Abstract

A finely stratified cylinder model (FSCM) for calculation of the scattered fields of infinitely long, radially inhomogeneous, circular cylinders normally illuminated by electromagnetic plane waves is introduced for the first time, to our knowledge. Because of its capability of using a very large number of layers (more than 80,000), the model is useful for both continuous and discontinuous refractive-index profiles. Numerical results agree well with published solutions for radially inhomogeneous cylinders; for cylinders with a dimensionless size parameter larger than 60, results obtained with the FSCM agree with the geometric optics for both continuous and discontinuous refractive-index profiles.

© 1995 Optical Society of America

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1994

1991

P. S. Swathi, T. W. Tong, G. R. Cunnington, “Scattering of electromagnetic waves by cylinders coated with a radially inhomogeneous layers,” J. Quant. Spectrosc. Radiat. Transfer 46, 281–292 (1991).
[CrossRef]

1990

1987

1986

J. R. Wait, “Impedance condition for a coated cylindrical conductor,” Radio Sci. 21, 623–626 (1986).
[CrossRef]

1985

1977

1972

1961

Aitenkirch, R. A.

Alexopoulos, N. G.

Barabás, M.

Bhandari, R.

Bohren, C. F.

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

Cunnington, G. R.

P. S. Swathi, T. W. Tong, G. R. Cunnington, “Scattering of electromagnetic waves by cylinders coated with a radially inhomogeneous layers,” J. Quant. Spectrosc. Radiat. Transfer 46, 281–292 (1991).
[CrossRef]

D'Alessio, A.

L. Kai, P. Massoli, A. D'Alessio, “Studying inhomogeneity of radially inhomogeneous spherical particles by light scattering,” in Proceedings of the Third Internal Congress on Optical Particle Sizing (Keio University, Yokohama, Japan, 1993), pp. 135–154.

Dong, J. -L.

J. -L. Dong, A Practical Handbook of Mathematical Formulas (China Hebei Science and Technology, Heibei, China, 1988).

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsy, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cam-bridge, 1986), pp. 170–183.

Huffman, D. R.

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

Kai, L.

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]

L. Kai, P. Massoli, A. D'Alessio, “Studying inhomogeneity of radially inhomogeneous spherical particles by light scattering,” in Proceedings of the Third Internal Congress on Optical Particle Sizing (Keio University, Yokohama, Japan, 1993), pp. 135–154.

Kerker, M.

Lindgren, G.

Mackowski, D. M.

Massoli, P.

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]

L. Kai, P. Massoli, A. D'Alessio, “Studying inhomogeneity of radially inhomogeneous spherical particles by light scattering,” in Proceedings of the Third Internal Congress on Optical Particle Sizing (Keio University, Yokohama, Japan, 1993), pp. 135–154.

Matijevic, E.

Menguc, M. P.

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsy, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cam-bridge, 1986), pp. 170–183.

Swathi, P. S.

P. S. Swathi, T. W. Tong, G. R. Cunnington, “Scattering of electromagnetic waves by cylinders coated with a radially inhomogeneous layers,” J. Quant. Spectrosc. Radiat. Transfer 46, 281–292 (1991).
[CrossRef]

Teukolsy, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsy, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cam-bridge, 1986), pp. 170–183.

Tong, T. W.

P. S. Swathi, T. W. Tong, G. R. Cunnington, “Scattering of electromagnetic waves by cylinders coated with a radially inhomogeneous layers,” J. Quant. Spectrosc. Radiat. Transfer 46, 281–292 (1991).
[CrossRef]

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsy, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cam-bridge, 1986), pp. 170–183.

Wait, J. R.

J. R. Wait, “Impedance condition for a coated cylindrical conductor,” Radio Sci. 21, 623–626 (1986).
[CrossRef]

Yeh, C.

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Quant. Spectrosc. Radiat. Transfer

P. S. Swathi, T. W. Tong, G. R. Cunnington, “Scattering of electromagnetic waves by cylinders coated with a radially inhomogeneous layers,” J. Quant. Spectrosc. Radiat. Transfer 46, 281–292 (1991).
[CrossRef]

Radio Sci.

J. R. Wait, “Impedance condition for a coated cylindrical conductor,” Radio Sci. 21, 623–626 (1986).
[CrossRef]

Other

M. Kerker, The Scattering of Light (Academic, New York, 1969).

L. Kai, P. Massoli, A. D'Alessio, “Studying inhomogeneity of radially inhomogeneous spherical particles by light scattering,” in Proceedings of the Third Internal Congress on Optical Particle Sizing (Keio University, Yokohama, Japan, 1993), pp. 135–154.

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

J. -L. Dong, A Practical Handbook of Mathematical Formulas (China Hebei Science and Technology, Heibei, China, 1988).

W. H. Press, B. P. Flannery, S. A. Teukolsy, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cam-bridge, 1986), pp. 170–183.

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Figures (10)

Fig. 1
Fig. 1

Cylindrical polar coordinate system for a cylinder of radius a normally illuminated by electromagnetic plane waves.

Fig. 2
Fig. 2

Cross section of the stratified cylinder model.

Fig. 3
Fig. 3

Angular intensity of the far-field scattered wave calculated with the FSCM. The cylinder is a doubly clad optical fiber (m1 = 1.62, m2 = 1.505, m3 = 1.56, r1 = 5.6 μm, r2 = 6.3 μm, r3 = 7.0 μm). The normally incident light is of wavelength λ = 0.633 μm and is plane polarized (a) in the (x, z) plane, (b) in the (x, y) plane.

Fig. 4
Fig. 4

Stratification method used in Fig. 3, where j is the number index of the jth layer and rj and mj are the radius and the relative refractive index of the jth layer, respectively.

Fig. 5
Fig. 5

Scattering patterns of two cylinders with variable refractive indices, where normalized scattered field I ̅ = | Σ n = b n cos n ϕ |, k = 2π/λ, and L = 2000. Incident electric fields are parallel to the z axis.

Fig. 6
Fig. 6

Ray trace of four cyclinders obtained with geometric optics: (a) homogeneous cylinder with m = 1.3, (b) homogeneous cylinder with m = 1.4, (c) cylinder with a homogeneous core (r1 = 0.5a, m1 = 1.3) and an inhomogeneous sheath with m(r) = 1.3 + 1.0(2r/a − 1)2, and (d) cylinder with a homogeneous core (rn = 0.5a, m1 = 1.4) and an inhomogeneous sheath with m(r) = 1.4 − 0.1(2r/a − 1)2. The incident angle is 45°.

Fig. 7
Fig. 7

Far-field angular patterns of T ̅ 11 ( ϕ ) = i ̅ ( ϕ ) + i ̅ ( ϕ ) calculated with the FSCM (L = 80,000) for the cylinders (each with a = 5 μm and λ = 0.488 μm) shown in Fig. 6 [curves (a)–(d) correspond to cylinders (a)–(d)]. The incident wave is 45° polarized, i.e., E 0 = E 0 = E 0. Arrows indicate the primary rainbows.

Fig. 8
Fig. 8

Far-field angular patterns of T ̅ 11 ( ϕ ) = i ̅ ( ϕ ) + i ̅ ( ϕ ) calculated with the FSCM (L = 80,000) (a) for a homogeneous cylinder (m = 1.5) and (b) for an inhomogeneous cylinder with an absorbent core (mcore = 1.5 + i0.1, rcore = 0.5a) and a transparent coating (mshell = 1.5). The incident wave is 45° polarized, i.e., E 0 = E 0 = E 0. For each cylinder, a = 5 μm and λ = 0.488 μm.

Fig. 9
Fig. 9

Ray tracing inside a coated cylinder.

Fig. 10
Fig. 10

Normalized electric density S = | E | 2 / ( | E 0 | 2 + | E 0 | 2 ) on the main diameter parallel to the incident wave direction inside a homogeneous cylinder with m = 1.33 (dashed curve) and inside an inhomogeneous cylinder with m(r) = 1.33[1 − 0.01(r/a)2] (solid curve). The incident wave is 45° polarized, i.e., E 0 = E 0 = E 0. For both cylinders, a = 0.2 μm and λ = 0.55 μm. The results were obtained by the use of the FSCM, with 1000 components of equal thickness.

Equations (81)

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scattering from inhomogeneous cylinder { normal incidence { single-layered cylinder multilayered cylinder 3 { homogeneous layer 1 inhomogeneous layer 2 oblique incidence { single-layered cylinder multilayered cylinder 6 { homogeneous layer 4 inhomogeneous layer 5
E 0 = E 0 + E 0 ,
E 0 = E 0 ê z exp ( ikr cos ϕ ) ,
E 0 = E 0 ê y exp ( ikr cos ϕ ) ,
M n ( ν ) = exp ( i n ϕ ) [ i n Z n ( ρ ) ρ ê r Z n ( ρ ) ê ϕ ] ,
N n ( ν ) = exp ( i n ϕ ) Z n ( ρ ) ê z ,
E 0 = E 0 ê z exp ( ikr cos ϕ )
E 0 = n = E n N n ( 1 ) ,
E j = n = E n [ a j n N n ( 1 ) + c j n N n ( 3 ) ] .
E s = n = E n b n N n ( 3 ) .
E 0 = E 0 ê y exp ( ikr cos ϕ )
E 0 = i n = E n M n ( 1 ) ,
E j = i n = E n [ b j n M n ( 1 ) + d j n M n ( 3 ) ] .
E s = i n = E n a n M n ( 3 ) .
E 0 = n = [ E n N n ( 1 ) i E n M n ( 1 ) ] ,
E j = n = { E n [ a j n N n ( 1 ) + c j n N n ( 3 ) ] i E n [ b j n M n ( 1 ) + d j n M n ( 3 ) ] } ,
E s = n = [ E n b n N n ( 3 ) + i E n a n M n ( 3 ) ] ,
E 0 r = 1 ρ n = exp ( i n ϕ ) E n n J n ( ρ ) ,
E 0 ϕ = i n exp ( i n ϕ ) E n J n ( ρ ) ,
E 0 z = n = exp ( i n ϕ ) E n J n ( ρ ) ,
E j r = 1 ρ n = E n exp ( i n ϕ ) n [ b j n J n ( ρ ) + d j n H n ( 2 ) ( ρ ) ] ,
E j ϕ = i n = E n exp ( i n ϕ ) [ b j n J n ( ρ ) + d j n H n ( 2 ) ( ρ ) ] ,
E j z = n = E n exp ( i n ϕ ) [ a j n J n ( ρ ) + c j n H n ( 2 ) ( ρ ) ] ,
E s r = 1 ρ n exp ( i n ϕ ) E n a n n H n ( 2 ) ( ρ ) ,
E s ϕ = i n = exp ( i n ϕ ) E n a n H n ( 2 ) ( ρ ) ,
E s z = n = exp ( i n ϕ ) E n b n H n ( 2 ) ( ρ ) .
( E s E s ) = exp ( i 3 π / 4 ) 2 π k r exp ( ikr ) [ T 1 0 0 T 2 ] ( E 0 E 0 ) ,
i = | T 1 | 2 , i = | T 2 | 2 .
( E j + 1 E j ) × ê r = 0 , ( H j + 1 H j ) × ê r = 0 ,
( E s + E 0 E L ) × ê r = 0 , ( H s + H 0 H L ) × ê r = 0 .
a j n * a j n J n ( m j x j ) ,
b j n * b j n J n ( m j x j ) ,
c j n * c j n H n ( 2 ) ( m j x j ) ,
d j n * d j n H n ( 2 ) ( m j x j ) ,
ā j n a j n * i c j n * ,
b ̅ j n b j n * i d j n * ,
c ̅ j n i c j n * ,
d ̅ j n i d j n * ,
ā j n = A ( j 1 ) n / A L n ,
b ̅ j n = B ( j 1 ) n / B L n ,
c ̅ j n = C ( j 1 ) n / A L n ,
d ̅ j n = D ( j 1 ) n / B L n ,
b n = C L n / A L n ,
a n = D L n / B L n ,
A j n = F j n ( 1 , 3 ) A ( j 1 ) n + F j n ( 3 , 3 ) C ( j 1 ) n Δ n m j R n ( 1 ) ( m j + 1 x j , m j + 1 x j + 1 ) ,
C j n = F j n ( 1 , 1 ) A ( j 1 ) n + F j n ( 3 , 1 ) C ( j 1 ) n Δ n m j R n ( 3 ) ( m j + 1 x j , m j + 1 x j + 1 ) ,
B j n = G j n ( 1 , 3 ) B ( j 1 ) n + G j n ( 3 , 3 ) D ( j 1 ) n Δ n m j R n ( 1 ) ( m j + 1 x j , m j + 1 x j + 1 ) ,
D j n = G j n ( 1 , 1 ) B ( j 1 ) n + G j n ( 3 , 1 ) D ( j 1 ) n Δ n m j R n ( 3 ) ( m j + 1 x j , m j + 1 x j + 1 ) ,
A L n = H n ( 2 ) ( 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 = J 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 = H n ( 2 ) ( 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 = J 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 j n ( ν , κ ) = m j D n ( ν ) ( m j x j ) m j + 1 D n ( κ ) ( m j + 1 x j ) ,
G j n ( ν , κ ) = m j + 1 D n ( ν ) ( m j x j ) m j D n ( κ ) ( m j + 1 x j ) ,
D n ( 1 ) ( ρ ) = J n ( ρ ) / J n ( ρ ) ,
D n ( 3 ) ( ρ ) = H n ( 2 ) ( ρ ) / H n ( 2 ) ( ρ ) ,
R n ( 1 ) ( ρ 1 , ρ 2 ) = J n ( ρ 1 ) / J n ( ρ 2 ) ,
R n ( 3 ) ( ρ 1 , ρ 2 ) = H n ( 2 ) ( ρ 1 ) / H n ( 2 ) ( ρ 2 ) ,
Δ n = D n ( 3 ) ( m j + 1 x j ) D n ( 1 ) ( m j + 1 x j ) ,
A 0 n = B 0 n = 1 , C 0 n = D 0 n = 0 .
D n 1 ( 1 ) ( ρ ) = n 1 ρ [ n ρ + D n ( 1 ) ( ρ ) ] 1 ,
J n ( ρ ) H n ( 2 ) ( ρ ) = J n 1 ( ρ ) H n 1 ( 2 ) ( ρ ) [ n 1 ρ D n 1 ( 1 ) ( ρ ) ] × [ n 1 ρ D n 1 ( 3 ) ( ρ ) ] ,
D n ( 3 ) ( ρ ) = D n ( 1 ) ( ρ ) + 2 [ i π ρ J n ( ρ ) H n ( 2 ) ( ρ ) ] 1 ,
R n ( ν ) ( ρ 1 , ρ 2 ) = R n 1 ( ν ) ( ρ 1 , ρ 2 ) D n ( ν ) ( ρ 2 ) + n / ρ 2 D n ( ν ) ( ρ 1 ) + n / ρ 1 ,
R n ( ν ) ( m j + 1 x j , m j + 1 x j + 1 ) [ 1 + ( x j + 1 x j ) m j ( m j + 1 m j ) x j + 1 / D n ( ν ) ( m j x j ) ] 1 .
i ̅ ( ϕ ) = | T 1 ( ϕ ) | 2 / | T 1 ( 0 ) | 2 , i ̅ ( ϕ ) = | T 2 ( ϕ ) | 2 / | T 2 ( 0 ) | 2 .
E j n = { 1 m j x j n = E n exp ( i n ϕ ) n [ B ( j 1 ) n + D ( j 1 ) n ] / B L n } ê r + { i n E n exp ( i n ϕ ) × [ B ( j 1 ) n D n ( 1 ) ( m j x j ) + D ( j 1 ) n D n ( 3 ) ( m j x j ) ] / B L n } ê ϕ + { n = E n exp ( i n ϕ ) [ A ( j 1 ) n + C ( j 1 ) n ] / A L n } ê z .
m 2 a 1 n J n ( m 1 x 1 ) = m 1 [ a 2 n J n ( m 2 x 1 ) + c 2 n H n ( 2 ) ( m 2 x 1 ) ] , a 1 n J n ( m 1 x 1 ) = a 2 n J n ( m 2 x 1 ) + c 2 n H n ( 2 ) ( m 2 x 1 ) , b 1 n J n ( m 1 x 1 ) = b 2 n J n ( m 2 x 1 ) + d 2 n H n ( 2 ) ( m 2 x 1 ) , m 2 b 1 n J n ( m 1 x 1 ) = m 1 [ b 2 n J n ( m 2 x 1 ) + d 2 n H n ( 2 ) ( m 2 x 1 ) ] ,
m j + 1 [ a j n J n ( m j x j ) + c j n H n ( 2 ) ( m j x j ) ] = m j [ a ( j + 1 ) n J n ( m j + 1 x j ) + c ( j + 1 ) n H n ( 2 ) ( m j + 1 x j ) ] , a j n J n ( m j x j ) + c j n H n ( 2 ) ( m j x j ) = a ( j + 1 ) n J n ( m j + 1 x j ) + c ( j + 1 ) n H n ( 2 ) ( m j + 1 x j ) , b j n J n ( m j x j ) + d j n H n ( 2 ) ( m j x j ) = b ( j + 1 ) n J n ( m j + 1 x j ) + d ( j + 1 ) n H n ( 2 ) ( m j + 1 x j ) , m j + 1 [ b j n J n ( m j x j ) + d j n H n ( 2 ) ( m j x j ) ] = m j [ b ( j + 1 ) n J n ( m j + 1 x j ) + d ( j + 1 ) n H n ( 2 ) ( m j + 1 x j ) ] ,
a L n J n ( m L x L ) + c L n H n ( 2 ) ( m L x L ) = m L [ J n ( x L ) b n H n ( 2 ) ( x L ) ] , a L n J n ( m L x L ) + c L n H n ( 2 ) ( m L x L ) = J n ( x L ) b n H n ( 2 ) ( x L ) , b L n J n ( m L x L ) + d L n H n ( 2 ) ( m L x L ) = J n ( x L ) a n H n ( 2 ) ( x L ) , b L n J n ( m L x L ) + d L n H n ( 2 ) ( m L x L ) = m L [ J n ( x L ) a n H n ( 2 ) ( x L ) ] .
m 2 ā 1 n = m 1 [ ā 2 n R n ( 1 ) ( m 2 x 1 , m 2 x 2 ) + c ̅ 2 n R n ( 3 ) ( m 2 x 1 , m 2 x 2 ) ] , ā 1 n D n ( 1 ) ( m 1 x 1 ) = ā 2 n D n ( 1 ) ( m 2 x 1 ) R n ( 1 ) ( m 2 x 1 , m 2 x 2 ) + c ̅ 2 n D n ( 3 ) ( m 2 x 1 ) R n ( 3 ) ( m 2 x 1 , m 2 x 2 ) , b ̅ 1 n = b ̅ 2 n R n ( 1 ) ( m 2 x 1 , m 2 x 2 ) + d ̅ 2 n R n ( 3 ) ( m 2 x 1 , m 2 x 2 ) , m 2 b ̅ 1 n D n ( 1 ) ( m 1 x 1 ) = m 1 [ b ̅ 2 n D n ( 1 ) ( m 2 x 1 ) R n ( 1 ) ( m 2 x 1 , m 2 x 2 ) + d ̅ 2 n D n ( 3 ) ( m 2 x 1 ) R n ( 3 ) ( m 2 x 1 , m 2 x 2 ) ] ,
m j + 1 ( ā j n + c ̅ j n ) = m j [ ā ( j + 1 ) n R n ( 1 ) ( m j + 1 x j , m j + 1 x j ) + c ̅ ( j + 1 ) n R n ( 3 ) ( m j + 1 x j , m j + 1 x j ) ] , ā j n D n ( 1 ) ( m j x j ) + c ̅ j n D n ( 3 ) ( m j x j ) = ā ( j + 1 ) n D n ( 1 ) ( m j + 1 x j ) R n ( 1 ) ( m j + 1 x j , m j + 1 x j + 1 ) + c ̅ ( j + 1 ) n D n ( 3 ) ( m j + 1 x j ) R n ( 3 ) ( m j + 1 x j , m j + 1 x j + 1 ) , b ̅ j n + d ̅ j n = b ̅ ( j + 1 ) n R n ( 1 ) ( m j + 1 x j , m j + 1 x j + 1 ) + c ̅ ( j + 1 ) n R n ( 3 ) ( m j + 1 x j , m j + 1 x j + 1 ) , m j + 1 [ b ̅ j n D n ( 1 ) ( m j x j ) + d ̅ j m D n ( 3 ) ( m j x j ) ] = m j [ b ̅ j n D n ( 1 ) ( m j + 1 x j ) R n ( 1 ) ( m j + 1 x j , m j + 1 x j ) + d ̅ j n D n ( 3 ) ( m j + 1 x j ) R n ( 3 ) ( m j + 1 x j , m j + 1 x j ) ] ,
ā L n + c ̅ L n = m L [ ψ n ( x L ) b n ξ n ( x L ) ] , ā L n D n ( 1 ) ( m L x L ) + c ̅ L n D n ( 3 ) ( m L x L ) = ψ n ( x L ) b n ξ n ( x L ) , b ̅ L n + d ̅ L n = ψ n ( x L ) a n ξ n ( x L ) , b ̅ L n D n ( 1 ) ( m L x L ) + d ̅ L n D n ( 2 ) ( m L x L ) = m L [ ψ n ( x L ) a n ξ n ( x L ) .
ā 2 n = ā 1 n [ m 2 D n ( 1 ) ( m 2 x 1 ) m 1 D n ( 3 ) ( m 1 x 1 ) ] m 1 R n ( 1 ) ( m 2 x 1 , m 2 x 2 ) [ D n ( 3 ) ( m 2 x 1 ) D n ( 1 ) ( m 2 x 1 ) ] , c ̅ 2 n = + ā 1 n [ m 1 D n ( 1 ) ( m 1 x 1 ) m 2 D n ( 1 ) ( m 2 x 1 ) ] m 1 R n ( 3 ) ( m 2 x 1 , m 2 x 2 ) [ D n ( 3 ) ( m 2 x 1 ) D n ( 1 ) ( m 2 x 1 ) ] , b ̅ 2 n = b ̅ 1 n [ m 2 D n ( 1 ) ( m 1 x 1 ) m 1 D n ( 3 ) ( m 2 x 1 ) ] m 1 R n ( 1 ) ( m 2 x 1 , m 2 x 2 ) [ D n ( 3 ) ( m 2 x 1 ) D n ( 1 ) ( m 2 x 1 ) ] , d ̅ 2 n = + b ̅ 1 n [ m 2 D n ( 1 ) ( m 1 x 1 ) m 1 D n ( 1 ) ( m 2 x 1 ) ] m 1 R n ( 3 ) ( m 2 x 1 , m 2 x 2 ) [ D n ( 3 ) ( m 2 x 1 ) D n ( 1 ) ( m 2 x 1 ) ] ,
ā ( j + 1 ) n = [ m j D n ( 1 ) ( m j x j ) m j + 1 D n ( 3 ) ( m j + 1 x j ) ] ā j n + [ m j D n ( 3 ) ( m j x j ) m j + 1 D n ( 3 ) ( m j + 1 x j ) ] c ̅ j n m j R n ( 1 ) ( m j + 1 x j , m j + 1 x j + 1 ) [ D n ( 3 ) ( m j + 1 x j ) D n ( 1 ) ( m j + 1 x j ) ] , c ̅ ( j + 1 ) n = + [ m j D n ( 1 ) ( m j x j ) m j + 1 D n ( 1 ) ( m j + 1 x j ) ] ā j n + [ m j D n ( 3 ) ( m j x j ) m j + 1 D n ( 1 ) ( m j + 1 x j ) ] c ̅ j n m j R n ( 3 ) ( m j + 1 x j , m j + 1 x j + 1 ) [ D n ( 3 ) ( m j + 1 x j ) D n ( 1 ) ( m j + 1 x j ) ] , b ̅ ( j + 1 ) n = [ m j th D n ( 1 ) ( m j x j ) m j D n ( 3 ) ( m j + 1 x j ) ] b ̅ j n + [ m j + 1 D n ( 3 ) ( m j x j ) m j D n ( 3 ) ( m j + 1 x j ) ] d ̅ j n m j R n ( 1 ) ( m j + 1 x j , m j + 1 x j + 1 ) [ D n ( 3 ) ( m j + 1 x j ) D n ( 1 ) ( m j + 1 x j ) ] , d ̅ ( j + 1 ) n = + [ m j + 1 D n ( 1 ) ( m j x j ) m j D n ( 1 ) ( m j + 1 x j ) ] b ̅ j n + [ m j + 1 D n ( 3 ) ( m j x j ) m j D n ( 1 ) ( m j + 1 x j ) ] d ̅ j n m j R n ( 3 ) ( m j + 1 x j , m j + 1 x j + 1 ) [ D n ( 3 ) ( m j + 1 x j ) D n ( 1 ) ( m j + 1 x j ) ] ,
b n = m L J n ( x L ) ā L n c ̅ L n m L H n ( 2 ) ( x L ) = J n ( x L ) ā L n D n ( 1 ) ( m L x L ) c ̅ L n D n ( 3 ) ( m L x L ) H n ( 2 ) ( x L ) , a n = J n ( x L ) b ̅ L n d ̅ L n H n ( 2 ) ( x L ) = m L J n ( x L ) b ̅ L n D n ( 1 ) ( m L x L ) d ̅ L n D n ( 3 ) ( m L x L ) m L H n ( 2 ) ( x L ) .
1 ā 1 n = H n ( 2 ) ( x L ) i m L × { [ m L D n ( 1 ) ( m L x L ) m L + 1 D n ( 3 ) ( m L + 1 x L ) ] ā L n ā 1 n + [ m L D n ( 3 ) ( m L x L ) m L + 1 D n ( 3 ) ( m L + 1 x L ) ] c ̅ L n ā 1 n } , 1 b ̅ 1 n = H n ( 2 ) ( x L ) i m L × { [ m L + 1 D n ( 1 ) ( m L x L ) m L D n ( 3 ) ( m L + 1 x L ) ] b ̅ L n b ̅ 1 n + [ m L + 1 D n ( 3 ) ( m L x L ) m L D n ( 3 ) ( m L + 1 x L ) ] d ̅ L n b ̅ 1 n } ,
b n = ā 1 n J n ( x L ) i m L × { [ m L D n ( 1 ) ( m L x L ) m L + 1 D n ( 1 ) ( m L + 1 x L ) ] ā L n ā 1 n + [ m L D n ( 3 ) ( m L x L ) m L + 1 D n ( 1 ) ( m L + 1 ) ] c ̅ L n ā 1 n } , a n = b ̅ 1 n J n ( x L ) i m L × { [ m L + 1 D n ( 1 ) ( m L x L ) m L D n ( 1 ) ( m L + 1 x L ) ] b ̅ L n b ̅ 1 n + [ m L + 1 D n ( 3 ) ( m L x L ) m L D n ( 1 ) ( m L + 1 ) ] d ̅ L n d ̅ 1 n } .
A L n = 1 / ā 1 n B L n = 1 / b ̅ 1 n C L n = b n A L n , D L n = a n B L n ,
ā j n = A ( j 1 ) n / A L n , c ̅ j n = C ( j 1 ) n / A L n , b ̅ j n = B ( j 1 ) n / B L n , d ̅ j n = D ( j 1 ) n / B L n ,
A 0 n = B 0 n = 1 , C 0 n = D 0 n = 0 .

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