Abstract

Coupling-induced resonance frequency shifts (CIFS) are theoretically described, and are found to be an important fundamental source of resonance frequency mismatch between coupled optical cavities that would be degenerate in isolation. Their deleterious effect on high-order resonant filter responses and complete correction by pre-distortion are described. Analysis of the physical effects contributing to CIFS shows that a positive index perturbation may bring about a resonance shift of either sign. Higher-order CIFS effects, the scaling of CIFS-caused impairment with finesse, FSR and index contrast, and the tolerability of frequency mismatch in telecom-grade filters are addressed. The results also suggest possible designs and applications for CIFS-free coupled-resonator systems.

© 2006 Optical Society of America

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  1. 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]
  2. S. Blair and Y. Chen, "Resonant-enhanced evanescent-wave fluorescence biosensing with cylindrical optical cavities," Appl. Opt. 40, 570-582 (2001).
    [CrossRef]
  3. P. P. Absil, J. V. Hryniewicz, B. E. Little, P. S. Cho, R. A. Wilson, L. G. Joneckis and P.-T. Ho, "Wavelength conversion in GaAs microring resonators," Opt. Lett. 25, 554-556 (2000).
    [CrossRef]
  4. B. Liu, A. Shakouri and J. E. Bowers, "Passive microring-resonator-coupled lasers," Appl. Phys. Lett. 79, 3561-3563 (2001).
    [CrossRef]
  5. M. Lončar, T. Yoshie, Y. Qiu, P. Gogna and A. Scherer, "Low-threshold photonic crystal laser," in Proc. SPIE5000, 16-26 (2003).
    [CrossRef]
  6. C. K. Madsen and J. H. Zhao, Optical filter design and analysis: a signal processing approach (Wiley, 1999).
  7. S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. J. Khan, C. Manolatou and H. A. Haus, "Theoretical analysis of channel drop tunneling processes," Phys. Rev. B 59, 15882-15892 (1999).
    [CrossRef]
  8. H. A. Haus, B. E. Little, M. A. Popović, S. T. Chu, M. R. Watts and C. Manolatou, "Optical resonators and filters," in Optical Microcavities, K. Vahala, ed. (World Scientific, Singapore, 2004).
  9. H. A. Haus, "Microwaves and Photonics," in OSA TOPS 23 Symposium on Electro-Optics: Present and Future, H.A. Haus, ed., (Optical Society of America, Washington, DC, 1998), pp. 2-8.
  10. M. J. Khan, C. Manolatou, S. Fan, P. R. Villeneuve, H. A. Haus and J. D. Joannopoulos, "Mode-coupling analysis of multipole symmetric resonant add/drop filters," IEEE J. Quantum Electron. 35, 1451-1460 (1999).
    [CrossRef]
  11. A. Melloni and M. Martinelli, "Synthesis of direct-coupled-resonators bandpass filters for WDM systems," J. Lightwave Technol. 20, 296-303 (2002).
    [CrossRef]
  12. R. Orta, P. Savi, R. Tascone, and D. Trinchero, "Synthesis of multiple-ring-resonator filters for optical systems," IEEE Photonics Technol. Lett. 7, 1447-1449 (1995).
    [CrossRef]
  13. C. Manolatou, M. A. Popović, P. T. Rakich, T. Barwicz, H. A. Haus and E. P. Ippen, "Spectral anomalies due to coupling-induced frequency shifts in dielectric coupled-resonator filters," in Proceedings of Optical Fiber Communication Conference on CD-ROM (Los Angeles, CA, February 2004), TuD5.
  14. T. Barwicz, M. A. Popović, P. T. Rakich, M. R. Watts, H. A. Haus, E. P. Ippen and H. I. Smith, "Microring-resonator-based add-drop filters in SiN: fabrication and analysis," Opt. Express 12, 1437-1442 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-7-1437.
    [CrossRef] [PubMed]
  15. M. A. Popović, M. R. Watts, T. Barwicz, P. T. Rakich, L. Socci, E. P. Ippen, F. X. Kärtner and H. I. Smith, "High-index-contrast, wide-FSR microring-resonator filter design and realization with frequency-shift compensation," in Proc. Optical Fiber Comm. Conf. (Optical Society of America, Washington, DC, 2005).
  16. H. A. Haus, Waves and fields in optoelectronics (Prentice-Hall, Englewood Cliffs, NJ, 1984).
  17. H. A. Haus and W.-P. Huang, "Coupled-mode theory," inProc. IEEE 79, 1505-1518 (1991).
  18. T. Barwicz, M. A. Popović, M. R. Watts, P. T. Rakich, E. P. Ippen and H. I. Smith, "Fabrication of add-drop filters based on frequency-matched microring resonators," submitted to J. Lightwave Technol.
  19. M. Popović, "Complex-frequency leaky mode computations using PML boundary layers for dielectric resonant structures," in Proceedings of Integrated Photonics Research (Washington, DC, June 17, 2003).
  20. A. D. Berk, "Variational principles for electromagnetic resonators and waveguides," IRE Trans. Antennas Propag., April 1956, pp. 104-111.
  21. H. A. Haus, W. P. Huang and A. W. Snyder, "Coupled-mode formulations," Opt. Lett. 14, 1222-1224 (1989).
    [CrossRef] [PubMed]
  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, 1322-1331 (1999).
    [CrossRef]
  23. M. J. Khan, M. Lim, C. Joyner, T. Murphy, H. A. Haus and H. I. Smith, "Integrated Bragg grating structures," in Digest of the LEOS Summer Topical Meeting on WDM Components (Copper Mountain, CO, 2001).
  24. J. Scheuer and A. Yariv, "Two-dimensional optical ring resonators based on radial Bragg resonance," Opt. Lett. 28, 1528-1530 (2003).
    [CrossRef] [PubMed]
  25. A. Yariv, "Universal relations for coupling of optical power between microresonators and dielectric waveguides," Electron. Lett. 36, 321-322 (2000).
    [CrossRef]
  26. S. V. Boriskina, T. M. Benson, P. Sewell and A. I. Nosich, "Effect of a layered environment on the complex natural frequencies of 2D WGM dielectric-ring resonators," J. Lightwave Technol. 20, 1563-1572 (2002).
    [CrossRef]
  27. B. E. Little, J.-P. Laine and S. T. Chu, "Surface-roughness-induced contradirectional coupling in ring and disk resonators," Opt. Lett. 22, 4-6 (1997).
    [CrossRef] [PubMed]
  28. M. M. Lee and M. C. Wu, "MEMS-actuated microdisk resonators with variable power coupling ratios," IEEE Photon. Technol. Lett. 17, 1034-1036 (2005).
    [CrossRef]

2005 (1)

M. M. Lee and M. C. Wu, "MEMS-actuated microdisk resonators with variable power coupling ratios," IEEE Photon. Technol. Lett. 17, 1034-1036 (2005).
[CrossRef]

2004 (1)

2003 (1)

2002 (2)

2001 (2)

B. Liu, A. Shakouri and J. E. Bowers, "Passive microring-resonator-coupled lasers," Appl. Phys. Lett. 79, 3561-3563 (2001).
[CrossRef]

S. Blair and Y. Chen, "Resonant-enhanced evanescent-wave fluorescence biosensing with cylindrical optical cavities," Appl. Opt. 40, 570-582 (2001).
[CrossRef]

2000 (2)

P. P. Absil, J. V. Hryniewicz, B. E. Little, P. S. Cho, R. A. Wilson, L. G. Joneckis and P.-T. Ho, "Wavelength conversion in GaAs microring resonators," Opt. Lett. 25, 554-556 (2000).
[CrossRef]

A. Yariv, "Universal relations for coupling of optical power between microresonators and dielectric waveguides," Electron. Lett. 36, 321-322 (2000).
[CrossRef]

1999 (3)

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, 1322-1331 (1999).
[CrossRef]

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. J. Khan, C. Manolatou and H. A. Haus, "Theoretical analysis of channel drop tunneling processes," Phys. Rev. B 59, 15882-15892 (1999).
[CrossRef]

M. J. Khan, C. Manolatou, S. Fan, P. R. Villeneuve, H. A. Haus and J. D. Joannopoulos, "Mode-coupling analysis of multipole symmetric resonant add/drop filters," IEEE J. Quantum Electron. 35, 1451-1460 (1999).
[CrossRef]

1997 (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]

B. E. Little, J.-P. Laine and S. T. Chu, "Surface-roughness-induced contradirectional coupling in ring and disk resonators," Opt. Lett. 22, 4-6 (1997).
[CrossRef] [PubMed]

1995 (1)

R. Orta, P. Savi, R. Tascone, and D. Trinchero, "Synthesis of multiple-ring-resonator filters for optical systems," IEEE Photonics Technol. Lett. 7, 1447-1449 (1995).
[CrossRef]

1991 (1)

H. A. Haus and W.-P. Huang, "Coupled-mode theory," inProc. IEEE 79, 1505-1518 (1991).

1989 (1)

Absil, P. P.

Barwicz, T.

Benson, T. M.

Blair, S.

Boriskina, S. V.

Bowers, J. E.

B. Liu, A. Shakouri and J. E. Bowers, "Passive microring-resonator-coupled lasers," Appl. Phys. Lett. 79, 3561-3563 (2001).
[CrossRef]

Chen, Y.

Cho, P. S.

Chu, S. T.

B. E. Little, J.-P. Laine and S. T. Chu, "Surface-roughness-induced contradirectional coupling in ring and disk resonators," Opt. Lett. 22, 4-6 (1997).
[CrossRef] [PubMed]

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]

Fan, S.

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, 1322-1331 (1999).
[CrossRef]

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. J. Khan, C. Manolatou and H. A. Haus, "Theoretical analysis of channel drop tunneling processes," Phys. Rev. B 59, 15882-15892 (1999).
[CrossRef]

M. J. Khan, C. Manolatou, S. Fan, P. R. Villeneuve, H. A. Haus and J. D. Joannopoulos, "Mode-coupling analysis of multipole symmetric resonant add/drop filters," IEEE J. Quantum Electron. 35, 1451-1460 (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]

Haus, H. A.

T. Barwicz, M. A. Popović, P. T. Rakich, M. R. Watts, H. A. Haus, E. P. Ippen and H. I. Smith, "Microring-resonator-based add-drop filters in SiN: fabrication and analysis," Opt. Express 12, 1437-1442 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-7-1437.
[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, 1322-1331 (1999).
[CrossRef]

M. J. Khan, C. Manolatou, S. Fan, P. R. Villeneuve, H. A. Haus and J. D. Joannopoulos, "Mode-coupling analysis of multipole symmetric resonant add/drop filters," IEEE J. Quantum Electron. 35, 1451-1460 (1999).
[CrossRef]

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. J. Khan, C. Manolatou and H. A. Haus, "Theoretical analysis of channel drop tunneling processes," Phys. Rev. B 59, 15882-15892 (1999).
[CrossRef]

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]

H. A. Haus and W.-P. Huang, "Coupled-mode theory," inProc. IEEE 79, 1505-1518 (1991).

H. A. Haus, W. P. Huang and A. W. Snyder, "Coupled-mode formulations," Opt. Lett. 14, 1222-1224 (1989).
[CrossRef] [PubMed]

Ho, P.-T.

Hryniewicz, J. V.

Huang, W. P.

Huang, W.-P.

H. A. Haus and W.-P. Huang, "Coupled-mode theory," inProc. IEEE 79, 1505-1518 (1991).

Ippen, E. P.

Joannopoulos, J. D.

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, 1322-1331 (1999).
[CrossRef]

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. J. Khan, C. Manolatou and H. A. Haus, "Theoretical analysis of channel drop tunneling processes," Phys. Rev. B 59, 15882-15892 (1999).
[CrossRef]

M. J. Khan, C. Manolatou, S. Fan, P. R. Villeneuve, H. A. Haus and J. D. Joannopoulos, "Mode-coupling analysis of multipole symmetric resonant add/drop filters," IEEE J. Quantum Electron. 35, 1451-1460 (1999).
[CrossRef]

Joneckis, L. G.

Khan, M. J.

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. J. Khan, C. Manolatou and H. A. Haus, "Theoretical analysis of channel drop tunneling processes," Phys. Rev. B 59, 15882-15892 (1999).
[CrossRef]

M. J. Khan, C. Manolatou, S. Fan, P. R. Villeneuve, H. A. Haus and J. D. Joannopoulos, "Mode-coupling analysis of multipole symmetric resonant add/drop filters," IEEE J. Quantum Electron. 35, 1451-1460 (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, 1322-1331 (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]

B. E. Little, J.-P. Laine and S. T. Chu, "Surface-roughness-induced contradirectional coupling in ring and disk resonators," Opt. Lett. 22, 4-6 (1997).
[CrossRef] [PubMed]

Lee, M. M.

M. M. Lee and M. C. Wu, "MEMS-actuated microdisk resonators with variable power coupling ratios," IEEE Photon. Technol. Lett. 17, 1034-1036 (2005).
[CrossRef]

Little, B. E.

Liu, B.

B. Liu, A. Shakouri and J. E. Bowers, "Passive microring-resonator-coupled lasers," Appl. Phys. Lett. 79, 3561-3563 (2001).
[CrossRef]

Manolatou, C.

M. J. Khan, C. Manolatou, S. Fan, P. R. Villeneuve, H. A. Haus and J. D. Joannopoulos, "Mode-coupling analysis of multipole symmetric resonant add/drop filters," IEEE J. Quantum Electron. 35, 1451-1460 (1999).
[CrossRef]

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. J. Khan, C. Manolatou and H. A. Haus, "Theoretical analysis of channel drop tunneling processes," Phys. Rev. B 59, 15882-15892 (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, 1322-1331 (1999).
[CrossRef]

Martinelli, M.

A. Melloni and M. Martinelli, "Synthesis of direct-coupled-resonators bandpass filters for WDM systems," J. Lightwave Technol. 20, 296-303 (2002).
[CrossRef]

Melloni, A.

A. Melloni and M. Martinelli, "Synthesis of direct-coupled-resonators bandpass filters for WDM systems," J. Lightwave Technol. 20, 296-303 (2002).
[CrossRef]

Nosich, A. I.

Orta, R.

R. Orta, P. Savi, R. Tascone, and D. Trinchero, "Synthesis of multiple-ring-resonator filters for optical systems," IEEE Photonics Technol. Lett. 7, 1447-1449 (1995).
[CrossRef]

Popovic, M. A.

Rakich, P. T.

Savi, P.

R. Orta, P. Savi, R. Tascone, and D. Trinchero, "Synthesis of multiple-ring-resonator filters for optical systems," IEEE Photonics Technol. Lett. 7, 1447-1449 (1995).
[CrossRef]

Scheuer, J.

Sewell, P.

Shakouri, A.

B. Liu, A. Shakouri and J. E. Bowers, "Passive microring-resonator-coupled lasers," Appl. Phys. Lett. 79, 3561-3563 (2001).
[CrossRef]

Smith, H. I.

Snyder, A. W.

Tascone, R.

R. Orta, P. Savi, R. Tascone, and D. Trinchero, "Synthesis of multiple-ring-resonator filters for optical systems," IEEE Photonics Technol. Lett. 7, 1447-1449 (1995).
[CrossRef]

Trinchero, D.

R. Orta, P. Savi, R. Tascone, and D. Trinchero, "Synthesis of multiple-ring-resonator filters for optical systems," IEEE Photonics Technol. Lett. 7, 1447-1449 (1995).
[CrossRef]

Villeneuve, P. R.

M. J. Khan, C. Manolatou, S. Fan, P. R. Villeneuve, H. A. Haus and J. D. Joannopoulos, "Mode-coupling analysis of multipole symmetric resonant add/drop filters," IEEE J. Quantum Electron. 35, 1451-1460 (1999).
[CrossRef]

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. J. Khan, C. Manolatou and H. A. Haus, "Theoretical analysis of channel drop tunneling processes," Phys. Rev. B 59, 15882-15892 (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, 1322-1331 (1999).
[CrossRef]

Watts, M. R.

Wilson, R. A.

Wu, M. C.

M. M. Lee and M. C. Wu, "MEMS-actuated microdisk resonators with variable power coupling ratios," IEEE Photon. Technol. Lett. 17, 1034-1036 (2005).
[CrossRef]

Yariv, A.

J. Scheuer and A. Yariv, "Two-dimensional optical ring resonators based on radial Bragg resonance," Opt. Lett. 28, 1528-1530 (2003).
[CrossRef] [PubMed]

A. Yariv, "Universal relations for coupling of optical power between microresonators and dielectric waveguides," Electron. Lett. 36, 321-322 (2000).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

B. Liu, A. Shakouri and J. E. Bowers, "Passive microring-resonator-coupled lasers," Appl. Phys. Lett. 79, 3561-3563 (2001).
[CrossRef]

Electron. Lett. (1)

A. Yariv, "Universal relations for coupling of optical power between microresonators and dielectric waveguides," Electron. Lett. 36, 321-322 (2000).
[CrossRef]

IEEE J. Quantum Electron. (2)

M. J. Khan, C. Manolatou, S. Fan, P. R. Villeneuve, H. A. Haus and J. D. Joannopoulos, "Mode-coupling analysis of multipole symmetric resonant add/drop filters," IEEE J. Quantum Electron. 35, 1451-1460 (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, 1322-1331 (1999).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

M. M. Lee and M. C. Wu, "MEMS-actuated microdisk resonators with variable power coupling ratios," IEEE Photon. Technol. Lett. 17, 1034-1036 (2005).
[CrossRef]

IEEE Photonics Technol. Lett. (1)

R. Orta, P. Savi, R. Tascone, and D. Trinchero, "Synthesis of multiple-ring-resonator filters for optical systems," IEEE Photonics Technol. Lett. 7, 1447-1449 (1995).
[CrossRef]

J. Lightwave Technol. (1)

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]

A. Melloni and M. Martinelli, "Synthesis of direct-coupled-resonators bandpass filters for WDM systems," J. Lightwave Technol. 20, 296-303 (2002).
[CrossRef]

Opt. Express (1)

Opt. Lett. (4)

Phys. Rev. B (1)

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. J. Khan, C. Manolatou and H. A. Haus, "Theoretical analysis of channel drop tunneling processes," Phys. Rev. B 59, 15882-15892 (1999).
[CrossRef]

Proc. IEEE (1)

H. A. Haus and W.-P. Huang, "Coupled-mode theory," inProc. IEEE 79, 1505-1518 (1991).

Other (11)

T. Barwicz, M. A. Popović, M. R. Watts, P. T. Rakich, E. P. Ippen and H. I. Smith, "Fabrication of add-drop filters based on frequency-matched microring resonators," submitted to J. Lightwave Technol.

M. Popović, "Complex-frequency leaky mode computations using PML boundary layers for dielectric resonant structures," in Proceedings of Integrated Photonics Research (Washington, DC, June 17, 2003).

A. D. Berk, "Variational principles for electromagnetic resonators and waveguides," IRE Trans. Antennas Propag., April 1956, pp. 104-111.

M. J. Khan, M. Lim, C. Joyner, T. Murphy, H. A. Haus and H. I. Smith, "Integrated Bragg grating structures," in Digest of the LEOS Summer Topical Meeting on WDM Components (Copper Mountain, CO, 2001).

H. A. Haus, B. E. Little, M. A. Popović, S. T. Chu, M. R. Watts and C. Manolatou, "Optical resonators and filters," in Optical Microcavities, K. Vahala, ed. (World Scientific, Singapore, 2004).

H. A. Haus, "Microwaves and Photonics," in OSA TOPS 23 Symposium on Electro-Optics: Present and Future, H.A. Haus, ed., (Optical Society of America, Washington, DC, 1998), pp. 2-8.

M. Lončar, T. Yoshie, Y. Qiu, P. Gogna and A. Scherer, "Low-threshold photonic crystal laser," in Proc. SPIE5000, 16-26 (2003).
[CrossRef]

C. K. Madsen and J. H. Zhao, Optical filter design and analysis: a signal processing approach (Wiley, 1999).

C. Manolatou, M. A. Popović, P. T. Rakich, T. Barwicz, H. A. Haus and E. P. Ippen, "Spectral anomalies due to coupling-induced frequency shifts in dielectric coupled-resonator filters," in Proceedings of Optical Fiber Communication Conference on CD-ROM (Los Angeles, CA, February 2004), TuD5.

M. A. Popović, M. R. Watts, T. Barwicz, P. T. Rakich, L. Socci, E. P. Ippen, F. X. Kärtner and H. I. Smith, "High-index-contrast, wide-FSR microring-resonator filter design and realization with frequency-shift compensation," in Proc. Optical Fiber Comm. Conf. (Optical Society of America, Washington, DC, 2005).

H. A. Haus, Waves and fields in optoelectronics (Prentice-Hall, Englewood Cliffs, NJ, 1984).

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Figures (7)

Fig. 1.
Fig. 1.

(a) Coupled-dielectric-resonator system in infinite perfect-conductor waveguide, with analytic mode solutions [17]; (b) TM10 supermode frequency splitting and CIFS vs.resonator spacing.

Fig. 2.
Fig. 2.

CIFS impairment of 3-cavity microring filter: (a) simulated structure using identical rings (dimensions shown); (b) ideal (synthesized) and simulated (complete-structure FDTD and model) frequency response showing apparent cavity mismatch due to CIFS.

Fig. 3.
Fig. 3.

Diagram of CIFS resonance mismatch and its correction: isolated resonator (a) attains CIFS when part of a coupled structure (b); the combination of nominally degenerate resonators in a coupled-cavity filter (c) results in effectively mismatched resonators and a distorted response (Fig. 2), which can be corrected by pre-distorting the resonators by the expected CIFS

Fig. 4.
Fig. 4.

CIFS in a single-ring cavity due to side-coupled bus waveguides: (a) mode-solver-simulated structure; (b) CIFS for TE, TM resonances and for two bus widths, vs. gap spacing. Phasor-amplitude plots (c), (d) of the standing-wave-mode resonant field show a partial standing wave in the coupling region and, respectively, a symmetric-like and antisymmetric-like supermode formed with the waveguide continuum corresponding to negative- and positive-frequency CIFS.

Fig. 5.
Fig. 5.

CIFS due to coupled bus waveguides in a square standing-wave cavity: (a) CIFS vs. gap spacing and waveguide width, and mode-solver simulated structure. Mode phasor magnitude plots show (b) symmetric and (c) anti-symmetric coupling, respectively, with a partial standing wave established in the waveguide, for negative- and positive-frequency CIFS. Case (b) shows significant mode shape modification caused by coupling.

Fig. 6.
Fig. 6.

(a) Lumped point-interaction model of coupling of traveling-wave resonator and access waveguide. FDTD simulation of the coupled structure with respect to ports 1-4, and of the two waveguides individually, leads to lumped point-interaction matrix U, describing the effect of the coupling on port-to-port scattering matrix element (b) phase, and (c) power coupling.

Fig. 7.
Fig. 7.

FDTD simulation of 3-cavity microring filter, with cavity resonance frequencies pre-compensated by design for CIFS: (a) simulated structure with lower-core-index middle ring; (b) FDTD simulation and model of the filter showing recovered ideal frequency responses.

Equations (15)

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d dt a = j ω ¯ ¯ a j μ ¯ ¯ a
ω s , a = ω 1 μ 11 + ω 2 μ 22 2 ( ( ω 1 μ 11 ) ( ω 2 μ 22 ) 2 ) 2 + μ 12 μ 21 .
d dt a = j ω ̿ a j W ̿ 1 M ̿ . a = j ω ̿ a j μ ̿ a
δ ω 1 = μ 11 = M 11 W 12 W 22 M 21 W 11 W 12 W 22 W 21 .
ω 2 = V E × μ ̿ 1 × E dv V E . ̿ . E dv
Δ f CIFS Δ f FSR Δ f CIFS n g L c Δ ϕ 2 π .
U ̿ u 11 e 11 u 12 e 12 u 21 e 21 u 22 e 22 = e o 1 κ e 1 i κ e 2 i κ e 2 1 κ e 1
ϕ 11 + ϕ 22 = ϕ 21 + ϕ 12 ± π
δ β 1 = K 11 P 12 P 22 K 21 P 11 P 12 P 22 P 21
Δ θ 1 = β 1 z + δ β 1 z + [ δz arg { cos ( β o z ) + j δ β o sin ( β o z ) } ]
CBR cavity cavity BW FSR fixed coupling geometry = 1 finesse
CBR ϕ 11 κ = 1 L eff α R eff fixed BW , FSR
× ε ̿ 1 × H = ω 2 μ ̿ H
W mn ω n ω m V ˜ H m T μ ̿ H n d v ˜
M mn ω n 2 V ˜ D m T Δ ε n ¯ ¯ 1 D n d v ˜

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