Abstract

By using a discontinuous spectral element method, we analyze evanescent wave coupling of whispering gallery modes (WGMs) in microcylinder coupled resonator optical waveguides (CROWs). We demonstrate successful light propagation by WGMs through a chain of coupled cylinder resonators, and that the speed of such propagation is strongly dependent on the inter-resonator gap sizes. Our simulations also show that light propagates slower by WGMs with bigger azimuthal numbers than by those with smaller azimuthal numbers. On the other hand, the light propagation by WGMs of the same azimuthal number appears to have the same speed in CROWs regardless of the size and the material of the resonators, indicating that the tail (the part of a WGM outside the resonator) determines inter-resonator coupling strength.

© 2004 Optical Society of America

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  4. J. Poon, J. Scheuer, S. Mookherjea, G. T. Paloczi, Y. Huang, and A. Yariv, “Matrix analysis of microring coupled-resonator optical waveguides,” Opt. Express 12, 90–103 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-1-90.
    [CrossRef] [PubMed]
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  6. J. E. Heebner, R. W. Boyd, and Q-H. Park, “SCISSOR solitons and other novel propagation effects in microresonator-modified waveguides,” J. Opt. Soc. Am. B 19, 722–731 (2002).
    [CrossRef]
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  32. A. H. Mahammadian, V. Shankar, and W. F. Hall, “Computation of electromagnetic scattering and radiation using a time-domain finite-volume discretization procedure,” Comput. Phys. Comm. 8, 175–196 (1991).
    [CrossRef]
  33. J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 14, 185–200 (1994).
    [CrossRef]
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    [CrossRef]
  35. B. Yang, D. Gottlieb, and J. S. Hesthaven, “Spectral simulation of electromagnetic wave scattering,” J. Comput. Phys. 34, 216–230 (1997).
    [CrossRef]

2004 (4)

B. M. Möller, U. Woggon, M. V. Artemyev, and R. Wannemacher, “Photonic molecules doped with semiconductor nanocrystals,” Phys. Rev. B 70, 115323 (2004).
[CrossRef]

V. N. Astratov, J. P. Franchak, and S. P. Ashili, “Optical coupling and transport phenomena in chains of spherical dielectric microresonators with size disorder,” Appl. Phys. Lett. 85, 5508–5510 (2004).
[CrossRef]

T. Lu, P. Zhang, and W. Cai, “Discontinuous Galerkin methods for dispersive and lossy Maxwell’s equations and PML boundary conditions,” J. Comput. Phys. 200, 549–580 (2004).
[CrossRef]

J. Poon, J. Scheuer, S. Mookherjea, G. T. Paloczi, Y. Huang, and A. Yariv, “Matrix analysis of microring coupled-resonator optical waveguides,” Opt. Express 12, 90–103 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-1-90.
[CrossRef] [PubMed]

2003 (4)

A. Martinez, A. Garcia, P. Sanchis, and J. Marti, “Group velocity and dispersion model of coupled-cavity waveguides in photonic crystals,” J. Opt. Soc. Am. A 20, 147–150 (2003).
[CrossRef]

A. Melloni, F. Morichetti, and M. Martinelli, “Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures,” Opt. and Quamtum Electron. 35, 365–379 (2003).
[CrossRef]

T. D. Happ, M. Kamp, A. Forchel, J.-L. Gentner, and L. Goldstein, “Two-dimensional photonic crystal coupled-defect laser diode,” Appl. Phys. Lett. 82, 4–6 (2003).
[CrossRef]

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

2002 (4)

S. Mookherjea and A. Yariv, “Coupled resonator optical waveguides,” IEEE J. Sel. Top. Quamtum Electron. 8, 448–456 (2002).
[CrossRef]

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

J. E. Heebner, R. W. Boyd, and Q-H. Park, “SCISSOR solitons and other novel propagation effects in microresonator-modified waveguides,” J. Opt. Soc. Am. B 19, 722–731 (2002).
[CrossRef]

J. E. Heebner, R. W. Boyd, and Q-H. Park, “Slow light, induced dispersion, enhanced nonlinearity, and optical solitons in a resonant-array waveguide,” Phys. Rev. E 65, 036619 (2002).
[CrossRef]

2001 (1)

2000 (1)

M. Bayindir and E. Ozbay, “Heavy photons at coupled-cavity waveguide band edges in a three-dimensional photonic crystal,” Phys. Rev. B 62, R2247–R2250 (2000).
[CrossRef]

1999 (3)

A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24, 711–713 (1999).
[CrossRef]

M. Bayer, T. Gutbrod, A. Forchel, T. L. Reinecke, P. A. Knipp, R. Werner, and J. P. Reithmaier, “Optical demonstration of a crystal band structure formation,” Phys. Rev. Lett. 83, 5374–5377 (1999).
[CrossRef]

T. Mukaiyama, K. Takeda, H. Miyazaki, Y. Jimba, and M. Kuwata-Gonokami, “Tight-binding photonic molecule modes of resonant bispheres,” Phys. Rev. Lett. 82, 4623–4626 (1999).
[CrossRef]

1998 (2)

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

N. Stefanou and A. Modinos, “Impurity bands in photonic insulators,” Phys. Rev. B 57, 12127–12133 (1998).
[CrossRef]

1997 (3)

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).
[CrossRef]

B. Yang, D. Gottlieb, and J. S. Hesthaven, “Spectral simulation of electromagnetic wave scattering,” J. Comput. Phys. 34, 216–230 (1997).
[CrossRef]

S. C. Hagness, D. Rafizadeh, S. T. Ho, and 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]

1995 (1)

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

1994 (1)

J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 14, 185–200 (1994).
[CrossRef]

1991 (1)

A. H. Mahammadian, V. Shankar, and W. F. Hall, “Computation of electromagnetic scattering and radiation using a time-domain finite-volume discretization procedure,” Comput. Phys. Comm. 8, 175–196 (1991).
[CrossRef]

1967 (1)

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

1966 (1)

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

Abarbanel, S.

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

Artemyev, M. V.

B. M. Möller, U. Woggon, M. V. Artemyev, and R. Wannemacher, “Photonic molecules doped with semiconductor nanocrystals,” Phys. Rev. B 70, 115323 (2004).
[CrossRef]

Ashili, S. P.

V. N. Astratov, J. P. Franchak, and S. P. Ashili, “Optical coupling and transport phenomena in chains of spherical dielectric microresonators with size disorder,” Appl. Phys. Lett. 85, 5508–5510 (2004).
[CrossRef]

Astratov, V. N.

V. N. Astratov, J. P. Franchak, and S. P. Ashili, “Optical coupling and transport phenomena in chains of spherical dielectric microresonators with size disorder,” Appl. Phys. Lett. 85, 5508–5510 (2004).
[CrossRef]

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

Babuska, I.

B. Szabo and I. Babuska, Finite Element Analysis (John Wiley & Sons, New York, 1991).

Bayer, M.

M. Bayer, T. Gutbrod, A. Forchel, T. L. Reinecke, P. A. Knipp, R. Werner, and J. P. Reithmaier, “Optical demonstration of a crystal band structure formation,” Phys. Rev. Lett. 83, 5374–5377 (1999).
[CrossRef]

Bayindir, M.

M. Bayindir and E. Ozbay, “Heavy photons at coupled-cavity waveguide band edges in a three-dimensional photonic crystal,” Phys. Rev. B 62, R2247–R2250 (2000).
[CrossRef]

Benisty, H.

Berenger, J.

J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 14, 185–200 (1994).
[CrossRef]

Boyd, R. W.

J. E. Heebner, R. W. Boyd, and Q-H. Park, “Slow light, induced dispersion, enhanced nonlinearity, and optical solitons in a resonant-array waveguide,” Phys. Rev. E 65, 036619 (2002).
[CrossRef]

J. E. Heebner, R. W. Boyd, and Q-H. Park, “SCISSOR solitons and other novel propagation effects in microresonator-modified waveguides,” J. Opt. Soc. Am. B 19, 722–731 (2002).
[CrossRef]

Bristow, A. D.

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

Cai, W.

T. Lu, P. Zhang, and W. Cai, “Discontinuous Galerkin methods for dispersive and lossy Maxwell’s equations and PML boundary conditions,” J. Comput. Phys. 200, 549–580 (2004).
[CrossRef]

X. Ji, T. Lu, W. Cai, and P. Zhang, “Discontinuous Galerkin time domain (DGTD) methods for the study of waveguide coupled microring resonators,” J. Lightwave Technol. (submitted, October 2004).

S. Deng and W. Cai, “Discontinuous spectral element method modelling of optical coupling by whispering gallery modes between microcylinders,” J. Opt. Soc. Am. A (to be published).

Canuto, C.

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

Chu, S. T.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).
[CrossRef]

Deng, S.

S. Deng and W. Cai, “Discontinuous spectral element method modelling of optical coupling by whispering gallery modes between microcylinders,” J. Opt. Soc. Am. A (to be published).

Forchel, A.

T. D. Happ, M. Kamp, A. Forchel, J.-L. Gentner, and L. Goldstein, “Two-dimensional photonic crystal coupled-defect laser diode,” Appl. Phys. Lett. 82, 4–6 (2003).
[CrossRef]

M. Bayer, T. Gutbrod, A. Forchel, T. L. Reinecke, P. A. Knipp, R. Werner, and J. P. Reithmaier, “Optical demonstration of a crystal band structure formation,” Phys. Rev. Lett. 83, 5374–5377 (1999).
[CrossRef]

Foresi, J.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).
[CrossRef]

Franchak, J. P.

V. N. Astratov, J. P. Franchak, and S. P. Ashili, “Optical coupling and transport phenomena in chains of spherical dielectric microresonators with size disorder,” Appl. Phys. Lett. 85, 5508–5510 (2004).
[CrossRef]

Garcia, A.

Gehring, G. A.

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

Gentner, J.-L.

T. D. Happ, M. Kamp, A. Forchel, J.-L. Gentner, and L. Goldstein, “Two-dimensional photonic crystal coupled-defect laser diode,” Appl. Phys. Lett. 82, 4–6 (2003).
[CrossRef]

Goldstein, L.

T. D. Happ, M. Kamp, A. Forchel, J.-L. Gentner, and L. Goldstein, “Two-dimensional photonic crystal coupled-defect laser diode,” Appl. Phys. Lett. 82, 4–6 (2003).
[CrossRef]

Gottlieb, D.

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

B. Yang, D. Gottlieb, and J. S. Hesthaven, “Spectral simulation of electromagnetic wave scattering,” J. Comput. Phys. 34, 216–230 (1997).
[CrossRef]

D. Gottlieb and S. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications (SIAM-CBMS, Philadelphia, 1977).
[CrossRef]

Groucher, M. P.

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

Gutbrod, T.

M. Bayer, T. Gutbrod, A. Forchel, T. L. Reinecke, P. A. Knipp, R. Werner, and J. P. Reithmaier, “Optical demonstration of a crystal band structure formation,” Phys. Rev. Lett. 83, 5374–5377 (1999).
[CrossRef]

Hagness, S. C.

S. C. Hagness, D. Rafizadeh, S. T. Ho, and 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]

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

Hall, W. F.

A. H. Mahammadian, V. Shankar, and W. F. Hall, “Computation of electromagnetic scattering and radiation using a time-domain finite-volume discretization procedure,” Comput. Phys. Comm. 8, 175–196 (1991).
[CrossRef]

Happ, T. D.

T. D. Happ, M. Kamp, A. Forchel, J.-L. Gentner, and L. Goldstein, “Two-dimensional photonic crystal coupled-defect laser diode,” Appl. Phys. Lett. 82, 4–6 (2003).
[CrossRef]

Haus, H. A.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).
[CrossRef]

Heebner, J. E.

J. E. Heebner, R. W. Boyd, and Q-H. Park, “Slow light, induced dispersion, enhanced nonlinearity, and optical solitons in a resonant-array waveguide,” Phys. Rev. E 65, 036619 (2002).
[CrossRef]

J. E. Heebner, R. W. Boyd, and Q-H. Park, “SCISSOR solitons and other novel propagation effects in microresonator-modified waveguides,” J. Opt. Soc. Am. B 19, 722–731 (2002).
[CrossRef]

Hesthaven, J. S.

B. Yang, D. Gottlieb, and J. S. Hesthaven, “Spectral simulation of electromagnetic wave scattering,” J. Comput. Phys. 34, 216–230 (1997).
[CrossRef]

Ho, S. T.

S. C. Hagness, D. Rafizadeh, S. T. Ho, and 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. Groucher, and G. A. Gehring, “Defect states and commensurability in dual-period AlxGa1-xAs photonic crystal waveguides,” Phys. Rev. B 68, 033303 (2003).
[CrossRef]

Houdre, R.

Huang, Y.

Hussaini, M. Y.

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

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

Ji, X.

X. Ji, T. Lu, W. Cai, and P. Zhang, “Discontinuous Galerkin time domain (DGTD) methods for the study of waveguide coupled microring resonators,” J. Lightwave Technol. (submitted, October 2004).

Jimba, Y.

T. Mukaiyama, K. Takeda, H. Miyazaki, Y. Jimba, and M. Kuwata-Gonokami, “Tight-binding photonic molecule modes of resonant bispheres,” Phys. Rev. Lett. 82, 4623–4626 (1999).
[CrossRef]

Kamp, M.

T. D. Happ, M. Kamp, A. Forchel, J.-L. Gentner, and L. Goldstein, “Two-dimensional photonic crystal coupled-defect laser diode,” Appl. Phys. Lett. 82, 4–6 (2003).
[CrossRef]

Karniadakis, G. E.

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

Knipp, P. A.

M. Bayer, T. Gutbrod, A. Forchel, T. L. Reinecke, P. A. Knipp, R. Werner, and J. P. Reithmaier, “Optical demonstration of a crystal band structure formation,” Phys. Rev. Lett. 83, 5374–5377 (1999).
[CrossRef]

Kopriva, D. A.

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

Krauss, T.

Krauss, T. F.

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

Kuwata-Gonokami, M.

T. Mukaiyama, K. Takeda, H. Miyazaki, Y. Jimba, and M. Kuwata-Gonokami, “Tight-binding photonic molecule modes of resonant bispheres,” Phys. Rev. Lett. 82, 4623–4626 (1999).
[CrossRef]

Laine, J. P.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).
[CrossRef]

Lee, R. K.

Little, B. E.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).
[CrossRef]

Lu, T.

T. Lu, P. Zhang, and W. Cai, “Discontinuous Galerkin methods for dispersive and lossy Maxwell’s equations and PML boundary conditions,” J. Comput. Phys. 200, 549–580 (2004).
[CrossRef]

X. Ji, T. Lu, W. Cai, and P. Zhang, “Discontinuous Galerkin time domain (DGTD) methods for the study of waveguide coupled microring resonators,” J. Lightwave Technol. (submitted, October 2004).

Mahammadian, A. H.

A. H. Mahammadian, V. Shankar, and W. F. Hall, “Computation of electromagnetic scattering and radiation using a time-domain finite-volume discretization procedure,” Comput. Phys. Comm. 8, 175–196 (1991).
[CrossRef]

Marti, J.

Martinelli, M.

A. Melloni, F. Morichetti, and M. Martinelli, “Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures,” Opt. and Quamtum Electron. 35, 365–379 (2003).
[CrossRef]

Martinez, A.

Melloni, A.

A. Melloni, F. Morichetti, and M. Martinelli, “Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures,” Opt. and Quamtum Electron. 35, 365–379 (2003).
[CrossRef]

Miyazaki, H.

T. Mukaiyama, K. Takeda, H. Miyazaki, Y. Jimba, and M. Kuwata-Gonokami, “Tight-binding photonic molecule modes of resonant bispheres,” Phys. Rev. Lett. 82, 4623–4626 (1999).
[CrossRef]

Modinos, A.

N. Stefanou and A. Modinos, “Impurity bands in photonic insulators,” Phys. Rev. B 57, 12127–12133 (1998).
[CrossRef]

Möller, B. M.

B. M. Möller, U. Woggon, M. V. Artemyev, and R. Wannemacher, “Photonic molecules doped with semiconductor nanocrystals,” Phys. Rev. B 70, 115323 (2004).
[CrossRef]

Mookherjea, S.

Morichetti, F.

A. Melloni, F. Morichetti, and M. Martinelli, “Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures,” Opt. and Quamtum Electron. 35, 365–379 (2003).
[CrossRef]

Mukaiyama, T.

T. Mukaiyama, K. Takeda, H. Miyazaki, Y. Jimba, and M. Kuwata-Gonokami, “Tight-binding photonic molecule modes of resonant bispheres,” Phys. Rev. Lett. 82, 4623–4626 (1999).
[CrossRef]

Oesterle, U.

Oliver, S.

Orszag, S.

D. Gottlieb and S. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications (SIAM-CBMS, Philadelphia, 1977).
[CrossRef]

Ozbay, E.

M. Bayindir and E. Ozbay, “Heavy photons at coupled-cavity waveguide band edges in a three-dimensional photonic crystal,” Phys. Rev. B 62, R2247–R2250 (2000).
[CrossRef]

Paloczi, G. T.

Park, Q-H.

J. E. Heebner, R. W. Boyd, and Q-H. Park, “SCISSOR solitons and other novel propagation effects in microresonator-modified waveguides,” J. Opt. Soc. Am. B 19, 722–731 (2002).
[CrossRef]

J. E. Heebner, R. W. Boyd, and Q-H. Park, “Slow light, induced dispersion, enhanced nonlinearity, and optical solitons in a resonant-array waveguide,” Phys. Rev. E 65, 036619 (2002).
[CrossRef]

Petropoulos, P. G.

A. Yefet and P. G. Petropoulos, “A non-dissipative staggered fourth-order accurate explicit finite difference scheme for the time-domain Maxwell’s equations,” ICASE Technical Report, No. 99-30, NASA/CR-1999-209514, 1999.

Poon, J.

Quarteroni, A.

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

Rafizadeh, D.

S. C. Hagness, D. Rafizadeh, S. T. Ho, and 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]

Rattier, M.

Reinecke, T. L.

M. Bayer, T. Gutbrod, A. Forchel, T. L. Reinecke, P. A. Knipp, R. Werner, and J. P. Reithmaier, “Optical demonstration of a crystal band structure formation,” Phys. Rev. Lett. 83, 5374–5377 (1999).
[CrossRef]

Reithmaier, J. P.

M. Bayer, T. Gutbrod, A. Forchel, T. L. Reinecke, P. A. Knipp, R. Werner, and J. P. Reithmaier, “Optical demonstration of a crystal band structure formation,” Phys. Rev. Lett. 83, 5374–5377 (1999).
[CrossRef]

Sanchis, P.

Scherer, A.

Scheuer, J.

Shankar, V.

A. H. Mahammadian, V. Shankar, and W. F. Hall, “Computation of electromagnetic scattering and radiation using a time-domain finite-volume discretization procedure,” Comput. Phys. Comm. 8, 175–196 (1991).
[CrossRef]

Sherwin, S. J.

S. J. Sherwin and G. E. Karniadakis, “A new triangular and tetrahedral basis for high-order (hp) finite element methods,” Int. J. Numer. Meth. 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. Groucher, and G. A. Gehring, “Defect states and commensurability in dual-period AlxGa1-xAs photonic crystal waveguides,” Phys. Rev. B 68, 033303 (2003).
[CrossRef]

Smith, C.

Stefanou, N.

N. Stefanou and A. Modinos, “Impurity bands in photonic insulators,” Phys. Rev. B 57, 12127–12133 (1998).
[CrossRef]

Stratton, J. A.

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

Szabo, B.

B. Szabo and I. Babuska, Finite Element Analysis (John Wiley & Sons, New York, 1991).

Taflove, A.

S. C. Hagness, D. Rafizadeh, S. T. Ho, and 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]

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

Tahraoui, A.

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

Takeda, K.

T. Mukaiyama, K. Takeda, H. Miyazaki, Y. Jimba, and M. Kuwata-Gonokami, “Tight-binding photonic molecule modes of resonant bispheres,” Phys. Rev. Lett. 82, 4623–4626 (1999).
[CrossRef]

Wait, J. R.

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

Wannemacher, R.

B. M. Möller, U. Woggon, M. V. Artemyev, and R. Wannemacher, “Photonic molecules doped with semiconductor nanocrystals,” Phys. Rev. B 70, 115323 (2004).
[CrossRef]

Weisbuch, C.

Werner, R.

M. Bayer, T. Gutbrod, A. Forchel, T. L. Reinecke, P. A. Knipp, R. Werner, and J. P. Reithmaier, “Optical demonstration of a crystal band structure formation,” Phys. Rev. Lett. 83, 5374–5377 (1999).
[CrossRef]

Whittaker, D. M.

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

Woggon, U.

B. M. Möller, U. Woggon, M. V. Artemyev, and R. Wannemacher, “Photonic molecules doped with semiconductor nanocrystals,” Phys. Rev. B 70, 115323 (2004).
[CrossRef]

Woodruff, S. L.

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

Xu, Y.

Yang, B.

B. Yang, D. Gottlieb, and J. S. Hesthaven, “Spectral simulation of electromagnetic wave scattering,” J. Comput. Phys. 34, 216–230 (1997).
[CrossRef]

Yariv, A.

Yee, K. S.

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

Yefet, A.

A. Yefet and P. G. Petropoulos, “A non-dissipative staggered fourth-order accurate explicit finite difference scheme for the time-domain Maxwell’s equations,” ICASE Technical Report, No. 99-30, NASA/CR-1999-209514, 1999.

Zhang, P.

T. Lu, P. Zhang, and W. Cai, “Discontinuous Galerkin methods for dispersive and lossy Maxwell’s equations and PML boundary conditions,” J. Comput. Phys. 200, 549–580 (2004).
[CrossRef]

X. Ji, T. Lu, W. Cai, and P. Zhang, “Discontinuous Galerkin time domain (DGTD) methods for the study of waveguide coupled microring resonators,” J. Lightwave Technol. (submitted, October 2004).

Zhang, T. A.

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

Appl. Numer. Math. (1)

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

Appl. Phys. Lett. (2)

T. D. Happ, M. Kamp, A. Forchel, J.-L. Gentner, and L. Goldstein, “Two-dimensional photonic crystal coupled-defect laser diode,” Appl. Phys. Lett. 82, 4–6 (2003).
[CrossRef]

V. N. Astratov, J. P. Franchak, and S. P. Ashili, “Optical coupling and transport phenomena in chains of spherical dielectric microresonators with size disorder,” Appl. Phys. Lett. 85, 5508–5510 (2004).
[CrossRef]

Comput. Phys. Comm. (1)

A. H. Mahammadian, V. Shankar, and W. F. Hall, “Computation of electromagnetic scattering and radiation using a time-domain finite-volume discretization procedure,” Comput. Phys. Comm. 8, 175–196 (1991).
[CrossRef]

IEEE J. Sel. Top. Quamtum Electron. (1)

S. Mookherjea and A. Yariv, “Coupled resonator optical waveguides,” IEEE J. Sel. Top. Quamtum Electron. 8, 448–456 (2002).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

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

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

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

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

J. Comput. Phys. (3)

T. Lu, P. Zhang, and W. Cai, “Discontinuous Galerkin methods for dispersive and lossy Maxwell’s equations and PML boundary conditions,” J. Comput. Phys. 200, 549–580 (2004).
[CrossRef]

J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 14, 185–200 (1994).
[CrossRef]

B. Yang, D. Gottlieb, and J. S. Hesthaven, “Spectral simulation of electromagnetic wave scattering,” J. Comput. Phys. 34, 216–230 (1997).
[CrossRef]

J. Lightwave Technol. (2)

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).
[CrossRef]

S. C. Hagness, D. Rafizadeh, S. T. Ho, and 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. Opt. Soc. Am. A (1)

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

Opt. and Quamtum Electron. (1)

A. Melloni, F. Morichetti, and M. Martinelli, “Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures,” Opt. and Quamtum Electron. 35, 365–379 (2003).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. B (4)

M. Bayindir and E. Ozbay, “Heavy photons at coupled-cavity waveguide band edges in a three-dimensional photonic crystal,” Phys. Rev. B 62, R2247–R2250 (2000).
[CrossRef]

N. Stefanou and A. Modinos, “Impurity bands in photonic insulators,” Phys. Rev. B 57, 12127–12133 (1998).
[CrossRef]

B. M. Möller, U. Woggon, M. V. Artemyev, and R. Wannemacher, “Photonic molecules doped with semiconductor nanocrystals,” Phys. Rev. B 70, 115323 (2004).
[CrossRef]

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

Phys. Rev. E (1)

J. E. Heebner, R. W. Boyd, and Q-H. Park, “Slow light, induced dispersion, enhanced nonlinearity, and optical solitons in a resonant-array waveguide,” Phys. Rev. E 65, 036619 (2002).
[CrossRef]

Phys. Rev. Lett. (2)

M. Bayer, T. Gutbrod, A. Forchel, T. L. Reinecke, P. A. Knipp, R. Werner, and J. P. Reithmaier, “Optical demonstration of a crystal band structure formation,” Phys. Rev. Lett. 83, 5374–5377 (1999).
[CrossRef]

T. Mukaiyama, K. Takeda, H. Miyazaki, Y. Jimba, and M. Kuwata-Gonokami, “Tight-binding photonic molecule modes of resonant bispheres,” Phys. Rev. Lett. 82, 4623–4626 (1999).
[CrossRef]

Radio Sci. (1)

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

Other (8)

D. Gottlieb and S. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications (SIAM-CBMS, Philadelphia, 1977).
[CrossRef]

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

X. Ji, T. Lu, W. Cai, and P. Zhang, “Discontinuous Galerkin time domain (DGTD) methods for the study of waveguide coupled microring resonators,” J. Lightwave Technol. (submitted, October 2004).

S. Deng and W. Cai, “Discontinuous spectral element method modelling of optical coupling by whispering gallery modes between microcylinders,” J. Opt. Soc. Am. A (to be published).

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

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

A. Yefet and P. G. Petropoulos, “A non-dissipative staggered fourth-order accurate explicit finite difference scheme for the time-domain Maxwell’s equations,” ICASE Technical Report, No. 99-30, NASA/CR-1999-209514, 1999.

B. Szabo and I. Babuska, Finite Element Analysis (John Wiley & Sons, New York, 1991).

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Figures (7)

Fig. 1.
Fig. 1.

DSEM simulation of light propagation through 6 coupled microcylinder resonators. The field plotted here is the Ez component. The radius of the cylinders is r=4.023µm, the material index is n=3.2, and the inter-resonator gap size is w=4%r. The initial state of the system is represented by a WGM of azimuthal number 16 in the most left cylinder, and the frequency of the WGM is assumed to be 100THz. The five sequential snapshots demonstrate the successful wave propagation all the way through the chain to the most right cylinder. (a) t=3520fs; (b) t=5500fs; (c) t=7480fs; (d) t=8800fs; and (e) t=12100fs.

Fig. 2.
Fig. 2.

Temporal history of energy distribution in individual microcylinders during the light propagation through the CROW described in Fig. 1. The energies are normalized to the electromagnetic energy of the initial WGM in the most left cylinder.

Fig. 3.
Fig. 3.

Temporal history of energy distribution in individual microcylinders during the light propagation by a WGM through a 10 cylinder long CROW. The radius of the cylinders is r=2.283µm, the material index is n=3.2, and the inter-resonator gap size is w=4%r. The initial WGM in the most left cylinder has azimuthal number 10 and time frequency 100THz. The energies are normalized to the electromagnetic energy of the initial WGM in the most left cylinder.

Fig. 4.
Fig. 4.

Temporal history of energy distribution in individual microcylinders during the light propagation through the CROW described in Fig. 3. However, light will be coupled out of the CROW via an output waveguide attached to the end of the CROW. The energies are normalized to the electromagnetic energy of the initial WGM in the most left cylinder.

Fig. 5.
Fig. 5.

The cross-center sections of the field component Ez of the two WGMs described in Figs. 1 and 3, respectively. The fields are normalized to the maximum field amplitude in the system. For the WGM described in Fig. 1, 98.5% of the electromagnetic energy is confined inside the cylinder, while for the WGM described in Fig. 3, only 96.2% of the energy is confined inside the cylinder.

Fig. 6.
Fig. 6.

The speed of light propagating through 10 cylinder long CROWs. The radius and the index of refraction of the cylinders are r=1.7325µm and n=3.2, respectively. The initial WGM’s azimuthal number is 8. And the inter-cylinder gap size varies from 15%r to 40%r.

Fig. 7.
Fig. 7.

The speed of light propagating through 10 cylinder long CROWs. The initial WGM’s azimuthal number and the inter-cylinder gap size are fixed at 8 and w=16%r, respectively. However, the material index of refraction varies from n=3.6 to n=1.6, and accordingly the radius of the cylinder varies from r=1.53µm to r=4.05µm.

Equations (23)

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 ih λ G n ( λ r ) ] F n ,
H θ = [ a n i k 2 μ ω λ G n ( λ r ) b n nh λ 2 r G n ( λ r ) ] F n ,
H z = b n G n ( λ r ) F n ,
E r = [ a n ih 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 ( ihz ) , E ( x , y , z , t ) = E ( x , y , t ) exp ( ihz ) .
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 = [ ih E y ih E x 0 ih H y ih 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 ,
ψ mn ( 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 )
Q ̂ ( ξ , η , t ) Q ̂ M ( ξ , η , t ) = m , n = 0 M Q ̂ mn ( t ) ψ mn ( ξ , η ) ,
( Q ̂ M t , ψ ij ) + Ω ψ ij F ̂ · n d s ( F ̂ · ψ ij ) = ( S ̂ , ψ ij ) , i , j = 0 , 1 , 2 , , M ,
F · n = [ n × ( Y E n × H ) + ( Y E + n × H ) + Y + Y + n × ( Z H + n × E ) + ( Z H n × E ) + Z + Z + ] .
Q t + A ( ε , μ ) Q x + B ( ε , μ ) Q y = S ( σ x + σ y ) Q ,
P = V ( ε E 2 2 + μ H 2 2 ) d v .

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