Abstract

The accuracy of finite-element-method techniques used for waveguide modal analysis has usually been assessed by testing the precision of the propagation constant. As a consequence, no useful criteria have been proposed for checking the spatial distribution of the evaluated unknown field. To overcome this lack, two error figures have been introduced and applied to different finite-element-method formulations. In particular, the following approaches have been compared: the one based on the transverse magnetic field, those based on the so-called edge elements, and a new one presented by the authors. The new approach, obtained by simply operating on the matrices of the original node-based formulation, directly solves for the propagation constant at a given frequency, preserves the matrix sparsity, and directly evaluates all of the unknown magnetic field components. Results demonstrate the applicability of the proposed approach and the usefulness of the introduced figures for a deep waveguide analysis.

© 1997 Optical Society of America

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References

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  1. W. C. Chew, M. A. Nasir, “A variational analysis of anisotropic inhomogeneous dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 37, 661–668 (1989).
    [Crossref]
  2. K. Hayata, K. Koshiba, M. Eguchi, M. Suzuki, “Vectorial finite element method without any spurious solution for dielectric waveguiding problems using transverse magnetic field components,” IEEE Trans. Microwave Theory Tech. MTT-34, 1120–1124 (1986).
    [Crossref]
  3. Y. Lu, F. A. Fernandez, “An efficient finite element solution of inhomogeneous anisotropic and lossy dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 41, 1215–1223 (1993).
    [Crossref]
  4. J. F. Lee, D. K. Sun, Z. J. Cendes, “Full wave analysis of dielectric waveguides using tangential vector finite elements,” IEEE Trans. Microwave Theory Tech. 39, 1262–1271 (1991).
    [Crossref]
  5. K. Hayata, K. Inoue, “Simple and efficient finite element analysis of microwave and optical waveguides,” IEEE Trans. Microwave Theory Tech. 40, 371–377 (1992).
    [Crossref]
  6. M. Koshiba, S. Maruyama, K. Hirayama, “A vector finite element method with the high order mixed interpolation type triangular elements for optical waveguiding problems,” J. Lightwave Technol. 12, 495–502 (1994).
    [Crossref]
  7. B. M. A. Rahman, J. B. Davies, “Penalty function improvement of waveguides solution by finite elements,” IEEE Trans. Microwave Theory Tech. MTT-32, 922–928 (1984).
    [Crossref]
  8. S. Selleri, M. Zoboli, “An improved finite element method formulation for the analysis of nonlinear anisotropic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 43, 887–892 (1995).
    [Crossref]
  9. C. Vassalo, “Finite difference analysis of vectorial transversal field in optical waveguides,” presented at the 3rd International Conference on Mathematical and Numerical Aspects of Wave Propagation Phenomena, INRIA-SIAM, Mandelien, France, April 11–12, 1995.
  10. M. Koshiba, K. Hayata, 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]
  11. M. Zoboli, P. Bassi, “The finite element method for anisotropic optical waveguides,” in Anisotropic and Nonlinear Optical Waveguides, G. Stegeman, C. G. Someda, eds. (Elsevier, Amsterdam, 1992), pp. 77–116.
  12. A. Fernandez, J. B. Davies, S. Zhu, Y. Lu, “Sparse matrix eigenvalue solver for finite element solution of dielectric waveguides,” Electron. Lett. 27, 1824–1826 (1991).
    [Crossref]
  13. B. M. Dillon, P. T. S. Liu, J. P. Webb, “Spurious modes in quadrilateral and triangular edge elements,” Int. J. Comput. Math. Electr. Electron. Eng. (COMPEL) 13 (Suppl. A), 311–316 (1994).
  14. S. Selleri, M. Zoboli, “A comparison of vector finite element formulations for waveguide analysis,” presented at Electrosoft 96, May 28–30, 1996, San Miniato, Italy.
  15. K. Hayata, M. Koshiba, M. Suzuki, “Vectorial wave analysis of stress-applied polarization-maintaining optical fibers by the finite element method,” J. Lightwave Technol. JLT-4, 133–139 (1986).
    [Crossref]
  16. A. Jay, “FEM modeling and preprocessing,” in Finite Element Handbook, H. Kardestuncer, D. H. Norrie, eds. (McGraw Hill, New York, 1987). The triangle aspect ratio measures the shape of the triangular element. It can be defined as the ratio of the length of the longest to the shortest element side. It can be used to test when an element becomes too elongated and the vertex angles too small or too large.

1995 (1)

S. Selleri, M. Zoboli, “An improved finite element method formulation for the analysis of nonlinear anisotropic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 43, 887–892 (1995).
[Crossref]

1994 (2)

M. Koshiba, S. Maruyama, K. Hirayama, “A vector finite element method with the high order mixed interpolation type triangular elements for optical waveguiding problems,” J. Lightwave Technol. 12, 495–502 (1994).
[Crossref]

B. M. Dillon, P. T. S. Liu, J. P. Webb, “Spurious modes in quadrilateral and triangular edge elements,” Int. J. Comput. Math. Electr. Electron. Eng. (COMPEL) 13 (Suppl. A), 311–316 (1994).

1993 (1)

Y. Lu, F. A. Fernandez, “An efficient finite element solution of inhomogeneous anisotropic and lossy dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 41, 1215–1223 (1993).
[Crossref]

1992 (1)

K. Hayata, K. Inoue, “Simple and efficient finite element analysis of microwave and optical waveguides,” IEEE Trans. Microwave Theory Tech. 40, 371–377 (1992).
[Crossref]

1991 (2)

A. Fernandez, J. B. Davies, S. Zhu, Y. Lu, “Sparse matrix eigenvalue solver for finite element solution of dielectric waveguides,” Electron. Lett. 27, 1824–1826 (1991).
[Crossref]

J. F. Lee, D. K. Sun, Z. J. Cendes, “Full wave analysis of dielectric waveguides using tangential vector finite elements,” IEEE Trans. Microwave Theory Tech. 39, 1262–1271 (1991).
[Crossref]

1989 (1)

W. C. Chew, M. A. Nasir, “A variational analysis of anisotropic inhomogeneous dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 37, 661–668 (1989).
[Crossref]

1986 (2)

K. Hayata, K. Koshiba, M. Eguchi, M. Suzuki, “Vectorial finite element method without any spurious solution for dielectric waveguiding problems using transverse magnetic field components,” IEEE Trans. Microwave Theory Tech. MTT-34, 1120–1124 (1986).
[Crossref]

K. Hayata, M. Koshiba, M. Suzuki, “Vectorial wave analysis of stress-applied polarization-maintaining optical fibers by the finite element method,” J. Lightwave Technol. JLT-4, 133–139 (1986).
[Crossref]

1985 (1)

M. Koshiba, K. Hayata, 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]

1984 (1)

B. M. A. Rahman, J. B. Davies, “Penalty function improvement of waveguides solution by finite elements,” IEEE Trans. Microwave Theory Tech. MTT-32, 922–928 (1984).
[Crossref]

Bassi, P.

M. Zoboli, P. Bassi, “The finite element method for anisotropic optical waveguides,” in Anisotropic and Nonlinear Optical Waveguides, G. Stegeman, C. G. Someda, eds. (Elsevier, Amsterdam, 1992), pp. 77–116.

Cendes, Z. J.

J. F. Lee, D. K. Sun, Z. J. Cendes, “Full wave analysis of dielectric waveguides using tangential vector finite elements,” IEEE Trans. Microwave Theory Tech. 39, 1262–1271 (1991).
[Crossref]

Chew, W. C.

W. C. Chew, M. A. Nasir, “A variational analysis of anisotropic inhomogeneous dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 37, 661–668 (1989).
[Crossref]

Davies, J. B.

A. Fernandez, J. B. Davies, S. Zhu, Y. Lu, “Sparse matrix eigenvalue solver for finite element solution of dielectric waveguides,” Electron. Lett. 27, 1824–1826 (1991).
[Crossref]

B. M. A. Rahman, J. B. Davies, “Penalty function improvement of waveguides solution by finite elements,” IEEE Trans. Microwave Theory Tech. MTT-32, 922–928 (1984).
[Crossref]

Dillon, B. M.

B. M. Dillon, P. T. S. Liu, J. P. Webb, “Spurious modes in quadrilateral and triangular edge elements,” Int. J. Comput. Math. Electr. Electron. Eng. (COMPEL) 13 (Suppl. A), 311–316 (1994).

Eguchi, M.

K. Hayata, K. Koshiba, M. Eguchi, M. Suzuki, “Vectorial finite element method without any spurious solution for dielectric waveguiding problems using transverse magnetic field components,” IEEE Trans. Microwave Theory Tech. MTT-34, 1120–1124 (1986).
[Crossref]

Fernandez, A.

A. Fernandez, J. B. Davies, S. Zhu, Y. Lu, “Sparse matrix eigenvalue solver for finite element solution of dielectric waveguides,” Electron. Lett. 27, 1824–1826 (1991).
[Crossref]

Fernandez, F. A.

Y. Lu, F. A. Fernandez, “An efficient finite element solution of inhomogeneous anisotropic and lossy dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 41, 1215–1223 (1993).
[Crossref]

Hayata, K.

K. Hayata, K. Inoue, “Simple and efficient finite element analysis of microwave and optical waveguides,” IEEE Trans. Microwave Theory Tech. 40, 371–377 (1992).
[Crossref]

K. Hayata, K. Koshiba, M. Eguchi, M. Suzuki, “Vectorial finite element method without any spurious solution for dielectric waveguiding problems using transverse magnetic field components,” IEEE Trans. Microwave Theory Tech. MTT-34, 1120–1124 (1986).
[Crossref]

K. Hayata, M. Koshiba, M. Suzuki, “Vectorial wave analysis of stress-applied polarization-maintaining optical fibers by the finite element method,” J. Lightwave Technol. JLT-4, 133–139 (1986).
[Crossref]

M. Koshiba, K. Hayata, 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]

Hirayama, K.

M. Koshiba, S. Maruyama, K. Hirayama, “A vector finite element method with the high order mixed interpolation type triangular elements for optical waveguiding problems,” J. Lightwave Technol. 12, 495–502 (1994).
[Crossref]

Inoue, K.

K. Hayata, K. Inoue, “Simple and efficient finite element analysis of microwave and optical waveguides,” IEEE Trans. Microwave Theory Tech. 40, 371–377 (1992).
[Crossref]

Jay, A.

A. Jay, “FEM modeling and preprocessing,” in Finite Element Handbook, H. Kardestuncer, D. H. Norrie, eds. (McGraw Hill, New York, 1987). The triangle aspect ratio measures the shape of the triangular element. It can be defined as the ratio of the length of the longest to the shortest element side. It can be used to test when an element becomes too elongated and the vertex angles too small or too large.

Koshiba, K.

K. Hayata, K. Koshiba, M. Eguchi, M. Suzuki, “Vectorial finite element method without any spurious solution for dielectric waveguiding problems using transverse magnetic field components,” IEEE Trans. Microwave Theory Tech. MTT-34, 1120–1124 (1986).
[Crossref]

Koshiba, M.

M. Koshiba, S. Maruyama, K. Hirayama, “A vector finite element method with the high order mixed interpolation type triangular elements for optical waveguiding problems,” J. Lightwave Technol. 12, 495–502 (1994).
[Crossref]

K. Hayata, M. Koshiba, M. Suzuki, “Vectorial wave analysis of stress-applied polarization-maintaining optical fibers by the finite element method,” J. Lightwave Technol. JLT-4, 133–139 (1986).
[Crossref]

M. Koshiba, K. Hayata, 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]

Lee, J. F.

J. F. Lee, D. K. Sun, Z. J. Cendes, “Full wave analysis of dielectric waveguides using tangential vector finite elements,” IEEE Trans. Microwave Theory Tech. 39, 1262–1271 (1991).
[Crossref]

Liu, P. T. S.

B. M. Dillon, P. T. S. Liu, J. P. Webb, “Spurious modes in quadrilateral and triangular edge elements,” Int. J. Comput. Math. Electr. Electron. Eng. (COMPEL) 13 (Suppl. A), 311–316 (1994).

Lu, Y.

Y. Lu, F. A. Fernandez, “An efficient finite element solution of inhomogeneous anisotropic and lossy dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 41, 1215–1223 (1993).
[Crossref]

A. Fernandez, J. B. Davies, S. Zhu, Y. Lu, “Sparse matrix eigenvalue solver for finite element solution of dielectric waveguides,” Electron. Lett. 27, 1824–1826 (1991).
[Crossref]

Maruyama, S.

M. Koshiba, S. Maruyama, K. Hirayama, “A vector finite element method with the high order mixed interpolation type triangular elements for optical waveguiding problems,” J. Lightwave Technol. 12, 495–502 (1994).
[Crossref]

Nasir, M. A.

W. C. Chew, M. A. Nasir, “A variational analysis of anisotropic inhomogeneous dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 37, 661–668 (1989).
[Crossref]

Rahman, B. M. A.

B. M. A. Rahman, J. B. Davies, “Penalty function improvement of waveguides solution by finite elements,” IEEE Trans. Microwave Theory Tech. MTT-32, 922–928 (1984).
[Crossref]

Selleri, S.

S. Selleri, M. Zoboli, “An improved finite element method formulation for the analysis of nonlinear anisotropic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 43, 887–892 (1995).
[Crossref]

S. Selleri, M. Zoboli, “A comparison of vector finite element formulations for waveguide analysis,” presented at Electrosoft 96, May 28–30, 1996, San Miniato, Italy.

Sun, D. K.

J. F. Lee, D. K. Sun, Z. J. Cendes, “Full wave analysis of dielectric waveguides using tangential vector finite elements,” IEEE Trans. Microwave Theory Tech. 39, 1262–1271 (1991).
[Crossref]

Suzuki, M.

K. Hayata, K. Koshiba, M. Eguchi, M. Suzuki, “Vectorial finite element method without any spurious solution for dielectric waveguiding problems using transverse magnetic field components,” IEEE Trans. Microwave Theory Tech. MTT-34, 1120–1124 (1986).
[Crossref]

K. Hayata, M. Koshiba, M. Suzuki, “Vectorial wave analysis of stress-applied polarization-maintaining optical fibers by the finite element method,” J. Lightwave Technol. JLT-4, 133–139 (1986).
[Crossref]

M. Koshiba, K. Hayata, 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]

Vassalo, C.

C. Vassalo, “Finite difference analysis of vectorial transversal field in optical waveguides,” presented at the 3rd International Conference on Mathematical and Numerical Aspects of Wave Propagation Phenomena, INRIA-SIAM, Mandelien, France, April 11–12, 1995.

Webb, J. P.

B. M. Dillon, P. T. S. Liu, J. P. Webb, “Spurious modes in quadrilateral and triangular edge elements,” Int. J. Comput. Math. Electr. Electron. Eng. (COMPEL) 13 (Suppl. A), 311–316 (1994).

Zhu, S.

A. Fernandez, J. B. Davies, S. Zhu, Y. Lu, “Sparse matrix eigenvalue solver for finite element solution of dielectric waveguides,” Electron. Lett. 27, 1824–1826 (1991).
[Crossref]

Zoboli, M.

S. Selleri, M. Zoboli, “An improved finite element method formulation for the analysis of nonlinear anisotropic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 43, 887–892 (1995).
[Crossref]

M. Zoboli, P. Bassi, “The finite element method for anisotropic optical waveguides,” in Anisotropic and Nonlinear Optical Waveguides, G. Stegeman, C. G. Someda, eds. (Elsevier, Amsterdam, 1992), pp. 77–116.

S. Selleri, M. Zoboli, “A comparison of vector finite element formulations for waveguide analysis,” presented at Electrosoft 96, May 28–30, 1996, San Miniato, Italy.

Electron. Lett. (1)

A. Fernandez, J. B. Davies, S. Zhu, Y. Lu, “Sparse matrix eigenvalue solver for finite element solution of dielectric waveguides,” Electron. Lett. 27, 1824–1826 (1991).
[Crossref]

IEEE Trans. Microwave Theory Tech. (8)

W. C. Chew, M. A. Nasir, “A variational analysis of anisotropic inhomogeneous dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 37, 661–668 (1989).
[Crossref]

K. Hayata, K. Koshiba, M. Eguchi, M. Suzuki, “Vectorial finite element method without any spurious solution for dielectric waveguiding problems using transverse magnetic field components,” IEEE Trans. Microwave Theory Tech. MTT-34, 1120–1124 (1986).
[Crossref]

Y. Lu, F. A. Fernandez, “An efficient finite element solution of inhomogeneous anisotropic and lossy dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 41, 1215–1223 (1993).
[Crossref]

J. F. Lee, D. K. Sun, Z. J. Cendes, “Full wave analysis of dielectric waveguides using tangential vector finite elements,” IEEE Trans. Microwave Theory Tech. 39, 1262–1271 (1991).
[Crossref]

K. Hayata, K. Inoue, “Simple and efficient finite element analysis of microwave and optical waveguides,” IEEE Trans. Microwave Theory Tech. 40, 371–377 (1992).
[Crossref]

B. M. A. Rahman, J. B. Davies, “Penalty function improvement of waveguides solution by finite elements,” IEEE Trans. Microwave Theory Tech. MTT-32, 922–928 (1984).
[Crossref]

S. Selleri, M. Zoboli, “An improved finite element method formulation for the analysis of nonlinear anisotropic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 43, 887–892 (1995).
[Crossref]

M. Koshiba, K. Hayata, 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]

Int. J. Comput. Math. Electr. Electron. Eng. (COMPEL) (1)

B. M. Dillon, P. T. S. Liu, J. P. Webb, “Spurious modes in quadrilateral and triangular edge elements,” Int. J. Comput. Math. Electr. Electron. Eng. (COMPEL) 13 (Suppl. A), 311–316 (1994).

J. Lightwave Technol. (2)

K. Hayata, M. Koshiba, M. Suzuki, “Vectorial wave analysis of stress-applied polarization-maintaining optical fibers by the finite element method,” J. Lightwave Technol. JLT-4, 133–139 (1986).
[Crossref]

M. Koshiba, S. Maruyama, K. Hirayama, “A vector finite element method with the high order mixed interpolation type triangular elements for optical waveguiding problems,” J. Lightwave Technol. 12, 495–502 (1994).
[Crossref]

Other (4)

M. Zoboli, P. Bassi, “The finite element method for anisotropic optical waveguides,” in Anisotropic and Nonlinear Optical Waveguides, G. Stegeman, C. G. Someda, eds. (Elsevier, Amsterdam, 1992), pp. 77–116.

C. Vassalo, “Finite difference analysis of vectorial transversal field in optical waveguides,” presented at the 3rd International Conference on Mathematical and Numerical Aspects of Wave Propagation Phenomena, INRIA-SIAM, Mandelien, France, April 11–12, 1995.

A. Jay, “FEM modeling and preprocessing,” in Finite Element Handbook, H. Kardestuncer, D. H. Norrie, eds. (McGraw Hill, New York, 1987). The triangle aspect ratio measures the shape of the triangular element. It can be defined as the ratio of the length of the longest to the shortest element side. It can be used to test when an element becomes too elongated and the vertex angles too small or too large.

S. Selleri, M. Zoboli, “A comparison of vector finite element formulations for waveguide analysis,” presented at Electrosoft 96, May 28–30, 1996, San Miniato, Italy.

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Figures (7)

Fig. 1
Fig. 1

Step-index fiber: relative error e=(n¯eff-neff)/n¯eff of the fundamental mode versus normalized frequency ν =rk0nco2-ncl2. The core and the cladding refractive indices are nco=1.57 and ncl=1.55, respectively, and the core radius is r=1 μm. The fiber cross section was divided into 688 triangles with 1393 nodal points.

Fig. 2
Fig. 2

Step-index fiber: H¯t shape error versus nodal point number for the HE11 mode at a normalized frequency ν=2. The core and the cladding refractive indices are nco=1.57 and ncl=1.55, respectively, and the core radius is r=1 μm.

Fig. 3
Fig. 3

Step-index fiber: Hz shape error versus nodal point number for the HE11 mode at a normalized frequency ν=2. The core and the cladding refractive indices are nco=1.57 and ncl=1.55, respectively, and the core radius is r=1 μm.

Fig. 4
Fig. 4

Step-index fiber: H¯t shape error versus nodal point number for the HE11 mode at a normalized frequency ν=4. The core and the cladding refractive indices are nco=2.0 and ncl=1.55, respectively, and the core radius is r=1 μm.

Fig. 5
Fig. 5

Relative error e=(n¯eff-neff)n¯eff versus frequency for the fundamental mode of the metallic rectangular waveguides.

Fig. 6
Fig. 6

Metallic rectangular waveguide: Hx shape error versus nodal point number for the TE10 mode at a frequency of 15 GHz.

Fig. 7
Fig. 7

Metallic rectangular waveguide: size error versus nodal point number with ϕ1=Hy and ϕ2=Hx, for the TE10 mode at a frequency of 15 GHz.

Equations (19)

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

¯×(^-1¯×H¯)-k02H=0inΩ,
H¯×nˆ=0onΓ1,
¯×H¯×nˆ=0onΓ2,
F(H¯)=12 Ω(-1|×H¯|2-k02|H|2)dΩ.
[S]{H}-k02[T]{H}=0,
hx=βHx,
hy=βHy,
hz=-jHz,
β2[A]+[B]+β[L]β[E]Tβ[E][C]+β2[D]{Ht}{Hz}
-k02[F]00[G]{Ht}{Hz}=0.
ˆ=xxxy0xyyy000zz,
[A]+[B]/β2[E]T[E][C]β{Ht}{Hz}
-k02[F]/β200[G]β{Ht}{Hz}=0.
[A]+[B]/β2[E]T[E][C]+β2[D]β{Ht}{Hz}
-k02[F]/β200[G]β{Ht}{Hz}=0,
β2[A]+[B][E]Tβ2[E][C]+β2[D]β{Ht}{Hz}
-k02[F]00[G]β{Ht}{Hz}=0.
k02[F]-[B]-[E]T0k02[G]-[C]β{Ht}{Hz}
-β2[A]0[E][D]β{Ht}{Hz}=0.

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