Abstract

The recurrence algorithm for calculating electromagnetic scattering from a multilayer sphere, which was described recently by Wu and Wang [Radio Sci. 26, 1393, (1991)], is derived in a slightly modified form and extended to include a calculation of the internal field and the absorption cross sections of the individual layers. The original algorithm calculates the scattering by a recurrence procedure that propagates the log derivatives of the Debye potentials outward from the core to the outer layer. The extended algorithm then continues the calculation by an inward recurrence procedure that propagates the Debye potentials from the outer layer to the core. Concurrent with the inward propagation, a separate algorithm calculates the absorption cross sections of the imbedded concentric spheres. The results of several example calculations are presented, including the differential cross section and internal electric field of a Luneburg lens.

© 1996 Optical Society of America

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References

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  1. A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
    [CrossRef]
  2. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  3. J. R. Wait, “Electromagnetic scattering from radially inhomogeneous spheres,” Appl. Sci. Res. B 10, 441–450 (1963).
    [CrossRef]
  4. R. Bhandari, “Scattering coefficients for a multilayered sphere: analytic expressions and algorithms,” Appl. Opt. 24, 1960–1967 (1985).
    [CrossRef] [PubMed]
  5. D. W. Mackowski, R. A. Altenkirch, M. P. Menguc, “Internal absorption cross sections in a stratified sphere,” Appl. Opt. 29, 1551–1559 (1990).
    [CrossRef] [PubMed]
  6. Z. S. Wu, Y. P. Y. Wang, “Electromagnetic scattering for multilayered sphere: recursive algorithms,” Radio Sci. 26, 1393–1401 (1991).
    [CrossRef]
  7. P. J. Wyatt, “Scattering of electromagnetic plane waves from inhomogeneous spherically symmetric objects,” Phys. Rev. 127, 1837–1843 (1962);P. J. Wyatt, “Scattering of electromagnetic plane waves from inhomogeneous spherically symmetric objects: erratum,” Phys. Rev., 134, AB1 (1964);
    [CrossRef]
  8. C. T. Tai, Dyadic Greens Functions in Electromagnetic Theory (Intext Educational, Scranton Pa., 1971).
  9. C. T. Tai, “The electromagnetic theory of the spherical Luneberg lens,” Appl. Sci. Res. B 7, 113–130 (1958).
    [CrossRef]
  10. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
  11. 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), Eqs. (24), (25).
    [CrossRef]
  12. R. K. Luneburg, Mathematical Theory of Optics (U. of California Press, Berkeley, Calif., 1964), Secs. 27–30;mimeographed lecture notes, Brown U. Press, Providence, R. I., 1944;reprinted in Selected papers on Gradient-Index Optics, D. T. Moore, ed. (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 45–68.
  13. M. Sitarski, “Internal heating of multilayered aerosol particles by electromagnetic radiation,” Langmuir 3, 85–93 (1987).
    [CrossRef]
  14. S. Levine, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres, when outer shell has a variable refractive index,” in Electromagnetic Scattering, M. Kerker, ed. (Pergamon, Oxford, 1963), pp. 37–46.
  15. W. Cheney, D. Kincaid, Numerical Mathematics and Computing, 2nd ed. (Brooks-Cole, Belmont, Mass., 1985), Chap. 2.
  16. L. Kai, P. Massoli, “Scattering of electromagnetic-plane waves by radially inhomogeneous spheres: a finely stratified sphere model,” Appl. Opt. 33, 501–511 (1994).
    [CrossRef] [PubMed]
  17. W H. Press, B. P. Plannery, S. A. Teukolsky, W T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, England, 1986), Chap. 4.
  18. R. C. Johnson, ed., Antenna Engineering Handbook, 3rd ed. (McGraw-Hill, New York, 1993), Sec. 18-2.
  19. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), p. 644.
  20. A. S. Gutman, “Modified Luneberg lens,” J. Appl. Phys. 25, 855–859 (1954).
    [CrossRef]
  21. H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys. 10, 82–124 (1969).
    [CrossRef]

1994 (1)

1991 (1)

Z. S. Wu, Y. P. Y. Wang, “Electromagnetic scattering for multilayered sphere: recursive algorithms,” Radio Sci. 26, 1393–1401 (1991).
[CrossRef]

1990 (1)

1988 (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), Eqs. (24), (25).
[CrossRef]

1987 (1)

M. Sitarski, “Internal heating of multilayered aerosol particles by electromagnetic radiation,” Langmuir 3, 85–93 (1987).
[CrossRef]

1985 (1)

1969 (1)

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys. 10, 82–124 (1969).
[CrossRef]

1963 (1)

J. R. Wait, “Electromagnetic scattering from radially inhomogeneous spheres,” Appl. Sci. Res. B 10, 441–450 (1963).
[CrossRef]

1962 (1)

P. J. Wyatt, “Scattering of electromagnetic plane waves from inhomogeneous spherically symmetric objects,” Phys. Rev. 127, 1837–1843 (1962);P. J. Wyatt, “Scattering of electromagnetic plane waves from inhomogeneous spherically symmetric objects: erratum,” Phys. Rev., 134, AB1 (1964);
[CrossRef]

1958 (1)

C. T. Tai, “The electromagnetic theory of the spherical Luneberg lens,” Appl. Sci. Res. B 7, 113–130 (1958).
[CrossRef]

1954 (1)

A. S. Gutman, “Modified Luneberg lens,” J. Appl. Phys. 25, 855–859 (1954).
[CrossRef]

1951 (1)

A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

Aden, A. L.

A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

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), Eqs. (24), (25).
[CrossRef]

Altenkirch, R. A.

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), Eqs. (24), (25).
[CrossRef]

Bhandari, R.

Cheney, W.

W. Cheney, D. Kincaid, Numerical Mathematics and Computing, 2nd ed. (Brooks-Cole, Belmont, Mass., 1985), Chap. 2.

Gutman, A. S.

A. S. Gutman, “Modified Luneberg lens,” J. Appl. Phys. 25, 855–859 (1954).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), p. 644.

Kai, L.

Kerker, M.

A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

S. Levine, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres, when outer shell has a variable refractive index,” in Electromagnetic Scattering, M. Kerker, ed. (Pergamon, Oxford, 1963), pp. 37–46.

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

Kincaid, D.

W. Cheney, D. Kincaid, Numerical Mathematics and Computing, 2nd ed. (Brooks-Cole, Belmont, Mass., 1985), Chap. 2.

Levine, S.

S. Levine, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres, when outer shell has a variable refractive index,” in Electromagnetic Scattering, M. Kerker, ed. (Pergamon, Oxford, 1963), pp. 37–46.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (U. of California Press, Berkeley, Calif., 1964), Secs. 27–30;mimeographed lecture notes, Brown U. Press, Providence, R. I., 1944;reprinted in Selected papers on Gradient-Index Optics, D. T. Moore, ed. (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 45–68.

Mackowski, D. W.

Massoli, P.

Menguc, M. P.

Nussenzveig, H. M.

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys. 10, 82–124 (1969).
[CrossRef]

Plannery, B. P.

W H. Press, B. P. Plannery, S. A. Teukolsky, W T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, England, 1986), Chap. 4.

Press, W H.

W H. Press, B. P. Plannery, S. A. Teukolsky, W T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, England, 1986), Chap. 4.

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), Eqs. (24), (25).
[CrossRef]

Sitarski, M.

M. Sitarski, “Internal heating of multilayered aerosol particles by electromagnetic radiation,” Langmuir 3, 85–93 (1987).
[CrossRef]

Stratton, J. A.

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

Tai, C. T.

C. T. Tai, “The electromagnetic theory of the spherical Luneberg lens,” Appl. Sci. Res. B 7, 113–130 (1958).
[CrossRef]

C. T. Tai, Dyadic Greens Functions in Electromagnetic Theory (Intext Educational, Scranton Pa., 1971).

Teukolsky, S. A.

W H. Press, B. P. Plannery, S. A. Teukolsky, W T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, England, 1986), Chap. 4.

Vetterling, W T.

W H. Press, B. P. Plannery, S. A. Teukolsky, W T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, England, 1986), Chap. 4.

Wait, J. R.

J. R. Wait, “Electromagnetic scattering from radially inhomogeneous spheres,” Appl. Sci. Res. B 10, 441–450 (1963).
[CrossRef]

Wang, Y. P. Y.

Z. S. Wu, Y. P. Y. Wang, “Electromagnetic scattering for multilayered sphere: recursive algorithms,” Radio Sci. 26, 1393–1401 (1991).
[CrossRef]

Wu, Z. S.

Z. S. Wu, Y. P. Y. Wang, “Electromagnetic scattering for multilayered sphere: recursive algorithms,” Radio Sci. 26, 1393–1401 (1991).
[CrossRef]

Wyatt, P. J.

P. J. Wyatt, “Scattering of electromagnetic plane waves from inhomogeneous spherically symmetric objects,” Phys. Rev. 127, 1837–1843 (1962);P. J. Wyatt, “Scattering of electromagnetic plane waves from inhomogeneous spherically symmetric objects: erratum,” Phys. Rev., 134, AB1 (1964);
[CrossRef]

Appl. Opt. (3)

Appl. Sci. Res. B (2)

C. T. Tai, “The electromagnetic theory of the spherical Luneberg lens,” Appl. Sci. Res. B 7, 113–130 (1958).
[CrossRef]

J. R. Wait, “Electromagnetic scattering from radially inhomogeneous spheres,” Appl. Sci. Res. B 10, 441–450 (1963).
[CrossRef]

J. Math. Phys. (1)

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys. 10, 82–124 (1969).
[CrossRef]

J. Appl. Phys. (3)

A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

A. S. Gutman, “Modified Luneberg lens,” J. Appl. Phys. 25, 855–859 (1954).
[CrossRef]

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), Eqs. (24), (25).
[CrossRef]

Langmuir (1)

M. Sitarski, “Internal heating of multilayered aerosol particles by electromagnetic radiation,” Langmuir 3, 85–93 (1987).
[CrossRef]

Phys. Rev. (1)

P. J. Wyatt, “Scattering of electromagnetic plane waves from inhomogeneous spherically symmetric objects,” Phys. Rev. 127, 1837–1843 (1962);P. J. Wyatt, “Scattering of electromagnetic plane waves from inhomogeneous spherically symmetric objects: erratum,” Phys. Rev., 134, AB1 (1964);
[CrossRef]

Radio Sci. (1)

Z. S. Wu, Y. P. Y. Wang, “Electromagnetic scattering for multilayered sphere: recursive algorithms,” Radio Sci. 26, 1393–1401 (1991).
[CrossRef]

Other (9)

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

W H. Press, B. P. Plannery, S. A. Teukolsky, W T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, England, 1986), Chap. 4.

R. C. Johnson, ed., Antenna Engineering Handbook, 3rd ed. (McGraw-Hill, New York, 1993), Sec. 18-2.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), p. 644.

C. T. Tai, Dyadic Greens Functions in Electromagnetic Theory (Intext Educational, Scranton Pa., 1971).

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

S. Levine, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres, when outer shell has a variable refractive index,” in Electromagnetic Scattering, M. Kerker, ed. (Pergamon, Oxford, 1963), pp. 37–46.

W. Cheney, D. Kincaid, Numerical Mathematics and Computing, 2nd ed. (Brooks-Cole, Belmont, Mass., 1985), Chap. 2.

R. K. Luneburg, Mathematical Theory of Optics (U. of California Press, Berkeley, Calif., 1964), Secs. 27–30;mimeographed lecture notes, Brown U. Press, Providence, R. I., 1944;reprinted in Selected papers on Gradient-Index Optics, D. T. Moore, ed. (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 45–68.

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Figures (7)

Fig. 1
Fig. 1

Index of refraction of a multilayer sphere, shown as a discontinuous function of the radial coordinate, r. The particle has N layers. The radius of the outer surface of the sphere is r = a. The region outside the sphere is labeled N + 1.

Fig. 2
Fig. 2

Absolute error of the scattering efficiency versus the number of layers in interval a/4 < ra of the model sphere. Size parameters are x = 2, 20, 200, and 500.

Fig. 3
Fig. 3

Reduced differential cross section of the model sphere for size parameter x = 20. The solid curve is calculated by the exact analytic method; the diamonds are calculated by the use of the multilayer model with N = 42.

Fig. 4
Fig. 4

Luneburg lens, showing parallel light rays brought to a focus on the back surface of the lens. Also shown are the impact parameter, b, and the scattering angle, θ, for a typical ray.

Fig. 5
Fig. 5

Reduced differential scattering cross section of a Luneburg lens for size parameter ka = 60. The solid curve is the wave optics result; the dotted curve is the geometric optics result.

Fig. 6
Fig. 6

Reduced differential cross section of a Luneburg lens for size parameter ka = 500, and comparison of wave optics and geometric optics results. In the lower graph the wave optics results have been averaged over a 4° window.

Fig. 7
Fig. 7

Two views of the electric field strength, |E|, in the xz plane of a Luneburg lens. The lens radius is a = 1. The top figure shows both internal and external fields; the bottom figure is a different view of the internal field.

Tables (2)

Tables Icon

Table 1 Scattering Efficiency Parameters for the Model Sphere

Tables Icon

Table 2 Absorption Efficiencies (λ = 1.06 μm)

Equations (56)

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× × E k 2 m 2 ( r ) E = 0 ,
M = × [ Ψ r ] ,
N = 1 k m 2 ( r ) × × [ Φ r ] ,
2 Ψ + k 2 m 2 ( r ) Ψ = 0 ,
2 Φ 2 m ( r ) d m ( r ) d r Φ r + k 2 m 2 ( r ) Φ = 0 .
Ψ o e , m , n = 1 k S n ( r ) P n m ( cos θ ) sin cos m ϕ ,
Φ o e , m , n = 1 kr T n ( r ) P n m ( cos θ ) sin cos m ϕ ,
d 2 S n ( r ) d r 2 + [ k 2 m 2 ( r ) n ( n + 1 ) r 2 ] S n ( r ) = 0 ,
d 2 T n ( r ) d r 2 2 m ( r ) d m ( r ) d r d T n ( r ) d r + [ k 2 m 2 ( r ) n ( n + 1 ) r 2 ] T n ( r ) = 0 .
E inc = exp ( ikz ) ê x = n = 1 i n 2 n + 1 n ( n + 1 ) [ M o , 1 , n ( 1 ) i N e , 1 , n ( 1 ) ] ,
E sca = n = 1 i n 2 n + 1 n ( n + 1 ) [ a n M o , 1 , n ( 3 ) i b n N e , 1 , n ( 3 ) ] .
E = n = 1 i n 2 n + 1 n ( n + 1 ) [ M o , 1 , n i N e , 1 , n ] .
S n ( r ) = ψ n ( kr ) + a n ξ n ( kr ) ,
T n ( r ) = ψ n ( kr ) + b n ξ n ( kr ) ,
E r = i cos ϕ [ m ( r ) kt ] 2 n = 1 i n ( 2 n + 1 ) T n ( r ) P n 1 ( cos θ ) ,
E θ = cos ϕ kr n = 1 i n ( 2 n + 1 ) n ( n + 1 ) × [ S n ( r ) π n ( θ ) i k m 2 ( r ) T n ( r ) τ n ( θ ) ] ,
E ϕ = sin ϕ kr n = 1 i n ( 2 n + 1 ) n ( n + 1 ) × [ S n ( r ) τ n ( θ ) i k m 2 ( r ) T n ( r ) π n ( θ ) ] ,
π n ( θ ) = 1 sin θ P n 1 ( cos θ ) ,
τ n ( θ ) = d P n 1 ( cos θ ) d θ .
H = k ω μ 0 n = 1 i n 2 n + 1 n ( n + 1 ) [ M e , 1 , n + i N o , 1 , n ] .
H r = k ω μ 0 i sin ϕ [ kr ] 2 n = 1 i n ( 2 n + 1 ) S n ( r ) P n 1 ( cos θ ) ,
H θ = k ω μ 0 sin ϕ kr n = 1 i n ( 2 n + 1 ) n ( n + 1 ) × [ T n ( r ) π n ( θ ) i k S n ( r ) τ n ( θ ) ] ,
H ϕ = k ω μ 0 cos ϕ kr n = 1 i n ( 2 n + 1 ) n ( n + 1 ) × [ T n ( r ) τ n ( θ ) i k S n ( r ) π n ( θ ) ] .
S n ( i ) ( r ) = A n ( i ) [ ψ n ( m i kr ) + a n ( i ) ξ n ( m i kr ) ] ,
T n ( i ) ( r ) = B n ( i ) [ ψ n ( m i kr ) + b n ( i ) ξ n ( m i kr ) ] ,
U n ( r ) = 1 k [ S n ( r ) / S n ( r ) ] ,
V n ( r ) = 1 k m 2 ( r ) [ T n ( r ) / T n ( r ) ] .
D n ( ρ ) = ψ n ( ρ ) / ψ n ( ρ ) ,
G n ( ρ ) = ξ n ( ρ ) / ξ n ( ρ ) ,
R n ( ρ ) = ψ n ( ρ ) / ξ n ( ρ ) .
U n ( r i ) = m i R n ( m i k r i ) D n ( m i k r i ) + a n ( i ) G n ( m i k r i ) R n ( m i k r i ) + a n ( i ) ,
V n ( r i ) = 1 m i R n ( m i k r i ) D n ( m i k r i ) + b n ( i ) G n ( m i k r i ) R n ( m i k r i ) + b n ( i ) .
a n ( i + 1 ) = R n ( m i + 1 k r i ) U n ( r i ) m i + 1 D n ( m i + 1 k r i ) U n ( r i ) m i + 1 G n ( m i + 1 k r i ) ,
b n ( i + 1 ) = R n ( m i + 1 k r i ) m i + 1 V n ( r i ) D n ( m i + 1 k r i ) m i + 1 V n ( r i ) G n ( m i + 1 k r i ) .
n max ( i ) = min [ η max , η ( | m i | k r i ) ]
n max ( i ) = max [ η max ( i ) , η max ( i 1 ) ]
U n ( r i ) = m i D ( m i k r i ) ,
V n ( r i ) = D ( m i k r i ) / m i .
S n ( r i ) = A n ( i + 1 ) ξ n ( m i + 1 k r i ) [ R n ( m i + 1 k r i ) + a n ( i + 1 ) ] ,
T n ( r i ) = B n ( i + 1 ) ξ n ( m i + 1 k r i ) [ R n ( m i + 1 k r i ) + b n ( i + 1 ) ] .
A n ( i ) = S n ( r i ) ξ n ( m i k r i ) [ R n ( m i k r i ) + a n ( i ) ] ,
B n ( i ) = T n ( r i ) ξ n ( m i k r i ) [ R n ( m i k r i ) + b n ( i ) ] .
C abs ( N ) = 2 π k 2 n = 1 ( 2 n + 1 ) [ Re ( a n + b n ) + ( | a n | 2 + | b n | 2 ) ] ,
ā n = R n ( k r i ) U n ( r i ) D n ( k r i ) U n ( r i ) G n ( k r i ) ,
b ¯ n = R n ( k r i ) V n ( r i ) D n ( k r i ) V n ( r i ) G n ( k r i ) .
Ā n = S n ( r i ) ξ n ( k r i ) [ R n ( k r i ) + ā n ] ,
B ¯ n = T n ( r i ) ξ n ( k r i ) [ R n ( k r i ) + b ¯ n ] .
C abs ( i ) = 2 π k 2 n = 1 ( 2 n + 1 ) × [ | Ā n | 2 ( Re ā n + | ā n | 2 ) + | B ¯ n | 2 ( Re b ¯ n + | b ¯ n | 2 ) ] .
e ( N , x ) = | Q sca ( ML ) ( N , x ) Q sca ( x ) | .
e ( N , x ) = γ ( x ) ( N 1 ) 2 ,
m ( r ) = [ 2 ( r / a ) 2 ] 1 / 2 .
d σ d Ω = b sin θ d b d θ ,
d σ d Ω = a 2 cos ( θ ) .
S n ( ρ ) = T n ( ρ ) = ψ n ( ρ ) ,
S n ( 2 ) ( ρ ) = A n ( 2 ) ρ 1 / 2 [ J μ ( 2 x 1 / 2 ρ 1 / 2 ) + a n ( 2 ) Y μ ( 2 x 1 / 2 ρ 1 / 2 ) ] ,
T n ( 2 ) ( ρ ) = B n ( 2 ) ρ 1 / 2 [ J υ ( 2 x 1 / 2 ρ 1 / 2 ) + b n ( 2 ) Y υ ( 2 x 1 / 2 ρ 1 / 2 ) ] ,

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