Abstract

The scattering formulation for a coated infinite cylinder in an absorbing medium is presented in this paper. The cylinder is subjected to an arbitrarily polarized plane wave propagating in a general direction at the cylinder. The refractive index and magnetic permeability of the host medium, as well as those for the core and coating of the cylinder, can be real or complex. The scattering and extinction efficiencies and the scattering amplitudes are derived for both the near field and the far field. As the medium is absorbing, the “true” extinction and scattering efficiencies are derived based on the radiative energy outflow at the surface of the cylinder. The radiative efficiencies in the far field are denoted as “apparent” properties because they include absorption by the intervening medium. The influence of the refractive index and permeability of the host medium on the scattering properties of a coated cylinder is illustrated by numerical examples.

© 2011 Optical Society of America

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References

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  1. B. J. Davis, P. S. Carney, and R. Bhargava, “Theory of infrared microspectroscopy for intact fibers,” Anal. Chem. 83, 525–532 (2011).
    [CrossRef]
  2. S. C. Lee and G. R. Cunnington, “Theoretical models for radiative transfer in fibrous media,” Annual Review of Heat Transfer, C.L.Tien, ed. (Begell House, 1998) Vol.  9, 159–212.
  3. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, 1969).
  4. H. C. van de Hulst, Light Scattering by Small Particles(Dover, 1981).
  5. M. K. Moaveni, A. A. Rizvi, and B. A. Kamran, “Plane-wave scattering by gratings of conducting cylinders in an inhomogeneous and lossy dielectric,” J. Opt. Soc. Am. A 5, 834–843 (1988).
    [CrossRef]
  6. R. Ruppin, “Extinction by a circular cylinder in an absorbing medium,” Opt. Commun. 211, 335–340 (2002).
    [CrossRef]
  7. W. Sun, N. G. Loeb, and B. Lin, “Light scattering by an infinite circular cylinder immersed in an absorbing medium,” Appl. Opt. 44, 2338–2342 (2005).
    [CrossRef] [PubMed]

2011

B. J. Davis, P. S. Carney, and R. Bhargava, “Theory of infrared microspectroscopy for intact fibers,” Anal. Chem. 83, 525–532 (2011).
[CrossRef]

2005

2002

R. Ruppin, “Extinction by a circular cylinder in an absorbing medium,” Opt. Commun. 211, 335–340 (2002).
[CrossRef]

1988

Bhargava, R.

B. J. Davis, P. S. Carney, and R. Bhargava, “Theory of infrared microspectroscopy for intact fibers,” Anal. Chem. 83, 525–532 (2011).
[CrossRef]

Carney, P. S.

B. J. Davis, P. S. Carney, and R. Bhargava, “Theory of infrared microspectroscopy for intact fibers,” Anal. Chem. 83, 525–532 (2011).
[CrossRef]

Cunnington, G. R.

S. C. Lee and G. R. Cunnington, “Theoretical models for radiative transfer in fibrous media,” Annual Review of Heat Transfer, C.L.Tien, ed. (Begell House, 1998) Vol.  9, 159–212.

Davis, B. J.

B. J. Davis, P. S. Carney, and R. Bhargava, “Theory of infrared microspectroscopy for intact fibers,” Anal. Chem. 83, 525–532 (2011).
[CrossRef]

Kamran, B. A.

Kerker, M.

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

Lee, S. C.

S. C. Lee and G. R. Cunnington, “Theoretical models for radiative transfer in fibrous media,” Annual Review of Heat Transfer, C.L.Tien, ed. (Begell House, 1998) Vol.  9, 159–212.

Lin, B.

Loeb, N. G.

Moaveni, M. K.

Rizvi, A. A.

Ruppin, R.

R. Ruppin, “Extinction by a circular cylinder in an absorbing medium,” Opt. Commun. 211, 335–340 (2002).
[CrossRef]

Sun, W.

van de Hulst, H. C.

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

Anal. Chem.

B. J. Davis, P. S. Carney, and R. Bhargava, “Theory of infrared microspectroscopy for intact fibers,” Anal. Chem. 83, 525–532 (2011).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Commun.

R. Ruppin, “Extinction by a circular cylinder in an absorbing medium,” Opt. Commun. 211, 335–340 (2002).
[CrossRef]

Other

S. C. Lee and G. R. Cunnington, “Theoretical models for radiative transfer in fibrous media,” Annual Review of Heat Transfer, C.L.Tien, ed. (Begell House, 1998) Vol.  9, 159–212.

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

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

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Figures (6)

Fig. 1
Fig. 1

A coated cylinder irradiated by a plane wave in an absorbing medium.

Fig. 2
Fig. 2

Variation of the true extinction and scattering efficiencies with the angle of incidence for a coated and homogeneous cylinder in an absorbing medium.

Fig. 3
Fig. 3

Effect of absorption and magnetic permeability of the host medium on a coated nonmagnetic cylinder.

Fig. 4
Fig. 4

Effect of absorption and permeability of the host medium on a coated magnetic cylinder.

Fig. 5
Fig. 5

Influence of the polarization of the incident wave on the u-mode scattering amplitude.

Fig. 6
Fig. 6

Influence of the polarization of the incident wave on the v-mode scattering amplitude.

Equations (52)

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E = × ( e z v ) + i k × × ( e z u ) ,
H = m ˜ μ × ( e z u ) + i μ k o × × ( e z v ) ,
( m ˜ , μ , k ) = { ( m ˜ 1 , μ 1 , k 1 ) , R P > r o , 2 ( m ˜ o , 2 , μ o , 2 , k o , 2 ) , r o , 2 > R P > r o , 1 ( m ˜ o , 1 , μ o , 1 , k o , 1 ) , R P < r o , 1 .
ψ inc = α ψ exp ( i k inc R P i h z ) ,
e inc = cos ϕ 1 e x + sin ϕ 1 e z .
α ψ = α ψ o exp ( i δ ψ ) ,
ψ inc = α ψ exp ( i h z ) n = ( i ) n J n ( 1 R P ) exp ( i n γ P ) ,
S inc = c o 8 π Re ( E inc × H inc * ) = S o exp ( 2 1 i R P cos γ P 2 h i z ) e inc ,
S o = c o | 1 | 2 8 π Re ( m ˜ 1 * μ 1 * ) ( | α u | 2 + | α v | 2 )
( u sca v sca ) = n = ( i ) n exp ( i n γ P i h z ) H n ( 1 R P ) ( b n a n ) .
( u sca v sca ) = n = ( i ) n exp ( i n γ P i h z ) { J n ( o , 2 R P ) [ B n ( 2 ) A n ( 2 ) ] H n ( o , 2 R P ) [ b n ( 2 ) a n ( 2 ) ] } ,
( u sca v sca ) = n = ( i ) n exp ( i n γ P i h z ) J n ( o , 1 R P ) [ B n ( 1 ) A n ( 1 ) ] ,
A Z = α u B 1 + α v B 2 ,
Z = [ b n a n b n ( 2 ) a n ( 2 ) B n ( 2 ) A n ( 2 ) B n ( 1 ) A n ( 1 ) ] T ,
B 1 = [ i n h k 1 r o , 2 J n ( 1 r o , 2 ) 1 2 k 1 J n ( 1 r o , 2 ) 1 J n ( 1 r o , 2 ) 0 0 0 0 0 ] T ,
B 2 = [ 1 J n ( 1 r o , 2 ) 0 i n h k 1 r o , 2 J n ( 1 r o , 2 ) 1 2 J n ( 1 r o , 2 ) 0 0 0 0 ] T ,
A = [ A 11 A 12 0 0 A 21 A 22 ] ,
A 11 ( 1 r o , 2 ) = [ i n h k 1 r o , 2 H n 1 H n 1 2 k 1 H n 0 1 H n i n h k 1 r o , 2 H n 0 1 2 H n ] , A 22 ( o , 1 r o , 1 ) = [ i n h k o , 1 r o , 1 J n o , 1 J n o , 1 2 k o , 1 J n 0 m ^ o , 1 μ ^ o , 1 o , 1 J n i n h μ ^ o , 1 k 1 r o , 1 J n 0 o , 1 2 μ ^ o , 1 J n ] ,
A 12 ( o , 2 r o , 2 ) = [ i n h k o , 2 r o , 2 H n o , 2 H n i n h k o , 2 r o , 2 J n o , 2 J n o , 2 2 k o , 2 H n 0 o , 2 2 k o , 2 J n 0 m ^ o , 2 μ ^ o , 2 o , 2 H n i n h μ ^ o , 2 k 1 r o , 2 H n m ^ o , 2 μ ^ o , 2 o , 2 J n i n h μ ^ o , 2 k 1 r o , 2 J n 0 o , 2 2 μ ^ o , 2 H n 0 o , 2 2 μ ^ o , 2 J n ] ,
A 21 ( o , 2 r o , 1 ) = [ i n h k o , 2 r o , 1 H n o , 2 H n i n h k o , 2 r j , 1 J n o , 2 J n o , 2 2 k o , 2 H n 0 o , 2 2 k o , 2 J n 0 m ^ o , 2 μ ^ o , 2 o , 2 H n i n h μ ^ o , 2 k 1 r o , 1 H n m ^ o , 2 μ ^ o , 2 o , 2 J n i n h μ ^ o , 2 k 1 r o , 1 J n 0 o , 2 2 μ ^ o , 2 H n 0 o , 2 2 μ ^ o , 2 J n ] ,
[ b n , a n , ... , B n ( 1 ) , A n ( 1 ) ] T = α u [ b n I , a n I , ... , B n I , ( 1 ) , A n I , ( 1 ) ] T + α v [ b n I I , a n I I , ... , B n I I , ( 1 ) , A n I I , ( 1 ) ] T .
b n I = b n I , a n I I = a n I I , a n I = a n I , b n I I = b n I I .
S = c o 8 π Re [ ( E inc + E sca ) × ( H inc + H sca ) * ] ,
S inc = c o 8 π Re ( E inc × H inc * ) ,
S = c o 8 π Re ( E inc × H sca * + E sca × H inc * ) ,
S sca = c o 8 π Re ( E sca × H sca * ) .
W abs = L / 2 L / 2 0 2 π ( S inc + S + S sca ) e R R P d γ P d z = W ext + W sca ,
W ext ( R P ) = R P L / 2 L / 2 0 2 π ( S inc + S ) e R d γ P d z = 2 π R P L e I 1 ( 2 1 i R P ) S o cos ϕ 1 c o R P L e 4 | 1 | 2 cos ϕ 1 · Re { m ˜ 1 * μ 1 * n = [ n 1 R P ( 1 1 1 * ) α u sin ϕ 1 J n H n * a n * + i ( α u J n H n ' * b n * α v J n H n * a n * ) + n 1 * R P ( 1 1 * 1 ) α v * sin ϕ 1 J n * H n b n + i ( α u * J n * H n b n α v * J n * H n a n ) ] } ,
L e = L / 2 L / 2 exp ( 2 h i z ) d z = sinh ( h i L ) / h i .
W sca ( R P ) = R P L / 2 L / 2 0 2 π S sca e R d γ P d z = c o R P L e 4 | 1 | 2 cos ϕ 1 · Re { m ˜ 1 * μ 1 * n = [ n 1 R P ( 1 1 1 * ) sin ϕ 1 | H n | 2 b n a n * + i ( H n H n * | a n | 2 - H n H n * | b n | 2 ) ] } .
W o ( R P = r o , 2 ) = r o , 2 L / 2 L / 2 π / 2 3 π / 2 S inc e R d γ P d z = 2 S o cos ϕ 1 L e r e ,
r e ( ϕ 1 ) = r o , 2 2 π / 2 3 π / 2 exp ( 2 1 i r o , 2 cos γ P ) cos γ P d γ P
Q ext ( R P ) = W ext ( R P ) W o ( ϕ 1 = 0 ) = π R P r e 0 I 1 ( 2 1 i R P ) cos ϕ 1 π R P cos ϕ 1 / r e 0 Re ( m ˜ 1 * / μ 1 * ) ( | α u | 2 + | α v | 2 ) · Re { m ˜ 1 * μ 1 * n = [ n 1 R P ( 1 1 1 * ) α u sin ϕ 1 J n H n * a n * + i ( α u J n H n * b n * α v J n H n * a n * ) + n 1 * R P ( 1 1 * 1 ) α v * sin ϕ 1 J n * H n b n + i ( α u * J n * H n b n α v * J n * H n a n ) ] } ,
Q sca ( R P ) = W sca ( R P ) W o ( ϕ 1 = 0 ) = π R P cos ϕ 1 / r e 0 Re ( m ˜ 1 * / μ 1 * ) ( | α u | 2 + | α v | 2 ) · Re { m ˜ 1 * μ 1 * n = [ n 1 R P ( 1 1 1 * ) sin ϕ 1 | H n | 2 b n a n * + i ( H n H n * | a n | 2 H n H n * | b n | 2 ) ] } ,
( Q tr ext , Q tr sca ) = [ Q ext ( R P = r o , 2 ) , Q sca ( R P = r o , 2 ) ] .
Q tr abs = Q tr ext Q tr sca .
C tr ν = 2 r e 0 Q tr ν
Q ff sca ( R P ) = 2 | k 1 | r e 0 n = ( | b n | 2 + | a n | 2 ) | α u | 2 + | α v | 2 exp ( 2 1 i R P ) .
Q ff ext ( R P ) = π R P r e 0 I 1 ( 2 1 i R P ) cos ϕ 1 + 2 | k 1 | r e 0 Re n = ( α u * b n + α v * a n ) ( | α u | 2 + | α v | 2 ) exp ( 2 1 i R P ) .
S ff sca = c o | 1 | 4 π 2 R P exp ( 2 1 i R P 2 h i z ) Re ( m ˜ 1 * μ 1 * ) [ | T u ( ϕ 1 , γ P ) | 2 + | T v ( ϕ 1 , γ P ) | 2 ] e s ,
[ T u ( ϕ 1 , γ P ) T u ( ϕ 1 , γ P ) ] = n = ( b n a n ) exp ( i n γ P )
e s = cos ϕ 1 e R + sin ϕ 1 e z
I ( ϕ 1 , γ P ) = S ff sca e R S inc ( γ P = 0 ) = 2 π | k 1 | R P [ i u ( ϕ 1 , γ P ) + i v ( ϕ 1 , γ P ) ] exp ( 2 1 i R P ) ,
i u ( ϕ 1 , γ P ) = | T u ( ϕ 1 , γ P ) | 2 / ( | α u | 2 + | α v | 2 ) ,
i v ( ϕ 1 , γ P ) = | T v ( ϕ 1 , γ P ) | 2 / ( | α u | 2 + | α v | 2 )
[ k 1 H n ( k 1 r o , 2 ) k o , 2 H n ( k o , 2 r o , 2 ) k o , 2 J n ( k o , 2 r o , 2 ) 0 k 1 H n ( k 1 r o , 2 ) m ^ o , 2 μ ^ o , 2 k o , 2 H n ( k o , 2 r o , 2 ) m ^ o , 2 μ ^ o , 2 k o , 2 J n ( k o , 2 r o , 2 ) 0 0 k o , 2 H n ( k o , 2 r o , 1 ) k o , 2 J n ( k o , 2 r o , 1 ) k o , 1 J n ( k o , 1 r o , 1 ) 0 m ^ o , 2 μ ^ o , 2 o , 2 H n ( k o , 2 r o , 1 ) m ^ o , 2 μ ^ o , 2 k o , 2 J n ( k o , 2 r o , 1 ) m ^ j , 1 μ ^ j , 1 k o , 1 J n ( k o , 1 r o , 1 ) ] [ b n b n ( 2 ) B n ( 2 ) B n ( 1 ) ] = [ k 1 J n ( k 1 r o , 2 ) k 1 J n ( k 1 r o , 2 ) 0 0 ] ,
[ k 1 H n ( k 1 r o , 2 ) k o , 2 H n ( k o , 2 r o , 2 ) k o , 2 J n ( k o , 2 r o , 2 ) 0 k 1 2 H n ( k 1 r o , 2 ) k o , 2 2 μ ^ o , 2 H n ( k o , 2 r o , 2 ) k o , 2 2 μ ^ o , 2 J n ( k o , 2 r o , 2 ) 0 0 k o , 2 H n ( k o , 2 r o , 1 ) k o , 2 J n ( k o , 2 r o , 1 ) k o , 1 J n ' ( k o , 1 r o , 1 ) 0 k o , 2 2 μ ^ o , 2 H n ( k o , 2 r o , 1 ) k o , 2 2 μ ^ o , 2 J n ( k o , 2 r o , 1 ) k o , 1 2 μ ^ o , 1 J n ( k o , 1 r o , 1 ) ] [ a j n a n ( 2 ) A n ( 2 ) A n ( 1 ) ] = [ k 1 J n ( k 1 r o , 2 ) k 1 2 J n ( k 1 r o , 2 ) 0 0 ]
Q I ext ( ϕ 1 = 0 , R P ) = π R P r e 0 I 1 ( 2 k 1 i R P ) + π R P / r e 0 Re ( m ˜ 1 * / μ 1 * ) Im { m ˜ 1 * μ 1 * n = ( J n H n * b n * + J n * H n b n ) } ,
Q I I ext ( ϕ 1 = 0 , R P ) = π R P r e 0 I 1 ( 2 k 1 i R P ) π R P / r e 0 Re ( m ˜ 1 * / μ 1 * ) Im { m ˜ 1 * μ 1 * n = ( J n H n * a n * + J n * H n a n ) } ,
Q I sca ( ϕ 1 = 0 , R P ) = π R P / r e 0 Re ( m ˜ 1 * / μ 1 * ) Im { m ˜ 1 * μ 1 * n = H n H n * | b n | 2 } ,
Q I I sca ( ϕ 1 = 0 , R P ) = π R P / r e 0 Re ( m ˜ 1 * / μ 1 * ) Im { m ˜ 1 * μ 1 * n = H n H n * | a n | 2 } ,
S ψ , ff sca = 2 | k 1 | R P S o exp ( 2 k 1 i R P ) | T ψ ( ϕ 1 = 0 , γ P ) | 2 e R

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