Abstract

The mapped Galerkin method based on the transverse electric fields (E formulation) for vectorial analysis of optical waveguides is described. The vector wave equation is solved by using nonlinear mapping of the x-y plane and subsequent Fourier decomposition. The modal propagation constants and field distributions for rectangular waveguides and optical rib waveguides are presented. The calculated results accord well with those published earlier, which proves the accuracy and validity of the mapped Galerkin method.

© 2004 Optical Society of America

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References

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  1. O. Wada, T. Sakurai, and T. Nakagami, “Recent progress in optoelectric integrated circuits (OEICs),” IEEE J. Quantum Electron. 22, 805–821 (1986).
    [CrossRef]
  2. R. Kaiser and H. Heidrich, “Optoelectronic–photonic integrated circuits on InP between technological feasibility and commercial success,” IEICE Trans. Electron. E85-C, 970–981 (2002).
  3. T. Goh, M. Yasu, K. Hattori, A. Himeno, M. Okuno, and Y. Ohmori, “Low loss and high extinction ratio strictly nonblocking 16×16 thermo-optic matrix switch on 6-in wafer using silica-based planar lightwave circuit technology,” J. Lightwave Technol. 9, 371–379 (2001).
    [CrossRef]
  4. C. Vassallo, “1993–1995 optical mode solvers,” Opt. Quantum Electron. 29, 95–114 (1997).
    [CrossRef]
  5. R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150–162 (2000).
    [CrossRef]
  6. R. Z. L. Ye and D. O. Yevick, “Noniterative calculation of complex propagation constants in planar waveguides,” J. Opt. Soc. Am. A 18, 2819–2822 (2001).
    [CrossRef]
  7. M. Koshiba, K. Hayata, and M. Suzuki, “Improved finite-element formulation in terms of the magnetic field vector for dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-33, 227–233 (1985).
    [CrossRef]
  8. C. H. Henry and B. H. Verbeek, “Solution of the scalar wave equation for arbitrarily shaped dielectric waveguides by two-dimensional Fourier analysis,” J. Lightwave Technol. 7, 308–313 (1989).
    [CrossRef]
  9. D. Marcuse, “Solution of the vector wave equation for general dielectric waveguides by the Galerkin method,” IEEE J. Quantum Electron. 28, 459–465 (1992).
    [CrossRef]
  10. A. Weisshaar, J. Li, R. I. Gallawa, and I. C. Goyal, “Vector and quasi-vector solutions for optical waveguide modes using efficient Galerkin’s method with Hermite-Gauss basis functions,” J. Lightwave Technol. 13, 1795–1800 (1995).
    [CrossRef]
  11. S. J. Hewlett and F. Ladouceur, “Fourier decomposition method applied to mapped infinite domains: scalar analysis of dielectric waveguides down to modal cutoff,” J. Lightwave Technol. 13, 375–383 (1995).
    [CrossRef]
  12. M. F. Forastiere and G. C. Righini, “Improved scalar analysis of integrated optical structures by the mapped Galerkin method and Arnoldi iteration,” J. Opt. Soc. Am. A 18, 966–974 (2001).
    [CrossRef]
  13. K. M. Lo and E. H. Li, “Solutions of the quasi-vector wave equation for optical waveguides in a mapped infinite domain by the Galerkin method,” J. Lightwave Technol. 16, 937–944 (1998).
    [CrossRef]
  14. M. S. Stern, “Semi-vectorial, polarized, finite-difference method for optical waveguides with arbitrary index profiles,” IEE Proc. J. Optoelectron. 135, 56–63 (1988).
    [CrossRef]
  15. G. M. Berry, S. V. Burke, C. J. Smartt, T. M. Benson, and P. C. Kendall, “Exact and variational Fourier transform methods for analysis of multilayered planar waveguides,” IEE Proc. J. Optoelectron. 142, 66–75 (1995).
    [CrossRef]
  16. S. Sujecki, T. M. Benson, P. Sewell, and P. C. Kendall, “Novel vectorial analysis of optical waveguides,” J. Lightwave Technol. 16, 1329–1335 (1998).
    [CrossRef]
  17. M. S. Stern, “Rayleigh quotient solution of semivectorial field problems for optical waveguides with arbitrary index profiles,” IEE Proc. J. Optoelectron. 138, 185–190 (1991).
    [CrossRef]
  18. M. S. Stern, P. C. Kendall, and P. W. A. Mcllroy, “Analysis of the spectral index method for vector modes of rib waveguides,” IEE Proc. J. Optoelectron. 137, 21–26 (1990).
    [CrossRef]
  19. P. L. Liu and B. J. Li, “Semivectorial beam-propagation method for analyzing polarized modes of rib waveguides,” IEEE J. Quantum Electron. 28, 778–782 (1992).
    [CrossRef]
  20. P. L. Liu and B. J. Li, “Full vectorial mode analysis of rib waveguides by iterative Lanczos reduction,” IEEE J. Quantum Electron. 29, 2859–2863 (1993).
    [CrossRef]

2002

R. Kaiser and H. Heidrich, “Optoelectronic–photonic integrated circuits on InP between technological feasibility and commercial success,” IEICE Trans. Electron. E85-C, 970–981 (2002).

2001

T. Goh, M. Yasu, K. Hattori, A. Himeno, M. Okuno, and Y. Ohmori, “Low loss and high extinction ratio strictly nonblocking 16×16 thermo-optic matrix switch on 6-in wafer using silica-based planar lightwave circuit technology,” J. Lightwave Technol. 9, 371–379 (2001).
[CrossRef]

R. Z. L. Ye and D. O. Yevick, “Noniterative calculation of complex propagation constants in planar waveguides,” J. Opt. Soc. Am. A 18, 2819–2822 (2001).
[CrossRef]

M. F. Forastiere and G. C. Righini, “Improved scalar analysis of integrated optical structures by the mapped Galerkin method and Arnoldi iteration,” J. Opt. Soc. Am. A 18, 966–974 (2001).
[CrossRef]

2000

R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150–162 (2000).
[CrossRef]

1998

1997

C. Vassallo, “1993–1995 optical mode solvers,” Opt. Quantum Electron. 29, 95–114 (1997).
[CrossRef]

1995

G. M. Berry, S. V. Burke, C. J. Smartt, T. M. Benson, and P. C. Kendall, “Exact and variational Fourier transform methods for analysis of multilayered planar waveguides,” IEE Proc. J. Optoelectron. 142, 66–75 (1995).
[CrossRef]

A. Weisshaar, J. Li, R. I. Gallawa, and I. C. Goyal, “Vector and quasi-vector solutions for optical waveguide modes using efficient Galerkin’s method with Hermite-Gauss basis functions,” J. Lightwave Technol. 13, 1795–1800 (1995).
[CrossRef]

S. J. Hewlett and F. Ladouceur, “Fourier decomposition method applied to mapped infinite domains: scalar analysis of dielectric waveguides down to modal cutoff,” J. Lightwave Technol. 13, 375–383 (1995).
[CrossRef]

1993

P. L. Liu and B. J. Li, “Full vectorial mode analysis of rib waveguides by iterative Lanczos reduction,” IEEE J. Quantum Electron. 29, 2859–2863 (1993).
[CrossRef]

1992

P. L. Liu and B. J. Li, “Semivectorial beam-propagation method for analyzing polarized modes of rib waveguides,” IEEE J. Quantum Electron. 28, 778–782 (1992).
[CrossRef]

D. Marcuse, “Solution of the vector wave equation for general dielectric waveguides by the Galerkin method,” IEEE J. Quantum Electron. 28, 459–465 (1992).
[CrossRef]

1991

M. S. Stern, “Rayleigh quotient solution of semivectorial field problems for optical waveguides with arbitrary index profiles,” IEE Proc. J. Optoelectron. 138, 185–190 (1991).
[CrossRef]

1990

M. S. Stern, P. C. Kendall, and P. W. A. Mcllroy, “Analysis of the spectral index method for vector modes of rib waveguides,” IEE Proc. J. Optoelectron. 137, 21–26 (1990).
[CrossRef]

1989

C. H. Henry and B. H. Verbeek, “Solution of the scalar wave equation for arbitrarily shaped dielectric waveguides by two-dimensional Fourier analysis,” J. Lightwave Technol. 7, 308–313 (1989).
[CrossRef]

1988

M. S. Stern, “Semi-vectorial, polarized, finite-difference method for optical waveguides with arbitrary index profiles,” IEE Proc. J. Optoelectron. 135, 56–63 (1988).
[CrossRef]

1986

O. Wada, T. Sakurai, and T. Nakagami, “Recent progress in optoelectric integrated circuits (OEICs),” IEEE J. Quantum Electron. 22, 805–821 (1986).
[CrossRef]

1985

M. Koshiba, K. Hayata, and M. Suzuki, “Improved finite-element formulation in terms of the magnetic field vector for dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-33, 227–233 (1985).
[CrossRef]

Benson, T. M.

S. Sujecki, T. M. Benson, P. Sewell, and P. C. Kendall, “Novel vectorial analysis of optical waveguides,” J. Lightwave Technol. 16, 1329–1335 (1998).
[CrossRef]

G. M. Berry, S. V. Burke, C. J. Smartt, T. M. Benson, and P. C. Kendall, “Exact and variational Fourier transform methods for analysis of multilayered planar waveguides,” IEE Proc. J. Optoelectron. 142, 66–75 (1995).
[CrossRef]

Berry, G. M.

G. M. Berry, S. V. Burke, C. J. Smartt, T. M. Benson, and P. C. Kendall, “Exact and variational Fourier transform methods for analysis of multilayered planar waveguides,” IEE Proc. J. Optoelectron. 142, 66–75 (1995).
[CrossRef]

Burke, S. V.

G. M. Berry, S. V. Burke, C. J. Smartt, T. M. Benson, and P. C. Kendall, “Exact and variational Fourier transform methods for analysis of multilayered planar waveguides,” IEE Proc. J. Optoelectron. 142, 66–75 (1995).
[CrossRef]

Forastiere, M. F.

Gallawa, R. I.

A. Weisshaar, J. Li, R. I. Gallawa, and I. C. Goyal, “Vector and quasi-vector solutions for optical waveguide modes using efficient Galerkin’s method with Hermite-Gauss basis functions,” J. Lightwave Technol. 13, 1795–1800 (1995).
[CrossRef]

Goh, T.

T. Goh, M. Yasu, K. Hattori, A. Himeno, M. Okuno, and Y. Ohmori, “Low loss and high extinction ratio strictly nonblocking 16×16 thermo-optic matrix switch on 6-in wafer using silica-based planar lightwave circuit technology,” J. Lightwave Technol. 9, 371–379 (2001).
[CrossRef]

Gopinath, A.

R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150–162 (2000).
[CrossRef]

Goyal, I. C.

A. Weisshaar, J. Li, R. I. Gallawa, and I. C. Goyal, “Vector and quasi-vector solutions for optical waveguide modes using efficient Galerkin’s method with Hermite-Gauss basis functions,” J. Lightwave Technol. 13, 1795–1800 (1995).
[CrossRef]

Hattori, K.

T. Goh, M. Yasu, K. Hattori, A. Himeno, M. Okuno, and Y. Ohmori, “Low loss and high extinction ratio strictly nonblocking 16×16 thermo-optic matrix switch on 6-in wafer using silica-based planar lightwave circuit technology,” J. Lightwave Technol. 9, 371–379 (2001).
[CrossRef]

Hayata, K.

M. Koshiba, K. Hayata, and M. Suzuki, “Improved finite-element formulation in terms of the magnetic field vector for dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-33, 227–233 (1985).
[CrossRef]

Heidrich, H.

R. Kaiser and H. Heidrich, “Optoelectronic–photonic integrated circuits on InP between technological feasibility and commercial success,” IEICE Trans. Electron. E85-C, 970–981 (2002).

Helfert, S.

R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150–162 (2000).
[CrossRef]

Henry, C. H.

C. H. Henry and B. H. Verbeek, “Solution of the scalar wave equation for arbitrarily shaped dielectric waveguides by two-dimensional Fourier analysis,” J. Lightwave Technol. 7, 308–313 (1989).
[CrossRef]

Hewlett, S. J.

S. J. Hewlett and F. Ladouceur, “Fourier decomposition method applied to mapped infinite domains: scalar analysis of dielectric waveguides down to modal cutoff,” J. Lightwave Technol. 13, 375–383 (1995).
[CrossRef]

Himeno, A.

T. Goh, M. Yasu, K. Hattori, A. Himeno, M. Okuno, and Y. Ohmori, “Low loss and high extinction ratio strictly nonblocking 16×16 thermo-optic matrix switch on 6-in wafer using silica-based planar lightwave circuit technology,” J. Lightwave Technol. 9, 371–379 (2001).
[CrossRef]

Kaiser, R.

R. Kaiser and H. Heidrich, “Optoelectronic–photonic integrated circuits on InP between technological feasibility and commercial success,” IEICE Trans. Electron. E85-C, 970–981 (2002).

Kendall, P. C.

S. Sujecki, T. M. Benson, P. Sewell, and P. C. Kendall, “Novel vectorial analysis of optical waveguides,” J. Lightwave Technol. 16, 1329–1335 (1998).
[CrossRef]

G. M. Berry, S. V. Burke, C. J. Smartt, T. M. Benson, and P. C. Kendall, “Exact and variational Fourier transform methods for analysis of multilayered planar waveguides,” IEE Proc. J. Optoelectron. 142, 66–75 (1995).
[CrossRef]

M. S. Stern, P. C. Kendall, and P. W. A. Mcllroy, “Analysis of the spectral index method for vector modes of rib waveguides,” IEE Proc. J. Optoelectron. 137, 21–26 (1990).
[CrossRef]

Koshiba, M.

M. Koshiba, K. Hayata, and M. Suzuki, “Improved finite-element formulation in terms of the magnetic field vector for dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-33, 227–233 (1985).
[CrossRef]

Ladouceur, F.

S. J. Hewlett and F. Ladouceur, “Fourier decomposition method applied to mapped infinite domains: scalar analysis of dielectric waveguides down to modal cutoff,” J. Lightwave Technol. 13, 375–383 (1995).
[CrossRef]

Li, B. J.

P. L. Liu and B. J. Li, “Full vectorial mode analysis of rib waveguides by iterative Lanczos reduction,” IEEE J. Quantum Electron. 29, 2859–2863 (1993).
[CrossRef]

P. L. Liu and B. J. Li, “Semivectorial beam-propagation method for analyzing polarized modes of rib waveguides,” IEEE J. Quantum Electron. 28, 778–782 (1992).
[CrossRef]

Li, E. H.

Li, J.

A. Weisshaar, J. Li, R. I. Gallawa, and I. C. Goyal, “Vector and quasi-vector solutions for optical waveguide modes using efficient Galerkin’s method with Hermite-Gauss basis functions,” J. Lightwave Technol. 13, 1795–1800 (1995).
[CrossRef]

Liu, P. L.

P. L. Liu and B. J. Li, “Full vectorial mode analysis of rib waveguides by iterative Lanczos reduction,” IEEE J. Quantum Electron. 29, 2859–2863 (1993).
[CrossRef]

P. L. Liu and B. J. Li, “Semivectorial beam-propagation method for analyzing polarized modes of rib waveguides,” IEEE J. Quantum Electron. 28, 778–782 (1992).
[CrossRef]

Lo, K. M.

Marcuse, D.

D. Marcuse, “Solution of the vector wave equation for general dielectric waveguides by the Galerkin method,” IEEE J. Quantum Electron. 28, 459–465 (1992).
[CrossRef]

Mcllroy, P. W. A.

M. S. Stern, P. C. Kendall, and P. W. A. Mcllroy, “Analysis of the spectral index method for vector modes of rib waveguides,” IEE Proc. J. Optoelectron. 137, 21–26 (1990).
[CrossRef]

Nakagami, T.

O. Wada, T. Sakurai, and T. Nakagami, “Recent progress in optoelectric integrated circuits (OEICs),” IEEE J. Quantum Electron. 22, 805–821 (1986).
[CrossRef]

Ohmori, Y.

T. Goh, M. Yasu, K. Hattori, A. Himeno, M. Okuno, and Y. Ohmori, “Low loss and high extinction ratio strictly nonblocking 16×16 thermo-optic matrix switch on 6-in wafer using silica-based planar lightwave circuit technology,” J. Lightwave Technol. 9, 371–379 (2001).
[CrossRef]

Okuno, M.

T. Goh, M. Yasu, K. Hattori, A. Himeno, M. Okuno, and Y. Ohmori, “Low loss and high extinction ratio strictly nonblocking 16×16 thermo-optic matrix switch on 6-in wafer using silica-based planar lightwave circuit technology,” J. Lightwave Technol. 9, 371–379 (2001).
[CrossRef]

Pregla, R.

R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150–162 (2000).
[CrossRef]

Righini, G. C.

Sakurai, T.

O. Wada, T. Sakurai, and T. Nakagami, “Recent progress in optoelectric integrated circuits (OEICs),” IEEE J. Quantum Electron. 22, 805–821 (1986).
[CrossRef]

Scarmozzino, R.

R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150–162 (2000).
[CrossRef]

Sewell, P.

Smartt, C. J.

G. M. Berry, S. V. Burke, C. J. Smartt, T. M. Benson, and P. C. Kendall, “Exact and variational Fourier transform methods for analysis of multilayered planar waveguides,” IEE Proc. J. Optoelectron. 142, 66–75 (1995).
[CrossRef]

Stern, M. S.

M. S. Stern, “Rayleigh quotient solution of semivectorial field problems for optical waveguides with arbitrary index profiles,” IEE Proc. J. Optoelectron. 138, 185–190 (1991).
[CrossRef]

M. S. Stern, P. C. Kendall, and P. W. A. Mcllroy, “Analysis of the spectral index method for vector modes of rib waveguides,” IEE Proc. J. Optoelectron. 137, 21–26 (1990).
[CrossRef]

M. S. Stern, “Semi-vectorial, polarized, finite-difference method for optical waveguides with arbitrary index profiles,” IEE Proc. J. Optoelectron. 135, 56–63 (1988).
[CrossRef]

Sujecki, S.

Suzuki, M.

M. Koshiba, K. Hayata, and M. Suzuki, “Improved finite-element formulation in terms of the magnetic field vector for dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-33, 227–233 (1985).
[CrossRef]

Vassallo, C.

C. Vassallo, “1993–1995 optical mode solvers,” Opt. Quantum Electron. 29, 95–114 (1997).
[CrossRef]

Verbeek, B. H.

C. H. Henry and B. H. Verbeek, “Solution of the scalar wave equation for arbitrarily shaped dielectric waveguides by two-dimensional Fourier analysis,” J. Lightwave Technol. 7, 308–313 (1989).
[CrossRef]

Wada, O.

O. Wada, T. Sakurai, and T. Nakagami, “Recent progress in optoelectric integrated circuits (OEICs),” IEEE J. Quantum Electron. 22, 805–821 (1986).
[CrossRef]

Weisshaar, A.

A. Weisshaar, J. Li, R. I. Gallawa, and I. C. Goyal, “Vector and quasi-vector solutions for optical waveguide modes using efficient Galerkin’s method with Hermite-Gauss basis functions,” J. Lightwave Technol. 13, 1795–1800 (1995).
[CrossRef]

Yasu, M.

T. Goh, M. Yasu, K. Hattori, A. Himeno, M. Okuno, and Y. Ohmori, “Low loss and high extinction ratio strictly nonblocking 16×16 thermo-optic matrix switch on 6-in wafer using silica-based planar lightwave circuit technology,” J. Lightwave Technol. 9, 371–379 (2001).
[CrossRef]

Ye, R. Z. L.

Yevick, D. O.

IEE Proc. J. Optoelectron.

M. S. Stern, “Rayleigh quotient solution of semivectorial field problems for optical waveguides with arbitrary index profiles,” IEE Proc. J. Optoelectron. 138, 185–190 (1991).
[CrossRef]

M. S. Stern, P. C. Kendall, and P. W. A. Mcllroy, “Analysis of the spectral index method for vector modes of rib waveguides,” IEE Proc. J. Optoelectron. 137, 21–26 (1990).
[CrossRef]

M. S. Stern, “Semi-vectorial, polarized, finite-difference method for optical waveguides with arbitrary index profiles,” IEE Proc. J. Optoelectron. 135, 56–63 (1988).
[CrossRef]

G. M. Berry, S. V. Burke, C. J. Smartt, T. M. Benson, and P. C. Kendall, “Exact and variational Fourier transform methods for analysis of multilayered planar waveguides,” IEE Proc. J. Optoelectron. 142, 66–75 (1995).
[CrossRef]

IEEE J. Quantum Electron.

P. L. Liu and B. J. Li, “Semivectorial beam-propagation method for analyzing polarized modes of rib waveguides,” IEEE J. Quantum Electron. 28, 778–782 (1992).
[CrossRef]

P. L. Liu and B. J. Li, “Full vectorial mode analysis of rib waveguides by iterative Lanczos reduction,” IEEE J. Quantum Electron. 29, 2859–2863 (1993).
[CrossRef]

D. Marcuse, “Solution of the vector wave equation for general dielectric waveguides by the Galerkin method,” IEEE J. Quantum Electron. 28, 459–465 (1992).
[CrossRef]

O. Wada, T. Sakurai, and T. Nakagami, “Recent progress in optoelectric integrated circuits (OEICs),” IEEE J. Quantum Electron. 22, 805–821 (1986).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150–162 (2000).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

M. Koshiba, K. Hayata, and M. Suzuki, “Improved finite-element formulation in terms of the magnetic field vector for dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-33, 227–233 (1985).
[CrossRef]

IEICE Trans. Electron.

R. Kaiser and H. Heidrich, “Optoelectronic–photonic integrated circuits on InP between technological feasibility and commercial success,” IEICE Trans. Electron. E85-C, 970–981 (2002).

J. Lightwave Technol.

T. Goh, M. Yasu, K. Hattori, A. Himeno, M. Okuno, and Y. Ohmori, “Low loss and high extinction ratio strictly nonblocking 16×16 thermo-optic matrix switch on 6-in wafer using silica-based planar lightwave circuit technology,” J. Lightwave Technol. 9, 371–379 (2001).
[CrossRef]

C. H. Henry and B. H. Verbeek, “Solution of the scalar wave equation for arbitrarily shaped dielectric waveguides by two-dimensional Fourier analysis,” J. Lightwave Technol. 7, 308–313 (1989).
[CrossRef]

A. Weisshaar, J. Li, R. I. Gallawa, and I. C. Goyal, “Vector and quasi-vector solutions for optical waveguide modes using efficient Galerkin’s method with Hermite-Gauss basis functions,” J. Lightwave Technol. 13, 1795–1800 (1995).
[CrossRef]

S. J. Hewlett and F. Ladouceur, “Fourier decomposition method applied to mapped infinite domains: scalar analysis of dielectric waveguides down to modal cutoff,” J. Lightwave Technol. 13, 375–383 (1995).
[CrossRef]

S. Sujecki, T. M. Benson, P. Sewell, and P. C. Kendall, “Novel vectorial analysis of optical waveguides,” J. Lightwave Technol. 16, 1329–1335 (1998).
[CrossRef]

K. M. Lo and E. H. Li, “Solutions of the quasi-vector wave equation for optical waveguides in a mapped infinite domain by the Galerkin method,” J. Lightwave Technol. 16, 937–944 (1998).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Quantum Electron.

C. Vassallo, “1993–1995 optical mode solvers,” Opt. Quantum Electron. 29, 95–114 (1997).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Cross section of (a) rectangular waveguide and (b) optical rib waveguide.

Fig. 2
Fig. 2

Field pattern of the transverse E field for the fundamental mode of a rectangular waveguide with w=4 µm, h=2 µm, n2=1.5, and n1=1.45 at λ=1.15 µm: (a) ux for the x-polarized mode, (b) uy for the x-polarized mode, (c) ux for the y-polarized mode, (d) uy for the y-polarized mode.

Fig. 3
Fig. 3

Field pattern of the transverse E field for E21x and E21y of a rectangular waveguide with w=4 µm, h=2 µm, n2=1.5, and n1=1.45 at λ=1.15 µm: (a) ux for the x-polarized mode, (b) uy for the x-polarized mode, (c) ux for the y-polarized mode, (d) uy for the y-polarized mode.

Fig. 4
Fig. 4

Field pattern of the transverse E field for E12x and E12y of a rectangular waveguide with w=4 µm, h=2 µm, n2=1.5, and n1=1.45 at λ=1.15 µm: (a) ux for the x-polarized mode, (b) uy for the x-polarized mode, (c) ux for the y-polarized mode, (d) uy for the y-polarized mode.

Fig. 5
Fig. 5

Field pattern of the transverse E field for the fundamental mode of structure 1 in Table 3: (a) ux for the x-polarized mode, (b) uy for the x-polarized mode, (c) ux for the y-polarized mode, (d) uy for the y-polarized mode.

Fig. 6
Fig. 6

Field pattern of the transverse E field for the fundamental mode of structure 2 in Table 3: (a) ux for the x-polarized mode, (b) uy for the x-polarized mode, (c) ux for the y-polarized mode, (d) uy for the y-polarized mode.

Fig. 7
Fig. 7

Field pattern of the transverse E field for the fundamental mode of structure 3 in Table 3: (a) ux for the x-polarized mode, (b) uy for the x-polarized mode, (c) ux for the y-polarized mode, (d) uy for the y-polarized mode.

Tables (4)

Tables Icon

Table 1 Normalized Propagation Constant B of the Rectangular Waveguide Versus Normalized Frequency Va

Tables Icon

Table 2 Normalized Propagation Constants B of the Rectangular Waveguide for Different Values of Refractive Index n1 in Claddinga

Tables Icon

Table 3 Optical Rib Waveguide Parameters

Tables Icon

Table 4 Effective Indices neff and Normalized Propagation Constants B of Optical Rib Waveguide Structures at λ=1.55 µm Computed by Different Methods

Equations (53)

Equations on this page are rendered with MathJax. Learn more.

×E=-jωμ0H,
×H=jωn20E,
2E+E·n2n2+n2k02E=0,
E(x, y, z)=u(x, y)exp(-jβz),
u(x, y)=ux(x, y)x+uy(x, y)y+uz(x, y)z,
2=2x2+2y2-β2.
2uxx2+2uxy2+k02(n2-neff2)ux
+2 xux  ln(n)x+uy  ln(n)y=0,
2uyx2+2uyy2+k02(n2-neff2)uy
+2 yux  ln(n)x+uy  ln(n)y=0.
x=αx  tanπξ-12,
y=αy  tanπη-12,
dξdx2 2uxξ2+d2ξdx2uxξ+dηdy2 2uxη2+d2ηdy2uxy
+k02(n2-neff2)ux+2dξdx2 ξux  ln(n)ξ
+d2ξdx2ux  ln(n)ξ+dξdxdηdyξuy  ln(n)η=0,
dξdx2 2uyξ2+d2ξdx2uyξ+dηdy2 2uyη2+d2ηdy2uyy
+k02(n2-neff2)uy+2dηdy2 ηuy  ln(n)η
+d2ηdy2uy  ln(n)η+dξdxdηdyηux  ln(n)ξ=0.
ux(ξ, η)=k=1Nx×Nyckxϕk(ξ, η)=p=1Nxq=1Nycpqxφp(ξ)φq(η),
uy(ξ, η)=k=1Nx×Nyckyϕk(ξ, η)=p=1Nxq=1Nycpqyφp(ξ)φq(η),
p=(k-1)div Ny+1,
q=(k-1)mod Ny+1.
φp(ξ)=2 sin(pπξ),
φq(η)=2 sin(qπη),
ξ=01η=01φp(ξ)φq(η)φp(ξ)φq(η)dξdη=δppδqq.
k=1Nx×Ny(Ak,kxxckx+Ak,kxycky)=β2ckx,
k=1Nx×Ny(Ak,kyycky+Ak,kyxckx)=β2cky,
Ak,kxx=i=14Ii+Pk,k+i=56Ii,
Ak,kxy=I7,
Ak,kyx=I8,
Ak,kyy=i=14Ii+Pk,k+i=910Ii,
Pk,k=k020101ϕk(ξ, η)n2(ξ, η)ϕk(ξ, η)dξdη.
Cx=[c1x, c2x, c3x,  , cNx×Nyx]T,
Cy=[c1y, c2y, c3y,  , cNx×Nyy]T,
AxxAxyAyxAyyCxCy=β2CxCy.
B=neff2-n12n22-n12,
V=hλ n22-n12.
I1=0101dξdx2 2ϕk(ξ, η)ξ2ϕk(ξ, η)dξdη=-p22αx23δp,p4-δp,p-22-δp,p+22+δp,2-p2+δp,p-48+δp,p+48-δp,4-p8δq,q,
I2=0101d2ξdx2 ϕk(ξ, η)ξϕk(ξ, η)dξdη=pαx2δp,2-p4+δp,p-24-δp,p+24-δp,4-p8-δp,p-48+δp,p+48δq,q,
I3=0101dηdy2 2ϕk(ξ, η)y2ϕk(ξ, η)dξdη=-q22αy23δq,q4-δq,q-22-δq,q+22+δq,2-q2+δq,q-48+δq,q+48-δq,4-q8δp,p,
I4=0101d2ηdy2 ϕk(ξ, η)yϕk(ξ, η)dξdη=qαy2δq,2-q4+δq,q-24-δq,q+24-δq,4-q8-δq,q-48+δq,q+48δp,p,
I5=01012dξdx2 ξϕk(ξ, η) ln(n)ξ×ϕk(ξ, η)dξdη=801dξ01dη{[ln(n)sin(qπη)sin(qπη)]×[f1(ξ)sin(pπξ)sin(pπξ)+pπf1(ξ)cos(pπξ)sin(pπξ)+2pπf1(ξ)sin(pπξ)cos(pπξ)×ppπ2f1(ξ)cos(pπξ)cos(pπξ)-p2π2f1(ξ)sin(pπξ)sin(pπξ)]},
I6=01012d2ξdx2ϕk(ξ, η) ln(n)ξϕk(ξ, η)dξdη=-801dξ01dη{[ln(n)sin(qπη)sin(qπη)]×[f2(ξ)sin(pπξ)sin(pπξ)+pπf2(ξ)cos(pπξ)sin(pπξ)+pπf2(ξ)sin(pπξ)cos(pπξ)]},
I7=01012dξdxdηdyξϕk(ξ, η) ln(n)η×ϕk(ξ, η)dξdη=801dξ01dη{ln(n)[f3(ξ)sin(pπξ)sin(pπξ)+pπf3(ξ)sin(pπξ)cos(pπξ)]×[f4(η)sin(qπη)sin(qπη)+qπf4(η)cos(qπη)sin(qπη)+qπf4(η)sin(qπη)cos(qπη)]},
I8=01012dξdxdηdyηϕk(ξ, η) ln(n)ξ×ϕk(ξ, η)dξdη=801dξ01dη{ln(n)[f4(η)sin(qπη)sin(qπη)+qπf4(η)sin(qπη)cos(qπη)]×[f3(ξ)sin(pπξ)sin(pπξ)+pπf3(ξ)cos(pπξ)sin(pπξ)+pπf3(ξ)sin(pπξ)cos(pπξ)]},
I9=01012dηdy2 ηϕk(ξ, η) ln(n)η×ϕk(ξ, η)dξdη=801dξ01dη{[ln(n)sin(pπξ)sin(pπξ)]×[f5(η)sin(qπη)sin(qπη)+qπf5(η)cos(qπη)sin(qπη)+2qπf5(η)sin(qπη)×cos(qπη)qqπ2f5(η)cos(qπη)cos(qπη)-q2π2f5(η)sin(qπη)sin(qπη)]},
I10=01012d2ηdy2ϕk(ξ, η) ln(n)ηϕk(ξ, η)dξdη=-801dξ01dη{[ln(n)sin(pπξ)sin(pπξ)]×[f6(η)sin(qπη)sin(qπη)+qπf6(η)cos(qπη)sin(qπη)+qπf6(η)sin(qπη)cos(qπη)]},
f1(ξ)=1/αx2π2  sin4(πξ),
f2(ξ)=1/αx2π sin(2πξ)sin2(πξ),
f3(ξ)=1/αxπ sin2(πξ),
f4(η)=1/αyπ sin2(πη),
f5(η)=1/αy2π2  sin4(πη),
f6(η)=1/αy2π sin(2πη)sin2(πη).

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