Abstract

We investigate the thermal and Kerr nonlinearity in a system of two optically-coupled silica microtoroid resonators experimentally and theoretically. A model for two coupled oscillators describing nonlinear resonance curves is developed. Stability of the static solutions is analyzed. It is shown that thermal nonlinearity is responsible for driving the eigenfrequencies of the two resonators apart, making the normal modes of the system unstable as the pump power grows. The red-detuned normal mode becomes unstable for certain pumping powers.

© 2009 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. V. S. Ilchenko, M. L. Gorodetsky, and S. P. Vyatchanin, “Coupling and tunability of optical whispering-gallery modes: a basis for coordinate meter,” Opt. Commun. 107(1-2), 41–48 (1994).
    [CrossRef]
  2. T. Mukaiyama, K. Takeda, H. Miyazaki, Y. Jimba, and M. Kuwata-Gonokami, “Tight-binding photonic molecule modes of resonant bispheres,” Phys. Rev. Lett. 82(23), 4623–4626 (1999).
    [CrossRef]
  3. A. A. Savchenkov, V. S. Ilchenko, T. Handley, and L. Maleki, “Second-order filter response with series-coupled silica microresonators,” IEEE Photon. Technol. Lett. 15(4), 543–544 (2003).
    [CrossRef]
  4. A. A. Savchenkov, V. S. Ilchenko, A. B. Matsko, and L. Maleki, “High-order tunable filters based on a chain of coupled crystalline whispering gallery-mode resonators,” IEEE Photon. Technol. Lett. 17(1), 136–138 (2005).
    [CrossRef]
  5. D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, and R. W. Boyd, “Coupled-resonator-induced transparency,” Phys. Rev. A 69(6), 063804 (2004).
    [CrossRef]
  6. A. Naweed, G. Farca, S. I. Shopova, and A. T. Rosenberger, “Induced transparency and absorption in coupled whispering-gallery microresonators,” Phys. Rev. A 71(4), 043804 (2005).
    [CrossRef]
  7. D. D. Smith and H. Chang, “Coherence phenomena in coupled optical resonators,” J. Mod. Opt. 51, 2503–2513 (2004).
  8. B. Möller, U. Woggon, and M. V. Artemyev, “Photons in coupled microsphere resonators,” J. Opt. A 8, S113–S121 (2006).
    [CrossRef]
  9. A. V. Kanaev, V. N. Astratov, and W. Cai, “Optical coupling at a distance between detuned spherical cavities,” Appl. Phys. Lett. 88(11), 111111 (2006).
    [CrossRef]
  10. A. Mazzei, S. Götzinger, Lde. S. Menezes, G. Zumofen, O. Benson, and V. Sandoghdar, “Controlled coupling of counterpropagating whispering-gallery modes by a single Rayleigh scatterer: a classical problem in a quantum optical light,” Phys. Rev. Lett. 99(17), 173603 (2007).
    [CrossRef] [PubMed]
  11. M. Benyoucef, S. Kiravittaya, Y. F. Mei, A. Rastelli, and O. G. Schmidt, “Strongly coupled semiconductor microcavities: A route to couple artificial atoms over micrometric distances,” Phys. Rev. B 77(3), 035108 (2008).
    [CrossRef]
  12. A. E. Fomin, M. L. Gorodetsky, I. S. Grudinin, and V. S. Ilchenko, “Nonstationary nonlinear effects in optical microspheres,” J. Opt. Soc. Am. B 22(2), 459–465 (2005).
    [CrossRef]
  13. V. S. Il’chenko and M. L. Gorodetskii, “Thermal nonlinear effects in optical whispering gallery microresonators,” Laser Phys. 2, 1004–1009 (1992).
  14. D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421(6926), 925–928 (2003).
    [CrossRef] [PubMed]
  15. V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, “Quality-factor and nonlinear properties of optical whispering-gallery modes,” Phys. Lett. A 137(7-8), 393–397 (1989).
    [CrossRef]

2008

M. Benyoucef, S. Kiravittaya, Y. F. Mei, A. Rastelli, and O. G. Schmidt, “Strongly coupled semiconductor microcavities: A route to couple artificial atoms over micrometric distances,” Phys. Rev. B 77(3), 035108 (2008).
[CrossRef]

2007

A. Mazzei, S. Götzinger, Lde. S. Menezes, G. Zumofen, O. Benson, and V. Sandoghdar, “Controlled coupling of counterpropagating whispering-gallery modes by a single Rayleigh scatterer: a classical problem in a quantum optical light,” Phys. Rev. Lett. 99(17), 173603 (2007).
[CrossRef] [PubMed]

2006

B. Möller, U. Woggon, and M. V. Artemyev, “Photons in coupled microsphere resonators,” J. Opt. A 8, S113–S121 (2006).
[CrossRef]

A. V. Kanaev, V. N. Astratov, and W. Cai, “Optical coupling at a distance between detuned spherical cavities,” Appl. Phys. Lett. 88(11), 111111 (2006).
[CrossRef]

2005

A. Naweed, G. Farca, S. I. Shopova, and A. T. Rosenberger, “Induced transparency and absorption in coupled whispering-gallery microresonators,” Phys. Rev. A 71(4), 043804 (2005).
[CrossRef]

A. A. Savchenkov, V. S. Ilchenko, A. B. Matsko, and L. Maleki, “High-order tunable filters based on a chain of coupled crystalline whispering gallery-mode resonators,” IEEE Photon. Technol. Lett. 17(1), 136–138 (2005).
[CrossRef]

A. E. Fomin, M. L. Gorodetsky, I. S. Grudinin, and V. S. Ilchenko, “Nonstationary nonlinear effects in optical microspheres,” J. Opt. Soc. Am. B 22(2), 459–465 (2005).
[CrossRef]

2004

D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, and R. W. Boyd, “Coupled-resonator-induced transparency,” Phys. Rev. A 69(6), 063804 (2004).
[CrossRef]

D. D. Smith and H. Chang, “Coherence phenomena in coupled optical resonators,” J. Mod. Opt. 51, 2503–2513 (2004).

2003

A. A. Savchenkov, V. S. Ilchenko, T. Handley, and L. Maleki, “Second-order filter response with series-coupled silica microresonators,” IEEE Photon. Technol. Lett. 15(4), 543–544 (2003).
[CrossRef]

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421(6926), 925–928 (2003).
[CrossRef] [PubMed]

1999

T. Mukaiyama, K. Takeda, H. Miyazaki, Y. Jimba, and M. Kuwata-Gonokami, “Tight-binding photonic molecule modes of resonant bispheres,” Phys. Rev. Lett. 82(23), 4623–4626 (1999).
[CrossRef]

1994

V. S. Ilchenko, M. L. Gorodetsky, and S. P. Vyatchanin, “Coupling and tunability of optical whispering-gallery modes: a basis for coordinate meter,” Opt. Commun. 107(1-2), 41–48 (1994).
[CrossRef]

1992

V. S. Il’chenko and M. L. Gorodetskii, “Thermal nonlinear effects in optical whispering gallery microresonators,” Laser Phys. 2, 1004–1009 (1992).

1989

V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, “Quality-factor and nonlinear properties of optical whispering-gallery modes,” Phys. Lett. A 137(7-8), 393–397 (1989).
[CrossRef]

Armani, D. K.

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421(6926), 925–928 (2003).
[CrossRef] [PubMed]

Artemyev, M. V.

B. Möller, U. Woggon, and M. V. Artemyev, “Photons in coupled microsphere resonators,” J. Opt. A 8, S113–S121 (2006).
[CrossRef]

Astratov, V. N.

A. V. Kanaev, V. N. Astratov, and W. Cai, “Optical coupling at a distance between detuned spherical cavities,” Appl. Phys. Lett. 88(11), 111111 (2006).
[CrossRef]

Benson, O.

A. Mazzei, S. Götzinger, Lde. S. Menezes, G. Zumofen, O. Benson, and V. Sandoghdar, “Controlled coupling of counterpropagating whispering-gallery modes by a single Rayleigh scatterer: a classical problem in a quantum optical light,” Phys. Rev. Lett. 99(17), 173603 (2007).
[CrossRef] [PubMed]

Benyoucef, M.

M. Benyoucef, S. Kiravittaya, Y. F. Mei, A. Rastelli, and O. G. Schmidt, “Strongly coupled semiconductor microcavities: A route to couple artificial atoms over micrometric distances,” Phys. Rev. B 77(3), 035108 (2008).
[CrossRef]

Boyd, R. W.

D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, and R. W. Boyd, “Coupled-resonator-induced transparency,” Phys. Rev. A 69(6), 063804 (2004).
[CrossRef]

Braginsky, V. B.

V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, “Quality-factor and nonlinear properties of optical whispering-gallery modes,” Phys. Lett. A 137(7-8), 393–397 (1989).
[CrossRef]

Cai, W.

A. V. Kanaev, V. N. Astratov, and W. Cai, “Optical coupling at a distance between detuned spherical cavities,” Appl. Phys. Lett. 88(11), 111111 (2006).
[CrossRef]

Chang, H.

D. D. Smith and H. Chang, “Coherence phenomena in coupled optical resonators,” J. Mod. Opt. 51, 2503–2513 (2004).

D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, and R. W. Boyd, “Coupled-resonator-induced transparency,” Phys. Rev. A 69(6), 063804 (2004).
[CrossRef]

Farca, G.

A. Naweed, G. Farca, S. I. Shopova, and A. T. Rosenberger, “Induced transparency and absorption in coupled whispering-gallery microresonators,” Phys. Rev. A 71(4), 043804 (2005).
[CrossRef]

Fomin, A. E.

Fuller, K. A.

D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, and R. W. Boyd, “Coupled-resonator-induced transparency,” Phys. Rev. A 69(6), 063804 (2004).
[CrossRef]

Gorodetskii, M. L.

V. S. Il’chenko and M. L. Gorodetskii, “Thermal nonlinear effects in optical whispering gallery microresonators,” Laser Phys. 2, 1004–1009 (1992).

Gorodetsky, M. L.

A. E. Fomin, M. L. Gorodetsky, I. S. Grudinin, and V. S. Ilchenko, “Nonstationary nonlinear effects in optical microspheres,” J. Opt. Soc. Am. B 22(2), 459–465 (2005).
[CrossRef]

V. S. Ilchenko, M. L. Gorodetsky, and S. P. Vyatchanin, “Coupling and tunability of optical whispering-gallery modes: a basis for coordinate meter,” Opt. Commun. 107(1-2), 41–48 (1994).
[CrossRef]

V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, “Quality-factor and nonlinear properties of optical whispering-gallery modes,” Phys. Lett. A 137(7-8), 393–397 (1989).
[CrossRef]

Götzinger, S.

A. Mazzei, S. Götzinger, Lde. S. Menezes, G. Zumofen, O. Benson, and V. Sandoghdar, “Controlled coupling of counterpropagating whispering-gallery modes by a single Rayleigh scatterer: a classical problem in a quantum optical light,” Phys. Rev. Lett. 99(17), 173603 (2007).
[CrossRef] [PubMed]

Grudinin, I. S.

Handley, T.

A. A. Savchenkov, V. S. Ilchenko, T. Handley, and L. Maleki, “Second-order filter response with series-coupled silica microresonators,” IEEE Photon. Technol. Lett. 15(4), 543–544 (2003).
[CrossRef]

Il’chenko, V. S.

V. S. Il’chenko and M. L. Gorodetskii, “Thermal nonlinear effects in optical whispering gallery microresonators,” Laser Phys. 2, 1004–1009 (1992).

Ilchenko, V. S.

A. E. Fomin, M. L. Gorodetsky, I. S. Grudinin, and V. S. Ilchenko, “Nonstationary nonlinear effects in optical microspheres,” J. Opt. Soc. Am. B 22(2), 459–465 (2005).
[CrossRef]

A. A. Savchenkov, V. S. Ilchenko, A. B. Matsko, and L. Maleki, “High-order tunable filters based on a chain of coupled crystalline whispering gallery-mode resonators,” IEEE Photon. Technol. Lett. 17(1), 136–138 (2005).
[CrossRef]

A. A. Savchenkov, V. S. Ilchenko, T. Handley, and L. Maleki, “Second-order filter response with series-coupled silica microresonators,” IEEE Photon. Technol. Lett. 15(4), 543–544 (2003).
[CrossRef]

V. S. Ilchenko, M. L. Gorodetsky, and S. P. Vyatchanin, “Coupling and tunability of optical whispering-gallery modes: a basis for coordinate meter,” Opt. Commun. 107(1-2), 41–48 (1994).
[CrossRef]

V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, “Quality-factor and nonlinear properties of optical whispering-gallery modes,” Phys. Lett. A 137(7-8), 393–397 (1989).
[CrossRef]

Jimba, Y.

T. Mukaiyama, K. Takeda, H. Miyazaki, Y. Jimba, and M. Kuwata-Gonokami, “Tight-binding photonic molecule modes of resonant bispheres,” Phys. Rev. Lett. 82(23), 4623–4626 (1999).
[CrossRef]

Kanaev, A. V.

A. V. Kanaev, V. N. Astratov, and W. Cai, “Optical coupling at a distance between detuned spherical cavities,” Appl. Phys. Lett. 88(11), 111111 (2006).
[CrossRef]

Kippenberg, T. J.

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421(6926), 925–928 (2003).
[CrossRef] [PubMed]

Kiravittaya, S.

M. Benyoucef, S. Kiravittaya, Y. F. Mei, A. Rastelli, and O. G. Schmidt, “Strongly coupled semiconductor microcavities: A route to couple artificial atoms over micrometric distances,” Phys. Rev. B 77(3), 035108 (2008).
[CrossRef]

Kuwata-Gonokami, M.

T. Mukaiyama, K. Takeda, H. Miyazaki, Y. Jimba, and M. Kuwata-Gonokami, “Tight-binding photonic molecule modes of resonant bispheres,” Phys. Rev. Lett. 82(23), 4623–4626 (1999).
[CrossRef]

Maleki, L.

A. A. Savchenkov, V. S. Ilchenko, A. B. Matsko, and L. Maleki, “High-order tunable filters based on a chain of coupled crystalline whispering gallery-mode resonators,” IEEE Photon. Technol. Lett. 17(1), 136–138 (2005).
[CrossRef]

A. A. Savchenkov, V. S. Ilchenko, T. Handley, and L. Maleki, “Second-order filter response with series-coupled silica microresonators,” IEEE Photon. Technol. Lett. 15(4), 543–544 (2003).
[CrossRef]

Matsko, A. B.

A. A. Savchenkov, V. S. Ilchenko, A. B. Matsko, and L. Maleki, “High-order tunable filters based on a chain of coupled crystalline whispering gallery-mode resonators,” IEEE Photon. Technol. Lett. 17(1), 136–138 (2005).
[CrossRef]

Mazzei, A.

A. Mazzei, S. Götzinger, Lde. S. Menezes, G. Zumofen, O. Benson, and V. Sandoghdar, “Controlled coupling of counterpropagating whispering-gallery modes by a single Rayleigh scatterer: a classical problem in a quantum optical light,” Phys. Rev. Lett. 99(17), 173603 (2007).
[CrossRef] [PubMed]

Mei, Y. F.

M. Benyoucef, S. Kiravittaya, Y. F. Mei, A. Rastelli, and O. G. Schmidt, “Strongly coupled semiconductor microcavities: A route to couple artificial atoms over micrometric distances,” Phys. Rev. B 77(3), 035108 (2008).
[CrossRef]

Menezes, Lde. S.

A. Mazzei, S. Götzinger, Lde. S. Menezes, G. Zumofen, O. Benson, and V. Sandoghdar, “Controlled coupling of counterpropagating whispering-gallery modes by a single Rayleigh scatterer: a classical problem in a quantum optical light,” Phys. Rev. Lett. 99(17), 173603 (2007).
[CrossRef] [PubMed]

Miyazaki, H.

T. Mukaiyama, K. Takeda, H. Miyazaki, Y. Jimba, and M. Kuwata-Gonokami, “Tight-binding photonic molecule modes of resonant bispheres,” Phys. Rev. Lett. 82(23), 4623–4626 (1999).
[CrossRef]

Möller, B.

B. Möller, U. Woggon, and M. V. Artemyev, “Photons in coupled microsphere resonators,” J. Opt. A 8, S113–S121 (2006).
[CrossRef]

Mukaiyama, T.

T. Mukaiyama, K. Takeda, H. Miyazaki, Y. Jimba, and M. Kuwata-Gonokami, “Tight-binding photonic molecule modes of resonant bispheres,” Phys. Rev. Lett. 82(23), 4623–4626 (1999).
[CrossRef]

Naweed, A.

A. Naweed, G. Farca, S. I. Shopova, and A. T. Rosenberger, “Induced transparency and absorption in coupled whispering-gallery microresonators,” Phys. Rev. A 71(4), 043804 (2005).
[CrossRef]

Rastelli, A.

M. Benyoucef, S. Kiravittaya, Y. F. Mei, A. Rastelli, and O. G. Schmidt, “Strongly coupled semiconductor microcavities: A route to couple artificial atoms over micrometric distances,” Phys. Rev. B 77(3), 035108 (2008).
[CrossRef]

Rosenberger, A. T.

A. Naweed, G. Farca, S. I. Shopova, and A. T. Rosenberger, “Induced transparency and absorption in coupled whispering-gallery microresonators,” Phys. Rev. A 71(4), 043804 (2005).
[CrossRef]

D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, and R. W. Boyd, “Coupled-resonator-induced transparency,” Phys. Rev. A 69(6), 063804 (2004).
[CrossRef]

Sandoghdar, V.

A. Mazzei, S. Götzinger, Lde. S. Menezes, G. Zumofen, O. Benson, and V. Sandoghdar, “Controlled coupling of counterpropagating whispering-gallery modes by a single Rayleigh scatterer: a classical problem in a quantum optical light,” Phys. Rev. Lett. 99(17), 173603 (2007).
[CrossRef] [PubMed]

Savchenkov, A. A.

A. A. Savchenkov, V. S. Ilchenko, A. B. Matsko, and L. Maleki, “High-order tunable filters based on a chain of coupled crystalline whispering gallery-mode resonators,” IEEE Photon. Technol. Lett. 17(1), 136–138 (2005).
[CrossRef]

A. A. Savchenkov, V. S. Ilchenko, T. Handley, and L. Maleki, “Second-order filter response with series-coupled silica microresonators,” IEEE Photon. Technol. Lett. 15(4), 543–544 (2003).
[CrossRef]

Schmidt, O. G.

M. Benyoucef, S. Kiravittaya, Y. F. Mei, A. Rastelli, and O. G. Schmidt, “Strongly coupled semiconductor microcavities: A route to couple artificial atoms over micrometric distances,” Phys. Rev. B 77(3), 035108 (2008).
[CrossRef]

Shopova, S. I.

A. Naweed, G. Farca, S. I. Shopova, and A. T. Rosenberger, “Induced transparency and absorption in coupled whispering-gallery microresonators,” Phys. Rev. A 71(4), 043804 (2005).
[CrossRef]

Smith, D. D.

D. D. Smith and H. Chang, “Coherence phenomena in coupled optical resonators,” J. Mod. Opt. 51, 2503–2513 (2004).

D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, and R. W. Boyd, “Coupled-resonator-induced transparency,” Phys. Rev. A 69(6), 063804 (2004).
[CrossRef]

Spillane, S. M.

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421(6926), 925–928 (2003).
[CrossRef] [PubMed]

Takeda, K.

T. Mukaiyama, K. Takeda, H. Miyazaki, Y. Jimba, and M. Kuwata-Gonokami, “Tight-binding photonic molecule modes of resonant bispheres,” Phys. Rev. Lett. 82(23), 4623–4626 (1999).
[CrossRef]

Vahala, K. J.

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421(6926), 925–928 (2003).
[CrossRef] [PubMed]

Vyatchanin, S. P.

V. S. Ilchenko, M. L. Gorodetsky, and S. P. Vyatchanin, “Coupling and tunability of optical whispering-gallery modes: a basis for coordinate meter,” Opt. Commun. 107(1-2), 41–48 (1994).
[CrossRef]

Woggon, U.

B. Möller, U. Woggon, and M. V. Artemyev, “Photons in coupled microsphere resonators,” J. Opt. A 8, S113–S121 (2006).
[CrossRef]

Zumofen, G.

A. Mazzei, S. Götzinger, Lde. S. Menezes, G. Zumofen, O. Benson, and V. Sandoghdar, “Controlled coupling of counterpropagating whispering-gallery modes by a single Rayleigh scatterer: a classical problem in a quantum optical light,” Phys. Rev. Lett. 99(17), 173603 (2007).
[CrossRef] [PubMed]

Appl. Phys. Lett.

A. V. Kanaev, V. N. Astratov, and W. Cai, “Optical coupling at a distance between detuned spherical cavities,” Appl. Phys. Lett. 88(11), 111111 (2006).
[CrossRef]

IEEE Photon. Technol. Lett.

A. A. Savchenkov, V. S. Ilchenko, T. Handley, and L. Maleki, “Second-order filter response with series-coupled silica microresonators,” IEEE Photon. Technol. Lett. 15(4), 543–544 (2003).
[CrossRef]

A. A. Savchenkov, V. S. Ilchenko, A. B. Matsko, and L. Maleki, “High-order tunable filters based on a chain of coupled crystalline whispering gallery-mode resonators,” IEEE Photon. Technol. Lett. 17(1), 136–138 (2005).
[CrossRef]

J. Mod. Opt.

D. D. Smith and H. Chang, “Coherence phenomena in coupled optical resonators,” J. Mod. Opt. 51, 2503–2513 (2004).

J. Opt. A

B. Möller, U. Woggon, and M. V. Artemyev, “Photons in coupled microsphere resonators,” J. Opt. A 8, S113–S121 (2006).
[CrossRef]

J. Opt. Soc. Am. B

Laser Phys.

V. S. Il’chenko and M. L. Gorodetskii, “Thermal nonlinear effects in optical whispering gallery microresonators,” Laser Phys. 2, 1004–1009 (1992).

Nature

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421(6926), 925–928 (2003).
[CrossRef] [PubMed]

Opt. Commun.

V. S. Ilchenko, M. L. Gorodetsky, and S. P. Vyatchanin, “Coupling and tunability of optical whispering-gallery modes: a basis for coordinate meter,” Opt. Commun. 107(1-2), 41–48 (1994).
[CrossRef]

Phys. Lett. A

V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, “Quality-factor and nonlinear properties of optical whispering-gallery modes,” Phys. Lett. A 137(7-8), 393–397 (1989).
[CrossRef]

Phys. Rev. A

D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, and R. W. Boyd, “Coupled-resonator-induced transparency,” Phys. Rev. A 69(6), 063804 (2004).
[CrossRef]

A. Naweed, G. Farca, S. I. Shopova, and A. T. Rosenberger, “Induced transparency and absorption in coupled whispering-gallery microresonators,” Phys. Rev. A 71(4), 043804 (2005).
[CrossRef]

Phys. Rev. B

M. Benyoucef, S. Kiravittaya, Y. F. Mei, A. Rastelli, and O. G. Schmidt, “Strongly coupled semiconductor microcavities: A route to couple artificial atoms over micrometric distances,” Phys. Rev. B 77(3), 035108 (2008).
[CrossRef]

Phys. Rev. Lett.

A. Mazzei, S. Götzinger, Lde. S. Menezes, G. Zumofen, O. Benson, and V. Sandoghdar, “Controlled coupling of counterpropagating whispering-gallery modes by a single Rayleigh scatterer: a classical problem in a quantum optical light,” Phys. Rev. Lett. 99(17), 173603 (2007).
[CrossRef] [PubMed]

T. Mukaiyama, K. Takeda, H. Miyazaki, Y. Jimba, and M. Kuwata-Gonokami, “Tight-binding photonic molecule modes of resonant bispheres,” Phys. Rev. Lett. 82(23), 4623–4626 (1999).
[CrossRef]

Supplementary Material (1)

» Media 1: AVI (326 KB)     

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1.
Fig. 1.

Schematic and a photograph of the typical coupled toroid system.

Fig. 2.
Fig. 2.

Schematic of the experimental setup

Fig. 3.
Fig. 3.

A: Transmission spectra of coupled resonators recorded for several air gap values. B: Static solutions provided by the model for several values of optical coupling constant. Optical Q used in the model is around 107. The asymmetry in the lowest trace was modeled by setting the eigenfrequencies of each resonator slightly mismatched.

Fig. 4.
Fig. 4.

A: Transmission spectra of coupled resonators recorded as the temperature of one resonator was gradually changing leading to the eigenfrequency shift and avoided crossing. Degenerate splitting is set at 40 MHz. The horizontal shift is due to unintended cross-heating of the resonators by the other resonator’s heating element. B: Static solutions obtained theoretically for conditions similar to the experiment.

Fig. 5.
Fig. 5.

Experimental transmission spectra and static solutions at low power. Optical pump power is below the nonlinearity threshold and the resulting resonance curve is represented by a double peak. A: transmission spectra from the experiment. B: total optical power circulating in the coupled resonator system. Pump power is 1μW. C: power in resonator A, D: power in resonator B. The partial modes (supermodes) are denoted by Ψ+ and Ψ-.

Fig. 6.
Fig. 6.

Experimental transmission spectra and static solutions at moderate power. With further increases in optical pump power (relative to Fig. 5) the red supermode becomes unstable, causing reversion to a transmission spectrum corresponding to a single resonator. A: transmission spectrum from the experiment. B: total optical power circulating in the coupled resonator system. Pump power is 4μW. C: power in resonator A, D: power in resonator B. The partial modes (supermodes) are denoted by Ψ+ and Ψ-.

Fig. 7.
Fig. 7.

Experimental transmission spectra and static solutions at high power. Under this condition the red supermode may become unstable and impossible to observe in the experiment. In this particular example the Q factor of resonator A is lower than that of resonator B. A: transmission spectrum from the experiment. B: total optical power circulating in the coupled resonator system. Pump power is 20μW. C: power in resonator A, D: power in resonator B. The partial modes (supermodes) are denoted by Ψ+ and Ψ-.

Fig. 8.
Fig. 8.

Nonlinear resonance curve of coupled resonator system for increasing pump power values. Note the sharp transition around 0.9μW pump power (Media 1)

Tables (1)

Tables Icon

Table 1. Parameters used in computation of resonance curve in Fig. 8.

Equations (9)

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

E(r,t)=a(t) E0 (r) eiωt .
E02dr=1.
{a˙+a(γa+i[ωωa+μaa2+ωaβaθa])=iF+ikbb˙+b(γb+i[ωωb+μbb2+ωbβbθb])=ikaθ˙a+δθaθa=vaδθaωaβaa2θ˙b+δθbθb=vbδθbωbβbb2
τ3=πRr2ln(R/R0)dD,
{a˙1=a1γa+a2(ωωa+μaa2+ωaβaθa)kb2a˙2=a2γaa1(ωωa+μaa2+ωaβaθa)+F+kb1b˙1=b1γb+b2(ωωb+μbb2+ωbβbθb)ka2.b˙2=b2γbb1(ωωb+μbb2+ωbβbθb)ka1θ˙a=δθaθa+vaδθaa2/(ωaβa)θ˙b=δθbθb+vbδθbb2/(ωbβb)
a1=y1ac,a2=y2ac,b1=y3bc,b2=y4bc,t=τtc,θa=y5θca,θb=y6θcbandΔωa,b=ωωa,b
{y˙1=y1+y2Δωaγa+y2y12+y23+y2y5y4kbcγaacy˙2=y2y1Δωaγay13y1y22y1y5+Fμaγa3+y3kbcγaacy˙3=y3+y4Δωbγb+y4y32+y43+y4y6y2kbcγbac.y˙4=y4y3Δωbγby33y3y43y3y6+y1kbcγbacy˙5=δθay5/γa+vaδθa(y12+y22)/(γaμa)y˙6=δθby6/γb+vbδθb(y12+y22)/(γbμb)
Pcirc=(y12+y22) ac2 ncVa(2π)2Ra ×107[Watt].
A=2y20y101Δa+2y2020kbcγaacy200(Δa+2y102)(1+2y10y20)kbcγaac0y1000kacγbac2y40y301Δb+2y400y40kacγbac0(Δb+2y302)(1+2y40y30)0y302vaδθaγaμay102vaδθaγaμay2000δθaγa0002vaδθbγbμby302vbδθbγbμby400δθbγb.Here

Metrics