Abstract

The linearly polarized approximation (LPA) is employed for the analysis of the propagation characteristics of the guided modes in parabolic-index fibers. Characteristic equations for fibers in both the infinite- and finite-cladding configurations are derived and the conditions for the core-mode cutoffs are thereby deduced. In particular, it is shown that the effects of the finite cladding on the properties of the guided modes can be accounted for by a direct generalization of the LPA that has been used for the analysis of infinite-cladding fibers. Furthermore, it is demonstrated that the LPA is applicable even when the cladding layer is thin compared with the core radius and the refractive index of the surrounding medium is much smaller than that of the cladding, provided the weakly guiding condition for the core modes is fulfilled and the propagating radiation frequency is far from the cladding-mode cutoff. The good agreement between the results obtained by the LPA and those calculated by the vector wave analysis, which is overwhelmingly complex for the finite-cladding fibers, demonstrates the appeal of the LPA by providing simple and yet sufficiently accurate formulas for the practical design calculations for parabolic-index fibers. It is also pointed out that the mathematical procedure employed in this paper can be adapted for other types of graded-index fibers.

© 1980 Optical Society of America

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References

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  1. D. Gloge and E. A. J. Marcatili, “Multimode Theory of Graded-Core Fibers,” Bell Syst. Tech. J. 52, 1563–1578 (1973).
    [Crossref]
  2. T. Okoshi and K. Okamoto, “Analysis of Wave Propagation in Inhomogeneous Optical Fibers using a Variational Method,” IEEE Trans. Microwave Theory Tech. MTT-22, 938–945 (1974).
    [Crossref]
  3. K. Okamoto and T. Okoshi, “Analysis of Wave Propagation in Optical Fibers Having Core with α-Power Refractive Index Distribution and Uniform Cladding,” IEEE Trans. Microwave Theory Tech. MTT-24, 416–421 (1976).
    [Crossref]
  4. M. Geshiro, M. Matsuhara, and N. Kumagai, “Truncated Parabolic-Index Fiber with Minimum Mode Dispersion,” IEEE Trans. Microwave Theory Tech. MTT-26, 115–119 (1978).
    [Crossref]
  5. P. J. B. Clarricoats and K. B. Chan, “Electromagnetic-Wave Propagation along Radially Inhomogeneous Dielectric Cylinders,” Electron. Lett. 6, 694–695 (1970).
    [Crossref]
  6. K. Okamoto and T. Okoshi, “Vectorial Wave Analysis of Inhomogeneous Optical Fibers using Finite Element Method,” IEEE Trans. Microwave Theory Tech. MTT-26, 109–114 (1978).
    [Crossref]
  7. W. Streifer and C. N. Kurtz, “Scalar Analysis of Radially Inhomogeneous Guiding Media,” J. Opt. Soc. Am. 57, 779–786 (1967).
    [Crossref]
  8. C. N. Kurtz and W. Streifer, “Guided Waves in Inhomogeneous Focussing Media, Part I: Formulation, Solution for Quadratic Inhomogeneity,” IEEE Trans. Microwave Theory Tech. MTT-17, 11–15 (1969).
    [Crossref]
  9. H. Kirchoff, “Optical Wave Propagation in Self-Focussing Fibres,” in Proceedings of Conference on Trunk Telecommunications by Guilded Waves, 1970, London (IEEE Conf. Publn. No. 71), pp. 69–72.
  10. R. Yamada and Y. Inabe, “Guided Waves Along Graded Index Dielectric Rod,” IEEE Trans. Microwave Theory Tech. MTT-22, 813–814 (1974).
    [Crossref]
  11. M. Hasimoto, S. Nemoto, and T. Makimoto, “Analysis of Guided Waves Along the Cladded Optical Fiber: Parabolic-Index Core and Homogeneous Cladding,”  MTT-25, 1–17 (1977).
  12. T. K. Lim, B. K. Garside, and J. P. Marton, “Guided Modes in Fibers with Parabolic-Index Core and Homogeneous Cladding,” Opt. Quantum Electron. 11, 329–344 (1979).
    [Crossref]
  13. D. Gloge, “Weakly guiding fibers,” Appl. Opt. 10, 2252–2258 (1971).
    [Crossref] [PubMed]
  14. A. W. Snyder and W. R. Young, “Modes of optical waveguides,” J. Opt. Soc. Am. 68, 297–309 (1978).
    [Crossref]
  15. G. L. Yip and Y. H. Ahmew, “Propagation Characteristics of Radially Inhomogeneous Optical Fiber,” Electron. Lett. 10, 37–38 (1974).
    [Crossref]
  16. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1964).
  17. T. I. Lukowski and F. P. Kapron, “Parabolic fiber cutoffs: A comparison of theories,” J. Opt. Soc. Am. 67, 1185–1187 (1977).
    [Crossref]

1979 (1)

T. K. Lim, B. K. Garside, and J. P. Marton, “Guided Modes in Fibers with Parabolic-Index Core and Homogeneous Cladding,” Opt. Quantum Electron. 11, 329–344 (1979).
[Crossref]

1978 (3)

A. W. Snyder and W. R. Young, “Modes of optical waveguides,” J. Opt. Soc. Am. 68, 297–309 (1978).
[Crossref]

M. Geshiro, M. Matsuhara, and N. Kumagai, “Truncated Parabolic-Index Fiber with Minimum Mode Dispersion,” IEEE Trans. Microwave Theory Tech. MTT-26, 115–119 (1978).
[Crossref]

K. Okamoto and T. Okoshi, “Vectorial Wave Analysis of Inhomogeneous Optical Fibers using Finite Element Method,” IEEE Trans. Microwave Theory Tech. MTT-26, 109–114 (1978).
[Crossref]

1977 (2)

T. I. Lukowski and F. P. Kapron, “Parabolic fiber cutoffs: A comparison of theories,” J. Opt. Soc. Am. 67, 1185–1187 (1977).
[Crossref]

M. Hasimoto, S. Nemoto, and T. Makimoto, “Analysis of Guided Waves Along the Cladded Optical Fiber: Parabolic-Index Core and Homogeneous Cladding,”  MTT-25, 1–17 (1977).

1976 (1)

K. Okamoto and T. Okoshi, “Analysis of Wave Propagation in Optical Fibers Having Core with α-Power Refractive Index Distribution and Uniform Cladding,” IEEE Trans. Microwave Theory Tech. MTT-24, 416–421 (1976).
[Crossref]

1974 (3)

T. Okoshi and K. Okamoto, “Analysis of Wave Propagation in Inhomogeneous Optical Fibers using a Variational Method,” IEEE Trans. Microwave Theory Tech. MTT-22, 938–945 (1974).
[Crossref]

R. Yamada and Y. Inabe, “Guided Waves Along Graded Index Dielectric Rod,” IEEE Trans. Microwave Theory Tech. MTT-22, 813–814 (1974).
[Crossref]

G. L. Yip and Y. H. Ahmew, “Propagation Characteristics of Radially Inhomogeneous Optical Fiber,” Electron. Lett. 10, 37–38 (1974).
[Crossref]

1973 (1)

D. Gloge and E. A. J. Marcatili, “Multimode Theory of Graded-Core Fibers,” Bell Syst. Tech. J. 52, 1563–1578 (1973).
[Crossref]

1971 (1)

1970 (1)

P. J. B. Clarricoats and K. B. Chan, “Electromagnetic-Wave Propagation along Radially Inhomogeneous Dielectric Cylinders,” Electron. Lett. 6, 694–695 (1970).
[Crossref]

1969 (1)

C. N. Kurtz and W. Streifer, “Guided Waves in Inhomogeneous Focussing Media, Part I: Formulation, Solution for Quadratic Inhomogeneity,” IEEE Trans. Microwave Theory Tech. MTT-17, 11–15 (1969).
[Crossref]

1967 (1)

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1964).

Ahmew, Y. H.

G. L. Yip and Y. H. Ahmew, “Propagation Characteristics of Radially Inhomogeneous Optical Fiber,” Electron. Lett. 10, 37–38 (1974).
[Crossref]

Chan, K. B.

P. J. B. Clarricoats and K. B. Chan, “Electromagnetic-Wave Propagation along Radially Inhomogeneous Dielectric Cylinders,” Electron. Lett. 6, 694–695 (1970).
[Crossref]

Clarricoats, P. J. B.

P. J. B. Clarricoats and K. B. Chan, “Electromagnetic-Wave Propagation along Radially Inhomogeneous Dielectric Cylinders,” Electron. Lett. 6, 694–695 (1970).
[Crossref]

Garside, B. K.

T. K. Lim, B. K. Garside, and J. P. Marton, “Guided Modes in Fibers with Parabolic-Index Core and Homogeneous Cladding,” Opt. Quantum Electron. 11, 329–344 (1979).
[Crossref]

Geshiro, M.

M. Geshiro, M. Matsuhara, and N. Kumagai, “Truncated Parabolic-Index Fiber with Minimum Mode Dispersion,” IEEE Trans. Microwave Theory Tech. MTT-26, 115–119 (1978).
[Crossref]

Gloge, D.

D. Gloge and E. A. J. Marcatili, “Multimode Theory of Graded-Core Fibers,” Bell Syst. Tech. J. 52, 1563–1578 (1973).
[Crossref]

D. Gloge, “Weakly guiding fibers,” Appl. Opt. 10, 2252–2258 (1971).
[Crossref] [PubMed]

Hasimoto, M.

M. Hasimoto, S. Nemoto, and T. Makimoto, “Analysis of Guided Waves Along the Cladded Optical Fiber: Parabolic-Index Core and Homogeneous Cladding,”  MTT-25, 1–17 (1977).

Inabe, Y.

R. Yamada and Y. Inabe, “Guided Waves Along Graded Index Dielectric Rod,” IEEE Trans. Microwave Theory Tech. MTT-22, 813–814 (1974).
[Crossref]

Kapron, F. P.

Kirchoff, H.

H. Kirchoff, “Optical Wave Propagation in Self-Focussing Fibres,” in Proceedings of Conference on Trunk Telecommunications by Guilded Waves, 1970, London (IEEE Conf. Publn. No. 71), pp. 69–72.

Kumagai, N.

M. Geshiro, M. Matsuhara, and N. Kumagai, “Truncated Parabolic-Index Fiber with Minimum Mode Dispersion,” IEEE Trans. Microwave Theory Tech. MTT-26, 115–119 (1978).
[Crossref]

Kurtz, C. N.

C. N. Kurtz and W. Streifer, “Guided Waves in Inhomogeneous Focussing Media, Part I: Formulation, Solution for Quadratic Inhomogeneity,” IEEE Trans. Microwave Theory Tech. MTT-17, 11–15 (1969).
[Crossref]

W. Streifer and C. N. Kurtz, “Scalar Analysis of Radially Inhomogeneous Guiding Media,” J. Opt. Soc. Am. 57, 779–786 (1967).
[Crossref]

Lim, T. K.

T. K. Lim, B. K. Garside, and J. P. Marton, “Guided Modes in Fibers with Parabolic-Index Core and Homogeneous Cladding,” Opt. Quantum Electron. 11, 329–344 (1979).
[Crossref]

Lukowski, T. I.

Makimoto, T.

M. Hasimoto, S. Nemoto, and T. Makimoto, “Analysis of Guided Waves Along the Cladded Optical Fiber: Parabolic-Index Core and Homogeneous Cladding,”  MTT-25, 1–17 (1977).

Marcatili, E. A. J.

D. Gloge and E. A. J. Marcatili, “Multimode Theory of Graded-Core Fibers,” Bell Syst. Tech. J. 52, 1563–1578 (1973).
[Crossref]

Marton, J. P.

T. K. Lim, B. K. Garside, and J. P. Marton, “Guided Modes in Fibers with Parabolic-Index Core and Homogeneous Cladding,” Opt. Quantum Electron. 11, 329–344 (1979).
[Crossref]

Matsuhara, M.

M. Geshiro, M. Matsuhara, and N. Kumagai, “Truncated Parabolic-Index Fiber with Minimum Mode Dispersion,” IEEE Trans. Microwave Theory Tech. MTT-26, 115–119 (1978).
[Crossref]

Nemoto, S.

M. Hasimoto, S. Nemoto, and T. Makimoto, “Analysis of Guided Waves Along the Cladded Optical Fiber: Parabolic-Index Core and Homogeneous Cladding,”  MTT-25, 1–17 (1977).

Okamoto, K.

K. Okamoto and T. Okoshi, “Vectorial Wave Analysis of Inhomogeneous Optical Fibers using Finite Element Method,” IEEE Trans. Microwave Theory Tech. MTT-26, 109–114 (1978).
[Crossref]

K. Okamoto and T. Okoshi, “Analysis of Wave Propagation in Optical Fibers Having Core with α-Power Refractive Index Distribution and Uniform Cladding,” IEEE Trans. Microwave Theory Tech. MTT-24, 416–421 (1976).
[Crossref]

T. Okoshi and K. Okamoto, “Analysis of Wave Propagation in Inhomogeneous Optical Fibers using a Variational Method,” IEEE Trans. Microwave Theory Tech. MTT-22, 938–945 (1974).
[Crossref]

Okoshi, T.

K. Okamoto and T. Okoshi, “Vectorial Wave Analysis of Inhomogeneous Optical Fibers using Finite Element Method,” IEEE Trans. Microwave Theory Tech. MTT-26, 109–114 (1978).
[Crossref]

K. Okamoto and T. Okoshi, “Analysis of Wave Propagation in Optical Fibers Having Core with α-Power Refractive Index Distribution and Uniform Cladding,” IEEE Trans. Microwave Theory Tech. MTT-24, 416–421 (1976).
[Crossref]

T. Okoshi and K. Okamoto, “Analysis of Wave Propagation in Inhomogeneous Optical Fibers using a Variational Method,” IEEE Trans. Microwave Theory Tech. MTT-22, 938–945 (1974).
[Crossref]

Snyder, A. W.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1964).

Streifer, W.

C. N. Kurtz and W. Streifer, “Guided Waves in Inhomogeneous Focussing Media, Part I: Formulation, Solution for Quadratic Inhomogeneity,” IEEE Trans. Microwave Theory Tech. MTT-17, 11–15 (1969).
[Crossref]

W. Streifer and C. N. Kurtz, “Scalar Analysis of Radially Inhomogeneous Guiding Media,” J. Opt. Soc. Am. 57, 779–786 (1967).
[Crossref]

Yamada, R.

R. Yamada and Y. Inabe, “Guided Waves Along Graded Index Dielectric Rod,” IEEE Trans. Microwave Theory Tech. MTT-22, 813–814 (1974).
[Crossref]

Yip, G. L.

G. L. Yip and Y. H. Ahmew, “Propagation Characteristics of Radially Inhomogeneous Optical Fiber,” Electron. Lett. 10, 37–38 (1974).
[Crossref]

Young, W. R.

Analysis of Guided Waves Along the Cladded Optical Fiber: Parabolic-Index Core and Homogeneous Cladding (1)

M. Hasimoto, S. Nemoto, and T. Makimoto, “Analysis of Guided Waves Along the Cladded Optical Fiber: Parabolic-Index Core and Homogeneous Cladding,”  MTT-25, 1–17 (1977).

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

D. Gloge and E. A. J. Marcatili, “Multimode Theory of Graded-Core Fibers,” Bell Syst. Tech. J. 52, 1563–1578 (1973).
[Crossref]

Electron. Lett. (2)

P. J. B. Clarricoats and K. B. Chan, “Electromagnetic-Wave Propagation along Radially Inhomogeneous Dielectric Cylinders,” Electron. Lett. 6, 694–695 (1970).
[Crossref]

G. L. Yip and Y. H. Ahmew, “Propagation Characteristics of Radially Inhomogeneous Optical Fiber,” Electron. Lett. 10, 37–38 (1974).
[Crossref]

IEEE Trans. Microwave Theory Tech. (6)

R. Yamada and Y. Inabe, “Guided Waves Along Graded Index Dielectric Rod,” IEEE Trans. Microwave Theory Tech. MTT-22, 813–814 (1974).
[Crossref]

K. Okamoto and T. Okoshi, “Vectorial Wave Analysis of Inhomogeneous Optical Fibers using Finite Element Method,” IEEE Trans. Microwave Theory Tech. MTT-26, 109–114 (1978).
[Crossref]

C. N. Kurtz and W. Streifer, “Guided Waves in Inhomogeneous Focussing Media, Part I: Formulation, Solution for Quadratic Inhomogeneity,” IEEE Trans. Microwave Theory Tech. MTT-17, 11–15 (1969).
[Crossref]

T. Okoshi and K. Okamoto, “Analysis of Wave Propagation in Inhomogeneous Optical Fibers using a Variational Method,” IEEE Trans. Microwave Theory Tech. MTT-22, 938–945 (1974).
[Crossref]

K. Okamoto and T. Okoshi, “Analysis of Wave Propagation in Optical Fibers Having Core with α-Power Refractive Index Distribution and Uniform Cladding,” IEEE Trans. Microwave Theory Tech. MTT-24, 416–421 (1976).
[Crossref]

M. Geshiro, M. Matsuhara, and N. Kumagai, “Truncated Parabolic-Index Fiber with Minimum Mode Dispersion,” IEEE Trans. Microwave Theory Tech. MTT-26, 115–119 (1978).
[Crossref]

J. Opt. Soc. Am. (3)

Opt. Quantum Electron. (1)

T. K. Lim, B. K. Garside, and J. P. Marton, “Guided Modes in Fibers with Parabolic-Index Core and Homogeneous Cladding,” Opt. Quantum Electron. 11, 329–344 (1979).
[Crossref]

Other (2)

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1964).

H. Kirchoff, “Optical Wave Propagation in Self-Focussing Fibres,” in Proceedings of Conference on Trunk Telecommunications by Guilded Waves, 1970, London (IEEE Conf. Publn. No. 71), pp. 69–72.

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Figures (1)

FIG. 1
FIG. 1

Variation of the core-mode cutoff frequency v with the relative cladding thickness f = c/a for several lowest-order LPαm modes. Each of the open circles for f = ∞ and joined to the solid curves by dashed lines denotes the value of v for the corresponding infinite-cladding fiber mode. Parameters: n1 = 1.51, δ = 0.01 and n2 = n2(1 − δ)1/2.

Tables (2)

Tables Icon

TABLE I Comparison of the cutoff frequencies v of the parabolic-index fiber modes as calculated from the infinite-core approximation (ICA), the linearly polarized approximation (LPA), and the vector-wave analysis (VWA).12 Parameters: n1 = 1.51, δ = 0.01, n = n1(1 − δ)1/2.

Tables Icon

TABLE II Core-mode cutoff frequencies of the parabolic-index fibers with finite-thickness cladding. Parameters: n1 = 1.53, n2 = 1.50, n3 = 1.0, δ = 1 − (n2/n1)2 = 0.0388 and f = c/a = 1.25

Equations (50)

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n ( r ) = { n 1 [ 1 δ ( r / a ) 2 ] 1 / 2 0 r a n 2 a r ,
n 2 = n 1 ( 1 δ ) 1 / 2 .
δ = 1 ( n 2 / n 1 ) 2 1.
Φ α ( ξ ) = ξ α / 2 e ξ / 2 M ( A , B , ξ ) ,
ξ = ( r / w 0 ) 2 , w 0 2 = a / n 1 k 0 δ 1 / 2 ,
M ( A , B , ξ ) = l = 0 ( A ) l ξ l ( B ) l l ! , | ξ | < , B 0 , 1 , 2 , ,
A = ( 1 + α ) / 2 b / 4 , B = 1 + α , b = k 0 r t n 1 χ 1 / 2 , χ = δ ( r t / a ) 2 = 1 β 2 / n 1 k 0 2 .
E x = H y = 0 ,
E y = H x { Z 0 / n 1 Z 0 / n 2 } = E 0 { Φ α ( ξ ) / Φ α ( σ ) K α ( w r / a ) / K α ( w ) } cos ( α ϕ ) ,
σ = ξ | r = a = k 0 a n 1 δ 1 / 2 , w = a ( β 2 n 2 2 k 0 2 ) 1 / 2 .
E z = i Z 0 k 0 { 1 / n 1 2 1 / n 1 2 } H x y ,
H z = ( i / k 0 Z 0 ) E y / x .
E ϕ = E y cos ϕ E x sin ϕ ,
H ϕ = H y cos ϕ H x sin ϕ .
E z = i E 0 2 k 0 { [ b 2 / n 1 r t Φ α ( σ ) ] [ G 1 ( ξ ) sin ( α + 1 ) ϕ + G 2 ( ξ ) sin ( α 1 ) ϕ ] [ w / n 2 a K α ( w ) ] [ K α + 1 ( w r / a ) sin ( α + 1 ) ϕ + K α 1 ( w r / a ) sin ( α 1 ) ϕ ]
H z = i E 0 2 k 0 Z 0 { [ b 2 / r t Φ α ( σ ) ] [ G 1 ( ξ ) cos ( α + 1 ) ϕ + G 2 ( ξ ) cos ( α 1 ) ϕ ] [ w / a K α ( w ) ] [ K α + 1 ( w r / a ) cos ( α + 1 ) ϕ + K α 1 ( w r / a ) cos ( α 1 ) ϕ ]
E ϕ = ( E 0 / 2 ) { Φ α ( ξ ) / Φ α ( σ ) K α ( w r / a ) / K α ( w ) } [ cos ( α + 1 ) ϕ + cos ( α 1 ) ϕ ] ,
H ϕ = ( E 0 / 2 Z 0 ) { n 1 Φ α ( ξ ) / Φ α ( σ ) n 2 K α ( w r / a ) / K α ( w ) } [ sin ( α + 1 ) ϕ sin ( α 1 ) ϕ ] .
G j = ( 1 b 2 ) ( d Φ α d ρ + ( 1 ) α j ρ Φ a ) , j = 1 , 2
ρ = r / r t = ( ξ / b ) 1 / 2 .
( b 2 / r t ) G 2 ( u ) / Φ α ( u ) = ( w / a ) K α 1 ( w ) / K α ( w ) .
b 2 n 1 r t G 1 ( u ) Φ α ( u ) = w n 2 a K α + 1 ( w ) K α ( w ) , for HE α + 1 , m modes
b 2 n 1 r t G 2 ( u ) Φ α ( u ) = w n 2 a K α 1 ( w ) K α ( w ) , for EH α 1 , m ( or TE 0 m , TM 0 m ) modes .
u = ( b σ ) 1 / 2 = a ( n 1 2 k 0 2 β 2 ) 1 / 2 .
υ = ( u 2 + w 2 ) 1 / 2 = a k 0 ( n 1 2 n 2 2 ) 1 / 2 ,
[ u 2 / b ) N α ( u ) + 2 q 1 α ] = ( n 1 / n 2 ) w K α 1 ( w ) / K α ( w ) ,
q 1 = { n 1 / n 2 for HE α + 1 , m modes 1 for EH α 1 , m ( o r TE 0 m , TM 0 m ) modes ,
N α ( u ) = 2 A B M ( A + 1 , B + 1 , u 2 / b ) M ( A , B , u 2 / b ) 1.
w c = 0 , u c = b c = σ c = υ = a k 0 ( n 1 2 n 2 2 ) 1 / 2 ,
υ N α ( υ ) + 2 q 1 α = 0 ,
n ( r ) = { n 1 [ 1 δ ( r / a ) 2 ] 1 / 2 0 r a n 2 a r c n 3 c r < ,
| ( β n 2 k 0 ) / n 2 k 0 | 1.
n 2 k 0 β n 1 k 0 , for core modes n 3 k 0 β n 2 k 0 , for cladding modes ,
E y = H x { Z 0 / n 1 Z 0 / n 2 Z 0 / n 3 } = { A Φ α ( ξ ) / Φ α ( σ ) B K α ( w r / a ) / K α ( w ) + C I α ( w r / a ) / I α ( w ) D K α ( s r / a ) / K α ( s f ) } cos ( α ϕ ) ,
s = ( β 2 n 3 2 k 0 2 ) 1 / 2 a ,
f = c / a .
[ F α ( u ) + ( n 1 / n 2 ) w K α 1 ( w ) K α ( w ) ] [ Q α ( s ) + w f I α 1 ( w f ) I α ( w f ) ] C α ( w ) [ F α ( u ) ( n 1 / n 2 ) w I α 1 ( w ) I α ( w ) ] × [ Q α ( s ) w f K α 1 ( w f ) K α ( w f ) ] = 0 ,
C α ( w ) = I α ( w ) I α ( w f ) K α ( w f ) K α ( w ) ,
F α ( u ) = ( u 2 / b ) N α ( u ) + 2 q 1 α ,
Q α ( s ) = ( n 2 / n 3 ) s f K α 1 ( s f ) / K α ( s f ) + 2 α ( q 2 1 ) ,
q 2 = { n 2 / n 3 for HE α + 1 , m modes 1 for EH α 1 , m ( o r TE 0 m , TM 0 m ) modes .
E y = Z 0 n 2 H x = [ B Y α ( p r / a ) Y α ( p ) + C J α ( p r / a ) J α ( p ) ] cos ( α ϕ ) ,
p = i w = a ( n 2 2 k 0 2 β 2 ) 1 / 2 .
[ F α ( u ) ( n 1 / n 2 ) p Y α 1 ( p ) Y α ( p ) ] [ Q α ( s ) + p f J α 1 ( p f ) J α ( p f ) ] D α ( p ) [ F α ( u ) ( n 1 / n 2 ) p J α 1 ( p ) J α ( p ) ] × [ Q α ( s ) + p f Y α 1 ( p f ) Y α ( p f ) ] = 0 ,
D α ( p ) = J α ( p ) J α ( p f ) Y α ( p f ) Y α ( p ) ,
F 0 ( υ ) [ ln f + 1 / Q 0 ( s c ) ] + ( n 1 / n 2 ) = 0 , for α = 0 ( HE 1 , m modes )
F α ( υ ) [ 1 + 2 α / Q α ( s c ) ] f 2 α [ F α ( υ ) 2 α n 1 / n 2 ] = 0 , for α 1
F α ( υ ) = υ N α ( υ ) + 2 q 1 α ,
Q α ( s c ) ( n 2 / n 3 ) s c f K α 1 ( s c f ) / K α ( s c f ) + 2 α ( q 2 1 ) ,
s c = a k 0 ( n 2 2 n 3 2 ) 1 / 2 .