Abstract

A method is developed for extracting the coupling and loss coefficients of ring resonators from the peak widths, depths, and spacings of the resonances of a single resonator. Although the formulas used do not distinguish which coefficient is coupling and which is loss, it is shown how these coefficients can be disentangled based on how they vary with wavelength or device parameters.

© 2009 Optical Society of America

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References

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  1. D. Po, S.F. Preble, and M. Lipson, “All-optical compact silicon comb switch,” Opt. Express 15, 9600–9605 (2007)
    [CrossRef]
  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]
  3. D.-X. Xu, A. Densmore, A. Delâge, P. Waldron, R. McKinnon, S. Janz, J. Lapointe, G. Lopinski, T. Mischki, E. Post, P. Cheben, and J. H. Schmid, “Folded cavity SOI microring sensors for high sensitivity and real time measurement of biomolecular binding,” Opt. Express 16, 15137–15148 (2008).
    [CrossRef] [PubMed]
  4. D.G. Rabus, Integrated Ring Resonators: the Compedium (Springer-Verlag, Berlin2007).
  5. J. E. Heebner, V. Wong, A. Schweinsberg, R. W. Boyd, and D. J. Jackson, “Optical Transmission Characteristics of Fiber Ring Resonators,” IEEE J. Quantum Electron. 40726–370 (2004).
    [CrossRef]
  6. F. Xia, L. Sekaric, and Y. A. Vlasov, “Mode conversion losses in silicon-on-insulator photonic wire based racetrack insulators,” Opt. Express 14, 3872–3886 (2006).
    [CrossRef] [PubMed]
  7. J. Scheuer, “Fabrication and Characterization of Low-Loss PolymericWaveguides and Micro-Resonators,” in Integrated Photonics and Nanophotonics Research and Applications, OSA Technical Digest (CD) (Optical Society of America, 2007), paper ITuA3.
  8. J. Niehusmann, A. Vorckel, P. H. Bolivar, T. Wahlbrink, W. Henschel, and H. Kurz, “Ultrahigh-quality-factor silicon-on-insulator microring resonator,” Opt. Lett. 29, 2861–2863 (2004).
    [CrossRef]
  9. A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36, 321–322 (2000).
    [CrossRef]
  10. A. Delâge, D.-X. Xu, W. R. McKinnon, E. Post, P. Waldron, J. Lapointe, C. Storey, A. Densmore, S. Janz, B. Lamontagne, P. Cheben, and J. H. Schmid, “Wavelength Dependent Model of a Ring Resonator Sensor Excited by a Directional Coupler,” J. Lightwave Tech. 27, 1172–80 (2009).
    [CrossRef]
  11. G. Gupta, Y.-H. Kuo, H. Tazawa, W. Steier, A. Stapleton, and J. O’Brien, “Analysis and Demonstration of Coupling Control in Polymer Microring Resonators Using Photobleaching,” Appl. Opt. (to be published).
  12. L. F. Stokes, M. Chodorow, and H. J. Shaw, “All-single-mode fiber resonator,” Opt. Lett. 7288–290 (1982).
    [CrossRef] [PubMed]
  13. R. Loudon, The quantum theory of light, 3rd ed (Oxford University Press, Oxford2000).
  14. D.-X. Xu, S. Janz, and P. Cheben, “Design of Polarization-Insensitive Ring Resonators in Silicon-on-Insulator Using MMI Couplers and Cladding Stress Engineering,” IEEE Photon. Tech. Lett. 18, 343–345 (2006).
    [CrossRef]
  15. A. Densmore and D.-X. Xu, unpublished results.

2009 (1)

A. Delâge, D.-X. Xu, W. R. McKinnon, E. Post, P. Waldron, J. Lapointe, C. Storey, A. Densmore, S. Janz, B. Lamontagne, P. Cheben, and J. H. Schmid, “Wavelength Dependent Model of a Ring Resonator Sensor Excited by a Directional Coupler,” J. Lightwave Tech. 27, 1172–80 (2009).
[CrossRef]

2008 (1)

2007 (1)

2006 (2)

F. Xia, L. Sekaric, and Y. A. Vlasov, “Mode conversion losses in silicon-on-insulator photonic wire based racetrack insulators,” Opt. Express 14, 3872–3886 (2006).
[CrossRef] [PubMed]

D.-X. Xu, S. Janz, and P. Cheben, “Design of Polarization-Insensitive Ring Resonators in Silicon-on-Insulator Using MMI Couplers and Cladding Stress Engineering,” IEEE Photon. Tech. Lett. 18, 343–345 (2006).
[CrossRef]

2004 (2)

J. E. Heebner, V. Wong, A. Schweinsberg, R. W. Boyd, and D. J. Jackson, “Optical Transmission Characteristics of Fiber Ring Resonators,” IEEE J. Quantum Electron. 40726–370 (2004).
[CrossRef]

J. Niehusmann, A. Vorckel, P. H. Bolivar, T. Wahlbrink, W. Henschel, and H. Kurz, “Ultrahigh-quality-factor silicon-on-insulator microring resonator,” Opt. Lett. 29, 2861–2863 (2004).
[CrossRef]

2000 (1)

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

1997 (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]

1982 (1)

Bolivar, P. H.

Boyd, R. W.

J. E. Heebner, V. Wong, A. Schweinsberg, R. W. Boyd, and D. J. Jackson, “Optical Transmission Characteristics of Fiber Ring Resonators,” IEEE J. Quantum Electron. 40726–370 (2004).
[CrossRef]

Cheben, P.

A. Delâge, D.-X. Xu, W. R. McKinnon, E. Post, P. Waldron, J. Lapointe, C. Storey, A. Densmore, S. Janz, B. Lamontagne, P. Cheben, and J. H. Schmid, “Wavelength Dependent Model of a Ring Resonator Sensor Excited by a Directional Coupler,” J. Lightwave Tech. 27, 1172–80 (2009).
[CrossRef]

D.-X. Xu, A. Densmore, A. Delâge, P. Waldron, R. McKinnon, S. Janz, J. Lapointe, G. Lopinski, T. Mischki, E. Post, P. Cheben, and J. H. Schmid, “Folded cavity SOI microring sensors for high sensitivity and real time measurement of biomolecular binding,” Opt. Express 16, 15137–15148 (2008).
[CrossRef] [PubMed]

D.-X. Xu, S. Janz, and P. Cheben, “Design of Polarization-Insensitive Ring Resonators in Silicon-on-Insulator Using MMI Couplers and Cladding Stress Engineering,” IEEE Photon. Tech. Lett. 18, 343–345 (2006).
[CrossRef]

Chodorow, M.

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]

Delâge, A.

A. Delâge, D.-X. Xu, W. R. McKinnon, E. Post, P. Waldron, J. Lapointe, C. Storey, A. Densmore, S. Janz, B. Lamontagne, P. Cheben, and J. H. Schmid, “Wavelength Dependent Model of a Ring Resonator Sensor Excited by a Directional Coupler,” J. Lightwave Tech. 27, 1172–80 (2009).
[CrossRef]

D.-X. Xu, A. Densmore, A. Delâge, P. Waldron, R. McKinnon, S. Janz, J. Lapointe, G. Lopinski, T. Mischki, E. Post, P. Cheben, and J. H. Schmid, “Folded cavity SOI microring sensors for high sensitivity and real time measurement of biomolecular binding,” Opt. Express 16, 15137–15148 (2008).
[CrossRef] [PubMed]

Densmore, A.

A. Delâge, D.-X. Xu, W. R. McKinnon, E. Post, P. Waldron, J. Lapointe, C. Storey, A. Densmore, S. Janz, B. Lamontagne, P. Cheben, and J. H. Schmid, “Wavelength Dependent Model of a Ring Resonator Sensor Excited by a Directional Coupler,” J. Lightwave Tech. 27, 1172–80 (2009).
[CrossRef]

D.-X. Xu, A. Densmore, A. Delâge, P. Waldron, R. McKinnon, S. Janz, J. Lapointe, G. Lopinski, T. Mischki, E. Post, P. Cheben, and J. H. Schmid, “Folded cavity SOI microring sensors for high sensitivity and real time measurement of biomolecular binding,” Opt. Express 16, 15137–15148 (2008).
[CrossRef] [PubMed]

A. Densmore and D.-X. Xu, unpublished results.

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]

Gupta, G.

G. Gupta, Y.-H. Kuo, H. Tazawa, W. Steier, A. Stapleton, and J. O’Brien, “Analysis and Demonstration of Coupling Control in Polymer Microring Resonators Using Photobleaching,” Appl. Opt. (to be published).

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, V. Wong, A. Schweinsberg, R. W. Boyd, and D. J. Jackson, “Optical Transmission Characteristics of Fiber Ring Resonators,” IEEE J. Quantum Electron. 40726–370 (2004).
[CrossRef]

Henschel, W.

Jackson, D. J.

J. E. Heebner, V. Wong, A. Schweinsberg, R. W. Boyd, and D. J. Jackson, “Optical Transmission Characteristics of Fiber Ring Resonators,” IEEE J. Quantum Electron. 40726–370 (2004).
[CrossRef]

Janz, S.

A. Delâge, D.-X. Xu, W. R. McKinnon, E. Post, P. Waldron, J. Lapointe, C. Storey, A. Densmore, S. Janz, B. Lamontagne, P. Cheben, and J. H. Schmid, “Wavelength Dependent Model of a Ring Resonator Sensor Excited by a Directional Coupler,” J. Lightwave Tech. 27, 1172–80 (2009).
[CrossRef]

D.-X. Xu, A. Densmore, A. Delâge, P. Waldron, R. McKinnon, S. Janz, J. Lapointe, G. Lopinski, T. Mischki, E. Post, P. Cheben, and J. H. Schmid, “Folded cavity SOI microring sensors for high sensitivity and real time measurement of biomolecular binding,” Opt. Express 16, 15137–15148 (2008).
[CrossRef] [PubMed]

D.-X. Xu, S. Janz, and P. Cheben, “Design of Polarization-Insensitive Ring Resonators in Silicon-on-Insulator Using MMI Couplers and Cladding Stress Engineering,” IEEE Photon. Tech. Lett. 18, 343–345 (2006).
[CrossRef]

Kuo, Y.-H.

G. Gupta, Y.-H. Kuo, H. Tazawa, W. Steier, A. Stapleton, and J. O’Brien, “Analysis and Demonstration of Coupling Control in Polymer Microring Resonators Using Photobleaching,” Appl. Opt. (to be published).

Kurz, H.

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]

Lamontagne, B.

A. Delâge, D.-X. Xu, W. R. McKinnon, E. Post, P. Waldron, J. Lapointe, C. Storey, A. Densmore, S. Janz, B. Lamontagne, P. Cheben, and J. H. Schmid, “Wavelength Dependent Model of a Ring Resonator Sensor Excited by a Directional Coupler,” J. Lightwave Tech. 27, 1172–80 (2009).
[CrossRef]

Lapointe, J.

A. Delâge, D.-X. Xu, W. R. McKinnon, E. Post, P. Waldron, J. Lapointe, C. Storey, A. Densmore, S. Janz, B. Lamontagne, P. Cheben, and J. H. Schmid, “Wavelength Dependent Model of a Ring Resonator Sensor Excited by a Directional Coupler,” J. Lightwave Tech. 27, 1172–80 (2009).
[CrossRef]

D.-X. Xu, A. Densmore, A. Delâge, P. Waldron, R. McKinnon, S. Janz, J. Lapointe, G. Lopinski, T. Mischki, E. Post, P. Cheben, and J. H. Schmid, “Folded cavity SOI microring sensors for high sensitivity and real time measurement of biomolecular binding,” Opt. Express 16, 15137–15148 (2008).
[CrossRef] [PubMed]

Lipson, M.

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]

Lopinski, G.

Loudon, R.

R. Loudon, The quantum theory of light, 3rd ed (Oxford University Press, Oxford2000).

McKinnon, R.

McKinnon, W. R.

A. Delâge, D.-X. Xu, W. R. McKinnon, E. Post, P. Waldron, J. Lapointe, C. Storey, A. Densmore, S. Janz, B. Lamontagne, P. Cheben, and J. H. Schmid, “Wavelength Dependent Model of a Ring Resonator Sensor Excited by a Directional Coupler,” J. Lightwave Tech. 27, 1172–80 (2009).
[CrossRef]

Mischki, T.

Niehusmann, J.

O’Brien, J.

G. Gupta, Y.-H. Kuo, H. Tazawa, W. Steier, A. Stapleton, and J. O’Brien, “Analysis and Demonstration of Coupling Control in Polymer Microring Resonators Using Photobleaching,” Appl. Opt. (to be published).

Po, D.

Post, E.

A. Delâge, D.-X. Xu, W. R. McKinnon, E. Post, P. Waldron, J. Lapointe, C. Storey, A. Densmore, S. Janz, B. Lamontagne, P. Cheben, and J. H. Schmid, “Wavelength Dependent Model of a Ring Resonator Sensor Excited by a Directional Coupler,” J. Lightwave Tech. 27, 1172–80 (2009).
[CrossRef]

D.-X. Xu, A. Densmore, A. Delâge, P. Waldron, R. McKinnon, S. Janz, J. Lapointe, G. Lopinski, T. Mischki, E. Post, P. Cheben, and J. H. Schmid, “Folded cavity SOI microring sensors for high sensitivity and real time measurement of biomolecular binding,” Opt. Express 16, 15137–15148 (2008).
[CrossRef] [PubMed]

Preble, S.F.

Rabus, D.G.

D.G. Rabus, Integrated Ring Resonators: the Compedium (Springer-Verlag, Berlin2007).

Scheuer, J.

J. Scheuer, “Fabrication and Characterization of Low-Loss PolymericWaveguides and Micro-Resonators,” in Integrated Photonics and Nanophotonics Research and Applications, OSA Technical Digest (CD) (Optical Society of America, 2007), paper ITuA3.

Schmid, J. H.

A. Delâge, D.-X. Xu, W. R. McKinnon, E. Post, P. Waldron, J. Lapointe, C. Storey, A. Densmore, S. Janz, B. Lamontagne, P. Cheben, and J. H. Schmid, “Wavelength Dependent Model of a Ring Resonator Sensor Excited by a Directional Coupler,” J. Lightwave Tech. 27, 1172–80 (2009).
[CrossRef]

D.-X. Xu, A. Densmore, A. Delâge, P. Waldron, R. McKinnon, S. Janz, J. Lapointe, G. Lopinski, T. Mischki, E. Post, P. Cheben, and J. H. Schmid, “Folded cavity SOI microring sensors for high sensitivity and real time measurement of biomolecular binding,” Opt. Express 16, 15137–15148 (2008).
[CrossRef] [PubMed]

Schweinsberg, A.

J. E. Heebner, V. Wong, A. Schweinsberg, R. W. Boyd, and D. J. Jackson, “Optical Transmission Characteristics of Fiber Ring Resonators,” IEEE J. Quantum Electron. 40726–370 (2004).
[CrossRef]

Sekaric, L.

Shaw, H. J.

Stapleton, A.

G. Gupta, Y.-H. Kuo, H. Tazawa, W. Steier, A. Stapleton, and J. O’Brien, “Analysis and Demonstration of Coupling Control in Polymer Microring Resonators Using Photobleaching,” Appl. Opt. (to be published).

Steier, W.

G. Gupta, Y.-H. Kuo, H. Tazawa, W. Steier, A. Stapleton, and J. O’Brien, “Analysis and Demonstration of Coupling Control in Polymer Microring Resonators Using Photobleaching,” Appl. Opt. (to be published).

Stokes, L. F.

Storey, C.

A. Delâge, D.-X. Xu, W. R. McKinnon, E. Post, P. Waldron, J. Lapointe, C. Storey, A. Densmore, S. Janz, B. Lamontagne, P. Cheben, and J. H. Schmid, “Wavelength Dependent Model of a Ring Resonator Sensor Excited by a Directional Coupler,” J. Lightwave Tech. 27, 1172–80 (2009).
[CrossRef]

Tazawa, H.

G. Gupta, Y.-H. Kuo, H. Tazawa, W. Steier, A. Stapleton, and J. O’Brien, “Analysis and Demonstration of Coupling Control in Polymer Microring Resonators Using Photobleaching,” Appl. Opt. (to be published).

Vlasov, Y. A.

Vorckel, A.

Wahlbrink, T.

Waldron, P.

A. Delâge, D.-X. Xu, W. R. McKinnon, E. Post, P. Waldron, J. Lapointe, C. Storey, A. Densmore, S. Janz, B. Lamontagne, P. Cheben, and J. H. Schmid, “Wavelength Dependent Model of a Ring Resonator Sensor Excited by a Directional Coupler,” J. Lightwave Tech. 27, 1172–80 (2009).
[CrossRef]

D.-X. Xu, A. Densmore, A. Delâge, P. Waldron, R. McKinnon, S. Janz, J. Lapointe, G. Lopinski, T. Mischki, E. Post, P. Cheben, and J. H. Schmid, “Folded cavity SOI microring sensors for high sensitivity and real time measurement of biomolecular binding,” Opt. Express 16, 15137–15148 (2008).
[CrossRef] [PubMed]

Wong, V.

J. E. Heebner, V. Wong, A. Schweinsberg, R. W. Boyd, and D. J. Jackson, “Optical Transmission Characteristics of Fiber Ring Resonators,” IEEE J. Quantum Electron. 40726–370 (2004).
[CrossRef]

Xia, F.

Xu, D.-X.

A. Delâge, D.-X. Xu, W. R. McKinnon, E. Post, P. Waldron, J. Lapointe, C. Storey, A. Densmore, S. Janz, B. Lamontagne, P. Cheben, and J. H. Schmid, “Wavelength Dependent Model of a Ring Resonator Sensor Excited by a Directional Coupler,” J. Lightwave Tech. 27, 1172–80 (2009).
[CrossRef]

D.-X. Xu, A. Densmore, A. Delâge, P. Waldron, R. McKinnon, S. Janz, J. Lapointe, G. Lopinski, T. Mischki, E. Post, P. Cheben, and J. H. Schmid, “Folded cavity SOI microring sensors for high sensitivity and real time measurement of biomolecular binding,” Opt. Express 16, 15137–15148 (2008).
[CrossRef] [PubMed]

D.-X. Xu, S. Janz, and P. Cheben, “Design of Polarization-Insensitive Ring Resonators in Silicon-on-Insulator Using MMI Couplers and Cladding Stress Engineering,” IEEE Photon. Tech. Lett. 18, 343–345 (2006).
[CrossRef]

A. Densmore and D.-X. Xu, unpublished results.

Yariv, A.

A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36, 321–322 (2000).
[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. (1)

J. E. Heebner, V. Wong, A. Schweinsberg, R. W. Boyd, and D. J. Jackson, “Optical Transmission Characteristics of Fiber Ring Resonators,” IEEE J. Quantum Electron. 40726–370 (2004).
[CrossRef]

IEEE Photon. Tech. Lett. (1)

D.-X. Xu, S. Janz, and P. Cheben, “Design of Polarization-Insensitive Ring Resonators in Silicon-on-Insulator Using MMI Couplers and Cladding Stress Engineering,” IEEE Photon. Tech. Lett. 18, 343–345 (2006).
[CrossRef]

J. Lightwave Tech. (1)

A. Delâge, D.-X. Xu, W. R. McKinnon, E. Post, P. Waldron, J. Lapointe, C. Storey, A. Densmore, S. Janz, B. Lamontagne, P. Cheben, and J. H. Schmid, “Wavelength Dependent Model of a Ring Resonator Sensor Excited by a Directional Coupler,” J. Lightwave Tech. 27, 1172–80 (2009).
[CrossRef]

J. Lightwave Technol. (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]

Opt. Express (3)

Opt. Lett. (2)

Other (5)

D.G. Rabus, Integrated Ring Resonators: the Compedium (Springer-Verlag, Berlin2007).

A. Densmore and D.-X. Xu, unpublished results.

G. Gupta, Y.-H. Kuo, H. Tazawa, W. Steier, A. Stapleton, and J. O’Brien, “Analysis and Demonstration of Coupling Control in Polymer Microring Resonators Using Photobleaching,” Appl. Opt. (to be published).

J. Scheuer, “Fabrication and Characterization of Low-Loss PolymericWaveguides and Micro-Resonators,” in Integrated Photonics and Nanophotonics Research and Applications, OSA Technical Digest (CD) (Optical Society of America, 2007), paper ITuA3.

R. Loudon, The quantum theory of light, 3rd ed (Oxford University Press, Oxford2000).

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Figures (9)

Fig. 1.
Fig. 1.

Ring resonator with a directional coupler (DC): (a) schematic of the DC-coupled resonator and (b) expanded view of the coupler, showing the notation used in the text for the fields (a, b, a′, and b), the self-coupling coefficients tc and t c , the cross-coupling coefficients κc and κ c , and the transmission t r around the ring.

Fig. 2.
Fig. 2.

Resonance spectrum for a folded resonator (shown schematically in the insert at the lower right) with cavity length 367.08 µm, coupler length 5 µm, and nominal separation 0.3 µm of the waveguides in the coupler. The minimum radius of curvature is 5 µm. The upper inserts show three expanded views over a 1-nm range centered at 1471 nm, 1522 nm, and 1571 nm.

Fig. 3.
Fig. 3.

Inverse of the extinction ratio ��=T max/T min (open squares) and inverse of the finesse ��λ FSRλ FWHM (closed circles), for the spectrum shown in Fig. 2.

Fig. 4.
Fig. 4.

Self-coupling coefficient t (open squares) and loss coefficient α (closed circles) for the resonator spectrum in Fig. 2. The solid line through the squares shows a fit to Eq. (27), with a 0=-5.52 and a 1=4.14 µm-1.

Fig. 5.
Fig. 5.

Resonance spectrum for a folded resonator (shown schematically in the insert on the lower right) with cavity length 1112.7 µm, and the same coupler dimensions and minimum radius of curvature as the resonator in Fig. 2. The insert on the lower left shows an expanded view in a 1-nm range about 1515 nm.

Fig. 6.
Fig. 6.

Self-coupling coefficient t (open squares) and loss coefficient α (closed circles) for the resonator spectrum in Fig. 5. The solid line through the squares shows a fit to Eq. (27), with a 0=-5.90 and a 1=4.40 µm-1.

Fig. 7.
Fig. 7.

Resonance spectrum for a resonator with racetrack geometry (shown schematically in the insert) with radius 50 µm, cavity length 334.16 µm, coupler length 10 µm, and nominal separation 0.2 µm of the waveguides in the coupler.

Fig. 8.
Fig. 8.

Self-coupling coefficient t (open squares) and loss coefficient α (closed circles) for the resonator spectrum in Fig. 7. The solid line through the squares shows a fit to Eq. (27), with a 0=-15.05 and a 1=11.61 µm-1.

Fig. 9.
Fig. 9.

(a) Resonance spectrum for six racetrack resonators. The three values in parentheses in each panel give the values of radius R, coupler length Lc , and separation Sc between the waveguides, all in micrometers. These devices were not covered with SU-8 photoresist. (b) Self-coupling coefficient t (open squares) and loss coefficient α (closed circles) for the resonator spectra in Fig. 9. The solid line through the squares shows fits to the expression in Eq. (27). From left to right, then top to bottom, the values of a 0 are -1.96, -6.03, -4.49, -1.86, -2.99, and -3.31; the values of a 1 are 1.44, 4.27, 3.22, 1.31, 2.15, and 2.33 µm-1.

Equations (28)

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

b = tc a + κc a ;
b = tc a + κc a .
a = tr b .
b=tc(tctcκcκc)tra.
b2 + b2 = αc2 a2 + αc2 a2 .
tc2 + κc2 = αc2 ;
tc2 + κc2 = αc2 ;
tc* κc + κc* tc = 0 .
tc tc κc κc = (tctc*+κcκc*)tctc*=αc2tctc*.
tr = trer;
tc = tcec ,
t tcαc;
κ κc/αc;
α trαc;
ϕ ϕc+ϕr.
ba = (tαe1αte)tctc*αcec.
T ba2 = tctc*2 αc2 (t2+α22αtcosϕ1+α2t22αtcosϕ)=tctc*2αc2 𝒯 ,
𝒯 (t2+α22αtcosϕ1+α2t22αtcosϕ).
𝓕 Δ λFSR / Δ λFWHM ,
𝓔 Tmax Tmin .
𝓔 = [(α+t)(αt)(1αt)(1+αt)]2 ;
cos (π𝓕)=2αt1+α2t2.
A cos(π𝓕)1+sin(π𝓕) ;
B 1 [1cos(π𝓕)1+cos(π𝓕)] 1𝓔 .
(α,t) = (AB)12 ± (ABA)12 .
t = cos(πLc2Lπ).
t cos(a0+a1λ),
αt=2AB1B1𝓔,

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