Abstract

Series expressions for the radially dependent absorption cross section and angle-averaged absorption heat source function within a stratified sphere are presented. A numerically stable and accurate algorithm for computation of the internal radiative properties, as well as the overall scattering and extinction of a stratified sphere having an arbitrary number of layers is developed. The results allow for direct estimation of the degree of penetration and intensity of radiative heating in radially inhomogeneous spherical particles, and also provide an estimate of the thermal emission coefficient of particles having a radial temperature distribution.

© 1990 Optical Society of America

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References

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  1. W. J. Wiscombe, “Improved Mie Scattering Algorithms,” Appl. Opt. 19, 1505–1509 (1980).
    [CrossRef] [PubMed]
  2. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  3. A. L. Aden, M. Kerker, “Scattering of Electromagnetic Waves from Two Concentric Spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
    [CrossRef]
  4. O. B. Toon, T. P. Ackerman, “Algorithms for the Calculation of Scattering by Stratified Spheres,” Appl. Opt. 20, 3657–3660 (1981).
    [CrossRef] [PubMed]
  5. R. Bhandari, “Scattering Coefficients for a Multilayered Sphere: Analytic Expressions and Algorithms,” Appl. Opt. 24, 1960–1967 (1985).
    [CrossRef] [PubMed]
  6. Yu. I. Yalamov, V. B. Kutukov, E. R. Shchukin, “Theory of the Photophoretic Motion of the Large-Size Volatile Aerosol Particle,” J. Coll. Int. Sci. 57, 564–571 (1976).
    [CrossRef]
  7. D. W. Mackowski, “Photophoresis of Aerosol Particles in the Free-Molecular and Slip-Flow Regimes,” Int. J. Heat Mass Trans. 32, 843–854 (1989).
    [CrossRef]
  8. M. Sitarski, “Absorption of Infrared Radiation Inside an Explosively Boiling Fine Coal–Water Particle,” Particulate Sci. Tech. 5, 193–205 (1987).
    [CrossRef]
  9. M. Sitarski, “On the Feasibility of Secondary Atomization of Small Slurry Droplets Exposed to Intense Thermal Radiation,” Combust. Sci. Tech. 54, 177–201 (1987).
    [CrossRef]
  10. S. Choi, C. H. Kruger, “Modeling Coal Particle Behavior Under Simultaneous Devolatilization and Combustion,” Combust. Flame 61, 131–144 (1985).
    [CrossRef]
  11. D. W. Mackowski, R. A. Altenkirch, M. P. Menguc, “A Comparison of Electromagnetic Wave and Radiative Transfer Analyses of a Coal Particle Surrounded by a Soot Cloud,” Combust. Flame, 76, 415–420 (1989).
    [CrossRef]
  12. M. Sitarski, “Internal Heating of Multilayered Aerosol Particles by Electromagnetic Radiation,” Langmuir 3, 85–93 (1987).
    [CrossRef]
  13. P. W. Dusel, M. Kerker, D. D. Cooke, “Distribution of Absorption Centers within Irradiated Spheres,” J. Opt. Soc. Amer. 69, 55–59 (1979).
    [CrossRef]
  14. G. W. Kattawar, M. Eisner, “Radiation from a Homogeneous Isothermal Sphere,” Appl. Opt. 9, 2685–269 (1970).
    [CrossRef] [PubMed]

1989 (2)

D. W. Mackowski, “Photophoresis of Aerosol Particles in the Free-Molecular and Slip-Flow Regimes,” Int. J. Heat Mass Trans. 32, 843–854 (1989).
[CrossRef]

D. W. Mackowski, R. A. Altenkirch, M. P. Menguc, “A Comparison of Electromagnetic Wave and Radiative Transfer Analyses of a Coal Particle Surrounded by a Soot Cloud,” Combust. Flame, 76, 415–420 (1989).
[CrossRef]

1987 (3)

M. Sitarski, “Internal Heating of Multilayered Aerosol Particles by Electromagnetic Radiation,” Langmuir 3, 85–93 (1987).
[CrossRef]

M. Sitarski, “Absorption of Infrared Radiation Inside an Explosively Boiling Fine Coal–Water Particle,” Particulate Sci. Tech. 5, 193–205 (1987).
[CrossRef]

M. Sitarski, “On the Feasibility of Secondary Atomization of Small Slurry Droplets Exposed to Intense Thermal Radiation,” Combust. Sci. Tech. 54, 177–201 (1987).
[CrossRef]

1985 (2)

S. Choi, C. H. Kruger, “Modeling Coal Particle Behavior Under Simultaneous Devolatilization and Combustion,” Combust. Flame 61, 131–144 (1985).
[CrossRef]

R. Bhandari, “Scattering Coefficients for a Multilayered Sphere: Analytic Expressions and Algorithms,” Appl. Opt. 24, 1960–1967 (1985).
[CrossRef] [PubMed]

1981 (1)

1980 (1)

1979 (1)

P. W. Dusel, M. Kerker, D. D. Cooke, “Distribution of Absorption Centers within Irradiated Spheres,” J. Opt. Soc. Amer. 69, 55–59 (1979).
[CrossRef]

1976 (1)

Yu. I. Yalamov, V. B. Kutukov, E. R. Shchukin, “Theory of the Photophoretic Motion of the Large-Size Volatile Aerosol Particle,” J. Coll. Int. Sci. 57, 564–571 (1976).
[CrossRef]

1970 (1)

1951 (1)

A. L. Aden, M. Kerker, “Scattering of Electromagnetic Waves from Two Concentric Spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

Ackerman, T. P.

Aden, A. L.

A. L. Aden, M. Kerker, “Scattering of Electromagnetic Waves from Two Concentric Spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

Altenkirch, R. A.

D. W. Mackowski, R. A. Altenkirch, M. P. Menguc, “A Comparison of Electromagnetic Wave and Radiative Transfer Analyses of a Coal Particle Surrounded by a Soot Cloud,” Combust. Flame, 76, 415–420 (1989).
[CrossRef]

Bhandari, R.

Bohren, C. F.

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

Choi, S.

S. Choi, C. H. Kruger, “Modeling Coal Particle Behavior Under Simultaneous Devolatilization and Combustion,” Combust. Flame 61, 131–144 (1985).
[CrossRef]

Cooke, D. D.

P. W. Dusel, M. Kerker, D. D. Cooke, “Distribution of Absorption Centers within Irradiated Spheres,” J. Opt. Soc. Amer. 69, 55–59 (1979).
[CrossRef]

Dusel, P. W.

P. W. Dusel, M. Kerker, D. D. Cooke, “Distribution of Absorption Centers within Irradiated Spheres,” J. Opt. Soc. Amer. 69, 55–59 (1979).
[CrossRef]

Eisner, M.

Huffman, D. R.

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

Kattawar, G. W.

Kerker, M.

P. W. Dusel, M. Kerker, D. D. Cooke, “Distribution of Absorption Centers within Irradiated Spheres,” J. Opt. Soc. Amer. 69, 55–59 (1979).
[CrossRef]

A. L. Aden, M. Kerker, “Scattering of Electromagnetic Waves from Two Concentric Spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

Kruger, C. H.

S. Choi, C. H. Kruger, “Modeling Coal Particle Behavior Under Simultaneous Devolatilization and Combustion,” Combust. Flame 61, 131–144 (1985).
[CrossRef]

Kutukov, V. B.

Yu. I. Yalamov, V. B. Kutukov, E. R. Shchukin, “Theory of the Photophoretic Motion of the Large-Size Volatile Aerosol Particle,” J. Coll. Int. Sci. 57, 564–571 (1976).
[CrossRef]

Mackowski, D. W.

D. W. Mackowski, R. A. Altenkirch, M. P. Menguc, “A Comparison of Electromagnetic Wave and Radiative Transfer Analyses of a Coal Particle Surrounded by a Soot Cloud,” Combust. Flame, 76, 415–420 (1989).
[CrossRef]

D. W. Mackowski, “Photophoresis of Aerosol Particles in the Free-Molecular and Slip-Flow Regimes,” Int. J. Heat Mass Trans. 32, 843–854 (1989).
[CrossRef]

Menguc, M. P.

D. W. Mackowski, R. A. Altenkirch, M. P. Menguc, “A Comparison of Electromagnetic Wave and Radiative Transfer Analyses of a Coal Particle Surrounded by a Soot Cloud,” Combust. Flame, 76, 415–420 (1989).
[CrossRef]

Shchukin, E. R.

Yu. I. Yalamov, V. B. Kutukov, E. R. Shchukin, “Theory of the Photophoretic Motion of the Large-Size Volatile Aerosol Particle,” J. Coll. Int. Sci. 57, 564–571 (1976).
[CrossRef]

Sitarski, M.

M. Sitarski, “Internal Heating of Multilayered Aerosol Particles by Electromagnetic Radiation,” Langmuir 3, 85–93 (1987).
[CrossRef]

M. Sitarski, “Absorption of Infrared Radiation Inside an Explosively Boiling Fine Coal–Water Particle,” Particulate Sci. Tech. 5, 193–205 (1987).
[CrossRef]

M. Sitarski, “On the Feasibility of Secondary Atomization of Small Slurry Droplets Exposed to Intense Thermal Radiation,” Combust. Sci. Tech. 54, 177–201 (1987).
[CrossRef]

Toon, O. B.

Wiscombe, W. J.

Yalamov, Yu. I.

Yu. I. Yalamov, V. B. Kutukov, E. R. Shchukin, “Theory of the Photophoretic Motion of the Large-Size Volatile Aerosol Particle,” J. Coll. Int. Sci. 57, 564–571 (1976).
[CrossRef]

Appl. Opt. (4)

Combust. Flame (2)

S. Choi, C. H. Kruger, “Modeling Coal Particle Behavior Under Simultaneous Devolatilization and Combustion,” Combust. Flame 61, 131–144 (1985).
[CrossRef]

D. W. Mackowski, R. A. Altenkirch, M. P. Menguc, “A Comparison of Electromagnetic Wave and Radiative Transfer Analyses of a Coal Particle Surrounded by a Soot Cloud,” Combust. Flame, 76, 415–420 (1989).
[CrossRef]

Combust. Sci. Tech. (1)

M. Sitarski, “On the Feasibility of Secondary Atomization of Small Slurry Droplets Exposed to Intense Thermal Radiation,” Combust. Sci. Tech. 54, 177–201 (1987).
[CrossRef]

Int. J. Heat Mass Trans. (1)

D. W. Mackowski, “Photophoresis of Aerosol Particles in the Free-Molecular and Slip-Flow Regimes,” Int. J. Heat Mass Trans. 32, 843–854 (1989).
[CrossRef]

J. Appl. Phys. (1)

A. L. Aden, M. Kerker, “Scattering of Electromagnetic Waves from Two Concentric Spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

J. Coll. Int. Sci. (1)

Yu. I. Yalamov, V. B. Kutukov, E. R. Shchukin, “Theory of the Photophoretic Motion of the Large-Size Volatile Aerosol Particle,” J. Coll. Int. Sci. 57, 564–571 (1976).
[CrossRef]

J. Opt. Soc. Amer. (1)

P. W. Dusel, M. Kerker, D. D. Cooke, “Distribution of Absorption Centers within Irradiated Spheres,” J. Opt. Soc. Amer. 69, 55–59 (1979).
[CrossRef]

Langmuir (1)

M. Sitarski, “Internal Heating of Multilayered Aerosol Particles by Electromagnetic Radiation,” Langmuir 3, 85–93 (1987).
[CrossRef]

Particulate Sci. Tech. (1)

M. Sitarski, “Absorption of Infrared Radiation Inside an Explosively Boiling Fine Coal–Water Particle,” Particulate Sci. Tech. 5, 193–205 (1987).
[CrossRef]

Other (1)

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

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Figures (7)

Fig. 1
Fig. 1

Stratified sphere radiative model.

Fig. 2
Fig. 2

Internal absorption efficiency Qabs(r) and average field intensity B(r) for a homogeneous sphere. Refractive index m = 1.7 + 0.3i, size parameter x = 5, 20, and 100.

Fig. 3
Fig. 3

Internal absorption efficiency Qabs(r) and average field intensity B(r) for a homogeneous sphere. Refractive index m = 1.7 + 0.03i, size parameter x = 5, 20, and 100.

Fig. 4
Fig. 4

Internal absorption efficiency Qabs(r) and average field intensity B(r) for a homogeneous sphere. Refractive index m = 1.7 + 0.0003i, size parameter x = 5, 20, and 100.

Fig. 5
Fig. 5

Average field intensity B(r) for a homogeneous sphere. Refractive index m = 1.33 + 0i, 2 + 0i, and 5 + 0i; size parameter x = 100.

Fig. 6
Fig. 6

Internal absorption efficiency Qabs(r), average field intensity B(r), and dimensionless heat source function q(r)λ/I0, for five different radial distributions of absorbing material in a water sphere.

Fig. 7
Fig. 7

Differential scattering cross sections CHH(θ) and CVV(θ) for the five distributions of absorbing material in a water sphere. P.I.: plotting increment.

Tables (2)

Tables Icon

Table I Comparison of Absorption Efficiencies Qabs,i Obtained from Present Analysis and Through Numerical Integration12

Tables Icon

Table II Extinction, Scattering, Absorption and Backscattering Efficiencies for Stratified-Absorption Particlesa

Equations (72)

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E i r = cos φ sin θ ρ 2 n = 1 ( 2 n + 1 ) E n π n ( θ ) [ b i n ψ n ( ρ ) + d i n χ n ( ρ ) ]
E i θ = 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 = 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 ) ] ] ,
τ n = d P n 1 ( cos θ ) d θ
π n = P n 1 ( cos θ ) sin θ ,
E s r = i cos φ sin θ ρ 2 n = 1 ( 2 n + 1 ) E n a n π n ( θ ) ξ n ( ρ )
E s θ = 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 = - sin φ ρ n = 1 E n ( i a n π n ( θ ) ξ n ( ρ ) - b n τ n ( θ ) ξ n ( ρ ) ] ,
E 0 r = - i cos φ sin θ ρ n = 1 ( 2 n + 1 ) E n π n ( θ ) ψ n ( ρ )
E 0 θ = - cos φ ρ n = 1 E n ( i τ n ( θ ) ψ n ( ρ ) - π n ( θ ) ψ n ( ρ ) ]
E 0 φ = sin φ ρ n = 1 E n ( i π n ( θ ) ψ n ( ρ ) - τ n ( θ ) ψ n ( ρ ) ] .
m 2 a 1 n ψ ( m 1 x 1 ) = m 1 [ a 2 n ψ n ( m 2 x 1 ) + c 2 n χ n ( m 2 x 1 ) ]
a 1 n ψ n ( m 1 x 1 ) = a 2 n ψ n ( m 2 x 1 ) + c 2 n χ n ( m 2 x 1 )
m 2 b 1 n ψ n ( m 1 x 1 ) = m 1 [ b 2 n ψ n ( m 2 x 1 ) + d 2 n χ n ( m 2 x 1 ) ]
b 1 n ψ n ( m 1 x 1 ) = b 2 n ψ n ( m 2 x 1 ) + d 2 n χ n ( m 2 x 1 )
m j [ a i n ψ n ( m i x i ) + c i n χ n ( m i x i ) ] = m i [ a j n ψ n ( m j x i ) + c j n χ n ( m j x i ) ]
a i n ψ n ( m i x i ) + c i n χ n ( m i x i ) = a j n ψ n ( m j x i ) + c j n χ n ( m j x i )
m j [ b i n ψ n ( m i x i ) + d i n χ n ( m i x i ) ] = m i [ b j n ψ n ( m j x i ) + d j n χ n ( m j x i ) ]
b i n ψ n ( m i x i ) + d i n χ n ( m i x i ) = b j n ψ n ( m j x i ) + d j n χ n ( m j x i )
a L n ψ n ( m L x L ) + c L n χ n ( m L x L ) = m L [ ψ n ( x L ) - b n ξ n ( x L ) ]
a L n ψ n ( m L x L ) + c L n χ n ( m L x L ) = ψ n ( x L ) - m i b n ξ n ( x L )
b L n ψ n ( m L x L ) + d L n χ n ( m L x L ) = m L [ ψ n ( x L ) - a n ξ n ( x L ) ]
b L n ψ n ( m L x L ) + d L n χ n ( m L x L ) = ψ n ( x L ) - a n ξ n ( x L ) ] .
S = 1 2 Re ( E × H * ) ,
W abs ( r ) = 1 2 Re [ 0 2 π 0 π ( E 1 φ H 1 θ * - E 1 θ H 1 φ * ) sin θ d θ d φ ] .
0 π ( π m π n + τ m τ n ) sin θ d θ = δ n m 2 n 2 ( n + 1 ) 2 n + 1
0 π ( π m τ n + τ m π n ) sin θ d θ = 0 ,
W abs ( r ) = E 0 2 λ 2 ω μ Re { i m i n = 1 ( 2 n + 1 ) × [ [ b i n ψ n ( ρ ) + d i n ψ n ( ρ ) ] [ b i n ψ n ( ρ ) + d i n χ n ( ρ ) ] * - [ a i n ψ n ( ρ ) + c i n χ n ( ρ ) ] [ a i n ψ n ( ρ ) + c i n χ n ( ρ ) ] * ] } ,
C abs ( a i ) = λ 2 2 π Re { i m i n = 1 ( 2 n + 1 ) × [ [ b i n ψ n ( m i x i ) + d i n χ n ( m i x i ) ] × [ b i n ψ n ( m i x i ) + d i n χ n ( m i x i ) ] * - [ a i n ψ n ( m i x i ) + c i n χ n ( m i x i ) ] × [ a i n ψ n ( m i x i ) + c i n χ n ( m i x i ) ] * ] } .
C abs ( a L ) = λ 2 2 π n = 1 ( 2 n + 1 ) [ Re ( a n + b n ) - ( a n 2 + b n 2 ) ] = C ext - C sca
q i = 3 I 0 C abs , i 4 π ( r i 3 - r i - 1 3 ) ,
q ( r ) = I 0 r 2 d d r C abs ( r ) .
q ( a i ) = 2 n k I 0 π λ m i x i 2 n = 1 ( 2 n + 1 ) × [ n ( n + 1 ) m i x i 2 b i n ψ n ( m i x i ) + d n χ n ( m i x i ) 2 + b i n ψ n ( m i x i ) + d i n χ n ( m i x i ) 2 + a i n ψ n ( m i x i ) + c i n χ n ( m i x i ) 2 ] .
q ( r , θ , φ ) = 4 π I 0 n k λ E ( r , θ , φ ) 2 E 0 2 = 4 π I 0 n k λ B ( r , θ , φ ) ,
B ( a i ) = 0 2 π 0 π B ( r , θ , φ ) sin θ d θ d φ = 1 2 π 2 m i x i 2 n = 1 ( 2 n + 1 ) × [ n ( n + 1 ) m i x i 2 b i n ψ n ( m x ) + d i n χ n ( m i x i ) 2 + b i n ψ n ( m i x i ) + d i n χ n ( m i x i ) 2 + a i n ψ n ( m i x i ) + c i n χ n ( m i x i ) 2 ] .
e λ = 1 r L 2 i = 1 L r i 2 Q abs , i e b λ ( T i ) ,
a ¯ i n = i A ( i - 1 ) n / A L n
c ¯ i n = i C ( i - 1 ) n / A L n
b ¯ i n = i B ( i - 1 ) n / B L n
d ¯ i n = i D ( i - 1 ) n / B L n
b n = - C L n / A L n
a n = - D L n / B L n ,
A 0 n = B 0 n = 1
C 0 n = D 0 n = 0.
A i n = - ψ n ( m i + 1 x i ) ξ n ( m i + 1 x i ) i m i R n ( 1 ) ( m i + 1 x i , m i + 1 x i + 1 ) × [ F i n ( 1 , 3 ) A i - 1 , n + F i n ( 3 , 3 ) C i - 1 , n ]
C i n = ψ n ( m i + 1 x i ) ξ n ( m i + 1 x i ) i m i R n ( 3 ) ( m i + 1 x i , m i + 1 x i + 1 ) × [ F i n ( 1 , 1 ) A i - 1 , n + F i n ( 3 , 1 ) C i - 1 , n ]
B i n = - ψ n ( m i + 1 x i ) ξ n ( m i + 1 x i ) i m i R n ( 1 ) ( m i + 1 x i , m i + 1 x i + 1 ) × [ G i n ( 1 , 3 ) B i - 1 , n + G i n ( 3 , 3 ) D i - 1 , n ]
D i n = ψ n ( m i + 1 x i ) ξ n ( m i + 1 x i ) i m i R n ( 3 ) ( m i + 1 x i , m i + 1 x i + 1 ) × [ G i n ( 1 , 1 ) B i - 1 , n + G i n ( 3 , 1 ) D i - 1 , n ]
A L n = - ξ n ( x L ) i m L [ F L n ( 1 , 3 ) A L - 1 , n + F L n ( 3 , 3 ) C L - 1 , n ]
C L n = ψ n ( x L ) i m L [ F L n ( 1 , 1 ) A L - 1 , n + F L n ( 3 , 1 ) C L - 1 , n ]
B L n = - ξ n ( x L ) i m L [ G L n ( 1 , 3 ) B L - 1 , n + G L n ( 3 , 3 ) D L - 1 , n ]
D L n = ψ n ( x L ) i m L [ G L n ( 1 , 1 ) B L - 1 , n + G L n ( 3 , 1 ) D L - 1 , n ] .
F i ( j , k ) = m i D n ( j ) ( m i x i ) - m i + 1 D n ( k ) ( m i + 1 x i )
G i ( j , k ) = m i + 1 D n ( j ) ( m i x i ) - m i D n ( k ) ( m i + 1 x i ) .
C abs ( a i ) = λ 2 2 π Re [ i m i n = 1 ( 2 n + 1 ) × [ [ b ¯ i n D n ( 1 ) ( m i x i ) + d ¯ i n D n ( 3 ) ( m i x i ) ] [ b ¯ i n + d ¯ i n ] * - [ a ¯ i n + c ¯ i n ] [ a ¯ i n D n ( 1 ) ( m i x i ) + c ¯ i n D n ( 3 ) ( m i x i ) ] * ] ,
B ( a i ) = 1 2 π 2 m i x i 2 n = 1 ( 2 n + 1 ) [ n ( n + 1 ) m i x i 2 b ¯ i n + d ¯ i n 2 + b ¯ i n D n ( 1 ) ( m i x i ) + d ¯ i n D n ( 3 ) ( m i x i ) 2 + a ¯ i n + c ¯ i n 2 ] .
Q sca = 2 x 2 n = 1 ( 2 n + 1 ) ( a n 2 + b n 2 )
Q ext = 2 x 2 n = 1 ( 2 n + 1 ) Re ( a n + b n )
S 1 = n = 1 2 n + 1 n ( n + 1 ) ( a n π n + b n τ n )
S 2 = n = 1 2 n + 1 n ( n + 1 ) ( a n τ n + b n π n )
C v v = λ 2 4 π S 1 2 , C h h = λ 2 4 π S 2 2 .
D n - 1 ( 1 ) ( z ) = n z - 1 D n ( 1 ) ( z ) + n z .
ψ n ( z ) ξ n ( z ) = ψ n - 1 ( z ) ξ n - 1 ( z ) [ n 2 - D n - 1 ( 1 ) ( z ) ] [ n z - D n - 1 ( 3 ) ( z ) ]
D n ( 3 ) ( z ) = D n ( 1 ) ( z ) + i ψ n ( z ) ξ n ( z )
R n ( j ) ( z 1 , z 2 ) = R n - 1 ( j ) ( z 1 , z 2 ) D n ( j ) ( z 2 ) + n / z 2 D n ( j ) ( z 1 ) + n / z 1 ,
ψ n ( x L ) = ψ n - 1 ( x L ) [ n x L - D n - 1 ( 1 ) ( x L ) ]
ξ n ( x L ) = ξ n - 1 ( x L ) [ n x L - D n - 1 ( 3 ) ( x L ) ]
A i n = A i - 1 , n / R n ( 1 ) ( m i x i , m i + 1 x i )
C i n = C i - 1 , n / R n ( 3 ) ( m i x i , m i + 1 x i )
B i n = B i - 1 , n / R n ( 1 ) ( m i x i , m i + 1 x i )
D i n = D i - 1 , n / R n ( 3 ) ( m i x i , m i + 1 x i )
m eff 2 ( r ) = m med 2 { 1 + 3 f v ( r ) m abs 2 - m med 2 m abs 2 + 2 m med 2 × [ 1 - f v m abs 2 - m med 2 m abs 2 + 2 m med 2 ] - 1 } .

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