Abstract

Using scattering matrices and the angular spectrum representation of waves, we develop the analytical theory of scattering of random scalar waves from random collections of particles, valid under the first Born approximation. We demonstrate that in the calculation of far-field statistics, such as the spectral density and the spectral degree of coherence, the knowledge of the pair-structure factor of the collection is crucial. We illustrate our analytical approach by considering a numerical example involving scattering of two partially correlated plane waves from a random distribution of spheres.

© 2009 Optical Society of America

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References

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  1. L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).
  2. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Multiple Scattering of Light By Particles (Cambridge U. Press, 2006).
  3. A. Dogariu and E. Wolf, Opt. Lett. 23, 1340 (1998).
    [CrossRef]
  4. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge U. Press, 2007).
  5. S. Sahin and O. Korotkova, Phys. Rev. A 78, 063815 (2008).
    [CrossRef]
  6. G. Gbur and E. Wolf, Opt. Commun. 168, 39 (1999).
    [CrossRef]
  7. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).
  8. O. Korotkova and E. Wolf, Phys. Rev. E 75, 056609 (2007).
    [CrossRef]
  9. J. M. Ziman, Models of Disorder (Cambridge U. Press, 1979).
  10. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
  11. C. Brosseau, Polarized Light: A Statistical Optics Approach (Wiley, 1998).
  12. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge U. Press, 2002).

2008 (1)

S. Sahin and O. Korotkova, Phys. Rev. A 78, 063815 (2008).
[CrossRef]

2007 (2)

O. Korotkova and E. Wolf, Phys. Rev. E 75, 056609 (2007).
[CrossRef]

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

2006 (1)

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Multiple Scattering of Light By Particles (Cambridge U. Press, 2006).

2002 (1)

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge U. Press, 2002).

1999 (2)

G. Gbur and E. Wolf, Opt. Commun. 168, 39 (1999).
[CrossRef]

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

1998 (2)

A. Dogariu and E. Wolf, Opt. Lett. 23, 1340 (1998).
[CrossRef]

C. Brosseau, Polarized Light: A Statistical Optics Approach (Wiley, 1998).

1995 (1)

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

1985 (1)

L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).

1979 (1)

J. M. Ziman, Models of Disorder (Cambridge U. Press, 1979).

Born, M.

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

Brosseau, C.

C. Brosseau, Polarized Light: A Statistical Optics Approach (Wiley, 1998).

Dogariu, A.

Gbur, G.

G. Gbur and E. Wolf, Opt. Commun. 168, 39 (1999).
[CrossRef]

Kong, J. A.

L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).

Korotkova, O.

S. Sahin and O. Korotkova, Phys. Rev. A 78, 063815 (2008).
[CrossRef]

O. Korotkova and E. Wolf, Phys. Rev. E 75, 056609 (2007).
[CrossRef]

Lacis, A. A.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Multiple Scattering of Light By Particles (Cambridge U. Press, 2006).

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge U. Press, 2002).

Mandel, L.

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

Mishchenko, M. I.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Multiple Scattering of Light By Particles (Cambridge U. Press, 2006).

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge U. Press, 2002).

Sahin, S.

S. Sahin and O. Korotkova, Phys. Rev. A 78, 063815 (2008).
[CrossRef]

Shin, R. T.

L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).

Travis, L. D.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Multiple Scattering of Light By Particles (Cambridge U. Press, 2006).

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge U. Press, 2002).

Tsang, L.

L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).

Wolf, E.

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

O. Korotkova and E. Wolf, Phys. Rev. E 75, 056609 (2007).
[CrossRef]

G. Gbur and E. Wolf, Opt. Commun. 168, 39 (1999).
[CrossRef]

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

A. Dogariu and E. Wolf, Opt. Lett. 23, 1340 (1998).
[CrossRef]

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

Ziman, J. M.

J. M. Ziman, Models of Disorder (Cambridge U. Press, 1979).

Opt. Commun. (1)

G. Gbur and E. Wolf, Opt. Commun. 168, 39 (1999).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (1)

S. Sahin and O. Korotkova, Phys. Rev. A 78, 063815 (2008).
[CrossRef]

Phys. Rev. E (1)

O. Korotkova and E. Wolf, Phys. Rev. E 75, 056609 (2007).
[CrossRef]

Other (8)

J. M. Ziman, Models of Disorder (Cambridge U. Press, 1979).

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

C. Brosseau, Polarized Light: A Statistical Optics Approach (Wiley, 1998).

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge U. Press, 2002).

L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Multiple Scattering of Light By Particles (Cambridge U. Press, 2006).

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 (2)

Fig. 1
Fig. 1

Contours of the spectral density of the far field: (a) δ = 10 7   m , (b) δ = 10 6   m . The other parameters are λ = 632.8   nm , Δ = 10 7   m , σ = 10 7   m , θ 1 = π / 4 , ϕ 1 = π / 3 , θ 2 = π / 6 , ϕ 2 = π / 5 , θ 2 = θ 1 , and ϕ 2 = ϕ 1 .

Fig. 2
Fig. 2

Modulus of the degree of coherence as a function of θ d = θ 1 when θ 2 = 0 : δ = 10 7   m (dashed curve), δ = 5 × 10 7   m (dotted curve), δ = 10 6   m (dotted–dashed curve). The other parameters are λ = 632.8   nm , σ = 10 7   m , Δ = 10 7   m , θ 1 = π / 4 , ϕ 1 = π / 3 , θ 2 = π / 6 , ϕ 2 = π / 5 , ϕ 1 = π / 2 , and ϕ 2 = π / 2 .

Equations (18)

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f ( r ; ω ) = 4 π 2 k [ n 2 ( r ) 1 ] ,
F ( r , ω ) = n = 1 N f ( r r n , ω ) ,
M ( u 1 , u 1 ; u 2 , u 2 ; ω ) = S ( u 1 , u 1 ; ω ) S ( u 2 , u 2 ; ω ) r m
S [ K ; ω ] = F ( r ; ω ) ¯ ,
S ( K ; ω ) = n = 1 N e i r n K f ¯ ( K , ω ) .
M ( K 1 , K 2 ; ω ) = f ¯ ( K 1 , ω ) f ¯ ( K 2 , ω ) Q ( K 1 , K 2 ; ω ) ,
Q ( K 1 , K 2 ; ω ) = n = 1 N m = 1 N e i [ r m K 2 r n K 1 ] r m ,
S ( K ; ω ) = n = 1 N m = 1 N e i K [ r m r n ] r m .
q ( K 1 , K 2 ; ω ) = Q ( K 1 , K 2 ; ω ) S ( K 1 ; ω ) S ( K 2 ; ω ) ,
Q ( K 1 , K 2 ; ω ) = S ( K 1 ) S ( K 2 ) q ( | K 2 K 1 | ) ,
q ( | K 2 K 1 | ) = exp [ | K 2 K 1 | 2 ( k δ ) 2 ] ,
W ( i ) ( r 1 , r 2 , ω ) = U ( i ) ( r 1 , ω ) U ( i ) ( r 2 , ω ) ,
A ( i ) ( u 1 , u 2 , ω ) = k 4 ( 2 π ) 4 W ( i ) ( r 1 , r 2 , ω ) × exp [ i ( u 1 r 1 + u 2 r 2 ) ] d 2 r 1 d 2 r 2 ,
W ( t ) ( r u 1 , r u 2 , ω ) = ± 4 π 2 k 2 r 2 u 1 z u 2 z f ¯ ( K 1 ) f ¯ ( K 2 ) × Q ( K 1 , K 2 ) A ( i ) ( u 1 , u 2 , ω ) d 2 u 1 d 2 u 2 ,
S ( t ) ( r u ; ω ) = W ( t ) ( r u , r u ; ω ) ,
μ ( t ) ( r u 1 , r u 2 ; ω ) = W ( t ) ( r u 1 , r u 2 ; ω ) S ( t ) ( r u 1 ; ω ) S ( t ) ( r u 2 ; ω ) .
a ( u p , u q ; ω ) = a p q e ( k 2 Δ 2 / 2 ) ( u q u p ) 2     ( p , q = 1 , 2 ) .
f ( r n ; ω ) = B   exp [ ( x x n ) 2 + ( y y n ) 2 + ( z z n ) 2 2 σ 2 ] ,

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