Abstract

The field generated by scattering of light from a quasi-homogeneous source on a quasi-homogeneous, random medium is investigated. It is found that, within the accuracy of the first-order Born approximation, the far field satisfies two reciprocity relations (sometimes called uncertainty relations). One of them implies that the spectral density (or spectral intensity) is proportional to the convolution of the spectral density of the source and the spatial Fourier transform of the correlation coefficient of the scattering potential. The other implies that the spectral degree of coherence of the far field is proportional to the convolution of the correlation coefficient of the source and the spatial Fourier transform of the strength of the scattering potential. While the case we consider might seem restrictive, it is actually quite general. For instance, the quasi-homogeneous source model can be used to describe the generation of beams with different coherence properties and different angular spreads. In addition, the quasi-homogeneous scattering model adequately describes a wide class of turbulent media, including a stratified, turbulent atmosphere and confined plasmas.

© 2006 Optical Society of America

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References

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  1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
  2. R. A. Silverman, 'Scattering of plane waves by locally homogeneous dielectric noise,' Proc. Cambridge Philos. Soc. 54, 530-537 (1958).
    [Crossref]
  3. J. Gozani, 'Effect of the intermittent atmosphere on laser scintillations,' Opt. Lett. 24, 436-438 (1999).
    [Crossref]
  4. J. Howard, 'Laser probing of random weakly scattering media,' J. Opt. Soc. Am. A 8, 1955-1963 (1991).
    [Crossref]
  5. D. F. V. James, 'The Wolf effect and the redshift of quasars,' Pure Appl. Opt. 7, 959-970 (1998).
    [Crossref]
  6. W. H. Carter and E. Wolf, 'Scattering from quasi-homogeneous media,' Opt. Commun. 67, 85-90 (1988).
    [Crossref]
  7. D. G. Fischer and E. Wolf, 'Inverse problems with quasi-homogeneous media,' J. Opt. Soc. Am. A 11, 1128-1135 (1994).
    [Crossref]
  8. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th (expanded) ed. (Cambridge U. Press, 1999).
    [PubMed]
  9. J. Jannson, T. Jannson, and E. Wolf, 'Spatial coherence discrimination in scattering,' Opt. Lett. 13, 1060-1062 (1988).
    [Crossref] [PubMed]
  10. J. Wu and A. D. Boardman, 'Coherence length of a Gaussian-Schell beam and atmospheric turbulence,' J. Mod. Opt. 38, 1355-1363 (1991).
    [Crossref]
  11. G. Gbur and E. Wolf, 'Spreading of partially coherent beams in random media,' J. Opt. Soc. Am. A 19, 1592-1598 (2002).
    [Crossref]
  12. H. Roychowdhury and E. Wolf, 'Invariance of spectrum of light generated by a class of quasi-homogeneous sources on propagation through turbulence,' Opt. Commun. 241, 11-15 (2004).
    [Crossref]

2004 (1)

H. Roychowdhury and E. Wolf, 'Invariance of spectrum of light generated by a class of quasi-homogeneous sources on propagation through turbulence,' Opt. Commun. 241, 11-15 (2004).
[Crossref]

2002 (1)

1999 (1)

1998 (1)

D. F. V. James, 'The Wolf effect and the redshift of quasars,' Pure Appl. Opt. 7, 959-970 (1998).
[Crossref]

1994 (1)

1991 (2)

J. Wu and A. D. Boardman, 'Coherence length of a Gaussian-Schell beam and atmospheric turbulence,' J. Mod. Opt. 38, 1355-1363 (1991).
[Crossref]

J. Howard, 'Laser probing of random weakly scattering media,' J. Opt. Soc. Am. A 8, 1955-1963 (1991).
[Crossref]

1988 (2)

W. H. Carter and E. Wolf, 'Scattering from quasi-homogeneous media,' Opt. Commun. 67, 85-90 (1988).
[Crossref]

J. Jannson, T. Jannson, and E. Wolf, 'Spatial coherence discrimination in scattering,' Opt. Lett. 13, 1060-1062 (1988).
[Crossref] [PubMed]

1958 (1)

R. A. Silverman, 'Scattering of plane waves by locally homogeneous dielectric noise,' Proc. Cambridge Philos. Soc. 54, 530-537 (1958).
[Crossref]

Boardman, A. D.

J. Wu and A. D. Boardman, 'Coherence length of a Gaussian-Schell beam and atmospheric turbulence,' J. Mod. Opt. 38, 1355-1363 (1991).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th (expanded) ed. (Cambridge U. Press, 1999).
[PubMed]

Carter, W. H.

W. H. Carter and E. Wolf, 'Scattering from quasi-homogeneous media,' Opt. Commun. 67, 85-90 (1988).
[Crossref]

Fischer, D. G.

Gbur, G.

Gozani, J.

Howard, J.

James, D. F. V.

D. F. V. James, 'The Wolf effect and the redshift of quasars,' Pure Appl. Opt. 7, 959-970 (1998).
[Crossref]

Jannson, J.

Jannson, T.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

Roychowdhury, H.

H. Roychowdhury and E. Wolf, 'Invariance of spectrum of light generated by a class of quasi-homogeneous sources on propagation through turbulence,' Opt. Commun. 241, 11-15 (2004).
[Crossref]

Silverman, R. A.

R. A. Silverman, 'Scattering of plane waves by locally homogeneous dielectric noise,' Proc. Cambridge Philos. Soc. 54, 530-537 (1958).
[Crossref]

Wolf, E.

H. Roychowdhury and E. Wolf, 'Invariance of spectrum of light generated by a class of quasi-homogeneous sources on propagation through turbulence,' Opt. Commun. 241, 11-15 (2004).
[Crossref]

G. Gbur and E. Wolf, 'Spreading of partially coherent beams in random media,' J. Opt. Soc. Am. A 19, 1592-1598 (2002).
[Crossref]

D. G. Fischer and E. Wolf, 'Inverse problems with quasi-homogeneous media,' J. Opt. Soc. Am. A 11, 1128-1135 (1994).
[Crossref]

J. Jannson, T. Jannson, and E. Wolf, 'Spatial coherence discrimination in scattering,' Opt. Lett. 13, 1060-1062 (1988).
[Crossref] [PubMed]

W. H. Carter and E. Wolf, 'Scattering from quasi-homogeneous media,' Opt. Commun. 67, 85-90 (1988).
[Crossref]

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th (expanded) ed. (Cambridge U. Press, 1999).
[PubMed]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

Wu, J.

J. Wu and A. D. Boardman, 'Coherence length of a Gaussian-Schell beam and atmospheric turbulence,' J. Mod. Opt. 38, 1355-1363 (1991).
[Crossref]

J. Mod. Opt. (1)

J. Wu and A. D. Boardman, 'Coherence length of a Gaussian-Schell beam and atmospheric turbulence,' J. Mod. Opt. 38, 1355-1363 (1991).
[Crossref]

J. Opt. Soc. Am. A (3)

Opt. Commun. (2)

W. H. Carter and E. Wolf, 'Scattering from quasi-homogeneous media,' Opt. Commun. 67, 85-90 (1988).
[Crossref]

H. Roychowdhury and E. Wolf, 'Invariance of spectrum of light generated by a class of quasi-homogeneous sources on propagation through turbulence,' Opt. Commun. 241, 11-15 (2004).
[Crossref]

Opt. Lett. (2)

Proc. Cambridge Philos. Soc. (1)

R. A. Silverman, 'Scattering of plane waves by locally homogeneous dielectric noise,' Proc. Cambridge Philos. Soc. 54, 530-537 (1958).
[Crossref]

Pure Appl. Opt. (1)

D. F. V. James, 'The Wolf effect and the redshift of quasars,' Pure Appl. Opt. 7, 959-970 (1998).
[Crossref]

Other (2)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th (expanded) ed. (Cambridge U. Press, 1999).
[PubMed]

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Figures (6)

Fig. 1
Fig. 1

Illustration of the notation.

Fig. 2
Fig. 2

Normalized spectral density S ( s ) ( r u , ω ) S ( s ) ( r s 0 , ω ) of the far field [Eq. (32)], as a function of the angle θ between the direction of incidence s 0 and the direction of scattering u, for selected values of the scaled correlation length k σ η .

Fig. 3
Fig. 3

Normalized time-averaged total scattered power P ( s ) k 3 A I ( i ) ( ω ) σ S 3 ( 2 π ) 4 , given by Eq. (34), as a function of the scaled correlation length k σ η of a Gaussian-correlated, random spherical scatterer.

Fig. 4
Fig. 4

Directions of observation u 1 and u 2 , located symmetrically with respect to the direction of incidence s 0 .

Fig. 5
Fig. 5

Spectral degree of coherence of the far field [Eq. (36)], for two symmetrically located directions of scattering, u 1 and u 2 , as a function of the angle ϕ between the vectors u 1 and u 2 , for selected values of the scaled effective radius k σ S of the sphere.

Fig. 6
Fig. 6

Illustration of the notation.

Equations (74)

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V ( i ) ( r , t ) = U ( i ) ( r , ω ) exp ( i ω t ) ,
U ( i ) ( r , ω ) = a ( ω ) exp ( i k s 0 r ) ( s 0 2 = 1 ) ,
U ( s ) ( r , ω ) = D F ( r , ω ) U ( i ) ( r , ω ) G ( r , r ; ω ) d 3 r ,
F ( r , ω ) = k 2 4 π [ n 2 ( r , ω ) 1 ]
G ( r , r ; ω ) = exp ( i k r r ) r r
C F ( r 1 , r 2 ; ω ) = F * ( r 1 , ω ) F ( r 2 , ω )
W ( s ) ( r 1 , r 2 ; ω ) = U ( s ) * ( r 1 , ω ) U ( s ) ( r 2 , ω ) .
W ( s ) ( r 1 , r 2 ; ω ) = D D W ( i ) ( r 1 , r 2 ; ω ) C F ( r 1 , r 2 ; ω ) G * ( r 1 , r 1 ; ω ) G ( r 2 , r 2 ; ω ) d 3 r 1 d 3 r 2 ,
W ( i ) ( r 1 , r 2 ; ω ) = U ( i ) * ( r 1 , ω ) U ( i ) ( r 2 , ω ) = I ( i ) ( ω ) exp [ i k s 0 ( r 2 , r 1 ) ] ,
G ( r u , r ; ω ) exp ( i k r ) r exp [ i k u r ] as k r ,
η F ( r 1 , r 2 ; ω ) = C F ( r 1 , r 2 ; ω ) S F ( r 1 , ω ) S F ( r 2 , ω ) ,
S F ( r , ω ) = C F ( r , r ; ω ) .
η F ( r 1 , r 2 ; ω ) η F ( r 2 r 1 ; ω ) .
S F ( r 1 , ω ) S F ( r 2 , ω ) S F [ ( r 1 + r 2 ) 2 , ω ] .
C F ( r 1 , r 2 ; ω ) = S F ( r 1 , ω ) S F ( r 2 , ω ) η F ( r 1 , r 2 ; ω ) ,
S F [ ( r 1 + r 2 ) 2 , ω ] η F ( r 2 r 1 ; ω ) .
R S = r 2 r 1 , R S + = ( r 1 + r 2 ) 2 .
W ( s ) ( r 1 u 1 , r 2 u 2 ; ω ) Λ ( r 1 , r 2 ; ω ) S ̃ F [ k ( u 1 u 2 ) , ω ] η ̃ F [ k ( s 0 ( u 1 + u 2 ) 2 ) ; ω ] ( u 1 2 = u 2 2 = 1 ) .
Λ ( r 1 , r 2 ; ω ) = I ( i ) ( ω ) exp [ i k ( r 2 r 1 ) ] r 1 r 2 ,
S ̃ F [ k u , ω ] = S F ( R S + , ω ) exp ( i k R S + u ) d 3 R S + ,
η ̃ F [ k u , ω ] = η F ( R S , ω ) exp ( i k R S + u ) d 3 R S
μ ( s ) ( r 1 , r 2 ; ω ) = W ( s ) ( r 1 , r 2 ; ω ) S ( s ) ( r 1 , ω ) S ( s ) ( r 2 , ω ) ,
S ( s ) ( r , ω ) = W ( s ) ( r , r ; ω ) .
S ( s ) ( r u , ω ) = I ( i ) ( ω ) S ̃ F ( 0 , ω ) r 2 η ̃ F [ k ( s 0 u ) , ω ] ;
μ ( s ) ( r 1 u 1 , r 2 u 2 ; ω )
= η ̃ F [ k ( s 0 ( u 1 + u 2 ) 2 ) , ω ] S ̃ F [ k ( u 1 u 2 ) , ω ] η ̃ F [ k ( s 0 u 1 ) , ω ] η ̃ F [ k ( s 0 u 2 ) , ω ] exp [ i k ( r 2 r 1 ) ] S ̃ F ( 0 , ω ) .
η ̃ F [ k ( s 0 u 1 ) , ω ] η ̃ F [ k ( s 0 u 2 ) , ω ] η ̃ F [ k ( s 0 ( u 1 + u 2 ) 2 ) , ω ] .
μ ( s ) ( r 1 u 1 , r 2 u 2 ; ω ) = S ̃ F [ k ( u 1 u 2 ) , ω ] S ̃ F ( 0 , ω ) exp [ i k ( r 2 r 1 ) ] .
η F ( R S ; ω ) = exp [ ( R S ) 2 2 σ η 2 ] ,
S F ( R S + , ω ) = A exp [ ( R S + ) 2 2 σ S 2 ] ,
η ̃ F ( k u , ω ) ( σ η 2 π ) 3 exp [ ( k σ η u ) 2 2 ] ,
S ̃ F ( k u , ω ) A ( σ S 2 π ) 3 exp [ ( k σ S u ) 2 2 ] .
S ( s ) ( r u , ω ) = ( σ η σ S 2 π ) 3 A I ( i ) ( ω ) r 2 exp [ 2 k 2 σ η 2 sin 2 ( θ 2 ) ] ,
P ( s ) = 0 π 0 2 π S ( s ) ( r u , ω ) r 2 sin θ d θ d ϕ ,
= A I ( i ) ( ω ) σ η 3 σ S 3 ( 2 π ) 4 [ 1 exp ( 2 k 2 σ η 2 ) k 2 σ η 2 ] .
μ ( s ) ( r 1 u 1 , r 2 u 2 ; ω ) = exp [ k 2 σ S 2 ( u 1 u 2 ) 2 2 ] exp [ i k ( r 2 r 1 ) ] .
μ ( s ) ( r u 1 , r u 2 ; ω ) = exp [ 2 k 2 σ S 2 sin 2 ( ϕ 2 ) ] ,
W Q ( r 1 , r 2 ; ω ) = U Q * ( r 1 , ω ) U Q ( r 2 , ω ) .
S Q ( r , ω ) = W Q ( r , r ; ω ) ,
μ Q ( r 1 , r 2 ; ω ) = W Q ( r 1 , r 2 ; ω ) S Q ( r 1 , ω ) S Q ( r 2 , ω ) .
μ Q ( r 1 , r 2 ) μ Q ( r 2 r 1 ) ;
W Q ( r 1 , r 2 ) S Q [ ( r 1 + r 2 ) 2 ] μ Q ( r 2 r 1 ) .
W ( i ) ( r 1 , r 2 ) = W Q ( r 1 , r 2 ) G * ( r 1 , r 1 ) G ( r 2 , r 2 ) d 3 r 1 d 3 r 2 ,
R Q = r 2 r 1 , R Q + = ( r 1 + r 2 ) 2
W ( i ) ( r 1 , r 2 ) exp [ i k ( r 2 r 1 ) ] r 1 r 2 S Q ( R Q + ) μ Q ( R Q ) exp [ i k r 1 ( R Q + R Q 2 ) r 1 ] exp [ i k r 2 ( R Q + + R Q 2 ) r 2 ] d 3 R Q + d 3 R Q ,
r 1 r 2 R 2 ,
Δ p k L Q L S 2 R ,
W ( ) ( r 1 , r 2 ) = D D W ( i ) ( r 1 , r 2 ) C F ( r 1 , r 2 ) G * ( r 1 , r 1 ) G ( r 2 , r 2 ) d 3 r 1 d 3 r 2 ,
C ¯ F ( r 1 , r 2 ) = C F ( r 1 , r 2 ) exp [ i k ( r 2 r 1 ) ] .
η ¯ F ( r 1 , r 2 ) = η F ( r 1 , r 2 ) exp [ i k ( r 2 r 1 ) ] .
r 2 r 1 ( r 2 r 1 ) r 2 r 2 ( r 2 r 1 ) R R .
C ¯ F ( r 1 r 2 ) = η ¯ F ( R S ) S F ( R S + ) ,
W ( ) ( r 1 s 1 , r 2 s 2 ) exp [ i k ( r 2 r 1 ) ] R 2 r 1 r 2 S Q ( R Q + ) μ Q ( R Q ) S F ( R S + ) η ¯ F ( R S ) exp ( i k Φ ) d 3 R Q + d 3 R Q d 3 R S + d 3 R S ,
Φ = r 1 ( R Q + R Q 2 ) R r 2 ( R Q + + R Q 2 ) R R S + ( s 2 s 1 ) R S ( s 1 s 2 ) 2 ,
= R S + [ ( s 2 s 1 ) + R Q R ] R S [ ( s 1 + s 2 ) 2 + R Q + R ] ,
W ( ) ( r 1 s 1 , r 2 s 2 ) = exp [ i k ( r 2 r 1 ) ] R 2 r 1 r 2 A ( s 2 s 1 ) B [ ( s 1 + s 2 ) 2 ] ,
A ( s 2 s 1 ) = μ Q ( R Q ) S ̃ F { k [ ( s 2 s 1 ) + R Q R ] } d 3 R Q ,
B [ ( s 1 + s 2 ) 2 ] = S Q ( R Q + ) η ̃ F { k [ ( s 1 + s 2 ) 2 + R Q + R ] } d 3 R Q + ,
S ̃ F ( k u ) = S F ( R S + ) exp ( i k R S + u ) d 3 R S + ,
η ̃ F ( k u ) = η ¯ F ( R S ) exp ( i k R S u ) d 3 R S ,
S ( ) ( r s ) = W ( ) ( r s , r s ) = 1 R 2 r 2 A ( 0 ) B ( s ) .
μ ( ) ( r 1 s 1 , r 2 s 2 ) = A ( s 2 s 1 ) B [ ( s 1 + s 2 ) 2 ] A ( 0 ) B ( s 1 ) B ( s 2 ) exp [ i k ( r 2 r 1 ) ] .
B ( s 1 ) B ( s 2 ) B [ ( s 1 + s 2 ) 2 ] .
μ ( ) ( r 1 s 1 , r 2 s 2 ) = A ( s 2 s 1 ) A ( 0 ) exp [ i k ( r 2 r 1 ) ] .
exp [ i k r 2 ( R Q + + R Q 2 ) r 2 ] = exp [ i k r 2 r 2 r 2 ] exp [ i k r 2 r 2 R ] .
Δ p = k r 2 r 2 [ 1 R 1 r 2 ] .
r 2 R ,
r 2 L Q ,
Δ p k R L Q 1 R 1 r 2 .
R L S ,
R L S 2 r 2 R + L S 2 .
[ 1 R 1 R ± L S 2 ] = ± L S 2 R 2 ± R L S 2 ± L S 2 R 2 ,
1 R 1 r 2 L S 2 R 2 .
Δ p k L Q L S 2 R ,

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