Abstract

A mathematical solution of the wave equation for coaxial fibers having four different refractive-index profiles is presented. The transcendental equations are obtained under LP approximation and calculated for comparison of HEmn-mode-dispersion characteristics. Attention is paid to the HE11 and HE12 modes because of their importance for modeling directional couplers, and calculations are carried out to obtain dispersion dependence on dimensional parameters of the fibers. The field expressions also are given, and the spatial distributions of the HE11 and HE12 modal fields of all coaxial structures for different conditions of propagation are calculated.

© 1996 Optical Society of America

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  1. J. R. Cozens, A. C. Boucouvalas, “Coaxial optical coupler,” Electron. Lett. 18, 138–140 (1982).
    [Crossref]
  2. J. R. Cozens, A. C. Boucouvalas, A. Al-Assam, M. J. Lee, D. G. Morris, “Optical coupling in coaxial fibres,” Electron. Lett. 18, 679–681 (1982).
    [Crossref]
  3. D. Papageorgiou, A. C. Boucouvalas, “Propagation constant of cylindrical dielectric waveguides with arbitrary refractive index profile using resonance technique,” Electron. Lett. 18, 788–796 (1982).
    [Crossref]
  4. A. C. Boucouvalas, “Coaxial optical fiber coupling,” IEEE J. Lightwave Technol. LT-3, 1151–1158 (1985).
    [Crossref]
  5. A. C. Boucouvalas, “Mode cutoff frequencies of coaxial optical couplers,” Opt. Lett. 10, 95–97 (1985).
    [Crossref] [PubMed]
  6. A. C. Boucouvalas, G. Georgiou, “Biconical taper coaxial optical fibre coupler,” Electron. Lett. 21, 864–865 (1985).
    [Crossref]
  7. A. C. Boucouvalas, G. Georgiou, “Biconical taper coaxial coupler filter,” Electron. Lett. 21, 1033–1034 (1985).
    [Crossref]
  8. A. C. Boucouvalas, G. Georgiou, “Concatenated tapered coaxial coupler filters,” IEE Proc. 134, 191–195 (1987).
  9. Y. W. Li, C. D. Hussey, T. A. Birks, “Triple-clad single mode fibers for dispersion shifting,” J. Lightwave Technol. 11, 1812–1819 (1993).
    [Crossref]
  10. J. D. Love, W. M. Henry, W. J. Stwewart, R. J. Black, S. Lacroix, F. Gonthier, “Tapered single-mode fibres and devices. Part I: Adiabatic criteria,” IEE Proc. 138, 343–354 (1991).
  11. S. Celaschi, J. T. Jesus, D. Dini, A. Juriollo, R. Arradi, “Narrow band all-fiber transmission spectral filters,” in Proceedings of the International Workshop on Advanced Materials for Multifunctional Waveguides (Ceramic Society Optoelectronic Industries and Technology Development Association, Chiba, Japan, 1995), pp. 76–79.
  12. M. J. Adams, An Introduction to Optical Waveguides (Wiley, New York, 1981), pp. 228–233.
  13. C. A. de S. Melo, “Study and modeling of coaxial fibers,” M.S. thesis (Department of Physics, Federal University of Ceara, Campus do Pici, Fortaleza, 60.550-476, Ceará, Brazil, 1994).
  14. S. Kawakami, S. Nishida, “Characteristics of doubly clad optical fiber with low-index inner cladding,” IEEE J. Quantum Electron. QE-10, 879–887 (1974).
    [Crossref]
  15. M. Abramowitz, Handbook of Mathematical Functions (Dover, New York, 1965), pp. 355–433.

1993 (1)

Y. W. Li, C. D. Hussey, T. A. Birks, “Triple-clad single mode fibers for dispersion shifting,” J. Lightwave Technol. 11, 1812–1819 (1993).
[Crossref]

1991 (1)

J. D. Love, W. M. Henry, W. J. Stwewart, R. J. Black, S. Lacroix, F. Gonthier, “Tapered single-mode fibres and devices. Part I: Adiabatic criteria,” IEE Proc. 138, 343–354 (1991).

1987 (1)

A. C. Boucouvalas, G. Georgiou, “Concatenated tapered coaxial coupler filters,” IEE Proc. 134, 191–195 (1987).

1985 (4)

A. C. Boucouvalas, “Coaxial optical fiber coupling,” IEEE J. Lightwave Technol. LT-3, 1151–1158 (1985).
[Crossref]

A. C. Boucouvalas, “Mode cutoff frequencies of coaxial optical couplers,” Opt. Lett. 10, 95–97 (1985).
[Crossref] [PubMed]

A. C. Boucouvalas, G. Georgiou, “Biconical taper coaxial optical fibre coupler,” Electron. Lett. 21, 864–865 (1985).
[Crossref]

A. C. Boucouvalas, G. Georgiou, “Biconical taper coaxial coupler filter,” Electron. Lett. 21, 1033–1034 (1985).
[Crossref]

1982 (3)

J. R. Cozens, A. C. Boucouvalas, “Coaxial optical coupler,” Electron. Lett. 18, 138–140 (1982).
[Crossref]

J. R. Cozens, A. C. Boucouvalas, A. Al-Assam, M. J. Lee, D. G. Morris, “Optical coupling in coaxial fibres,” Electron. Lett. 18, 679–681 (1982).
[Crossref]

D. Papageorgiou, A. C. Boucouvalas, “Propagation constant of cylindrical dielectric waveguides with arbitrary refractive index profile using resonance technique,” Electron. Lett. 18, 788–796 (1982).
[Crossref]

1974 (1)

S. Kawakami, S. Nishida, “Characteristics of doubly clad optical fiber with low-index inner cladding,” IEEE J. Quantum Electron. QE-10, 879–887 (1974).
[Crossref]

Abramowitz, M.

M. Abramowitz, Handbook of Mathematical Functions (Dover, New York, 1965), pp. 355–433.

Adams, M. J.

M. J. Adams, An Introduction to Optical Waveguides (Wiley, New York, 1981), pp. 228–233.

Al-Assam, A.

J. R. Cozens, A. C. Boucouvalas, A. Al-Assam, M. J. Lee, D. G. Morris, “Optical coupling in coaxial fibres,” Electron. Lett. 18, 679–681 (1982).
[Crossref]

Arradi, R.

S. Celaschi, J. T. Jesus, D. Dini, A. Juriollo, R. Arradi, “Narrow band all-fiber transmission spectral filters,” in Proceedings of the International Workshop on Advanced Materials for Multifunctional Waveguides (Ceramic Society Optoelectronic Industries and Technology Development Association, Chiba, Japan, 1995), pp. 76–79.

Birks, T. A.

Y. W. Li, C. D. Hussey, T. A. Birks, “Triple-clad single mode fibers for dispersion shifting,” J. Lightwave Technol. 11, 1812–1819 (1993).
[Crossref]

Black, R. J.

J. D. Love, W. M. Henry, W. J. Stwewart, R. J. Black, S. Lacroix, F. Gonthier, “Tapered single-mode fibres and devices. Part I: Adiabatic criteria,” IEE Proc. 138, 343–354 (1991).

Boucouvalas, A. C.

A. C. Boucouvalas, G. Georgiou, “Concatenated tapered coaxial coupler filters,” IEE Proc. 134, 191–195 (1987).

A. C. Boucouvalas, G. Georgiou, “Biconical taper coaxial coupler filter,” Electron. Lett. 21, 1033–1034 (1985).
[Crossref]

A. C. Boucouvalas, “Coaxial optical fiber coupling,” IEEE J. Lightwave Technol. LT-3, 1151–1158 (1985).
[Crossref]

A. C. Boucouvalas, “Mode cutoff frequencies of coaxial optical couplers,” Opt. Lett. 10, 95–97 (1985).
[Crossref] [PubMed]

A. C. Boucouvalas, G. Georgiou, “Biconical taper coaxial optical fibre coupler,” Electron. Lett. 21, 864–865 (1985).
[Crossref]

J. R. Cozens, A. C. Boucouvalas, “Coaxial optical coupler,” Electron. Lett. 18, 138–140 (1982).
[Crossref]

J. R. Cozens, A. C. Boucouvalas, A. Al-Assam, M. J. Lee, D. G. Morris, “Optical coupling in coaxial fibres,” Electron. Lett. 18, 679–681 (1982).
[Crossref]

D. Papageorgiou, A. C. Boucouvalas, “Propagation constant of cylindrical dielectric waveguides with arbitrary refractive index profile using resonance technique,” Electron. Lett. 18, 788–796 (1982).
[Crossref]

Celaschi, S.

S. Celaschi, J. T. Jesus, D. Dini, A. Juriollo, R. Arradi, “Narrow band all-fiber transmission spectral filters,” in Proceedings of the International Workshop on Advanced Materials for Multifunctional Waveguides (Ceramic Society Optoelectronic Industries and Technology Development Association, Chiba, Japan, 1995), pp. 76–79.

Cozens, J. R.

J. R. Cozens, A. C. Boucouvalas, “Coaxial optical coupler,” Electron. Lett. 18, 138–140 (1982).
[Crossref]

J. R. Cozens, A. C. Boucouvalas, A. Al-Assam, M. J. Lee, D. G. Morris, “Optical coupling in coaxial fibres,” Electron. Lett. 18, 679–681 (1982).
[Crossref]

de S. Melo, C. A.

C. A. de S. Melo, “Study and modeling of coaxial fibers,” M.S. thesis (Department of Physics, Federal University of Ceara, Campus do Pici, Fortaleza, 60.550-476, Ceará, Brazil, 1994).

Dini, D.

S. Celaschi, J. T. Jesus, D. Dini, A. Juriollo, R. Arradi, “Narrow band all-fiber transmission spectral filters,” in Proceedings of the International Workshop on Advanced Materials for Multifunctional Waveguides (Ceramic Society Optoelectronic Industries and Technology Development Association, Chiba, Japan, 1995), pp. 76–79.

Georgiou, G.

A. C. Boucouvalas, G. Georgiou, “Concatenated tapered coaxial coupler filters,” IEE Proc. 134, 191–195 (1987).

A. C. Boucouvalas, G. Georgiou, “Biconical taper coaxial coupler filter,” Electron. Lett. 21, 1033–1034 (1985).
[Crossref]

A. C. Boucouvalas, G. Georgiou, “Biconical taper coaxial optical fibre coupler,” Electron. Lett. 21, 864–865 (1985).
[Crossref]

Gonthier, F.

J. D. Love, W. M. Henry, W. J. Stwewart, R. J. Black, S. Lacroix, F. Gonthier, “Tapered single-mode fibres and devices. Part I: Adiabatic criteria,” IEE Proc. 138, 343–354 (1991).

Henry, W. M.

J. D. Love, W. M. Henry, W. J. Stwewart, R. J. Black, S. Lacroix, F. Gonthier, “Tapered single-mode fibres and devices. Part I: Adiabatic criteria,” IEE Proc. 138, 343–354 (1991).

Hussey, C. D.

Y. W. Li, C. D. Hussey, T. A. Birks, “Triple-clad single mode fibers for dispersion shifting,” J. Lightwave Technol. 11, 1812–1819 (1993).
[Crossref]

Jesus, J. T.

S. Celaschi, J. T. Jesus, D. Dini, A. Juriollo, R. Arradi, “Narrow band all-fiber transmission spectral filters,” in Proceedings of the International Workshop on Advanced Materials for Multifunctional Waveguides (Ceramic Society Optoelectronic Industries and Technology Development Association, Chiba, Japan, 1995), pp. 76–79.

Juriollo, A.

S. Celaschi, J. T. Jesus, D. Dini, A. Juriollo, R. Arradi, “Narrow band all-fiber transmission spectral filters,” in Proceedings of the International Workshop on Advanced Materials for Multifunctional Waveguides (Ceramic Society Optoelectronic Industries and Technology Development Association, Chiba, Japan, 1995), pp. 76–79.

Kawakami, S.

S. Kawakami, S. Nishida, “Characteristics of doubly clad optical fiber with low-index inner cladding,” IEEE J. Quantum Electron. QE-10, 879–887 (1974).
[Crossref]

Lacroix, S.

J. D. Love, W. M. Henry, W. J. Stwewart, R. J. Black, S. Lacroix, F. Gonthier, “Tapered single-mode fibres and devices. Part I: Adiabatic criteria,” IEE Proc. 138, 343–354 (1991).

Lee, M. J.

J. R. Cozens, A. C. Boucouvalas, A. Al-Assam, M. J. Lee, D. G. Morris, “Optical coupling in coaxial fibres,” Electron. Lett. 18, 679–681 (1982).
[Crossref]

Li, Y. W.

Y. W. Li, C. D. Hussey, T. A. Birks, “Triple-clad single mode fibers for dispersion shifting,” J. Lightwave Technol. 11, 1812–1819 (1993).
[Crossref]

Love, J. D.

J. D. Love, W. M. Henry, W. J. Stwewart, R. J. Black, S. Lacroix, F. Gonthier, “Tapered single-mode fibres and devices. Part I: Adiabatic criteria,” IEE Proc. 138, 343–354 (1991).

Morris, D. G.

J. R. Cozens, A. C. Boucouvalas, A. Al-Assam, M. J. Lee, D. G. Morris, “Optical coupling in coaxial fibres,” Electron. Lett. 18, 679–681 (1982).
[Crossref]

Nishida, S.

S. Kawakami, S. Nishida, “Characteristics of doubly clad optical fiber with low-index inner cladding,” IEEE J. Quantum Electron. QE-10, 879–887 (1974).
[Crossref]

Papageorgiou, D.

D. Papageorgiou, A. C. Boucouvalas, “Propagation constant of cylindrical dielectric waveguides with arbitrary refractive index profile using resonance technique,” Electron. Lett. 18, 788–796 (1982).
[Crossref]

Stwewart, W. J.

J. D. Love, W. M. Henry, W. J. Stwewart, R. J. Black, S. Lacroix, F. Gonthier, “Tapered single-mode fibres and devices. Part I: Adiabatic criteria,” IEE Proc. 138, 343–354 (1991).

Electron. Lett. (5)

J. R. Cozens, A. C. Boucouvalas, “Coaxial optical coupler,” Electron. Lett. 18, 138–140 (1982).
[Crossref]

J. R. Cozens, A. C. Boucouvalas, A. Al-Assam, M. J. Lee, D. G. Morris, “Optical coupling in coaxial fibres,” Electron. Lett. 18, 679–681 (1982).
[Crossref]

D. Papageorgiou, A. C. Boucouvalas, “Propagation constant of cylindrical dielectric waveguides with arbitrary refractive index profile using resonance technique,” Electron. Lett. 18, 788–796 (1982).
[Crossref]

A. C. Boucouvalas, G. Georgiou, “Biconical taper coaxial optical fibre coupler,” Electron. Lett. 21, 864–865 (1985).
[Crossref]

A. C. Boucouvalas, G. Georgiou, “Biconical taper coaxial coupler filter,” Electron. Lett. 21, 1033–1034 (1985).
[Crossref]

IEE Proc. (2)

A. C. Boucouvalas, G. Georgiou, “Concatenated tapered coaxial coupler filters,” IEE Proc. 134, 191–195 (1987).

J. D. Love, W. M. Henry, W. J. Stwewart, R. J. Black, S. Lacroix, F. Gonthier, “Tapered single-mode fibres and devices. Part I: Adiabatic criteria,” IEE Proc. 138, 343–354 (1991).

IEEE J. Lightwave Technol. (1)

A. C. Boucouvalas, “Coaxial optical fiber coupling,” IEEE J. Lightwave Technol. LT-3, 1151–1158 (1985).
[Crossref]

IEEE J. Quantum Electron. (1)

S. Kawakami, S. Nishida, “Characteristics of doubly clad optical fiber with low-index inner cladding,” IEEE J. Quantum Electron. QE-10, 879–887 (1974).
[Crossref]

J. Lightwave Technol. (1)

Y. W. Li, C. D. Hussey, T. A. Birks, “Triple-clad single mode fibers for dispersion shifting,” J. Lightwave Technol. 11, 1812–1819 (1993).
[Crossref]

Opt. Lett. (1)

Other (4)

M. Abramowitz, Handbook of Mathematical Functions (Dover, New York, 1965), pp. 355–433.

S. Celaschi, J. T. Jesus, D. Dini, A. Juriollo, R. Arradi, “Narrow band all-fiber transmission spectral filters,” in Proceedings of the International Workshop on Advanced Materials for Multifunctional Waveguides (Ceramic Society Optoelectronic Industries and Technology Development Association, Chiba, Japan, 1995), pp. 76–79.

M. J. Adams, An Introduction to Optical Waveguides (Wiley, New York, 1981), pp. 228–233.

C. A. de S. Melo, “Study and modeling of coaxial fibers,” M.S. thesis (Department of Physics, Federal University of Ceara, Campus do Pici, Fortaleza, 60.550-476, Ceará, Brazil, 1994).

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Figures (8)

Fig. 1
Fig. 1

Refractive-index profiles of four coaxial fibers: (a) WI, (b) WII, (c) MI, (d) MII.

Fig. 2
Fig. 2

Universal dispersion curves for different coaxial fibers: (a) WI, (b) WII, (c) MI, (d) MII.

Fig. 3
Fig. 3

Influence of the parameter P on dispersion curves of the HE11 mode of coaxial fibers: (a) WI, (b) WII, (c) MI, (d) MII.

Fig. 4
Fig. 4

Influence of the parameter P on dispersion curves of the HE12 mode of coaxial fibers: (a) WI, (b) WII, (c) MI, (d) MII.

Fig. 5
Fig. 5

Influence of the parameter Q on dispersion curves of the HE11 mode of coaxial fibers: (a) WI, (b) WII, (c) MI, (d) MII.

Fig. 6
Fig. 6

Influence of the parameter Q on dispersion curves of the HE12 mode of coaxial fibers: (a) WI, (b) WII, (c) MI, (d) MII.

Fig. 7
Fig. 7

Field distributions of the HE11 modes in coaxial fiber MI for different points of the universal dispersion curves.

Fig. 8
Fig. 8

Field distributions of the HE12 modes in coaxial fiber MII for different points of the universal dispersion curves.

Tables (6)

Tables Icon

Table 1 Waveguide Solutions of Structures WI and WII for the Different Regions and Effective-Refractive-Index Ranges with Their Transversal Propagation Constant

Tables Icon

Table 2 Transcendental Equations for Structures WI and WII in Different Ranges of ne

Tables Icon

Table 3 Waveguide Solutions and Adequate Transversal Propagation Constants for Each Region and Effective-Refractive-Index Range of Structures MI and MII

Tables Icon

Table 4 Transcendental Equations for Structures MI and MII

Tables Icon

Table 5 Expressions of the Coefficients (A–F) Corresponding to Structures WI and WII in the Effective-Refractive-Index Range

Tables Icon

Table 6 Expressions of the Coefficients (A–F) as in Table 5 but for the Other Ranges of All the Structures

Equations (48)

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0 < ρ < a rod n = n 1 , a < ρ < b gap n = n 2 , b < ρ < c tube n = n 3 , ρ > c clad n = n 4 ,
ψ = φ ( ρ , θ ) exp ( i β z ) ,
φ ( ρ , θ ) = R ( ρ ) [ cos ( m θ ) sin ( m θ ) ] , m = 0 , 1 , 2 ,
n e = β k 0 ,
W 3 = W 2 ( b a ) ,
U 4 = U 3 ( c b ) .
P = b c , Q = a c .
υ = ( n 1 2 n 4 2 ) 1 / 2 k 0 c , B = n e 2 n 4 2 n 1 2 n 4 2 for W structures , υ = ( n 3 2 n 4 2 ) 1 / 2 k 0 c , B = n e 2 n 4 2 n 3 2 n 4 2 for M structures .
A J m ( U 1 ρ a ) cos ( m θ ) U 1 = ( k 0 2 n 1 2 β 2 ) 1 / 2 a
A J m ( U 1 ρ a ) cos ( m θ ) U 1 = ( k 0 2 n 1 2 β 2 ) 1 / 2 a
A J m ( U 1 ρ a ) cos ( m θ ) U 1 = ( k 0 2 n 1 2 β 2 ) 1 / 2 a
[ B J m ( W 2 ρ a ) + C Y m ( W 2 ρ a ) ] cos ( m θ ) W 2 = ( k 0 2 n 2 2 β 2 ) 1 / 2 a
[ B I m ( W 2 ρ a ) + C K m ( W 2 ρ a ) ] cos ( m θ ) W 2 = ( k 0 2 n 2 2 β 2 ) 1 / 2 a
[ B I m ( W 2 ρ a ) + C K m ( W 2 ρ a ) ] cos ( m θ ) W 2 = ( β 2 k 0 2 n 2 2 ) 1 / 2 a
[ D J m ( U 3 ρ b ) + E Y m ( U 3 ρ b ) ] cos ( m θ ) , U 3 = ( k 0 2 n 3 2 β 2 ) 1 / 2 b
[ D J m ( U 3 ρ b ) + E Y m ( U 3 ρ b ) ] cos ( m θ ) U 3 = ( k 0 2 n 3 2 β 2 ) 1 / 2 b
[ D I m ( U 3 ρ b ) + E K m ( U 3 ρ b ) ] cos ( m θ ) U 3 = ( β 2 k 0 2 n 3 2 ) 1 / 2 b
F K m ( W 4 ρ b ) cos ( m θ ) W 4 = ( β 2 k 0 2 n 4 2 ) 1 / 2 c
F K m ( W 4 ρ b ) cos ( m θ ) W 4 = ( β 2 k 0 2 n 4 2 ) 1 / 2 c
F K m ( W 4 ρ b ) cos ( m θ ) W 4 = ( β 2 k 0 2 n 4 2 ) 1 / 2 c
J m ± 1 ( U 4 ) Y m ± 1 ( U 4 ) [ J ( U 4 ) K ( W 4 ) Y ( U 4 ) K ( W 4 ) ] { Y m ± 1 ( W 3 ) Y m ± 1 ( W 2 ) [ Y ( U 3 ) Y ( W 3 ) J ( U 1 ) Y ( W 2 ) ] J m ± 1 ( W 3 ) J m ± 1 ( W 2 ) [ Y ( U 3 ) J ( W 3 ) J ( U 1 ) J ( W 2 ) ] }
= J m ± 1 ( U 3 ) Y m ± 1 ( U 3 ) { Y m ± 1 ( W 3 ) Y m ± 1 ( W 2 ) [ J ( U 3 ) Y ( W 3 ) J ( U 1 ) Y ( W 2 ) ] J m ± 1 ( W 3 ) J m ± 1 ( W 2 ) [ J ( U 3 ) J ( W 3 ) J ( U 1 ) J ( W 2 ) ] }
J m ± 1 ( U 4 ) Y m ± 1 ( U 4 ) [ J ( U 4 ) K ( W 4 ) Y ( U 4 ) K ( W 4 ) ] { ± K m ± 1 ( W 3 ) K m ± 1 ( W 2 ) [ Y ( U 3 ) K ( W 3 ) J ( U 1 ) K ( W 2 ) ] I m ± 1 ( W 3 ) I m ± 1 ( W 2 ) [ Y ( U 3 ) ± I ( W 3 ) J ( U 1 ) ± I ( W 2 ) ] }
= J m ± 1 ( U 3 ) Y m ± 1 ( U 3 ) { K m ± 1 ( W 3 ) K m ± 1 ( W 2 ) [ ± J ( U 3 ) K ( W 3 ) J ( U 1 ) + K ( W 2 ) ] I m ± 1 ( W 3 ) I m ± 1 ( W 2 ) [ J ( U 3 ) ± I ( W 3 ) J ( U 1 ) ± I ( W 2 ) ] }
I m ± 1 ( U 4 ) K m ± 1 ( U 4 ) [ I ( U 4 ) + K ( W 4 ) K ( U 4 ) K ( W 4 ) ] { K m ± 1 ( W 3 ) K m ± 1 ( W 2 ) [ K ( W 3 ) K ( U 3 ) K ( W 2 ) J ( U 1 ) ] + I m ± 1 ( W 3 ) I m ± 1 ( W 2 ) [ I ( W 3 ) K ( U 3 ) I ( W 2 ) J ( U 1 ) ] }
= I m ± 1 ( U 3 ) K m ± 1 ( U 3 ) { K m ± 1 ( W 3 ) K m ± 1 ( W 2 ) [ K ( W 3 ) I ( U 3 ) K ( W 2 ) ± J ( U 1 ) ] + I m ± 1 ( W 3 ) I m ± 1 ( W 2 ) [ I ( W 3 ) I ( U 3 ) I ( W 2 ) ± J ( U 1 ) ] }
A J m ( U 1 ρ a ) cos ( m θ ) U 1 = ( k 0 2 n 1 2 β 2 ) 1 / 2 a
A J m ( U 1 ρ a ) cos ( m θ ) U 1 = ( k 0 2 n 1 2 β 2 ) 1 / 2 a
A I m ( U 1 ρ a ) cos ( m θ ) U 1 = ( β 2 k 0 2 n 1 2 ) 1 / 2 a
[ B J m ( W 2 ρ a ) + C Y m ( W 2 ρ a ) ] cos ( m θ ) W 2 = ( k 0 2 n 2 2 β 2 ) 1 / 2 a
[ B I m ( W 2 ρ a ) + C K m ( W 2 ρ a ) ] cos ( m θ ) W 2 = ( β 2 k 0 2 n 2 2 ) 1 / 2 a
[ B I m ( W 2 ρ a ) + C K m ( W 2 ρ a ) ] cos ( m θ ) W 2 = ( β 2 k 0 2 n 2 2 ) 1 / 2 a
[ D J m ( U 3 ρ b ) + E Y m ( U 3 ρ b ) ] cos ( m θ ) , U 3 = ( k 0 2 n 3 2 β 2 ) 1 / 2 b
[ D J m ( U 3 ρ b ) + E Y m ( U 3 ρ b ) ] cos ( m θ ) U 3 = ( k 0 2 n 3 2 β 2 ) 1 / 2 b
[ D J m ( U 3 ρ b ) + E Y m ( U 3 ρ b ) ] cos ( m θ ) U 3 = ( k 0 2 n 3 2 β 2 ) 1 / 2 b
F K m ( W 4 ρ c ) cos ( m θ ) W 4 = ( β 2 k 0 2 n 4 2 ) 1 / 2 c
F K m ( W 4 ρ c ) cos ( m θ ) W 4 = ( β 2 k 0 2 n 4 2 ) 1 / 2 c
F K m ( W 4 ρ c ) cos ( m θ ) W 4 = ( β 2 k 0 2 n 4 2 ) 1 / 2 c
J m ± 1 ( U 4 ) Y m ± 1 ( U 4 ) [ J ( U 4 ) K ( W 4 ) Y ( U 4 ) K ( W 4 ) ] { Y m ± 1 ( W 3 ) Y m ± 1 ( W 2 ) [ Y ( U 3 ) Y ( W 3 ) J ( U 1 ) Y ( W 2 ) ] J m ± 1 ( W 3 ) J m ± 1 ( W 2 ) [ Y ( U 3 ) J ( W 3 ) J ( U 1 ) J ( W 2 ) ] }
= J m ± 1 ( U 3 ) Y m ± 1 ( U 3 ) { Y m ± 1 ( W 3 ) Y m ± 1 ( W 2 ) [ J ( U 3 ) Y ( W 3 ) J ( U 1 ) Y ( W 2 ) ] J m ± 1 ( W 3 ) J m ± 1 ( W 2 ) [ J ( U 3 ) J ( W 3 ) J ( U 1 ) J ( W 2 ) ] }
J m ± 1 ( U 4 ) Y m ± 1 ( U 4 ) [ J ( U 4 ) K ( W 4 ) Y ( U 4 ) K ( W 4 ) ] { ± K m ± 1 ( W 3 ) K m ± 1 ( W 2 ) [ Y ( U 3 ) K ( W 3 ) J ( U 1 ) K ( W 2 ) ] I m ± 1 ( W 3 ) I m ± 1 ( W 2 ) [ Y ( U 3 ) ± I ( W 3 ) J ( U 1 ) ± I ( W 2 ) ] }
= J m ± 1 ( U 3 ) Y m ± 1 ( U 3 ) { K m ± 1 ( W 3 ) K m ± 1 ( W 2 ) [ ± J ( U 3 ) K ( W 3 ) J ( U 1 ) + K ( W 2 ) ] I m ± 1 ( W 3 ) I m ± 1 ( W 2 ) [ J ( U 3 ) ± I ( W 3 ) J ( U 1 ) ± I ( W 2 ) ] }
J m ± 1 ( U 4 ) Y m ± 1 ( U 4 ) [ J ( U 4 ) K ( W 4 ) Y ( U 4 ) K ( W 4 ) ] { K m ± 1 ( W 3 ) K m ± 1 ( W 2 ) [ ± Y ( U 3 ) K ( W 3 ) I ( U 1 ) + K ( W 1 ) ] I m ± 1 ( W 3 ) I m ± 1 ( W 2 ) [ ± Y ( U 3 ) + I ( W 3 ) I ( U 1 ) I ( W 2 ) ] }
= J m ± 1 ( U 3 ) Y m ± 1 ( U 3 ) { K m ± 1 ( W 3 ) K m ± 1 ( W 2 ) [ ± J ( U 3 ) K ( W 3 ) I ( U 1 ) + K ( W 2 ) ] I m ± 1 ( W 3 ) I m ± 1 ( W 2 ) [ ± J ( U 3 ) + I ( W 3 ) I ( U 1 ) I ( W 2 ) ] }
B = W 2 K m ± 1 ( W 2 ) J m ( U 1 ) U 1 J m ± 1 ( U 1 ) K m ( W 2 ) W 2 [ K m ± 1 ( W 2 ) I m ( W 2 ) + I m ± 1 ( W 2 ) K m ( W 2 ) ] C = W 2 I m ± 1 ( W 2 ) J m ( U 1 ) ± U 1 J m ± 1 ( U 1 ) I m ( W 2 ) W 2 [ I m ± 1 ( W 2 ) K m ( W 2 ) + K m ± 1 ( W 2 ) I m ( W 2 ) ] D = [ U 3 K m ± 1 ( U 3 ) I m ( W 3 ) + W 3 I m ± 1 ( W 3 ) K m ( U 3 ) ] B + [ U 3 K m ± 1 ( U 3 ) K m ( W 3 ) W 3 K m ± 1 ( W 3 ) K m ( U 3 ) ] C U 3 [ K m ± 1 ( U 3 ) I m ( U 3 ) + I m ± 1 ( U 3 ) K m ( U 3 ) ] E = [ U 3 I m ± 1 ( U 3 ) I m ( W 3 ) W 3 I m ± 1 ( W 3 ) I m ( U 3 ) ] B + [ U 3 I m ± 1 ( U 3 ) K m ( W 3 ) + W 3 K m ± 1 ( W 3 ) I m ( U 3 ) ] C U 3 [ K m ± 1 ( U 3 ) I m ( U 3 ) + I m ± 1 ( U 3 ) K m ( U 3 ) ] F = [ I m ( U 4 ) + U 4 I m ± 1 ( U 4 ) ] D + [ K m ( U 4 ) U 4 K m ± 1 ( U 4 ) ] E [ K m ( W 4 ) W 4 K m ± 1 ( W 4 ) ]
B = W 2 K m ± 1 ( W 2 ) J m ( U 1 ) U 1 J m ± 1 ( U 1 ) K m ( W 2 ) W 2 [ K m ± 1 ( W 2 ) I m ( W 2 ) + I m ± 1 ( W 2 ) K m ( W 2 ) ] C = W 2 I m ± 1 ( W 2 ) J m ( U 1 ) ± U 1 J m ± 1 ( U 1 ) I m ( W 2 ) W 2 [ K m ± 1 ( W 2 ) I m ( W 2 ) + I m ± 1 ( W 2 ) K m ( W 2 ) ] D = [ U 3 K m ± 1 ( U 3 ) I m ( W 3 ) + W 3 I m ± 1 ( W 3 ) K m ( U 3 ) ] B + [ U 3 K m ± 1 ( U 3 ) K m ( W 3 ) W 3 K m ± 1 ( W 3 ) K m ( U 3 ) ] C U 3 [ K m ± 1 ( U 3 ) I m ( U 3 ) + I m ± 1 ( U 3 ) K m ( U 3 ) ] E = [ U 3 I m ± 1 ( U 3 ) I m ( W 3 ) W 3 I m ± 1 ( W 3 ) I m ( U 3 ) ] B + [ U 3 I m ± 1 ( U 3 ) K m ( W 3 ) + W 3 K m ± 1 ( W 3 ) I m ( U 3 ) ] C U 3 [ I m ± 1 ( U 3 ) K m ( U 3 ) + K m ± 1 ( U 3 ) I m ( U 3 ) ] F = [ I m ( U 4 ) + U 4 I m ± 1 ( U 4 ) ] D + [ K m ( U 4 ) U 4 K m ± 1 ( U 4 ) ] E [ K m ( W 4 ) W 4 K m ± 1 ( W 4 ) ]
B = W 2 K m ± 1 ( W 2 ) J m ( U 1 ) U 1 J m ± 1 ( U 1 ) K m ( W 2 ) W 2 [ K m ± 1 ( W 2 ) I m ( W 2 ) + I m ± 1 ( W 2 ) K m ( W 2 ) ] C = W 2 I m ± 1 ( W 2 ) J m ( U 1 ) ± U 1 J m ± 1 ( U 1 ) I m ( W 2 ) W 2 [ I m ± 1 ( W 2 ) K m ( W 2 ) + K m ± 1 ( W 2 ) I m ( W 2 ) ] D = [ U 3 Y m ± 1 ( U 3 ) I m ( W 3 ) ± W 3 I m ± 1 ( W 3 ) Y m ( U 3 ) ] B + [ U 3 Y m ± 1 ( U 3 ) K m ( W 3 ) W 3 K m ± 1 ( W 3 ) Y m ( U 3 ) ] C U 3 [ Y m ± 1 ( U 3 ) J m ( U 3 ) J m ± 1 ( U 3 ) Y m ( U 3 ) ] E = [ U 3 J m ± 1 ( U 3 ) I m ( W 3 ) ± W 3 I m ± 1 ( W 3 ) J m ( U 3 ) ] B + [ U 3 J m ± 1 ( U 3 ) K m ( W 3 ) W 3 K m ± 1 ( W 3 ) J m ( U 3 ) ] C U 3 [ J m ± 1 ( U 3 ) Y m ( U 3 ) Y m ± 1 ( U 3 ) J m ( U 3 ) ] F = [ J m ( U 4 ) + U 4 J m ± 1 ( U 4 ) ] D + [ Y m ( U 4 ) + U 4 Y m ± 1 ( U 4 ) ] E [ K m ( W 4 ) ± W 4 K m ± 1 ( W 4 ) ]
B = W 2 Y m ± 1 ( W 2 ) J m ( U 1 ) U 1 J m ± 1 ( U 1 ) Y m ( W 2 ) W 2 [ Y m ± 1 ( W 2 ) J m ( W 2 ) J m ± 1 ( W 2 ) Y m ( W 2 ) ] C = W 2 J m ± 1 ( W 2 ) J m ( U 1 ) U 1 J m ± 1 ( U 1 ) J m ( W 2 ) W 2 [ J m ± 1 ( W 2 ) Y m ( W 2 ) Y m ± 1 ( W 2 ) J m ( W 2 ) ] D = [ U 3 Y m ± 1 ( U 3 ) J m ( W 3 ) W 3 J m ± 1 ( W 3 ) Y m ( U 3 ) ] B + [ U 3 Y m ± 1 ( U 3 ) Y m ( W 3 ) W 3 Y m ± 1 ( W 3 ) Y m ( U 3 ) ] C U 3 [ Y m ± 1 ( U 3 ) J m ( U 3 ) J m ± 1 ( U 3 ) Y m ( U 3 ) ] E = [ U 3 J m ± 1 ( U 3 ) J m ( W 3 ) W 3 J m ± 1 ( W 3 ) J m ( U 3 ) ] B + [ U 3 J m ± 1 ( U 3 ) Y m ( W 3 ) W 3 Y m ± 1 ( W 3 ) J m ( U 3 ) ] C U 3 [ J m ± 1 ( U 3 ) Y m ( U 3 ) Y m ± 1 ( U 3 ) J m ( U 3 ) ] F = [ J m ( U 4 ) + U 4 J m ± 1 ( U 4 ) ] D + [ Y m ( U 4 ) ± U 4 Y m ± 1 ( U 4 ) ] E [ K m ( W 4 ) + W 4 K m ± 1 ( W 4 ) ]

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