Abstract

We propose and analyze a novel design of a hybrid micro-ring resonator and photonic crystal device. The proposed device is based on a micro-ring resonator with the addition of a series of periodic defects that are introduced to the microring. When the wavelength of operation approaches the band-gap of the periodic structure, the modal dispersion is significantly increased. The huge dispersion leads to narrowing of the spectral linewidth of the resonator. We predict an order of magnitude linewidth narrowing for a microring radius of the order of 10μm. The proposed hybrid device is analyzed theoretically and numerically using finite-elements calculations and finite-difference-time-domain calculations. We also present as well as the design and analysis of add-drop and notch filters based on the highly dispersive ring resonator.

© 2007 Optical Society of America

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References

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  1. Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, "Micrometre-scale silicon electro-optic modulator," Nature 435, 325-327 (2005).
    [CrossRef] [PubMed]
  2. Y. Akahane, T. Asano, B. Song, and S. Noda, "High-Q photonic nanocavity in a two-dimensional photonic crystal," Nature 425, 944-947 (2003).
    [CrossRef] [PubMed]
  3. J. S. Forsei, P. R. Villeneuve, J. Ferrara, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L.C Kimerling, H. I. Smith, and E. P. Ippen, "Photonic-bandgap microcavities in optical waveguides," Nature 390, 143-145 (1997).
    [CrossRef]
  4. A. A. Savchenkov, A. B. Matsko, and L. Maleki, "White light whispering gallery mode resonators," Opt. Lett. 31, 92-94 (2006).
    [CrossRef] [PubMed]
  5. G. Muller, M. Muller, A. Wicht, R. H. Rinkleff, and K. Danzmann, "Optical resonator with steep internal dispersion," Phys. Rev. A 56, 2385-2389 (1997).
    [CrossRef]
  6. M. Soljacic, E. Lidorikis, L. Vestergaard Hau, J. D. Joannopoulus, "Enhancement of microcavities lifetime using highly dispersive materials," Phys. Rev. E. 71, 026602 (2005).
    [CrossRef]
  7. U. Levy, and Y. Fainman, "Dispersion properties of inhomogeneous nanostructures," J. Opt. Soc. Am. A 21, 881-889 (2004).
    [CrossRef]
  8. S. Mookherjea, "Dispersion characteristics of coupled-resonator optical waveguide," Opt. Lett. 30, 2406-2408 (2005).
    [CrossRef] [PubMed]
  9. Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, "Active control of slow light on a chip with photonic crystal waveguides," Nature 438, 65-69 (2005).
    [CrossRef] [PubMed]
  10. S. Chang, H. Cao, S. Ho, "Cavity formation and light propagation in partially ordered and completely random one-dimensional systems," IEEE J. Quantum Electron. 39, 364-374 (2003).
    [CrossRef]
  11. J. Scheuer and A. Yariv, "Coupled-waves approach to the design and analysis of Bragg and photonic crystal annular resonators"IEEE J. Quantum Electron. 39, 1555-1562 (2003).
    [CrossRef]
  12. All the calculations in this paper are in 2D. Since we examine only channel waveguides (as opposed to 2D PhC, for example) the 2D approximation gives satisfying and qualitative and sometimes even quantitative results.
  13. M. Povinelli, S. Johnson, and J. Joannopoulos, "Slow-light, band-edge waveguides for tunable time delays," Opt. Express 13, 7145-7159 (2005).
    [CrossRef] [PubMed]
  14. C. Manolatu, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulus, "Coupling of modes analysis of resonant channel add-drop filters," IEEE J. Quantum Electron. 35, 1322-1331 (1999).
    [CrossRef]
  15. Y. Akahane, T. Asano, H. Takano, B. S. Song, Y. Takana, and S. Noda, "Two-dimensional photonic-crystal-slab channel drop filter with flat-top response," Opt. Express 13, 2512-2530 (2005)
    [CrossRef] [PubMed]
  16. C. Sauvan, G. Lecamp, P. Lalanne, and J. P. Hugonin, "Modal-reflectivity enhancement by geometry tuning in Photonic Crystal microcavities," Opt. Express 13, 245-255 (2005).
    [CrossRef] [PubMed]
  17. J. Scheuer and A. Yariv, "Sagnac effect in coupled-resonator slow-light waveguide structures," Phys. Rev. Lett. 96, 053901 (2006).
    [CrossRef] [PubMed]
  18. A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, and L. Maleki, "Optical gyroscope with whispering gallery mode optical cavities," Opt. Commun. 233, 107-112 (2004).
    [CrossRef]

2006 (2)

A. A. Savchenkov, A. B. Matsko, and L. Maleki, "White light whispering gallery mode resonators," Opt. Lett. 31, 92-94 (2006).
[CrossRef] [PubMed]

J. Scheuer and A. Yariv, "Sagnac effect in coupled-resonator slow-light waveguide structures," Phys. Rev. Lett. 96, 053901 (2006).
[CrossRef] [PubMed]

2005 (7)

2004 (2)

U. Levy, and Y. Fainman, "Dispersion properties of inhomogeneous nanostructures," J. Opt. Soc. Am. A 21, 881-889 (2004).
[CrossRef]

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, and L. Maleki, "Optical gyroscope with whispering gallery mode optical cavities," Opt. Commun. 233, 107-112 (2004).
[CrossRef]

2003 (3)

S. Chang, H. Cao, S. Ho, "Cavity formation and light propagation in partially ordered and completely random one-dimensional systems," IEEE J. Quantum Electron. 39, 364-374 (2003).
[CrossRef]

J. Scheuer and A. Yariv, "Coupled-waves approach to the design and analysis of Bragg and photonic crystal annular resonators"IEEE J. Quantum Electron. 39, 1555-1562 (2003).
[CrossRef]

Y. Akahane, T. Asano, B. Song, and S. Noda, "High-Q photonic nanocavity in a two-dimensional photonic crystal," Nature 425, 944-947 (2003).
[CrossRef] [PubMed]

1999 (1)

C. Manolatu, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulus, "Coupling of modes analysis of resonant channel add-drop filters," IEEE J. Quantum Electron. 35, 1322-1331 (1999).
[CrossRef]

1997 (2)

J. S. Forsei, P. R. Villeneuve, J. Ferrara, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L.C Kimerling, H. I. Smith, and E. P. Ippen, "Photonic-bandgap microcavities in optical waveguides," Nature 390, 143-145 (1997).
[CrossRef]

G. Muller, M. Muller, A. Wicht, R. H. Rinkleff, and K. Danzmann, "Optical resonator with steep internal dispersion," Phys. Rev. A 56, 2385-2389 (1997).
[CrossRef]

Akahane, Y.

Asano, T.

Cao, H.

S. Chang, H. Cao, S. Ho, "Cavity formation and light propagation in partially ordered and completely random one-dimensional systems," IEEE J. Quantum Electron. 39, 364-374 (2003).
[CrossRef]

Chang, S.

S. Chang, H. Cao, S. Ho, "Cavity formation and light propagation in partially ordered and completely random one-dimensional systems," IEEE J. Quantum Electron. 39, 364-374 (2003).
[CrossRef]

Danzmann, K.

G. Muller, M. Muller, A. Wicht, R. H. Rinkleff, and K. Danzmann, "Optical resonator with steep internal dispersion," Phys. Rev. A 56, 2385-2389 (1997).
[CrossRef]

Fainman, Y.

Fan, S.

C. Manolatu, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulus, "Coupling of modes analysis of resonant channel add-drop filters," IEEE J. Quantum Electron. 35, 1322-1331 (1999).
[CrossRef]

J. S. Forsei, P. R. Villeneuve, J. Ferrara, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L.C Kimerling, H. I. Smith, and E. P. Ippen, "Photonic-bandgap microcavities in optical waveguides," Nature 390, 143-145 (1997).
[CrossRef]

Ferrara, J.

J. S. Forsei, P. R. Villeneuve, J. Ferrara, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L.C Kimerling, H. I. Smith, and E. P. Ippen, "Photonic-bandgap microcavities in optical waveguides," Nature 390, 143-145 (1997).
[CrossRef]

Forsei, J. S.

J. S. Forsei, P. R. Villeneuve, J. Ferrara, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L.C Kimerling, H. I. Smith, and E. P. Ippen, "Photonic-bandgap microcavities in optical waveguides," Nature 390, 143-145 (1997).
[CrossRef]

Hamann, H. F.

Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, "Active control of slow light on a chip with photonic crystal waveguides," Nature 438, 65-69 (2005).
[CrossRef] [PubMed]

Haus, H. A.

C. Manolatu, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulus, "Coupling of modes analysis of resonant channel add-drop filters," IEEE J. Quantum Electron. 35, 1322-1331 (1999).
[CrossRef]

Ho, S.

S. Chang, H. Cao, S. Ho, "Cavity formation and light propagation in partially ordered and completely random one-dimensional systems," IEEE J. Quantum Electron. 39, 364-374 (2003).
[CrossRef]

Hugonin, J. P.

Ilchenko, V. S.

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, and L. Maleki, "Optical gyroscope with whispering gallery mode optical cavities," Opt. Commun. 233, 107-112 (2004).
[CrossRef]

Ippen, E. P.

J. S. Forsei, P. R. Villeneuve, J. Ferrara, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L.C Kimerling, H. I. Smith, and E. P. Ippen, "Photonic-bandgap microcavities in optical waveguides," Nature 390, 143-145 (1997).
[CrossRef]

Joannopoulos, J.

Joannopoulos, J. D.

J. S. Forsei, P. R. Villeneuve, J. Ferrara, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L.C Kimerling, H. I. Smith, and E. P. Ippen, "Photonic-bandgap microcavities in optical waveguides," Nature 390, 143-145 (1997).
[CrossRef]

Joannopoulus, J. D.

M. Soljacic, E. Lidorikis, L. Vestergaard Hau, J. D. Joannopoulus, "Enhancement of microcavities lifetime using highly dispersive materials," Phys. Rev. E. 71, 026602 (2005).
[CrossRef]

C. Manolatu, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulus, "Coupling of modes analysis of resonant channel add-drop filters," IEEE J. Quantum Electron. 35, 1322-1331 (1999).
[CrossRef]

Johnson, S.

Khan, M. J.

C. Manolatu, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulus, "Coupling of modes analysis of resonant channel add-drop filters," IEEE J. Quantum Electron. 35, 1322-1331 (1999).
[CrossRef]

Kimerling, L.C

J. S. Forsei, P. R. Villeneuve, J. Ferrara, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L.C Kimerling, H. I. Smith, and E. P. Ippen, "Photonic-bandgap microcavities in optical waveguides," Nature 390, 143-145 (1997).
[CrossRef]

Lalanne, P.

Lecamp, G.

Levy, U.

Lidorikis, E.

M. Soljacic, E. Lidorikis, L. Vestergaard Hau, J. D. Joannopoulus, "Enhancement of microcavities lifetime using highly dispersive materials," Phys. Rev. E. 71, 026602 (2005).
[CrossRef]

Lipson, M.

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, "Micrometre-scale silicon electro-optic modulator," Nature 435, 325-327 (2005).
[CrossRef] [PubMed]

Maleki, L.

A. A. Savchenkov, A. B. Matsko, and L. Maleki, "White light whispering gallery mode resonators," Opt. Lett. 31, 92-94 (2006).
[CrossRef] [PubMed]

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, and L. Maleki, "Optical gyroscope with whispering gallery mode optical cavities," Opt. Commun. 233, 107-112 (2004).
[CrossRef]

Manolatu, C.

C. Manolatu, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulus, "Coupling of modes analysis of resonant channel add-drop filters," IEEE J. Quantum Electron. 35, 1322-1331 (1999).
[CrossRef]

Matsko, A. B.

A. A. Savchenkov, A. B. Matsko, and L. Maleki, "White light whispering gallery mode resonators," Opt. Lett. 31, 92-94 (2006).
[CrossRef] [PubMed]

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, and L. Maleki, "Optical gyroscope with whispering gallery mode optical cavities," Opt. Commun. 233, 107-112 (2004).
[CrossRef]

McNab, S. J.

Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, "Active control of slow light on a chip with photonic crystal waveguides," Nature 438, 65-69 (2005).
[CrossRef] [PubMed]

Mookherjea, S.

Muller, G.

G. Muller, M. Muller, A. Wicht, R. H. Rinkleff, and K. Danzmann, "Optical resonator with steep internal dispersion," Phys. Rev. A 56, 2385-2389 (1997).
[CrossRef]

Muller, M.

G. Muller, M. Muller, A. Wicht, R. H. Rinkleff, and K. Danzmann, "Optical resonator with steep internal dispersion," Phys. Rev. A 56, 2385-2389 (1997).
[CrossRef]

Noda, S.

O’Boyle, M.

Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, "Active control of slow light on a chip with photonic crystal waveguides," Nature 438, 65-69 (2005).
[CrossRef] [PubMed]

Povinelli, M.

Pradhan, S.

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, "Micrometre-scale silicon electro-optic modulator," Nature 435, 325-327 (2005).
[CrossRef] [PubMed]

Rinkleff, R. H.

G. Muller, M. Muller, A. Wicht, R. H. Rinkleff, and K. Danzmann, "Optical resonator with steep internal dispersion," Phys. Rev. A 56, 2385-2389 (1997).
[CrossRef]

Sauvan, C.

Savchenkov, A. A.

A. A. Savchenkov, A. B. Matsko, and L. Maleki, "White light whispering gallery mode resonators," Opt. Lett. 31, 92-94 (2006).
[CrossRef] [PubMed]

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, and L. Maleki, "Optical gyroscope with whispering gallery mode optical cavities," Opt. Commun. 233, 107-112 (2004).
[CrossRef]

Scheuer, J.

J. Scheuer and A. Yariv, "Sagnac effect in coupled-resonator slow-light waveguide structures," Phys. Rev. Lett. 96, 053901 (2006).
[CrossRef] [PubMed]

J. Scheuer and A. Yariv, "Coupled-waves approach to the design and analysis of Bragg and photonic crystal annular resonators"IEEE J. Quantum Electron. 39, 1555-1562 (2003).
[CrossRef]

Schmidt, B.

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, "Micrometre-scale silicon electro-optic modulator," Nature 435, 325-327 (2005).
[CrossRef] [PubMed]

Smith, H. I.

J. S. Forsei, P. R. Villeneuve, J. Ferrara, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L.C Kimerling, H. I. Smith, and E. P. Ippen, "Photonic-bandgap microcavities in optical waveguides," Nature 390, 143-145 (1997).
[CrossRef]

Soljacic, M.

M. Soljacic, E. Lidorikis, L. Vestergaard Hau, J. D. Joannopoulus, "Enhancement of microcavities lifetime using highly dispersive materials," Phys. Rev. E. 71, 026602 (2005).
[CrossRef]

Song, B.

Y. Akahane, T. Asano, B. Song, and S. Noda, "High-Q photonic nanocavity in a two-dimensional photonic crystal," Nature 425, 944-947 (2003).
[CrossRef] [PubMed]

Song, B. S.

Steinmeyer, G.

J. S. Forsei, P. R. Villeneuve, J. Ferrara, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L.C Kimerling, H. I. Smith, and E. P. Ippen, "Photonic-bandgap microcavities in optical waveguides," Nature 390, 143-145 (1997).
[CrossRef]

Takana, Y.

Takano, H.

Thoen, E. R.

J. S. Forsei, P. R. Villeneuve, J. Ferrara, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L.C Kimerling, H. I. Smith, and E. P. Ippen, "Photonic-bandgap microcavities in optical waveguides," Nature 390, 143-145 (1997).
[CrossRef]

Vestergaard Hau, L.

M. Soljacic, E. Lidorikis, L. Vestergaard Hau, J. D. Joannopoulus, "Enhancement of microcavities lifetime using highly dispersive materials," Phys. Rev. E. 71, 026602 (2005).
[CrossRef]

Villeneuve, P. R.

C. Manolatu, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulus, "Coupling of modes analysis of resonant channel add-drop filters," IEEE J. Quantum Electron. 35, 1322-1331 (1999).
[CrossRef]

J. S. Forsei, P. R. Villeneuve, J. Ferrara, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L.C Kimerling, H. I. Smith, and E. P. Ippen, "Photonic-bandgap microcavities in optical waveguides," Nature 390, 143-145 (1997).
[CrossRef]

Vlasov, Y. A.

Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, "Active control of slow light on a chip with photonic crystal waveguides," Nature 438, 65-69 (2005).
[CrossRef] [PubMed]

Wicht, A.

G. Muller, M. Muller, A. Wicht, R. H. Rinkleff, and K. Danzmann, "Optical resonator with steep internal dispersion," Phys. Rev. A 56, 2385-2389 (1997).
[CrossRef]

Xu, Q.

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, "Micrometre-scale silicon electro-optic modulator," Nature 435, 325-327 (2005).
[CrossRef] [PubMed]

Yariv, A.

J. Scheuer and A. Yariv, "Sagnac effect in coupled-resonator slow-light waveguide structures," Phys. Rev. Lett. 96, 053901 (2006).
[CrossRef] [PubMed]

J. Scheuer and A. Yariv, "Coupled-waves approach to the design and analysis of Bragg and photonic crystal annular resonators"IEEE J. Quantum Electron. 39, 1555-1562 (2003).
[CrossRef]

IEEE J. Quantum Electron. (3)

S. Chang, H. Cao, S. Ho, "Cavity formation and light propagation in partially ordered and completely random one-dimensional systems," IEEE J. Quantum Electron. 39, 364-374 (2003).
[CrossRef]

J. Scheuer and A. Yariv, "Coupled-waves approach to the design and analysis of Bragg and photonic crystal annular resonators"IEEE J. Quantum Electron. 39, 1555-1562 (2003).
[CrossRef]

C. Manolatu, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulus, "Coupling of modes analysis of resonant channel add-drop filters," IEEE J. Quantum Electron. 35, 1322-1331 (1999).
[CrossRef]

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

Nature (4)

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, "Micrometre-scale silicon electro-optic modulator," Nature 435, 325-327 (2005).
[CrossRef] [PubMed]

Y. Akahane, T. Asano, B. Song, and S. Noda, "High-Q photonic nanocavity in a two-dimensional photonic crystal," Nature 425, 944-947 (2003).
[CrossRef] [PubMed]

J. S. Forsei, P. R. Villeneuve, J. Ferrara, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L.C Kimerling, H. I. Smith, and E. P. Ippen, "Photonic-bandgap microcavities in optical waveguides," Nature 390, 143-145 (1997).
[CrossRef]

Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, "Active control of slow light on a chip with photonic crystal waveguides," Nature 438, 65-69 (2005).
[CrossRef] [PubMed]

Opt. Commun. (1)

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, and L. Maleki, "Optical gyroscope with whispering gallery mode optical cavities," Opt. Commun. 233, 107-112 (2004).
[CrossRef]

Opt. Express (3)

Opt. Lett. (2)

Phys. Rev. A (1)

G. Muller, M. Muller, A. Wicht, R. H. Rinkleff, and K. Danzmann, "Optical resonator with steep internal dispersion," Phys. Rev. A 56, 2385-2389 (1997).
[CrossRef]

Phys. Rev. Lett. (1)

J. Scheuer and A. Yariv, "Sagnac effect in coupled-resonator slow-light waveguide structures," Phys. Rev. Lett. 96, 053901 (2006).
[CrossRef] [PubMed]

Phys. Review E. (1)

M. Soljacic, E. Lidorikis, L. Vestergaard Hau, J. D. Joannopoulus, "Enhancement of microcavities lifetime using highly dispersive materials," Phys. Rev. E. 71, 026602 (2005).
[CrossRef]

Other (1)

All the calculations in this paper are in 2D. Since we examine only channel waveguides (as opposed to 2D PhC, for example) the 2D approximation gives satisfying and qualitative and sometimes even quantitative results.

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Figures (11)

Fig. 1.
Fig. 1.

(a). Two types of dielectric structures (single cell) embedded in the waveguide. Red corresponds to the waveguide core. (b). Dispersion curve of the 1D PhC waveguide with circular-hole periodic defect. The inset depicts the mode profile (Hy field - in plane polarization) of the waveguide’s lowest mode.

Fig. 2.
Fig. 2.

(a). Enlarged view of the dispersion diagram presented in Fig. 1 around the band edge of the first mode. (b) The corresponding calculated effective index (continuous line) and group index (dashed line).

Fig. 3.
Fig. 3.

Micro ring radius values that satisfy the resonance conditions as a function of propagation constant.

Fig. 4.
Fig. 4.

Schematic description of the two highly dispersive micro ring configurations. (a) Add-drop configuration that includes two coupling waveguides, a modified microring and two reflectors providing maximum drop efficiency. (b) Notch filter configuration, that includes a single coupling waveguide and a single reflector. The signal is detected in reflection mode.

Fig. 5.
Fig. 5.

Highly dispersive MRR based add-drop filter spectral response compared to standard MRR. (a) Continuous green lines represent the dispersive MRR drop channel for different values of Kn . Dashed red lines stand for the standard ring with similar parameters. (b) Curved line shows FWHM of the drop channel (normalized to regular ring) vs. Kn . Dashed line denotes the standard MRR.

Fig. 6.
Fig. 6.

Spectral response of a highly dispersive MRR based notch filter compared to a standard MRR. (a) Continuous green lines represent the dispersive MRR for different values of Kn, dashed red line stand for the standard MRR with similar parameters. (b) Curved blue line shows slope of the notch (normalized to standard MRR) vs. Kn. Dashed black line denotes the standard MRR.

Fig. 7.
Fig. 7.

(a). Continuous blue lines - simulated reflection intensity vs. wavelength for the notch configuration. Dashed green lines represent the location of resonances according to the approximated model. (b) Comparison between calculated group index of 1D PhC waveguide and the group index of the simulated modified MRR.

Fig. 8.
Fig. 8.

(a). Magnified view of Fig. 7(a), where all deeps are superposed on each other for comparison purposes. (b) Maximum slope calculated for each of the deeps in Fig. 7(a).

Fig. 9.
Fig. 9.

Power flow in the modified MRR.

Fig. 10.
Fig. 10.

Frequency response of standard MRRs and modified MRRs based add-drop filters.

Fig. 11.
Fig. 11.

Power flow in a modified MRR based add-drop filter. The power flows into the system from the upper waveguide and is transferred to the lower waveguide through the MRR.

Tables (1)

Tables Icon

Table 1. Parameters for calculation of the frequency response obtained by the modified MRR filter

Equations (16)

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2 π R K z = m 2 π
2 π R = q P
d a ad dt = ( j ω 0 1 τ 0 2 τ e ) a ad + κ [ exp ( j β d 1 ) s + 1 + ( s + 2 + s + 4 ) exp ( j β d 2 ) ]
d a nt dt = ( j ω 0 1 τ 0 1 τ e ) a nt + κ [ exp ( j β d 1 ) s + 1 + exp ( j β d 2 ) s + 2 ]
s 1,3 = exp [ j β ( d 1 + d 2 ) ] [ s + 2,4 κ * exp ( j β d 2 ) a ]
s 2,4 = exp [ j β ( d 1 + d 2 ) ] [ s + 1,3 κ * exp ( j β d 1 ) a ]
κ = 1 τ e
s + 2,4 = s 2,4 e j Δ
a ad = 1 τ e exp ( j β d 1 ) [ 1 + exp ( j θ ) ] s + 1 j ( ω ω 0 ) + 1 τ 0 + 2 τ e [ 1 + exp ( j θ ) ]
a nt = 1 τ e exp ( j β d 1 ) [ 1 + exp ( j θ ) ] s + 1 j ( ω ω 0 ) + 1 τ 0 + 1 τ e [ 1 + exp ( j θ ) ]
θ = 2 β d 2 + Δ
ω ˜ 0 = ω 0 ( 1 n eff ( ω ) n eff ( ω 0 ) n eff ( ω 0 ) )
s 3 s + 1 = 1 τ e [ 1 + exp ( j θ ) ] exp ( j β d 1 ) a s + 1
T s 3 s + 1 2 = [ 2 τ e ( 1 + cos θ ) ] 2 [ 1 τ 0 + 2 τ e ( 1 + cos θ ) ] 2 + [ ω ω ˜ 0 + 2 τ e sin θ ] 2
R s 1 s + 1 2 = 1 + T + 2 Re { s 3 s + 1 }
R s 1 s + 1 2 = [ 1 τ 0 1 τ e ( 1 + cos θ ) ] 2 + [ ω ω ˜ 0 + 1 τ e sin θ ] 2 [ 1 τ 0 + 1 τ e ( 1 + cos θ ) ] 2 + [ ω ω ˜ 0 + 1 τ e sin θ ] 2

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