Abstract

We introduce a high-order time-domain discontinuous spectral element method for the study of the optical coupling by evanescent whispering gallery modes between two microcylinders, the building blocks of coupled resonator optical waveguide devices. By using the discontinuous spectral element method with a Dubiner orthogonal polynomial basis on triangles and a Legendre nodal orthogonal basis on quadrilaterals, we conduct a systematic study of the optical coupling by whispering gallery modes between two microcylinders and demonstrate the successful coupling between the microcylinders and also the dependence of such a coupling on the separation and the size variation of the microcylinders.

© 2005 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Yariv, Y. Xu, R. K. Lee, A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24, 711–713 (1999).
    [CrossRef]
  2. K. Sakoda, Optical Properties of Photonic Crystals (Springer-Verlag, New York, 2001).
    [CrossRef]
  3. B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley-Interscience, New York, 1991).
    [CrossRef]
  4. J. Joannopoulos, R. Meade, J. Winn, Photonic Crystals, Modeling the Flow of Light (Princeton U. Press, Princeton, N.J., 1995).
  5. M. L. Gorodetsky, A. A. Savchenkov, V. S. Ilchenko, “Ultimate Q of optical microsphere resonators,” Opt. Lett. 21, 453–455 (1996).
    [CrossRef] [PubMed]
  6. N. W. Ashcroft, D. Mermin, Solid State Physics (Saunders, Philadelphia, Pa., 1976).
  7. D. Gottlieb, S. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1977).
    [CrossRef]
  8. C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zhang, Spectral Methods in Fluid Dynamics (Springer-Verlag, New York, 1987).
  9. S. J. Sherwin, G. E. Karniadakis, “A new triangular and tetrahedral basis for high-order (hp) finite element methods,” Int. J. Numer. Methods Eng. 38, 3775–3802 (1995).
    [CrossRef]
  10. D. A. Kopriva, S. L. Woodruff, M. Y. Hussaini, “Computation of electromagnetic scattering with a non-conforming discontinuous spectral element method,” Int. J. Numer. Methods Eng. 53, 105–122 (2002).
    [CrossRef]
  11. S. C. Hagness, D. Rafizadeh, S. T. Ho, A. Taflove, “FDTD microcavity simulations: design and experimental realization of waveguide-coupled single-mode ring and whispering-gallery-mode disk resonators,” J. Lightwave Technol. 15, 2154–2165 (1997).
    [CrossRef]
  12. L. Rayleigh, “Further applications of Bessel functions of high order to the whispering gallery and allied problems,” Philos. Mag. 27, 100 (1914).
    [CrossRef]
  13. X. H. Jiao, P. Guillon, L. Bermudez, P. Auxemery, “Whispering gallery modes of dielectric structures: applications to millimeter wave band filters,” IEEE Trans. Microwave Theory Tech. MTT-35, 1169–1175 (1972).
  14. S. L. McCall, A. F. J. Lewis, R. E. Slusher, S. J. Pearton, R. A. Logan, “Whispering gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
    [CrossRef]
  15. A. D. Bristow, D. M. Whittaker, V. N. Astratov, M. S. Skolnick, A. Tahraoui, T. F. Krauss, M. Hopkinson, M. P. Croucher, G. A. Gehring, “Defect state and commensurability in dual-period AlxGa1−xAs photonic crystal waveguides,” Phys. Rev. B 68, 033303 (2003).
    [CrossRef]
  16. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
  17. J. R. Wait, “Electromagnetic whispering gallery modes in a dielectric rod,” Radio Sci. 2, 1005–1017 (1967).
  18. B. Yang, D. Gottlieb, J. S. Hesthaven, “Spectral simulation of electromagnetic wave scattering,” J. Comput. Phys. 134, 216–230 (1997).
    [CrossRef]
  19. B. Szabo, I. Babuska, Finite Element Analysis (Wiley, New York, 1991).
  20. A. H. Mahammadian, V. Shankar, W. F. Hall, “Computation of electromagnetic scattering and radiation using a time-domain finite-volume discretization procedure,” Comput. Phys. Commun. 68, 175–196 (1991).
    [CrossRef]
  21. M. Dubiner, “Spectral methods on triangles and other domains,” J. Sci. Comput. 6, 345–390 (1991).
    [CrossRef]
  22. J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
    [CrossRef]
  23. S. Abarbanel, D. Gottlieb, “On the construction and analysis of absorbing layers in CEM,” Appl. Numer. Math. 27, 331–340 (1998).
    [CrossRef]

2003 (1)

A. D. Bristow, D. M. Whittaker, V. N. Astratov, M. S. Skolnick, A. Tahraoui, T. F. Krauss, M. Hopkinson, M. P. Croucher, G. A. Gehring, “Defect state and commensurability in dual-period AlxGa1−xAs photonic crystal waveguides,” Phys. Rev. B 68, 033303 (2003).
[CrossRef]

2002 (1)

D. A. Kopriva, S. L. Woodruff, M. Y. Hussaini, “Computation of electromagnetic scattering with a non-conforming discontinuous spectral element method,” Int. J. Numer. Methods Eng. 53, 105–122 (2002).
[CrossRef]

1999 (1)

1998 (1)

S. Abarbanel, D. Gottlieb, “On the construction and analysis of absorbing layers in CEM,” Appl. Numer. Math. 27, 331–340 (1998).
[CrossRef]

1997 (2)

S. C. Hagness, D. Rafizadeh, S. T. Ho, A. Taflove, “FDTD microcavity simulations: design and experimental realization of waveguide-coupled single-mode ring and whispering-gallery-mode disk resonators,” J. Lightwave Technol. 15, 2154–2165 (1997).
[CrossRef]

B. Yang, D. Gottlieb, J. S. Hesthaven, “Spectral simulation of electromagnetic wave scattering,” J. Comput. Phys. 134, 216–230 (1997).
[CrossRef]

1996 (1)

1995 (1)

S. J. Sherwin, G. E. Karniadakis, “A new triangular and tetrahedral basis for high-order (hp) finite element methods,” Int. J. Numer. Methods Eng. 38, 3775–3802 (1995).
[CrossRef]

1994 (1)

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

1992 (1)

S. L. McCall, A. F. J. Lewis, R. E. Slusher, S. J. Pearton, R. A. Logan, “Whispering gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

1991 (2)

A. H. Mahammadian, V. Shankar, W. F. Hall, “Computation of electromagnetic scattering and radiation using a time-domain finite-volume discretization procedure,” Comput. Phys. Commun. 68, 175–196 (1991).
[CrossRef]

M. Dubiner, “Spectral methods on triangles and other domains,” J. Sci. Comput. 6, 345–390 (1991).
[CrossRef]

1972 (1)

X. H. Jiao, P. Guillon, L. Bermudez, P. Auxemery, “Whispering gallery modes of dielectric structures: applications to millimeter wave band filters,” IEEE Trans. Microwave Theory Tech. MTT-35, 1169–1175 (1972).

1967 (1)

J. R. Wait, “Electromagnetic whispering gallery modes in a dielectric rod,” Radio Sci. 2, 1005–1017 (1967).

1914 (1)

L. Rayleigh, “Further applications of Bessel functions of high order to the whispering gallery and allied problems,” Philos. Mag. 27, 100 (1914).
[CrossRef]

Abarbanel, S.

S. Abarbanel, D. Gottlieb, “On the construction and analysis of absorbing layers in CEM,” Appl. Numer. Math. 27, 331–340 (1998).
[CrossRef]

Ashcroft, N. W.

N. W. Ashcroft, D. Mermin, Solid State Physics (Saunders, Philadelphia, Pa., 1976).

Astratov, V. N.

A. D. Bristow, D. M. Whittaker, V. N. Astratov, M. S. Skolnick, A. Tahraoui, T. F. Krauss, M. Hopkinson, M. P. Croucher, G. A. Gehring, “Defect state and commensurability in dual-period AlxGa1−xAs photonic crystal waveguides,” Phys. Rev. B 68, 033303 (2003).
[CrossRef]

Auxemery, P.

X. H. Jiao, P. Guillon, L. Bermudez, P. Auxemery, “Whispering gallery modes of dielectric structures: applications to millimeter wave band filters,” IEEE Trans. Microwave Theory Tech. MTT-35, 1169–1175 (1972).

Babuska, I.

B. Szabo, I. Babuska, Finite Element Analysis (Wiley, New York, 1991).

Berenger, J.

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

Bermudez, L.

X. H. Jiao, P. Guillon, L. Bermudez, P. Auxemery, “Whispering gallery modes of dielectric structures: applications to millimeter wave band filters,” IEEE Trans. Microwave Theory Tech. MTT-35, 1169–1175 (1972).

Bristow, A. D.

A. D. Bristow, D. M. Whittaker, V. N. Astratov, M. S. Skolnick, A. Tahraoui, T. F. Krauss, M. Hopkinson, M. P. Croucher, G. A. Gehring, “Defect state and commensurability in dual-period AlxGa1−xAs photonic crystal waveguides,” Phys. Rev. B 68, 033303 (2003).
[CrossRef]

Canuto, C.

C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zhang, Spectral Methods in Fluid Dynamics (Springer-Verlag, New York, 1987).

Croucher, M. P.

A. D. Bristow, D. M. Whittaker, V. N. Astratov, M. S. Skolnick, A. Tahraoui, T. F. Krauss, M. Hopkinson, M. P. Croucher, G. A. Gehring, “Defect state and commensurability in dual-period AlxGa1−xAs photonic crystal waveguides,” Phys. Rev. B 68, 033303 (2003).
[CrossRef]

Dubiner, M.

M. Dubiner, “Spectral methods on triangles and other domains,” J. Sci. Comput. 6, 345–390 (1991).
[CrossRef]

Gehring, G. A.

A. D. Bristow, D. M. Whittaker, V. N. Astratov, M. S. Skolnick, A. Tahraoui, T. F. Krauss, M. Hopkinson, M. P. Croucher, G. A. Gehring, “Defect state and commensurability in dual-period AlxGa1−xAs photonic crystal waveguides,” Phys. Rev. B 68, 033303 (2003).
[CrossRef]

Gorodetsky, M. L.

Gottlieb, D.

S. Abarbanel, D. Gottlieb, “On the construction and analysis of absorbing layers in CEM,” Appl. Numer. Math. 27, 331–340 (1998).
[CrossRef]

B. Yang, D. Gottlieb, J. S. Hesthaven, “Spectral simulation of electromagnetic wave scattering,” J. Comput. Phys. 134, 216–230 (1997).
[CrossRef]

D. Gottlieb, S. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1977).
[CrossRef]

Guillon, P.

X. H. Jiao, P. Guillon, L. Bermudez, P. Auxemery, “Whispering gallery modes of dielectric structures: applications to millimeter wave band filters,” IEEE Trans. Microwave Theory Tech. MTT-35, 1169–1175 (1972).

Hagness, S. C.

S. C. Hagness, D. Rafizadeh, S. T. Ho, A. Taflove, “FDTD microcavity simulations: design and experimental realization of waveguide-coupled single-mode ring and whispering-gallery-mode disk resonators,” J. Lightwave Technol. 15, 2154–2165 (1997).
[CrossRef]

Hall, W. F.

A. H. Mahammadian, V. Shankar, W. F. Hall, “Computation of electromagnetic scattering and radiation using a time-domain finite-volume discretization procedure,” Comput. Phys. Commun. 68, 175–196 (1991).
[CrossRef]

Hesthaven, J. S.

B. Yang, D. Gottlieb, J. S. Hesthaven, “Spectral simulation of electromagnetic wave scattering,” J. Comput. Phys. 134, 216–230 (1997).
[CrossRef]

Ho, S. T.

S. C. Hagness, D. Rafizadeh, S. T. Ho, A. Taflove, “FDTD microcavity simulations: design and experimental realization of waveguide-coupled single-mode ring and whispering-gallery-mode disk resonators,” J. Lightwave Technol. 15, 2154–2165 (1997).
[CrossRef]

Hopkinson, M.

A. D. Bristow, D. M. Whittaker, V. N. Astratov, M. S. Skolnick, A. Tahraoui, T. F. Krauss, M. Hopkinson, M. P. Croucher, G. A. Gehring, “Defect state and commensurability in dual-period AlxGa1−xAs photonic crystal waveguides,” Phys. Rev. B 68, 033303 (2003).
[CrossRef]

Hussaini, M. Y.

D. A. Kopriva, S. L. Woodruff, M. Y. Hussaini, “Computation of electromagnetic scattering with a non-conforming discontinuous spectral element method,” Int. J. Numer. Methods Eng. 53, 105–122 (2002).
[CrossRef]

C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zhang, Spectral Methods in Fluid Dynamics (Springer-Verlag, New York, 1987).

Ilchenko, V. S.

Jiao, X. H.

X. H. Jiao, P. Guillon, L. Bermudez, P. Auxemery, “Whispering gallery modes of dielectric structures: applications to millimeter wave band filters,” IEEE Trans. Microwave Theory Tech. MTT-35, 1169–1175 (1972).

Joannopoulos, J.

J. Joannopoulos, R. Meade, J. Winn, Photonic Crystals, Modeling the Flow of Light (Princeton U. Press, Princeton, N.J., 1995).

Karniadakis, G. E.

S. J. Sherwin, G. E. Karniadakis, “A new triangular and tetrahedral basis for high-order (hp) finite element methods,” Int. J. Numer. Methods Eng. 38, 3775–3802 (1995).
[CrossRef]

Kopriva, D. A.

D. A. Kopriva, S. L. Woodruff, M. Y. Hussaini, “Computation of electromagnetic scattering with a non-conforming discontinuous spectral element method,” Int. J. Numer. Methods Eng. 53, 105–122 (2002).
[CrossRef]

Krauss, T. F.

A. D. Bristow, D. M. Whittaker, V. N. Astratov, M. S. Skolnick, A. Tahraoui, T. F. Krauss, M. Hopkinson, M. P. Croucher, G. A. Gehring, “Defect state and commensurability in dual-period AlxGa1−xAs photonic crystal waveguides,” Phys. Rev. B 68, 033303 (2003).
[CrossRef]

Lee, R. K.

Lewis, A. F. J.

S. L. McCall, A. F. J. Lewis, R. E. Slusher, S. J. Pearton, R. A. Logan, “Whispering gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

Logan, R. A.

S. L. McCall, A. F. J. Lewis, R. E. Slusher, S. J. Pearton, R. A. Logan, “Whispering gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

Mahammadian, A. H.

A. H. Mahammadian, V. Shankar, W. F. Hall, “Computation of electromagnetic scattering and radiation using a time-domain finite-volume discretization procedure,” Comput. Phys. Commun. 68, 175–196 (1991).
[CrossRef]

McCall, S. L.

S. L. McCall, A. F. J. Lewis, R. E. Slusher, S. J. Pearton, R. A. Logan, “Whispering gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

Meade, R.

J. Joannopoulos, R. Meade, J. Winn, Photonic Crystals, Modeling the Flow of Light (Princeton U. Press, Princeton, N.J., 1995).

Mermin, D.

N. W. Ashcroft, D. Mermin, Solid State Physics (Saunders, Philadelphia, Pa., 1976).

Orszag, S.

D. Gottlieb, S. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1977).
[CrossRef]

Pearton, S. J.

S. L. McCall, A. F. J. Lewis, R. E. Slusher, S. J. Pearton, R. A. Logan, “Whispering gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

Quarteroni, A.

C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zhang, Spectral Methods in Fluid Dynamics (Springer-Verlag, New York, 1987).

Rafizadeh, D.

S. C. Hagness, D. Rafizadeh, S. T. Ho, A. Taflove, “FDTD microcavity simulations: design and experimental realization of waveguide-coupled single-mode ring and whispering-gallery-mode disk resonators,” J. Lightwave Technol. 15, 2154–2165 (1997).
[CrossRef]

Rayleigh, L.

L. Rayleigh, “Further applications of Bessel functions of high order to the whispering gallery and allied problems,” Philos. Mag. 27, 100 (1914).
[CrossRef]

Sakoda, K.

K. Sakoda, Optical Properties of Photonic Crystals (Springer-Verlag, New York, 2001).
[CrossRef]

Saleh, B. E. A.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley-Interscience, New York, 1991).
[CrossRef]

Savchenkov, A. A.

Scherer, A.

Shankar, V.

A. H. Mahammadian, V. Shankar, W. F. Hall, “Computation of electromagnetic scattering and radiation using a time-domain finite-volume discretization procedure,” Comput. Phys. Commun. 68, 175–196 (1991).
[CrossRef]

Sherwin, S. J.

S. J. Sherwin, G. E. Karniadakis, “A new triangular and tetrahedral basis for high-order (hp) finite element methods,” Int. J. Numer. Methods Eng. 38, 3775–3802 (1995).
[CrossRef]

Skolnick, M. S.

A. D. Bristow, D. M. Whittaker, V. N. Astratov, M. S. Skolnick, A. Tahraoui, T. F. Krauss, M. Hopkinson, M. P. Croucher, G. A. Gehring, “Defect state and commensurability in dual-period AlxGa1−xAs photonic crystal waveguides,” Phys. Rev. B 68, 033303 (2003).
[CrossRef]

Slusher, R. E.

S. L. McCall, A. F. J. Lewis, R. E. Slusher, S. J. Pearton, R. A. Logan, “Whispering gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

Szabo, B.

B. Szabo, I. Babuska, Finite Element Analysis (Wiley, New York, 1991).

Taflove, A.

S. C. Hagness, D. Rafizadeh, S. T. Ho, A. Taflove, “FDTD microcavity simulations: design and experimental realization of waveguide-coupled single-mode ring and whispering-gallery-mode disk resonators,” J. Lightwave Technol. 15, 2154–2165 (1997).
[CrossRef]

Tahraoui, A.

A. D. Bristow, D. M. Whittaker, V. N. Astratov, M. S. Skolnick, A. Tahraoui, T. F. Krauss, M. Hopkinson, M. P. Croucher, G. A. Gehring, “Defect state and commensurability in dual-period AlxGa1−xAs photonic crystal waveguides,” Phys. Rev. B 68, 033303 (2003).
[CrossRef]

Teich, M. C.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley-Interscience, New York, 1991).
[CrossRef]

Wait, J. R.

J. R. Wait, “Electromagnetic whispering gallery modes in a dielectric rod,” Radio Sci. 2, 1005–1017 (1967).

Whittaker, D. M.

A. D. Bristow, D. M. Whittaker, V. N. Astratov, M. S. Skolnick, A. Tahraoui, T. F. Krauss, M. Hopkinson, M. P. Croucher, G. A. Gehring, “Defect state and commensurability in dual-period AlxGa1−xAs photonic crystal waveguides,” Phys. Rev. B 68, 033303 (2003).
[CrossRef]

Winn, J.

J. Joannopoulos, R. Meade, J. Winn, Photonic Crystals, Modeling the Flow of Light (Princeton U. Press, Princeton, N.J., 1995).

Woodruff, S. L.

D. A. Kopriva, S. L. Woodruff, M. Y. Hussaini, “Computation of electromagnetic scattering with a non-conforming discontinuous spectral element method,” Int. J. Numer. Methods Eng. 53, 105–122 (2002).
[CrossRef]

Xu, Y.

Yang, B.

B. Yang, D. Gottlieb, J. S. Hesthaven, “Spectral simulation of electromagnetic wave scattering,” J. Comput. Phys. 134, 216–230 (1997).
[CrossRef]

Yariv, A.

Zhang, T. A.

C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zhang, Spectral Methods in Fluid Dynamics (Springer-Verlag, New York, 1987).

Appl. Numer. Math. (1)

S. Abarbanel, D. Gottlieb, “On the construction and analysis of absorbing layers in CEM,” Appl. Numer. Math. 27, 331–340 (1998).
[CrossRef]

Appl. Phys. Lett. (1)

S. L. McCall, A. F. J. Lewis, R. E. Slusher, S. J. Pearton, R. A. Logan, “Whispering gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

Comput. Phys. Commun. (1)

A. H. Mahammadian, V. Shankar, W. F. Hall, “Computation of electromagnetic scattering and radiation using a time-domain finite-volume discretization procedure,” Comput. Phys. Commun. 68, 175–196 (1991).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

X. H. Jiao, P. Guillon, L. Bermudez, P. Auxemery, “Whispering gallery modes of dielectric structures: applications to millimeter wave band filters,” IEEE Trans. Microwave Theory Tech. MTT-35, 1169–1175 (1972).

Int. J. Numer. Methods Eng. (2)

S. J. Sherwin, G. E. Karniadakis, “A new triangular and tetrahedral basis for high-order (hp) finite element methods,” Int. J. Numer. Methods Eng. 38, 3775–3802 (1995).
[CrossRef]

D. A. Kopriva, S. L. Woodruff, M. Y. Hussaini, “Computation of electromagnetic scattering with a non-conforming discontinuous spectral element method,” Int. J. Numer. Methods Eng. 53, 105–122 (2002).
[CrossRef]

J. Comput. Phys. (2)

B. Yang, D. Gottlieb, J. S. Hesthaven, “Spectral simulation of electromagnetic wave scattering,” J. Comput. Phys. 134, 216–230 (1997).
[CrossRef]

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

J. Lightwave Technol. (1)

S. C. Hagness, D. Rafizadeh, S. T. Ho, A. Taflove, “FDTD microcavity simulations: design and experimental realization of waveguide-coupled single-mode ring and whispering-gallery-mode disk resonators,” J. Lightwave Technol. 15, 2154–2165 (1997).
[CrossRef]

J. Sci. Comput. (1)

M. Dubiner, “Spectral methods on triangles and other domains,” J. Sci. Comput. 6, 345–390 (1991).
[CrossRef]

Opt. Lett. (2)

Philos. Mag. (1)

L. Rayleigh, “Further applications of Bessel functions of high order to the whispering gallery and allied problems,” Philos. Mag. 27, 100 (1914).
[CrossRef]

Phys. Rev. B (1)

A. D. Bristow, D. M. Whittaker, V. N. Astratov, M. S. Skolnick, A. Tahraoui, T. F. Krauss, M. Hopkinson, M. P. Croucher, G. A. Gehring, “Defect state and commensurability in dual-period AlxGa1−xAs photonic crystal waveguides,” Phys. Rev. B 68, 033303 (2003).
[CrossRef]

Radio Sci. (1)

J. R. Wait, “Electromagnetic whispering gallery modes in a dielectric rod,” Radio Sci. 2, 1005–1017 (1967).

Other (8)

B. Szabo, I. Babuska, Finite Element Analysis (Wiley, New York, 1991).

K. Sakoda, Optical Properties of Photonic Crystals (Springer-Verlag, New York, 2001).
[CrossRef]

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley-Interscience, New York, 1991).
[CrossRef]

J. Joannopoulos, R. Meade, J. Winn, Photonic Crystals, Modeling the Flow of Light (Princeton U. Press, Princeton, N.J., 1995).

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

N. W. Ashcroft, D. Mermin, Solid State Physics (Saunders, Philadelphia, Pa., 1976).

D. Gottlieb, S. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1977).
[CrossRef]

C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zhang, Spectral Methods in Fluid Dynamics (Springer-Verlag, New York, 1987).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

Exponential decay of the average normalized L error of H x , H y , and E z with increasing expansion order for the simulation of WGM 8 , 1 , 0 in a single dielectric cylinder.

Fig. 2
Fig. 2

Optical energy transport by WGMs between two identical microcylinders in contact. The four sequential snapshots at t = 2 , 6, 8, and 10 illustrate generation of a clockwise WGM in the right cylinder as a result of resonant optical coupling.

Fig. 3
Fig. 3

Time history of the total energy in the system, the energy in the left cylinder, and the energy in the right cylinder. Energies are normalized by the initial total energy, including energy stored in the electromagnetic fields outside the cylinders.

Fig. 4
Fig. 4

Optical coupling and energy transport history between two separated identical microcylinders. Panel (a) is the final state of E z at t = 20 for a separation w = 0.04 r 1 , while (b) shows the time history of the energy transport. Similarly, (c) and (d) are for w = 0.12 r 1 , and (e) and (f) are for w = 0.20 r 1 .

Fig. 5
Fig. 5

Percentage of energy coupled into the right microcylinder from the left microcylinder as a function of the separation width w and the size variation of microcylinders, r 2 r 1 . The energy is measured at t = 20 .

Fig. 6
Fig. 6

Optical coupling between two microcylinders of different sizes. The separation between the cylinders is 5% of the left cylinder radius. Panel (a) is the final state of E z at t = 20 for r 2 = 1.02 r 1 , while (b) shows the time history of the energy transport. Similarly, (c) and (d) are for r 2 = 1.06 r 1 , and (e) and (f) are for r 2 = 1.10 r 1 .

Fig. 7
Fig. 7

Percentage of energy coupled into the right cylinder from the left cylinder as a function of the ratio of the perimeter L of the right cylinder to the azimuthal wavelength λ of the WGM in the left cylinder. The two cylinders are either in contact or separated by w = 0.05 r 1 , and the energy is measured at t = 20 .

Equations (40)

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

H r = [ a n n k 2 μ ω λ 2 r G n ( λ r ) + b n i h λ G n ( λ r ) ] F n ,
H θ = [ a n i k 2 μ ω λ G n ( λ r ) b n n h λ 2 r G n ( λ r ) ] F n ,
H z = b n G n ( λ r ) F n ,
E r = [ a n i h λ G n ( λ r ) b n μ ω n λ 2 r G n ( λ r ) ] F n ,
E θ = [ a n n h λ 2 r G n ( λ r ) + b n i μ ω λ G n ( λ r ) ] F n ,
E z = a n G n ( λ r ) F n ,
[ μ 1 u J n ( u ) J n ( u ) μ 2 v H n ( 1 ) ( v ) H n ( 1 ) ( v ) ] [ k 1 2 μ 1 u J n ( u ) J n ( u ) k 2 2 μ 2 v H n ( 1 ) ( v ) H n ( 1 ) ( v ) ]
= n 2 h 2 ( 1 v 2 1 u 2 ) 2 ,
H ( x , y , z , t ) = H ( x , y , t ) exp ( i h z ) ,
E ( x , y , z , t ) = E ( x , y , t ) exp ( i h z ) .
μ H t = × E , ϵ E t = × H ,
Q t + A ( ϵ , μ ) Q x + B ( ϵ , μ ) Q y = S ,
Q = ( μ H ϵ E ) ,
A ( ϵ , μ ) = [ 0 0 0 0 0 0 0 0 0 0 0 1 ϵ 0 0 0 0 1 ϵ 0 0 0 0 0 0 0 0 0 1 μ 0 0 0 0 1 μ 0 0 0 0 ] ,
B ( ϵ , μ ) = [ 0 0 0 0 0 1 ϵ 0 0 0 0 0 0 0 0 0 1 ϵ 0 0 0 0 1 μ 0 0 0 0 0 0 0 0 0 1 μ 0 0 0 0 0 ] ,
S = ( i h E y i h E x 0 i h H y i h H x 0 ) .
Q t + F = S ,
Q ̂ t + ξ F ̂ = S ̂ .
Q ̂ = J Q , S ̂ = J S ,
F ̂ 1 = ( y η , x η ) F , F ̂ 2 = ( y ξ , x ξ ) F ,
Q ̂ ( ξ , η , t ) Q ̂ N ( ξ , η , t ) = j = 1 N Q ̂ j ( t ) ψ j ( ξ , η ) ,
( Q ̂ N t , ψ i ) + Ω ψ i ( F ̂ n ) d s ( F ̂ , ξ ψ i )
= ( S ̂ , ψ i ) , i = 1 , 2 , , N ,
F n = ( n × ( Y E n × H ) + ( Y E + n × H ) + Y + Y + n × ( Z H + n × E ) + ( Z H n × E ) + Z + Z + ) .
ψ m n ( ξ , η ) = ϕ m ( ξ ) ϕ n ( η ) , 0 m , n M ,
ϕ i ( ξ ) = j = 0 , j i M ξ τ j τ i τ j ,
P M = span { ξ m η n , 0 m , n , m + n M } .
g m n ( ξ , η ) = 2 m P m 0 , 0 ( 2 ξ 1 η 1 ) ( 1 η ) m P n 2 m + 1 , 0 ( 2 η 1 ) ,
0 m , n , m + n M ,
1 1 ( 1 x ) α ( 1 + x ) β P l α , β ( x ) P m α , β ( x ) d x = δ l m .
Q t + A ( ϵ , μ ) Q x + B ( ϵ , μ ) Q y = S ( σ x + σ y ) Q ,
σ x ( x ) = { C x x a m , x a 0 , x < a } ,
σ y ( y ) = { C y y b m , y b 0 , y < b } ,
J m ( x , t ) = J m 0 ( x ) exp ( α t ) ,
J e ( x , t ) = J e 0 ( x ) exp ( α t ) ,
J m 0 ( x ) = n × [ E + ( x , 0 ) E ( x , 0 ) ] ,
J e 0 ( x ) = n × [ H + ( x , 0 ) H ( x , 0 ) ] .
( F n ) = ( n × ( Y E n × H ) + ( Y E + n × H ) + J e Y + Y + Y + Y + Y + J m n × ( Z H + n × E ) + ( Z H n × E ) + + J m Z + Z + + Z + Z + Z + J e )
( F n ) + = ( n × ( Y E n × H ) + ( Y E + n × H ) + J e Y + Y + + Y Y + Y + J m n × ( Z H + n × E ) + ( Z H n × E ) + + J m Z + Z + Z Z + Z + J e )
P = V ( ϵ E 2 2 + μ H 2 2 ) d v .

Metrics