Abstract

The theoretical formulation for the scattering of electromagnetic waves by a finite configuration of closely spaced parallel nonhomogeneous infinite circular cylinders in an absorbing/magnetic medium is presented in this paper. The source radiation is an arbitrarily polarized plane wave propagating in a general direction at the cylinders. Each circular cylinder is radially stratified with an arbitrary number of layers, and the refractive index and magnetic permeability in each layer are distinct. The refractive index and permeability of the host medium and cylinders are in general complex. Closed form solutions are derived for the extinction and scattering cross sections and scat tering intensity distribution in the far field. Numerical results on the scattering properties of a specific configuration of coated cylinders in an absorbing/magnetic medium are presented for the purpose of illustration.

© 2011 Optical Society of America

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  1. M. I. Mishchenko, G. Videen, V. A. Babenko, N. G. Khlebtsov, and T. Wriedt, “T-matrix theory of electromagnetic scattering by particles and its applications: a comprehensive reference database,” J. Quant. Spectrosc. Radiat. Transfer 88, 357–406(2004).
    [CrossRef]
  2. S. C. Lee and G. R. Cunnington, “Theoretical models for radiative transfer in fibrous media,” in Annual Review of Heat Transfer, Vol. 9, C.L.Tien, ed. (Begell House, 1998), pp. 159–212.
  3. S. C. Lee, “A quasi-dependent scattering radiative properties model for high density fiber composites,” J. Heat Transfer 132, 023303 (2010).
    [CrossRef]
  4. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, 1969).
  5. H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).
  6. M. Barabas, “Scattering of a plane wave by a radially stratified tilted cylinder,” J. Opt. Soc. Am. A 4, 2240–2248 (1987).
    [CrossRef]
  7. V. Twersky, “Multiple scattering of radiation by an arbitrary configuration of parallel cylinders,” J. Acoust. Soc. Am. 24, 42–46(1952).
    [CrossRef]
  8. G. O. Oloafe, “Scattering by an arbitrary configuration of parallel cylinders,” J. Opt. Soc. Am. 60, 1233–1236 (1970).
    [CrossRef]
  9. 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]
  10. S. C. Lee, “Scattering by closely-spaced radially stratified parallel cylinders,” J. Quant. Spectrosc. Radiat. Transfer 48, 119–130(1992).
    [CrossRef]
  11. 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]
  12. S. C. Lee and J. A. Grzesik, “Light scattering by closely spaced parallel cylinders embedded in a semi-infinite dielectric medium,” J. Opt. Soc. Am. A 15, 163–173 (1998).
    [CrossRef]
  13. S. C. Lee, “Light scattering by closely spaced parallel cylinders embedded in a finite dielectric slab,” J. Opt. Soc. Am. A 16, 1350–1361 (1999).
    [CrossRef]
  14. M. Di Vico, F. Frezza, L. Pajewski, and G. Schettini, “Scattering by a finite set of perfectly conducting cylinders buried in a dielectric half-space: A spectral-domain solution,” IEEE Trans. Antennas Propag. 53, 719–727 (2005).
    [CrossRef]
  15. F. Frezza, L. Pajewski, C. Ponti, and G. Schettini, “Scattering by perfectly conducting circular cylinders buried in a dielectric slab through the cylindrical wave approach,” IEEE Trans. Antennas Propag. 57, 1208–1217 (2009).
    [CrossRef]
  16. 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]
  17. R. Ruppin, “Extinction by a circular cylinder in an absorbing medium,” Opt. Commun. 211, 335–340 (2002).
    [CrossRef]
  18. 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]
  19. S. C. Lee, “Scattering by an infinite coated cylinder in an absorbing medium at oblique incidence,” J. Opt. Soc. Am. A 28, 1067–1075 (2011).
    [CrossRef]

2011 (1)

2010 (1)

S. C. Lee, “A quasi-dependent scattering radiative properties model for high density fiber composites,” J. Heat Transfer 132, 023303 (2010).
[CrossRef]

2009 (1)

F. Frezza, L. Pajewski, C. Ponti, and G. Schettini, “Scattering by perfectly conducting circular cylinders buried in a dielectric slab through the cylindrical wave approach,” IEEE Trans. Antennas Propag. 57, 1208–1217 (2009).
[CrossRef]

2005 (2)

M. Di Vico, F. Frezza, L. Pajewski, and G. Schettini, “Scattering by a finite set of perfectly conducting cylinders buried in a dielectric half-space: A spectral-domain solution,” IEEE Trans. Antennas Propag. 53, 719–727 (2005).
[CrossRef]

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]

2004 (1)

M. I. Mishchenko, G. Videen, V. A. Babenko, N. G. Khlebtsov, and T. Wriedt, “T-matrix theory of electromagnetic scattering by particles and its applications: a comprehensive reference database,” J. Quant. Spectrosc. Radiat. Transfer 88, 357–406(2004).
[CrossRef]

2002 (1)

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

1999 (1)

1998 (1)

1994 (1)

1992 (1)

S. C. Lee, “Scattering by closely-spaced radially stratified parallel cylinders,” J. Quant. Spectrosc. Radiat. Transfer 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]

1988 (1)

1987 (1)

1970 (1)

1952 (1)

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

Babenko, V. A.

M. I. Mishchenko, G. Videen, V. A. Babenko, N. G. Khlebtsov, and T. Wriedt, “T-matrix theory of electromagnetic scattering by particles and its applications: a comprehensive reference database,” J. Quant. Spectrosc. Radiat. Transfer 88, 357–406(2004).
[CrossRef]

Barabas, M.

Cunnington, G. R.

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

Di Vico, M.

M. Di Vico, F. Frezza, L. Pajewski, and G. Schettini, “Scattering by a finite set of perfectly conducting cylinders buried in a dielectric half-space: A spectral-domain solution,” IEEE Trans. Antennas Propag. 53, 719–727 (2005).
[CrossRef]

Felbacq, D.

Frezza, F.

F. Frezza, L. Pajewski, C. Ponti, and G. Schettini, “Scattering by perfectly conducting circular cylinders buried in a dielectric slab through the cylindrical wave approach,” IEEE Trans. Antennas Propag. 57, 1208–1217 (2009).
[CrossRef]

M. Di Vico, F. Frezza, L. Pajewski, and G. Schettini, “Scattering by a finite set of perfectly conducting cylinders buried in a dielectric half-space: A spectral-domain solution,” IEEE Trans. Antennas Propag. 53, 719–727 (2005).
[CrossRef]

Grzesik, J. A.

Kamran, B. A.

Kerker, M.

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

Khlebtsov, N. G.

M. I. Mishchenko, G. Videen, V. A. Babenko, N. G. Khlebtsov, and T. Wriedt, “T-matrix theory of electromagnetic scattering by particles and its applications: a comprehensive reference database,” J. Quant. Spectrosc. Radiat. Transfer 88, 357–406(2004).
[CrossRef]

Lee, S. C.

S. C. Lee, “Scattering by an infinite coated cylinder in an absorbing medium at oblique incidence,” J. Opt. Soc. Am. A 28, 1067–1075 (2011).
[CrossRef]

S. C. Lee, “A quasi-dependent scattering radiative properties model for high density fiber composites,” J. Heat Transfer 132, 023303 (2010).
[CrossRef]

S. C. Lee, “Light scattering by closely spaced parallel cylinders embedded in a finite dielectric slab,” J. Opt. Soc. Am. A 16, 1350–1361 (1999).
[CrossRef]

S. C. Lee and J. A. Grzesik, “Light scattering by closely spaced parallel cylinders embedded in a semi-infinite dielectric medium,” J. Opt. Soc. Am. A 15, 163–173 (1998).
[CrossRef]

S. C. Lee, “Scattering by closely-spaced radially stratified parallel cylinders,” J. Quant. Spectrosc. Radiat. Transfer 48, 119–130(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]

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

Lin, B.

Loeb, N. G.

Maystre, D.

Mishchenko, M. I.

M. I. Mishchenko, G. Videen, V. A. Babenko, N. G. Khlebtsov, and T. Wriedt, “T-matrix theory of electromagnetic scattering by particles and its applications: a comprehensive reference database,” J. Quant. Spectrosc. Radiat. Transfer 88, 357–406(2004).
[CrossRef]

Moaveni, M. K.

Oloafe, G. O.

Pajewski, L.

F. Frezza, L. Pajewski, C. Ponti, and G. Schettini, “Scattering by perfectly conducting circular cylinders buried in a dielectric slab through the cylindrical wave approach,” IEEE Trans. Antennas Propag. 57, 1208–1217 (2009).
[CrossRef]

M. Di Vico, F. Frezza, L. Pajewski, and G. Schettini, “Scattering by a finite set of perfectly conducting cylinders buried in a dielectric half-space: A spectral-domain solution,” IEEE Trans. Antennas Propag. 53, 719–727 (2005).
[CrossRef]

Ponti, C.

F. Frezza, L. Pajewski, C. Ponti, and G. Schettini, “Scattering by perfectly conducting circular cylinders buried in a dielectric slab through the cylindrical wave approach,” IEEE Trans. Antennas Propag. 57, 1208–1217 (2009).
[CrossRef]

Rizvi, A. A.

Ruppin, R.

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

Schettini, G.

F. Frezza, L. Pajewski, C. Ponti, and G. Schettini, “Scattering by perfectly conducting circular cylinders buried in a dielectric slab through the cylindrical wave approach,” IEEE Trans. Antennas Propag. 57, 1208–1217 (2009).
[CrossRef]

M. Di Vico, F. Frezza, L. Pajewski, and G. Schettini, “Scattering by a finite set of perfectly conducting cylinders buried in a dielectric half-space: A spectral-domain solution,” IEEE Trans. Antennas Propag. 53, 719–727 (2005).
[CrossRef]

Sun, W.

Tayeb, G.

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).

Videen, G.

M. I. Mishchenko, G. Videen, V. A. Babenko, N. G. Khlebtsov, and T. Wriedt, “T-matrix theory of electromagnetic scattering by particles and its applications: a comprehensive reference database,” J. Quant. Spectrosc. Radiat. Transfer 88, 357–406(2004).
[CrossRef]

Wriedt, T.

M. I. Mishchenko, G. Videen, V. A. Babenko, N. G. Khlebtsov, and T. Wriedt, “T-matrix theory of electromagnetic scattering by particles and its applications: a comprehensive reference database,” J. Quant. Spectrosc. Radiat. Transfer 88, 357–406(2004).
[CrossRef]

Appl. Opt. (1)

IEEE Trans. Antennas Propag. (2)

M. Di Vico, F. Frezza, L. Pajewski, and G. Schettini, “Scattering by a finite set of perfectly conducting cylinders buried in a dielectric half-space: A spectral-domain solution,” IEEE Trans. Antennas Propag. 53, 719–727 (2005).
[CrossRef]

F. Frezza, L. Pajewski, C. Ponti, and G. Schettini, “Scattering by perfectly conducting circular cylinders buried in a dielectric slab through the cylindrical wave approach,” IEEE Trans. Antennas Propag. 57, 1208–1217 (2009).
[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. Heat Transfer (1)

S. C. Lee, “A quasi-dependent scattering radiative properties model for high density fiber composites,” J. Heat Transfer 132, 023303 (2010).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (6)

J. Quant. Spectrosc. Radiat. Transfer (2)

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

M. I. Mishchenko, G. Videen, V. A. Babenko, N. G. Khlebtsov, and T. Wriedt, “T-matrix theory of electromagnetic scattering by particles and its applications: a comprehensive reference database,” J. Quant. Spectrosc. Radiat. Transfer 88, 357–406(2004).
[CrossRef]

Opt. Commun. (1)

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

Other (3)

S. C. Lee and G. R. Cunnington, “Theoretical models for radiative transfer in fibrous media,” in Annual Review of Heat Transfer, Vol. 9, C.L.Tien, ed. (Begell House, 1998), pp. 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 (5)

Fig. 1
Fig. 1

Schematic diagrams depicting the present problem: (a) isometric view, (b) side view.

Fig. 2
Fig. 2

Scaled extinction and scattering cross sections of seven nonmagnetic cylinders in an absorbing medium at (a) normal incidence ( ϕ 1 = 0 ° ), (b) oblique incidence ( ϕ 1 = 30 ° ).

Fig. 3
Fig. 3

Scattered intensity distribution for a hexagonal configuration of seven nonmagnetic cylinders in an absorbing medium at normal incidence ( ϕ 1 = 0 ° ) in (a) dielectric medium, (b) absorbing medium.

Fig. 4
Fig. 4

Scattered intensity distribution for a hexagonal configuration of seven nonmagnetic cylinders at oblique incidence ( ϕ 1 = 30 ° ) in (a) dielectric medium, (b) absorbing medium.

Fig. 5
Fig. 5

Scattered intensity distribution for a hexagonal configuration of seven magnetic cylinders at normal incidence ( ϕ 1 = 0 ° ) in (a) absorbing/magnetic medium, (b) absorbing/nonmagnetic medium.

Equations (59)

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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 ( ε i 4 π σ / ω ) ,
( m ˜ , μ , k ) = { ( m ˜ 1 , μ 1 , k 1 ) , R j P > r j , Q j ( m ˜ j , q , μ j , q , k j , q ) , r j , q 1 < R j P < r j , q , ( m ˜ j , 1 , μ j , 1 , k j , 1 ) , R j P < r j , 1 q = 2 , Q j ,
ψ j ( R P ) = ψ j inc ( R j P ) + ψ j sca ( R j P ) + k j N ψ k j sca ( R k P ) ,
ψ j , q ( R j P ) = ψ j , q inc ( R j P ) + ψ j , q sca ( R j P )
ψ j inc = ε j α ψ exp ( i k inc R j P i h z ) ,
e inc = cos ϕ 1 cos θ 1 e x cos ϕ 1 sin θ 1 e y + sin ϕ 1 e z .
α ψ = ε O O α ψ o exp ( i δ ψ ) ,
ψ j inc ( R j P > r j , Q j ) = ε j α ψ exp ( i h z ) n = ( i ) n J n ( 1 R j P ) exp [ i n ( γ j P + θ 1 ) ] ,
( u j , q inc v j , q inc ) = exp ( i h z ) n = ( i ) n J n ( j , q R j P ) exp ( i n γ j P ) ( B j n ( q ) A j n ( q ) ) ,
( u j sca v j sca ) = exp ( i h z ) n = ( i ) n exp ( i n γ j P ) H n ( 1 R j P ) ( b j n a j n ) .
( u j , q sca v j , q sca ) = exp ( i h z ) n = ( i ) n exp ( i n γ j P ) H n ( j , q R j P ) ( b j n ( q ) a j n ( q ) ) ,
( u k j sca v k j sca ) = exp ( i h z ) n , s = ( i ) n exp ( i n γ j P ) J n ( 1 R j P ) G k s j n ( b k s a k s ) ,
G k s j n = ( i ) s n exp [ i ( s n ) γ k j ] H s n ( 1 R j k ) .
M j Z j = U j B j + V j A j ,
( U j V j ) = ε j exp ( i n θ 1 ) ( α u α v ) k j N s = G k s j n ( b k s a k s ) .
M j = [ M j , 00 M j , 01 0 0 0 M j , 11 M j , 12 0 0 0 M j , 22 M j , 23 0 0 0 M j , Q j 2 , Q j 2 M j , Q j 2 , Q j 1 0 0 0 M j , Q j 1 , Q j 1 M j , Q j 1 , Q j ] ,
M j , 00 = [ i n h k 1 r j , Q j H n ( 1 r j , Q j ) 1 H n ( 1 r j , Q j ) 1 2 k 1 H n ( 1 r j , Q j ) 0 1 H n ( 1 r j , Q j ) i n h k 1 r j , Q j H n ( 1 r j , Q j ) 0 1 2 H n ( 1 r j , Q j ) ] , M j , Q j 1 , Q j = [ i n h k j , 1 r j , 1 J n ( j , 1 r j , 1 ) j , 1 J n ( j , 1 r j , 1 ) j , 1 2 k j , 1 J n ( j , 1 r j , 1 ) 0 m ^ j , 1 μ ^ j , 1 j , 1 J n ( j , 1 r j , 1 ) i n h μ ^ j , 1 k 1 r j , 1 J n ( j , 1 r j , 1 ) 0 j , 1 2 μ ^ j , 1 J n ( j , 1 r j , 1 ) ] ,
M j , p q = ( 1 ) p + q [ i n h k j , α r j , β H n ( j , α r j , β ) j , α H n ( j , α r j , β ) i n h k j , α r j , β J n ( j , α r j , β ) j , α J n ( j , α r j , β ) j , α 2 k j , α H n ( j , α r j , β ) 0 j , α 2 k j , α J n ( j , α r j , β ) 0 m ^ j , α j , α μ ^ j , α H n ( j , α r j , β ) i n h μ ^ j , α k 1 r j , α H n ( j , α r j , β ) m ^ j , α j , α μ ^ j , α J n ( j , α r j , β ) i n h μ ^ j , α k 1 r j , α J n ( j , α r j , β ) 0 j , q + 1 2 μ ^ j , α H n ( j , α r j , β ) 0 j , α 2 μ ^ j , α J n ( j , α r j , β ) ] .
Z j = [ b j n a j n b j n ( Q j ) a j n ( Q j ) B j n ( Q j ) A j n ( Q j ) b j n ( 2 ) a j n ( 2 ) B j n ( 2 ) A j n ( 2 ) B j n ( 1 ) A j n ( 1 ) ] T ,
B j = [ i n h k 1 r j , Q j J n ( 1 r j , Q j ) 1 2 k 1 J n ( 1 r j , Q j ) 1 J n ( 1 r j , Q j ) 0 ] T ,
A j = [ 1 J n ( 1 r j , Q j ) 0 i n h k 1 r j , Q j J n ( 1 r j , Q j ) 1 2 J n ( 1 r j , Q j ) 0 ] T .
M j = [ M j , 00 M j , 01 ] ,
M j = [ M j , 00 M j , 01 0 0 M j , 11 M j , 12 ] ,
M j = [ M j , 00 M j , 01 0 0 0 M j , 11 M j , 12 0 0 0 M j , 22 M j , 23 ]
Z o = α u Z o , I + α v Z o , II
Z j = U j Z j o , I + V j Z j o , II .
[ δ j k δ n s + ( 1 δ j k ) G k s j n b j n o , I ( 1 δ j k ) G k s j n b j n o , II ( 1 δ j k ) G k s j n a j n o , I δ j k δ n s + ( 1 δ j k ) G k s j n a j n o , II ] [ b k s a k s ] = ε j exp ( i n θ 1 ) [ b j n o a j n o ] ,
S = c o 8 π Re [ ( E inc × H inc * ) + ( E sca × H sca * ) + ( E inc × H sca * + E sca × H inc * ) ] = S inc + S sca + S si ,
S inc ( R P ) = S o exp [ 2 1 i R P cos ( γ P + θ 1 ) 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 ) .
S sca = 2 S o π | 1 | R P i s ( ϕ 1 , γ P ) exp ( 2 1 i R P 2 h i z ) e s ,
i s ( ϕ 1 , γ P ) = ( | T u | 2 + | T v | 2 ) / ( | α u | 2 + | α v | 2 )
[ T u ( ϕ 1 , γ P ) T v ( ϕ 1 , γ P ) ] = j = 1 N n = [ b j n a j n ] exp [ i n γ P + i 1 R j cos ( γ P γ j ) ]
S si = S o exp ( 2 h i z ) Re ( m ˜ 1 * / μ 1 * ) ( | α u | 2 + | α v | 2 ) Re [ m ˜ 1 * μ 1 * ( S o si V 1 + S o si * V 2 ) ] ,
S o si = ( 2 i π 1 R P ) 1 / 2 exp [ i 1 R P + i 1 * R P cos ( γ P + θ 1 ) ] ,
V 1 ( γ P ) = ( H z o inc * T v + cos ϕ 1 H γ o inc * T u ) e R + ( cos ϕ 1 H R o inc * sin ϕ 1 H z o inc * ) T u e γ + ( sin ϕ 1 H γ o inc * T u H R o inc * T v ) e z ,
V 2 ( γ P ) = ( E z o inc T u * + cos ϕ 1 E γ o inc T v * ) e R + ( cos ϕ 1 E R o inc + sin ϕ 1 E z o inc ) T v * e γ + ( E R o inc T u * sin ϕ 1 E γ o inc T v * ) e z ,
( E R o inc H R o inc ) = ( α v α u ) sin ( γ P + θ 1 ) + ( α u α v ) sin ϕ 1 cos ( γ P + θ 1 ) ,
( E γ o inc H γ o inc ) = ( α v α u ) cos ( γ P + θ 1 ) + ( α u α v ) sin ϕ 1 sin ( γ P + θ 1 ) ,
( E z o inc , H z o inc ) = ( α u , α v ) cos ϕ 1 .
W abs ( R P ) = L / 2 L / 2 0 2 π ( S inc + S si ) e R R P d γ P d z + L / 2 L / 2 0 2 π S sca e R R P d γ P d z .
W ext = R P L / 2 L / 2 0 2 π ( S inc + S si ) e R d γ P d z = W inc + W si ,
W sca = R P L / 2 L / 2 0 2 π S sca e R d γ P d z
W inc ( R P ) = 2 S o L e r e cos ϕ 1 ,
L e = L / 2 L / 2 exp ( 2 h i z ) d z = sinh ( h i L ) h i ,
r e ( ϕ 1 ) = R 0 2 0 2 π exp [ 2 1 i R 0 cos ( γ P + θ 1 ) ] cos ( γ P + θ 1 ) d γ P = π R 0 I 1 ( 2 1 i R 0 ) ,
lim 1 R P W si = 4 | k 1 | W o si S o L e exp ( 2 1 i R P ) ,
W o si = Re [ α u * T u ( γ P = θ 1 ) + α v * T v ( γ P = θ 1 ) ] | α u | 2 + | α v | 2 .
lim 1 R P W sca = 4 | k 1 | W o sca S o L e exp ( 2 1 i R P ) ,
W o sca = j , k = 1 N n , s = exp [ i ( n s ) ( Γ j k π / 2 ) ] J s n ( 1 R ¯ j k ) b j n b k s * + a j n a k s * | α u | 2 + | α v | 2 ,
exp ( i Γ j k ) = [ 1 R j exp ( i γ j ) 1 * R k exp ( i γ k ) ] / ( 1 R ¯ j k ) ,
1 R ¯ j k = ( 1 R j ) 2 2 | 1 | 2 R j R k cos ( γ j γ k ) + ( 1 * R k ) 2 .
C ext = 2 π R P I 1 ( 2 1 i R P ) cos ϕ 1 + 4 | k 1 | W o si exp ( 2 1 i R P ) ,
C sca = 4 | k 1 | W o sca exp ( 2 1 i R P ) ,
Q ν = C ν / ( 2 r e 0 ) ,
[ δ j k δ n s + ( 1 δ j k ) G k s j n b j n o , I ] [ b k s ] = ε j exp ( i n θ 1 ) [ b j n o ] ,
[ δ j k δ n s + ( 1 δ j k ) G k s j n a j n o , I I ] [ a k s ] = ε j exp ( i n θ 1 ) [ a j n o ] .

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