Abstract

The solution for scattering by a layer of densely distributed infinite cylinders is presented. The layer is irradiated by an arbitrarily polarized plane wave that propagates in the plane perpendicular to the axes of the cylinders. The theoretical formulation utilized the effective field and quasi-crystalline approximation to treat the multiple scattering interactions in the dense finite medium. Governing equations for the propagation constants and amplitudes of the effective fields are derived for TM and TE mode incident waves, from which the scattered intensity distribution and scattering cross section for arbitrary polarization are obtained. The dense medium gives rise to coherent and incoherent scattered radiation that propagates in the plane normal to the axes of the cylinders. The coherent scattered radiation includes the forward component in the direction of the incident wave and the backward component in the direction of specular reflection. The incoherent scattered intensity distribution shows a pronounced forward peak that coincides with the angle of refraction of the effective waves inside the medium. Numerical results are presented to illustrate the scattering characteristics of a dense layer of cylinders as a function of layer thickness for a given solid volume fraction.

© 2008 Optical Society of America

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References

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  1. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, 1969).
  2. H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).
  3. S. C. Lee and G. R. Cunnington, “Conduction and radiation heat transfer in high-porosity fiber thermal insulation,” J. Thermophys. Heat Transfer 14, 121-136 (2000).
    [CrossRef]
  4. V. Twersky, “Multiple scattering of radiation by an arbitrary configuration of parallel cylinders,” J. Acoust. Soc. Am. 24, 42-46 (1952).
    [CrossRef]
  5. G. O. Oloafe, “Scattering by an arbitrary configuration of parallel cylinders,” J. Opt. Soc. Am. 60, 1233-1236 (1970).
    [CrossRef]
  6. D. Felbacq, G. Tayeb, and D. Maystre, “Scattering by a random set of parallel cylinders,” J. Opt. Soc. Am. A 11, 2526-2538 (1994).
    [CrossRef]
  7. S. C. Lee, “Dependent scattering of an obliquely incident plane wave by a collection of parallel cylinders,” J. Appl. Phys. 68, 4952-4957 (1990).
    [CrossRef]
  8. S. C. Lee, “Scattering by closely-spaced radially stratified parallel cylinders,” J. Quant. Spectrosc. Radiat. Transf. 48, 119-130 (1992).
    [CrossRef]
  9. S. C. Lee, “Scattering of polarized radiation by an arbitrary collection of closely spaced parallel nonhomogeneous tilted cylinders,” J. Opt. Soc. Am. A 13, 2256-2265 (1996).
    [CrossRef]
  10. L. L. Foldy, “The multiple scattering of waves. I. General theory of isotropic scattering by randomly distributed scatterers,” Phys. Rev. 67, 107-119 (1945).
    [CrossRef]
  11. M. Lax, “Multiple scattering of waves. II. The effective field in dense systems,” Phys. Rev. 85, 621-629 (1952).
    [CrossRef]
  12. L. Tsang and J. A. Kong, Scattering of Electromagnetic Waves. Advanced Topics (Wiley, 2001).
  13. S. C. Lee, “Effective propagation constant of fibrous media containing parallel fibers in the dependent scattering regime,” J. Heat Transfer 114, 473-478 (1992).
    [CrossRef]
  14. S. C. Lee, “Propagation of radiation in high-density fibrous composites containing coated fibers,” J. Thermophys. Heat Transfer 7, 637-643 (1993).
    [CrossRef]
  15. S. C. Lee, “Scattering characteristics of fibrous media containing closely spaced parallel fibers,” J. Thermophys. Heat Transfer 9, 403-409 (1995).
    [CrossRef]
  16. M. Born and A. Wolf, Principles of Optics (Pergamon, 1993).
  17. W. W. Wood, “NpT-Ensemble Monte Carlo calculations for the hard-disk fluid,” J. Chem. Phys. 52, 729-741 (1970).
    [CrossRef]

2000 (1)

S. C. Lee and G. R. Cunnington, “Conduction and radiation heat transfer in high-porosity fiber thermal insulation,” J. Thermophys. Heat Transfer 14, 121-136 (2000).
[CrossRef]

1996 (1)

1995 (1)

S. C. Lee, “Scattering characteristics of fibrous media containing closely spaced parallel fibers,” J. Thermophys. Heat Transfer 9, 403-409 (1995).
[CrossRef]

1994 (1)

1993 (1)

S. C. Lee, “Propagation of radiation in high-density fibrous composites containing coated fibers,” J. Thermophys. Heat Transfer 7, 637-643 (1993).
[CrossRef]

1992 (2)

S. C. Lee, “Effective propagation constant of fibrous media containing parallel fibers in the dependent scattering regime,” J. Heat Transfer 114, 473-478 (1992).
[CrossRef]

S. C. Lee, “Scattering by closely-spaced radially stratified parallel cylinders,” J. Quant. Spectrosc. Radiat. Transf. 48, 119-130 (1992).
[CrossRef]

1990 (1)

S. C. Lee, “Dependent scattering of an obliquely incident plane wave by a collection of parallel cylinders,” J. Appl. Phys. 68, 4952-4957 (1990).
[CrossRef]

1970 (2)

W. W. Wood, “NpT-Ensemble Monte Carlo calculations for the hard-disk fluid,” J. Chem. Phys. 52, 729-741 (1970).
[CrossRef]

G. O. Oloafe, “Scattering by an arbitrary configuration of parallel cylinders,” J. Opt. Soc. Am. 60, 1233-1236 (1970).
[CrossRef]

1952 (2)

M. Lax, “Multiple scattering of waves. II. The effective field in dense systems,” Phys. Rev. 85, 621-629 (1952).
[CrossRef]

V. Twersky, “Multiple scattering of radiation by an arbitrary configuration of parallel cylinders,” J. Acoust. Soc. Am. 24, 42-46 (1952).
[CrossRef]

1945 (1)

L. L. Foldy, “The multiple scattering of waves. I. General theory of isotropic scattering by randomly distributed scatterers,” Phys. Rev. 67, 107-119 (1945).
[CrossRef]

Born, M.

M. Born and A. Wolf, Principles of Optics (Pergamon, 1993).

Cunnington, G. R.

S. C. Lee and G. R. Cunnington, “Conduction and radiation heat transfer in high-porosity fiber thermal insulation,” J. Thermophys. Heat Transfer 14, 121-136 (2000).
[CrossRef]

Felbacq, D.

Foldy, L. L.

L. L. Foldy, “The multiple scattering of waves. I. General theory of isotropic scattering by randomly distributed scatterers,” Phys. Rev. 67, 107-119 (1945).
[CrossRef]

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, 1969).

Kong, J. A.

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

Lax, M.

M. Lax, “Multiple scattering of waves. II. The effective field in dense systems,” Phys. Rev. 85, 621-629 (1952).
[CrossRef]

Lee, S. C.

S. C. Lee and G. R. Cunnington, “Conduction and radiation heat transfer in high-porosity fiber thermal insulation,” J. Thermophys. Heat Transfer 14, 121-136 (2000).
[CrossRef]

S. C. Lee, “Scattering of polarized radiation by an arbitrary collection of closely spaced parallel nonhomogeneous tilted cylinders,” J. Opt. Soc. Am. A 13, 2256-2265 (1996).
[CrossRef]

S. C. Lee, “Scattering characteristics of fibrous media containing closely spaced parallel fibers,” J. Thermophys. Heat Transfer 9, 403-409 (1995).
[CrossRef]

S. C. Lee, “Propagation of radiation in high-density fibrous composites containing coated fibers,” J. Thermophys. Heat Transfer 7, 637-643 (1993).
[CrossRef]

S. C. Lee, “Scattering by closely-spaced radially stratified parallel cylinders,” J. Quant. Spectrosc. Radiat. Transf. 48, 119-130 (1992).
[CrossRef]

S. C. Lee, “Effective propagation constant of fibrous media containing parallel fibers in the dependent scattering regime,” J. Heat Transfer 114, 473-478 (1992).
[CrossRef]

S. C. Lee, “Dependent scattering of an obliquely incident plane wave by a collection of parallel cylinders,” J. Appl. Phys. 68, 4952-4957 (1990).
[CrossRef]

Maystre, D.

Oloafe, G. O.

Tayeb, G.

Tsang, L.

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

Twersky, V.

V. Twersky, “Multiple scattering of radiation by an arbitrary configuration of parallel cylinders,” J. Acoust. Soc. Am. 24, 42-46 (1952).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).

Wolf, A.

M. Born and A. Wolf, Principles of Optics (Pergamon, 1993).

Wood, W. W.

W. W. Wood, “NpT-Ensemble Monte Carlo calculations for the hard-disk fluid,” J. Chem. Phys. 52, 729-741 (1970).
[CrossRef]

J. Acoust. Soc. Am. (1)

V. Twersky, “Multiple scattering of radiation by an arbitrary configuration of parallel cylinders,” J. Acoust. Soc. Am. 24, 42-46 (1952).
[CrossRef]

J. Appl. Phys. (1)

S. C. Lee, “Dependent scattering of an obliquely incident plane wave by a collection of parallel cylinders,” J. Appl. Phys. 68, 4952-4957 (1990).
[CrossRef]

J. Chem. Phys. (1)

W. W. Wood, “NpT-Ensemble Monte Carlo calculations for the hard-disk fluid,” J. Chem. Phys. 52, 729-741 (1970).
[CrossRef]

J. Heat Transfer (1)

S. C. Lee, “Effective propagation constant of fibrous media containing parallel fibers in the dependent scattering regime,” J. Heat Transfer 114, 473-478 (1992).
[CrossRef]

J. Opt. Soc. Am. (1)

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

J. Quant. Spectrosc. Radiat. Transf. (1)

S. C. Lee, “Scattering by closely-spaced radially stratified parallel cylinders,” J. Quant. Spectrosc. Radiat. Transf. 48, 119-130 (1992).
[CrossRef]

J. Thermophys. Heat Transfer (3)

S. C. Lee and G. R. Cunnington, “Conduction and radiation heat transfer in high-porosity fiber thermal insulation,” J. Thermophys. Heat Transfer 14, 121-136 (2000).
[CrossRef]

S. C. Lee, “Propagation of radiation in high-density fibrous composites containing coated fibers,” J. Thermophys. Heat Transfer 7, 637-643 (1993).
[CrossRef]

S. C. Lee, “Scattering characteristics of fibrous media containing closely spaced parallel fibers,” J. Thermophys. Heat Transfer 9, 403-409 (1995).
[CrossRef]

Phys. Rev. (2)

L. L. Foldy, “The multiple scattering of waves. I. General theory of isotropic scattering by randomly distributed scatterers,” Phys. Rev. 67, 107-119 (1945).
[CrossRef]

M. Lax, “Multiple scattering of waves. II. The effective field in dense systems,” Phys. Rev. 85, 621-629 (1952).
[CrossRef]

Other (4)

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

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, 1969).

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).

M. Born and A. Wolf, Principles of Optics (Pergamon, 1993).

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Figures (9)

Fig. 1
Fig. 1

Schematic diagram of the scattering problem.

Fig. 2
Fig. 2

Effective propagation constant as a function of solid volume fraction.

Fig. 3
Fig. 3

Extinction efficiency as a function of solid volume fraction.

Fig. 4
Fig. 4

Coherent forward-scattered intensity as a function of thickness.

Fig. 5
Fig. 5

Collimated transmission as a function of thickness.

Fig. 6
Fig. 6

Coherent backscattered intensity as a function of thickness.

Fig. 7
Fig. 7

Incoherent scattered intensity distribution for α D = 30 .

Fig. 8
Fig. 8

Scattering cross sections ( T M mode) as a function of thickness.

Fig. 9
Fig. 9

Scattering cross sections ( T E mode) as a function of thickness.

Equations (50)

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( u i v i ) = ( α I o α II o ) exp ( i k i R p ) ,
S i = c o 8 π m i k i 2 e i ,
e i = cos θ i e x sin θ i e y .
u s = j = 1 N n = ( i ) n exp ( i n γ j p ) H n ( k i R j p ) b j n I ,
b j n I = ε j exp ( i n θ i ) b j n o , I k j N s = G k s j n b j n o , I b k s I ,
G k s j n = ( i ) s n exp [ i ( s n ) γ k j ] H s n ( k i R j k )
ψ s j k = ψ s p ( R 1 R 2 R N ) d N r d V 1 d V n d V N , n j , k ,
p ( R j , R k ) = { g ( R j k ) V R j R k > R o 0 R j R k R o }
b j n I j k = b j n I j .
b j n I j τ = ± X n τ exp ( i K τ R j ) ,
K sin θ t = k i sin θ i .
k i 2 K 2 = k i x 2 K x 2 ,
τ = ± s = exp ( i K τ R j ) { δ s n + n o F s n τ b n o , I } X s τ = ε j exp ( i n θ i ) b n o , I ,
F s n τ = A o G k s j n Ψ j k τ [ g ( R j k ) 1 ] d A k + A o G k s j n Ψ j k τ d A k .
b n o , I = b n o , I ( m ̃ , r ) d 2 F ,
F s n , 1 τ = 2 π τ s n exp [ τ i ( s n ) θ t ] R o H s n ( k i R ) J s n ( K R ) [ g ( R ) 1 ] R d R ,
F s n , 2 τ = 1 ( k i 2 K 2 ) s [ G s n j k Ψ j k τ Ψ j k τ G s n j k ] d S k .
F s n , 2 τ ( S c ) = 2 π τ s n exp [ τ i ( s n ) θ t ] ( k i 2 K 2 ) { k i R o H s n ( k i R o ) J s n ( K R o ) K R o H s n ( k i R o ) J s n ( K R o ) } ,
F s n , 2 τ ( S b ) = 2 i k i x { k i x τ K x k i x 2 K x 2 ( 1 ) s n exp [ i ( s n ) θ i ] exp [ i ( k i x + τ K x ) ( D x j ) ] + k i x + τ K x k i x 2 K x 2 exp [ i ( s n ) θ i ] exp [ i ( k i x τ K x ) x j ] }
2 n o i τ = ± ( k i x + τ K x ) k i x ( k i x 2 K x 2 ) X τ ( θ i ) = 1 ,
τ = ± ( k i x τ K x ) exp ( i τ K x D ) X τ ( π + θ i ) = 0 ,
[ δ s n + n o f s n b n o , I ] [ τ s X s τ exp ( τ i s θ t ) ] = 0 ,
X τ ( γ ) = s = X s τ exp ( i s γ ) ,
f s n = 2 π { R o H s n ( k i R ) J s n ( K R ) [ g ( R ) 1 ] R d R + [ k i R o H s n ( k i R o ) J s n ( K R o ) K R o H s n ( k i R o ) J s n ( K R o ) ] ( k i 2 K 2 ) } .
δ s n + n o f s n b n o , I = 0 ,
E s = i m i k o × × ( e z u s ) + × ( e z v s ) ,
H s = m i × ( e z u s ) + i k o × × ( e z v s ) ,
S c = c o 8 π Re E s × H s * ,
u s = n o τ = ± n = ( i ) n X n τ 0 D exp ( i n γ j p ) H n ( k i R j p ) exp ( i K τ R j ) d y j d x j .
u f s = exp ( i k i R p + ) { 1 + τ = ± Ω f τ X τ ( θ i ) }
u b s = exp ( i k i R p ) τ = ± Ω b τ X τ ( π + θ i )
Ω f τ = 2 n o [ i k i x ( k i x τ K x ) ] exp [ i ( k i x τ K x ) D ] ,
Ω b τ = 2 n o [ i k i x ( k i x + τ K x ) ] .
I f c = { 1 + 2 Re ( τ = ± Ω f τ X τ ( θ i ) ) * + τ = ± Ω f τ X τ ( θ i ) 2 } e i ,
I b c = τ = ± Ω b τ X τ ( π + θ i ) 2 e r ,
I t r c = τ = ± Ω f τ X τ ( θ i ) 2 e i ,
S i c = c o 8 π Re ( E s E ¯ s ) × ( H s H ¯ s ) * .
S i c = c o 8 π m i k i 2 { u s u s * e R u s u s * e i } ,
b j n I b k s I * k j b j n I j b k s I * k .
S i c S i = e R 2 π k i R p n o { n = s = exp [ i ( n s ) γ ] A o b j n I j b j s I * j d A j + n o n = s = exp [ i ( s n ) γ p ] A o A j k b j n I j b k s I * k exp ( i k i R j k e p ) [ g ( R j k ) 1 ] d A j k d A j } ,
I i c = e R 2 n o π k i R p τ , τ s , n 1 exp [ i ( τ K x τ K x * ) D ] i ( τ K x τ K x * ) X n τ X s τ * exp [ i ( n s ) γ ] { 1 + 2 π n o m τ m exp [ i m ( γ + τ θ t * ) ] R o J m ( K * R ) J m ( k i R ) [ g ( R ) 1 ] R d R } ,
C s w = n o 4 k i τ , τ n , s 1 exp [ i ( τ K x τ K x * ) D ] i ( τ K x τ K x * ) X n τ X s τ * { δ s n + 8 f v τ s n exp [ i τ ( s n ) θ t * ] 1 J s n ( 2 K * r o R ) J s n ( 2 k i r o R ) [ g ( R ) 1 ] R d R } ,
v s = j = 1 N n = ( i ) n exp ( i n γ j p ) H n ( k i R j p ) a j n II ,
a j n II j τ = ± Y n τ exp ( i K τ R j ) .
2 n o i τ = ± ( k i x + τ K x ) k i x ( k i x 2 K x 2 ) Y τ ( θ i ) = 1 ,
τ = ± ( k i x τ K x ) exp ( i τ K x D ) Y τ ( π + θ i ) = 0 ,
[ δ s n + n o f s n a n o , II ] [ τ s Y s τ exp ( τ i s θ t ) ] = 0 ,
Y τ ( γ ) = s = Y s τ exp ( i s γ ) .
δ s n + n o f s n a n o , II = 0 .
Ψ = α I o 2 Ψ TM + α II o 2 Ψ TE ,

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