Abstract

The equivalent-step-index fiber method rests on the concept that properties of the fundamental mode are not very sensitive to refractive-index profile details. We show how this idea may be incorporated into the original mathematical scheme of Snyder and Sammut [ J. Opt. Soc. Am. 69, 1663 ( 1979)] in order to simplify greatly the calculations and to provide wavelength-independent equivalent step-index parameters. The new approach uses the effective waveguide parameter and the moments of the profile-shape function.

© 1984 Optical Society of America

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References

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  1. A. W. Snyder, R. A. Sammut, “Fundamental (HE11) modes of graded optical fibers,” J. Opt. Soc. Am. 69, 1663–1671 (1979).
    [CrossRef]
  2. E. Brinkmeyer, “Spot size of graded-index single-mode fibers: profile-independent representation and new determination method,” Appl. Opt. 18, 932–937 (1979).
    [CrossRef] [PubMed]
  3. C. Pask, R. A. Sammut, “Experimental characterization of graded-index single-mode fibers,” Electron. Lett. 16, 310–311 (1980).
    [CrossRef]
  4. H. Matsumura, T. Suganuma, “Normalization of single-mode fibers having an arbitrary index profile,” Appl. Opt. 19, 3151–3159 (1980).
    [CrossRef] [PubMed]
  5. C. A. Millar, “Direct method of determining equivalent-step-index profiles for monomode fibers,” Electron. Lett. 17, 458–460 (1981).
    [CrossRef]
  6. F. Alard, L. Jeunhomme, P. Sansonetti, “Fundamental mode spot-size measurements in single-mode optical fibers,” Electron. Lett. 17, 958–960 (1981).
    [CrossRef]
  7. C. D. Hussey, C. Pask, “Theory of the profile-moments description of single-mode fibers,” Proc. Inst. Electr. Eng. Part H 129, 123–134 (1982).
  8. W. A. Gambling, H. Matsumura, C. M. Ragdale, “Wave propagation in a single-mode fiber with dip in the refractive index,” Opt. Quantum Electron. 10, 301–309 (1978).
    [CrossRef]
  9. W. J. Stewart, “Simplified parameter based analysis of single-mode optical guides,” Electron. Lett. 16, 380–382 (1980).
    [CrossRef]
  10. A. W. Snyder, “Understanding monomode optical fibers,” Proc. IEEE 69, 6–13 (1981).
    [CrossRef]
  11. C. Yeh, “Noncircular dielectric waveguides,” in Optical Waveguides, N. S. Kapany, J. J. Burke, eds. (Academic, New York, 1972), App. A.
    [CrossRef]
  12. L. Eyges, “Fiber optics guides of noncircular cross section,” Appl. Opt. 17, 1673–1674 (1978).
    [CrossRef] [PubMed]
  13. R. J. Black, C. Pask, “Equivalent optical waveguides,” IEEE J. Lightwave Technol. LT-2, 268–276 (1984).
    [CrossRef]
  14. A. W. Snyder, “Asymptotic expressions for eigenfunctions and eigenvalues of a dielectric optical waveguide,” IEEE Trans. Microwave Theory Tech. MTT-17, 1130–1138 (1969).
    [CrossRef]
  15. D. Gloge, “Weakly guiding fibers,” Appl. Opt. 10, 2252–2258 (1971).
    [CrossRef] [PubMed]
  16. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C., 1964).

1984 (1)

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

1982 (1)

C. D. Hussey, C. Pask, “Theory of the profile-moments description of single-mode fibers,” Proc. Inst. Electr. Eng. Part H 129, 123–134 (1982).

1981 (3)

C. A. Millar, “Direct method of determining equivalent-step-index profiles for monomode fibers,” Electron. Lett. 17, 458–460 (1981).
[CrossRef]

F. Alard, L. Jeunhomme, P. Sansonetti, “Fundamental mode spot-size measurements in single-mode optical fibers,” Electron. Lett. 17, 958–960 (1981).
[CrossRef]

A. W. Snyder, “Understanding monomode optical fibers,” Proc. IEEE 69, 6–13 (1981).
[CrossRef]

1980 (3)

W. J. Stewart, “Simplified parameter based analysis of single-mode optical guides,” Electron. Lett. 16, 380–382 (1980).
[CrossRef]

C. Pask, R. A. Sammut, “Experimental characterization of graded-index single-mode fibers,” Electron. Lett. 16, 310–311 (1980).
[CrossRef]

H. Matsumura, T. Suganuma, “Normalization of single-mode fibers having an arbitrary index profile,” Appl. Opt. 19, 3151–3159 (1980).
[CrossRef] [PubMed]

1979 (2)

1978 (2)

W. A. Gambling, H. Matsumura, C. M. Ragdale, “Wave propagation in a single-mode fiber with dip in the refractive index,” Opt. Quantum Electron. 10, 301–309 (1978).
[CrossRef]

L. Eyges, “Fiber optics guides of noncircular cross section,” Appl. Opt. 17, 1673–1674 (1978).
[CrossRef] [PubMed]

1971 (1)

1969 (1)

A. W. Snyder, “Asymptotic expressions for eigenfunctions and eigenvalues of a dielectric optical waveguide,” IEEE Trans. Microwave Theory Tech. MTT-17, 1130–1138 (1969).
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C., 1964).

Alard, F.

F. Alard, L. Jeunhomme, P. Sansonetti, “Fundamental mode spot-size measurements in single-mode optical fibers,” Electron. Lett. 17, 958–960 (1981).
[CrossRef]

Black, R. J.

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

Brinkmeyer, E.

Eyges, L.

Gambling, W. A.

W. A. Gambling, H. Matsumura, C. M. Ragdale, “Wave propagation in a single-mode fiber with dip in the refractive index,” Opt. Quantum Electron. 10, 301–309 (1978).
[CrossRef]

Gloge, D.

Hussey, C. D.

C. D. Hussey, C. Pask, “Theory of the profile-moments description of single-mode fibers,” Proc. Inst. Electr. Eng. Part H 129, 123–134 (1982).

Jeunhomme, L.

F. Alard, L. Jeunhomme, P. Sansonetti, “Fundamental mode spot-size measurements in single-mode optical fibers,” Electron. Lett. 17, 958–960 (1981).
[CrossRef]

Matsumura, H.

H. Matsumura, T. Suganuma, “Normalization of single-mode fibers having an arbitrary index profile,” Appl. Opt. 19, 3151–3159 (1980).
[CrossRef] [PubMed]

W. A. Gambling, H. Matsumura, C. M. Ragdale, “Wave propagation in a single-mode fiber with dip in the refractive index,” Opt. Quantum Electron. 10, 301–309 (1978).
[CrossRef]

Millar, C. A.

C. A. Millar, “Direct method of determining equivalent-step-index profiles for monomode fibers,” Electron. Lett. 17, 458–460 (1981).
[CrossRef]

Pask, C.

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

C. D. Hussey, C. Pask, “Theory of the profile-moments description of single-mode fibers,” Proc. Inst. Electr. Eng. Part H 129, 123–134 (1982).

C. Pask, R. A. Sammut, “Experimental characterization of graded-index single-mode fibers,” Electron. Lett. 16, 310–311 (1980).
[CrossRef]

Ragdale, C. M.

W. A. Gambling, H. Matsumura, C. M. Ragdale, “Wave propagation in a single-mode fiber with dip in the refractive index,” Opt. Quantum Electron. 10, 301–309 (1978).
[CrossRef]

Sammut, R. A.

C. Pask, R. A. Sammut, “Experimental characterization of graded-index single-mode fibers,” Electron. Lett. 16, 310–311 (1980).
[CrossRef]

A. W. Snyder, R. A. Sammut, “Fundamental (HE11) modes of graded optical fibers,” J. Opt. Soc. Am. 69, 1663–1671 (1979).
[CrossRef]

Sansonetti, P.

F. Alard, L. Jeunhomme, P. Sansonetti, “Fundamental mode spot-size measurements in single-mode optical fibers,” Electron. Lett. 17, 958–960 (1981).
[CrossRef]

Snyder, A. W.

A. W. Snyder, “Understanding monomode optical fibers,” Proc. IEEE 69, 6–13 (1981).
[CrossRef]

A. W. Snyder, R. A. Sammut, “Fundamental (HE11) modes of graded optical fibers,” J. Opt. Soc. Am. 69, 1663–1671 (1979).
[CrossRef]

A. W. Snyder, “Asymptotic expressions for eigenfunctions and eigenvalues of a dielectric optical waveguide,” IEEE Trans. Microwave Theory Tech. MTT-17, 1130–1138 (1969).
[CrossRef]

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C., 1964).

Stewart, W. J.

W. J. Stewart, “Simplified parameter based analysis of single-mode optical guides,” Electron. Lett. 16, 380–382 (1980).
[CrossRef]

Suganuma, T.

Yeh, C.

C. Yeh, “Noncircular dielectric waveguides,” in Optical Waveguides, N. S. Kapany, J. J. Burke, eds. (Academic, New York, 1972), App. A.
[CrossRef]

Appl. Opt. (4)

Electron. Lett. (4)

W. J. Stewart, “Simplified parameter based analysis of single-mode optical guides,” Electron. Lett. 16, 380–382 (1980).
[CrossRef]

C. Pask, R. A. Sammut, “Experimental characterization of graded-index single-mode fibers,” Electron. Lett. 16, 310–311 (1980).
[CrossRef]

C. A. Millar, “Direct method of determining equivalent-step-index profiles for monomode fibers,” Electron. Lett. 17, 458–460 (1981).
[CrossRef]

F. Alard, L. Jeunhomme, P. Sansonetti, “Fundamental mode spot-size measurements in single-mode optical fibers,” Electron. Lett. 17, 958–960 (1981).
[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 Trans. Microwave Theory Tech. (1)

A. W. Snyder, “Asymptotic expressions for eigenfunctions and eigenvalues of a dielectric optical waveguide,” IEEE Trans. Microwave Theory Tech. MTT-17, 1130–1138 (1969).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Quantum Electron. (1)

W. A. Gambling, H. Matsumura, C. M. Ragdale, “Wave propagation in a single-mode fiber with dip in the refractive index,” Opt. Quantum Electron. 10, 301–309 (1978).
[CrossRef]

Proc. IEEE (1)

A. W. Snyder, “Understanding monomode optical fibers,” Proc. IEEE 69, 6–13 (1981).
[CrossRef]

Proc. Inst. Electr. Eng. Part H (1)

C. D. Hussey, C. Pask, “Theory of the profile-moments description of single-mode fibers,” Proc. Inst. Electr. Eng. Part H 129, 123–134 (1982).

Other (2)

C. Yeh, “Noncircular dielectric waveguides,” in Optical Waveguides, N. S. Kapany, J. J. Burke, eds. (Academic, New York, 1972), App. A.
[CrossRef]

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C., 1964).

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Tables (1)

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Table 1 Percentage Errors in the Modal Eigenvalue U as Given by Schemes 1–5 for a Parabolic-Index Fiber and Radius of ESI Fibers

Equations (22)

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n 2 ( r ) = { n cl 2 + ( n 0 2 - n cl 2 ) s ( r / ρ ) r ρ n cl 2 r > ρ ,
V = k ρ ( n 0 2 - n cl 2 ) 1 / 2 ,
β 2 f ( V s , ρ s ) β s 2 + k 2 0 [ n 2 ( r ) - n s 2 ( r ) ] ψ s 2 r d r 0 ψ s 2 r d r ,
V eff = V 2 Ω 0 ,
Ω m = 0 1 s ( R ) R m + 1 d R ,
V s = V eff , g .
ρ s 2 ( n 0 s 2 - n cl 2 ) = 2 Ω 0 ρ 2 ( n 0 2 - n cl 2 ) ,
β 2 f ( V eff , ρ s ) ,
W 2 = ρ 2 ( β 2 - k 2 n cl 2 )
b = W 2 / V 2 ,
b eff = W 2 / V eff 2 .
α = ρ / ρ s ,
b eff F ( V s , α ) = α 2 [ b s ( V s ) - C ( α , V s ) ] ,
C ( α , V s ) = η s ( V s ) { 1 - 0 1 s ( R ) ψ s 2 ( V s , α R ) R d R / Ω 0 × [ J 0 2 ( U s ) + J 1 2 ( U s ) ] } .
ψ s ( V s , X ) = { J 0 ( U s X ) X 1 , K 0 ( W s X ) J 0 ( U s ) / K 0 ( W s ) X > 1 ,
ψ s ( V s , X ) = m = 0 a m ( V s ) X 2 m ,
C ( α , V s ) = η s ( V s ) { 1 - m = 0 a m ( V s ) α 2 m Ω 2 m / Ω 0 × [ J 0 2 ( U s ) + J 1 2 ( U s ) ] } .
α 2 = [ J 0 2 ( U s ) - a 0 ] Ω 0 / 2 a 1 Ω 2 .
α 2 = Ω 0 / 2 Ω 2
ρ s = ρ / α = ρ ( Ω 0 / 2 Ω 2 ) 1 / 2 .
b eff ( V eff ) α 2 b s ( V s ) = α 2 b s ( V eff )
β β s ,

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