Abstract

Expressions are derived for the radiant intensity generated by a planar, anisotropic, Gaussian, Schell-model source of any state of coherence. It is found that with an appropriate choice of the source parameters the radiant intensity may be rotationally symmetric about the source plane and may even become identical with the radiant intensity produced by a completely coherent and rotationally symmetric laser source.

© 1982 Optical Society of America

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References

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  1. H. P. Baltes, B. Steinle, G. Antes, “Radiometric and correlation properties of bounded planar sources,” in Coherence and Quantum Optics, IV, L. Mandel, E. Wolf, eds. (Plenum, New York, 1978), pp. 431–441.
  2. W. H. Carter, M. Bertolotti, “An analysis of the far-field coherence and radiant intensity of light scattered from liquid crystals,” J. Opt. Soc. Am. 68, 329–333 (1978).
    [Crossref]
  3. E. Wolf, E. Collett, “Partially coherence sources which produce the same far-field intensity distribution as a laser,” Opt. Commun. 25, 293–296 (1978).
    [Crossref]
  4. J. T. Foley, M. S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. 26, 297–300 (1978).
    [Crossref]
  5. F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
    [Crossref]
  6. L. Mandel, E. Wolf, “Spectral coherence and the concept of cross-spectral purity,” J. Opt. Soc. Am. 66, 529–535 (1976).
    [Crossref]
  7. E. W. Marchand, E. Wolf, “Radiometry with sources of any state of coherence,” J. Opt. Soc. Am. 64, 1219–1226 (1974), Eq. (41).
    [Crossref]
  8. The appearance of the negative sign in front of the first rather than the second argument in W(0) in Eq. (2.2) is due to slightly different definitions of the cross-spectral densities employed in some of the earlier publications and in this Letter.
  9. To simplify the notation we do not display from now on the dependence of the degree of spatial coherence and of various other quantities on the frequency ω.
  10. W. H. Carter, E. Wolf, “Correlation theory of wave-fields generated by fluctuating, three-dimensional, scalar sources, II. Radiation from isotropic model sources,” Opt. Acta 28, 245–259 (1981).
    [Crossref]

1981 (1)

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

1980 (1)

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[Crossref]

1978 (3)

W. H. Carter, M. Bertolotti, “An analysis of the far-field coherence and radiant intensity of light scattered from liquid crystals,” J. Opt. Soc. Am. 68, 329–333 (1978).
[Crossref]

E. Wolf, E. Collett, “Partially coherence sources which produce the same far-field intensity distribution as a laser,” Opt. Commun. 25, 293–296 (1978).
[Crossref]

J. T. Foley, M. S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. 26, 297–300 (1978).
[Crossref]

1976 (1)

1974 (1)

Antes, G.

H. P. Baltes, B. Steinle, G. Antes, “Radiometric and correlation properties of bounded planar sources,” in Coherence and Quantum Optics, IV, L. Mandel, E. Wolf, eds. (Plenum, New York, 1978), pp. 431–441.

Baltes, H. P.

H. P. Baltes, B. Steinle, G. Antes, “Radiometric and correlation properties of bounded planar sources,” in Coherence and Quantum Optics, IV, L. Mandel, E. Wolf, eds. (Plenum, New York, 1978), pp. 431–441.

Bertolotti, M.

Carter, W. H.

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

W. H. Carter, M. Bertolotti, “An analysis of the far-field coherence and radiant intensity of light scattered from liquid crystals,” J. Opt. Soc. Am. 68, 329–333 (1978).
[Crossref]

Collett, E.

E. Wolf, E. Collett, “Partially coherence sources which produce the same far-field intensity distribution as a laser,” Opt. Commun. 25, 293–296 (1978).
[Crossref]

Foley, J. T.

J. T. Foley, M. S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. 26, 297–300 (1978).
[Crossref]

Gori, F.

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[Crossref]

Mandel, L.

Marchand, E. W.

Steinle, B.

H. P. Baltes, B. Steinle, G. Antes, “Radiometric and correlation properties of bounded planar sources,” in Coherence and Quantum Optics, IV, L. Mandel, E. Wolf, eds. (Plenum, New York, 1978), pp. 431–441.

Wolf, E.

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

E. Wolf, E. Collett, “Partially coherence sources which produce the same far-field intensity distribution as a laser,” Opt. Commun. 25, 293–296 (1978).
[Crossref]

L. Mandel, E. Wolf, “Spectral coherence and the concept of cross-spectral purity,” J. Opt. Soc. Am. 66, 529–535 (1976).
[Crossref]

E. W. Marchand, E. Wolf, “Radiometry with sources of any state of coherence,” J. Opt. Soc. Am. 64, 1219–1226 (1974), Eq. (41).
[Crossref]

Zubairy, M. S.

J. T. Foley, M. S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. 26, 297–300 (1978).
[Crossref]

J. Opt. Soc. Am. (3)

Opt. Acta (1)

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

Opt. Commun. (3)

E. Wolf, E. Collett, “Partially coherence sources which produce the same far-field intensity distribution as a laser,” Opt. Commun. 25, 293–296 (1978).
[Crossref]

J. T. Foley, M. S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. 26, 297–300 (1978).
[Crossref]

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[Crossref]

Other (3)

The appearance of the negative sign in front of the first rather than the second argument in W(0) in Eq. (2.2) is due to slightly different definitions of the cross-spectral densities employed in some of the earlier publications and in this Letter.

To simplify the notation we do not display from now on the dependence of the degree of spatial coherence and of various other quantities on the frequency ω.

H. P. Baltes, B. Steinle, G. Antes, “Radiometric and correlation properties of bounded planar sources,” in Coherence and Quantum Optics, IV, L. Mandel, E. Wolf, eds. (Plenum, New York, 1978), pp. 431–441.

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Figures (3)

Fig. 1
Fig. 1

Illustrating the notation relating to formula (2.2) for the radiant intensity. s is the projection, considered as a two-dimensional vector, of the unit vector s onto the source plane z = 0.

Fig. 2
Fig. 2

Illustrating two-dimensional, anisotropic, Gaussian, Schell-model sources, all with the same intensity distribution (solid line), but with different degrees of coherence (dashed lines), that will generate rotationally symmetric distributions of radiant intensity. For all these sources σx = 5 mm, σy = 2 mm. (a) δx = 1 mm, δy = 1.03 mm; (b) δx = 2 mm, δy = 2.25 mm; (c) δx = 3 mm, δy = 4.13 mm; (d) δx = 4.36 mm, δy → ∞. The curves represent the loci defined by Eqs. (3.16) and (3.17).

Fig. 3
Fig. 3

Illustrating two-dimensional, anisotropic, Gaussian, Schell-model sources, all with the same degree of coherence (dashed line) but with different intensity distributions (solid lines), that will generate rotationally symmetric distributions of radiant intensity. For all these sources δx = 3 mm, δy = 4.13 mm. (a) δx = 1.13 mm, σy = 1 mm; (b) σx = 2.07 mm, σy = 1.50 mm; (c) σx = 5 mm, σy = 2 mm, (d) σx → ∞, σy = 2.18 mm. The curves represent the loci defined by Eqs. (3.16) and (3.17).

Equations (25)

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W ( 0 ) ( ρ 1 , ρ 2 , ω ) = [ I ( 0 ) ( ρ 1 , ω ) ] 1 / 2 [ I ( 0 ) ( ρ 2 , ω ) ] 1 / 2 × μ ( 0 ) ( ρ 1 , ρ 2 , ω ) ,
J ( s , ω ) = ( 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 ( z = 0 ) W ( 0 ) ( ρ 1 , ρ 2 , ω ) × exp [ i ( f 1 · ρ 1 + f 2 · ρ 2 ) ] d 2 ρ 1 d 2 ρ 2
k = ω / c 1 ,
μ ( 0 ) ( ρ 1 , ρ 2 ) = g ( 0 ) ( ρ 1 ρ 2 ) .
I ( 0 ) ( ρ ) = A exp ( x 2 2 σ x 2 y 2 2 σ y 2 ) ,
g ( 0 ) ( ρ ) = exp ( x 2 2 δ x 2 y 2 2 δ y 2 ) ,
W ( ρ 1 , ρ 2 ) = A exp [ a x ( x 1 2 + x 2 2 ) a y ( y 1 2 + y 2 2 ) + 2 b x x 1 x 2 + 2 b y y 1 , y 2 ] ,
a x = 1 2 ( 1 2 σ x 2 + 1 δ x 2 ) , b x = 1 2 δ x 2 ,
W ˜ ( 0 ) ( f 1 , f 2 ) = A ( 2 π ) 2 γ x γ y σ x σ y [ α x ( f 1 x 2 + f 2 x 2 ) α y ( f 1 y 2 + f 2 y 2 ) 2 β x f 1 x f 2 x 2 β y f 1 y f 2 y ] ,
1 γ x 2 = 1 ( 2 σ x ) 2 + 1 δ x 2 ,
α x = γ x 2 σ x 2 a x , β x = 1 2 ( γ x σ x δ x ) 2 ,
s sin θ cos ϕ , sin θ sin ϕ , cos θ
J ( s ) = J ( 0 ) cos 2 θ exp [ ( k 2 / 2 ) ( γ x 2 cos 2 ϕ + γ y 2 sin 2 ϕ ) sin 2 θ ] ,
J ( 0 ) = A k 2 γ x γ y σ x σ y
J ( s ) J ( θ , ϕ ) = K ( θ , ϕ ) [ J x ( θ ) J y ( θ ) ] 1 / 2 ,
J x ( θ ) = J x ( 0 ) cos 2 θ exp [ ( k 2 / 2 ) γ x 2 sin 2 θ ] ,
J x ( 0 ) = A ( k γ x σ x ) 2 ,
K ( θ , ϕ ) = exp [ ( k 2 / 4 ) ( γ x 2 γ y 2 ) cos 2 ϕ sin 2 θ ] .
γ x = γ y ,
1 ( 2 σ x ) 2 + 1 δ x 2 = 1 ( 2 σ y ) 2 + 1 δ y 2 ,
x 2 2 σ x 2 + y 2 2 σ y 2 = 1
x 2 2 δ x 2 + y 2 2 δ y 2 = 1
I L ( ρ ) = A L exp ( ρ 2 / 2 Δ L 2 ) ,
A L = 8 σ x σ y γ x 2 + γ y 2 A , Δ L 2 = ( 1 / 8 ) ( γ x 2 + γ j 2 ) ,

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