Abstract

This paper presents the scattering solution for a finite dense layer of cylinders irradiated by an arbitrarily polarized plane wave at a general incident direction. The theoretical formulation utilizes the effective field approach and quasi-crystalline approximation to derive the governing equations for the propagation constant and amplitudes of the effective waves. The finite layer thickness gives rise to effective waves propagating in both the forward and backward directions inside the dense medium. Formulas are developed for the far-field coherent and incoherent scattered intensities, as well as the extinction and scattering cross sections of the dense layer. The forward peak of the incoherent scattered intensity is shown to be shifted to the propagating direction of the effective waves. The influence of incident direction, layer thickness, and solid volume fraction on the scattering properties is illustrated by means of a numerical example.

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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. S. K. Bose and A. K. Mal, “Longitudinal shear waves in fiber-reinforced composite,” Int. J. Solids Struct. 9, 1075-1085 (1973).
    [CrossRef]
  11. A. K. Mal and A. K. Chatterjee, “The elastic moduli of a fiber-reinforced composite,” J. Appl. Mech. 44, 61-67 (1977).
    [CrossRef]
  12. V. K. Varadan, V. V. Varadan, and Y.-H. Pao, “Multiple scattering of elastic waves by cylinders of arbitrary cross section. I. SH waves,” J. Acoust. Soc. Am. 63, 1310-1319 (1978).
    [CrossRef]
  13. V. K. Varadan, Y. Ma, and V. V. Varadan, “Multiple scattering of compressional and shear waves by fiber-reinforced composite materials,” J. Acoust. Soc. Am. 80, 333-339 (1986).
    [CrossRef]
  14. 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]
  15. S. C. Lee, “Propagation of radiation in high-density fibrous composites containing coated fibers,” J. Thermophys. Heat Transfer 7, 637-643 (1993).
    [CrossRef]
  16. S. C. Lee, “Scattering characteristics of fibrous media containing closely spaced parallel fibers,” J. Thermophys. Heat Transfer 9, 403-409 (1995).
    [CrossRef]
  17. S. C. Lee, “Scattering by a dense finite medium of infinite cylinders at normal incidence,” J. Opt. Soc. Am. A 25, 1022-1029 (2008).
    [CrossRef]
  18. 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]
  19. M. Lax, “Multiple scattering of waves. II. The effective field in dense systems,” Phys. Rev. 85, 621-629 (1952).
    [CrossRef]
  20. M. Born and E. Wolf, Principles of Optics (Pergamon, 1993).
  21. W. W. Wood, “NpT-ensemble Monte Carlo calculations for the hard disk fluid,” J. Chem. Phys. 52, 729-741 (1970).
    [CrossRef]

2008 (1)

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]

1986 (1)

V. K. Varadan, Y. Ma, and V. V. Varadan, “Multiple scattering of compressional and shear waves by fiber-reinforced composite materials,” J. Acoust. Soc. Am. 80, 333-339 (1986).
[CrossRef]

1978 (1)

V. K. Varadan, V. V. Varadan, and Y.-H. Pao, “Multiple scattering of elastic waves by cylinders of arbitrary cross section. I. SH waves,” J. Acoust. Soc. Am. 63, 1310-1319 (1978).
[CrossRef]

1977 (1)

A. K. Mal and A. K. Chatterjee, “The elastic moduli of a fiber-reinforced composite,” J. Appl. Mech. 44, 61-67 (1977).
[CrossRef]

1973 (1)

S. K. Bose and A. K. Mal, “Longitudinal shear waves in fiber-reinforced composite,” Int. J. Solids Struct. 9, 1075-1085 (1973).
[CrossRef]

1970 (2)

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

W. W. Wood, “NpT-ensemble Monte Carlo calculations for the hard disk fluid,” J. Chem. Phys. 52, 729-741 (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 E. Wolf, Principles of Optics (Pergamon, 1993).

Bose, S. K.

S. K. Bose and A. K. Mal, “Longitudinal shear waves in fiber-reinforced composite,” Int. J. Solids Struct. 9, 1075-1085 (1973).
[CrossRef]

Chatterjee, A. K.

A. K. Mal and A. K. Chatterjee, “The elastic moduli of a fiber-reinforced composite,” J. Appl. Mech. 44, 61-67 (1977).
[CrossRef]

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).

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, “Scattering by a dense finite medium of infinite cylinders at normal incidence,” J. Opt. Soc. Am. A 25, 1022-1029 (2008).
[CrossRef]

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, “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]

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]

Ma, Y.

V. K. Varadan, Y. Ma, and V. V. Varadan, “Multiple scattering of compressional and shear waves by fiber-reinforced composite materials,” J. Acoust. Soc. Am. 80, 333-339 (1986).
[CrossRef]

Mal, A. K.

A. K. Mal and A. K. Chatterjee, “The elastic moduli of a fiber-reinforced composite,” J. Appl. Mech. 44, 61-67 (1977).
[CrossRef]

S. K. Bose and A. K. Mal, “Longitudinal shear waves in fiber-reinforced composite,” Int. J. Solids Struct. 9, 1075-1085 (1973).
[CrossRef]

Maystre, D.

Oloafe, G. O.

Pao, Y.-H.

V. K. Varadan, V. V. Varadan, and Y.-H. Pao, “Multiple scattering of elastic waves by cylinders of arbitrary cross section. I. SH waves,” J. Acoust. Soc. Am. 63, 1310-1319 (1978).
[CrossRef]

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).

Varadan, V. K.

V. K. Varadan, Y. Ma, and V. V. Varadan, “Multiple scattering of compressional and shear waves by fiber-reinforced composite materials,” J. Acoust. Soc. Am. 80, 333-339 (1986).
[CrossRef]

V. K. Varadan, V. V. Varadan, and Y.-H. Pao, “Multiple scattering of elastic waves by cylinders of arbitrary cross section. I. SH waves,” J. Acoust. Soc. Am. 63, 1310-1319 (1978).
[CrossRef]

Varadan, V. V.

V. K. Varadan, Y. Ma, and V. V. Varadan, “Multiple scattering of compressional and shear waves by fiber-reinforced composite materials,” J. Acoust. Soc. Am. 80, 333-339 (1986).
[CrossRef]

V. K. Varadan, V. V. Varadan, and Y.-H. Pao, “Multiple scattering of elastic waves by cylinders of arbitrary cross section. I. SH waves,” J. Acoust. Soc. Am. 63, 1310-1319 (1978).
[CrossRef]

Wolf, E.

M. Born and E. 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]

Int. J. Solids Struct. (1)

S. K. Bose and A. K. Mal, “Longitudinal shear waves in fiber-reinforced composite,” Int. J. Solids Struct. 9, 1075-1085 (1973).
[CrossRef]

J. Acoust. Soc. Am. (3)

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

V. K. Varadan, V. V. Varadan, and Y.-H. Pao, “Multiple scattering of elastic waves by cylinders of arbitrary cross section. I. SH waves,” J. Acoust. Soc. Am. 63, 1310-1319 (1978).
[CrossRef]

V. K. Varadan, Y. Ma, and V. V. Varadan, “Multiple scattering of compressional and shear waves by fiber-reinforced composite materials,” J. Acoust. Soc. Am. 80, 333-339 (1986).
[CrossRef]

J. Appl. Mech. (1)

A. K. Mal and A. K. Chatterjee, “The elastic moduli of a fiber-reinforced composite,” J. Appl. Mech. 44, 61-67 (1977).
[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 (3)

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 (3)

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

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 (10)

Fig. 1
Fig. 1

Irradiation of a layer of densely distributed cylinders by a plane wave at a general incident direction.

Fig. 2
Fig. 2

Real part of the effective propagation constant as a function of solid volume fraction.

Fig. 3
Fig. 3

Imaginary part of the effective propagation constant as a function of solid volume fraction: (a) TM mode, (b) TE mode.

Fig. 4
Fig. 4

Variation of the extinction efficiency per cylinder with solid volume fraction: (a) TM mode, (b) TE mode.

Fig. 5
Fig. 5

Coherent forward scattered intensity as a function of layer thickness.

Fig. 6
Fig. 6

Coherent scattered intensity as a function of the polar angle of incidence.  

Fig. 7
Fig. 7

Coherent scattered intensity as a function of the azimuth angle of incidence.  

Fig. 8
Fig. 8

Incoherent scattered intensity distribution for various azimuth angles of incidence.

Fig. 9
Fig. 9

Variation of the scattering cross section with the polar angles of incidence.

Fig. 10
Fig. 10

Variation of the scattering cross section with the azimuth angle of incidence

Equations (73)

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

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