Abstract

This paper presents a systematic analysis of the problem of multiple scattering by a finite group of arbitrarily sized, shaped, and oriented particles embedded in an absorbing, homogeneous, isotropic, and unbounded medium. The volume integral equation is used to derive generalized Foldy—Lax equations and their order-of-scattering form. The far-field version of the Foldy—Lax equations is used to derive the transport equation for the so-called coherent field generated by a large group of sparsely, randomly, and uniformly distributed particles. The differences between the generalized equations and their counterparts describing multiple scattering by particles embedded in a non-absorbing medium are highlighted and discussed.

©2008 Optical Society of America

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References

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  1. F. T. Ulaby and C. Elachi, eds., Radar Polarimetry for Geoscience Applications (Artech House, Norwood, Mass., 1990).
  2. L. Tsang and J. A. Kong, Scattering of Electromagnetic Waves: Advanced Topics (Wiley, New York, 2001).
  3. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering (Cambridge U. Press, Cambridge, UK, 2006).
  4. P. A. Martin, Multiple Scattering: Interaction of Time-Harmonic Waves with N Obstacles (Cambridge U. Press, Cambridge, UK, 2006).
    [Crossref]
  5. M. I. Mishchenko, L. Liu, D. W. Mackowski, B. Cairns, and G. Videen, “Multiple scattering by random particulate media: exact 3D results,” Opt. Express 15, 2822–2836 (2007).
    [Crossref] [PubMed]
  6. S. Durant, O. Calvo-Perez, N. Vukadinovic, and J.-J. Greffet, “Light scattering by a random distribution of particles embedded in absorbing media: diagrammatic expansion of the extinction coefficient,” J. Opt. Soc. Am. A 24, 2943–2952 (2007).
    [Crossref]
  7. M. I. Mishchenko, “Multiple scattering, radiative transfer, and weak localization in discrete random media: the unified microphysical approach,” Rev. Geophys.46, doi:10.1029/2007RG000230.
  8. P. Yang, B.-C. Gao, W. J. Wiscombe, M. I. Mishchenko, S. E. Platnick, H.-L. Huang, B. A. Baum, Y. X. Hu, D. M. Winker, S.-C. Tsay, and S. K. Park, “Inherent and apparent scattering properties of coated or uncoated spheres embedded in an absorbing host medium,” Appl. Opt. 41, 2740–2759 (2002).
    [Crossref] [PubMed]
  9. S. Durant, O. Calvo-Perez, N. Vukadinovic, and J.-J. Greffet, “Light scattering by a random distribution of particles embedded in absorbing media: full-wave Monte Carlo solutions of the extinction coefficient,” J. Opt. Soc. Am. A 24, 2953–2962 (2007).
    [Crossref]
  10. M. I. Mishchenko, “Electromagnetic scattering by a fixed finite object embedded in an absorbing medium,” Opt. Express 15, 13188–13202 (2007).
    [Crossref] [PubMed]
  11. V. Twersky, “On propagation in random media of discrete scatterers,” Proc. Symp. Appl. Math. 16, 84–116 (1964).
  12. L. L. Foldy, “The multiple scattering of waves,” Phys. Rev. 67, 107–119 (1945).
    [Crossref]
  13. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).

2007 (4)

2002 (1)

1964 (1)

V. Twersky, “On propagation in random media of discrete scatterers,” Proc. Symp. Appl. Math. 16, 84–116 (1964).

1945 (1)

L. L. Foldy, “The multiple scattering of waves,” Phys. Rev. 67, 107–119 (1945).
[Crossref]

Baum, B. A.

Cairns, B.

Calvo-Perez, O.

Durant, S.

Foldy, L. L.

L. L. Foldy, “The multiple scattering of waves,” Phys. Rev. 67, 107–119 (1945).
[Crossref]

Gao, B.-C.

Greffet, J.-J.

Hu, Y. X.

Huang, H.-L.

Kong, J. A.

L. Tsang and J. A. Kong, Scattering of Electromagnetic Waves: Advanced Topics (Wiley, New York, 2001).

Lacis, A. A.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering (Cambridge U. Press, Cambridge, UK, 2006).

Liu, L.

Mackowski, D. W.

Mandel, L.

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

Martin, P. A.

P. A. Martin, Multiple Scattering: Interaction of Time-Harmonic Waves with N Obstacles (Cambridge U. Press, Cambridge, UK, 2006).
[Crossref]

Mishchenko, M. I.

Park, S. K.

Platnick, S. E.

Travis, L. D.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering (Cambridge U. Press, Cambridge, UK, 2006).

Tsang, L.

L. Tsang and J. A. Kong, Scattering of Electromagnetic Waves: Advanced Topics (Wiley, New York, 2001).

Tsay, S.-C.

Twersky, V.

V. Twersky, “On propagation in random media of discrete scatterers,” Proc. Symp. Appl. Math. 16, 84–116 (1964).

Videen, G.

Vukadinovic, N.

Winker, D. M.

Wiscombe, W. J.

Wolf, E.

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

Yang, P.

Appl. Opt. (1)

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

Opt. Express (2)

Phys. Rev. (1)

L. L. Foldy, “The multiple scattering of waves,” Phys. Rev. 67, 107–119 (1945).
[Crossref]

Proc. Symp. Appl. Math. (1)

V. Twersky, “On propagation in random media of discrete scatterers,” Proc. Symp. Appl. Math. 16, 84–116 (1964).

Other (6)

F. T. Ulaby and C. Elachi, eds., Radar Polarimetry for Geoscience Applications (Artech House, Norwood, Mass., 1990).

L. Tsang and J. A. Kong, Scattering of Electromagnetic Waves: Advanced Topics (Wiley, New York, 2001).

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering (Cambridge U. Press, Cambridge, UK, 2006).

P. A. Martin, Multiple Scattering: Interaction of Time-Harmonic Waves with N Obstacles (Cambridge U. Press, Cambridge, UK, 2006).
[Crossref]

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

M. I. Mishchenko, “Multiple scattering, radiative transfer, and weak localization in discrete random media: the unified microphysical approach,” Rev. Geophys.46, doi:10.1029/2007RG000230.

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Figures (6)

Fig. 1.
Fig. 1. Scattering by a fixed group of N finite particles.

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Fig. 2.
Fig. 2. Diagrammatic representation of Eq. (14).

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Fig. 3.
Fig. 3. Scattering by widely separated particles. The local origins Oi and Oj are chosen arbitrarily inside particles i and j, respectively.

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Fig. 4.
Fig. 4. Electromagnetic scattering by a large group of particles sparsely distributed throughout a macroscopic volume V.

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Fig. 5.
Fig. 5. Diagrammatic representation of the Twersky expansion.

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Fig. 6.
Fig. 6. Computation of the coherent field.

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Equations (91)

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V INT = ∪ i = 1 N V i ,
E ( r ) = E inc ( r ) + ∫ ℜ 3 d r ′ G ↔ ( r , r ′ ) · E ( r ′ ) U ( r ′ )   ,             r ∈ ℜ 3 ,
G ↔ ( r , r ′ ) = ( I ↔ + 1 k 1 2 ∇ ⊗ ∇ ) exp ⁡ ( i k 1 ∣ r − r ′ ∣ ) 4 π ∣ r − r ′ ∣
U ( r ) = ∑ i = 1 N U i ( r ) ,           r ∈ ℜ 3
U i ( r ) = { 0 ,   r ∉ V i ,   k 1 2 [ m i 2 ( r ) − 1 ] , r ∈ V i ,
E ( r ) = E inc ( r ) + ∑ i = 1 N ∫ V i d r ′ G ↔ ( r , r ′ ) · ∫ V i d r ″ T ↔ i ( r ′ , r ″ ) · E i ( r ″ ) , r ∈ ℜ 3 ,
E i ( r ) = E inc ( r ) + ∑ j ( ≠ i ) = 1 N E ij exc ( r ) ,
E ij exc ( r ) = ∫ V j d r ′ G ↔ ( r , r ′ ) · ∫ V j d r ″ T ↔ j ( r ′ , r ″ ) · E j ( r ″ ) , r ∈ V i ,
T ↔ i ( r , r ′ ) = U i ( r ) δ ( r − r ′ ) I ↔ + U i ( r ) ∫ V i d r ″ G ↔ ( r , r ″ ) · T ↔ i ( r ″ , r ′ ) , r , r ′ ∈ V i
E = E inc + ∑ i = 1 N G ̂ T ̂ i E i ,
E i = E inc + ∑ j ( ≠ i ) = 1 N G ̂ T ̂ j E j ,
G ̂ T ̂ j E j = ∫ V j d r ′ G ↔ ( r , r ′ ) · ∫ V j d r ″ T ↔ j ( r ′ , r ″ ) · E j ( r ″ ) .
E i = E inc + ∑ j ( ≠ i ) = 1 N G ̂ T ̂ j E inc + ∑ j ( ≠ i ) = 1 l ( ≠ j ) = 1 N G ̂ T ̂ j G ̂ T ̂ l E inc + ∑ j ( ≠ i ) = 1 l ( ≠ j ) = 1 m ( ≠ l ) = 1 N G ̂ T ̂ j G ̂ T ̂ l G ̂ T ̂ m E inc + … ,
E = E inc + ∑ i = 1 N G ̂ T ̂ i E inc + ∑ i = 1 j ( ≠ i ) = 1 N G ̂ T ̂ i G ̂ T ̂ j E inc + ∑ i = 1 j ( ≠ i ) = 1 l ( ≠ j ) = 1 N G ̂ T ̂ i G ̂ T ̂ j G ̂ T ̂ l E inc + ··· .
E ij exc ( r ) ≈ G ( r j ) E 1 ij ( r ̂ j )
≈ exp ⁡ ( − i k 1 R ̂ ij · R i ) E ij exp ⁡ ( i k 1 R ̂ ij · r ) , r ∈ V i ,
G ( r ) = exp ⁡ ( i k 1 r ) r ,
E ij = G ( R ij ) E 1 ij ( R ̂ ij ) , E ij · R ̂ ij = 0 ,
r ̂ j = r j r j , R ̂ ij = R ij R ij ,
r j = ∣ R ij + r − R i ∣ ≈ R ij + R ̂ ij · ( r − R i ) + ∣ r − R i ∣ 2 2 R ij ,
E i ( r ) ≈ E 0 inc exp ( i k 1 n ̂ inc · r ) + ∑ j ( ≠ i ) = 1 N exp ( − i k 1 R ̂ ij · R i ) E ij exp ( i k 1 R ̂ ij · r ) , r ∊ V i ,
E inc ( r ) = E 0 inc exp ( i k 1 n ̂ inc · r ) , E 0 inc · n ̂ inc = 0 .
E j ( r ) ≈ E inc ( R j ) exp ( i k 1 n ̂ inc · r j ) + ∑ l ( ≠ j ) = 1 N E jl exp ( − i k 1 R ̂ jl · r j ) , r ∊ V j .
G ( R ij ) A ↔ j ( R ̂ ij , n ̂ inc ) · E inc ( R j ) + G ( R ij ) ∑ l ( ≠ j ) = 1 N A ↔ j ( R ̂ ij , R ̂ jl ) · E jl .
E ij = G ( R ij ) A ↔ j ( R ̂ ij , n ̂ inc ) · E inc ( R j ) + G ( R ij ) ∑ l ( ≠ j ) = 1 N A ↔ j ( R ̂ ij , R ̂ jl ) · E jl , i , j = 1 , … , N , j ≠ i .
E i ( r ″ ) ≈ E inc ( R i ) exp ⁡ ( i k 1 n ̂ inc · r i ″ ) + ∑ j ( ≠ i ) = 1 N E ij exp ⁡ ( i k 1 R ̂ ij · r i ″ )
E ( r ) = E inc ( r ) + Σ i = 1 N G ( r i ) A ↔ i ( r ̂ i , n ̂ inc ) · E inc ( R i ) + Σ i = 1 N G ( r i ) Σ j ( ≠ i ) = 1 N A ↔ i ( r ̂ i , R ̂ ij ) · E ij ,
E = E inc + ∑ i = 1 N B ↔ ri 0 · E i inc + ∑ i = 1 N ∑ j ( ≠ i ) = 1 N B ↔ rij · B ↔ ij 0 · E j inc + ∑ i = 1 N ∑ j ( ≠ i ) = 1 N ∑ l ( ≠ j ) = 1 N B ↔ rij · B ↔ ijl · B ↔ jl 0 · E l inc
+ … ,
E = E ( r ) ,   E inc = E inc ( r ) , E i inc = E inc ( R i ) ,
B ⃡ ri 0 = G ( r i ) A i ⃡ ( r i ̂ , n ̂ inc ) ,
B ⃡ rij = G ( r i ) A ⃡ i ( r ̂ i , R ̂ ij ) ,
B ⃡ ij 0 = G ( R ij ) A j ⃡ ( R ij ̂ , n ̂ inc ) ,
B ⃡ ijl = G ( R ij ) A ⃡ j ( R ̂ ij , R ̂ jl ) ,
E ≈ E inc + ∑ i = 1 N B ⃡ ri 0 · E i inc + ∑ i = 1 N ∑ j = 1 j ≠ i N B ⃡ rij · B ⃡ ij 0 · E j inc + ∑ i = 1 N ∑ j = 1 j ≠ i N ∑ l = 1 l ≠ i l ≠ j N B ⃡ rij · B ⃡ ijl · B ⃡ jl 0 · E l inc
+ ⋯ .
E ( r ) = E c ( r ) + E f ( r ) .
E c ( r ) = 〈 E ( r ) 〉 t = 〈 E ( r ) 〉 R , ξ ,
〈 E f ( r ) 〉 t = 〈 E f ( r ) 〉 R , ξ = 0 ,
+ ∑ i = 1 N ∑ j = 1 j ≠ i N ∑ l = 1 l ≠ i l ≠ j N 〈 B ↔ rij · B ↔ ijl · B ↔ jl 0 · E l inc 〉 R , ξ + ⋯
= E inc + ∑ i = 1 N ∫ d R i d ξ i p R ( R i ) p ξ ( ξ i ) B ↔ ri 0 · E i inc
+ ∑ i = 1 N ∑ j = 1 j ≠ i N ∫ d R i d ξ i d R j d ξ j p R ( R i ) p ξ ( ξ i ) p R ( R j ) p ξ ( ξ j ) B ↔ rij · B ↔ ij 0 · E j inc
+ ∑ i = 1 N ∑ j = 1 j ≠ i N ∑ l = 1 l ≠ i l ≠ j N ∫ d R i d ξ i d R j d ξ j d R l d ξ l p R ( R i ) p ξ ( ξ i ) p R ( R j )
                                                × p ξ ( ξ j ) p R ( R l ) p ξ ( ξ l ) B ↔ rij · B ↔ ijl · B ↔ jl 0 · E l inc
+ ⋯ ,
E c = E inc + ∑ i = 1 N ∫ V d R i p R ( R i ) G ( r i ) 〈 A ↔ ( r ̂ i , n ̂ inc ) 〉 ξ · E i inc
          + ∑ i = 1 N ∑ j = 1 j ≠ i N ∫ V d R i d R j p R ( R i ) p R ( R i ) G ( r i ) G ( R ij ) 〈 A ↔ ( r ̂ i , R ̂ ij ) 〉 ξ · 〈 A ↔ ( R ̂ ij , n ̂ inc ) 〉 ξ · E j inc
            + ∑ i = 1 N ∑ j = 1 j ≠ i N ∑ l = 1 l ≠ i l ≠ j N ∫ V d R i d R j d R l p R ( R i ) p R ( R j ) p R ( R l ) G ( r i ) G ( R ij ) G ( R jl ) 〈 A ↔ ( r ̂ i , R ̂ ij ) 〉 ξ
                                                                    · 〈 A ↔ ( R ̂ ij , R ̂ jl ) 〉 ξ · 〈 A ↔ ( R ̂ jl , n ̂ inc ) 〉 ξ · E l inc
          + ⋯ ,  
E c = N → ∞ E inc + ∫ V d R i n 0 ( R i ) G ( r i ) 〈 A ↔ ( r ̂ i , n ̂ inc ) 〉 ξ · E i inc
+ ∫ V d R i d R j n 0 ( R i ) n 0 ( R j ) G ( r i ) G ( R ij ) 〈 A ↔ ( r ̂ i , R ̂ ij ) 〉 ξ · 〈 A ↔ ( R ̂ ij , n ̂ inc ) 〉 ξ · E j inc
+ ∫ V d R i d R j d R l n 0 ( R i ) n 0 ( R j ) n 0 ( R l ) G ( r i ) G ( R ij ) G ( R jl )
× 〈 A ↔ ( r ̂ i , R ̂ ij ) 〉 ξ · 〈 A ↔ ( R ̂ ij , R ̂ jl ) 〉 ξ · 〈 A ↔ ( R ̂ jl , n ̂ inc ) 〉 ξ · E l inc
+ … .
I 1 = n 0 ∫ V d R i G ( r i ) 〈 A ↔ ( r ̂ i , n ̂ inc ) 〉 ξ · E i inc
= n 0 ∫ V d R i ′ exp ( i k 1 n ̂ inc · R i ′ ) exp ( i k 1 R i ′ ) R i ′ 〈 A ↔ ( − R ̂ i ′ , n ̂ inc ) 〉 ξ · E inc ( r ) ,
exp ⁡ ( i k 1 n ̂ inc · R i ′ ) = k 1 ′ R i ′ → ∞ exp ⁡ ( − k 1 ″ n ̂ inc · R i ′ ) i 2 π k 1 ′ R i ′ [ δ ( n ̂ inc + R ̂ i ′ ) exp ⁡ ( − i k 1 ′ R i ′ ) − δ ( n ̂ inc − R ̂ i ′ ) exp ⁡ ( i k 1 ′ R i ′ ) ] .
I 1 = i 2 π n 0 k 1 ′ ∫ 4 π d R ̂ i ′ ∫ d R i ′ 〈 A ↔ ( − R ̂ i ′ n ̂ inc ) 〉 ξ · E inc ( r ) [ δ ( n ̂ inc + R ̂ i ′ ) − δ ( n ̂ inc − R ̂ i ′ ) exp ( 2 i k 1 R i ′ ) ]
= i 2 π n 0 k 1 ′ s ( r ) 〈 A ↔ ( n ̂ inc , n ̂ inc ) 〉 ξ · E inc ( r ) − π n 0 k 1 k 1 ′ exp { i 2 k 1 [ s ( B ) − s ( r ) ] } 〈 A ↔ ( − n ̂ inc , n ̂ inc ) 〉 ξ · E inc ( r ) .
I 1 = i 2 π n 0 k 1 ′ s ( r ) 〈 A ↔ ( n ̂ inc , n ̂ inc ) 〉 ξ · E inc ( r ) .
I 2 = n 0 2 ∫ d R i ′ R i 2 G ( R i ′ ) ∫ 4 π d R ̂ i ′ ∫ d R ji R ji 2 G ( R ji ) ∫ 4 π d R ̂ ji
× 〈 A ↔ ( − R ̂ i ′ , − R ̂ ji ) 〉 ξ · 〈 A ↔ ( − R ̂ ji , n ̂ inc ) 〉 ξ · E j inc ,
E j inc = ( i 2 π k 1 ′ ) 2 1 R i ′ exp ( − k 1 ″ n ̂ inc · R i ′ ) [ δ ( n ̂ inc + R ̂ i ′ ) exp ( − i k 1 ′ R i ′ ) − δ ( n ̂ inc − R ̂ i ′ ) exp ( i k 1 ′ R i ′ ) ]
× 1 R ji exp ( − k 1 ″ n ̂ inc · R ji ) [ δ ( n ̂ inc + R ̂ ji ) exp ( − i k 1 ′ R ji ) − δ ( n ̂ inc − R ̂ ji ) exp ( i k 1 ′ R ji ) ] E inc ( r ) .
I 2 = 1 2 [ i 2 π n 0 k 1 ′ s ( r ) ] 2 〈 A ↔ ( n ̂ inc , n ̂ inc ) 〉 ξ · 〈 A ↔ ( n ̂ inc , n ̂ inc ) 〉 ξ · E inc ( r ) .
E c ( r ) = exp [ i 2 π n 0 k 1 ′ s ( n ̂ inc ) 〈 A ↔ ( n ̂ inc , n ̂ inc ) 〉 ξ ] · E inc ( r ) .
E c ( r ) = exp [ i κ ↔ ( n ̂ inc ) s ( r ) ] · E inc ( r A )
E c ( s ) = η ↔ ( n ̂ inc , s ) · E c ( s = 0 ) ,
κ ↔ ( n ̂ inc ) = k 1 I ↔ + 2 π n 0 k 1 ′ 〈 A ↔ ( n ̂ inc , n ̂ inc ) 〉 ξ
η ↔ ( n ̂ inc , s ) = exp [ i κ ↔ ( n ̂ inc ) s ]
d E c ( r ) d s = i κ ↔ ( n ̂ inc ) · E c ( r ) .
E c ( r ) = E c θ ( r ) θ ̂ ( n ̂ inc ) + E c φ ( r ) φ ̂ ( n ̂ inc ) , n ̂ inc = θ ̂ ( n ̂ inc ) × φ ̂ ( n ̂ inc ) .
E c ( r ) = [ E c θ ( r ) E c φ ( r ) ] ,
d E c ( r ) d s = i k ( n ̂ inc ) E c ( r ) ,
k 11 ( n ̂ inc ) = θ ̂ ( n ̂ inc ) · κ ↔ ( n ̂ inc ) · θ ̂ ( n ̂ inc ) ,
k 12 ( n ̂ inc ) = θ ̂ ( n ̂ inc ) · κ ↔ ( n ̂ inc ) · φ ̂ ( n ̂ inc ) ,
k 21 ( n ̂ inc ) = φ ̂ ( n ̂ inc ) · κ ↔ ( n ̂ inc ) · θ ̂ ( n ̂ inc ) ,
k 22 ( n ̂ inc ) = φ ̂ ( n ̂ inc ) · κ ↔ ( n ̂ inc ) · φ ̂ ( n ̂ inc ) .
k ( n ̂ inc ) = k 1 [ 1 0 0 1 ] + 2 π n 0 k 1 ′ 〈 S ( n ̂ inc , n ̂ inc ) 〉 ξ ,
E c ( s ) = h ( n ̂ inc , s ) E c ( s = 0 ) ,
h ( n ̂ inc , s ) = exp [ i s k ( n ̂ inc ) ]
η ↔ ( − n ̂ inc , s ) = [ η ↔ ( n ̂ inc , s ) ] T ,
h ( − n ̂ inc , s ) = [ 1 0 0 − 1 ] [ h ( n ̂ inc , s ) ] T [ 1 0 0 − 1 ] ,
J c = Re ( k 1 2 ω μ 1 ) [ E c θ E c θ * E c θ E c φ * E c φ E c θ * E c φ E c φ * ] ,
d J c ( r ) d s = − 2 k 1 ″ J c ( r ) − n 0 〈 K J ( n ̂ inc ) 〉 ξ J c ( r ) ,
I c = D J c = Re ( k 1 2 ω μ 1 ) [ E c θ E c θ * + E c φ E c φ * E c θ E c θ * − E c φ E c φ * − 2 Re ( E c θ E c φ * ) 2 Im ( E c θ E c φ * ) ]
d I c ( r ) d s = − 2 k 1 ″ I c ( r ) − n 0 〈 K ( n ̂ inc ) 〉 ξ I c ( r ) ,
I c ( r ) = H [ n ̂ inc , s ( r ) ] I c ( r A ) ,
H ( n ̂ inc , s ) = exp [ − 2 k 1 ″ s − n 0 s 〈 K ( n ̂ inc ) 〉 ξ ]
H ( − n ̂ inc , s ) = Δ 3 [ H ( n ̂ inc , s ) ] T Δ 3 ,

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