Abstract

Axisymmetric light beams are defined as light beams for which the component of the Poynting vector in the direction of propagation does not depend on the azimuthal angle in suitably chosen coordinate systems to reveal the symmetric property of the beam. It is shown that such beams are encoded in a set of beam-shape coefficients g n that are, however, defined in a more general way than usual in the case of Gaussian beams. Partial-wave expansions and properties of such beams are studied.

© 1996 Optical Society of America

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  1. G. Gouesbet, “Generalized Lorenz–Mie theory and applications,” Part. Part. Syst. Charact. 11, 22–34 (1994).
  2. G. Gouesbet, G. Gréhan, B. Maheu, “Generalized Lorenz–Mie theory and applications to optical sizing,” in Combustion Measurements, N. Chigier, ed. (Hemisphere, New York, 1991), pp. 339–384.
  3. G. Gouesbet, B. Maheu, 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).
  4. J. P. Barton, D. R. Alexander, 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).
  5. J. A. Lock, “Contribution of high-order rainbows to the scattering of a Gaussian laser beam by a spherical particle,” J. Opt. Soc. Am. 10, 693–706 (1993).
  6. G. Gouesbet, G. Gréhan, B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center, using a Bromwich formulation,” J. Opt. (Paris) 16, 83–93 (1985).Republished in Selected Papers on Light Scattering, Vol. 951 of SPIE Milestone series, M. Kerker, ed. (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1988), part I, pp. 361–371.
  7. G. Gouesbet, G. Gréhan, “Sur la généralisation de la théorie de Lorenz–Mie,” J. Opt. (Paris) 13, 97–103 (1982).
  8. K. F. Ren, G. Gréhan, G. Gouesbet, “Electromagnetic field expression of a laser sheet and the order of approximation,” J. Opt. (Paris) 25, 165–176 (1994).
  9. K. F. Ren, G. Gréhan, G. Gouesbet, “Evaluation of laser sheet beam shape coefficients in generalized Lorenz–Mie theory by using a localized approximation,” J. Opt. Soc. Am. A 11, 2072–2079 (1994).
  10. G. Gouesbet, J. A. Lock, G. Gréhan, “Partial-wave representations of laser beams for use in light scattering calculations,” Appl. Opt. 34, 2133–2143 (1995).

1995 (1)

1994 (3)

K. F. Ren, G. Gréhan, G. Gouesbet, “Electromagnetic field expression of a laser sheet and the order of approximation,” J. Opt. (Paris) 25, 165–176 (1994).

K. F. Ren, G. Gréhan, G. Gouesbet, “Evaluation of laser sheet beam shape coefficients in generalized Lorenz–Mie theory by using a localized approximation,” J. Opt. Soc. Am. A 11, 2072–2079 (1994).

G. Gouesbet, “Generalized Lorenz–Mie theory and applications,” Part. Part. Syst. Charact. 11, 22–34 (1994).

1993 (1)

1988 (2)

G. Gouesbet, B. Maheu, 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).

J. P. Barton, D. R. Alexander, 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).

1985 (1)

G. Gouesbet, G. Gréhan, B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center, using a Bromwich formulation,” J. Opt. (Paris) 16, 83–93 (1985).Republished in Selected Papers on Light Scattering, Vol. 951 of SPIE Milestone series, M. Kerker, ed. (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1988), part I, pp. 361–371.

1982 (1)

G. Gouesbet, G. Gréhan, “Sur la généralisation de la théorie de Lorenz–Mie,” J. Opt. (Paris) 13, 97–103 (1982).

Alexander, D. R.

J. P. Barton, D. R. Alexander, 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).

Barton, J. P.

J. P. Barton, D. R. Alexander, 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).

Gouesbet, G.

G. Gouesbet, J. A. Lock, G. Gréhan, “Partial-wave representations of laser beams for use in light scattering calculations,” Appl. Opt. 34, 2133–2143 (1995).

K. F. Ren, G. Gréhan, G. Gouesbet, “Electromagnetic field expression of a laser sheet and the order of approximation,” J. Opt. (Paris) 25, 165–176 (1994).

K. F. Ren, G. Gréhan, G. Gouesbet, “Evaluation of laser sheet beam shape coefficients in generalized Lorenz–Mie theory by using a localized approximation,” J. Opt. Soc. Am. A 11, 2072–2079 (1994).

G. Gouesbet, “Generalized Lorenz–Mie theory and applications,” Part. Part. Syst. Charact. 11, 22–34 (1994).

G. Gouesbet, B. Maheu, 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).

G. Gouesbet, G. Gréhan, B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center, using a Bromwich formulation,” J. Opt. (Paris) 16, 83–93 (1985).Republished in Selected Papers on Light Scattering, Vol. 951 of SPIE Milestone series, M. Kerker, ed. (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1988), part I, pp. 361–371.

G. Gouesbet, G. Gréhan, “Sur la généralisation de la théorie de Lorenz–Mie,” J. Opt. (Paris) 13, 97–103 (1982).

G. Gouesbet, G. Gréhan, B. Maheu, “Generalized Lorenz–Mie theory and applications to optical sizing,” in Combustion Measurements, N. Chigier, ed. (Hemisphere, New York, 1991), pp. 339–384.

Gréhan, G.

G. Gouesbet, J. A. Lock, G. Gréhan, “Partial-wave representations of laser beams for use in light scattering calculations,” Appl. Opt. 34, 2133–2143 (1995).

K. F. Ren, G. Gréhan, G. Gouesbet, “Evaluation of laser sheet beam shape coefficients in generalized Lorenz–Mie theory by using a localized approximation,” J. Opt. Soc. Am. A 11, 2072–2079 (1994).

K. F. Ren, G. Gréhan, G. Gouesbet, “Electromagnetic field expression of a laser sheet and the order of approximation,” J. Opt. (Paris) 25, 165–176 (1994).

G. Gouesbet, B. Maheu, 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).

G. Gouesbet, G. Gréhan, B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center, using a Bromwich formulation,” J. Opt. (Paris) 16, 83–93 (1985).Republished in Selected Papers on Light Scattering, Vol. 951 of SPIE Milestone series, M. Kerker, ed. (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1988), part I, pp. 361–371.

G. Gouesbet, G. Gréhan, “Sur la généralisation de la théorie de Lorenz–Mie,” J. Opt. (Paris) 13, 97–103 (1982).

G. Gouesbet, G. Gréhan, B. Maheu, “Generalized Lorenz–Mie theory and applications to optical sizing,” in Combustion Measurements, N. Chigier, ed. (Hemisphere, New York, 1991), pp. 339–384.

Lock, J. A.

Maheu, B.

G. Gouesbet, B. Maheu, 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).

G. Gouesbet, G. Gréhan, B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center, using a Bromwich formulation,” J. Opt. (Paris) 16, 83–93 (1985).Republished in Selected Papers on Light Scattering, Vol. 951 of SPIE Milestone series, M. Kerker, ed. (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1988), part I, pp. 361–371.

G. Gouesbet, G. Gréhan, B. Maheu, “Generalized Lorenz–Mie theory and applications to optical sizing,” in Combustion Measurements, N. Chigier, ed. (Hemisphere, New York, 1991), pp. 339–384.

Ren, K. F.

K. F. Ren, G. Gréhan, G. Gouesbet, “Electromagnetic field expression of a laser sheet and the order of approximation,” J. Opt. (Paris) 25, 165–176 (1994).

K. F. Ren, G. Gréhan, G. Gouesbet, “Evaluation of laser sheet beam shape coefficients in generalized Lorenz–Mie theory by using a localized approximation,” J. Opt. Soc. Am. A 11, 2072–2079 (1994).

Schaub, S. A.

J. P. Barton, D. R. Alexander, 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).

Appl. Opt. (1)

J. Appl. Phys. (1)

J. P. Barton, D. R. Alexander, 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).

J. Opt. (Paris) (3)

G. Gouesbet, G. Gréhan, B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center, using a Bromwich formulation,” J. Opt. (Paris) 16, 83–93 (1985).Republished in Selected Papers on Light Scattering, Vol. 951 of SPIE Milestone series, M. Kerker, ed. (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1988), part I, pp. 361–371.

G. Gouesbet, G. Gréhan, “Sur la généralisation de la théorie de Lorenz–Mie,” J. Opt. (Paris) 13, 97–103 (1982).

K. F. Ren, G. Gréhan, G. Gouesbet, “Electromagnetic field expression of a laser sheet and the order of approximation,” J. Opt. (Paris) 25, 165–176 (1994).

J. Opt. Soc. Am. (1)

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

Part. Part. Syst. Charact. (1)

G. Gouesbet, “Generalized Lorenz–Mie theory and applications,” Part. Part. Syst. Charact. 11, 22–34 (1994).

Other (1)

G. Gouesbet, G. Gréhan, B. Maheu, “Generalized Lorenz–Mie theory and applications to optical sizing,” in Combustion Measurements, N. Chigier, ed. (Hemisphere, New York, 1991), pp. 339–384.

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Equations (150)

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U TM = E 0 r n = 1 m = n + n c n pw g n , TM m            × exp ( i m φ ) ψ n 1 ( k r ) P n m ( cos  θ ) ,
U TE = H 0 r n = 1 m = n + n c n pw g n , TE m           × exp ( i m φ ) ψ n 1 ( k r ) P n m ( cos  θ ) ,
c n pw = 1 i k ( i ) n 2 n + 1 n ( n + 1 ) .
ψ n ( k r ) = k r ψ n 1 ( k r ) .
U TM = E 0 k n = 1 m = n + n c n pw g n , TM m   exp ( i m φ ) ψ n P n m ,
U TE = H 0 k n = 1 m = n + n c n pw g n , TM m   exp ( i m φ ) ψ n P n m .
E r = k E 0 n = 1 m = n + n c n pw g n , TM m ( ψ n + ψ n )         × P n m exp ( i m φ ) ,
E θ = E 0 r n = 1 m = n + n c n pw exp ( i m φ ) ( g n , TM m ψ n τ n m          + m g n , TE m ψ n π n m ) ,
E φ = i E 0 r n = 1 m = n + n c n pw exp ( i m φ ) ( m g n , TM m ψ n π n m    + g n , TM m ψ n τ n m ) ,
H r = k H 0 n = 1 m = n + n c n pw g n , TE m ( ψ n + ψ n )    × P n m exp ( i m φ ) ,
H θ = H 0 r n = 1 m = n + n c n pw exp ( i m φ ) ( g n , TE m ψ n τ n m    m g n , TM m ψ n π n m ) ,
H φ = i H 0 r n = 1 m = n + n c n pw exp ( i m φ ) ( m g n , TE m ψ n π n m    g n , TM m ψ n τ n m ) ,
π n i ( cos  θ ) = p n i sin  θ = ( 1 ) i ( sin  θ) i 1 d i p n ( cos  θ) ( d  cos  θ ) i ,
τ n i ( cos  θ ) = d p n i = d ( 1 ) i ( sin  θ ) i d i p n ( cos  θ ) ( d  cos  θ ) i ,
V x = V r   sin  θ  cos  φ + V θ   cos  θ  cos  φ V φ  sin  φ ,
V y = V r   sin  θ  sin  φ + V θ   cos  θ  sin  φ V φ  cos  φ ,
V z = V r   cos  θ V θ   sin  θ ,
E x = E 0 n = 1 m = n + n c n pw   exp ( i m φ )    × [ k   sin  θ  cos  φ g n , TM m ( ψ n + ψ n ) p n m + 1 r cos  θ    × cos  φ ( g n , TM m ψ n τ n m + m g n , TE m ψ n π n m )    i sin  φ r ( m g n , TM m ψ n π n m + g n , TE m ψ n τ n m ) ] ,
E y = E 0 n = 1 m = n + n c n pw   exp ( i m φ )    × [ k   sin  θ  sin  φ g n , TM m ( ψ n + ψ n ) p n m + 1 r cos  θ    × sin  φ ( g n , TM m ψ n τ n m + m g n , TE m ψ n π n m )    + i cos  φ r ( m g n , TM m ψ n π n m + g n , TE m ψ n τ n m ) ] ,
E z = E 0 n = 1 m = n + n c n pw exp ( i m φ )    × [ k   cos  θ g n , TM m ( ψ n + ψ n ) p n m    sin  θ r ( g n , TM m ψ n τ n m + m g n , TE m ψ n π n m ) ] ,
H x = H 0 n = 1 m = n + n c n pw exp ( i m φ )    × [ k   sin  θ  cos  φ g n , TE m ( ψ n + ψ n ) p n m + 1 r cos  θ    × cos  φ ( g n , TE m ψ n τ n m m g n , TM m ψ n π n m )    i sin  φ r ( m g n , TE m ψ n π n m g n , TM m ψ n τ n m ) ] ,
H y = H 0 n = 1 m = n + n c n pw   exp ( i m φ )    × [ k   sin  θ  sin  φ g n , TE m ( ψ n + ψ n ) p n m + 1 r cos  θ    × sin  φ ( g n , TE m ψ n τ n m m g n , TM m ψ n π n m )    + i cos  φ r ( m g n , TE m ψ n π n m g n , TM m ψ n τ n m ) ] ,
H z = H 0 n = 1 m = n + n c n pw   exp ( i m φ )    × [ k   cos  θ g n , TE m ( ψ n + ψ n ) p n m    sin  θ r ( g n , TE m ψ n τ n m m g n , TM m ψ n π n m ) ] .
( S x S y S z ) = 1 2 Re ( E y H z * E z H y * E z H x * E x H z * E x H y * E y H x * ) ,
S z ( θ = 0 ) = 2 E 0 H 0 * r 2 n = 1 p = 1 i c n pw c p pw * Γ n Γ p × [ ( g n , TM 1 ψ n + g n , TE 1 ψ n ) ( g p , TE 1 * ψ p g p , TM 1 * ψ p ) ( g n , TM 1 ψ n g n , TE 1 ψ n ) ( g p , TM 1 * ψ p + g p , TE 1 * ψ p ) ] ,
S z ± 1 = E 0 H 0 * 2 r 2 Re n = 1 p = 1 i c n pw c p pw*    × [ exp ( 2 i φ ) ( S n p 1 1   sin  θ + C n p 1 1   cos  θ )    + exp ( 2 i φ ) ( S n p 11   sin  θ + C n p 11   cos  θ ) ] = 0 ,
S z ± 1 = E 0 H 0 * 2 r 2 Re { i   exp ( 2 i φ) n = 1 p = 1    × [ c n pw c p pw * ( S n p 1 1   sin  θ + C n p 1 1   cos  θ )    c n pw * c p pw ( S n p 11 *   sin  θ + C n p 11 *   cos  θ ) ] } = 0.
g n 2 g n , TM 1 = 1 K g n , TM 1 = i g n , TE 1 = i K g n , TE 1 ,
g n m = 0 , m 1 , g n 2 g n , TM 1 = 1 K g n , TM 1 = i g n , TE 1 = i K g n , TE 1 .
S x = E 0 H 0 * 2 r 2 Re n = 1 p = 1 m = n + n q = p + p c n pw c p pw *         × exp   i ( m q ) φ [ sin  φ ( B n p m q    cos 2   θ        + D n p m q   sin  θ  cos  θ + E n p m q    sin 2   θ )         + cos  φ ( F n p m q   sin  θ + G n p m q   cos  θ ) ] ,
B n p m q    = k r [ g n , TM m g p , TE q * ( ψ n ψ p τ n m p p q ψ p ψ n τ p q p n m )       + q g n , TM m g p , TM q * ( ψ n ψ p π p q p n m + ψ p ψ n π p q p n m )       + g n , TM m g p , TE q * ( ψ n ψ p τ p m p p q ψ p ψ n τ p q p p m )       + m g n , TE m g p , TE q * ( ψ n ψ p π p m p n q + ψ p ψ n π n m p p q ) ] ,
D n p m q     = g n , TM m τ n m ( ψ n ψ n )        × ( g p , TE q * ψ p τ p q q g p , TM q * ψ p π p q ) ,
E n p m q    = k r [ g n , TM m g p , TE q * ( ψ n ψ p τ n m p p q + ψ n ψ p τ n m p p q )       + m g n , TE m g p , TE q * ( ψ n ψ p π n m p p q + ψ n ψ p π n m p p q )       g n , TM m g p , TE q * ( ψ n ψ p τ p q p n m + ψ n ψ p τ p q p n m )       + q g n , TM m g p , TM q * ( ψ n ψ p π p q p n m + ψ n ψ p π p q p n m ) ] ,
F n p m q    = i [ g n , TM m g p , TM q * ( ψ n ψ p τ n m τ p q + m q ψ p ψ n π n m π p q )       + g n , TM m g p , TE q * ( m ψ n ψ p π n m τ p q + q ψ n ψ p τ n m π p q )       g n , TE m g p , TM q * ( m ψ n ψ p π n m τ p q + q ψ p ψ n τ n m π p q )       + g n , TE m g p , TE q * ( ψ n ψ p τ n m τ p q + m q ψ n ψ p π n m π p q ) ] ,
G n p m q     = i S n p m q ,
S x c = E 0 H 0 * 2 r 2 Re n = 1 p = 1 m = n + n q = p + p c n pw c p pw *          × exp [ i ( m q ) φ ] ( F n p m q   sin   θ+G n p m q   cos  θ )
S x , 1 c = E 0 H 0 * 2 r 2 Re n = 1 p = 1 m = n + n q = p + p c n pw c p pw *          × exp [ i ( m q ) φ ] F n p m q ,
S x , 2 c = E 0 H 0 * 2 r 2 Re n = 1 p = 1 m = n + n q = p + p c n pw c p pw *          × exp [ i ( m q ) φ ] G n p m q .
K = ± 1.
S y = E 0 H 0 * 2 r 2 Re n = 1 p = 1 m = n + n q = p + p c n pw c p pw *         × exp [ i ( m q ) φ ] [ sin  φ ( F n p m q   sin  θ + G n p m q   cos  θ )         cos  φ ( B n p m q    cos 2   θ + D n p m q   sin  θ  cos  θ         + E n p m q sin 2   θ ) ] = 0 ,
X = x , Y = y , Z = z ,
r = r , θ = π θ , φ = 2 π Φ .
U TM = E 0 r 2 [ exp ( i φ ) + K exp ( i φ ) ]            × n = 1 c n pw g n ψ n 1 p n 1 ( cos  θ ) ,
U TE = i H 0 r 2 [ exp ( i φ ) K exp ( i φ ) ]            × n = 1 c n pw g n ψ n 1 p n 1 ( cos  θ ) .
U TM = E 0 r 2 [ exp ( i Φ ) + K ̅ exp ( i Φ ) ]            × n = 1 c n pw g ¯ n ψ n 1 p n 1 ( cos  θ ) ,
U TE = i ¯ H 0 r 2 [ exp ( i Φ ) K ¯ exp ( i Φ ) ]           × n = 1 c n pw g ¯ n ψ n 1 p n 1 ( cos  θ ) ,
K ¯ = 1 / K ,
g ¯ n = K g n ( 1 ) n 1 ,
¯ = ,
p n 1 ( cos  θ ) = ( 1 ) n 1 p n 1 ( cos  θ ) .
U TM = E 0 r 2 [ exp ( i φ ) + K exp ( i φ ) ] n = 1 c n pw g n ψ n 1 p n 1 ,
U TE = i H 0 r 2 [ exp ( i φ ) K exp ( i φ ) ] n = 1 c n pw g n ψ n 1 p n 1 .
φ  = ϕ 0
A = 1 2 [ exp ( i φ 0 ) + K exp ( i φ 0 ) ] ,
B = 1 2 [ exp ( i φ 0 ) K exp ( i φ 0 ) ] ,
exp ( i φ 0 ) = A + B ,
K = A 2 B 2 R .
U TM = A E 0 r   cos  ϕ n = 1 c n pw g n ψ n 1 p n 1    + i B E 0 r   sin  ϕ n = 1 c n pw g n ψ n 1 p n 1 ,
U TE = A H 0 r   sin  ϕ n = 1 c n pw g n ψ n 1 p n 1    + i B H 0 r   cos  ϕ n = 1 c n pw g n ψ n 1 p n 1 ,
U TM = E 0 r   cos  ϕ n = 1 c n pw g n ψ n 1 p n 1 ,
U TE = H 0 r   sin  ϕ n = 1 c n pw g n ψ n 1 p n 1 ,
g n 2 = g n , TM 1 = g n , TM 1 = i g n , TE 1 = i g n , TE 1 .
E r = E 0 2 r [ exp ( i φ ) + K exp ( i φ ) ] n = 1    × c n pw g n n ( n + 1 ) ψ n 1 p n 1 ,
E θ = E 0 2 r [ exp ( i φ ) + K exp ( i φ ) ] n = 1 c n pw g n A n ,
E φ = i E 0 2 r [ exp ( i φ ) K exp ( i φ ) ] n = 1 c n pw g n B n ,
H r = i H 0 2 r [ exp ( i φ ) K exp ( i φ ) ]    × n = 1 c n pw g n n ( n + 1 ) ψ n 1 p n 1 ,
H θ = i H 0 2 r [ exp ( i φ ) K exp ( i φ ) ] n = 1 c n pw g n A n ,
H φ = H 0 2 r [ exp ( i φ ) + K exp ( i φ ) ] n = 1 c n pw g n B n ,
E x = E 0 2 r n = 1 c n pw g n { sin  θ  cos  φ [ exp ( i φ ) + K exp ( i φ ) ]    × n ( n + 1 ) ψ n 1 p n 1 + cos  θ  cos  φ[exp( i φ))   + K exp ( i φ ) ] A n i   sin  φ[exp( i φ)    K exp ( i φ ) ] B n } ,
E y = E 0 2 r n = 1 c n pw g n { sin  θ  sin  φ [ exp ( i φ ) + K exp ( i φ ) ]    × n ( n + 1 ) ψ n 1 p n 1 + cos  θ  sin  φ[exp( i φ )   + K exp ( i φ ) ] A n + i   cos  φ[exp( i φ)    K exp ( i φ ) ] B n } ,
E z = E 0 2 r [ exp ( i φ ) + K exp ( i φ ) ] n = 1 c n pw g n × [ cos  θ   n ( n + 1 ) ψ n 1 p n 1 sin  θ A n ] ,
H x = i E 0 2 r n = 1 c n pw g n { sin  θ  cos  φ [ exp ( i φ ) K exp ( i φ ) ]    × n ( n + 1 ) ψ n 1 p n 1 + cos  θ  cos  φ[exp( i φ)    K exp ( i φ ) ] A n i   sin  φ[exp( i φ)    + K exp ( i φ ) ] B n } ,
H y = i E 0 2 r n = 1 c n pw g n { sin  θ  sin  φ [ exp ( i φ ) K exp ( i φ ) ]    × n ( n + 1 ) ψ n 1 p n 1 + cos  θ  sin  φ[exp( i φ)    K exp ( i φ ) ] A n + i   cos  φ[exp( i φ)    + K exp ( i φ ) ] B n } ,
H z = i H 0 2 r [ exp ( i φ ) K exp ( i φ ) ] n = 1 c n pw g n     × [ cos  θ n ( n + 1 ) ψ n 1 p n 1 sin  θ A n ] .
i [ exp ( i φ ) K exp ( i φ ) ] E r E 0           = [ exp ( i φ ) + K exp ( i φ ) ] H r H 0 ,
i [ exp ( i φ ) K exp ( i φ ) ] E θ E 0           = [ exp ( i φ ) + K exp ( i φ ) ] H θ H 0 ,
i [ exp ( i φ ) + K exp ( i φ ) ] E φ E 0           = [ exp ( i φ ) K exp ( i φ ) ] H φ H 0 .
[ exp ( i φ ) K exp ( i φ ) ] E z E 0           = i [ exp ( i φ ) + K exp ( i φ ) ] H z H 0 .
E r = E 0 r cos  φ n = 1 c n pw g n n ( n + 1 ) ψ n 1 p n 1 ,
E θ = E 0 r cos  φ n = 1 c n pw g n A n ,
E φ = E 0 r sin  φ n = 1 c n pw g n B n ,
H r = H 0 r sin  φ n = 1 c n pw g n n ( n + 1 ) ψ n 1 p n 1 ,
H θ = H 0 r sin  φ n = 1 c n pw g n A n ,
H φ = H 0 r cos  φ n = 1 c n pw g n B n ,
A n = d r ψ n 1 d r τ n i k r ψ n 1 π n ,
B n = d r ψ n 1 d r π n i k r ψ n 1 τ n .
E x = E 0 r n = 1 c n pw g n [ cos  θ   cos 2   φ A n + sin 2   φ B n + sin  θ   cos 2   φ n ( n + 1 ) ψ n 1 p n 1 ] ,
E y = E 0 r sin  φ  cos  φ n = 1 c n pw g n [ cos  θ A n B n    + sin  θ n ( n + 1 ) ψ n 1 p n 1 ] ,
E z = E 0 r cos  φ n = 1 c n pw g n [ sin  θ A n    cos  θ n ( n + 1 ) ψ n 1 p n 1 ] ,
H x = H 0 r sin  φ  cos  φ n = 1 c n pw g n [ cos  θ A n B n     + sin  θ n ( n + 1 ) ψ n 1 p n 1 ] ,
H y = H 0 r n = 1 c n pw g n [ cos  θ   sin 2   φ A n + cos 2   φ B n + sin  θ   sin 2   φ n ( n + 1 ) ψ n 1 p n 1 ] ,
H z = H 0 r sin  φ n = 1 c n pw g n [ sin  θ A n    cos  θ n ( n + 1 ) ψ n 1 p n 1 ] .
E r E 0 sin  φ = H r H 0 cos  φ ,
E θ E 0 sin  φ = H θ H 0 cos  φ ,
E φ E 0 cos  φ = H φ H 0 sin  φ .
E z E 0 sin  φ = H z H 0 cos  φ ,
( E x E 0 H y H 0 ) sin  φ  cos  φ = H x H 0 ( cos 2   φ sin 2   φ ) ,
E y E 0 = H x H 0 .
S z = E 0 H 0 * 2 r 2 Re n = 1 m = n + n p = 1 q = p + p i c n pw c p pw *          × exp [ i ( m q ) ] φ ( S n p m q   sin  θ+ C n p m q   cos  θ ) ,
S n p m q = k r [ g n , TM m g p , TM q * ψ p n + ψ n ) p n m τ p q    + g n , TE m g p , TE q * ψ n ( ψ p + ψ p ) p p q τ n m    + q g n , TM m g p , TE q * ψ p ( ψ n + ψ n ) p n m π p q    + m g n , TM m g p , TE q * ψ n p + ψ p ) p p q π n m ] ,
C n p m q = g n , TM m g p , TM q * ψ p ψ n n m τ p q + m q π n m π p q )    + g n , TM m g p , TE q * ψ n ψ p ( m π n m τ p q + q π p q τ n m )    g n , TE m g p , TM q * ψ p ψ n ( m π n m τ p q + q π p q τ n m )    + g n , TE m g p , TE q * ψ n ψ p ( m q π n m π p q + τ n m τ p q ) .
k π n k ( 1 ) = τ n k ( 1 ) = 0 , k = 0 ,
π n 1 ( 1 ) = τ n 1 ( 1 ) 0 ,
π n k ( 1 ) = τ n k ( 1 ) = 0 , k 2.
Γ n = π n 1 ( 1 ) = τ n 1 ( 1 ) ,
A n p = 2 n + 1 n ( n + 1 ) 2 p + 1 p ( p + 1 ) ,
S z ± 1 = E 0 H 0 * 2 k 2 r 2 n = 1 p = 1 A n p    × Re { i exp ( 2 i φ) [ ( i ) n i p ( S n p 1 1   sin  θ + C n p 1 1   cos  θ )    i n ( i ) p ( S n p 11 *   sin  θ + C n p 11 *   cos  θ ) ] } .
Re { i exp ( 2 i φ) [ ( i ) n i p ( S n p 1 1   sin  θ + C n p 1 1   cos  θ )    i n ( i ) p ( S n p 11 *   sin  θ + C n p 11 *   cos  θ )    + ( i ) p i n ( S p n 1 1   sin  θ + C p n 1 1   cos  θ )    i p ( i ) n ( S p n 11 *   sin  θ + C p n 11 *   cos  θ ) ] }    = 0 ,
Re ( z z ) = 0 ,
( i ) n i p S n p 1 1 i n ( i ) p S n p 11 *      + ( i ) p i n S p n 1 1 i p ( i ) n S p n 11 * = 0 ,
( i ) n i p C n p 1 1 i n ( i ) p C n p 11 *      + ( i ) p i n C p n 1 1 i p ( i ) n C p n 11 * = 0.
( i ) n i p = i n ( i ) p ,
S n p 1 1 S n p 11 * + S p n 1 1 S p n 11 * = 0 ,
C n p 1 1 C n p 11 * + C p n 1 1 C p n 11 * = 0.
g n , TM 1 g p , TM 1 * g n , TM 1 * g p , TM 1 g p , TE 1 g n , TE 1 *            + g p , TE 1 * g n , TE 1 = 0 ,
g n , TM 1 g p , TM 1 * g n , TM 1 * g p , TM 1 g p , TE 1 g n , TE 1 *            + g p , TE 1 * g n , TE 1 = 0 ,
g n , TM 1 g p , TE 1 * + g n , TM 1 * g p , TE 1 g p , TM 1 g n , TE 1 *            + g p , TM 1 * g n , TE 1 = 0 ,
g n , TM 1 g p , TM 1 * g n , TM 1 * g p , TM 1 g p , TE 1 g n , TE 1 *            + g p , TE 1 * g n , TE 1 = 0 ,
g n , TM 1 g p , TE 1 * + g n , TM 1 * g p , TE 1 g p , TM 1 g n , TE 1 *            + g p , TM 1 * g n , TE 1 = 0.
g n , TM 1 * g p , TM 1 = g n , TM 1 * g p , TM 1 ,
g n , TM 1 = K g n , TM 1 ,
g n , TM 1 * g p , TM 1 ( K * K ) = 0.
g n , TE 1 = K g n , TE 1 .
g n , TE 1 = exp ( i γ ) g n , TM 1 ,
γ = ± π 2 .
g n 2 g n , TM 1 = 1 K g n , TM 1 = i g n , TE 1 = i K g n , TE 1 ,
S ¯ z = E 0 H 0 * 2 r 2 Re n = 1 p = 1 i c n pw c p pw*    × { exp [ i ( 1 + u ) φ ] ( S n p 1 u   sin  θ + C n p 1 u   cos  θ )    + exp [ i ( 1 + u ) φ ] ( S n p 1 u   sin  θ + C n p 1 u   cos  θ ) } = 0 ,
S n p 1 u S n p 1 u * + S p n 1 u S p n 1 u * = 0 ,
C n p 1 u C n p 1 u * + C p n 1 u C p n 1 u * = 0.
g n g p , TE u * K g n * g p , TE u = 0 ,
g n g p , TE u * + K g n * g p , TE u = 0 ,
g n g p , TM u * K g n * g p , TM u = 0 ,
g n g p , TM u * + K g n * g p , TM u = 0 ,
g n m = 0 , m 1 , g n 2 g n , TM 1 = 1 K g n , TM 1 = i g n , TE 1 = i K g n , TE 1 .
S z = E 0 H 0 * 8 r 2 Re { cos  θ n = 1 m = 1 c n pw c m pw * g n g m *    × [ ( K 2 + 1 + 2 K   cos   2 φ ) A n B m *    + ( K 2 + 1 2 K   cos   2 φ ) B n A m * ]    + sin  θ n = 1 m = 1 c n pw c m pw * g n g m *    × [ ( K 2 + 1 + 2 K   cos   2 φ ) n ( n + 1 ) B m * p n 1 ψ n 1    + ( K 2 + 1 2 K   cos   2 φ ) m ( m + 1 ) B n p m 1 ψ m 1 ] } ,
A n = d r ψ n 1 d r τ n + i k r ψ n 1 π n ,
B n = d r ψ n 1 d r π n + i k r ψ n 1 τ n ,
S z = E 0 H 0 * 4 r 2 ( K 2 + 1 ) Re [ cos  θ n = 1 m = 1 c n pw c m pw *    × g n g m * A n B m * + sin  θ n = 1 m = 1 c n pw c m pw * g n g m *    × m ( m + 1 ) B n ψ m 1 p m 1 ] ;
S x s = E 0 H 0 * 2 r 2 Re n = 1 p = 1 m = n + n q = p + p c n pw c p pw *    × exp [ i ( m q ) φ ] ( B n p m q    cos 2   θ+ D n p m q   sin  θ  cos  θ    + E n p m q    sin 2   θ ) = 0.
Re n = 1 p = 1 c n pw c p pw * [ ( B n p 11 + B n p 1 1 ) cos 2   θ    + ( D n p 11 + D n p 1 1 ) sin  θ  cos  θ + ( E n p 11 + E n p 1 1 )    × sin 2   θ + exp ( 2 i φ ) ( B n p 1 1 cos 2   θ    + D n p 1 1 sin  θ  cos  θ    + E n p 1 1    sin 2   θ) + exp ( 2 i φ ) ( B n p 11    cos 2   θ    + D n p 11   sin  θ  cos  θ+ E n p 11    sin 2   θ ) ] = 0.
Re { c n pw c p pw * [ ( B n p 11 + B n p 1 1 ) cos 2   θ + ( D n p 11 + D n p 1 1 )    × sin  θ  cos  θ + ( E n p 11 + E n p 1 1 ) sin 2   θ   +exp ( 2 i φ ) ( B n p 1 1    cos 2   θ    + D n p 1 1   sin  θ  cos  θ + E n p 1 1    sin 2   θ )   +exp ( 2 i φ ) ( B n p 11    cos 2   θ    + D n p 11   sin  θ  cosθ + E n p 11    sin 2   θ ) ]    + c n pw c p pw * [ ( B p n 11 + B p n 1 1 )    × cos 2   θ + ( D p n 11 + D p n 1 1 ) sin  θ  cos  θ    + ( E p n 11 + E p n 1 1 )    × sin 2   θ + exp ( 2 i φ ) ( B p n 1 1    cos 2   θ+ D p n 1 1   sin  θ    × cos  θ + E p n 1 1    sin 2   θ ) + exp ( 2 i φ ) ( B p n 11    × cos 2   θ + D p n 11   sin  θ  cos  θ + E p n 11    sin 2   θ ) ] } = 0.
Re [ ( B n p 11 + B p n 11 + B n p 1 1 + B p n 1 1 ) cos 2   θ    + ( D n p 11 + D p n 11 + D n p 1 1 + D p n 1 1 ) sin  θ  cos  θ    + ( E n p 11 + E p n 11 + E n p 1 1 + E p n 1 1 ) sin 2   θ    + exp ( 2 i φ ) ( B n p 1 1 * + B p n 1 1 * + B n p 11 + B p n 11 )    × cos 2   θ + exp ( 2 i φ ) ( D n p 1 1 * + D p n 1 1 *    + D n p 11 + D p n 11 ) sin  θ  cos  θ    + exp ( 2 i φ ) ( E n p 1 1 * + E p n 1 1 * + E n p 11    + E p n 11 ) sin 2   θ ] = 0.
n = 1 m = n + n c n pw exp ( i m φ ) ( ψ n + ψ n ) p n m     × { i [ exp ( i φ ) K exp ( i φ ) ] g n , TM m     [ exp ( i φ ) + K exp ( i φ ) ] g n , TE m } = 0 ,
m = n + n p n m { exp [ i ( m + 1 ) φ ] ( g n , TM m + i g n , TE m ) } + m = n + n p n m { exp [ i ( m 1 ) φ ] K ( i g n , TE m g n , TM m ) } = 0 ,
p n 1 ( g n , TM 1 + i g n , TE 1 + i K g n , TE 1 K g n , TM 1 )    + m I exp ( i m φ ) [ p n m 1 ( g n , TM m 1 + i g n , TE m 1 )    + p n m + 1 K ( i g n , TE m + 1 g n , TM m + 1 ) ]    + p n n 1 [ exp ( i n φ ) K ( i g n , TE n + 1 g n , TM n + 1 )    + exp ( i n φ ) ( g n , TM n 1 + i g n , TE n 1 ) ]    + p n n { exp [ i ( n + 1 ) φ] K ( i g n , TE n g n , TM n )    + exp [ i ( n + 1 ) φ]( g n , TM n + i g n , TE n ) } ,
n = 1 c n pw m = n + n { exp [ i ( m + 1 ) φ ] ( i g n , TM m ψ n τ n m    + i m g n , TE m ψ n π n m g n , TE m ψ n τ n m    + m g n , TM m ψ n π n m ) + exp [ i ( m 1 ) φ ]    × K ( i g n , TM m ψ n τ n m i m g n , TE m ψ n π n m    + g n , TE m ψ n τ n m } + m g n , TM m ψ n π n m } = 0 ,
n = 1 c n pw { m I 1 exp ( i m φ ) [ i g n , TM m 1 ψ n τ n m 1    + i ( m 1 ) g n , TE m 1 ψ n π n m 1 g n , TE m 1 ψ n τ n m 1    + ( m 1 ) g n , TM m 1 ψ n π n m 1    i K g n , TM m + 1 ψ n τ n m + 1    i K ( m + 1 ) g n , TE m + 1 ψ n π n m 1    K g n , TE m + 1 ψ n τ n m + 1 + K ( m + 1 )    × g n , TM m + 1 ψ n π n m 1 ]    + m I 2 exp ( i m φ ) [ i g n , TM m 1 ψ n τ n m + 1    i ( m 1 ) g n , TE m 1 ψ n π n m 1    g n , TE m 1 ψ n τ n m 1 + ( m 1 ) g n , TM m 1 ψ n π n m 1 ]    + m I 3 exp ( i m φ ) [ i K g n , TM m + 1 ψ n τ n m + 1    i K ( m + 1 ) g n , TE m + 1 ψ n π n m + 1    K g n , TE m + 1 ψ n τ n m + 1    + K ( m + 1 ) g n , TM m + 1 ψ n π n m 1 ] } = 0 ,
I 1 = { n + 1 , , 3 , + 3 , , n 1 } ,
I 2 = { 2 , 1 , 0 , 1 , 2 , n , n + 1 } ,
I 3 = { n 1 , n , 2 , 1 , 0 , 1 , 2 } .

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