Abstract

We demonstrate that the three-dimensional vectorial transmission line matrix (TLM) method is applicable to the analysis of lossy multilayer optical waveguiding structures. Any lossy multilayer waveguide geometry, including sharp discontinuities in the transverse plane, can be treated taking into account the coupling between all optical field components. The complex propagation constants (propagation constants and the attenuation coefficients) for the fundamental TE-like and TM-like modes can be determined. These parameters of the fundamental TM-like mode of a typical lossy multilayer rib dielectric waveguide are obtained as functions of free-space wavelength. Calculation of the electric-field pattern is also included. Numerical comparisons with the argument principle method (for the case of lossy multilayer slab waveguides) and the spectral-index technique (for the case of lossy multilayer rib waveguides) are also included, and it is shown that the application of the TLM method to lossy multilayer optical waveguides is accurate.

© 1996 Optical Society of America

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  1. S. Kawakami, S. Nishida, “Perturbation theory of a doubly clad optical fiber with a low-index inner cladding,” IEEE J. Quantum Electron. 11, 130–138 (1975).
    [CrossRef]
  2. E. Anemogiannis, E. N. Glytsis, “Multilayer waveguides: efficient numerical analysis of general structures,” J. Lightwave Technol. 10, 1344–1351 (1992).
    [CrossRef]
  3. A. K. Ghatak, K. Thyagarajan, M. R. Shanoy, “Numerical analysis of planar optical waveguides using matrix approach,” J. Lightwave Technol. 5, 660–667 (1991).
    [CrossRef]
  4. L. Sun, E. Marhic, “Numerical study of attenuation in multilayer infrared waveguides by the chain convergence method,” J. Opt. Soc. Am. B 8, 478–483 (1991).
    [CrossRef]
  5. K. H. Schlereth, M. Tacke, “The complex propagation constant of multilayer waveguides: an algorithm for a personal computer,” IEEE J. Quantum Electron. 26, 626–630 (1990).
    [CrossRef]
  6. S. Ruschin, E. Marom, “Coupling effects in symmetrical three-guide structures,” J. Opt. Soc. Am. A 1, 1120–1128 (1984).
    [CrossRef]
  7. J. Chilwell, I. Hodgkinson, “Thin-films field-transfer matrix theory of planar multilayer waveguides and reflection from prism-loaded waveguides,” J. Opt. Soc. Am. A 1, 742–753 (1984).
    [CrossRef]
  8. L. M. Walpita, “Solutions for planar optical waveguide equations by selecting zero elements in a characteristics matrix,” J. Opt. Soc. Am. A 2, 595–602 (1985).
    [CrossRef]
  9. L. Sun, G. L. Yip, “Analysis of metal-clad optical waveguide polarizers by the vector beam propagation method,” Appl. Opt. 33, 1047–1050 (1994).
    [CrossRef] [PubMed]
  10. C. Themistos, B. M. A. Rahman, K. T. V. Grattan, “Finite element analysis for lossy optical waveguides by using perturbation techniques,” Photon. Technol. Lett. 6, 537–539 (1994).
    [CrossRef]
  11. B. M. A. Rahman, J. B. Davies, “Finite element analysis of optical and microwave waveguide problems,” IEEE Trans. Microwave Theory Tech. MTT-32, 20–28 (1984).
    [CrossRef]
  12. B. M. A. Rahman, F. A. Fernandez, J. B. Davis, “Review of finite element methods for microwave and optical waveguides,” Proc. IEEE 79, 1442–1448 (1991).
    [CrossRef]
  13. K. Hayata, M. Koshiba, M. Suzuki, “Lateral mode analysis of buried heterostructure diode lasers by the finite element method,” IEEE J: Quantum Electron. 22, 781–788 (1986).
    [CrossRef]
  14. A. D. McAulay, “Variational finite-element solution for dissipative waveguides and transportation application,” IEEE Trans. Microwave Theory Tech. 25, 382–392 (1977).
    [CrossRef]
  15. K. Hayata, K. Miura, M. Koshiba, “Finite element formulation for lossy waveguides,” IEEE Trans. Microwave Theory Tech. 36, 268–276 (1988).
    [CrossRef]
  16. G. M. Berry, S. V. Burke, J. M. Heaton, D. R. Wight, “Analysis of multilayer semiconductor rib waveguides with high refractive index substrates,” Electron. Lett. 29, 1941–1942 (1993).
    [CrossRef]
  17. S. M. Moniri-Ardakani, E. N. Glytsis, “Application of the transmission line matrix method to the analysis of slab and channel optical waveguides,” Appl. Opt. 34, 2704–2711 (1995).
    [CrossRef] [PubMed]
  18. P. B. Johns, R. L. Beurle, “Numerical solution of 2-dimensional scattering problems using a transmission-line matrix,” Proc. Inst. Electr. Eng. 118, 1203–1208 (1971).
    [CrossRef]
  19. P. B. Johns, “Application of the transmission-line matrix method to homogenous waveguides of arbitrary cross-section,” Proc. Inst. Electr. Eng. 119, 1086–1091 (1972).
    [CrossRef]
  20. S. Akhtarzad, P. B. Johns, “Solutions of Maxwell’s equations in three space dimensions by the t.l.m. method of numerical analysis,” Proc. Inst. Electr. Eng. 122, 1344–1348 (1975).
    [CrossRef]
  21. S. Akhtarzad, P. B. Johns, “Three dimensional transmission-line matrix computer analysis of microstrip resonators,” IEEE Trans. Microwave Theory Tech. MTT-23, 990–997 (1975).
    [CrossRef]
  22. S. Akhtarzad, P. B. Johns, “Generalized element for t.l.m. method of analysis,” Proc. Inst. Electr. Eng. 122, 1349–1352 (1975).
    [CrossRef]
  23. W. J. R. Hoefer, “The transmission line matrix method theory and applications,” IEEE Trans. Microwave Theory Tech. MTT-33, 882–893 (1985).
    [CrossRef]
  24. P. B. Johns, “A symmetrical condensed node for TLM method,” IEEE Trans. Microwave Theory Tech. MTT-35, 370–377 (1987).
    [CrossRef]
  25. J. S. Nielsen, W. J. R. Hoefer, “Generalized dispersion analysis and spurious modes of 2-D and 3-D TLM formulations,” IEEE Trans. Microwave Theory Tech. 41, 1375–1384 (1993).
    [CrossRef]
  26. A. A. Oliner, S. T. Peng, “Guidance and leakage properties of a class of open dielectric waveguides. Part II. New physical effects,” IEEE Trans. Microwave Theory Tech. MTT-29, 855–869 (1981).
    [CrossRef]
  27. R. E. Collin, Field Theory of Guided Waves (IEEE Press, New York, 1991).
  28. H. Jin, R. Vahldieck, “Full-wave analysis of guiding structures using a 2-D array of 3-D TLM nodes,” IEEE Trans. Microwave Theory Tech. 41, 472–477 (1993).
    [CrossRef]
  29. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes: the Art of Scientific Computing (Cambridge U. Press, New York, 1989).
  30. M. S. Stern, P. C. Kendall, P. W. A. Mcllroy, “Analysis of the spectral index method for vector modes of rib waveguides,” Proc. Inst. Electr. Eng. Part J 137, 21–26 (1990).

1995 (1)

1994 (2)

L. Sun, G. L. Yip, “Analysis of metal-clad optical waveguide polarizers by the vector beam propagation method,” Appl. Opt. 33, 1047–1050 (1994).
[CrossRef] [PubMed]

C. Themistos, B. M. A. Rahman, K. T. V. Grattan, “Finite element analysis for lossy optical waveguides by using perturbation techniques,” Photon. Technol. Lett. 6, 537–539 (1994).
[CrossRef]

1993 (3)

G. M. Berry, S. V. Burke, J. M. Heaton, D. R. Wight, “Analysis of multilayer semiconductor rib waveguides with high refractive index substrates,” Electron. Lett. 29, 1941–1942 (1993).
[CrossRef]

J. S. Nielsen, W. J. R. Hoefer, “Generalized dispersion analysis and spurious modes of 2-D and 3-D TLM formulations,” IEEE Trans. Microwave Theory Tech. 41, 1375–1384 (1993).
[CrossRef]

H. Jin, R. Vahldieck, “Full-wave analysis of guiding structures using a 2-D array of 3-D TLM nodes,” IEEE Trans. Microwave Theory Tech. 41, 472–477 (1993).
[CrossRef]

1992 (1)

E. Anemogiannis, E. N. Glytsis, “Multilayer waveguides: efficient numerical analysis of general structures,” J. Lightwave Technol. 10, 1344–1351 (1992).
[CrossRef]

1991 (3)

A. K. Ghatak, K. Thyagarajan, M. R. Shanoy, “Numerical analysis of planar optical waveguides using matrix approach,” J. Lightwave Technol. 5, 660–667 (1991).
[CrossRef]

L. Sun, E. Marhic, “Numerical study of attenuation in multilayer infrared waveguides by the chain convergence method,” J. Opt. Soc. Am. B 8, 478–483 (1991).
[CrossRef]

B. M. A. Rahman, F. A. Fernandez, J. B. Davis, “Review of finite element methods for microwave and optical waveguides,” Proc. IEEE 79, 1442–1448 (1991).
[CrossRef]

1990 (2)

K. H. Schlereth, M. Tacke, “The complex propagation constant of multilayer waveguides: an algorithm for a personal computer,” IEEE J. Quantum Electron. 26, 626–630 (1990).
[CrossRef]

M. S. Stern, P. C. Kendall, P. W. A. Mcllroy, “Analysis of the spectral index method for vector modes of rib waveguides,” Proc. Inst. Electr. Eng. Part J 137, 21–26 (1990).

1988 (1)

K. Hayata, K. Miura, M. Koshiba, “Finite element formulation for lossy waveguides,” IEEE Trans. Microwave Theory Tech. 36, 268–276 (1988).
[CrossRef]

1987 (1)

P. B. Johns, “A symmetrical condensed node for TLM method,” IEEE Trans. Microwave Theory Tech. MTT-35, 370–377 (1987).
[CrossRef]

1986 (1)

K. Hayata, M. Koshiba, M. Suzuki, “Lateral mode analysis of buried heterostructure diode lasers by the finite element method,” IEEE J: Quantum Electron. 22, 781–788 (1986).
[CrossRef]

1985 (2)

L. M. Walpita, “Solutions for planar optical waveguide equations by selecting zero elements in a characteristics matrix,” J. Opt. Soc. Am. A 2, 595–602 (1985).
[CrossRef]

W. J. R. Hoefer, “The transmission line matrix method theory and applications,” IEEE Trans. Microwave Theory Tech. MTT-33, 882–893 (1985).
[CrossRef]

1984 (3)

1981 (1)

A. A. Oliner, S. T. Peng, “Guidance and leakage properties of a class of open dielectric waveguides. Part II. New physical effects,” IEEE Trans. Microwave Theory Tech. MTT-29, 855–869 (1981).
[CrossRef]

1977 (1)

A. D. McAulay, “Variational finite-element solution for dissipative waveguides and transportation application,” IEEE Trans. Microwave Theory Tech. 25, 382–392 (1977).
[CrossRef]

1975 (4)

S. Akhtarzad, P. B. Johns, “Solutions of Maxwell’s equations in three space dimensions by the t.l.m. method of numerical analysis,” Proc. Inst. Electr. Eng. 122, 1344–1348 (1975).
[CrossRef]

S. Akhtarzad, P. B. Johns, “Three dimensional transmission-line matrix computer analysis of microstrip resonators,” IEEE Trans. Microwave Theory Tech. MTT-23, 990–997 (1975).
[CrossRef]

S. Akhtarzad, P. B. Johns, “Generalized element for t.l.m. method of analysis,” Proc. Inst. Electr. Eng. 122, 1349–1352 (1975).
[CrossRef]

S. Kawakami, S. Nishida, “Perturbation theory of a doubly clad optical fiber with a low-index inner cladding,” IEEE J. Quantum Electron. 11, 130–138 (1975).
[CrossRef]

1972 (1)

P. B. Johns, “Application of the transmission-line matrix method to homogenous waveguides of arbitrary cross-section,” Proc. Inst. Electr. Eng. 119, 1086–1091 (1972).
[CrossRef]

1971 (1)

P. B. Johns, R. L. Beurle, “Numerical solution of 2-dimensional scattering problems using a transmission-line matrix,” Proc. Inst. Electr. Eng. 118, 1203–1208 (1971).
[CrossRef]

Akhtarzad, S.

S. Akhtarzad, P. B. Johns, “Solutions of Maxwell’s equations in three space dimensions by the t.l.m. method of numerical analysis,” Proc. Inst. Electr. Eng. 122, 1344–1348 (1975).
[CrossRef]

S. Akhtarzad, P. B. Johns, “Three dimensional transmission-line matrix computer analysis of microstrip resonators,” IEEE Trans. Microwave Theory Tech. MTT-23, 990–997 (1975).
[CrossRef]

S. Akhtarzad, P. B. Johns, “Generalized element for t.l.m. method of analysis,” Proc. Inst. Electr. Eng. 122, 1349–1352 (1975).
[CrossRef]

Anemogiannis, E.

E. Anemogiannis, E. N. Glytsis, “Multilayer waveguides: efficient numerical analysis of general structures,” J. Lightwave Technol. 10, 1344–1351 (1992).
[CrossRef]

Berry, G. M.

G. M. Berry, S. V. Burke, J. M. Heaton, D. R. Wight, “Analysis of multilayer semiconductor rib waveguides with high refractive index substrates,” Electron. Lett. 29, 1941–1942 (1993).
[CrossRef]

Beurle, R. L.

P. B. Johns, R. L. Beurle, “Numerical solution of 2-dimensional scattering problems using a transmission-line matrix,” Proc. Inst. Electr. Eng. 118, 1203–1208 (1971).
[CrossRef]

Burke, S. V.

G. M. Berry, S. V. Burke, J. M. Heaton, D. R. Wight, “Analysis of multilayer semiconductor rib waveguides with high refractive index substrates,” Electron. Lett. 29, 1941–1942 (1993).
[CrossRef]

Chilwell, J.

Collin, R. E.

R. E. Collin, Field Theory of Guided Waves (IEEE Press, New York, 1991).

Davies, J. B.

B. M. A. Rahman, J. B. Davies, “Finite element analysis of optical and microwave waveguide problems,” IEEE Trans. Microwave Theory Tech. MTT-32, 20–28 (1984).
[CrossRef]

Davis, J. B.

B. M. A. Rahman, F. A. Fernandez, J. B. Davis, “Review of finite element methods for microwave and optical waveguides,” Proc. IEEE 79, 1442–1448 (1991).
[CrossRef]

Fernandez, F. A.

B. M. A. Rahman, F. A. Fernandez, J. B. Davis, “Review of finite element methods for microwave and optical waveguides,” Proc. IEEE 79, 1442–1448 (1991).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes: the Art of Scientific Computing (Cambridge U. Press, New York, 1989).

Ghatak, A. K.

A. K. Ghatak, K. Thyagarajan, M. R. Shanoy, “Numerical analysis of planar optical waveguides using matrix approach,” J. Lightwave Technol. 5, 660–667 (1991).
[CrossRef]

Glytsis, E. N.

S. M. Moniri-Ardakani, E. N. Glytsis, “Application of the transmission line matrix method to the analysis of slab and channel optical waveguides,” Appl. Opt. 34, 2704–2711 (1995).
[CrossRef] [PubMed]

E. Anemogiannis, E. N. Glytsis, “Multilayer waveguides: efficient numerical analysis of general structures,” J. Lightwave Technol. 10, 1344–1351 (1992).
[CrossRef]

Grattan, K. T. V.

C. Themistos, B. M. A. Rahman, K. T. V. Grattan, “Finite element analysis for lossy optical waveguides by using perturbation techniques,” Photon. Technol. Lett. 6, 537–539 (1994).
[CrossRef]

Hayata, K.

K. Hayata, K. Miura, M. Koshiba, “Finite element formulation for lossy waveguides,” IEEE Trans. Microwave Theory Tech. 36, 268–276 (1988).
[CrossRef]

K. Hayata, M. Koshiba, M. Suzuki, “Lateral mode analysis of buried heterostructure diode lasers by the finite element method,” IEEE J: Quantum Electron. 22, 781–788 (1986).
[CrossRef]

Heaton, J. M.

G. M. Berry, S. V. Burke, J. M. Heaton, D. R. Wight, “Analysis of multilayer semiconductor rib waveguides with high refractive index substrates,” Electron. Lett. 29, 1941–1942 (1993).
[CrossRef]

Hodgkinson, I.

Hoefer, W. J. R.

J. S. Nielsen, W. J. R. Hoefer, “Generalized dispersion analysis and spurious modes of 2-D and 3-D TLM formulations,” IEEE Trans. Microwave Theory Tech. 41, 1375–1384 (1993).
[CrossRef]

W. J. R. Hoefer, “The transmission line matrix method theory and applications,” IEEE Trans. Microwave Theory Tech. MTT-33, 882–893 (1985).
[CrossRef]

Jin, H.

H. Jin, R. Vahldieck, “Full-wave analysis of guiding structures using a 2-D array of 3-D TLM nodes,” IEEE Trans. Microwave Theory Tech. 41, 472–477 (1993).
[CrossRef]

Johns, P. B.

P. B. Johns, “A symmetrical condensed node for TLM method,” IEEE Trans. Microwave Theory Tech. MTT-35, 370–377 (1987).
[CrossRef]

S. Akhtarzad, P. B. Johns, “Generalized element for t.l.m. method of analysis,” Proc. Inst. Electr. Eng. 122, 1349–1352 (1975).
[CrossRef]

S. Akhtarzad, P. B. Johns, “Three dimensional transmission-line matrix computer analysis of microstrip resonators,” IEEE Trans. Microwave Theory Tech. MTT-23, 990–997 (1975).
[CrossRef]

S. Akhtarzad, P. B. Johns, “Solutions of Maxwell’s equations in three space dimensions by the t.l.m. method of numerical analysis,” Proc. Inst. Electr. Eng. 122, 1344–1348 (1975).
[CrossRef]

P. B. Johns, “Application of the transmission-line matrix method to homogenous waveguides of arbitrary cross-section,” Proc. Inst. Electr. Eng. 119, 1086–1091 (1972).
[CrossRef]

P. B. Johns, R. L. Beurle, “Numerical solution of 2-dimensional scattering problems using a transmission-line matrix,” Proc. Inst. Electr. Eng. 118, 1203–1208 (1971).
[CrossRef]

Kawakami, S.

S. Kawakami, S. Nishida, “Perturbation theory of a doubly clad optical fiber with a low-index inner cladding,” IEEE J. Quantum Electron. 11, 130–138 (1975).
[CrossRef]

Kendall, P. C.

M. S. Stern, P. C. Kendall, P. W. A. Mcllroy, “Analysis of the spectral index method for vector modes of rib waveguides,” Proc. Inst. Electr. Eng. Part J 137, 21–26 (1990).

Koshiba, M.

K. Hayata, K. Miura, M. Koshiba, “Finite element formulation for lossy waveguides,” IEEE Trans. Microwave Theory Tech. 36, 268–276 (1988).
[CrossRef]

K. Hayata, M. Koshiba, M. Suzuki, “Lateral mode analysis of buried heterostructure diode lasers by the finite element method,” IEEE J: Quantum Electron. 22, 781–788 (1986).
[CrossRef]

Marhic, E.

Marom, E.

McAulay, A. D.

A. D. McAulay, “Variational finite-element solution for dissipative waveguides and transportation application,” IEEE Trans. Microwave Theory Tech. 25, 382–392 (1977).
[CrossRef]

Mcllroy, P. W. A.

M. S. Stern, P. C. Kendall, P. W. A. Mcllroy, “Analysis of the spectral index method for vector modes of rib waveguides,” Proc. Inst. Electr. Eng. Part J 137, 21–26 (1990).

Miura, K.

K. Hayata, K. Miura, M. Koshiba, “Finite element formulation for lossy waveguides,” IEEE Trans. Microwave Theory Tech. 36, 268–276 (1988).
[CrossRef]

Moniri-Ardakani, S. M.

Nielsen, J. S.

J. S. Nielsen, W. J. R. Hoefer, “Generalized dispersion analysis and spurious modes of 2-D and 3-D TLM formulations,” IEEE Trans. Microwave Theory Tech. 41, 1375–1384 (1993).
[CrossRef]

Nishida, S.

S. Kawakami, S. Nishida, “Perturbation theory of a doubly clad optical fiber with a low-index inner cladding,” IEEE J. Quantum Electron. 11, 130–138 (1975).
[CrossRef]

Oliner, A. A.

A. A. Oliner, S. T. Peng, “Guidance and leakage properties of a class of open dielectric waveguides. Part II. New physical effects,” IEEE Trans. Microwave Theory Tech. MTT-29, 855–869 (1981).
[CrossRef]

Peng, S. T.

A. A. Oliner, S. T. Peng, “Guidance and leakage properties of a class of open dielectric waveguides. Part II. New physical effects,” IEEE Trans. Microwave Theory Tech. MTT-29, 855–869 (1981).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes: the Art of Scientific Computing (Cambridge U. Press, New York, 1989).

Rahman, B. M. A.

C. Themistos, B. M. A. Rahman, K. T. V. Grattan, “Finite element analysis for lossy optical waveguides by using perturbation techniques,” Photon. Technol. Lett. 6, 537–539 (1994).
[CrossRef]

B. M. A. Rahman, F. A. Fernandez, J. B. Davis, “Review of finite element methods for microwave and optical waveguides,” Proc. IEEE 79, 1442–1448 (1991).
[CrossRef]

B. M. A. Rahman, J. B. Davies, “Finite element analysis of optical and microwave waveguide problems,” IEEE Trans. Microwave Theory Tech. MTT-32, 20–28 (1984).
[CrossRef]

Ruschin, S.

Schlereth, K. H.

K. H. Schlereth, M. Tacke, “The complex propagation constant of multilayer waveguides: an algorithm for a personal computer,” IEEE J. Quantum Electron. 26, 626–630 (1990).
[CrossRef]

Shanoy, M. R.

A. K. Ghatak, K. Thyagarajan, M. R. Shanoy, “Numerical analysis of planar optical waveguides using matrix approach,” J. Lightwave Technol. 5, 660–667 (1991).
[CrossRef]

Stern, M. S.

M. S. Stern, P. C. Kendall, P. W. A. Mcllroy, “Analysis of the spectral index method for vector modes of rib waveguides,” Proc. Inst. Electr. Eng. Part J 137, 21–26 (1990).

Sun, L.

Suzuki, M.

K. Hayata, M. Koshiba, M. Suzuki, “Lateral mode analysis of buried heterostructure diode lasers by the finite element method,” IEEE J: Quantum Electron. 22, 781–788 (1986).
[CrossRef]

Tacke, M.

K. H. Schlereth, M. Tacke, “The complex propagation constant of multilayer waveguides: an algorithm for a personal computer,” IEEE J. Quantum Electron. 26, 626–630 (1990).
[CrossRef]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes: the Art of Scientific Computing (Cambridge U. Press, New York, 1989).

Themistos, C.

C. Themistos, B. M. A. Rahman, K. T. V. Grattan, “Finite element analysis for lossy optical waveguides by using perturbation techniques,” Photon. Technol. Lett. 6, 537–539 (1994).
[CrossRef]

Thyagarajan, K.

A. K. Ghatak, K. Thyagarajan, M. R. Shanoy, “Numerical analysis of planar optical waveguides using matrix approach,” J. Lightwave Technol. 5, 660–667 (1991).
[CrossRef]

Vahldieck, R.

H. Jin, R. Vahldieck, “Full-wave analysis of guiding structures using a 2-D array of 3-D TLM nodes,” IEEE Trans. Microwave Theory Tech. 41, 472–477 (1993).
[CrossRef]

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes: the Art of Scientific Computing (Cambridge U. Press, New York, 1989).

Walpita, L. M.

Wight, D. R.

G. M. Berry, S. V. Burke, J. M. Heaton, D. R. Wight, “Analysis of multilayer semiconductor rib waveguides with high refractive index substrates,” Electron. Lett. 29, 1941–1942 (1993).
[CrossRef]

Yip, G. L.

Appl. Opt. (2)

Electron. Lett. (1)

G. M. Berry, S. V. Burke, J. M. Heaton, D. R. Wight, “Analysis of multilayer semiconductor rib waveguides with high refractive index substrates,” Electron. Lett. 29, 1941–1942 (1993).
[CrossRef]

IEEE J. Quantum Electron. (2)

S. Kawakami, S. Nishida, “Perturbation theory of a doubly clad optical fiber with a low-index inner cladding,” IEEE J. Quantum Electron. 11, 130–138 (1975).
[CrossRef]

K. H. Schlereth, M. Tacke, “The complex propagation constant of multilayer waveguides: an algorithm for a personal computer,” IEEE J. Quantum Electron. 26, 626–630 (1990).
[CrossRef]

IEEE J: Quantum Electron. (1)

K. Hayata, M. Koshiba, M. Suzuki, “Lateral mode analysis of buried heterostructure diode lasers by the finite element method,” IEEE J: Quantum Electron. 22, 781–788 (1986).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (9)

A. D. McAulay, “Variational finite-element solution for dissipative waveguides and transportation application,” IEEE Trans. Microwave Theory Tech. 25, 382–392 (1977).
[CrossRef]

K. Hayata, K. Miura, M. Koshiba, “Finite element formulation for lossy waveguides,” IEEE Trans. Microwave Theory Tech. 36, 268–276 (1988).
[CrossRef]

S. Akhtarzad, P. B. Johns, “Three dimensional transmission-line matrix computer analysis of microstrip resonators,” IEEE Trans. Microwave Theory Tech. MTT-23, 990–997 (1975).
[CrossRef]

W. J. R. Hoefer, “The transmission line matrix method theory and applications,” IEEE Trans. Microwave Theory Tech. MTT-33, 882–893 (1985).
[CrossRef]

P. B. Johns, “A symmetrical condensed node for TLM method,” IEEE Trans. Microwave Theory Tech. MTT-35, 370–377 (1987).
[CrossRef]

J. S. Nielsen, W. J. R. Hoefer, “Generalized dispersion analysis and spurious modes of 2-D and 3-D TLM formulations,” IEEE Trans. Microwave Theory Tech. 41, 1375–1384 (1993).
[CrossRef]

A. A. Oliner, S. T. Peng, “Guidance and leakage properties of a class of open dielectric waveguides. Part II. New physical effects,” IEEE Trans. Microwave Theory Tech. MTT-29, 855–869 (1981).
[CrossRef]

H. Jin, R. Vahldieck, “Full-wave analysis of guiding structures using a 2-D array of 3-D TLM nodes,” IEEE Trans. Microwave Theory Tech. 41, 472–477 (1993).
[CrossRef]

B. M. A. Rahman, J. B. Davies, “Finite element analysis of optical and microwave waveguide problems,” IEEE Trans. Microwave Theory Tech. MTT-32, 20–28 (1984).
[CrossRef]

J. Lightwave Technol. (2)

E. Anemogiannis, E. N. Glytsis, “Multilayer waveguides: efficient numerical analysis of general structures,” J. Lightwave Technol. 10, 1344–1351 (1992).
[CrossRef]

A. K. Ghatak, K. Thyagarajan, M. R. Shanoy, “Numerical analysis of planar optical waveguides using matrix approach,” J. Lightwave Technol. 5, 660–667 (1991).
[CrossRef]

J. Opt. Soc. Am. A (3)

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

Photon. Technol. Lett. (1)

C. Themistos, B. M. A. Rahman, K. T. V. Grattan, “Finite element analysis for lossy optical waveguides by using perturbation techniques,” Photon. Technol. Lett. 6, 537–539 (1994).
[CrossRef]

Proc. IEEE (1)

B. M. A. Rahman, F. A. Fernandez, J. B. Davis, “Review of finite element methods for microwave and optical waveguides,” Proc. IEEE 79, 1442–1448 (1991).
[CrossRef]

Proc. Inst. Electr. Eng. (4)

P. B. Johns, R. L. Beurle, “Numerical solution of 2-dimensional scattering problems using a transmission-line matrix,” Proc. Inst. Electr. Eng. 118, 1203–1208 (1971).
[CrossRef]

P. B. Johns, “Application of the transmission-line matrix method to homogenous waveguides of arbitrary cross-section,” Proc. Inst. Electr. Eng. 119, 1086–1091 (1972).
[CrossRef]

S. Akhtarzad, P. B. Johns, “Solutions of Maxwell’s equations in three space dimensions by the t.l.m. method of numerical analysis,” Proc. Inst. Electr. Eng. 122, 1344–1348 (1975).
[CrossRef]

S. Akhtarzad, P. B. Johns, “Generalized element for t.l.m. method of analysis,” Proc. Inst. Electr. Eng. 122, 1349–1352 (1975).
[CrossRef]

Proc. Inst. Electr. Eng. Part J (1)

M. S. Stern, P. C. Kendall, P. W. A. Mcllroy, “Analysis of the spectral index method for vector modes of rib waveguides,” Proc. Inst. Electr. Eng. Part J 137, 21–26 (1990).

Other (2)

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes: the Art of Scientific Computing (Cambridge U. Press, New York, 1989).

R. E. Collin, Field Theory of Guided Waves (IEEE Press, New York, 1991).

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Figures (13)

Fig. 1
Fig. 1

Cross-sectional view of a lossy multilayer optical rib waveguide.

Fig. 2
Fig. 2

(a) Shunt node incorporating loss and permittivity; (b) the lumped model equivalent circuit of the shunt node.

Fig. 3
Fig. 3

Cross-sectional view of a lossy multilayer optical waveguide.

Fig. 4
Fig. 4

Comparison of the effective index Neff (dispersion curve) and the TM0 (fundamental mode) of the multilayer lossy waveguide shown in Fig. 3 determined by the TLM method (filled triangle) and the APM (filled circle). The open triangle represents the effective index determined by the TLM when we reduced Δl to 67% of its initial value. The open square represents the effective index determined by the TLM when we placed the substrate ABC 30 Δl further away.

Fig. 5
Fig. 5

Comparison of the normalized effective attenuation coefficient α(λ0/2π) of the TM0 fundamental mode of the multilayer lossy waveguide shown in Fig. 3 obtained by the TLM method (filled triangle) and the APM (filled circle). The open triangle and open square were obtained as described in Fig. 4.

Fig. 6
Fig. 6

Percentage error of the effective index comparing the TLM results with the exact results.

Fig. 7
Fig. 7

Percentage error of the effective attenuation coefficient comparing the TLM results with the exact results.

Fig. 8
Fig. 8

Comparison of the electric-field pattern Ex of the fundamental TM mode of the waveguide shown in Fig. 3 determined by the TLM method (dashed line) and exact theory (solid curve) at λ0 = 0.6328 μm. The discontinuities are expected because only the Dx and not the Ex field component is continuous across the interfaces.

Fig. 9
Fig. 9

Comparison of normalized effective attenuation coefficient determined by linear extrapolation, the APM (exact), and the TLM method for linear scaling of the conductivity for the multilayer lossy waveguide shown in Fig. 3 at λ0 = 0.6328 μm.

Fig. 10
Fig. 10

Cross-sectional view of a lossy optical rib waveguide.

Fig. 11
Fig. 11

Dispersion curve of the TM fundamental mode of the multilayer lossy rib waveguide shown in Fig. 1 determined by the TLM method.

Fig. 12
Fig. 12

Effective attenuation coefficient curve of the TM fundamental mode of the multilayer lossy rib waveguide shown in Fig. 1 determined by the TLM method.

Fig. 13
Fig. 13

Electric field pattern Ex of the fundamental TM mode of the waveguide shown in Fig. 1 determined by the TLM method at λ0 = 0.7383 μm. The discontinuities are expected because only the Dx and not the Ex field component is continuous across the interfaces.

Tables (2)

Tables Icon

Table 1 Effective Indices and the Normalized Attenuation Coefficient of the Lowest-Order TM Mode of the Waveguide Shown in Fig. 3a

Tables Icon

Table 2 Effective Indices and the Normalized Attenuation Coefficient of the Lowest-Order TE-like and TM-like Modes of the Waveguide Shown in Fig. 10a

Equations (17)

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H z y - H z y = [ σ x x ( x , y ) + 0 ɛ x x ( x , y ) t ] E x ,
H x z - H z x = [ σ y y ( x , y ) + 0 ɛ y y ( x , y ) t ] E y ,
H y x - H x y = [ σ z z ( x , y ) + 0 ɛ z z ( x , y ) t ] E z ,
E z y - E y z = - μ 0 μ x x ( x , y ) H x t ,
E x z - E z x = - μ 0 μ y y ( x , y ) H y t ,
E y x - E x y = - μ 0 μ z z ( x , y ) H z t ,
¯ = 0 [ ɛ x x ( x , y ) 0 0 0 ɛ y y ( x , y ) 0 0 0 ɛ z z ( x , y ) ] ,
μ ¯ = μ 0 [ μ x x ( x , y ) 0 0 0 μ y y ( x , y ) 0 0 0 μ z z ( x , y ) ] ,
σ ¯ = [ σ x x ( x , y ) 0 0 0 σ y y ( x , y ) 0 0 0 σ z z ( x , y ) ] ,
I x x - I z z = - [ G y y Z 0 Δ l + 2 C ( 1 + Y y y / 4 ) t ] V y ,
E i ( k Δ t ) = E i ( 0 ) exp ( j ω i k Δ t ) ,
E T ( k Δ t ) = i = 1 m E i ( k Δ t ) .
k = 0 E T ( k Δ t ) exp ( - j ω i k Δ t ) = k = 0 l = 1 m E l ( 0 ) exp [ j ( ω l - ω i ) k Δ t ] = k = 0 E i ( 0 ) K = 0 N E i ( 0 ) = N E i ( 0 ) ,
E i ( k Δ t ) = E i ( 0 ) exp ( j ω i k Δ t ) exp ( - a i k Δ t ) .
k = 0 E T ( k Δ t ) exp ( - j ω i k Δ t ) = k = 0 { l = 1 m E l ( 0 ) exp [ j ( ω l - ω i ) k Δ t ] exp ( - α i k Δ t ) } k = 0 E i ( 0 ) exp ( - α i k Δ t ) = E i ( 0 ) k = 0 exp ( - α i k Δ t ) E i ( 0 ) k = 0 N exp ( - α i k Δ t ) = E ˜ i ( N Δ t ) ,
E ˜ i ( N Δ t ) = E i ( 0 ) 1 - exp [ - ( N + 1 ) α i Δ t ] 1 - exp ( - α i Δ t ) .
α i , g = α i υ i , g ,

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