Abstract

We present a method for full-wave characterization of optical waveguide structures. The method computes mode-propagation constants and cross-sectional field profiles from a straightforward discretization of Maxwell’s equations. These modes are directly excited in a three-dimensional finite-difference time-domain simulation to obtain optical field transmission and reflection coefficients for arbitrary waveguide discontinuities. The implementation uses the perfectly-matched-layer technique to absorb both guided modes and radiated fields. A scattered-field formulation is also utilized to allow accurate determination of weak scattered-field strengths. Individual three-dimensional waveguide sections are cascaded by S-parameter analysis. A complete 104-section Bragg resonator is successfully simulated with the method.

© 2002 Optical Society of America

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  1. D. Marcuse, “Tilt, offset, and end-separation loss of lowest-order slab waveguide mode,” J. Lightwave Technol. LT-4, 1647–1650 (1986).
    [CrossRef]
  2. S. I. Hosain, J.-P. Meunier, Z. H. Wang, “Coupling efficiency of butt-jointed planar waveguides with simultaneous tilt and transverse offset,” J. Lightwave Technol. 14, 901–907 (1996).
    [CrossRef]
  3. M. D. Feit, J. A. Fleck, “Computation of mode properties in optical fiber waveguides by a propagating beam method,” Appl. Opt. 19, 1154–1164 (1980).
    [CrossRef] [PubMed]
  4. W. Huang, C. Xu, S.-T. Chu, S. K. Chaudhuri, “The finite-difference vector beam propagation method: analysis and assessment,” J. Lightwave Technol. 10, 295–305 (1992).
    [CrossRef]
  5. P. Lüsse, P. Stuwe, J. Schüle, H. Unger, “Analysis of vectorial mode fields in optical waveguides by a new finite difference method,” J. Lightwave Technol. 12, 487–493 (1994).
    [CrossRef]
  6. C. L. Xu, W. P. Huang, M. S. Stern, S. K. Chaudhuri, “Full-vectorial mode calculations by finite difference method,” IEE Proc.: Optoelectron. 141, 281–285 (1994).
  7. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. AP-14, 302–307 (1966).
  8. A. Taflove, S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Boston, 2000).
  9. J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
    [CrossRef]
  10. H. E. Hernández-Figueroa, F. A. Fernández, J. B. Davies, “Finite element approach for the modal analysis of open-boundary waveguides,” Electron. Lett. 30, 2031–2032 (1994).
    [CrossRef]
  11. J. W. Wallace, M. A. Jensen, “An FD/FDTD method for optical waveguide modeling,” in Proceedings of the 2000 IEEE Antennas and Propagation Society International Symposium (Institute of Electrical and Electronics Engineers, New York, 2000), Vol. 3, pp. 1380–1383.
  12. F. Girardin, G.-H. Duan, A. Talneau, “Modeling and measurement of spatial-hole-burning applied to amplitude modulated coupling distributed feedback lasers,” IEEE J. Quantum Electron. 31, 834–841 (1995).
    [CrossRef]
  13. S. Adachi, “Model dielectric constants of GaP, GaAs, GaSb, InP, InAs, and InSb,” Phys. Rev. B 35, 7454–7463 (1987).
    [CrossRef]
  14. D. E. Aspnes, A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985–1009 (1983).
    [CrossRef]
  15. B. Jensen, A. Torabi, “Refractive index of quaternary In1-xGaxAsyP1-y lattice matched to InP,” J. Appl. Phys. 54, 3623–3625 (1983).
    [CrossRef]
  16. G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field equations,” IEEE Trans. Electromagn. Compat. EMC-23, 377–382 (1981).
    [CrossRef]
  17. K. K. Mei, J. Fang, “Superabsorption—a method to improve absorbing boundary conditions,” IEEE Trans. Antennas Propag. 40, 1001–1010 (1992).
    [CrossRef]
  18. S. D. Gedney, “Anisotropic perfectly matched layer absorbing medium for the truncation of FDTD lattices,” IEEE Trans. Antennas Propag. 44, 1630–1639 (1996).
    [CrossRef]
  19. C. M. Rappaport, “Perfectly matched absorbing boundary conditions based on anisotropic lossy mapping of space,” IEEE Microwave Guid. Wave Lett. 5, 90–92 (1995).
    [CrossRef]
  20. M. Okoniewski, M. A. Stuchly, M. Mrozowski, J. DeMoerloose, “Modal PML,” IEEE Microwave Guid. Wave Lett. 7, 33–35 (1997).
    [CrossRef]
  21. S. L. Dvorak, “Exact, closed-form expressions for transient fields in homogeneously filled waveguides,” IEEE Trans. Microwave Theory Tech. 42, 2164–2170 (1994).
    [CrossRef]
  22. M.-J. Park, S. Nam, “FDTD modal field generation and absorption in homogeneous waveguides with arbitrary cross section,” in Proceedings of the 1996 IEEE Antennas and Propagation Society International Symposium (Institute of Electrical and Electronics Engineers, New York, 1996), Vol. 2, pp. 1308–1311.
  23. C.-N. Kuo, T. Itoh, B. Houshmand, “Synthesis of absorbing boundary condition with digital filter bank,” in Proceedings of the 1996 IEEE Microwave Theory and Techniques Society International Symposium (Institute of Electrical and Electronics Engineers, New York, 1996), Vol. 2, pp. 1043–1046.
  24. M. Chen, B. Houshmand, T. Itoh, “FDTD analysis of a metal-strip-loaded dielectric leaky-wave antenna,” IEEE Trans. Antennas Propag. 45, 1294–1301 (1997).
    [CrossRef]
  25. M. A. Gribbons, W. P. Pinello, A. C. Cangellaris, “Stretched coordinate technique for numerical absorption of evanescent and propagating waves in planar waveguiding structures,” in Proceedings of the 1995 IEEE Microwave Theory and Techniques Society International Symposium (Institute of Electrical and Electronics Engineers, New York, 1995), Vol. 1, pp. 31–34.
  26. G. Gonzalez, Microwave Transistor Amplifiers (Prentice-Hall, Upper Saddle River, N.J., 1997).

1997 (2)

M. Okoniewski, M. A. Stuchly, M. Mrozowski, J. DeMoerloose, “Modal PML,” IEEE Microwave Guid. Wave Lett. 7, 33–35 (1997).
[CrossRef]

M. Chen, B. Houshmand, T. Itoh, “FDTD analysis of a metal-strip-loaded dielectric leaky-wave antenna,” IEEE Trans. Antennas Propag. 45, 1294–1301 (1997).
[CrossRef]

1996 (2)

S. D. Gedney, “Anisotropic perfectly matched layer absorbing medium for the truncation of FDTD lattices,” IEEE Trans. Antennas Propag. 44, 1630–1639 (1996).
[CrossRef]

S. I. Hosain, J.-P. Meunier, Z. H. Wang, “Coupling efficiency of butt-jointed planar waveguides with simultaneous tilt and transverse offset,” J. Lightwave Technol. 14, 901–907 (1996).
[CrossRef]

1995 (2)

C. M. Rappaport, “Perfectly matched absorbing boundary conditions based on anisotropic lossy mapping of space,” IEEE Microwave Guid. Wave Lett. 5, 90–92 (1995).
[CrossRef]

F. Girardin, G.-H. Duan, A. Talneau, “Modeling and measurement of spatial-hole-burning applied to amplitude modulated coupling distributed feedback lasers,” IEEE J. Quantum Electron. 31, 834–841 (1995).
[CrossRef]

1994 (5)

S. L. Dvorak, “Exact, closed-form expressions for transient fields in homogeneously filled waveguides,” IEEE Trans. Microwave Theory Tech. 42, 2164–2170 (1994).
[CrossRef]

P. Lüsse, P. Stuwe, J. Schüle, H. Unger, “Analysis of vectorial mode fields in optical waveguides by a new finite difference method,” J. Lightwave Technol. 12, 487–493 (1994).
[CrossRef]

C. L. Xu, W. P. Huang, M. S. Stern, S. K. Chaudhuri, “Full-vectorial mode calculations by finite difference method,” IEE Proc.: Optoelectron. 141, 281–285 (1994).

J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

H. E. Hernández-Figueroa, F. A. Fernández, J. B. Davies, “Finite element approach for the modal analysis of open-boundary waveguides,” Electron. Lett. 30, 2031–2032 (1994).
[CrossRef]

1992 (2)

W. Huang, C. Xu, S.-T. Chu, S. K. Chaudhuri, “The finite-difference vector beam propagation method: analysis and assessment,” J. Lightwave Technol. 10, 295–305 (1992).
[CrossRef]

K. K. Mei, J. Fang, “Superabsorption—a method to improve absorbing boundary conditions,” IEEE Trans. Antennas Propag. 40, 1001–1010 (1992).
[CrossRef]

1987 (1)

S. Adachi, “Model dielectric constants of GaP, GaAs, GaSb, InP, InAs, and InSb,” Phys. Rev. B 35, 7454–7463 (1987).
[CrossRef]

1986 (1)

D. Marcuse, “Tilt, offset, and end-separation loss of lowest-order slab waveguide mode,” J. Lightwave Technol. LT-4, 1647–1650 (1986).
[CrossRef]

1983 (2)

D. E. Aspnes, A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985–1009 (1983).
[CrossRef]

B. Jensen, A. Torabi, “Refractive index of quaternary In1-xGaxAsyP1-y lattice matched to InP,” J. Appl. Phys. 54, 3623–3625 (1983).
[CrossRef]

1981 (1)

G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field equations,” IEEE Trans. Electromagn. Compat. EMC-23, 377–382 (1981).
[CrossRef]

1980 (1)

1966 (1)

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. AP-14, 302–307 (1966).

Adachi, S.

S. Adachi, “Model dielectric constants of GaP, GaAs, GaSb, InP, InAs, and InSb,” Phys. Rev. B 35, 7454–7463 (1987).
[CrossRef]

Aspnes, D. E.

D. E. Aspnes, A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985–1009 (1983).
[CrossRef]

Berenger, J.-P.

J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

Cangellaris, A. C.

M. A. Gribbons, W. P. Pinello, A. C. Cangellaris, “Stretched coordinate technique for numerical absorption of evanescent and propagating waves in planar waveguiding structures,” in Proceedings of the 1995 IEEE Microwave Theory and Techniques Society International Symposium (Institute of Electrical and Electronics Engineers, New York, 1995), Vol. 1, pp. 31–34.

Chaudhuri, S. K.

C. L. Xu, W. P. Huang, M. S. Stern, S. K. Chaudhuri, “Full-vectorial mode calculations by finite difference method,” IEE Proc.: Optoelectron. 141, 281–285 (1994).

W. Huang, C. Xu, S.-T. Chu, S. K. Chaudhuri, “The finite-difference vector beam propagation method: analysis and assessment,” J. Lightwave Technol. 10, 295–305 (1992).
[CrossRef]

Chen, M.

M. Chen, B. Houshmand, T. Itoh, “FDTD analysis of a metal-strip-loaded dielectric leaky-wave antenna,” IEEE Trans. Antennas Propag. 45, 1294–1301 (1997).
[CrossRef]

Chu, S.-T.

W. Huang, C. Xu, S.-T. Chu, S. K. Chaudhuri, “The finite-difference vector beam propagation method: analysis and assessment,” J. Lightwave Technol. 10, 295–305 (1992).
[CrossRef]

Davies, J. B.

H. E. Hernández-Figueroa, F. A. Fernández, J. B. Davies, “Finite element approach for the modal analysis of open-boundary waveguides,” Electron. Lett. 30, 2031–2032 (1994).
[CrossRef]

DeMoerloose, J.

M. Okoniewski, M. A. Stuchly, M. Mrozowski, J. DeMoerloose, “Modal PML,” IEEE Microwave Guid. Wave Lett. 7, 33–35 (1997).
[CrossRef]

Duan, G.-H.

F. Girardin, G.-H. Duan, A. Talneau, “Modeling and measurement of spatial-hole-burning applied to amplitude modulated coupling distributed feedback lasers,” IEEE J. Quantum Electron. 31, 834–841 (1995).
[CrossRef]

Dvorak, S. L.

S. L. Dvorak, “Exact, closed-form expressions for transient fields in homogeneously filled waveguides,” IEEE Trans. Microwave Theory Tech. 42, 2164–2170 (1994).
[CrossRef]

Fang, J.

K. K. Mei, J. Fang, “Superabsorption—a method to improve absorbing boundary conditions,” IEEE Trans. Antennas Propag. 40, 1001–1010 (1992).
[CrossRef]

Feit, M. D.

Fernández, F. A.

H. E. Hernández-Figueroa, F. A. Fernández, J. B. Davies, “Finite element approach for the modal analysis of open-boundary waveguides,” Electron. Lett. 30, 2031–2032 (1994).
[CrossRef]

Fleck, J. A.

Gedney, S. D.

S. D. Gedney, “Anisotropic perfectly matched layer absorbing medium for the truncation of FDTD lattices,” IEEE Trans. Antennas Propag. 44, 1630–1639 (1996).
[CrossRef]

Girardin, F.

F. Girardin, G.-H. Duan, A. Talneau, “Modeling and measurement of spatial-hole-burning applied to amplitude modulated coupling distributed feedback lasers,” IEEE J. Quantum Electron. 31, 834–841 (1995).
[CrossRef]

Gonzalez, G.

G. Gonzalez, Microwave Transistor Amplifiers (Prentice-Hall, Upper Saddle River, N.J., 1997).

Gribbons, M. A.

M. A. Gribbons, W. P. Pinello, A. C. Cangellaris, “Stretched coordinate technique for numerical absorption of evanescent and propagating waves in planar waveguiding structures,” in Proceedings of the 1995 IEEE Microwave Theory and Techniques Society International Symposium (Institute of Electrical and Electronics Engineers, New York, 1995), Vol. 1, pp. 31–34.

Hagness, S. C.

A. Taflove, S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Boston, 2000).

Hernández-Figueroa, H. E.

H. E. Hernández-Figueroa, F. A. Fernández, J. B. Davies, “Finite element approach for the modal analysis of open-boundary waveguides,” Electron. Lett. 30, 2031–2032 (1994).
[CrossRef]

Hosain, S. I.

S. I. Hosain, J.-P. Meunier, Z. H. Wang, “Coupling efficiency of butt-jointed planar waveguides with simultaneous tilt and transverse offset,” J. Lightwave Technol. 14, 901–907 (1996).
[CrossRef]

Houshmand, B.

M. Chen, B. Houshmand, T. Itoh, “FDTD analysis of a metal-strip-loaded dielectric leaky-wave antenna,” IEEE Trans. Antennas Propag. 45, 1294–1301 (1997).
[CrossRef]

C.-N. Kuo, T. Itoh, B. Houshmand, “Synthesis of absorbing boundary condition with digital filter bank,” in Proceedings of the 1996 IEEE Microwave Theory and Techniques Society International Symposium (Institute of Electrical and Electronics Engineers, New York, 1996), Vol. 2, pp. 1043–1046.

Huang, W.

W. Huang, C. Xu, S.-T. Chu, S. K. Chaudhuri, “The finite-difference vector beam propagation method: analysis and assessment,” J. Lightwave Technol. 10, 295–305 (1992).
[CrossRef]

Huang, W. P.

C. L. Xu, W. P. Huang, M. S. Stern, S. K. Chaudhuri, “Full-vectorial mode calculations by finite difference method,” IEE Proc.: Optoelectron. 141, 281–285 (1994).

Itoh, T.

M. Chen, B. Houshmand, T. Itoh, “FDTD analysis of a metal-strip-loaded dielectric leaky-wave antenna,” IEEE Trans. Antennas Propag. 45, 1294–1301 (1997).
[CrossRef]

C.-N. Kuo, T. Itoh, B. Houshmand, “Synthesis of absorbing boundary condition with digital filter bank,” in Proceedings of the 1996 IEEE Microwave Theory and Techniques Society International Symposium (Institute of Electrical and Electronics Engineers, New York, 1996), Vol. 2, pp. 1043–1046.

Jensen, B.

B. Jensen, A. Torabi, “Refractive index of quaternary In1-xGaxAsyP1-y lattice matched to InP,” J. Appl. Phys. 54, 3623–3625 (1983).
[CrossRef]

Jensen, M. A.

J. W. Wallace, M. A. Jensen, “An FD/FDTD method for optical waveguide modeling,” in Proceedings of the 2000 IEEE Antennas and Propagation Society International Symposium (Institute of Electrical and Electronics Engineers, New York, 2000), Vol. 3, pp. 1380–1383.

Kuo, C.-N.

C.-N. Kuo, T. Itoh, B. Houshmand, “Synthesis of absorbing boundary condition with digital filter bank,” in Proceedings of the 1996 IEEE Microwave Theory and Techniques Society International Symposium (Institute of Electrical and Electronics Engineers, New York, 1996), Vol. 2, pp. 1043–1046.

Lüsse, P.

P. Lüsse, P. Stuwe, J. Schüle, H. Unger, “Analysis of vectorial mode fields in optical waveguides by a new finite difference method,” J. Lightwave Technol. 12, 487–493 (1994).
[CrossRef]

Marcuse, D.

D. Marcuse, “Tilt, offset, and end-separation loss of lowest-order slab waveguide mode,” J. Lightwave Technol. LT-4, 1647–1650 (1986).
[CrossRef]

Mei, K. K.

K. K. Mei, J. Fang, “Superabsorption—a method to improve absorbing boundary conditions,” IEEE Trans. Antennas Propag. 40, 1001–1010 (1992).
[CrossRef]

Meunier, J.-P.

S. I. Hosain, J.-P. Meunier, Z. H. Wang, “Coupling efficiency of butt-jointed planar waveguides with simultaneous tilt and transverse offset,” J. Lightwave Technol. 14, 901–907 (1996).
[CrossRef]

Mrozowski, M.

M. Okoniewski, M. A. Stuchly, M. Mrozowski, J. DeMoerloose, “Modal PML,” IEEE Microwave Guid. Wave Lett. 7, 33–35 (1997).
[CrossRef]

Mur, G.

G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field equations,” IEEE Trans. Electromagn. Compat. EMC-23, 377–382 (1981).
[CrossRef]

Nam, S.

M.-J. Park, S. Nam, “FDTD modal field generation and absorption in homogeneous waveguides with arbitrary cross section,” in Proceedings of the 1996 IEEE Antennas and Propagation Society International Symposium (Institute of Electrical and Electronics Engineers, New York, 1996), Vol. 2, pp. 1308–1311.

Okoniewski, M.

M. Okoniewski, M. A. Stuchly, M. Mrozowski, J. DeMoerloose, “Modal PML,” IEEE Microwave Guid. Wave Lett. 7, 33–35 (1997).
[CrossRef]

Park, M.-J.

M.-J. Park, S. Nam, “FDTD modal field generation and absorption in homogeneous waveguides with arbitrary cross section,” in Proceedings of the 1996 IEEE Antennas and Propagation Society International Symposium (Institute of Electrical and Electronics Engineers, New York, 1996), Vol. 2, pp. 1308–1311.

Pinello, W. P.

M. A. Gribbons, W. P. Pinello, A. C. Cangellaris, “Stretched coordinate technique for numerical absorption of evanescent and propagating waves in planar waveguiding structures,” in Proceedings of the 1995 IEEE Microwave Theory and Techniques Society International Symposium (Institute of Electrical and Electronics Engineers, New York, 1995), Vol. 1, pp. 31–34.

Rappaport, C. M.

C. M. Rappaport, “Perfectly matched absorbing boundary conditions based on anisotropic lossy mapping of space,” IEEE Microwave Guid. Wave Lett. 5, 90–92 (1995).
[CrossRef]

Schüle, J.

P. Lüsse, P. Stuwe, J. Schüle, H. Unger, “Analysis of vectorial mode fields in optical waveguides by a new finite difference method,” J. Lightwave Technol. 12, 487–493 (1994).
[CrossRef]

Stern, M. S.

C. L. Xu, W. P. Huang, M. S. Stern, S. K. Chaudhuri, “Full-vectorial mode calculations by finite difference method,” IEE Proc.: Optoelectron. 141, 281–285 (1994).

Stuchly, M. A.

M. Okoniewski, M. A. Stuchly, M. Mrozowski, J. DeMoerloose, “Modal PML,” IEEE Microwave Guid. Wave Lett. 7, 33–35 (1997).
[CrossRef]

Studna, A. A.

D. E. Aspnes, A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985–1009 (1983).
[CrossRef]

Stuwe, P.

P. Lüsse, P. Stuwe, J. Schüle, H. Unger, “Analysis of vectorial mode fields in optical waveguides by a new finite difference method,” J. Lightwave Technol. 12, 487–493 (1994).
[CrossRef]

Taflove, A.

A. Taflove, S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Boston, 2000).

Talneau, A.

F. Girardin, G.-H. Duan, A. Talneau, “Modeling and measurement of spatial-hole-burning applied to amplitude modulated coupling distributed feedback lasers,” IEEE J. Quantum Electron. 31, 834–841 (1995).
[CrossRef]

Torabi, A.

B. Jensen, A. Torabi, “Refractive index of quaternary In1-xGaxAsyP1-y lattice matched to InP,” J. Appl. Phys. 54, 3623–3625 (1983).
[CrossRef]

Unger, H.

P. Lüsse, P. Stuwe, J. Schüle, H. Unger, “Analysis of vectorial mode fields in optical waveguides by a new finite difference method,” J. Lightwave Technol. 12, 487–493 (1994).
[CrossRef]

Wallace, J. W.

J. W. Wallace, M. A. Jensen, “An FD/FDTD method for optical waveguide modeling,” in Proceedings of the 2000 IEEE Antennas and Propagation Society International Symposium (Institute of Electrical and Electronics Engineers, New York, 2000), Vol. 3, pp. 1380–1383.

Wang, Z. H.

S. I. Hosain, J.-P. Meunier, Z. H. Wang, “Coupling efficiency of butt-jointed planar waveguides with simultaneous tilt and transverse offset,” J. Lightwave Technol. 14, 901–907 (1996).
[CrossRef]

Xu, C.

W. Huang, C. Xu, S.-T. Chu, S. K. Chaudhuri, “The finite-difference vector beam propagation method: analysis and assessment,” J. Lightwave Technol. 10, 295–305 (1992).
[CrossRef]

Xu, C. L.

C. L. Xu, W. P. Huang, M. S. Stern, S. K. Chaudhuri, “Full-vectorial mode calculations by finite difference method,” IEE Proc.: Optoelectron. 141, 281–285 (1994).

Yee, K. S.

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. AP-14, 302–307 (1966).

Appl. Opt. (1)

Electron. Lett. (1)

H. E. Hernández-Figueroa, F. A. Fernández, J. B. Davies, “Finite element approach for the modal analysis of open-boundary waveguides,” Electron. Lett. 30, 2031–2032 (1994).
[CrossRef]

IEE Proc.: Optoelectron. (1)

C. L. Xu, W. P. Huang, M. S. Stern, S. K. Chaudhuri, “Full-vectorial mode calculations by finite difference method,” IEE Proc.: Optoelectron. 141, 281–285 (1994).

IEEE J. Quantum Electron. (1)

F. Girardin, G.-H. Duan, A. Talneau, “Modeling and measurement of spatial-hole-burning applied to amplitude modulated coupling distributed feedback lasers,” IEEE J. Quantum Electron. 31, 834–841 (1995).
[CrossRef]

IEEE Microwave Guid. Wave Lett. (2)

C. M. Rappaport, “Perfectly matched absorbing boundary conditions based on anisotropic lossy mapping of space,” IEEE Microwave Guid. Wave Lett. 5, 90–92 (1995).
[CrossRef]

M. Okoniewski, M. A. Stuchly, M. Mrozowski, J. DeMoerloose, “Modal PML,” IEEE Microwave Guid. Wave Lett. 7, 33–35 (1997).
[CrossRef]

IEEE Trans. Antennas Propag. (4)

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. AP-14, 302–307 (1966).

K. K. Mei, J. Fang, “Superabsorption—a method to improve absorbing boundary conditions,” IEEE Trans. Antennas Propag. 40, 1001–1010 (1992).
[CrossRef]

S. D. Gedney, “Anisotropic perfectly matched layer absorbing medium for the truncation of FDTD lattices,” IEEE Trans. Antennas Propag. 44, 1630–1639 (1996).
[CrossRef]

M. Chen, B. Houshmand, T. Itoh, “FDTD analysis of a metal-strip-loaded dielectric leaky-wave antenna,” IEEE Trans. Antennas Propag. 45, 1294–1301 (1997).
[CrossRef]

IEEE Trans. Electromagn. Compat. (1)

G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field equations,” IEEE Trans. Electromagn. Compat. EMC-23, 377–382 (1981).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

S. L. Dvorak, “Exact, closed-form expressions for transient fields in homogeneously filled waveguides,” IEEE Trans. Microwave Theory Tech. 42, 2164–2170 (1994).
[CrossRef]

J. Appl. Phys. (1)

B. Jensen, A. Torabi, “Refractive index of quaternary In1-xGaxAsyP1-y lattice matched to InP,” J. Appl. Phys. 54, 3623–3625 (1983).
[CrossRef]

J. Comput. Phys. (1)

J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

J. Lightwave Technol. (4)

W. Huang, C. Xu, S.-T. Chu, S. K. Chaudhuri, “The finite-difference vector beam propagation method: analysis and assessment,” J. Lightwave Technol. 10, 295–305 (1992).
[CrossRef]

P. Lüsse, P. Stuwe, J. Schüle, H. Unger, “Analysis of vectorial mode fields in optical waveguides by a new finite difference method,” J. Lightwave Technol. 12, 487–493 (1994).
[CrossRef]

D. Marcuse, “Tilt, offset, and end-separation loss of lowest-order slab waveguide mode,” J. Lightwave Technol. LT-4, 1647–1650 (1986).
[CrossRef]

S. I. Hosain, J.-P. Meunier, Z. H. Wang, “Coupling efficiency of butt-jointed planar waveguides with simultaneous tilt and transverse offset,” J. Lightwave Technol. 14, 901–907 (1996).
[CrossRef]

Phys. Rev. B (2)

S. Adachi, “Model dielectric constants of GaP, GaAs, GaSb, InP, InAs, and InSb,” Phys. Rev. B 35, 7454–7463 (1987).
[CrossRef]

D. E. Aspnes, A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985–1009 (1983).
[CrossRef]

Other (6)

M.-J. Park, S. Nam, “FDTD modal field generation and absorption in homogeneous waveguides with arbitrary cross section,” in Proceedings of the 1996 IEEE Antennas and Propagation Society International Symposium (Institute of Electrical and Electronics Engineers, New York, 1996), Vol. 2, pp. 1308–1311.

C.-N. Kuo, T. Itoh, B. Houshmand, “Synthesis of absorbing boundary condition with digital filter bank,” in Proceedings of the 1996 IEEE Microwave Theory and Techniques Society International Symposium (Institute of Electrical and Electronics Engineers, New York, 1996), Vol. 2, pp. 1043–1046.

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Figures (9)

Fig. 1
Fig. 1

Waveguide geometry used to assess error in the propagation constant and field profile found with the FD method. Also shown is the assignment of FD grid indices.

Fig. 2
Fig. 2

Error in the mode Ex field profile for the zero-field boundary condition. The dashed line gives the dimensions of the smallest domain for comparison.

Fig. 3
Fig. 3

Error in the mode Ex field profile for the decay boundary condition. The dashed line gives the dimensions of the smallest domain for comparison.

Fig. 4
Fig. 4

Error in the mode Ex field profile for the decay boundary condition for various grid resolutions.

Fig. 5
Fig. 5

Waveguide cross section for the Bragg resonator. n3 takes on discrete values in the longitudinal direction.

Fig. 6
Fig. 6

Section of the Bragg resonator to be modeled with 3D full-wave analysis. The dashed lines denote the extents of the unit section, lt and ls are physical lengths of the tooth and Bragg section, and θ is the angular length of the section in radians at λ0=1550 nm.

Fig. 7
Fig. 7

FDTD simulation volume used to assess performance of propagation and transverse PML.

Fig. 8
Fig. 8

Reflection and transmission of the Bragg resonator, ignoring any sources of error.

Fig. 9
Fig. 9

Transmission and reflection amplitude and phase of the Bragg resonator resulting from random phase shifts on A0 and A120 over 32 realizations. The thick gray curves are the mean and the thin black curves are deviation of the individual realizations from the mean.

Tables (4)

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Table 1 Propagation Constant Value and Fractional Error for Various Sizes of the Simulation Domain

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Table 2 Propagation Constant Value and Error for Various Simulation Cell Sizes

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Table 3 Propagation Constants and Transmission and Reflection Coefficients for the Bragg Section at Various Wavelengths

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Table 4 Sources of Error in the Bragg-Resonator Simulation and Approximate Values

Equations (29)

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Hx=1k0jEzy-βzEy,
Ex=1rk0βzHy-jHzy,
Hy=1k0βzEx-jEzx,
Ey=1rk0jHzx-βzHx,
Hz=jk0Eyx-Exy,Ez=jrk0Hxy-Hyx,
Hx,ij=1k0jEz, ij-Ez, i, j-1Δy-βzEy, ij,
Hy,ij=1k0βzEx, ij-jEz, ij-Ez, i-1, jΔx,
jHz,ij=1k0Ex, ij-Ex, i, j-1Δy-Ey, ij-Ey, i-1, jΔx,
Ex,ij=1rk0βzHy, ij-jHz, i, j+1-Hz, ijΔy,
Ey,ij=1rk0jHz, i+1, j-Hz, ijΔx-βzHx, ij,
jEz,ij=1rk0Hy, i+1, j-Hy, ijΔx-Hx, i, j+1-Hx, ijΔy,
βzEx, ij=1k0ΔxHy,i+1, jr,ijzΔx-Hx,i,j+1-Hx,ijr,ijzΔy+Hy,i-1, jr,i-1, jzΔx+Hx,i-1, j+1-Hx,i-1, jr,i-1, jzΔy+k0-1k0(Δx)21r,ijz+1r,i-1, jzHy,ij,
βzEy, ij=1k0ΔyHy,i+1, j-Hy,ijr,ijzΔx-Hx,i,j+1r,ijzΔy-Hy,i+1, j-1-Hy,i,j-1r,i,j-1zΔx-Hx,i,j-1r,i,j-1zΔy+1k0(Δy)21r,ijz+1r,i,j-1z-k0Hx,ij,
βzHx, ij=1k0ΔxEx,i+1, j-Ex,i+1, j-1-Ex,ij+Ex,i,j-1Δy-Ey,i+1, j+Ey,i-1, jΔx+2k0(Δx)2-r,ijyk0Ey,ij,
βzHy, ij=1k0ΔyEx,i,j+1+Ex,i,j-1Δy+Ey,i-1, j+1-Ey,i,j+1+Ey,ij-Ey,i-1, jΔx+r,ijxk0-2k0(Δy)2Ex,ij,
×E¯s=-μH¯st-(μ-μi)H¯inct,
×H¯s=E¯st+σE¯s+(-i)E¯inct+(σ-σi)E¯inc,
Ex,ijkn+1/2=ΔtHz,i,j+1/2,kn-Hz,i,j-1/2,kΔy-Hy,i,j,k+1/2n-Hy,i,j,k-1/2Δz+Ex,ijkn-1/2
G.
Ex,ijkn+1/2=G+(i-)Δt Ex,ijkinctt=nΔt
=G+(i-)(Ex,ijkinc, n+1/2-Ex,ijkinc, n-1/2),
axbx2x2+ayby2y2+azbz2z2-1Hp
=bppaxHxx+ayHyy+azHzz,
1ω2μψ2x2+ψ2y2+1μzrzrψ2z2+ψ=0,
2x2+2y2-βz2+ω2μψ(x, y)=0.
-jωμH¯(x, y)exp(-jβzz)=[T×E¯(x, y)]exp(-jβzz)+1qzzzˆ×[E¯(x, y)exp(-jβzz)]
-jωμH¯(x, y)=[T×E¯(x, y)]-βzqzqz[zˆ×E¯(x, y)],
σmaxr=-(n+1)ln R0ω2βzΔz,
Az=ijEij,zMij*ij|Mij|2.

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