Abstract

The temporal-coupled-mode theory is directly applied to the design of devices that feature a resonator with a high quality factor. For the temporal-coupled-mode theory we calculate the decay rate of the resonator to determine the transmission properties of the device. The analysis using the decay rates requires little computational effort, and therefore the optimum device properties can be determined quickly. Two examples, a wavelength filter and a resonator crossing, are presented to illustrate the use of the analysis.

© 2004 Optical Society of America

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References

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  1. S. Fan, P. R. Villeneuve, J. D. Joannopoulos, H. A. Haus, “Channel drop tunneling through localized states,” Phys. Rev. Lett. 80, 960–963 (1998).
    [CrossRef]
  2. M. Imada, S. Noda, A. Chutinan, M. Mochizuki, T. Tanaka, “Channel drop filter using a single defect in a 2-D photonic crystal slab waveguide,” J. Lightwave Technol. 20, 845–850 (2002).
    [CrossRef]
  3. Y. Xu, R. K. Lee, A. Yariv, “Propagation and second-harmonic generation of electromagnetic waves in a coupled-resonator optical waveguide,” J. Opt. Soc. Am. B 17, 387–400 (2000).
    [CrossRef]
  4. N. Stefanou, A. Modinos, “Impurity bands in photonic insulators,” Phys. Rev. B 57, 12127–12133 (1998).
    [CrossRef]
  5. A. Yariv, Y. Xu, R. K. Lee, A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24, 711–713 (1999).
    [CrossRef]
  6. S. Mookherjea, A. Yariv, “Coupled resonator optical waveguides,” IEEE J. Sel. Top. Quantum Electron. 8, 448–456 (2002).
    [CrossRef]
  7. S. G. Johnson, S. Fan, A. Mekis, J. D. Joannopoulos, “Multipole-cancellation mechanism for high-Q cavities in the absence of a complete photonic band gap,” Appl. Phys. Lett. 78, 3388–3390 (2001).
    [CrossRef]
  8. S. Lan, S. Nishikawa, Y. Sugimoto, N. Ikeda, K. Asakawa, H. Ishikawa, “Analysis of defect coupling in one- and two-dimensional photonic crystals,” Phys. Rev. B 65, 165208 (2002).
    [CrossRef]
  9. http://www.fdtd.org .
  10. H. A. Haus, Waves and Fields in Optoelectronics (Prentice Hall, Englewood Cliffs, N.J., 1984), Chap. 7.
  11. S. G. Johnson, J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173–190 (2001).
    [CrossRef] [PubMed]
  12. S. G. Johnson, C. Manolatou, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, H. A. Haus, “Elimination of cross talk in waveguide intersections,” Opt. Lett. 23, 1855–1857 (1998).
    [CrossRef]

2002 (3)

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

S. Lan, S. Nishikawa, Y. Sugimoto, N. Ikeda, K. Asakawa, H. Ishikawa, “Analysis of defect coupling in one- and two-dimensional photonic crystals,” Phys. Rev. B 65, 165208 (2002).
[CrossRef]

M. Imada, S. Noda, A. Chutinan, M. Mochizuki, T. Tanaka, “Channel drop filter using a single defect in a 2-D photonic crystal slab waveguide,” J. Lightwave Technol. 20, 845–850 (2002).
[CrossRef]

2001 (2)

S. G. Johnson, J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173–190 (2001).
[CrossRef] [PubMed]

S. G. Johnson, S. Fan, A. Mekis, J. D. Joannopoulos, “Multipole-cancellation mechanism for high-Q cavities in the absence of a complete photonic band gap,” Appl. Phys. Lett. 78, 3388–3390 (2001).
[CrossRef]

2000 (1)

1999 (1)

1998 (3)

S. G. Johnson, C. Manolatou, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, H. A. Haus, “Elimination of cross talk in waveguide intersections,” Opt. Lett. 23, 1855–1857 (1998).
[CrossRef]

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, H. A. Haus, “Channel drop tunneling through localized states,” Phys. Rev. Lett. 80, 960–963 (1998).
[CrossRef]

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

Asakawa, K.

S. Lan, S. Nishikawa, Y. Sugimoto, N. Ikeda, K. Asakawa, H. Ishikawa, “Analysis of defect coupling in one- and two-dimensional photonic crystals,” Phys. Rev. B 65, 165208 (2002).
[CrossRef]

Chutinan, A.

Fan, S.

S. G. Johnson, S. Fan, A. Mekis, J. D. Joannopoulos, “Multipole-cancellation mechanism for high-Q cavities in the absence of a complete photonic band gap,” Appl. Phys. Lett. 78, 3388–3390 (2001).
[CrossRef]

S. G. Johnson, C. Manolatou, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, H. A. Haus, “Elimination of cross talk in waveguide intersections,” Opt. Lett. 23, 1855–1857 (1998).
[CrossRef]

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, H. A. Haus, “Channel drop tunneling through localized states,” Phys. Rev. Lett. 80, 960–963 (1998).
[CrossRef]

Haus, H. A.

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, H. A. Haus, “Channel drop tunneling through localized states,” Phys. Rev. Lett. 80, 960–963 (1998).
[CrossRef]

S. G. Johnson, C. Manolatou, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, H. A. Haus, “Elimination of cross talk in waveguide intersections,” Opt. Lett. 23, 1855–1857 (1998).
[CrossRef]

H. A. Haus, Waves and Fields in Optoelectronics (Prentice Hall, Englewood Cliffs, N.J., 1984), Chap. 7.

Ikeda, N.

S. Lan, S. Nishikawa, Y. Sugimoto, N. Ikeda, K. Asakawa, H. Ishikawa, “Analysis of defect coupling in one- and two-dimensional photonic crystals,” Phys. Rev. B 65, 165208 (2002).
[CrossRef]

Imada, M.

Ishikawa, H.

S. Lan, S. Nishikawa, Y. Sugimoto, N. Ikeda, K. Asakawa, H. Ishikawa, “Analysis of defect coupling in one- and two-dimensional photonic crystals,” Phys. Rev. B 65, 165208 (2002).
[CrossRef]

Joannopoulos, J. D.

S. G. Johnson, S. Fan, A. Mekis, J. D. Joannopoulos, “Multipole-cancellation mechanism for high-Q cavities in the absence of a complete photonic band gap,” Appl. Phys. Lett. 78, 3388–3390 (2001).
[CrossRef]

S. G. Johnson, J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173–190 (2001).
[CrossRef] [PubMed]

S. G. Johnson, C. Manolatou, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, H. A. Haus, “Elimination of cross talk in waveguide intersections,” Opt. Lett. 23, 1855–1857 (1998).
[CrossRef]

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, H. A. Haus, “Channel drop tunneling through localized states,” Phys. Rev. Lett. 80, 960–963 (1998).
[CrossRef]

Johnson, S. G.

Lan, S.

S. Lan, S. Nishikawa, Y. Sugimoto, N. Ikeda, K. Asakawa, H. Ishikawa, “Analysis of defect coupling in one- and two-dimensional photonic crystals,” Phys. Rev. B 65, 165208 (2002).
[CrossRef]

Lee, R. K.

Manolatou, C.

Mekis, A.

S. G. Johnson, S. Fan, A. Mekis, J. D. Joannopoulos, “Multipole-cancellation mechanism for high-Q cavities in the absence of a complete photonic band gap,” Appl. Phys. Lett. 78, 3388–3390 (2001).
[CrossRef]

Mochizuki, M.

Modinos, A.

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

Mookherjea, S.

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

Nishikawa, S.

S. Lan, S. Nishikawa, Y. Sugimoto, N. Ikeda, K. Asakawa, H. Ishikawa, “Analysis of defect coupling in one- and two-dimensional photonic crystals,” Phys. Rev. B 65, 165208 (2002).
[CrossRef]

Noda, S.

Scherer, A.

Stefanou, N.

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

Sugimoto, Y.

S. Lan, S. Nishikawa, Y. Sugimoto, N. Ikeda, K. Asakawa, H. Ishikawa, “Analysis of defect coupling in one- and two-dimensional photonic crystals,” Phys. Rev. B 65, 165208 (2002).
[CrossRef]

Tanaka, T.

Villeneuve, P. R.

S. G. Johnson, C. Manolatou, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, H. A. Haus, “Elimination of cross talk in waveguide intersections,” Opt. Lett. 23, 1855–1857 (1998).
[CrossRef]

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, H. A. Haus, “Channel drop tunneling through localized states,” Phys. Rev. Lett. 80, 960–963 (1998).
[CrossRef]

Xu, Y.

Yariv, A.

Appl. Phys. Lett. (1)

S. G. Johnson, S. Fan, A. Mekis, J. D. Joannopoulos, “Multipole-cancellation mechanism for high-Q cavities in the absence of a complete photonic band gap,” Appl. Phys. Lett. 78, 3388–3390 (2001).
[CrossRef]

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

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

J. Lightwave Technol. (1)

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

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. B (2)

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

S. Lan, S. Nishikawa, Y. Sugimoto, N. Ikeda, K. Asakawa, H. Ishikawa, “Analysis of defect coupling in one- and two-dimensional photonic crystals,” Phys. Rev. B 65, 165208 (2002).
[CrossRef]

Phys. Rev. Lett. (1)

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, H. A. Haus, “Channel drop tunneling through localized states,” Phys. Rev. Lett. 80, 960–963 (1998).
[CrossRef]

Other (2)

http://www.fdtd.org .

H. A. Haus, Waves and Fields in Optoelectronics (Prentice Hall, Englewood Cliffs, N.J., 1984), Chap. 7.

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Figures (9)

Fig. 1
Fig. 1

Basic structure of the devices investigated in this paper. The conventional waveguides on the left and on the right side serve as input and output, respectively, and the resonator is a defect in a 1D photonic crystal.

Fig. 2
Fig. 2

Band diagram for TM polarization of an infinitely long row of Si rods in air. Rod spacing is a, and rod radius is 0.2a.

Fig. 3
Fig. 3

Q (solid curve) and corresponding resonance frequencies (dashed curve, dipole-resonance modes; dotted curve, quadrupole resonances) as a function of the defect radius for the resonator with an infinite photonic crystal.

Fig. 4
Fig. 4

Two resonating structures showing the Ez field. (a) Defect radius 0.375a, which results in a dipole resonance. (b) Defect radius of 0.88a, which results in a quadrupole field with two nodes. The Ez field is oriented parallel to the rod axis. The intensity of the gray tones indicates the magnitude of the Ez field.

Fig. 5
Fig. 5

(a) Transmission by FDTD and transmission by the temporal-coupled-mode theory as a function of resonator radius with Nrods=2. (b) The same transmissions with Nrods=3.

Fig. 6
Fig. 6

(a) τ, τe, and transmission by FDTD as a function of resonator radius for Nrods=2. (b) The same quantities for Nrods=3.

Fig. 7
Fig. 7

Transmission spectra of the devices with Nrods=2 and Nrods=3. The device with Nrods=2 has two transmission maxima, one where the radius of the defect rod is 0.38a that has a dipole-resonance mode, the other where the radius is 0.88a and that has a quadrupole-resonance mode.

Fig. 8
Fig. 8

Wavelength filter with Nrods=3 and radius 0.58a in resonance at a frequency of 0.2655c/a. The gray tones indicate the Ez field magnitude.

Fig. 9
Fig. 9

Crossing with Nrods=3 in resonance at a frequency of 0.2655c/a. The radius of the resonator is 0.58a. The gray tones indicate the Ez field magnitude.

Equations (5)

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T=4τ2τe2.
1τ=1τ0+2τe,
T=1-ττ02.
T1-ττiso2.
Q=τω0.

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