Abstract

An equivalence theorem is formulated that provides conditions under which planar sources of different states of spatial coherence will generate optical fields that have identical far-zone intensity distributions. As an example, a partially coherent source whose linear dimensions are large compared with the correlation length of the light across the source is described that will generate a field whose far-zone intensity distribution is identical with that of a Gaussian laser beam.

© 1978 Optical Society of America

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References

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  1. For a definition of the cross-spectral density function see, for example, L. Mandel, E. Wolf, “Spectral coherence and the concept of cross-spectral purity,” Opt. J. Soc. Am. 66, 529–535 (1976), Eqs. (2.4) and (2.5a).
    [CrossRef]
  2. The radiant intensity Jω(s) represents the rate at which energy, at the temporal frequency ω, is radiated by the source per unit solid angle around the s direction. It is related to the optical intensity Iω(Rs) at the point in the far zone, specified by position vector Rs, by the formula (to be understood in the asymptotic sense as kR → ∞) Iω(Rs) = [Jω(s)]/R2.
  3. E. W. Marchand, E. Wolf, “Radiometry with sources of any state of coherence,” J. Opt. Soc. Am. 64, 1219–1226 (1974), Eq. (41).
    [CrossRef]
  4. E. W. Marchand, E. Wolf, “Angular correlation and the far-zone behavior of partially coherent fields,” J. Opt. Soc. Am. 62, 379–385 (1972), Eq. (34).
    [CrossRef]
  5. W. H. Carter, E. Wolf, “Coherence and radiometry with quasihomogeneous planar sources,” J. Opt. Soc. Am. 67, 785–796 (1977).
    [CrossRef]
  6. F. Scudieri, M. Bertolotti, R. Bartolino, “Light scattered by a liquid crystal: a new quasi-thermal source,” Appl. Opt. 13, 181–185 (1974).
    [CrossRef] [PubMed]
  7. M. Bertolotti, F. Scudieri, S. Verginelli, “Spatial coherence of light scattered by media with large correlation length of refractive index fluctuations,” Appl. Opt. 15, 1842–1844 (1976).
    [CrossRef] [PubMed]

1977 (1)

1976 (2)

For a definition of the cross-spectral density function see, for example, L. Mandel, E. Wolf, “Spectral coherence and the concept of cross-spectral purity,” Opt. J. Soc. Am. 66, 529–535 (1976), Eqs. (2.4) and (2.5a).
[CrossRef]

M. Bertolotti, F. Scudieri, S. Verginelli, “Spatial coherence of light scattered by media with large correlation length of refractive index fluctuations,” Appl. Opt. 15, 1842–1844 (1976).
[CrossRef] [PubMed]

1974 (2)

1972 (1)

Bartolino, R.

Bertolotti, M.

Carter, W. H.

Mandel, L.

For a definition of the cross-spectral density function see, for example, L. Mandel, E. Wolf, “Spectral coherence and the concept of cross-spectral purity,” Opt. J. Soc. Am. 66, 529–535 (1976), Eqs. (2.4) and (2.5a).
[CrossRef]

Marchand, E. W.

Scudieri, F.

Verginelli, S.

Wolf, E.

Appl. Opt. (2)

J. Opt. Soc. Am. (3)

Opt. J. Soc. Am. (1)

For a definition of the cross-spectral density function see, for example, L. Mandel, E. Wolf, “Spectral coherence and the concept of cross-spectral purity,” Opt. J. Soc. Am. 66, 529–535 (1976), Eqs. (2.4) and (2.5a).
[CrossRef]

Other (1)

The radiant intensity Jω(s) represents the rate at which energy, at the temporal frequency ω, is radiated by the source per unit solid angle around the s direction. It is related to the optical intensity Iω(Rs) at the point in the far zone, specified by position vector Rs, by the formula (to be understood in the asymptotic sense as kR → ∞) Iω(Rs) = [Jω(s)]/R2.

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Figures (2)

Fig. 1
Fig. 1

Illustrating the notation.

Fig. 2
Fig. 2

Optical intensity distribution (a) and the spectral degree of spatial coherence (c) of a laser source and of a quasi-homogeneous source (b) and (d) that produce fields with identical distributions of the radiant intensity. The curves pertain to sources with σL = 1 mm, σQ = 10 mm. The optical intensity of the laser output is normalized to unity at the center of the output mirror. The vertical scale in (b) is ten times that of (a).

Equations (12)

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J ω ( s ) = ( 2 π k ) 2 cos 2 θ W ˜ ( 0 ) ( k s , k s ; ω ) .
W ˜ ( 0 ) ( f 1 , f 2 ; ω ) = 1 ( 2 π ) 4 W ( 0 ) ( r 1 , r 2 ; ω ) × exp [ i ( f 1 r 1 + f 2 r 2 ) ] d 2 r 1 d 2 r 2 ,
f 1 = f 2 = k s .
| k s | | k s | = k ,
μ ( 0 ) ( r 1 , r 2 ; ω ) = W ( 0 ) ( r 1 , r 2 ; ω ) I ( 0 ) ( r 1 ; ω ) I ( 0 ) ( r 2 ; ω ) ,
I ( 0 ) ( r j ; ω ) = W ( 0 ) ( r j , r j ; ω ) , ( j = 1 , 2 )
I L ( 0 ) ( r ; ω ) = A L exp ( r 2 / 2 σ L 2 ) ,
μ L ( 0 ) ( r 1 , r 2 ; ω ) g L ( 0 ) ( r 1 r 2 ; ω ) = 1
J ( s ) = J ( 0 ) cos 2 θ exp [ 2 ( k σ L ) 2 sin 2 θ ] ,
g Q ( 0 ) ( r 1 r 2 ; ω ) = exp [ ( r 1 r 2 ) 2 / 8 σ L 2 ]
I Q ( 0 ) ( r ; ω ) = ( σ L / σ Q ) 2 A L exp ( r 2 / 2 σ Q 2 ) ,
σ Q 2 σ L ,

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