Abstract

A new recursive algorithm to calculate the internal and scattered fields of finely stratified inhomogeneous spheres has been developed. No restriction on the number of layers—the thicknesses of which can be arbitrarily small—is imposed by this method. The number of layers is restricted only by the computer’s capability: calculations with spheres with more than 10,000 layers were successfully performed with a HP work station. The new algorithm circumvents the limitations introduced by the numerical round-off errors encountered when using the previously developed recursive relations to calculate the ratios of Riccati–Bessel functions. Tests and calculations show that the method is stable and accurate for a large range of size parameters and optical properties. By employing the proposed algorithm, the problems encountered in analyzing the scattering by spheres with continuous-profile refractive indices can be solved with good accuracy.

© 1994 Optical Society of America

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References

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  1. P. J. Wyatt, “Scattering of electromagnetic-plane waves from inhomogeneous spherically symetric objects,” Phys. Rev. 127, 1837–1843 (1962); Errata, 134, AB1 (1964).
    [CrossRef]
  2. D. Q. Chowdhury, S. C. Hill, P. W. Barber, “Morphology-dependent resonances in radially inhomogeneous spheres,” J. Opt. Soc. Am. A 8, 1702–1705 (1991).
    [CrossRef]
  3. A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
    [CrossRef]
  4. A. L. Aden, “Electromagnetic scattering from spheres with sizes comparable to the wavelength,” J. Appl. Phys. 22, 601–606 (1951).
    [CrossRef]
  5. R. W. Fenn, H. Oser, “Scattering properties of concentric soot-water spheres for visible and infrared light,” Appl. Opt. 4, 1504–1509 (1965).
    [CrossRef]
  6. G. W. Kattawar, D. A. Hood, “Electromagnetic scattering from a spherical polydispersion of coated spheres,” Appl. Opt. 15, 1996–1999 (1976).
    [CrossRef] [PubMed]
  7. T. P. Ackerman, O. B. Toon, “Absorption of visible radiation in atmosphere containing mixtures of absorbing and nonabsorbing particles,” Appl. Opt. 20, 3661–3667 (1981).
    [CrossRef] [PubMed]
  8. O. B. Toon, T. P. Ackerman, “Algorithms for the calculation of scattering by stratified spheres,” Appl. Opt. 20, 3657–3660 (1981).
    [CrossRef] [PubMed]
  9. R. Bhandari, “Scattering coefficients for a multilayered sphere: analytic expressions and algorithms,” Appl. Opt. 24, 1060–1967 (1985).
    [CrossRef]
  10. R. Bhandari, “Tiny core or thin layer as a perturbation in scattering by a single-layered sphere,” J. Opt. Soc. Am. A 3, 319–328 (1986).
    [CrossRef]
  11. M. Sitarski, “Internal heating of multilayered aerosol particles by electromagnetic radiation,” Langmuir 3, 85–93 (1987).
    [CrossRef]
  12. D. M. Mackowski, R. A. Aitenkirch, M. P. Menguc, “Internal absorption cross sections in a stratified sphere,” Appl. Opt. 29, 1551–1559 (1990).
    [CrossRef] [PubMed]
  13. P. Massoli, F. Beretta, A. D’Alessio, M. Lazzaro, “Temperature and size of single droplets by light scattering in the forward and rainbow regions,” Appl. Opt. 32, 3295–3301 (1993).
    [CrossRef] [PubMed]
  14. P. Massoli, F. Beretta, A. D’Alessio, “Pyrolysis in the liquid phase inside single droplets of light oil studied with laser light-scattering methods,” Combust. Sci. Tech. 72, 271–282 (1990).
    [CrossRef]
  15. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chap. 4, p. 89.
  16. W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19, 1505–1509 (1980).
    [CrossRef] [PubMed]
  17. M. Kerker, The Scattering of Light (Academic, New York, 1969), Chap. 5, p. 21.

1993 (1)

1991 (1)

1990 (2)

P. Massoli, F. Beretta, A. D’Alessio, “Pyrolysis in the liquid phase inside single droplets of light oil studied with laser light-scattering methods,” Combust. Sci. Tech. 72, 271–282 (1990).
[CrossRef]

D. M. Mackowski, R. A. Aitenkirch, M. P. Menguc, “Internal absorption cross sections in a stratified sphere,” Appl. Opt. 29, 1551–1559 (1990).
[CrossRef] [PubMed]

1987 (1)

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

1986 (1)

1985 (1)

R. Bhandari, “Scattering coefficients for a multilayered sphere: analytic expressions and algorithms,” Appl. Opt. 24, 1060–1967 (1985).
[CrossRef]

1981 (2)

1980 (1)

1976 (1)

1965 (1)

1962 (1)

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

1951 (2)

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

A. L. Aden, “Electromagnetic scattering from spheres with sizes comparable to the wavelength,” J. Appl. Phys. 22, 601–606 (1951).
[CrossRef]

Ackerman, T. P.

Aden, A. L.

A. L. Aden, “Electromagnetic scattering from spheres with sizes comparable to the wavelength,” J. Appl. Phys. 22, 601–606 (1951).
[CrossRef]

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

Aitenkirch, R. A.

Barber, P. W.

Beretta, F.

P. Massoli, F. Beretta, A. D’Alessio, M. Lazzaro, “Temperature and size of single droplets by light scattering in the forward and rainbow regions,” Appl. Opt. 32, 3295–3301 (1993).
[CrossRef] [PubMed]

P. Massoli, F. Beretta, A. D’Alessio, “Pyrolysis in the liquid phase inside single droplets of light oil studied with laser light-scattering methods,” Combust. Sci. Tech. 72, 271–282 (1990).
[CrossRef]

Bhandari, R.

R. Bhandari, “Tiny core or thin layer as a perturbation in scattering by a single-layered sphere,” J. Opt. Soc. Am. A 3, 319–328 (1986).
[CrossRef]

R. Bhandari, “Scattering coefficients for a multilayered sphere: analytic expressions and algorithms,” Appl. Opt. 24, 1060–1967 (1985).
[CrossRef]

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chap. 4, p. 89.

Chowdhury, D. Q.

D’Alessio, A.

P. Massoli, F. Beretta, A. D’Alessio, M. Lazzaro, “Temperature and size of single droplets by light scattering in the forward and rainbow regions,” Appl. Opt. 32, 3295–3301 (1993).
[CrossRef] [PubMed]

P. Massoli, F. Beretta, A. D’Alessio, “Pyrolysis in the liquid phase inside single droplets of light oil studied with laser light-scattering methods,” Combust. Sci. Tech. 72, 271–282 (1990).
[CrossRef]

Fenn, R. W.

Hill, S. C.

Hood, D. A.

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chap. 4, p. 89.

Kattawar, G. W.

Kerker, M.

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

M. Kerker, The Scattering of Light (Academic, New York, 1969), Chap. 5, p. 21.

Lazzaro, M.

Mackowski, D. M.

Massoli, P.

P. Massoli, F. Beretta, A. D’Alessio, M. Lazzaro, “Temperature and size of single droplets by light scattering in the forward and rainbow regions,” Appl. Opt. 32, 3295–3301 (1993).
[CrossRef] [PubMed]

P. Massoli, F. Beretta, A. D’Alessio, “Pyrolysis in the liquid phase inside single droplets of light oil studied with laser light-scattering methods,” Combust. Sci. Tech. 72, 271–282 (1990).
[CrossRef]

Menguc, M. P.

Oser, H.

Sitarski, M.

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

Toon, O. B.

Wiscombe, W. J.

Wyatt, P. J.

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

Appl. Opt. (8)

Combust. Sci. Tech. (1)

P. Massoli, F. Beretta, A. D’Alessio, “Pyrolysis in the liquid phase inside single droplets of light oil studied with laser light-scattering methods,” Combust. Sci. Tech. 72, 271–282 (1990).
[CrossRef]

J. Appl. Phys. (2)

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

A. L. Aden, “Electromagnetic scattering from spheres with sizes comparable to the wavelength,” J. Appl. Phys. 22, 601–606 (1951).
[CrossRef]

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

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 symetric objects,” Phys. Rev. 127, 1837–1843 (1962); Errata, 134, AB1 (1964).
[CrossRef]

Other (2)

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chap. 4, p. 89.

M. Kerker, The Scattering of Light (Academic, New York, 1969), Chap. 5, p. 21.

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Figures (11)

Fig. 1
Fig. 1

Spherical polar-coordinate system centered on a sphere of radius R.

Fig. 2
Fig. 2

The stratified-sphere model.

Fig. 3
Fig. 3

Scattering (Q sca) and extinction (Q ext) efficiencies versus the number of components (L). When x i = x 1 + (x L x 1)(i − 1)/(L − 1), n 1 = 1.43, n L = 1.33, n i = n 1 + 0.5(n L n 1)(1 − cos tπ), k i = 0.0, where t = (i − 1)/(L − 1), i = 1, 2, …, L: (a), x 1 = 0.995x L , x L = 3.13; (b), x 1 = 0.005x L , x L = 3.13; (c), x 1 = 0.995x L , x L = 31.3; (d), x 1 = 0.5x L , x L = 31.3. Curves a and b represent the scattering and extinction efficiencies for x = x L , when m = m 1, and m = m L , respectively.

Fig. 4
Fig. 4

Scattering efficiency (Q sca) versus the relative size parameter of the particle core (x 1/x L ) obtained with r elation (44) (top), and relation (50) (bottom): x L = 3.13, x i = x 1 + (x L x 1)(i − 1)/(L − 1), n 1 = 1.43, n L = 1.33, n i = n 1 + 0.5(n L n 1)(1 − cos tπ), k i = 0.0, where t = (i − 1)/(L − 1), i = 1, 2, …, L, the values of L being indicated on the plot for each case. Curves a and b represent the scattering and extinction efficiencies for x = x L , then m = m 1 and m = m L , respectively.

Fig. 5
Fig. 5

Angular patterns of the polarization ratio (γ) for a coated sphere with x core = 0.358, x coat = 13.121, m core = 1.59 + i0.66, m coat = 1.409 + i0.1747. A comparison of the results obtained by Bohren and Huffman,15 with a coated sphere model, and by Mackowski et al.12 with the multilayered model: (L = 2), and by the new appproach (L = 6000, p = 2000).

Fig. 6
Fig. 6

Vertically polarized intensity function i 1 versus the scattering angle for x sphere = 5.0: A comparison of the rigorous method17 with the stratified-sphere approximation (SSA). For the rigorous method: in case 1, m core = 1.0834 + i0.0, m(x) = x sphere/x; and in case 2, m core = 1.0779 + i0.0, m(x) = 1.01(x sphere/x)2. For the stratified-sphere approximation, with x L = x sphere, x i = x 1 + (x L x 1)(i − 1)/(L − 1): in case 1, m 1 = 1.0834 + i0.0, m i = x L /x i ; and in case 2, m 1 = 1.0779 + i0.0, m i = 1.01(x L /x i )2, i = 1,2, …, L, and L = 1500.

Fig. 7
Fig. 7

Angular pattern of the horizontally (C HH) and vertically (C VV) polarized cross sections for multilayered spheres for x 1 = 0.001x L , x L = 59.4, x i = x 1 + (x L x 1)(i − 1)/(L − 1), m i = m 1 + 0.5(m L m 1)(1 − cos tπ), where t = (i − 1)/(L − 1), i = 1, 2, …, L, L = 6000. For curve 1, m 1 = 1.43 + i0.0, m L = 1.33 + i0.0 (see curve b in Fig. 9). For curve 2, m 1 = 1.43 + i0.1, m L = 1.33 + i0.0 (see curves b and d in Fig. 9).

Fig. 8
Fig. 8

Polarization ratio (γ) versus the scattering angle ϑ for x = 3.13: A comparison of the completely recursive algorithm that uses relation (50) with the partially recursive algorithm: Curve 1, m = 1.43 + i0.0; curve 5, m = 1.33 + i0.0; curves 2, to 3, and 4, x L = 3.13, x i = x 1 + (x L x 1)(i − 1)/(L − 1), m L = 1.33 + i0.0, m i = m 1 + 0.5(m L m 1)(1 − cos tπ), where t = (i − 1)/(L − 1), i = 1, 2, …, L, L = 50. For curve 2, m 1 = 1.43 + i0.0, x 1 = 0.75x L (see curve in a Fig. 9); for curve 3, m 1 = 1.43 + i0.0, x 1 = 0.001x L (see curve b in Fig. 9); for curve 4, m 1 = 1.38 + i0.0, x 1 = 0.001x L (see curve c in Fig. 9).

Fig. 9
Fig. 9

Profiles of the refractive indices used in the calculations.

Fig. 10
Fig. 10

Minimum number of components L required to obtain converged-field solutions versus the size parameter (x L ): x 1 = 0.001x L , x i = x 1 + (x L x 1)(i − 1)/(L − 1), curves 1, 2, and 3 refer to the spheres with n 1 = 1.43, n 1 = 1.53, and n 1 = 1.63, respectively; n L = 1.33, n i = n 1 + 0.5(n L n 1)(1 − cos tπ), k i = 0.0, where t = (i − 1)/(L − 1), i = 1, 2, …, L.

Fig. 11
Fig. 11

Working domain (left and lower part of the dotted line) of particle size and relative refractive index (m = n + ik) for the FORTRAN77 code used by the authors. Hence, wavelength λ = 0.5 μm, n L = n, n 1 = 1.01n L , n i = n 1 + 0.5(n L n 1)(1 − cos tπ), or, k L = k, k 1 = 1.01k L , k i = k 1 + 0.5(k L k 1)(1 − cos tπ), t = (i − 1)/(L − 1), i = 1, 2, …, L.

Equations (63)

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1 / r 2 ( r 2 Ψ / r ) / r + 1 / ( r 2 sin θ ) ( sin θ Ψ / θ ) / θ + 1 / ( r 2 sin θ ) 2 Ψ / ϕ 2 + k 2 ( r ) Ψ = 0 ,
E 0 = E 0 exp ( i k r cos θ ) e ^ x ,
E 0 r ( ρ , θ , ϕ ) = - i cos ϕ sin θ / ρ 2 n = 1 n ( n + 1 ) E n π n ( θ ) Ψ n ( ρ ) ,
E 0 θ ( ρ , θ , ϕ ) = cos ϕ / ρ n = 1 E n [ π n ( θ ) Ψ n ( ρ ) - i π n ( θ ) Ψ n ( ρ ) ] ,
E 0 ϕ ( ρ , θ , ϕ ) = sin ϕ / ρ n = 1 E n [ i π n ( θ ) Ψ n ( ρ ) - τ n ( θ ) Ψ n ( ρ ) ] ,
E s r ( ρ , θ , ϕ ) = i cos ϕ sin θ / ρ 2 n = 1 n ( n + 1 ) E n a n π n ( θ ) ξ n ( ρ ) ,
E s θ ( ρ , θ , ϕ ) = cos ϕ / ρ n = 1 E n [ i a n τ n ( θ ) ξ n ( ρ ) - b n π n ( θ ) ξ n ( ρ ) ] ,
E s ϕ ( ρ , θ , ϕ ) = sin ϕ / ρ n = 1 E n [ b n τ n ( θ ) ξ n ( ρ ) - i a n π n ( θ ) ξ n ( ρ ) ] ,
E i r ( ρ , θ , ϕ ) = - i cos ϕ sin θ / ρ 2 n = 1 n ( n + 1 ) E n π n ( θ ) × [ b i n Ψ n ( ρ ) + d i n χ n ( ρ ) ] ,
E i θ ( ρ , θ , ϕ ) = cos ϕ / ρ n = 1 E n { π n ( θ ) [ a i n Ψ n ( ρ ) + c i n χ n ( ρ ) ] - i τ n ( θ ) [ b i n Ψ n ( ρ ) + d i n χ n ( ρ ) ] } ,
E i ϕ ( ρ , θ , ϕ ) = sin ϕ / ρ n = 1 E n { - τ n ( θ ) [ a i n Ψ n ( ρ ) + c i n χ n ( ρ ) ] + i π n ( θ ) [ b i n Ψ n ( ρ ) + d i n χ n ( ρ ) ] } ,
( E i + 1 - E i ) × e ˜ r = 0 ,
( H i + 1 - H i ) × e ^ r = 0 ,
( E s + E 0 - E L ) × e ^ r = 0.
( H s + H 0 - H L ) × e ^ r = 0.
D n ( 1 ) ( ρ ) = Ψ n ( ρ ) / Ψ n ( ρ ) ,
D n ( 3 ) ( ρ ) = ξ n ( ρ ) / ξ n ( ρ ) ,
R n ( 1 ) ( ρ 1 , ρ 2 ) = Ψ n ( ρ 1 ) / Ψ n ( ρ 2 ) ,
R n ( 3 ) ( ρ 1 , ρ 2 ) = ξ n ( ρ 1 ) / ξ n ( ρ 2 ) ,
a i n = i ( A ( i - 1 ) n - C ( i - 1 ) n ) / A L n / Ψ n ( m i x i ) ,
b i n = i ( B ( i - 1 ) n - D ( i - 1 ) n ) / B L n / Ψ n ( m i x i ) ,
c i n = C ( i - 1 ) n / A L n / χ n ( m i x i ) ,
d i n = D ( i - 1 ) n / B L n / χ n ( m i x i ) ,
b n = - C L n / A L n ,
a n = - D L n / B L n ,
A i n = - Ψ n ( m i + 1 x i ) ξ n ( m i + 1 x i ) × [ F i n ( 1 , 3 ) A ( i - 1 ) n + F i n ( 3 , 3 ) C ( i - 1 ) n ] / [ i m i R n ( 1 ) ( m i + 1 x i , m i + 1 x i + 1 ) ] ,
C i n = Ψ n ( m i + 1 x i ) ξ n ( m i + 1 x i ) × [ F i n ( 1 , 1 ) A ( i - 1 ) n + F i n ( 3 , 1 ) C ( i - 1 ) n ] / [ i m i R n ( 3 ) ( m i + 1 x i , m i + 1 x i + 1 ) ] ,
B i n = - Ψ n ( m i + 1 x i ) ξ n ( m i + 1 x i ) × [ G i n ( 1 , 3 ) B ( i - 1 ) n + G i n ( 3 , 3 ) D ( i - 1 ) n ] / [ i m i R n ( 1 ) ( m i + 1 x i , m i + 1 x i + 1 ) ] ,
D i n = Ψ n ( m i + 1 x i ) ξ n ( m i + 1 x i ) × [ G i n ( 1 , 1 ) B ( i - 1 ) n + G i n ( 3 , 1 ) D ( i - 1 ) n ] / [ i m i R n ( 3 ) ( m i + 1 x i , m i + 1 x i + 1 ) ] ,
A L n = - ξ n ( x L ) [ F L n ( 1 , 3 ) A ( L - 1 ) n + F L n ( 3 , 3 ) C ( L - 1 ) n ] / [ i m L ] ,
C L n = Ψ n ( x L ) [ F L n ( 1 , 1 ) A ( L - 1 ) n + F L n ( 3 , 1 ) C ( L - 1 ) n ] / [ i m L ] ,
B L n = - ξ n ( x L ) [ G L n ( 1 , 3 ) B ( L - 1 ) n + G L n ( 3 , 3 ) D ( L - 1 ) n ] / [ i m L ] ,
D L n = Ψ n ( x L ) [ G L n ( 1 , 1 ) B ( L - 1 ) n + G L n ( 3 , 1 ) D ( L - 1 ) n ] / [ i m L ] ,
F i n ( j , k ) = m i D n ( j ) ( m i x i ) - m i + 1 D n ( k ) ( m i + 1 x i ) ,
G i n ( j , k ) = m i + 1 D n ( j ) ( m i x i ) - m i D n ( k ) ( m i + 1 x i ) ,
A 0 n = B 0 n = 1 ,
C 0 n = D 0 n = 0 ,
Ψ n + 1 ( ρ ) = ( 2 n + 1 ) / ρ Ψ n ( ρ ) - Ψ n + 1 ( ρ ) ,
ξ n + 1 ( ρ ) = ( 2 n + 1 ) / ρ ξ n ( ρ ) - ξ n + 1 ( ρ ) ,
Ψ n ( ρ ) = Ψ n - 1 ( ρ ) - n Ψ n ( ρ ) / ρ ,
ξ n ( ρ ) = ξ n - 1 ( ρ ) - n ξ n ( ρ ) / ρ ,
Ψ - 1 ( ρ ) = cos ρ , Ψ 0 ( ρ ) = sin ρ , χ - 1 ( ρ ) = sin ρ , χ 0 ( ρ ) = cos ρ ,
ξ n ( ρ ) = Ψ n ( ρ ) - i χ n ( ρ ) .
D n - 1 ( 1 ) ( ρ ) = n / ρ - [ D n ( 1 ) ( ρ ) + n / ρ ] - 1 ,
D N max + 15 ( 1 ) = 0 + i 0 ,
N max = ρ + 4.05 ρ 1 / 3 + 2.
Ψ n ( ρ ) ξ n ( ρ ) = Ψ n - 1 ( ρ ) ξ n - 1 ( ρ ) [ n / ρ - D n - 1 ( 1 ) ( ρ ) ] × [ n / ρ - D n - 1 ( 3 ) ( ρ ) ] ,
D n ( 3 ) ( ρ ) = D n ( 1 ) ( ρ ) + i / Ψ n ( ρ ) ξ n ( ρ ) ,
D 0 ( 3 ) ( ρ ) = i , Ψ 0 ( ρ ) ξ 0 ( ρ ) = - i e i ρ sin ρ .
R n ( j ) ( ρ 1 , ρ 2 ) = R n - 1 ( j ) ( ρ 1 , ρ 2 ) [ D n ( j ) ( ρ 2 ) + n / ρ 2 ] / [ D n ( j ) ( ρ 1 ) + n / ρ 1 ] ,
R 0 ( 1 ) ( ρ 1 , ρ 2 ) = Ψ 0 ( ρ 1 ) / Ψ 0 ( ρ 2 ) , R 0 ( 3 ) ( ρ 1 , ρ 2 ) = ξ 0 ( ρ 1 ) / ξ 0 ( ρ 2 ) ,
Ψ 0 ( ρ 1 ) = sin ρ 1 , Ψ 0 ( ρ 2 ) = sin ρ 2 , ξ 0 ( ρ 1 ) = sin ρ 1 - i cos ρ 1 , ξ 0 ( ρ 2 ) = sin ρ 2 - i cos ρ 2 .
R n ( 3 ) ( ρ 1 , ρ 2 ) = R n - 1 ( 3 ) ( ρ 1 , ρ 2 ) [ D n - 1 ( 3 ) ( ρ 1 ) - n / ρ 1 ] / [ D n - 1 ( 3 ) ( ρ 2 ) - n / ρ 2 ] .
0 < Re [ D n ( j ) ( ρ 1 ) + n / ρ 1 ] 1 , 0 < Im [ D n ( j ) ( ρ 1 ) + n / ρ 1 ] 1 ,
0 < Re [ D n ( j ) ( ρ 2 ) + n / ρ 2 ] 1 , 0 < Im [ D n ( j ) ( ρ 2 ) + n / ρ 2 ] 1 ,
F n ( j ) ( m i + 1 x i + 1 ) = F n ( j ) ( m i x i ) + F n ( j ) ( m i x i ) m i ( x i + 1 - x i ) + F n ( j ) ( m i x i ) x i ( m i + 1 - m i ) + O ( x i + 1 - x i ) + O ( m i + 1 - m i ) ,
F n ( j ) ( m i + 1 x i + 1 ) / F n ( j ) ( m i x i ) = 1 + D n ( j ) ( m i x i ) ( m i x i + 1 + x i m i + 1 - 2 m i x i ) + O ( x i + 1 - x i ) + O ( m i + 1 - m i ) .
F n ( j ) ( m i + 1 x i + 1 ) / F n ( j ) ( m i x i ) = [ F n ( j ) ( m i + 1 x i + 1 ) / F n ( j ) ( m i + 1 x i ) ] [ F n ( j ) ( m i + 1 x i ) / F n ( j ) ( m i x i ) ] , = [ R n ( j ) ( m i + 1 x i , m i + 1 x i + 1 ) ] - 1 × [ F n ( j ) ( m i x i ) + F n ( j ) ( m i x i ) x i ( m i + 1 - m i ) + O ( m i + 1 - m i ) ] / F n ( j ) ( m i x i ) , = [ R n ( j ) ( m i + 1 x i , m i + 1 x i + 1 ) ] - 1 [ 1 + D n ( j ) ( m i x i ) ( m i + 1 - m i ) ] + O ( m i + 1 - m i ) .
R n ( i ) ( m i + 1 x i , m i + 1 x i + 1 ) = { 1 + ( x i + 1 - x i ) m i / [ x i ( m i + 1 - m i ) + 1 / D n ( j ) ( m i x i ) ] } - 1 + O ( x i + 1 - x i ) + O ( m i + 1 - m i ) .
R n ( j ) ( m i + 1 x i , m i + 1 x i + 1 ) { 1 + ( x i + 1 - x i ) m i / [ x i ( m i + 1 - m i ) + 1 / D n ( j ) ( m i x i ) ] } - 1 ,
m A = 1.5 + i 0.0 ( r 0.1 μ m ) , m B = 1.4 + i 0.1 ( r > 0.1 μ m ) ,
m i + 1 x i + 1 - m i x i = ɛ 1 ,
m i x i + 1 - m i + 1 x i = ɛ 2 ,

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