Abstract

On the basis of the generalized Lorenz–Mie theory, a description of expansion of the incident shaped beam in terms of cylindrical vector wave functions natural to an infinite cylinder of arbitrary orientation is presented. The expansion coefficients are derived by using an addition theorem for spherical vector wave functions under coordinate rotations. For the special cases of the cylinder axis intersecting the shaped beam axis and plane-wave illumination, the simplified expressions are given. The convergence of the beam shape coefficients is discussed.

© 2007 Optical Society of America

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References

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  1. G. Gouesbet, B. Maheu, and G. Gréhan, "Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation," J. Opt. Soc. Am. A 5, 1427-1443 (1988).
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  2. G. Gouesbet, G. Gréhan, and B. Maheu, "Computations of the gn coefficients in the generalized Lorenz-Mie theory using three different methods," Appl. Opt. 27, 4874-4883 (1988).
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    [CrossRef] [PubMed]
  5. Y. Han and Z. Wu, "Scattering of a spheroidal particle illuminated by a Gaussian beam," Appl. Opt. 40, 2501-2509 (2001).
    [CrossRef]
  6. Y. Han, H. Zhang, and X. Sun, "Scattering of shaped beam by an arbitrarily oriented spheroid having layers with non-confocal boundaries," Appl. Phys. B 84, 485-492 (2006).
    [CrossRef]
  7. Y. P. Han, H. Y. Zhang, and G. X. Han, "Expansion of shaped beam with respect to an arbitrarily oriented spheroidal particle," Opt. Express (to be published).
  8. J. P. Barton, D. R. Alexander, and S. A. Schaub, "Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam," J. Appl. Phys. 64, 1632-1639 (1988).
    [CrossRef]
  9. J. A. Lock, "Scattering of a diagonally incident focused Gaussian beam by an infinitely long homogeneous circular cylinder," J. Opt. Soc. Am. A 14, 640-652 (1997).
    [CrossRef]
  10. N. G. Alexopoulos and P. K. Park, "Scattering of waves with normal amplitude distribution from cylinders," IEEE Trans. Antennas Propag. AP-20, 216-217 (1972).
    [CrossRef]
  11. S. Kozaki, "A new expression for the scattering of a Gaussian beam by a conducting cylinder," IEEE Trans. Antennas Propag. AP-30, 881-887 (1982).
    [CrossRef]
  12. T. Kojima and Y. Yanagiuchi, "Scattering of an offset two-dimensional Gaussian beam wave by a cylinder," J. Appl. Phys. 50, 41-46 (1979).
    [CrossRef]
  13. S. Kozaki, "Scattering of a Gaussian beam by a homogeneous dielectric cylinder," J. Appl. Phys. 53, 7195-7200 (1982).
    [CrossRef]
  14. S. Kozaki, "Scattering of a Gaussian beam by an inhomogeneous dielectric cylinder," J. Opt. Soc. Am. 72, 1470-1474 (1982).
    [CrossRef]
  15. E. Zimmerman, R. Dandliker, N. Souli, and B. Krattiger, "Scattering of an off-axis Gaussian beam by a dielectric cylinder compared with a rigorous electromagnetic approach," J. Opt. Soc. Am. A 12, 398-403 (1995).
    [CrossRef]
  16. G. Gouesbet, "Interaction between Gaussian beams and infinite cylinders, by using the theory of distributions," J. Opt. (Paris) 26, 225-239 (1995).
    [CrossRef]
  17. G. Gouesbet and G. Gréhan, "Interaction between a Gaussian beam and an infinite cylinder with the use of non-∑-separable potentials," J. Opt. Soc. Am. A 11, 3261-3273 (1994).
    [CrossRef]
  18. K. F. Ren, G. Gréhan, and G. Gouesbet, "Scattering of a Gaussian beam by an infinite cylinder in the framework of generalized Lorenz-Mie theory formulation and numerical results," J. Opt. Soc. Am. A 14, 3014-3025 (1997).
    [CrossRef]
  19. A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton U. Press, 1957), Chap. 4.
  20. A. Doicu and T. Wriedt, "Computation of the beam-shape coefficients in the generalized Lorenz-Mie theory by using the translational addition theorem for spherical vector wave functions," Appl. Opt. 36, 2971-2978 (1997).
    [CrossRef] [PubMed]
  21. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).
  22. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
  23. L. W. Davis, "Theory of electromagnetic beam," Phys. Rev. A 19, 1177-1179 (1979).
    [CrossRef]

2006 (1)

Y. Han, H. Zhang, and X. Sun, "Scattering of shaped beam by an arbitrarily oriented spheroid having layers with non-confocal boundaries," Appl. Phys. B 84, 485-492 (2006).
[CrossRef]

2001 (1)

1997 (4)

1995 (2)

1994 (1)

1990 (1)

1988 (3)

1982 (3)

S. Kozaki, "A new expression for the scattering of a Gaussian beam by a conducting cylinder," IEEE Trans. Antennas Propag. AP-30, 881-887 (1982).
[CrossRef]

S. Kozaki, "Scattering of a Gaussian beam by a homogeneous dielectric cylinder," J. Appl. Phys. 53, 7195-7200 (1982).
[CrossRef]

S. Kozaki, "Scattering of a Gaussian beam by an inhomogeneous dielectric cylinder," J. Opt. Soc. Am. 72, 1470-1474 (1982).
[CrossRef]

1979 (2)

T. Kojima and Y. Yanagiuchi, "Scattering of an offset two-dimensional Gaussian beam wave by a cylinder," J. Appl. Phys. 50, 41-46 (1979).
[CrossRef]

L. W. Davis, "Theory of electromagnetic beam," Phys. Rev. A 19, 1177-1179 (1979).
[CrossRef]

1972 (1)

N. G. Alexopoulos and P. K. Park, "Scattering of waves with normal amplitude distribution from cylinders," IEEE Trans. Antennas Propag. AP-20, 216-217 (1972).
[CrossRef]

Alexander, D. R.

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam," J. Appl. Phys. 64, 1632-1639 (1988).
[CrossRef]

Alexopoulos, N. G.

N. G. Alexopoulos and P. K. Park, "Scattering of waves with normal amplitude distribution from cylinders," IEEE Trans. Antennas Propag. AP-20, 216-217 (1972).
[CrossRef]

Barton, J. P.

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam," J. Appl. Phys. 64, 1632-1639 (1988).
[CrossRef]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Dandliker, R.

Davis, L. W.

L. W. Davis, "Theory of electromagnetic beam," Phys. Rev. A 19, 1177-1179 (1979).
[CrossRef]

Doicu, A.

Edmonds, A. R.

A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton U. Press, 1957), Chap. 4.

Gouesbet, G.

Gréhan, G.

Guo, L. X.

Han, G. X.

Y. P. Han, H. Y. Zhang, and G. X. Han, "Expansion of shaped beam with respect to an arbitrarily oriented spheroidal particle," Opt. Express (to be published).

Han, Y.

Y. Han, H. Zhang, and X. Sun, "Scattering of shaped beam by an arbitrarily oriented spheroid having layers with non-confocal boundaries," Appl. Phys. B 84, 485-492 (2006).
[CrossRef]

Y. Han and Z. Wu, "Scattering of a spheroidal particle illuminated by a Gaussian beam," Appl. Opt. 40, 2501-2509 (2001).
[CrossRef]

Han, Y. P.

Y. P. Han, H. Y. Zhang, and G. X. Han, "Expansion of shaped beam with respect to an arbitrarily oriented spheroidal particle," Opt. Express (to be published).

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Kojima, T.

T. Kojima and Y. Yanagiuchi, "Scattering of an offset two-dimensional Gaussian beam wave by a cylinder," J. Appl. Phys. 50, 41-46 (1979).
[CrossRef]

Kozaki, S.

S. Kozaki, "Scattering of a Gaussian beam by an inhomogeneous dielectric cylinder," J. Opt. Soc. Am. 72, 1470-1474 (1982).
[CrossRef]

S. Kozaki, "A new expression for the scattering of a Gaussian beam by a conducting cylinder," IEEE Trans. Antennas Propag. AP-30, 881-887 (1982).
[CrossRef]

S. Kozaki, "Scattering of a Gaussian beam by a homogeneous dielectric cylinder," J. Appl. Phys. 53, 7195-7200 (1982).
[CrossRef]

Krattiger, B.

Lock, J. A.

Maheu, B.

Park, P. K.

N. G. Alexopoulos and P. K. Park, "Scattering of waves with normal amplitude distribution from cylinders," IEEE Trans. Antennas Propag. AP-20, 216-217 (1972).
[CrossRef]

Ren, K. F.

Schaub, S. A.

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam," J. Appl. Phys. 64, 1632-1639 (1988).
[CrossRef]

Souli, N.

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

Sun, X.

Y. Han, H. Zhang, and X. Sun, "Scattering of shaped beam by an arbitrarily oriented spheroid having layers with non-confocal boundaries," Appl. Phys. B 84, 485-492 (2006).
[CrossRef]

Wriedt, T.

Wu, Z.

Wu, Z. S.

Yanagiuchi, Y.

T. Kojima and Y. Yanagiuchi, "Scattering of an offset two-dimensional Gaussian beam wave by a cylinder," J. Appl. Phys. 50, 41-46 (1979).
[CrossRef]

Zhang, H.

Y. Han, H. Zhang, and X. Sun, "Scattering of shaped beam by an arbitrarily oriented spheroid having layers with non-confocal boundaries," Appl. Phys. B 84, 485-492 (2006).
[CrossRef]

Zhang, H. Y.

Y. P. Han, H. Y. Zhang, and G. X. Han, "Expansion of shaped beam with respect to an arbitrarily oriented spheroidal particle," Opt. Express (to be published).

Zimmerman, E.

Appl. Opt. (4)

Appl. Phys. B (1)

Y. Han, H. Zhang, and X. Sun, "Scattering of shaped beam by an arbitrarily oriented spheroid having layers with non-confocal boundaries," Appl. Phys. B 84, 485-492 (2006).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

N. G. Alexopoulos and P. K. Park, "Scattering of waves with normal amplitude distribution from cylinders," IEEE Trans. Antennas Propag. AP-20, 216-217 (1972).
[CrossRef]

S. Kozaki, "A new expression for the scattering of a Gaussian beam by a conducting cylinder," IEEE Trans. Antennas Propag. AP-30, 881-887 (1982).
[CrossRef]

J. Appl. Phys. (3)

T. Kojima and Y. Yanagiuchi, "Scattering of an offset two-dimensional Gaussian beam wave by a cylinder," J. Appl. Phys. 50, 41-46 (1979).
[CrossRef]

S. Kozaki, "Scattering of a Gaussian beam by a homogeneous dielectric cylinder," J. Appl. Phys. 53, 7195-7200 (1982).
[CrossRef]

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam," J. Appl. Phys. 64, 1632-1639 (1988).
[CrossRef]

J. Opt. (Paris) (1)

G. Gouesbet, "Interaction between Gaussian beams and infinite cylinders, by using the theory of distributions," J. Opt. (Paris) 26, 225-239 (1995).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Phys. Rev. A (1)

L. W. Davis, "Theory of electromagnetic beam," Phys. Rev. A 19, 1177-1179 (1979).
[CrossRef]

Other (4)

Y. P. Han, H. Y. Zhang, and G. X. Han, "Expansion of shaped beam with respect to an arbitrarily oriented spheroidal particle," Opt. Express (to be published).

A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton U. Press, 1957), Chap. 4.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

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Figures (2)

Fig. 1
Fig. 1

Cylinder axis of an arbitrarily oriented infinite cylinder coincides with the O z axis of O x y z . The x y z axes are obtained by a rigid-body rotation of the x y z axes through Euler angles α , β , γ . The infinite cylinder is illuminated by a shaped beam propagating along the O z axis with the middle of its beam waist located at origin O . O z is parallel to O z , and similar conditions hold for the other axes. The Cartesian coordinates of O in the system O x y z are ( x 0 , y 0 , z 0 ) .

Fig. 2
Fig. 2

Geometry of an infinite cylinder in the Cartesian coordinate system O x y z . The cylinder axis is along the O z axis, and its orientation in space is specified by the Euler angles α , β , γ of the x y z axes with respect to the x y z axes.

Equations (75)

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E i = E 0 n = 1 m = n n C n m [ i g n , T E m m m n r ( 1 ) ( k R , θ , ϕ ) + g n , T M m n m n r ( 1 ) ( k R , θ , ϕ ) ] ,
C n m = { C n m 0 ( 1 ) m ( n + m ) ! ( n m ) ! C n m < 0 } ,
C n = i n 1 2 n + 1 n ( n + 1 ) ,
P n m ( cos θ ) e i m ϕ = s = n n ρ ( m , s , n ) P n s ( cos θ ) e i s ϕ .
ρ ( m , s , n ) = ( 1 ) s + m e i s γ [ ( n + m ) ! ( n s ) ! ( n m ) ! ( n + s ) ! ] 1 2 u s m ( n ) ( β ) e i m α ,
u s m ( n ) ( β ) = [ ( n + s ) ! ( n s ) ! ( n + m ) ! ( n m ) ! ] 1 2 σ ( n + m n s σ ) ( n m σ ) ( 1 ) n s σ ( cos β 2 ) 2 σ + s + m ( sin β 2 ) 2 n 2 σ s m .
w m n r ( 1 ) ( k R , θ , ϕ ) = s = n n ρ ( m , s , n ) w s n r ( 1 ) ( k R , θ , ϕ ) ,
E i = E 0 n = 1 m = n n s = n n ρ ( m , s , n ) C n m [ i g n , T E m m s n r ( 1 ) ( k R , θ , ϕ ) + g n , T M m n s n r ( 1 ) ( k R , θ , ϕ ) ] .
E i = E 0 m = n = m [ i G n , T E m m m n r ( 1 ) ( k R , θ , ϕ ) + G n , T M m n m n r ( 1 ) ( k R , θ , ϕ ) ] ,
G n , T E m = s = n n ρ ( s , m , n ) C n s g n , T E s ,
G n , T E m = s = n n ρ ( s , m , n ) C n s g n , T M s .
m m n r ( 1 ) ( k R , θ , ϕ ) = 0 π [ c m n ( ζ ) m m λ ( 1 ) + a m n ( ζ ) n m λ ( 1 ) ] e i h z sin ζ d ζ ,
n m n r ( 1 ) ( k R , θ , ϕ ) = 0 π [ c m n ( ζ ) n m λ ( 1 ) + a m n ( ζ ) m m λ ( 1 ) ] e i h z sin ζ d ζ ,
c m n ( ζ ) = i m n + 1 2 k d P n m ( cos ζ ) d ( cos ζ ) ,
a m n ( ζ ) = m k λ 2 i m n 1 2 P n m ( cos ζ ) .
E i = E 0 m = 0 π [ I m , T E ( ζ ) m m λ ( 1 ) + I m , T M ( ζ ) n m λ ( 1 ) ] e i h z sin ζ d ζ ,
I m , T E = n = m [ i G n , T E m c m n ( ζ ) + G n , T M m a m n ( ζ ) ] ,
I m , T M = n = m [ i G n , T E m a m n ( ζ ) + G n , T M m c m n ( ζ ) ] .
E s = E 0 m = 0 π [ α m ( ζ ) m m λ ( 3 ) + β m ( ζ ) n m λ ( 3 ) ] e i h z sin ζ d ζ ,
E w = E 0 m = 0 π [ χ m ( ζ ) m m λ ( 1 ) + τ m ( ζ ) n m λ ( 1 ) ] e i h z sin ζ d ζ ,
H = 1 i w μ × E , m m λ e i h z = 1 k × ( n m λ e i h z ) , n m λ e i h z = 1 k × ( m m λ e i h z ) .
g n , T E 1 = g n , T E 1 = 1 2 g n , g n , T M 1 = g n , T M 1 = i 2 g n ,
g n , T E 1 = g n , T E 1 = i 2 g n , g n , T M 1 = g n , T M 1 = 1 2 g n .
G n , T E m = ( 1 ) m 1 ( n m ) ! ( n + m ) ! C n g n e i m γ [ d P n m ( cos β ) d β cos α i m P n m ( cos β ) sin β sin α ] ,
G n , T M m = ( 1 ) m 1 ( n m ) ! ( n + m ) ! C n g n e i m γ [ d P n m ( cos β ) d β sin α i m P n m ( cos β ) sin β cos α ] .
I m , T E = ( 1 ) m 1 i m + 1 2 k sin 2 ζ n = m ( n m ) ! ( n + m ) ! 2 n + 1 n ( n + 1 ) g n [ m 2 P n m ( cos β ) sin β P n m ( cos ζ ) + d P n m ( cos β ) d β d P n m ( cos ζ ) d ζ sin ζ ] ,
I m , T M = ( 1 ) m 1 m i m + 1 2 k sin 2 ζ n = m ( n m ) ! ( n + m ) ! 2 n + 1 n ( n + 1 ) g n [ P n m ( cos β ) sin β d P n m ( cos ζ ) d ζ sin ζ + d P n m ( cos β ) d β P n m ( cos ζ ) ] .
n = m 2 n + 1 n ( n + 1 ) ( n m ) ! ( n + m ) ! [ d P n m ( cos β ) d β d P n m ( cos ζ ) d ζ sin ζ + m 2 P n m ( cos β ) sin β P n m ( cos ζ ) ] = 2 δ ( ζ β ) ,
n = m ( n m ) ! ( n + m ) ! 2 n + 1 n ( n + 1 ) [ P n m ( cos β ) sin β d P n m ( cos ζ ) d ζ sin ζ + d P n m ( cos β ) d β P n m ( cos ζ ) ] = 0 ,
e ̑ y exp [ i k ( x sin ζ + z cos ζ ) ] = m = i m + 1 k sin ζ m m λ e i h z .
g n = 1 1 + 2 i s z 0 w 0 exp ( i k z 0 ) exp [ s 2 ( n + 1 2 ) 2 1 + 2 i s z 0 w 0 ] ,
m m n r ( 1 ) ( k R , θ , ϕ ) = m m n ¯ e i m ϕ = [ i m sin θ P n m ( cos θ ) j n ( k R ) e ̑ θ d P n m ( cos θ ) d θ j n ( k R ) e ̑ ϕ ] e i m ϕ ,
m m λ ( 1 ) e i h z = m m λ ¯ e i h z e i m ϕ = [ i m r J m ( λ r ) e ̂ r r J m ( λ r ) e ̂ ϕ ] e i h z e i m ϕ ,
n m λ ( 1 ) e i h z = n m λ ¯ e i h z e i m ϕ = [ i h k r J m ( λ r ) e ̂ r h m k r J m ( λ r ) e ̂ ϕ + λ 2 k J m ( λ r ) e ̂ z ] e i h z e i m ϕ .
c m n ( ζ ) = k 2 π m m n ¯ m m λ ¯ e i h z d z m m λ ¯ m m λ ¯ ,
a m n ( ζ ) = k 2 π m m n ¯ n m λ ¯ e i h z d z n m λ ¯ n m λ ¯ .
m m n ¯ m m λ ¯ e i h z d z = m sin θ P n m ( cos θ ) j n ( k R ) m r J m ( λ r ) cos θ e i h z d z + [ d P n m ( cos θ ) d θ j n ( k R ) r J m ( λ r ) ] e i h z d z ,
m m λ ¯ m m λ ¯ = λ 2 J m + 1 ( λ r ) J m 1 ( λ r ) .
m m n ¯ n m λ ¯ e i h z d z = m P n m ( cos θ ) j n ( k R ) h z k r r J m ( λ r ) e i h z d z i m P n m ( cos θ ) j n ( k R ) λ λ k J m ( λ r ) e i h z d z + d P n m ( cos θ ) d θ j n ( k R ) h m k r J m ( λ r ) e i h z d z ,
n m λ ¯ n m λ ¯ = h 2 k 2 λ 2 J m + 1 ( λ r ) J m 1 ( λ r ) + λ 2 k 2 λ 2 [ J m ( λ r ) ] 2 .
j n ( k R ) P n m ( cos θ ) = i m n 2 0 π e i k z cos α J m ( k r sin α ) P n m ( cos α ) sin α d α ,
j n ( k R ) d P n m ( cos θ ) d θ = i m n 2 [ i k r 0 π e i k z cos α J m ( k r sin α ) P n m ( cos α ) sin α cos α d α + k z 0 π e i k z cos α J m ( k r sin α ) P n m ( cos α ) sin α sin α d α ] .
e i k ( γ γ ) z d z = 2 π k δ ( γ γ ) ,
z e i k ( γ γ ) z d z = i 2 π k 2 δ ( 1 ) ( γ γ ) ,
i m n 2 m 2 r 2 J m ( λ r ) 0 π J m ( k r sin α ) P n m ( cos α ) sin α d α z e i k ( cos α cos ζ ) z d z = i m n + 1 2 2 π m 2 k 2 r 2 J m ( λ r ) 0 π δ ( 1 ) ( cos α cos ζ ) J m ( k r sin α ) P n m ( cos α ) sin α d α ,
i i m n 2 k r λ J m ( λ r ) 0 π J m ( k r sin α ) P n m ( cos α ) sin α cos α d α e i k ( cos α cos ζ ) z d z = i i m n 2 2 π r λ J m ( λ r ) 0 π δ ( cos α cos ζ ) J m ( k r sin α ) P n m ( cos α ) sin α cos α d α ,
i m n 2 k λ J m ( λ r ) 0 π J m ( k r sin α ) P n m ( cos α ) sin α sin α d α z e i k ( cos α cos ζ ) z d z = i i m n 2 2 π 1 k λ J m ( λ r ) 0 π δ ( 1 ) ( cos α cos ζ ) J m ( k r sin α ) P n m ( cos α ) sin α sin α d α .
f ( γ ) δ ( n ) ( γ γ ) d γ = ( 1 ) n [ f ( γ ) ] γ = γ ( n ) ,
π i m n + 1 m 2 k 2 r 2 J m ( λ r ) [ J m ( k r 1 x 2 ) P n m ( x ) ] x = cos ζ ( 1 ) = π i m n + 1 m 2 k 2 r 2 J m ( λ r ) [ k r J m ( k r sin ζ ) cos ζ sin ζ P n m ( cos ζ ) + J m ( k r sin ζ ) d P n m ( cos ζ ) d ( cos ζ ) ] ,
i m n 1 π r λ J m ( λ r ) [ J m ( k r sin α ) P n m ( cos α ) cos α ] cos α = cos ζ = i m n 1 π r λ J m ( λ r ) J m ( k r sin ζ ) P n m ( cos ζ ) cos ζ ,
i m n + 1 1 k π λ J m ( λ r ) d d x [ J m ( k r 1 x 2 ) P n m ( x ) 1 x 2 ] x = cos z ( 1 ) = i m n + 1 1 k π λ J m ( λ r ) [ J m ( k r sin ζ ) k r P n m ( cos ζ ) cos ζ m 2 k r sin 2 ζ J m ( k r sin ζ ) P n m ( cos ζ ) cos ζ + J m ( k r sin ζ ) d P n m ( cos ζ ) d ( cos ζ ) sin ζ ] .
m m n ¯ m m λ ¯ e i h z d z = π i m n + 1 d P n m ( cos ζ ) d ( cos ζ ) sin 2 ζ [ J m 1 ( λ r ) J m + 1 ( λ r ) ] .
m P n m ( cos θ ) j n ( k R ) h z k r r J m ( λ r ) e i h z d z = i m n 2 h k r m λ J m ( k r ) 0 π J m ( k r sin α ) P n m ( cos α ) sin α d α z e i k ( cos α cos ζ ) z d z ,
i m P n m ( cos θ ) j n ( k R ) λ λ k J m ( λ r ) e i h z d z = i i m n 2 λ λ k J m ( λ r ) m 0 π J m ( k r sin α ) P n m ( cos α ) sin α d α e i k ( cos α cos ζ ) z d z ,
i m n 2 h m k r J m ( λ r ) ( i k r ) 0 π J m ( k r sin α ) P n m ( cos α ) sin α cos α d α e i k ( cos α cos ζ ) z d z = i m n + 1 π h m k J m ( λ r ) 0 π δ ( cos α cos ζ ) J m ( k r sin α ) P n m ( cos α ) sin α cos α d α ,
i m n 2 h m k r J m ( λ r ) k 0 π J m ( k r sin α ) P n m ( cos α ) sin α sin α d α z e i k ( cos α cos ζ ) d z = i m n + 1 π h m k 2 r J m ( λ r ) 0 π δ ( 1 ) ( cos α cos ζ ) J m ( k r sin α ) P n m ( cos α ) sin α sin α d α .
i m n 2 2 π k 2 h k r m λ J m ( λ r ) ( i ) 0 π δ ( 1 ) ( cos α cos ζ ) J m ( k r sin α ) P n m ( cos α ) sin α d α = π i m n + 1 h k r 1 k 2 m λ J m ( λ r ) [ k r J m ( k r sin ζ ) cos ζ sin ζ P n m ( cos ζ ) + J m ( k r sin ζ ) d P n m ( cos ζ ) d ( cos ζ ) ] ,
i m n + 1 π λ 2 k 2 J m ( λ r ) m 0 π δ ( cos α cos ζ ) J m ( k r sin α ) P n m ( cos α ) sin α d α = i m n + 1 π λ 2 k 2 J m ( λ r ) m J m ( k r sin ζ ) P n m ( cos ζ )
i m n + 1 π h m k J m ( λ r ) [ J m ( k r sin α ) P n m ( cos α ) cos α ] cos α = cos ζ
= i m n + 1 π h m k J m ( λ r ) J m ( k r sin ζ ) P n m ( cos ζ ) cos ζ
π i m n + 1 k h m k r J m ( λ r ) [ J m ( k r 1 x 2 ) P n m ( x ) 1 x 2 ] x = cos ζ ( 1 )
= π i m n + 1 k h m k r J m ( λ r ) [ J m ( k r sin ζ ) k r P n m ( cos ζ ) cos ζ m 2 k r sin 2 ζ J m ( k r sin ζ ) P n m ( cos ζ ) cos ζ + J m ( k r sin ζ ) d P n m ( cos ζ ) d ( cos ζ ) sin ζ ] .
π i m n + 1 m P n m ( cos ζ ) { h 2 k 2 m 2 k 2 r 2 sin 2 ζ [ J m ( k r sin ζ ) ] 2 h 2 k 2 [ J m ( k r sin ζ ) ] 2 + λ 2 k 2 [ J m ( k r sin ζ ) ] 2 } = π i m n + 1 m P n m ( cos ζ ) { h 2 k 2 J m 1 ( λ r ) J m + 1 ( λ r ) + λ 2 k 2 [ J m ( λ r ) ] 2 } .
M = 1 k × N , N = 1 k × M .
f ( cos ζ ) = l = m a l P l m ( cos ζ ) ,
a l = 2 l + 1 2 ( l m ) ! ( l + m ) ! 0 π f ( cos ζ ) P l m ( cos ζ ) sin ζ d ζ .
0 π f ( cos ζ ) n = m 2 n + 1 n ( n + 1 ) ( n m ) ! ( n + m ) ! [ d P n m ( cos β ) d β d P n m ( cos ζ ) d ζ sin ζ + m 2 P n m ( cos β ) sin β P n m ( cos ζ ) ] sin ζ d ζ = l = m a l n = m 0 π 2 n + 1 n ( n + 1 ) ( n m ) ! ( n + m ) ! P l m ( cos ζ ) [ d P n m ( cos β ) d β d P n m ( cos ζ ) d ζ sin ζ + m 2 P n m ( cos β ) sin β P n m ( cos ζ ) ] sin ζ d ζ .
sin ζ d d ζ P n m ( cos ζ ) = 1 2 n + 1 [ ( n + 1 ) ( n + m ) P n 1 m ( cos ζ ) n ( n m + 1 ) P n + 1 m ( cos ζ ) ] ,
0 π P n m ( cos ζ ) P l m ( cos ζ ) sin ζ d ζ = { 0 n l 2 2 l + 1 ( l + m ) ! ( l m ) ! n = l } .
l = m a l n = m 2 n + 1 n ( n + 1 ) ( n m ) ! ( n + m ) ! 0 π P l m ( cos ζ ) d P n m ( cos β ) d β d P n m ( cos ζ ) d ζ sin ζ sin ζ d ζ = l = m a l n = m ( n m ) ! n ( n + 1 ) ( n + m ) ! d P n m ( cos β ) d β 0 π P l m ( cos ζ ) [ n ( n m + 1 ) P n + 1 m ( cos ζ ) ( n + 1 ) ( n + m ) P n 1 m ( cos ζ ) ] sin ζ d ζ = l = m a l [ 2 sin β P l m ( cos β ) 2 m 2 l ( l + 1 ) P l m ( cos β ) sin β ] ,
l = m a l n = m m 2 P n m ( cos β ) sin β 0 π 2 n + 1 n ( n + 1 ) ( n m ) ! ( n + m ) ! P l m ( cos ζ ) P n m ( cos ζ ) sin ζ d ζ = 2 l = m a l m 2 l ( l + 1 ) p l m ( cos β ) sin β .
2 sin β l = m a l P l m ( cos β ) = 2 sin β f ( cos β ) .
0 π f ( cos ζ ) δ ( ζ β ) sin ζ d ζ = f ( cos β ) sin β ,
0 π f ( cos ζ ) n = m ( n m ) ! ( n + m ) ! 2 n + 1 n ( n + 1 ) [ P n m ( cos β ) sin β d P n m ( cos ζ ) d ζ sin ζ + d P n m ( cos β ) d β P n m ( cos ζ ) ] sin ζ d ζ = l = m a l n = m 0 π P l m ( cos ζ ) ( n m ) ! ( n + m ) ! 2 n + 1 n ( n + 1 ) [ P n m ( cos β ) sin β d P n m ( cos ζ ) d ζ sin ζ + d P n m ( cos β ) d β P n m ( cos ζ ) ] sin ζ d ζ = 0 ;
P n m ( cos θ ) = ( 1 ) m ( n m ) ! ( n + m ) ! P n m ( cos θ ) , m > 0 ,

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