Abstract

An approximate semi-analytical iterative method is presented to find vector modes of high-index contrast single mode waveguides. Present method is developed to provide improvement over scalar analysis of Vopt method. To illustrate the accuracy and efficiency of this method, modal properties of silicon strip nanoscale waveguide are studied in detail and compared with other approximate and rigorous numerical analysis.

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References

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  1. M. Lipson, “Guiding, modulating, and emitting light on silicon-challenges and opportunities,” J. Lightwave Technol.23, 4222–4238 (2005).
    [CrossRef]
  2. Y. Vlasov and S. McNab, “Losses in single-mode silicon-on-insulator strip waveguides and bends,” Opt. Express12, 1622–1631 (2004).
    [CrossRef] [PubMed]
  3. D. M. H. Leung, N. Kejalakshmy, B. M. A. Rahman, and K. T. V. Grattan, “Rigorous modal analysis of silicon strip nanoscale waveguides,” Opt. Express18, 8528–8539 (2010).
    [CrossRef] [PubMed]
  4. E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J.48, 2071–2102 (1969).
  5. R. M. Knox and P. P. Toulios, “Integrated circuits for the millimeter through optical frequency range,” in Proceedings of the Symposium on Submillimeter Waves, M. H. Schlam and J. Fox, (Polytechnic Press, 1970), pp. 497–516.
  6. K. S. Chiang, “Dual effective-index method for the analysis of rectangular dielectric waveguides,” Appl. Opt.25, 2169–2174 (1986).
    [CrossRef] [PubMed]
  7. P. C. Kendall, M. J. Adams, S. Ritchie, and M. J. Robertson, “Theory for calculating approximate values for the propagation constants of an optical rib waveguide by weighting the refractive indices,” IEE Proc. A134, 699–702 (1987).
  8. A. Sharma, “On approximate theories of rectangular waveguides,” Opt. Quant. Electron.21, 517–520 (1989).
    [CrossRef]
  9. K. S. Chiang, “Analysis of the effective-index method for the vector modes of rectangular-core dielectric waveguides,” IEEE Trans. Microwave Theory Tech.44, 692–700 (1996).
    [CrossRef]
  10. A. Sharma, “Analysis of integrated optical waveguides: variational method and effective-index method with built-in perturbation correction,” J. Opt. Soc. Am. A18, 1383–1387 (2001).
    [CrossRef]

2010

2005

2004

2001

1996

K. S. Chiang, “Analysis of the effective-index method for the vector modes of rectangular-core dielectric waveguides,” IEEE Trans. Microwave Theory Tech.44, 692–700 (1996).
[CrossRef]

1989

A. Sharma, “On approximate theories of rectangular waveguides,” Opt. Quant. Electron.21, 517–520 (1989).
[CrossRef]

1987

P. C. Kendall, M. J. Adams, S. Ritchie, and M. J. Robertson, “Theory for calculating approximate values for the propagation constants of an optical rib waveguide by weighting the refractive indices,” IEE Proc. A134, 699–702 (1987).

1986

1969

E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J.48, 2071–2102 (1969).

Adams, M. J.

P. C. Kendall, M. J. Adams, S. Ritchie, and M. J. Robertson, “Theory for calculating approximate values for the propagation constants of an optical rib waveguide by weighting the refractive indices,” IEE Proc. A134, 699–702 (1987).

Chiang, K. S.

K. S. Chiang, “Analysis of the effective-index method for the vector modes of rectangular-core dielectric waveguides,” IEEE Trans. Microwave Theory Tech.44, 692–700 (1996).
[CrossRef]

K. S. Chiang, “Dual effective-index method for the analysis of rectangular dielectric waveguides,” Appl. Opt.25, 2169–2174 (1986).
[CrossRef] [PubMed]

Grattan, K. T. V.

Kejalakshmy, N.

Kendall, P. C.

P. C. Kendall, M. J. Adams, S. Ritchie, and M. J. Robertson, “Theory for calculating approximate values for the propagation constants of an optical rib waveguide by weighting the refractive indices,” IEE Proc. A134, 699–702 (1987).

Knox, R. M.

R. M. Knox and P. P. Toulios, “Integrated circuits for the millimeter through optical frequency range,” in Proceedings of the Symposium on Submillimeter Waves, M. H. Schlam and J. Fox, (Polytechnic Press, 1970), pp. 497–516.

Leung, D. M. H.

Lipson, M.

Marcatili, E. A. J.

E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J.48, 2071–2102 (1969).

McNab, S.

Rahman, B. M. A.

Ritchie, S.

P. C. Kendall, M. J. Adams, S. Ritchie, and M. J. Robertson, “Theory for calculating approximate values for the propagation constants of an optical rib waveguide by weighting the refractive indices,” IEE Proc. A134, 699–702 (1987).

Robertson, M. J.

P. C. Kendall, M. J. Adams, S. Ritchie, and M. J. Robertson, “Theory for calculating approximate values for the propagation constants of an optical rib waveguide by weighting the refractive indices,” IEE Proc. A134, 699–702 (1987).

Sharma, A.

Toulios, P. P.

R. M. Knox and P. P. Toulios, “Integrated circuits for the millimeter through optical frequency range,” in Proceedings of the Symposium on Submillimeter Waves, M. H. Schlam and J. Fox, (Polytechnic Press, 1970), pp. 497–516.

Vlasov, Y.

Appl. Opt.

Bell Syst. Tech. J.

E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J.48, 2071–2102 (1969).

IEE Proc. A

P. C. Kendall, M. J. Adams, S. Ritchie, and M. J. Robertson, “Theory for calculating approximate values for the propagation constants of an optical rib waveguide by weighting the refractive indices,” IEE Proc. A134, 699–702 (1987).

IEEE Trans. Microwave Theory Tech.

K. S. Chiang, “Analysis of the effective-index method for the vector modes of rectangular-core dielectric waveguides,” IEEE Trans. Microwave Theory Tech.44, 692–700 (1996).
[CrossRef]

J. Lightwave Technol.

J. Opt. Soc. Am. A

Opt. Express

Opt. Quant. Electron.

A. Sharma, “On approximate theories of rectangular waveguides,” Opt. Quant. Electron.21, 517–520 (1989).
[CrossRef]

Other

R. M. Knox and P. P. Toulios, “Integrated circuits for the millimeter through optical frequency range,” in Proceedings of the Symposium on Submillimeter Waves, M. H. Schlam and J. Fox, (Polytechnic Press, 1970), pp. 497–516.

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Figures (4)

Fig. 1
Fig. 1

Variation of effective index, neff with core width, W. The inset shows the variation of error in neff of fundamental mode with W for SV-Vopt and VEIM results. Error is defined as difference in neff obtained by SV-Vopt/VEIM and VFEM.

Fig. 2
Fig. 2

Variation of effective mode area, Aeff with core width, W.

Fig. 3
Fig. 3

Variation of power confinement factor in core, Γcore with core width, W.

Fig. 4
Fig. 4

Variation of neff for E 11 x and E 11 y mode (left scale) and modal birefringence (right scale) with core width, W.

Equations (10)

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x { 1 n 2 ( n 2 E x ) x } + 2 E x y 2 + ( k 2 n 2 β 2 ) E x = 0
β 2 = k 2 n 2 n x 2 χ 2 ϕ 2 d x d y + n x 2 χ x { 1 n 2 ( n 2 χ ) x } ϕ 2 d x d y + ϕ d 2 ϕ d y 2 d y
n x 2 χ 2 ( x ) d x = 1 = ϕ 2 ( y ) d y
β 2 = k 2 ( n x 2 χ ) 2 d x + n x 2 χ d d x { 1 n x 2 d ( n x 2 χ ) d x } d x + ϕ d 2 ϕ d y 2 d y + k 2 [ ( n 2 n x 2 ) n x 2 χ 2 d x + 1 k 2 n x 2 χ x { χ ln ( n 2 / n x 2 ) x } d x ] ϕ 2 d y
n y 2 ( y ) = ( n 2 n x 2 ) n x 2 χ 2 d x 1 k 2 χ d ( n x 2 χ ) d x ln ( n 2 / n x 2 ) x d x
β 2 = k 2 n y 2 ϕ 2 d x + ϕ d 2 ϕ d y 2 d y + n x 2 χ d d x { 1 n x 2 d ( n x 2 χ ) d x } d x + k 2 [ ( n 2 n y 2 ) ϕ 2 d y + 1 k 2 n x 2 χ x { χ ln ( n 2 / n x 2 ) x } d x ] n x 2 χ 2 d x
n x 2 ( x ) = ( n 2 n y 2 ) ϕ 2 d y 1 k 2 [ χ ( n x 2 χ ) x ln ( n 2 / n x 2 ) x d x ] ϕ 2 d y
β 2 = β x 2 + β y 2
A eff = ( | E t | 2 d x d y ) 2 | E t | 4 d x d y
Γ core = core { E * × H } z d x d y

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