Abstract

A new model is developed for the effects of the Kerr optical nonlinearity in a microring (racetrack) resonator coupled to input and output waveguides, which takes into account the nonlinearity in the couplers as well as the microring sections. It is shown how the nonlinear microring add-drop filter can be used as a self-adjusting all-optical beam splitter to extend the dynamic range of photodetectors and protect optoelectronic circuitry from high input powers.

© 2008 Optical Society of America

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References

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  1. J. E. Heebner and R. W. Boyd, "Enhanced all-optical switching by use of a nonlinear fiber ring resonator," Opt. Lett. 24, 847-849 (1999).
    [CrossRef]
  2. V. Van, T. Ibrahim, K. Ritter, P. Absil, F. Johnson, R. Grover, J. Goldhar, and P.-T. Ho, "All-optical nonlinear switching in GaAs-AlGaAs microring resonators," IEEE Photon. Technol. Lett. 14, 74-76 (2002).
    [CrossRef]
  3. V. Van, T. Ibrahim, P. Absil, F. Johnson, R. Grover, and P.-T. Ho, "Optical signal processing using nonlinear semiconductor microring resonators," IEEE J. Sel. Top. Quantum Electron. 8, 705-713 (2002).
    [CrossRef]
  4. S. Pereira, P. Chak, and J. E. Sipe, "All-optical AND gate by use of a Kerr nonlinear microresonator structure," Opt. Lett. 28, 444-446 (2003).
    [CrossRef] [PubMed]
  5. J. E. Heebner and R.W. Boyd, "Strong Dispersive and Nonlinear Optical Properties of Microresonator-Modified Optical Waveguides," Proc. SPIE 4969, 185-194 (2003).
    [CrossRef]
  6. V. Van, P. P. Absil, J. V. Hryniewicz, and P. T. Ho, "Propagation loss in single mode GaAs-AlGaAs microring resonators: measurement and model," J. Lightwave Technol. 19, 1734-1739 (2001).
    [CrossRef]
  7. S. M. Jensen, "The nonlinear directional coupler," IEEE Trans. Microwave Theory Tech. MTT-30, 1568-1571 (1982).
    [CrossRef]
  8. A. Yariv, Optical Electronics in Modern Communications, 5th ed. (Oxford, New York, 1997).
  9. S. Blair, J. E. Heebner, and R. W. Boyd, "Beyond the absorption-limited nonlinear phase shift with microring resonators," Opt. Lett. 27, 357-359 (2002).
    [CrossRef]
  10. H. Nakatsuka, S. Asaka, H. Itoh, K. Ikeda, and M. Matsuoka, "Observation of Bifurcation to Chaos in an All-Optical Bistable System," Phys. Rev. Lett. 50, 109-112 (1983).
    [CrossRef]
  11. K. Ikeda and M. Mizuno, "Frustrated Instabilities in Nonlinear Optical Resonators," Phys. Rev. Lett. 53, 1340-1343 (1984).
    [CrossRef]
  12. B. Crosignani, B. Daino, P. D. Porto, and S. Wabnitz, "Optical multistability in a fiber-optic passive-loop resonator," Opt. Commun. 59, 309-312 (1986).
    [CrossRef]
  13. A. Yariv, "Universal relations for coupling of optical power between microresonators and dielectric waveguides," Electron. Lett. 36, 321-322 (2000).
    [CrossRef]
  14. P. Li Kam Wa, J. E. Stitch, J. J. Mason, J. S. Roberts, and P. N. Robson, "All optical multiple quantum well waveguide switch," Electron. Lett. 21, 26-28 (1985).
    [CrossRef]
  15. A. Villeneuve, C. C. Yang, P. G. J. Wigley, G. I. Stegeman, J. S. Aitchison, and C. N. Ironside, "Ultrafast alloptical switching in semiconductor nonlinear directional couplers at half the band gap," Appl. Phys. Lett. 61, 147-149 (1992).
    [CrossRef]
  16. A. D. Bristow, R. Iyer, J. S. Aitchison, H. M. van Driel, and A. L. Smirl, "Switchable Al[sub x]Ga[sub 1 - x]As all-optical delay line at 1.55 ?m," Appl. Phys. Lett. 90, 101112 (2007).
    [CrossRef]

2007

A. D. Bristow, R. Iyer, J. S. Aitchison, H. M. van Driel, and A. L. Smirl, "Switchable Al[sub x]Ga[sub 1 - x]As all-optical delay line at 1.55 ?m," Appl. Phys. Lett. 90, 101112 (2007).
[CrossRef]

2003

S. Pereira, P. Chak, and J. E. Sipe, "All-optical AND gate by use of a Kerr nonlinear microresonator structure," Opt. Lett. 28, 444-446 (2003).
[CrossRef] [PubMed]

J. E. Heebner and R.W. Boyd, "Strong Dispersive and Nonlinear Optical Properties of Microresonator-Modified Optical Waveguides," Proc. SPIE 4969, 185-194 (2003).
[CrossRef]

2002

V. Van, T. Ibrahim, K. Ritter, P. Absil, F. Johnson, R. Grover, J. Goldhar, and P.-T. Ho, "All-optical nonlinear switching in GaAs-AlGaAs microring resonators," IEEE Photon. Technol. Lett. 14, 74-76 (2002).
[CrossRef]

V. Van, T. Ibrahim, P. Absil, F. Johnson, R. Grover, and P.-T. Ho, "Optical signal processing using nonlinear semiconductor microring resonators," IEEE J. Sel. Top. Quantum Electron. 8, 705-713 (2002).
[CrossRef]

S. Blair, J. E. Heebner, and R. W. Boyd, "Beyond the absorption-limited nonlinear phase shift with microring resonators," Opt. Lett. 27, 357-359 (2002).
[CrossRef]

2001

2000

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

1999

1992

A. Villeneuve, C. C. Yang, P. G. J. Wigley, G. I. Stegeman, J. S. Aitchison, and C. N. Ironside, "Ultrafast alloptical switching in semiconductor nonlinear directional couplers at half the band gap," Appl. Phys. Lett. 61, 147-149 (1992).
[CrossRef]

1986

B. Crosignani, B. Daino, P. D. Porto, and S. Wabnitz, "Optical multistability in a fiber-optic passive-loop resonator," Opt. Commun. 59, 309-312 (1986).
[CrossRef]

1985

P. Li Kam Wa, J. E. Stitch, J. J. Mason, J. S. Roberts, and P. N. Robson, "All optical multiple quantum well waveguide switch," Electron. Lett. 21, 26-28 (1985).
[CrossRef]

1984

K. Ikeda and M. Mizuno, "Frustrated Instabilities in Nonlinear Optical Resonators," Phys. Rev. Lett. 53, 1340-1343 (1984).
[CrossRef]

1983

H. Nakatsuka, S. Asaka, H. Itoh, K. Ikeda, and M. Matsuoka, "Observation of Bifurcation to Chaos in an All-Optical Bistable System," Phys. Rev. Lett. 50, 109-112 (1983).
[CrossRef]

1982

S. M. Jensen, "The nonlinear directional coupler," IEEE Trans. Microwave Theory Tech. MTT-30, 1568-1571 (1982).
[CrossRef]

Absil, P.

V. Van, T. Ibrahim, K. Ritter, P. Absil, F. Johnson, R. Grover, J. Goldhar, and P.-T. Ho, "All-optical nonlinear switching in GaAs-AlGaAs microring resonators," IEEE Photon. Technol. Lett. 14, 74-76 (2002).
[CrossRef]

V. Van, T. Ibrahim, P. Absil, F. Johnson, R. Grover, and P.-T. Ho, "Optical signal processing using nonlinear semiconductor microring resonators," IEEE J. Sel. Top. Quantum Electron. 8, 705-713 (2002).
[CrossRef]

Absil, P. P.

Aitchison, J. S.

A. D. Bristow, R. Iyer, J. S. Aitchison, H. M. van Driel, and A. L. Smirl, "Switchable Al[sub x]Ga[sub 1 - x]As all-optical delay line at 1.55 ?m," Appl. Phys. Lett. 90, 101112 (2007).
[CrossRef]

A. Villeneuve, C. C. Yang, P. G. J. Wigley, G. I. Stegeman, J. S. Aitchison, and C. N. Ironside, "Ultrafast alloptical switching in semiconductor nonlinear directional couplers at half the band gap," Appl. Phys. Lett. 61, 147-149 (1992).
[CrossRef]

Asaka, S.

H. Nakatsuka, S. Asaka, H. Itoh, K. Ikeda, and M. Matsuoka, "Observation of Bifurcation to Chaos in an All-Optical Bistable System," Phys. Rev. Lett. 50, 109-112 (1983).
[CrossRef]

Blair, S.

Boyd, R. W.

Boyd, R.W.

J. E. Heebner and R.W. Boyd, "Strong Dispersive and Nonlinear Optical Properties of Microresonator-Modified Optical Waveguides," Proc. SPIE 4969, 185-194 (2003).
[CrossRef]

Bristow, A. D.

A. D. Bristow, R. Iyer, J. S. Aitchison, H. M. van Driel, and A. L. Smirl, "Switchable Al[sub x]Ga[sub 1 - x]As all-optical delay line at 1.55 ?m," Appl. Phys. Lett. 90, 101112 (2007).
[CrossRef]

Chak, P.

Crosignani, B.

B. Crosignani, B. Daino, P. D. Porto, and S. Wabnitz, "Optical multistability in a fiber-optic passive-loop resonator," Opt. Commun. 59, 309-312 (1986).
[CrossRef]

Daino, B.

B. Crosignani, B. Daino, P. D. Porto, and S. Wabnitz, "Optical multistability in a fiber-optic passive-loop resonator," Opt. Commun. 59, 309-312 (1986).
[CrossRef]

Goldhar, J.

V. Van, T. Ibrahim, K. Ritter, P. Absil, F. Johnson, R. Grover, J. Goldhar, and P.-T. Ho, "All-optical nonlinear switching in GaAs-AlGaAs microring resonators," IEEE Photon. Technol. Lett. 14, 74-76 (2002).
[CrossRef]

Grover, R.

V. Van, T. Ibrahim, K. Ritter, P. Absil, F. Johnson, R. Grover, J. Goldhar, and P.-T. Ho, "All-optical nonlinear switching in GaAs-AlGaAs microring resonators," IEEE Photon. Technol. Lett. 14, 74-76 (2002).
[CrossRef]

V. Van, T. Ibrahim, P. Absil, F. Johnson, R. Grover, and P.-T. Ho, "Optical signal processing using nonlinear semiconductor microring resonators," IEEE J. Sel. Top. Quantum Electron. 8, 705-713 (2002).
[CrossRef]

Heebner, J. E.

Ho, P. T.

Ho, P.-T.

V. Van, T. Ibrahim, P. Absil, F. Johnson, R. Grover, and P.-T. Ho, "Optical signal processing using nonlinear semiconductor microring resonators," IEEE J. Sel. Top. Quantum Electron. 8, 705-713 (2002).
[CrossRef]

V. Van, T. Ibrahim, K. Ritter, P. Absil, F. Johnson, R. Grover, J. Goldhar, and P.-T. Ho, "All-optical nonlinear switching in GaAs-AlGaAs microring resonators," IEEE Photon. Technol. Lett. 14, 74-76 (2002).
[CrossRef]

Hryniewicz, J. V.

Ibrahim, T.

V. Van, T. Ibrahim, K. Ritter, P. Absil, F. Johnson, R. Grover, J. Goldhar, and P.-T. Ho, "All-optical nonlinear switching in GaAs-AlGaAs microring resonators," IEEE Photon. Technol. Lett. 14, 74-76 (2002).
[CrossRef]

V. Van, T. Ibrahim, P. Absil, F. Johnson, R. Grover, and P.-T. Ho, "Optical signal processing using nonlinear semiconductor microring resonators," IEEE J. Sel. Top. Quantum Electron. 8, 705-713 (2002).
[CrossRef]

Ikeda, K.

K. Ikeda and M. Mizuno, "Frustrated Instabilities in Nonlinear Optical Resonators," Phys. Rev. Lett. 53, 1340-1343 (1984).
[CrossRef]

H. Nakatsuka, S. Asaka, H. Itoh, K. Ikeda, and M. Matsuoka, "Observation of Bifurcation to Chaos in an All-Optical Bistable System," Phys. Rev. Lett. 50, 109-112 (1983).
[CrossRef]

Ironside, C. N.

A. Villeneuve, C. C. Yang, P. G. J. Wigley, G. I. Stegeman, J. S. Aitchison, and C. N. Ironside, "Ultrafast alloptical switching in semiconductor nonlinear directional couplers at half the band gap," Appl. Phys. Lett. 61, 147-149 (1992).
[CrossRef]

Itoh, H.

H. Nakatsuka, S. Asaka, H. Itoh, K. Ikeda, and M. Matsuoka, "Observation of Bifurcation to Chaos in an All-Optical Bistable System," Phys. Rev. Lett. 50, 109-112 (1983).
[CrossRef]

Iyer, R.

A. D. Bristow, R. Iyer, J. S. Aitchison, H. M. van Driel, and A. L. Smirl, "Switchable Al[sub x]Ga[sub 1 - x]As all-optical delay line at 1.55 ?m," Appl. Phys. Lett. 90, 101112 (2007).
[CrossRef]

Jensen, S. M.

S. M. Jensen, "The nonlinear directional coupler," IEEE Trans. Microwave Theory Tech. MTT-30, 1568-1571 (1982).
[CrossRef]

Johnson, F.

V. Van, T. Ibrahim, P. Absil, F. Johnson, R. Grover, and P.-T. Ho, "Optical signal processing using nonlinear semiconductor microring resonators," IEEE J. Sel. Top. Quantum Electron. 8, 705-713 (2002).
[CrossRef]

V. Van, T. Ibrahim, K. Ritter, P. Absil, F. Johnson, R. Grover, J. Goldhar, and P.-T. Ho, "All-optical nonlinear switching in GaAs-AlGaAs microring resonators," IEEE Photon. Technol. Lett. 14, 74-76 (2002).
[CrossRef]

Li Kam Wa, P.

P. Li Kam Wa, J. E. Stitch, J. J. Mason, J. S. Roberts, and P. N. Robson, "All optical multiple quantum well waveguide switch," Electron. Lett. 21, 26-28 (1985).
[CrossRef]

Mason, J. J.

P. Li Kam Wa, J. E. Stitch, J. J. Mason, J. S. Roberts, and P. N. Robson, "All optical multiple quantum well waveguide switch," Electron. Lett. 21, 26-28 (1985).
[CrossRef]

Matsuoka, M.

H. Nakatsuka, S. Asaka, H. Itoh, K. Ikeda, and M. Matsuoka, "Observation of Bifurcation to Chaos in an All-Optical Bistable System," Phys. Rev. Lett. 50, 109-112 (1983).
[CrossRef]

Mizuno, M.

K. Ikeda and M. Mizuno, "Frustrated Instabilities in Nonlinear Optical Resonators," Phys. Rev. Lett. 53, 1340-1343 (1984).
[CrossRef]

Nakatsuka, H.

H. Nakatsuka, S. Asaka, H. Itoh, K. Ikeda, and M. Matsuoka, "Observation of Bifurcation to Chaos in an All-Optical Bistable System," Phys. Rev. Lett. 50, 109-112 (1983).
[CrossRef]

Pereira, S.

Porto, P. D.

B. Crosignani, B. Daino, P. D. Porto, and S. Wabnitz, "Optical multistability in a fiber-optic passive-loop resonator," Opt. Commun. 59, 309-312 (1986).
[CrossRef]

Ritter, K.

V. Van, T. Ibrahim, K. Ritter, P. Absil, F. Johnson, R. Grover, J. Goldhar, and P.-T. Ho, "All-optical nonlinear switching in GaAs-AlGaAs microring resonators," IEEE Photon. Technol. Lett. 14, 74-76 (2002).
[CrossRef]

Roberts, J. S.

P. Li Kam Wa, J. E. Stitch, J. J. Mason, J. S. Roberts, and P. N. Robson, "All optical multiple quantum well waveguide switch," Electron. Lett. 21, 26-28 (1985).
[CrossRef]

Robson, P. N.

P. Li Kam Wa, J. E. Stitch, J. J. Mason, J. S. Roberts, and P. N. Robson, "All optical multiple quantum well waveguide switch," Electron. Lett. 21, 26-28 (1985).
[CrossRef]

Sipe, J. E.

Smirl, A. L.

A. D. Bristow, R. Iyer, J. S. Aitchison, H. M. van Driel, and A. L. Smirl, "Switchable Al[sub x]Ga[sub 1 - x]As all-optical delay line at 1.55 ?m," Appl. Phys. Lett. 90, 101112 (2007).
[CrossRef]

Stegeman, G. I.

A. Villeneuve, C. C. Yang, P. G. J. Wigley, G. I. Stegeman, J. S. Aitchison, and C. N. Ironside, "Ultrafast alloptical switching in semiconductor nonlinear directional couplers at half the band gap," Appl. Phys. Lett. 61, 147-149 (1992).
[CrossRef]

Stitch, J. E.

P. Li Kam Wa, J. E. Stitch, J. J. Mason, J. S. Roberts, and P. N. Robson, "All optical multiple quantum well waveguide switch," Electron. Lett. 21, 26-28 (1985).
[CrossRef]

Van, V.

V. Van, T. Ibrahim, K. Ritter, P. Absil, F. Johnson, R. Grover, J. Goldhar, and P.-T. Ho, "All-optical nonlinear switching in GaAs-AlGaAs microring resonators," IEEE Photon. Technol. Lett. 14, 74-76 (2002).
[CrossRef]

V. Van, T. Ibrahim, P. Absil, F. Johnson, R. Grover, and P.-T. Ho, "Optical signal processing using nonlinear semiconductor microring resonators," IEEE J. Sel. Top. Quantum Electron. 8, 705-713 (2002).
[CrossRef]

V. Van, P. P. Absil, J. V. Hryniewicz, and P. T. Ho, "Propagation loss in single mode GaAs-AlGaAs microring resonators: measurement and model," J. Lightwave Technol. 19, 1734-1739 (2001).
[CrossRef]

van Driel, H. M.

A. D. Bristow, R. Iyer, J. S. Aitchison, H. M. van Driel, and A. L. Smirl, "Switchable Al[sub x]Ga[sub 1 - x]As all-optical delay line at 1.55 ?m," Appl. Phys. Lett. 90, 101112 (2007).
[CrossRef]

Villeneuve, A.

A. Villeneuve, C. C. Yang, P. G. J. Wigley, G. I. Stegeman, J. S. Aitchison, and C. N. Ironside, "Ultrafast alloptical switching in semiconductor nonlinear directional couplers at half the band gap," Appl. Phys. Lett. 61, 147-149 (1992).
[CrossRef]

Wabnitz, S.

B. Crosignani, B. Daino, P. D. Porto, and S. Wabnitz, "Optical multistability in a fiber-optic passive-loop resonator," Opt. Commun. 59, 309-312 (1986).
[CrossRef]

Wigley, P. G. J.

A. Villeneuve, C. C. Yang, P. G. J. Wigley, G. I. Stegeman, J. S. Aitchison, and C. N. Ironside, "Ultrafast alloptical switching in semiconductor nonlinear directional couplers at half the band gap," Appl. Phys. Lett. 61, 147-149 (1992).
[CrossRef]

Yang, C. C.

A. Villeneuve, C. C. Yang, P. G. J. Wigley, G. I. Stegeman, J. S. Aitchison, and C. N. Ironside, "Ultrafast alloptical switching in semiconductor nonlinear directional couplers at half the band gap," Appl. Phys. Lett. 61, 147-149 (1992).
[CrossRef]

Yariv, A.

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

Appl. Phys. Lett.

A. Villeneuve, C. C. Yang, P. G. J. Wigley, G. I. Stegeman, J. S. Aitchison, and C. N. Ironside, "Ultrafast alloptical switching in semiconductor nonlinear directional couplers at half the band gap," Appl. Phys. Lett. 61, 147-149 (1992).
[CrossRef]

A. D. Bristow, R. Iyer, J. S. Aitchison, H. M. van Driel, and A. L. Smirl, "Switchable Al[sub x]Ga[sub 1 - x]As all-optical delay line at 1.55 ?m," Appl. Phys. Lett. 90, 101112 (2007).
[CrossRef]

Electron. Lett.

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

P. Li Kam Wa, J. E. Stitch, J. J. Mason, J. S. Roberts, and P. N. Robson, "All optical multiple quantum well waveguide switch," Electron. Lett. 21, 26-28 (1985).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

V. Van, T. Ibrahim, P. Absil, F. Johnson, R. Grover, and P.-T. Ho, "Optical signal processing using nonlinear semiconductor microring resonators," IEEE J. Sel. Top. Quantum Electron. 8, 705-713 (2002).
[CrossRef]

IEEE Photon. Technol. Lett.

V. Van, T. Ibrahim, K. Ritter, P. Absil, F. Johnson, R. Grover, J. Goldhar, and P.-T. Ho, "All-optical nonlinear switching in GaAs-AlGaAs microring resonators," IEEE Photon. Technol. Lett. 14, 74-76 (2002).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

S. M. Jensen, "The nonlinear directional coupler," IEEE Trans. Microwave Theory Tech. MTT-30, 1568-1571 (1982).
[CrossRef]

J. Lightwave Technol.

Opt. Commun.

B. Crosignani, B. Daino, P. D. Porto, and S. Wabnitz, "Optical multistability in a fiber-optic passive-loop resonator," Opt. Commun. 59, 309-312 (1986).
[CrossRef]

Opt. Lett.

Phys. Rev. Lett.

H. Nakatsuka, S. Asaka, H. Itoh, K. Ikeda, and M. Matsuoka, "Observation of Bifurcation to Chaos in an All-Optical Bistable System," Phys. Rev. Lett. 50, 109-112 (1983).
[CrossRef]

K. Ikeda and M. Mizuno, "Frustrated Instabilities in Nonlinear Optical Resonators," Phys. Rev. Lett. 53, 1340-1343 (1984).
[CrossRef]

Proc. SPIE

J. E. Heebner and R.W. Boyd, "Strong Dispersive and Nonlinear Optical Properties of Microresonator-Modified Optical Waveguides," Proc. SPIE 4969, 185-194 (2003).
[CrossRef]

Other

A. Yariv, Optical Electronics in Modern Communications, 5th ed. (Oxford, New York, 1997).

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Figures (5)

Fig. 1.
Fig. 1.

(a) A microring add-drop filter comprises two coupler sections (C 1 and C 2) and two half-ring sections (R 1 and R 2). A photoreceiver labeled “D” is located at the drop port. (b) Linear detection over a wide dynamic range may be achieved by cascading a number of such sections. The input light automatically split among one (D 1) or several (D 1,D 2,…) photodiodes, depending on whether the input intensity is low or high.

Fig. 2.
Fig. 2.

Calculation of the drop and through transfer functions of the microring add-drop filter. The directional coupler is labeled at the input and output planes with the field amplitudes in the two waveguides, whose evolution along the length of the coupler is described by Eqs. (1a)(1b).

Fig. 3.
Fig. 3.

The dropped power fraction (solid blue line) and through-transmitted power fraction (dashed red line) at the resonance wavelength. At low intensities, all the light is diverted to the drop port whereas at higher intensities, a greater fraction of the light remains in the input waveguide and reaches the through port. Two-photon absorption eventually dominates at high intensities.

Fig. 4.
Fig. 4.

Accounting for Kerr nonlinearity in the add-drop filter shows that the power available at the drop port is much more limited (blue line) than predicted by the linear model (black line). In the nonlinear case, the remaining power which is not dropped is shown by the shaded region, and is available at the through port for detection by subsequent detectors, as shown in Fig. 1(b). Above 100 MW/cm2, the “critical power” of the directional coupler [7] is exceeded, and the output oscillates but does not exceed the dotted blue line. The additional power that can be detected by the subsequent detectors represents a gain in receiver dynamic range upto 40 dB over a single detector.

Fig. 5.
Fig. 5.

Comparison of the drop coefficient vs. input intensity, as predicted by the fully-nonlinear model (blue) with a simplified model (red) which assumes that the ring is nonlinear, but the coupler is linear. Ignoring the nonlinearity of the coupler overestimates the switching intensity threshold by 30 dB. (Both models account for two-photon absorption.)

Equations (5)

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

i a z = M a a + κ ab b + ( Q 3 a 2 + 2 Q 4 b 2 ) a + i α 2 a + i β 2 2 a 2 a ,
i b z = M b b + κ ba a + ( Q 3 b 2 + 2 Q 4 a 2 ) b + i α 2 b + i β 2 2 b 2 b ,
b 2 b 1 = [ exp ( α L hr ) 1 + 2 k 0 n 2 K b 1 2 L hr eff ] 1 2 e i ( k L hr + ϕ hr ) ,
ϕ hr 1 2 K log [ 1 + 2 k 0 n 2 K b 1 2 L hr eff ] , L hr eff [ 1 exp ( α L hr ) ] α ,
where K β 2 2 k 0 n 2 , k 2 π n λ , k 0 2 π λ .

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