Abstract

We investigate the case when light is scattered by Gaussian-correlated, quasi-homogeneous, anisotropic media. The analytical expression for the cross-spectral density function of the scattered field that is produced by scattering of a polychromatic plane wave incident upon a Gaussian-correlated, quasi-homogeneous, anisotropic medium is derived by use of a tensor method. Numerical examples are given to illustrate the normalized spectral density and the spectral degree of coherence of the field scattered by the anisotropic scatterer in contrast with that scattered by the isotropic scatterer.

© 2010 Optical Society of America

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References

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  1. W. H. Carter and E. Wolf, Opt. Commun. 67, 85 (1988).
    [CrossRef]
  2. D. G. Fischer and E. Wolf, J. Opt. Soc. Am. A 11, 1128 (1994).
    [CrossRef]
  3. D. G. Fischer and E. Wolf, Opt. Commun. 133, 17 (1997).
    [CrossRef]
  4. T. D. Visser, D. G. Fischer, and E. Wolf, J. Opt. Soc. Am. A 23, 1631 (2006).
    [CrossRef]
  5. M. Lahiri, E. Wolf, D. G. Fischer, and T. Shirai, Phys. Rev. Lett. 102, 123901 (2009).
    [CrossRef] [PubMed]
  6. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge U. Press, 2007).
  7. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

2009 (1)

M. Lahiri, E. Wolf, D. G. Fischer, and T. Shirai, Phys. Rev. Lett. 102, 123901 (2009).
[CrossRef] [PubMed]

2006 (1)

1997 (1)

D. G. Fischer and E. Wolf, Opt. Commun. 133, 17 (1997).
[CrossRef]

1994 (1)

1988 (1)

W. H. Carter and E. Wolf, Opt. Commun. 67, 85 (1988).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

Carter, W. H.

W. H. Carter and E. Wolf, Opt. Commun. 67, 85 (1988).
[CrossRef]

Fischer, D. G.

M. Lahiri, E. Wolf, D. G. Fischer, and T. Shirai, Phys. Rev. Lett. 102, 123901 (2009).
[CrossRef] [PubMed]

T. D. Visser, D. G. Fischer, and E. Wolf, J. Opt. Soc. Am. A 23, 1631 (2006).
[CrossRef]

D. G. Fischer and E. Wolf, Opt. Commun. 133, 17 (1997).
[CrossRef]

D. G. Fischer and E. Wolf, J. Opt. Soc. Am. A 11, 1128 (1994).
[CrossRef]

Lahiri, M.

M. Lahiri, E. Wolf, D. G. Fischer, and T. Shirai, Phys. Rev. Lett. 102, 123901 (2009).
[CrossRef] [PubMed]

Shirai, T.

M. Lahiri, E. Wolf, D. G. Fischer, and T. Shirai, Phys. Rev. Lett. 102, 123901 (2009).
[CrossRef] [PubMed]

Visser, T. D.

Wolf, E.

M. Lahiri, E. Wolf, D. G. Fischer, and T. Shirai, Phys. Rev. Lett. 102, 123901 (2009).
[CrossRef] [PubMed]

T. D. Visser, D. G. Fischer, and E. Wolf, J. Opt. Soc. Am. A 23, 1631 (2006).
[CrossRef]

D. G. Fischer and E. Wolf, Opt. Commun. 133, 17 (1997).
[CrossRef]

D. G. Fischer and E. Wolf, J. Opt. Soc. Am. A 11, 1128 (1994).
[CrossRef]

W. H. Carter and E. Wolf, Opt. Commun. 67, 85 (1988).
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge U. Press, 2007).

J. Opt. Soc. Am. A (2)

Opt. Commun. (2)

W. H. Carter and E. Wolf, Opt. Commun. 67, 85 (1988).
[CrossRef]

D. G. Fischer and E. Wolf, Opt. Commun. 133, 17 (1997).
[CrossRef]

Phys. Rev. Lett. (1)

M. Lahiri, E. Wolf, D. G. Fischer, and T. Shirai, Phys. Rev. Lett. 102, 123901 (2009).
[CrossRef] [PubMed]

Other (2)

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge U. Press, 2007).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

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Figures (4)

Fig. 1
Fig. 1

Illustrating the notation relating to scattering.

Fig. 2
Fig. 2

Normalized spectral density of the scattered field in the far zone as a function of the direction of scattering s. The parameters are chosen as follows: (a) σ μ x = σ μ y = σ μ z = 2 λ ; (b) σ μ x = λ , σ μ y = 2 λ , and σ μ z = 3 λ .

Fig. 3
Fig. 3

Spectral degree of coherence of the scattered field in the far zone, for two directions of scattering s and incidence s 0 , as a function of s. The parameters are chosen as follows: (a) σ I x = σ I y = σ I z = 20 λ and σ μ x = σ μ y = σ μ z = 2 λ ; (b) σ I x = 10 λ , σ I y = 20 λ , σ I z = 30 λ , σ μ x = λ , σ μ y = 2 λ , and σ μ z = 3 λ .

Fig. 4
Fig. 4

Illustrating the isotropic case ( σ I x = σ I y = σ I z = 20 λ and σ μ x = σ μ y = σ μ z = 2 λ ) and the anisotropic case ( σ I x = 10 λ , σ I y = 20 λ , σ I z = 30 λ , σ μ x = λ , σ μ y = 2 λ , and σ μ z = 3 λ ) of scattering when s y = 0 . (a) Normalized spectral density as a function of s x . (b) Spectral degree of coherence as a function of s x .

Equations (19)

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W ( i ) ( r 1 , r 2 , ω ) = U ( i ) * ( r 1 , ω ) U ( i ) ( r 2 , ω ) ,
U ( i ) ( r , ω ) = a ( ω ) exp ( i k s 0 r ) ,
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 ,
C F ( r 1 , r 2 , ω ) = F * ( r 1 , ω ) F ( r 2 , ω ) m
W ( s ) ( r s 1 , r s 2 , ω ) = 1 r 2 S ( i ) ( ω ) C ̃ F [ k ( s 1 s 0 ) , k ( s 2 s 0 ) , ω ] ,
C ̃ F ( K 1 , K 2 , ω ) = D D C F ( r 1 , r 2 , ω ) exp [ i ( K 1 r 1 + K 2 r 2 ) ] d 3 r 1 d 3 r 2
C F ( r 1 , r 2 , ω ) = C 0 exp ( | R + | 2 2 σ I 2 ) exp ( | R | 2 2 σ μ 2 ) ,
C F ( x 1 , y 1 , z 1 , x 2 , y 2 , z 2 , ω ) = C 0 exp [ ( x 1 + x 2 ) 2 8 σ I x 2 ( y 1 + y 2 ) 2 8 σ I y 2 ( z 1 + z 2 ) 2 8 σ I z 2 ] × exp [ ( x 1 x 2 ) 2 2 σ μ x 2 ( y 1 y 2 ) 2 2 σ μ y 2 ( z 1 z 2 ) 2 2 σ μ z 2 ] .
C F ( r 12 , ω ) = C 0 exp ( r 12 T M ¯ r 12 ) ,
M ¯ = [ M + M M M + ] ,
M ± = [ 1 8 σ I x 2 ± 1 2 σ μ x 2 0 0 0 1 8 σ I y 2 ± 1 2 σ μ y 2 0 0 0 1 8 σ I z 2 ± 1 2 σ μ z 2 ] .
C ̃ F ( K 12 , ω ) = C F ( r 12 , ω ) exp ( i r 12 T K 12 ) d 6 r 12 ,
C ̃ F ( K 12 , ω ) = π 3 C 0 [ Det ( M ¯ ) ] 1 2 exp ( 1 4 K 12 T M ¯ 1 K 12 ) ,
K 12 = k I ¯ ( s 12 s 00 ) ,
I ¯ = [ I 0 0 I ] ,
W ( s ) ( r s 12 , ω ) = π 3 C 0 r 2 S ( i ) ( ω ) [ Det ( M ¯ ) ] 1 2 × exp [ 1 4 k 2 ( s 12 s 00 ) T I ¯ T M ¯ 1 I ¯ ( s 12 s 00 ) ] .
W ( s ) ( r s 12 , ω ) = 8 π 3 C 0 r 2 S ( i ) ( ω ) σ I x σ I y σ I z σ μ x σ μ y σ μ z × exp { 1 2 k 2 [ ( s 1 x s 2 x ) 2 σ I x 2 + ( s 1 y s 2 y ) 2 σ I y 2 + ( s 1 z s 2 z ) 2 σ I z 2 + ( s 1 x + s 2 x 2 s 0 x ) 2 σ μ x 2 + ( s 1 y + s 2 y 2 s 0 y ) 2 σ μ y 2 + ( s 1 z + s 2 z 2 s 0 z ) 2 σ μ z 2 ] } .
S ( s ) ( r s , ω ) = 8 π 3 C 0 r 2 S ( i ) ( ω ) σ I x σ I y σ I z σ μ x σ μ y σ μ z × exp { 1 2 k 2 [ ( s x s 0 x ) 2 σ μ x 2 + ( s y s 0 y ) 2 σ μ y 2 + ( s z s 0 z ) 2 σ μ z 2 ] } .
μ ( s ) ( r s 12 , ω ) = W ( s ) ( r s 12 , ω ) S ( s ) ( r s 1 , ω ) S ( s ) ( r s 2 , ω ) .

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