π ( ξ 1 · x 1 + ξ 2 · x 2 ) ] .
I α ( r ) = p 2 + q 2 1 m α exp [ i k ( p x + q y ) ] d p d q ,
x = r cos θ , p = ρ cos ϕ , y = r sin θ , q = ρ sin ϕ
0 2 π exp [ i k r ρ cos ( θ - ϕ ) ] d ϕ = 2 π J 0 ( k r ρ ) ,
I α ( r ) = 2 π 0 1 [ ( 1 - ρ 2 ) 1 / 2 ] α J 0 ( k ρ r ) ρ d ρ .
ρ = sin ψ , ( 1 - ρ 2 ) 1 / 2 = cos ψ
I α ( r ) = 2 π 0 π / 2 cos α + 1 ψ sin ψ J 0 ( k r sin ψ ) d ψ .
I α ( r ) = 2 π 2 α / 2 Γ [ ( α / 2 ) + 1 ] J α / 2 + 1 ( k r ) ( k r ) α / 2 + 1 ,
Γ ( l + 1 ) = l ! ,
ρ 2 + q 2 1 ( 1 - p 2 - q 2 ) l exp [ i k ( p x + q y ) ] d p d q = 2 l + 1 π l ! J l + 1 ( k r ) ( k r ) l + 1
Γ ( l + 1 2 ) = 1 × 3 × 5 × × ( 2 l - 1 ) 2 l π
J l + 1 / 2 ( k r ) = ( 2 k r π ) 1 / 2 j l ( k r ) ,
ρ 2 + q 2 = 1 [ ( 1 - p 2 - q 2 ) 1 / 2 ] 2 l - 1 exp [ i k ( p x + q y ) ] d p d q = 2 π [ 1 × 3 × 5 × × ( 2 l - 1 ) ] j l ( k r ) ( k r )
Π ( ξ ) = { 1 if ξ 1 0 otherwise .
- J l + 1 ( k r ) ( k r ) l + 1 d x d y = λ 2 2 π 2 l l ! ,             l > - 1
- j l ( k r ) ( k r ) l d x d y = λ 2 2 π ( 1 × 3 × 5 × × ( 2 l - 1 ) ) ,             l > - 1

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