Abstract

By invoking Debye potentials, we formulate exact eigenvalue equations and the corresponding field distributions for general, three-layered, radially stratified, dielectric, and nonferromagnetic metal, optical fibers. By using cross products of Bessel functions, which may be regarded as the basic functional elements of the eigenvalue equations, a comparison is made between the properties of a three-layer structure and a simple step-index profile, and a simple graphical solution is obtained. The technique is applied to several practical structures, including two-layer fibers having a central index depression in the core, ring-core fibers, W fibers, and progressively stepped three-layered structures. The mathematical procedure is simple, and the results are of interest to optical fiber designers.

© 1989 Optical Society of America

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References

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  1. S. Kawakami, S. Nishida, “Characteristics of a doubly clad optical fibre with a low-index inner cladding,” IEEE J. Quantum Electron. QE-10, 879–887 (1974).
    [CrossRef]
  2. S. Kawakami, “Low dispersion/low loss single mode optical fibres—topics related with W-fibres,”J. Inst. Electron. Commun. Eng. Jpn. (Japan) 68, 860–865 (1985).
  3. C. Yeh, G. Lindgren, “Computing the propagation characteristics of radially stratified fibers: an efficient method,” Appl. Opt. 16, 483–493 (1977).
    [CrossRef] [PubMed]
  4. P. L. Franqois, M. J. Adams, R. J. Mansfield, R. D. Birch, E. J. Tarbox, “Equivalent step-index profile for general W-fibres: application to TE01mode cut-off,” Opt. Quantum Electron. 14, 483–499 (1982).
    [CrossRef]
  5. P. L. Franqois, C. Vassallo, “Finite cladding effects in W fibers: a new interpretation of leaking modes,” Appl. Opt. 22, 3109–3120 (1983).
    [CrossRef]
  6. R. A. Sammut, “Range of monomode operation of W-fibres,” Opt. Quantum Electron. 10, 509–514 (1978).
    [CrossRef]
  7. M. Monerie, “Propagation in doubly clad single-mode fibers,” IEEE J. Quantum Electron. QE-18, 535–542 (1982).
    [CrossRef]
  8. A. S. Belanov, E. M. Dianov, G. I. Ezhov, A. M. Prokhorov, “Propagation of normal modes in multilayered optical waveguides. I. Component fields and dispersion characteristics,” Sov. J. Quantum Electron. 6, 43–50 (1976).
    [CrossRef]
  9. A. S. Belanov, E. M. Dianov, G. I. Ezhov, A. M. Prokhorov, “Propagation of normal modes in multilayered optical waveguides. II. Energy characteristics,” Sov. J. Quantum Electron. 6, 915–920 (1976).
    [CrossRef]
  10. C. D. Hussey, F. De Fornel, “Effective refractive index and range of monomode operation for W-fibres,” Electron. Lett. 20, 346–347 (1984).
    [CrossRef]
  11. H. M. Barlow, “A cladded tubular glass-fibre guide for single-mode transmission,”J. Phys. D 14, 405–412 (1981).
    [CrossRef]
  12. H. M. Barlow, “A large diameter optical-fibre waveguide for exclusive transmission in the HE11mode,”J. Phys. D 16, 1439–1451 (1983).
    [CrossRef]
  13. H. M. Barlow, “A new large-diameter optical fibre waveguide for operation in the TE01mode,” J. Phys. D 18, 1511–1520 (1985).
    [CrossRef]
  14. Th. Niemeier, S. B. Poole, R. Ulrich, “Self-imaging by ring-core fibers,” in Digest of Topical Meeting on Optical Fiber Communication, I. D. Aggarwal, ed. (Optical Society of America, Washington, D.C., 1985), pp. 122–124.
  15. R. J. Black, C. Pask, “Equivalent optical waveguides,” IEEE J. Lightwave Technol. LT-2, 268–276 (1984).
    [CrossRef]
  16. A. Majewski, “Numerical analysis of double-step-index single-mode optical fibres,” Arch. Electrotech. 33, 257–265 (1984).
  17. M. Kuroda, “Transmission characteristics of polarization-maintaining optical fiber with three-layer elliptical cross-section,” Trans. Inst. Electron. Commun. Eng. Jpn. B 68, 93–100 (1985).
  18. G. Coppa, P. Di Vita, “Cut-off condition of the fundamental mode in monomode fibres,” Opt. Commun. 49, 409–412 (1984).
    [CrossRef]
  19. W. R. Smythe, C. Yeh, “Formulas,” in American Institute of Physics Handbook, 3rd ed., D. E. Gray, ed. (McGraw-Hill, New York, 1972), Sec. 5, pp. 5–45.
  20. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1965), Chap. 9.
  21. E. T. Goodwin, “Recurrence relations for cross-products of Bessel functions,”Q. J. Mech. Appl. Math. 11, 72–74 (1949).
  22. M. J. Adams, An Introduction to Optical Waveguides (Wiley, Chichester, UK, 1981), p. 225.

1985 (3)

S. Kawakami, “Low dispersion/low loss single mode optical fibres—topics related with W-fibres,”J. Inst. Electron. Commun. Eng. Jpn. (Japan) 68, 860–865 (1985).

H. M. Barlow, “A new large-diameter optical fibre waveguide for operation in the TE01mode,” J. Phys. D 18, 1511–1520 (1985).
[CrossRef]

M. Kuroda, “Transmission characteristics of polarization-maintaining optical fiber with three-layer elliptical cross-section,” Trans. Inst. Electron. Commun. Eng. Jpn. B 68, 93–100 (1985).

1984 (4)

G. Coppa, P. Di Vita, “Cut-off condition of the fundamental mode in monomode fibres,” Opt. Commun. 49, 409–412 (1984).
[CrossRef]

R. J. Black, C. Pask, “Equivalent optical waveguides,” IEEE J. Lightwave Technol. LT-2, 268–276 (1984).
[CrossRef]

A. Majewski, “Numerical analysis of double-step-index single-mode optical fibres,” Arch. Electrotech. 33, 257–265 (1984).

C. D. Hussey, F. De Fornel, “Effective refractive index and range of monomode operation for W-fibres,” Electron. Lett. 20, 346–347 (1984).
[CrossRef]

1983 (2)

H. M. Barlow, “A large diameter optical-fibre waveguide for exclusive transmission in the HE11mode,”J. Phys. D 16, 1439–1451 (1983).
[CrossRef]

P. L. Franqois, C. Vassallo, “Finite cladding effects in W fibers: a new interpretation of leaking modes,” Appl. Opt. 22, 3109–3120 (1983).
[CrossRef]

1982 (2)

M. Monerie, “Propagation in doubly clad single-mode fibers,” IEEE J. Quantum Electron. QE-18, 535–542 (1982).
[CrossRef]

P. L. Franqois, M. J. Adams, R. J. Mansfield, R. D. Birch, E. J. Tarbox, “Equivalent step-index profile for general W-fibres: application to TE01mode cut-off,” Opt. Quantum Electron. 14, 483–499 (1982).
[CrossRef]

1981 (1)

H. M. Barlow, “A cladded tubular glass-fibre guide for single-mode transmission,”J. Phys. D 14, 405–412 (1981).
[CrossRef]

1978 (1)

R. A. Sammut, “Range of monomode operation of W-fibres,” Opt. Quantum Electron. 10, 509–514 (1978).
[CrossRef]

1977 (1)

1976 (2)

A. S. Belanov, E. M. Dianov, G. I. Ezhov, A. M. Prokhorov, “Propagation of normal modes in multilayered optical waveguides. I. Component fields and dispersion characteristics,” Sov. J. Quantum Electron. 6, 43–50 (1976).
[CrossRef]

A. S. Belanov, E. M. Dianov, G. I. Ezhov, A. M. Prokhorov, “Propagation of normal modes in multilayered optical waveguides. II. Energy characteristics,” Sov. J. Quantum Electron. 6, 915–920 (1976).
[CrossRef]

1974 (1)

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

1949 (1)

E. T. Goodwin, “Recurrence relations for cross-products of Bessel functions,”Q. J. Mech. Appl. Math. 11, 72–74 (1949).

Adams, M. J.

P. L. Franqois, M. J. Adams, R. J. Mansfield, R. D. Birch, E. J. Tarbox, “Equivalent step-index profile for general W-fibres: application to TE01mode cut-off,” Opt. Quantum Electron. 14, 483–499 (1982).
[CrossRef]

M. J. Adams, An Introduction to Optical Waveguides (Wiley, Chichester, UK, 1981), p. 225.

Barlow, H. M.

H. M. Barlow, “A new large-diameter optical fibre waveguide for operation in the TE01mode,” J. Phys. D 18, 1511–1520 (1985).
[CrossRef]

H. M. Barlow, “A large diameter optical-fibre waveguide for exclusive transmission in the HE11mode,”J. Phys. D 16, 1439–1451 (1983).
[CrossRef]

H. M. Barlow, “A cladded tubular glass-fibre guide for single-mode transmission,”J. Phys. D 14, 405–412 (1981).
[CrossRef]

Belanov, A. S.

A. S. Belanov, E. M. Dianov, G. I. Ezhov, A. M. Prokhorov, “Propagation of normal modes in multilayered optical waveguides. II. Energy characteristics,” Sov. J. Quantum Electron. 6, 915–920 (1976).
[CrossRef]

A. S. Belanov, E. M. Dianov, G. I. Ezhov, A. M. Prokhorov, “Propagation of normal modes in multilayered optical waveguides. I. Component fields and dispersion characteristics,” Sov. J. Quantum Electron. 6, 43–50 (1976).
[CrossRef]

Birch, R. D.

P. L. Franqois, M. J. Adams, R. J. Mansfield, R. D. Birch, E. J. Tarbox, “Equivalent step-index profile for general W-fibres: application to TE01mode cut-off,” Opt. Quantum Electron. 14, 483–499 (1982).
[CrossRef]

Black, R. J.

R. J. Black, C. Pask, “Equivalent optical waveguides,” IEEE J. Lightwave Technol. LT-2, 268–276 (1984).
[CrossRef]

Coppa, G.

G. Coppa, P. Di Vita, “Cut-off condition of the fundamental mode in monomode fibres,” Opt. Commun. 49, 409–412 (1984).
[CrossRef]

De Fornel, F.

C. D. Hussey, F. De Fornel, “Effective refractive index and range of monomode operation for W-fibres,” Electron. Lett. 20, 346–347 (1984).
[CrossRef]

Di Vita, P.

G. Coppa, P. Di Vita, “Cut-off condition of the fundamental mode in monomode fibres,” Opt. Commun. 49, 409–412 (1984).
[CrossRef]

Dianov, E. M.

A. S. Belanov, E. M. Dianov, G. I. Ezhov, A. M. Prokhorov, “Propagation of normal modes in multilayered optical waveguides. II. Energy characteristics,” Sov. J. Quantum Electron. 6, 915–920 (1976).
[CrossRef]

A. S. Belanov, E. M. Dianov, G. I. Ezhov, A. M. Prokhorov, “Propagation of normal modes in multilayered optical waveguides. I. Component fields and dispersion characteristics,” Sov. J. Quantum Electron. 6, 43–50 (1976).
[CrossRef]

Ezhov, G. I.

A. S. Belanov, E. M. Dianov, G. I. Ezhov, A. M. Prokhorov, “Propagation of normal modes in multilayered optical waveguides. I. Component fields and dispersion characteristics,” Sov. J. Quantum Electron. 6, 43–50 (1976).
[CrossRef]

A. S. Belanov, E. M. Dianov, G. I. Ezhov, A. M. Prokhorov, “Propagation of normal modes in multilayered optical waveguides. II. Energy characteristics,” Sov. J. Quantum Electron. 6, 915–920 (1976).
[CrossRef]

Franqois, P. L.

P. L. Franqois, C. Vassallo, “Finite cladding effects in W fibers: a new interpretation of leaking modes,” Appl. Opt. 22, 3109–3120 (1983).
[CrossRef]

P. L. Franqois, M. J. Adams, R. J. Mansfield, R. D. Birch, E. J. Tarbox, “Equivalent step-index profile for general W-fibres: application to TE01mode cut-off,” Opt. Quantum Electron. 14, 483–499 (1982).
[CrossRef]

Goodwin, E. T.

E. T. Goodwin, “Recurrence relations for cross-products of Bessel functions,”Q. J. Mech. Appl. Math. 11, 72–74 (1949).

Hussey, C. D.

C. D. Hussey, F. De Fornel, “Effective refractive index and range of monomode operation for W-fibres,” Electron. Lett. 20, 346–347 (1984).
[CrossRef]

Kawakami, S.

S. Kawakami, “Low dispersion/low loss single mode optical fibres—topics related with W-fibres,”J. Inst. Electron. Commun. Eng. Jpn. (Japan) 68, 860–865 (1985).

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

Kuroda, M.

M. Kuroda, “Transmission characteristics of polarization-maintaining optical fiber with three-layer elliptical cross-section,” Trans. Inst. Electron. Commun. Eng. Jpn. B 68, 93–100 (1985).

Lindgren, G.

Majewski, A.

A. Majewski, “Numerical analysis of double-step-index single-mode optical fibres,” Arch. Electrotech. 33, 257–265 (1984).

Mansfield, R. J.

P. L. Franqois, M. J. Adams, R. J. Mansfield, R. D. Birch, E. J. Tarbox, “Equivalent step-index profile for general W-fibres: application to TE01mode cut-off,” Opt. Quantum Electron. 14, 483–499 (1982).
[CrossRef]

Monerie, M.

M. Monerie, “Propagation in doubly clad single-mode fibers,” IEEE J. Quantum Electron. QE-18, 535–542 (1982).
[CrossRef]

Niemeier, Th.

Th. Niemeier, S. B. Poole, R. Ulrich, “Self-imaging by ring-core fibers,” in Digest of Topical Meeting on Optical Fiber Communication, I. D. Aggarwal, ed. (Optical Society of America, Washington, D.C., 1985), pp. 122–124.

Nishida, S.

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

Pask, C.

R. J. Black, C. Pask, “Equivalent optical waveguides,” IEEE J. Lightwave Technol. LT-2, 268–276 (1984).
[CrossRef]

Poole, S. B.

Th. Niemeier, S. B. Poole, R. Ulrich, “Self-imaging by ring-core fibers,” in Digest of Topical Meeting on Optical Fiber Communication, I. D. Aggarwal, ed. (Optical Society of America, Washington, D.C., 1985), pp. 122–124.

Prokhorov, A. M.

A. S. Belanov, E. M. Dianov, G. I. Ezhov, A. M. Prokhorov, “Propagation of normal modes in multilayered optical waveguides. II. Energy characteristics,” Sov. J. Quantum Electron. 6, 915–920 (1976).
[CrossRef]

A. S. Belanov, E. M. Dianov, G. I. Ezhov, A. M. Prokhorov, “Propagation of normal modes in multilayered optical waveguides. I. Component fields and dispersion characteristics,” Sov. J. Quantum Electron. 6, 43–50 (1976).
[CrossRef]

Sammut, R. A.

R. A. Sammut, “Range of monomode operation of W-fibres,” Opt. Quantum Electron. 10, 509–514 (1978).
[CrossRef]

Smythe, W. R.

W. R. Smythe, C. Yeh, “Formulas,” in American Institute of Physics Handbook, 3rd ed., D. E. Gray, ed. (McGraw-Hill, New York, 1972), Sec. 5, pp. 5–45.

Tarbox, E. J.

P. L. Franqois, M. J. Adams, R. J. Mansfield, R. D. Birch, E. J. Tarbox, “Equivalent step-index profile for general W-fibres: application to TE01mode cut-off,” Opt. Quantum Electron. 14, 483–499 (1982).
[CrossRef]

Ulrich, R.

Th. Niemeier, S. B. Poole, R. Ulrich, “Self-imaging by ring-core fibers,” in Digest of Topical Meeting on Optical Fiber Communication, I. D. Aggarwal, ed. (Optical Society of America, Washington, D.C., 1985), pp. 122–124.

Vassallo, C.

Yeh, C.

C. Yeh, G. Lindgren, “Computing the propagation characteristics of radially stratified fibers: an efficient method,” Appl. Opt. 16, 483–493 (1977).
[CrossRef] [PubMed]

W. R. Smythe, C. Yeh, “Formulas,” in American Institute of Physics Handbook, 3rd ed., D. E. Gray, ed. (McGraw-Hill, New York, 1972), Sec. 5, pp. 5–45.

Appl. Opt. (2)

Arch. Electrotech. (1)

A. Majewski, “Numerical analysis of double-step-index single-mode optical fibres,” Arch. Electrotech. 33, 257–265 (1984).

Electron. Lett. (1)

C. D. Hussey, F. De Fornel, “Effective refractive index and range of monomode operation for W-fibres,” Electron. Lett. 20, 346–347 (1984).
[CrossRef]

IEEE J. Lightwave Technol. (1)

R. J. Black, C. Pask, “Equivalent optical waveguides,” IEEE J. Lightwave Technol. LT-2, 268–276 (1984).
[CrossRef]

IEEE J. Quantum Electron. (2)

M. Monerie, “Propagation in doubly clad single-mode fibers,” IEEE J. Quantum Electron. QE-18, 535–542 (1982).
[CrossRef]

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

J. Inst. Electron. Commun. Eng. Jpn. (Japan) (1)

S. Kawakami, “Low dispersion/low loss single mode optical fibres—topics related with W-fibres,”J. Inst. Electron. Commun. Eng. Jpn. (Japan) 68, 860–865 (1985).

J. Phys. D (3)

H. M. Barlow, “A cladded tubular glass-fibre guide for single-mode transmission,”J. Phys. D 14, 405–412 (1981).
[CrossRef]

H. M. Barlow, “A large diameter optical-fibre waveguide for exclusive transmission in the HE11mode,”J. Phys. D 16, 1439–1451 (1983).
[CrossRef]

H. M. Barlow, “A new large-diameter optical fibre waveguide for operation in the TE01mode,” J. Phys. D 18, 1511–1520 (1985).
[CrossRef]

Opt. Commun. (1)

G. Coppa, P. Di Vita, “Cut-off condition of the fundamental mode in monomode fibres,” Opt. Commun. 49, 409–412 (1984).
[CrossRef]

Opt. Quantum Electron. (2)

P. L. Franqois, M. J. Adams, R. J. Mansfield, R. D. Birch, E. J. Tarbox, “Equivalent step-index profile for general W-fibres: application to TE01mode cut-off,” Opt. Quantum Electron. 14, 483–499 (1982).
[CrossRef]

R. A. Sammut, “Range of monomode operation of W-fibres,” Opt. Quantum Electron. 10, 509–514 (1978).
[CrossRef]

Q. J. Mech. Appl. Math. (1)

E. T. Goodwin, “Recurrence relations for cross-products of Bessel functions,”Q. J. Mech. Appl. Math. 11, 72–74 (1949).

Sov. J. Quantum Electron. (2)

A. S. Belanov, E. M. Dianov, G. I. Ezhov, A. M. Prokhorov, “Propagation of normal modes in multilayered optical waveguides. I. Component fields and dispersion characteristics,” Sov. J. Quantum Electron. 6, 43–50 (1976).
[CrossRef]

A. S. Belanov, E. M. Dianov, G. I. Ezhov, A. M. Prokhorov, “Propagation of normal modes in multilayered optical waveguides. II. Energy characteristics,” Sov. J. Quantum Electron. 6, 915–920 (1976).
[CrossRef]

Trans. Inst. Electron. Commun. Eng. Jpn. B (1)

M. Kuroda, “Transmission characteristics of polarization-maintaining optical fiber with three-layer elliptical cross-section,” Trans. Inst. Electron. Commun. Eng. Jpn. B 68, 93–100 (1985).

Other (4)

W. R. Smythe, C. Yeh, “Formulas,” in American Institute of Physics Handbook, 3rd ed., D. E. Gray, ed. (McGraw-Hill, New York, 1972), Sec. 5, pp. 5–45.

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1965), Chap. 9.

Th. Niemeier, S. B. Poole, R. Ulrich, “Self-imaging by ring-core fibers,” in Digest of Topical Meeting on Optical Fiber Communication, I. D. Aggarwal, ed. (Optical Society of America, Washington, D.C., 1985), pp. 122–124.

M. J. Adams, An Introduction to Optical Waveguides (Wiley, Chichester, UK, 1981), p. 225.

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Figures (9)

Fig. 1
Fig. 1

Fiber cross section and index profiles for a three-layered radially stratified structure. (a) Cross section, (b) stepped three-layer fiber, (c) W fiber, (d) fiber with a central index dip, and (e) ring-core fiber.

Fig. 2
Fig. 2

Cross products of Bessel functions pν, qν, rν, and sν, (a = 1.5) for (a) ν = 0 and (b) ν = 1.

Fig. 3
Fig. 3

Graphical method of solving Eq. (7): n1 = 1.47, n2 = 1.462, n3 = 1.458, r1 = 3.0 μm, r2 = 5.0 μm, and ν = 0. rhs, Right-hand side; lhs, left-hand side; sol 1 and sol 2, solutions for cases 1 and 2, respectively.

Fig. 4
Fig. 4

(a) Measured refractive-index profile, showing a central index dip; (b) equivalent index ne of the fundamental mode as a function of the radius of the index dip.

Fig. 5
Fig. 5

Self-imaging effects in a ring-core fiber with (a) a single light spot at the input end face, (b) two light spots at the end face after a half interference length L/2, and (c) an inverted light spot at the end face after the interference length L.

Fig. 6
Fig. 6

Light spots as a function of ϕ are superposed to include sin ϕ and sin 2ϕ: (a) sin ϕ and sin 2ϕ at the input end face, (b) π/2 phase shift in sin ϕ over L/2, and (c) π phase shift in sin ϕ at L.

Fig. 7
Fig. 7

Number of ν modes m are determined by (m − 1)π < Vr < . (a) Single-ν-mode operation (m = 1) for n1 = n3 = 1.458, n2 = 1.461, r1 = 20 μm, r2 = 23 μm; (b) double-ν-mode operation (m = 2) for n1 = n3 = 1.458, n2 = 1.464, r1 = 20 μm, r2 = 24 μm. rhc, Right-hand core; lhc, left-hand core.

Fig. 8
Fig. 8

The parameter U as a function of r2 in a W fiber for which n1 = 1.46, n2 = 1.44, and r1 = 3.0 μm:(a) n3 = 1.45, no cutoff for HE11–TE01 mode; (b) n3 = 1.4558, TE01-mode cutoffs; (c) n3 = 1.4585, both TE01- and HE11-mode cutoffs.

Fig. 9
Fig. 9

Equivalent refractive indices as functions of the geometry and the index profile for a stepped three-layer fiber: (a) n2 dependence, (b) r1 dependence, and (c) r2 dependence. In (c) the thick curves are for n2 = 1.464, the thin curves are for n2 = 1.461, and the dashed curves are for the n2/n3 SSP’s.

Equations (30)

Equations on this page are rendered with MathJax. Learn more.

e = ( Ψ / r ϕ - β Φ / ω r ) r ^ 0 + ( - Ψ / r - β Φ / ω r ϕ ) ϕ ^ 0 - ( k 2 n 2 - β 2 ) Φ z ^ 0 / j ω ,
h = ( Φ / r ϕ + β Ψ / ω μ r ) r ^ 0 + ( - Φ / r + β Ψ / ω μ r ϕ ) ϕ ^ 0 + ( k 2 n 2 - β 2 ) Ψ z ^ 0 / j ω μ .
[ Δ t + ( k 2 n i 2 - β 2 ) ] Ψ i = 0
[ Δ t + ( k 2 n i 2 - β 2 ) ] Φ i = 0
Ψ 1 = A 1 J ν ( U 1 r / r 1 ) f ν ( ν ϕ ) ,
Ψ 2 = { A 2 J ν ( U 2 r / r 1 ) + B 2 Y ν ( U 2 r / r 1 ) } f ν ( ν ϕ ) ,
Ψ 3 = A 3 K ν ( W 3 r / r 2 ) f ν ( ν ϕ ) ,
e ϕ 1 = - A 1 ( U 1 / r 1 ) J ν ( U 1 r / r 1 ) ,
e ϕ 2 = - ( U 2 / r 1 ) [ A 2 J ν ( U 2 r / r 1 ) + B 2 Y ν ( U 2 r / r 1 ) ] ,
e ϕ 3 = - A 3 ( W 3 / r 2 ) K ν ( W 3 r / r 2 ) ,
h z 1 = - ( U 1 2 / r 1 2 j ω μ ) A 1 J ν ( U 1 r / r 1 ) ,
h z 2 = - ( U 2 2 / r 1 2 j ω μ ) [ A 2 J ν ( U 2 r / r 1 ) + B 2 Y ν ( U 2 r / r 1 ) ] ,
h z 3 = ( W 3 2 / r 2 2 j ω μ ) A 3 K ν ( W 3 r / r 2 ) .
J ( r ν / a U 2 + K p ν ) = ( K q ν / U 2 + s ν / a U 2 2 ) .
J ( s 23 r ν / a U 2 + K p ν ) = s 21 ( K q ν / U 2 + s 23 s ν / a U 2 2 ) .
P ν 2 = 2 x 1 x 2 ( n 2 2 / n 1 n 3 ) ( 2 / π a U 2 2 ) 2 + x 1 2 x 2 2 [ J ( r ν / a U 2 + K p ν ) - ( K q ν / U 2 + s ν / a U 2 2 ) ] × [ J ( s 23 r ν / a U 2 + K p ν ) - s 21 ( K q ν / U 2 + s 23 s ν / a U 2 2 ) ] = x 1 2 ( J p ν - s 21 q ν / U 2 ) ( J p ν - q ν / U 2 ) + x 2 2 ( K p ν + s 23 r ν / a U 2 ) ( K p ν + s 23 r ν / a U 2 ) .
J ν ( U 1 ) / U 1 J ν ( U 1 ) = - K ( W 2 ) / W 2 K ( W 2 ) ,
J ν ( U 1 ) / U 1 J ν ( U 1 ) = - K ν / W 3 K ν ( W 3 )
K = - ( 1 / a U 2 ) [ r ν - s ν J ν ( U 2 ) / J ν ( U 2 ) ] / [ p ν - q ν J ν ( U 2 ) / J ν ( U 2 ) ] ,
K ν ( W 3 ) / W 3 K ν ( W 3 ) = - J ν ( a U 2 ) / a U 2 J ν ( a U 2 ) .
J = ( q ν / U 2 + s ν / K a U 2 ) / ( p ν + r ν / K a U 2 )
J ν ( U 1 ) / U 1 J ν ( U 1 ) = - K ν ( W 2 ) / W 2 K ν ( W 2 ) .
- K ν ( W 3 ) / W 3 K ν ( W 3 ) = [ J ν ( a U 2 ) / a U 2 J ν ( a U 2 ) ] × ( 1 - a - 2 ν + 1 ) / ( 1 + a - 2 ν ) .
- K ν ( W 3 ) / W 3 K ν ( W 3 ) = [ J ν ( a U 2 ) / a U 2 J ν ( a U 2 ) ] × ( 1 + a - 2 ν ) / ( 1 - a - 2 ν ) .
K [ ( a + 1 ) a U 2 ] q ν = ( K 2 - 1 / a U 2 2 ) p ν .
p ν - [ 2 / π ( a U 2 ) 1 / 2 ] sin ( a - 1 ) U 2 , q ν [ 2 / π ( a U 2 ) 1 / 2 ] cos ( a - 1 ) U 2 .
p ν ( 2 / π ) d , q ν 2 / π U 2 , r ν - 2 / π U 2 , s ν ( 2 d / π ) ( ν 2 / U 2 2 - 1 ) .
p ν = s ν = - ( 2 / π U 2 a 1 / 2 ) sin ( a - 1 ) U 2 , q ν = - r ν = ( 2 / π U 2 a 1 / 2 ) cos ( a - 1 ) U 2 .
p ν = - ( 2 / π ) [ I ν ( a W 2 ) K ν ( W 2 ) - I ν ( W 2 ) K ν ( a W 2 ) ] q ν = j ( 2 / π ) [ I ν ( a W 2 ) K ν ( W 2 ) - I ν ( W 2 ) K ν ( a W 2 ) ] , r ν = j ( 2 / π ) [ I ν ( a W 2 ) K ν ( W 2 ) - I ν ( W 2 ) K ν ( a W 2 ) ] , s ν = ( 2 / π ) [ I ν ( a W 2 ) K ν ( W 2 ) - I ν ( W 2 ) K ν ( a W 2 ) ] .
p ν = - ( 2 / π ) I ν ( a W 2 ) K ν ( W 2 ) , q ν = j ( 2 / π ) I ν ( a W 2 ) K ν ( W 2 ) , r ν = j ( 2 / π ) I ν ( a W 2 ) K ν ( W 2 ) , s ν = ( 2 / π ) I ν ( a W 2 ) K ν ( W 2 ) .

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