Abstract

We have proposed a three-port high efficient channel-drop filter (CDF) with a coupled cavity-based wavelength-selective reflector, which can be used in wavelength division multiplexing (WDM) optical communication systems. The coupling mode theory (CMT) is employed to drive the necessary conditions for achieving 100% drop efficiency. The finite-difference time-domain (FDTD) simulation results of proposed CDF which is implemented in two dimensional photonic crystals (2D-PC), show that the analysis is valid. In the designed CDF, the drop efficiency larger than 0.95% and the spectral line-width 0.78nm at the center wavelength 1550nm have been achieved.

© 2009 Optical Society of America

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References

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  1. J. D. Joannopoulos, S. G. Johnson, J. N. Winn and R. D. Meade, Photonic Crystal: Molding the Flow of Light (Princeton, Princeton Univ. Press, 2008).
  2. M. F. Yanik, H. Altug, J. Vuckovic, and S. Fan, "Submicrometer All-Optical Digital Memory and Integration of Nanoscale Photonic Devices without Isolator," J. Lightwave Technol. 22, 2316-2322 (2004).
    [CrossRef]
  3. M. Koshiba, "Wavelength Division Multiplexing and Demultiplexing With Photonic Crystal Waveguide Coupler," J. Lightwave Technol. 19, 1970-1975 (2001).
    [CrossRef]
  4. A. Mekis, M. Meier, A. Dodabalapur, R. E. Slusher, and J. D. Joannopoulos, "Lasing mechanism in two-dimensional photonic crystal lasers," Appl. Phys. A: Mat. Sci. Proc. 69, 111-114 (1999).
    [CrossRef]
  5. M. F. Yanik, S. Fan, M. Soljacic, and J. D. Joannopoulos, "All-optical transistor action with bistable switching in a photonic crystal cross-waveguide geometry," Opt. Lett. 28, 2506-2508 (2003).
    [CrossRef] [PubMed]
  6. Z. Zhang and M. Qiu, "Compact in-plane channel drop filter design using a single cavity with two degenerate modes in 2D photonic crystal slabs," Opt. Express 13, 2596-2604 (2005).
    [CrossRef] [PubMed]
  7. S. Kim, I. park, H. Lim, and C. Kee, "Highly efficient photonic crystal-based multi-channel drop filters of three-port system with reflection feedback," Opt. Express 12, 5518-5525 (2004).
    [CrossRef] [PubMed]
  8. S. Noda, A. Chutinan, and M. Imada, "Trapping and emission of photons by a single defect in a photonic bandgap structure," Nature 407, 608-610 (2000).
    [CrossRef] [PubMed]
  9. B. Song, T. Asano, Y. Akahane, and S. Noda, "Role of interfaces in hetero photonic crystals for manipulation of photons," Phys. Rev. B 71, 195101-195101 (2005).
    [CrossRef]
  10. S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, "Channel drop filters in photonic crystals," Opt. Express 3, 4-11 (1998).
    [CrossRef] [PubMed]
  11. B. K. Min, J. E. Kim, and H. Y. Park, "Channel drop filters using resonant tunneling processes in two dimensional triangular lattice photonic crystal slabs," Opt. Commun. 237, 59-63 (2004).
    [CrossRef]
  12. Z. Zhang and M. Qiu, "Coupled-mode analysis of a resonant channel drop filter using waveguides with mirror boundaries," J. Opt. Soc. Am. B 23, 104-113 (2004).
    [CrossRef]
  13. M. Y. Tekeste and J. M. Yarrison-Rice, "High efficiency photonic crystal based wavelength demultiplexer," Opt. Express 14, 7931-7942 (2006).
    [CrossRef] [PubMed]
  14. M. Notomi, A. Shinya, S. Mitsugi, E. Kuramochi, and H. Ryu, "Waveguides, resonators and their coupled elements in photonic crystal slabs," Opt. Express 12, 1551-1561 (2004).
    [CrossRef] [PubMed]
  15. C. Jin, S. Fan, S. Han, and D. Zhang, "Reflectionless multichannel wavelength demultiplexer in a transmission resonator configuration," J. Quantum Electron. 39, 160-165 (2003).
    [CrossRef]
  16. C. W. Kuo, C. F. Chang, M. H. Chen, S. Y. Chen, and Y. D. Wu, "A new approach of planar multi-channel wavelength division multiplexing system using asymmetric super-cell photonic crystal structures," Opt. Express 15, 198-206 (2006).
    [CrossRef]
  17. H. Ren, C. Jiang, W. Hu, M. Gao, and J. Wang, "Photonic crystal channel drop filter with a wavelength-selective reflection micro-cavity," Opt. Express 14, 2446-2458 (2006).
    [CrossRef] [PubMed]
  18. P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, "Microcavities in photonic crystals: Mode symmetry, tunability, and coupling efficiency," Phys. Rev. B 54, 7837-7842 (1996).
    [CrossRef]
  19. A. Yariv, Y. Xu, R. Lee and A. Scherer, "Coupled-resonator optical waveguide: a proposal and analysis," Opt. Lett. 24, 711-713 (1999).
    [CrossRef]
  20. W. Ding, L. Chen and S. Liu, "Localization properties and the effects on multi-mode switching in discrete mode CCWs," Opt. Commun. 248, 479-484 (2004).
    [CrossRef]
  21. H. A. Haus, Waves and Field in Optoelectronics (Prentice-Hall, 1984).
  22. C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, "Coupling of modes analysis of resonant channel add-drop filters," IEEE J. Quantum Electron. 35, 1322 -1333 (1999).
    [CrossRef]
  23. T. Fujisawa and M. Koshiba, "Finite-Element Modeling of Nonlinear Mach-Zehnder Interferometers Based on Photonic- Crystal Waveguides for All-optical Signal Processing," J. Lightwave Technol. 24, 617-623 (2006).
    [CrossRef]
  24. C. C. Chen, C. Y. Chen, W. K. Wang, F. H. Huang, C. K. Lin, W. Y. Chiu, and Y. J. Chan, "Photonic crystal directional couplers formed by InAlGaAs nano-rods," Opt. Express 13, 38-43 (2005).
    [CrossRef] [PubMed]

2006 (4)

2005 (3)

2004 (6)

2003 (2)

M. F. Yanik, S. Fan, M. Soljacic, and J. D. Joannopoulos, "All-optical transistor action with bistable switching in a photonic crystal cross-waveguide geometry," Opt. Lett. 28, 2506-2508 (2003).
[CrossRef] [PubMed]

C. Jin, S. Fan, S. Han, and D. Zhang, "Reflectionless multichannel wavelength demultiplexer in a transmission resonator configuration," J. Quantum Electron. 39, 160-165 (2003).
[CrossRef]

2001 (1)

2000 (1)

S. Noda, A. Chutinan, and M. Imada, "Trapping and emission of photons by a single defect in a photonic bandgap structure," Nature 407, 608-610 (2000).
[CrossRef] [PubMed]

1999 (3)

A. Mekis, M. Meier, A. Dodabalapur, R. E. Slusher, and J. D. Joannopoulos, "Lasing mechanism in two-dimensional photonic crystal lasers," Appl. Phys. A: Mat. Sci. Proc. 69, 111-114 (1999).
[CrossRef]

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

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, "Coupling of modes analysis of resonant channel add-drop filters," IEEE J. Quantum Electron. 35, 1322 -1333 (1999).
[CrossRef]

1998 (1)

1996 (1)

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, "Microcavities in photonic crystals: Mode symmetry, tunability, and coupling efficiency," Phys. Rev. B 54, 7837-7842 (1996).
[CrossRef]

Akahane, Y.

B. Song, T. Asano, Y. Akahane, and S. Noda, "Role of interfaces in hetero photonic crystals for manipulation of photons," Phys. Rev. B 71, 195101-195101 (2005).
[CrossRef]

Altug, H.

Asano, T.

B. Song, T. Asano, Y. Akahane, and S. Noda, "Role of interfaces in hetero photonic crystals for manipulation of photons," Phys. Rev. B 71, 195101-195101 (2005).
[CrossRef]

Chan, Y. J.

Chang, C. F.

Chen, C. C.

Chen, C. Y.

Chen, L.

W. Ding, L. Chen and S. Liu, "Localization properties and the effects on multi-mode switching in discrete mode CCWs," Opt. Commun. 248, 479-484 (2004).
[CrossRef]

Chen, M. H.

Chen, S. Y.

Chiu, W. Y.

Chutinan, A.

S. Noda, A. Chutinan, and M. Imada, "Trapping and emission of photons by a single defect in a photonic bandgap structure," Nature 407, 608-610 (2000).
[CrossRef] [PubMed]

Ding, W.

W. Ding, L. Chen and S. Liu, "Localization properties and the effects on multi-mode switching in discrete mode CCWs," Opt. Commun. 248, 479-484 (2004).
[CrossRef]

Dodabalapur, A.

A. Mekis, M. Meier, A. Dodabalapur, R. E. Slusher, and J. D. Joannopoulos, "Lasing mechanism in two-dimensional photonic crystal lasers," Appl. Phys. A: Mat. Sci. Proc. 69, 111-114 (1999).
[CrossRef]

Fan, S.

M. F. Yanik, H. Altug, J. Vuckovic, and S. Fan, "Submicrometer All-Optical Digital Memory and Integration of Nanoscale Photonic Devices without Isolator," J. Lightwave Technol. 22, 2316-2322 (2004).
[CrossRef]

M. F. Yanik, S. Fan, M. Soljacic, and J. D. Joannopoulos, "All-optical transistor action with bistable switching in a photonic crystal cross-waveguide geometry," Opt. Lett. 28, 2506-2508 (2003).
[CrossRef] [PubMed]

C. Jin, S. Fan, S. Han, and D. Zhang, "Reflectionless multichannel wavelength demultiplexer in a transmission resonator configuration," J. Quantum Electron. 39, 160-165 (2003).
[CrossRef]

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, "Coupling of modes analysis of resonant channel add-drop filters," IEEE J. Quantum Electron. 35, 1322 -1333 (1999).
[CrossRef]

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, "Channel drop filters in photonic crystals," Opt. Express 3, 4-11 (1998).
[CrossRef] [PubMed]

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, "Microcavities in photonic crystals: Mode symmetry, tunability, and coupling efficiency," Phys. Rev. B 54, 7837-7842 (1996).
[CrossRef]

Fujisawa, T.

Gao, M.

Han, S.

C. Jin, S. Fan, S. Han, and D. Zhang, "Reflectionless multichannel wavelength demultiplexer in a transmission resonator configuration," J. Quantum Electron. 39, 160-165 (2003).
[CrossRef]

Haus, H. A.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, "Coupling of modes analysis of resonant channel add-drop filters," IEEE J. Quantum Electron. 35, 1322 -1333 (1999).
[CrossRef]

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, "Channel drop filters in photonic crystals," Opt. Express 3, 4-11 (1998).
[CrossRef] [PubMed]

Hu, W.

Huang, F. H.

Imada, M.

S. Noda, A. Chutinan, and M. Imada, "Trapping and emission of photons by a single defect in a photonic bandgap structure," Nature 407, 608-610 (2000).
[CrossRef] [PubMed]

Jiang, C.

Jin, C.

C. Jin, S. Fan, S. Han, and D. Zhang, "Reflectionless multichannel wavelength demultiplexer in a transmission resonator configuration," J. Quantum Electron. 39, 160-165 (2003).
[CrossRef]

Joannopoulos, J. D.

M. F. Yanik, S. Fan, M. Soljacic, and J. D. Joannopoulos, "All-optical transistor action with bistable switching in a photonic crystal cross-waveguide geometry," Opt. Lett. 28, 2506-2508 (2003).
[CrossRef] [PubMed]

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, "Coupling of modes analysis of resonant channel add-drop filters," IEEE J. Quantum Electron. 35, 1322 -1333 (1999).
[CrossRef]

A. Mekis, M. Meier, A. Dodabalapur, R. E. Slusher, and J. D. Joannopoulos, "Lasing mechanism in two-dimensional photonic crystal lasers," Appl. Phys. A: Mat. Sci. Proc. 69, 111-114 (1999).
[CrossRef]

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, "Channel drop filters in photonic crystals," Opt. Express 3, 4-11 (1998).
[CrossRef] [PubMed]

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, "Microcavities in photonic crystals: Mode symmetry, tunability, and coupling efficiency," Phys. Rev. B 54, 7837-7842 (1996).
[CrossRef]

Khan, M. J.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, "Coupling of modes analysis of resonant channel add-drop filters," IEEE J. Quantum Electron. 35, 1322 -1333 (1999).
[CrossRef]

Kim, J. E.

B. K. Min, J. E. Kim, and H. Y. Park, "Channel drop filters using resonant tunneling processes in two dimensional triangular lattice photonic crystal slabs," Opt. Commun. 237, 59-63 (2004).
[CrossRef]

Kim, S.

Koshiba, M.

Kuo, C. W.

Kuramochi, E.

Lee, R.

Lin, C. K.

Liu, S.

W. Ding, L. Chen and S. Liu, "Localization properties and the effects on multi-mode switching in discrete mode CCWs," Opt. Commun. 248, 479-484 (2004).
[CrossRef]

Manolatou, C.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, "Coupling of modes analysis of resonant channel add-drop filters," IEEE J. Quantum Electron. 35, 1322 -1333 (1999).
[CrossRef]

Meier, M.

A. Mekis, M. Meier, A. Dodabalapur, R. E. Slusher, and J. D. Joannopoulos, "Lasing mechanism in two-dimensional photonic crystal lasers," Appl. Phys. A: Mat. Sci. Proc. 69, 111-114 (1999).
[CrossRef]

Mekis, A.

A. Mekis, M. Meier, A. Dodabalapur, R. E. Slusher, and J. D. Joannopoulos, "Lasing mechanism in two-dimensional photonic crystal lasers," Appl. Phys. A: Mat. Sci. Proc. 69, 111-114 (1999).
[CrossRef]

Min, B. K.

B. K. Min, J. E. Kim, and H. Y. Park, "Channel drop filters using resonant tunneling processes in two dimensional triangular lattice photonic crystal slabs," Opt. Commun. 237, 59-63 (2004).
[CrossRef]

Mitsugi, S.

Noda, S.

B. Song, T. Asano, Y. Akahane, and S. Noda, "Role of interfaces in hetero photonic crystals for manipulation of photons," Phys. Rev. B 71, 195101-195101 (2005).
[CrossRef]

S. Noda, A. Chutinan, and M. Imada, "Trapping and emission of photons by a single defect in a photonic bandgap structure," Nature 407, 608-610 (2000).
[CrossRef] [PubMed]

Notomi, M.

Park, H. Y.

B. K. Min, J. E. Kim, and H. Y. Park, "Channel drop filters using resonant tunneling processes in two dimensional triangular lattice photonic crystal slabs," Opt. Commun. 237, 59-63 (2004).
[CrossRef]

Qiu, M.

Ren, H.

Ryu, H.

Scherer, A.

Shinya, A.

Slusher, R. E.

A. Mekis, M. Meier, A. Dodabalapur, R. E. Slusher, and J. D. Joannopoulos, "Lasing mechanism in two-dimensional photonic crystal lasers," Appl. Phys. A: Mat. Sci. Proc. 69, 111-114 (1999).
[CrossRef]

Soljacic, M.

Song, B.

B. Song, T. Asano, Y. Akahane, and S. Noda, "Role of interfaces in hetero photonic crystals for manipulation of photons," Phys. Rev. B 71, 195101-195101 (2005).
[CrossRef]

Tekeste, M. Y.

Villeneuve, P. R.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, "Coupling of modes analysis of resonant channel add-drop filters," IEEE J. Quantum Electron. 35, 1322 -1333 (1999).
[CrossRef]

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, "Channel drop filters in photonic crystals," Opt. Express 3, 4-11 (1998).
[CrossRef] [PubMed]

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, "Microcavities in photonic crystals: Mode symmetry, tunability, and coupling efficiency," Phys. Rev. B 54, 7837-7842 (1996).
[CrossRef]

Vuckovic, J.

Wang, J.

Wang, W. K.

Wu, Y. D.

Xu, Y.

Yanik, M. F.

Yariv, A.

Yarrison-Rice, J. M.

Zhang, D.

C. Jin, S. Fan, S. Han, and D. Zhang, "Reflectionless multichannel wavelength demultiplexer in a transmission resonator configuration," J. Quantum Electron. 39, 160-165 (2003).
[CrossRef]

Zhang, Z.

Appl. Phys. A: Mat. Sci. Proc. (1)

A. Mekis, M. Meier, A. Dodabalapur, R. E. Slusher, and J. D. Joannopoulos, "Lasing mechanism in two-dimensional photonic crystal lasers," Appl. Phys. A: Mat. Sci. Proc. 69, 111-114 (1999).
[CrossRef]

IEEE J. Quantum Electron. (1)

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, "Coupling of modes analysis of resonant channel add-drop filters," IEEE J. Quantum Electron. 35, 1322 -1333 (1999).
[CrossRef]

J. Lightwave Technol. (3)

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

J. Quantum Electron. (1)

C. Jin, S. Fan, S. Han, and D. Zhang, "Reflectionless multichannel wavelength demultiplexer in a transmission resonator configuration," J. Quantum Electron. 39, 160-165 (2003).
[CrossRef]

Nature (1)

S. Noda, A. Chutinan, and M. Imada, "Trapping and emission of photons by a single defect in a photonic bandgap structure," Nature 407, 608-610 (2000).
[CrossRef] [PubMed]

Opt. Commun. (2)

B. K. Min, J. E. Kim, and H. Y. Park, "Channel drop filters using resonant tunneling processes in two dimensional triangular lattice photonic crystal slabs," Opt. Commun. 237, 59-63 (2004).
[CrossRef]

W. Ding, L. Chen and S. Liu, "Localization properties and the effects on multi-mode switching in discrete mode CCWs," Opt. Commun. 248, 479-484 (2004).
[CrossRef]

Opt. Express (8)

S. Kim, I. park, H. Lim, and C. Kee, "Highly efficient photonic crystal-based multi-channel drop filters of three-port system with reflection feedback," Opt. Express 12, 5518-5525 (2004).
[CrossRef] [PubMed]

C. C. Chen, C. Y. Chen, W. K. Wang, F. H. Huang, C. K. Lin, W. Y. Chiu, and Y. J. Chan, "Photonic crystal directional couplers formed by InAlGaAs nano-rods," Opt. Express 13, 38-43 (2005).
[CrossRef] [PubMed]

Z. Zhang and M. Qiu, "Compact in-plane channel drop filter design using a single cavity with two degenerate modes in 2D photonic crystal slabs," Opt. Express 13, 2596-2604 (2005).
[CrossRef] [PubMed]

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, "Channel drop filters in photonic crystals," Opt. Express 3, 4-11 (1998).
[CrossRef] [PubMed]

M. Notomi, A. Shinya, S. Mitsugi, E. Kuramochi, and H. Ryu, "Waveguides, resonators and their coupled elements in photonic crystal slabs," Opt. Express 12, 1551-1561 (2004).
[CrossRef] [PubMed]

H. Ren, C. Jiang, W. Hu, M. Gao, and J. Wang, "Photonic crystal channel drop filter with a wavelength-selective reflection micro-cavity," Opt. Express 14, 2446-2458 (2006).
[CrossRef] [PubMed]

M. Y. Tekeste and J. M. Yarrison-Rice, "High efficiency photonic crystal based wavelength demultiplexer," Opt. Express 14, 7931-7942 (2006).
[CrossRef] [PubMed]

C. W. Kuo, C. F. Chang, M. H. Chen, S. Y. Chen, and Y. D. Wu, "A new approach of planar multi-channel wavelength division multiplexing system using asymmetric super-cell photonic crystal structures," Opt. Express 15, 198-206 (2006).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. B (2)

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, "Microcavities in photonic crystals: Mode symmetry, tunability, and coupling efficiency," Phys. Rev. B 54, 7837-7842 (1996).
[CrossRef]

B. Song, T. Asano, Y. Akahane, and S. Noda, "Role of interfaces in hetero photonic crystals for manipulation of photons," Phys. Rev. B 71, 195101-195101 (2005).
[CrossRef]

Other (2)

J. D. Joannopoulos, S. G. Johnson, J. N. Winn and R. D. Meade, Photonic Crystal: Molding the Flow of Light (Princeton, Princeton Univ. Press, 2008).

H. A. Haus, Waves and Field in Optoelectronics (Prentice-Hall, 1984).

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Figures (15)

Fig. 1.
Fig. 1.

(a) Structure of a HW2 in a PC of square lattice. (b) Schematic diagram for HW2 based on CMT.

Fig. 2.
Fig. 2.

The modified HW2 with Lorentzian transmission spectrum in a PC of square lattice.

Fig. 3.
Fig. 3.

The phase-shift between two coupled cavities versus the cavities radius of the modified HW2 in a PC of square lattice.

Fig. 4.
Fig. 4.

The simulation results of transmission spectrum of the modified HW2 obtained by the FDTD and CMT method.

Fig. 5.
Fig. 5.

The basic structure of a three-port CDF which is based on two coupled cavities.

Fig. 6.
Fig. 6.

The curve of the three-port CDF transmission peaks versus τ 2 /τ 1.

Fig. 7.
Fig. 7.

The basic structure of a wavelength-selective reflector which is based on two coupled cavities.

Fig. 8.
Fig. 8.

The basic structure of the proposed three-port CDF with coupled cavity based wavelength selective-reflector.

Fig. 9.
Fig. 9.

(a) Dependence of drop efficiency at resonant frequencies on the ratio of decay rates τ 2/τ 1 when ρ = (2n + 1)π. (b) Dependence of the maximum of drop efficiency on ρ.

Fig. 10.
Fig. 10.

Dependence of the maximum of drop efficiency on the frequency detuning factor (ω Resb /ω Resa )

Fig. 11.
Fig. 11.

The structure of three-port CDF with coupled cavitiy-based wavelength-selective reflection feedback, in 2D-PC of square lattice composed of dielectric rods in air. The dashed-line and the dotted-line rectangulars are the drop and the reflector sections, respectively.

Fig. 12.
Fig. 12.

Dispersion curve of the bus/drop line-defect waveguides versus the wave vector component k along the defect.

Fig. 13.
Fig. 13.

Dependence of the resonant frequencies of the coupled cavities in the HW2 structure on the coupled cavities radii.

Fig. 14.
Fig. 14.

Transmission spectra for the designed CDF calculated using the 2D-FDTD method.(a) The drop port (the dashed curve) and the bus port transmission spectrum (the solid curve).(b) The drop port transmission spectrum in dB.

Fig. 15.
Fig. 15.

The steady state wave propagation at the resonant frequencies of the designed CDF. (a) ω Res 1 =0.36076 (2πc/a). (b) ω Res 2 =0.36573 (2πc/a).

Equations (64)

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

da1dt=jωResa1a11τ1a11τ′1+ejθ12τ1S+1+2τ′1S′+1
da2dt=jωResa2a21τ2a21τ′2+2τ2S+2+ejθ22τ′2S′+2
S1=S+1+2τ1a1
S′1=S′+1+2τ′1a1
S2=S+2+2τ2a2
S′2=S′+2+2τ′2a2
S′+1=S2e
S+2=S′1e
S′+1=2τ2(a2a1e2jsinφ)
S+2=2τ2(a1a2e2jsinφ).
S+1=ja2ejθ1sinφ2/τ [1(γjsinφ)2]
T=S′2S+12=4sin2φ(1+sin2φγ2)2+(2γsinφ)2.
ωiT=1/2=ωRes+1τtanφ±(1τtanφ)2±12τsinφ,i=1,2,3,4.
FWHM=ω2ω1=ω4ω32τ=ωRes2Q .
da1dt=jωResa1a11τ1a11τ2+ejθ12τ1S+1+2τ2S″+ejθ12τ1S′+1
da2dt=jωResa2a22τ2+2τ2S′+2+ejθ22τ2S+2
S′1=S+1ejθ12τ1a1
S1=S′+1ejθ12τ1a1
S2=S+2ejθ22τ2a2
S′′1=S′′+1+2τ1a1
S′2=S′+2+2τ2a2.
S′+2=S′′1e
S′′+1=S′2e.
S′′+1=2τ2(a2a1e2jsinφ).
S′+2=2τ2(a1a2e2jsinφ).
S+1=a2(1γ2)jsinφ2τ22τ1ejθ1
D=S2S+12=(sin2φ(4τ2/τ1)((γsinφ)(1+2τ2/τ1))2+(1+(2τ2/τ1)sin2φγ2)2).
PT=4(τ2/τ1)(1+2(τ2/τ1))2,
ωRes1,2=ωRes+1τ2tanφ±1τ2tanφ(112sin2φ(1+(2τ2/τ1)2)1/2).
R=S1S+12=14 , T=S′1S+12=14
da1dt=jωResa1a11τ1a11τ2+ejθ12τ1S+1+2τ2S′′+1
da2dt=jωResa2a21τ2+2τ2S′+2.
S+1= a2(γ1/γj2τ2/τ1sinφ)jsinφ2τ22/τ1ejθ1
R=S1S+11=jsinφ2τ2/τ1(γ1/γj2τ2/τ1sinφ)2=γ2(2τ2/τ1)2sin2φ(γ21)2+γ2(2τ2/τ1)2sin2φ.
ωRes1,2=ωRes+1τ2(1tanφ±1sinφ)
da1dt=jωResaa1a11τ0a11τ1a11τ2+ejθ12τ1S+1+2τ2S″+1+ejθ12τ1S′+1
da2dt=jωResaa2a22τ2a21τ0+2τ2S′+2+ejθ22τ2S+2
db1dt=jωResbb1b11τ0b12τ3b11τ4+ejθ12τ3S+3+2τ4S″+1
db2dt=jωResbb2b21τ4b21τ0+2τ4R+2
S′3=S+32τ3ejθ3b1
S3=S′+32τ3ejθ3b1
S′1=S+12τ1ejθ1a1
S1=S′+12τ1ejθ1a1
S2=S+2+2τ2ejθ2a1
S″1=S″+1+2τ2a1
S′2=S′+2+2τ2a2
R1=R+1+2τ4b1
R2=R+2+2τ4b2
S+3=S′1ejβd
S′+1=S3ejβd.
S′+2=S″1e
S″+1=S′2e
R+2=R1e
R+1=R2e.
R+1=2τ4(b2b1e2jsinφ)
R+2=2τ4(b1b2e2jsinφ).
S+3=ejθ3[γ2τ4τ0(2τ4τ3+τ4τ0)sin2φ1jγsinφ(2τ4τ3+τ4τ0)]j2τ42τ3sinφ[γj(τ4τ0)sinφ]b1
S3S+3=r=2τ4τ3sinφ[γj(τ4τ0)sinφ][γ2τ4τ0(2τ4τ3+τ4τ0)sin2φ1]j+γsinφ(2τ4τ3+τ4τ0)
S′+1=re(S+12τ1ejθ1a1)
ωRes1,2=ωResb+1τ4(1tanφ±1sinφ).
D=S2S+1=(2/τ1τ2)ej(θ1θ2)[(1rcosρ)+j(rsinρ)](jτ4/τ0sinφγ)1j[(ωωResa)+2rτ1sinρ1τ2tanφγατ2sinφ]+[1τ0+2τ12rτ1cosρ+(τ2/τ0+1)ατ2]
Dφ0=(2/τ1τ2)ej(θ1θ2)[(1rcosρ)+j(rsinρ)]j[(ωωResa)+2rτ1sinρ]+[1τ0+2τ1+1τ22rτ1cosρ].
ηω=ωRes1,2=D2ω=ωRes1,2=8k(1cosρ)8k2(1cosρ)+4k(1cosρ)+1
ηmaxωRes1,2=64τ12(ωResbωResa)2+64=4Q12(ωResbωResa1)2+4 .

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