Abstract

The effective refractive index as a function of vacuum wavelength is approximated by Lagrange interpolation polynomials. The rms value of the chromatic dispersion is then calculated analytically. It is demonstrated that use of fourth-degree polynomials is far more efficient than the use of second-degree polynomials. The rms value of the chromatic dispersion over the wavelength range (1.25 μm, 1.60 μm) is calculated and minimized for step-index fibers, triangular index fibers, and α-power fibers. The full vector solution of Maxwell’s equations is used. The error induced by the approximate refractive-index model is found to be negligible at the point of minimum dispersion.

© 1993 Optical Society of America

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References

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  1. B. J. Ainslie, C. R. Day, “A review of single-mode fibers with modified dispersion characteristics,” J. Lightwave Technol. LT-4, 967–979 (1986).
    [CrossRef]
  2. D. Marcuse, “Pulse distortion in single-mode fibers,” Appl. Opt. 19, 1653–1660 (1980).
    [CrossRef] [PubMed]
  3. J. W. Fleming, “Material dispersion in lightguide glasses,” Electron. Lett. 14, 326–328 (1978).
    [CrossRef]
  4. M. J. Adams, An Introduction to Optical Waveguides (Wiley, New York, 1981), pp. 243–245.
  5. G. L. Yip, J. J. Jiang, “Dispersion studies of a single-mode triangular-index fiber with a trench by the vector mode analysis,” Appl. Opt. 29, 5343–5352 (1990).
    [CrossRef] [PubMed]
  6. A. Safaai-Jazi, L. J. Lu, “Accuracy of approximate methods for the evaluation of chromatic dispersion in dispersion-flattened fibers,” J. Lightwave Technol. 8, 1145–1150 (1990).
    [CrossRef]
  7. S. J. Garth, “Effect of bending on zero dispersion operation of single-mode optical fibers,” Appl. Opt. 30, 1048–1051 (1991).
    [CrossRef] [PubMed]
  8. R. Lundin, “A general power-series expansion method for exact analysis of the guided modes in an optical fiber,” J. Lightwave Technol. LT-4, 1617–1625 (1986).
    [CrossRef]
  9. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C., 1964), pp. 878–879.
  10. K. Morishita, “Hybrid modes in circular cylindrical optical fibers,” IEEE Trans. Microwave Theory Tech. MTT-31, 344–350 (1983).
    [CrossRef]
  11. G. Keiser, Optical Fiber Communications (McGraw-Hill, 1983), p. 39.
  12. K. Oyamada, T. Okoshi, “High-accuracy numerical data on propagation characteristics of α-power graded core fibers,” IEEE Trans. Microwave Theory Tech. MTT-28, 1113–1118 (1980).
    [CrossRef]

1991

1990

G. L. Yip, J. J. Jiang, “Dispersion studies of a single-mode triangular-index fiber with a trench by the vector mode analysis,” Appl. Opt. 29, 5343–5352 (1990).
[CrossRef] [PubMed]

A. Safaai-Jazi, L. J. Lu, “Accuracy of approximate methods for the evaluation of chromatic dispersion in dispersion-flattened fibers,” J. Lightwave Technol. 8, 1145–1150 (1990).
[CrossRef]

1986

B. J. Ainslie, C. R. Day, “A review of single-mode fibers with modified dispersion characteristics,” J. Lightwave Technol. LT-4, 967–979 (1986).
[CrossRef]

R. Lundin, “A general power-series expansion method for exact analysis of the guided modes in an optical fiber,” J. Lightwave Technol. LT-4, 1617–1625 (1986).
[CrossRef]

1983

K. Morishita, “Hybrid modes in circular cylindrical optical fibers,” IEEE Trans. Microwave Theory Tech. MTT-31, 344–350 (1983).
[CrossRef]

1980

K. Oyamada, T. Okoshi, “High-accuracy numerical data on propagation characteristics of α-power graded core fibers,” IEEE Trans. Microwave Theory Tech. MTT-28, 1113–1118 (1980).
[CrossRef]

D. Marcuse, “Pulse distortion in single-mode fibers,” Appl. Opt. 19, 1653–1660 (1980).
[CrossRef] [PubMed]

1978

J. W. Fleming, “Material dispersion in lightguide glasses,” Electron. Lett. 14, 326–328 (1978).
[CrossRef]

Adams, M. J.

M. J. Adams, An Introduction to Optical Waveguides (Wiley, New York, 1981), pp. 243–245.

Ainslie, B. J.

B. J. Ainslie, C. R. Day, “A review of single-mode fibers with modified dispersion characteristics,” J. Lightwave Technol. LT-4, 967–979 (1986).
[CrossRef]

Day, C. R.

B. J. Ainslie, C. R. Day, “A review of single-mode fibers with modified dispersion characteristics,” J. Lightwave Technol. LT-4, 967–979 (1986).
[CrossRef]

Fleming, J. W.

J. W. Fleming, “Material dispersion in lightguide glasses,” Electron. Lett. 14, 326–328 (1978).
[CrossRef]

Garth, S. J.

Jiang, J. J.

Keiser, G.

G. Keiser, Optical Fiber Communications (McGraw-Hill, 1983), p. 39.

Lu, L. J.

A. Safaai-Jazi, L. J. Lu, “Accuracy of approximate methods for the evaluation of chromatic dispersion in dispersion-flattened fibers,” J. Lightwave Technol. 8, 1145–1150 (1990).
[CrossRef]

Lundin, R.

R. Lundin, “A general power-series expansion method for exact analysis of the guided modes in an optical fiber,” J. Lightwave Technol. LT-4, 1617–1625 (1986).
[CrossRef]

Marcuse, D.

Morishita, K.

K. Morishita, “Hybrid modes in circular cylindrical optical fibers,” IEEE Trans. Microwave Theory Tech. MTT-31, 344–350 (1983).
[CrossRef]

Okoshi, T.

K. Oyamada, T. Okoshi, “High-accuracy numerical data on propagation characteristics of α-power graded core fibers,” IEEE Trans. Microwave Theory Tech. MTT-28, 1113–1118 (1980).
[CrossRef]

Oyamada, K.

K. Oyamada, T. Okoshi, “High-accuracy numerical data on propagation characteristics of α-power graded core fibers,” IEEE Trans. Microwave Theory Tech. MTT-28, 1113–1118 (1980).
[CrossRef]

Safaai-Jazi, A.

A. Safaai-Jazi, L. J. Lu, “Accuracy of approximate methods for the evaluation of chromatic dispersion in dispersion-flattened fibers,” J. Lightwave Technol. 8, 1145–1150 (1990).
[CrossRef]

Yip, G. L.

Appl. Opt.

Electron. Lett.

J. W. Fleming, “Material dispersion in lightguide glasses,” Electron. Lett. 14, 326–328 (1978).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

K. Morishita, “Hybrid modes in circular cylindrical optical fibers,” IEEE Trans. Microwave Theory Tech. MTT-31, 344–350 (1983).
[CrossRef]

K. Oyamada, T. Okoshi, “High-accuracy numerical data on propagation characteristics of α-power graded core fibers,” IEEE Trans. Microwave Theory Tech. MTT-28, 1113–1118 (1980).
[CrossRef]

J. Lightwave Technol.

B. J. Ainslie, C. R. Day, “A review of single-mode fibers with modified dispersion characteristics,” J. Lightwave Technol. LT-4, 967–979 (1986).
[CrossRef]

R. Lundin, “A general power-series expansion method for exact analysis of the guided modes in an optical fiber,” J. Lightwave Technol. LT-4, 1617–1625 (1986).
[CrossRef]

A. Safaai-Jazi, L. J. Lu, “Accuracy of approximate methods for the evaluation of chromatic dispersion in dispersion-flattened fibers,” J. Lightwave Technol. 8, 1145–1150 (1990).
[CrossRef]

Other

M. J. Adams, An Introduction to Optical Waveguides (Wiley, New York, 1981), pp. 243–245.

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C., 1964), pp. 878–879.

G. Keiser, Optical Fiber Communications (McGraw-Hill, 1983), p. 39.

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Figures (4)

Fig. 1
Fig. 1

Number of significant figures in the rms values f of the material dispersion as a function of the number of quadrature points, i.e., as a function of the number of refractive-index evaluations.

Fig. 2
Fig. 2

Root-mean-square value f of the chromatic dispersion over the vacuum wavelength range (1.25 μm, 1.60 μ) for a step-index fiber as a function of the relative refractive-index increase N1 in the core.

Fig. 3
Fig. 3

Chromatic dispersion of the step-index profile that minimizes the rms value f of the chromatic dispersion over the vacuum wavelength range (1.25 μm, 1.60 μm).

Fig. 4
Fig. 4

Root-mean-square value f of the chromatic dispersion over the vacuum wavelength range (1.25 μm, 1.60 μm) for different α-power fibers as a function of the relative refractive-index increase N1 in the core center.

Equations (27)

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exp [ j ( ω t - β z ) ] ,
v g = d ω d β ,
τ g = 1 v g .
C = d τ g d λ 0 ,
σ out 2 = σ in 2 + ( Δ λ 0 L C ) 2 ,
n e = β / k 0 ,
C = - λ 0 c d 2 n e d λ 0 2 ,
C m = - λ 0 c d 2 n s d λ 0 2 .
n ( r , λ 0 ) = N ( r , λ 0 ) n s ( λ 0 ) ,
n ( r , λ 0 ) = N ( r ) n s ( λ 0 ) .
f = ( 1 λ 2 - λ 1 λ 1 λ 2 C 2 ( λ 0 ) d λ 0 ) 1 / 2 .
5 , 9 , , n , n + n - 1 , .
N ( r ) = N 1 ,             0 r < a , = 1 ,             r > a ,
n ( r , λ 0 ) = N 1 n s ( λ 0 ) ,             0 r < a , = n s ( λ 0 ) ,             r > a .
λ c = 1.25 × 10 - 6 m ,
V = j 01 = 2.405 ,
V = 2 π λ 0 a ( N 1 2 - 1 ) 1 / 2 n s .
a ( N 1 2 - 1 ) 1 / 2 = j 01 λ c 2 π n s ( λ c ) = 3.3 × 10 - 7 m .
f min = 4.8 ps / ( km nm ) , N 1 = 1.0103 , a = 2.3 × 10 - 6 m ,
N ( r ) = N 1 + ( 1 - N 1 ) ( r / a ) , 0 r a , = 1 , r a .
f min = 6.7 ps / ( km nm ) , N 1 = 1.0110 , a = 4.0 × 10 - 6 m .
N ( r ) = [ N 1 2 - ( N 1 2 - 1 ) ( r / a ) α ] 1 / 2 , 0 r a , = 1 , r a .
a b x 2 ( d 2 f d x 2 ) 2 d x ,
x p = a + b 2 + p b - a 4 ,             p = - 2 , - 1 , 0 , 1 , 2.
d 2 f d x 2 = ( 4 b - a ) 2 d 2 f d p 2 ,
d 2 f d p 2 = - f - 2 + 16 f - 1 - 30 f 0 + 16 f 1 - f 2 12 + - f - 2 + 2 f - 1 - 2 f 1 + f 2 2 p + f - 2 - 4 f - 1 + 6 f 0 - 4 f 1 + 6 f 2 2 p 2 .
a b x 2 ( d 2 f d x 2 ) 2 d x = ( a + b 2 ) 2 ( 4 b - a ) 3 - 2 2 ( d 2 f d p 2 ) 2 d p + ( a + b 2 ) ( 4 b - a ) 2 - 2 2 2 p ( d 2 f d p 2 ) 2 d p + ( 4 b - a ) - 2 2 p 2 ( d 2 f d p 2 ) 2 d p .

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