Abstract

An asymptotic study of the effective-index method for the analysis of rectangular-core dielectric waveguides is given. Two ways of applying the effective-index method, depending on how the effective index is calculated, are considered, and expressions for the errors in the calculation of the propagation constant are derived. These expressions show explicitly how the accuracy of the method varies with the normalized frequency, the mode orders, the dimensions of the waveguide, and the relative refractive indices of the core and the surrounding media. Many novel properties of the method are revealed by these expressions. For example, it can be shown that the effective-index method can underestimate the propagation constant for a strip waveguide.

© 1991 Optical Society of America

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References

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  1. R. M. Knox, P. P. Toulios, in Proceedings of MRI Symposium on Submillimeter Waves, J. Fox, ed. (Polytechnic, New York, 1970), p. 497.
  2. K. S. Chiang, Appl. Opt. 25, 2169 (1986).
    [Crossref] [PubMed]
  3. A. Kumar, D. F. Clark, B. Culshaw, Opt. Lett. 13, 1129 (1988).
    [Crossref] [PubMed]
  4. A. W. Snyder, J. D. Love, Optical Wavegide Theory (Chapman & Hall, London, 1983), Chap. 18.
  5. A. W. Snyder, X.-H. Zheng, J. Opt. Soc. Am. A 3, 600 (1986).
    [Crossref]
  6. B. M. A. Rahman, J. B. Davies, IEE Proc. Pt. J 132, 349 (1985).
  7. M. J. Robertson, P. C. Kendall, S. Ritchie, P. W. A. McIlory, M. J. Adams, IEEE J. Lightwave Technol. 7, 2105 (1989).
    [Crossref]
  8. K. S. Chiang, Appl. Opt. 25, 348 (1986).
    [Crossref] [PubMed]
  9. L. Eyges, P. Gianino, P. Wintersteiner, J. Opt. Soc. Am. 69, 1226 (1979).
    [Crossref]
  10. K. S. Chiang, IEEE J. Lightwave Technol. 9, 62 (1991).
    [Crossref]

1991 (1)

K. S. Chiang, IEEE J. Lightwave Technol. 9, 62 (1991).
[Crossref]

1989 (1)

M. J. Robertson, P. C. Kendall, S. Ritchie, P. W. A. McIlory, M. J. Adams, IEEE J. Lightwave Technol. 7, 2105 (1989).
[Crossref]

1988 (1)

1986 (3)

1985 (1)

B. M. A. Rahman, J. B. Davies, IEE Proc. Pt. J 132, 349 (1985).

1979 (1)

Adams, M. J.

M. J. Robertson, P. C. Kendall, S. Ritchie, P. W. A. McIlory, M. J. Adams, IEEE J. Lightwave Technol. 7, 2105 (1989).
[Crossref]

Chiang, K. S.

Clark, D. F.

Culshaw, B.

Davies, J. B.

B. M. A. Rahman, J. B. Davies, IEE Proc. Pt. J 132, 349 (1985).

Eyges, L.

Gianino, P.

Kendall, P. C.

M. J. Robertson, P. C. Kendall, S. Ritchie, P. W. A. McIlory, M. J. Adams, IEEE J. Lightwave Technol. 7, 2105 (1989).
[Crossref]

Knox, R. M.

R. M. Knox, P. P. Toulios, in Proceedings of MRI Symposium on Submillimeter Waves, J. Fox, ed. (Polytechnic, New York, 1970), p. 497.

Kumar, A.

Love, J. D.

A. W. Snyder, J. D. Love, Optical Wavegide Theory (Chapman & Hall, London, 1983), Chap. 18.

McIlory, P. W. A.

M. J. Robertson, P. C. Kendall, S. Ritchie, P. W. A. McIlory, M. J. Adams, IEEE J. Lightwave Technol. 7, 2105 (1989).
[Crossref]

Rahman, B. M. A.

B. M. A. Rahman, J. B. Davies, IEE Proc. Pt. J 132, 349 (1985).

Ritchie, S.

M. J. Robertson, P. C. Kendall, S. Ritchie, P. W. A. McIlory, M. J. Adams, IEEE J. Lightwave Technol. 7, 2105 (1989).
[Crossref]

Robertson, M. J.

M. J. Robertson, P. C. Kendall, S. Ritchie, P. W. A. McIlory, M. J. Adams, IEEE J. Lightwave Technol. 7, 2105 (1989).
[Crossref]

Snyder, A. W.

A. W. Snyder, X.-H. Zheng, J. Opt. Soc. Am. A 3, 600 (1986).
[Crossref]

A. W. Snyder, J. D. Love, Optical Wavegide Theory (Chapman & Hall, London, 1983), Chap. 18.

Toulios, P. P.

R. M. Knox, P. P. Toulios, in Proceedings of MRI Symposium on Submillimeter Waves, J. Fox, ed. (Polytechnic, New York, 1970), p. 497.

Wintersteiner, P.

Zheng, X.-H.

Appl. Opt. (2)

IEE Proc. Pt. J (1)

B. M. A. Rahman, J. B. Davies, IEE Proc. Pt. J 132, 349 (1985).

IEEE J. Lightwave Technol. (2)

M. J. Robertson, P. C. Kendall, S. Ritchie, P. W. A. McIlory, M. J. Adams, IEEE J. Lightwave Technol. 7, 2105 (1989).
[Crossref]

K. S. Chiang, IEEE J. Lightwave Technol. 9, 62 (1991).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Lett. (1)

Other (2)

R. M. Knox, P. P. Toulios, in Proceedings of MRI Symposium on Submillimeter Waves, J. Fox, ed. (Polytechnic, New York, 1970), p. 497.

A. W. Snyder, J. D. Love, Optical Wavegide Theory (Chapman & Hall, London, 1983), Chap. 18.

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Figures (4)

Fig. 1
Fig. 1

Rectangular-core dielectric waveguide.

Fig. 2
Fig. 2

Waveguide that is exactly analyzed by the effective-index method that results in an x-dependent index profile.

Fig. 3
Fig. 3

Waveguide that is exactly analyzed by the effective-index method that results in a y-dependent index profile.

Fig. 4
Fig. 4

Difference in the calculated normalized propagation constants Py2Px2 as a function of V for the E11 mode in a channel waveguide with an aspect ratio R and indices n1 = 1.5, n2 = n4 = 1.48492, and n3 = 1.0. Solid curves, numerical results; dotted curves, asymptotic results.

Tables (1)

Tables Icon

Table 1 Calculated Normalized Propagation Constants for the First 13 Modes of a Rectangular-Core Fiber with R = 2 at V = 2π

Equations (7)

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x = ( n 1 2 - n x 2 ) k 2 P x 1 - ( n x 2 - n 4 2 ) k 2 P x 2 ,
P x 1 = 2 0 2 b - 0 ψ ψ ˜ d x d y - + - + ψ ψ ˜ d x d y ,
P x 2 = 2 - 0 - 0 ψ ψ ˜ d x d y + 2 2 b + - 0 ψ ψ ˜ d x d y - + - + ψ ψ ˜ d x d y ,
ψ ψ ˜ { sin m π x 2 a sin n π y 2 b             0 < x < 2 a , 0 < y < 2 b m π b 2 a V ( Δ 2 Δ 4 ) 1 / 2 sin n π y 2 b exp [ V b ( Δ 4 Δ 2 ) 1 / 2 x ]             - < x 0 , 0 < y < 2 b , ( - 1 ) n + 1 m n π 2 b 4 a V 2 ( Δ 2 Δ 4 ) 1 / 2 exp ( V b y ) exp [ V b ( Δ 4 Δ 2 ) 1 / 2 x ]             - < x 0 , - < y 0 , m n π 2 b 4 a V 2 ( Δ 2 Δ 3 ) 1 / 2 ( Δ 2 Δ 4 ) 1 / 2 exp [ - V b ( Δ 3 Δ 2 ) 1 / 2 ( y - 2 b ) ] exp [ V b ( Δ 4 Δ 2 ) 1 / 2 x ]             - < x 0 , 2 b y + .
x = π 4 m 2 n 2 16 R 3 b 2 V 3 ( Δ 2 Δ 4 ) 1 / 2 { Δ 2 Δ 4 - 1 2 V [ 1 + ( Δ 2 Δ 3 ) 3 / 2 ] } ,
y = π 4 m 2 n 2 32 R 2 b 2 V 3 [ 1 + ( Δ 2 Δ 3 ) 3 / 2 ] [ 1 - 1 R V ( Δ 2 Δ 4 ) 1 / 2 ] ,
β 2 = R β x 2 - β y 2 R - 1 .

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