Abstract

Thermal nonlinearity is known to cause bistability in Whispering Gallery Mode (WGM) resonators and to destabilize the red slope of the Lorentzian resonant curve. We demonstrate an optical technique that allows compensation of the thermal effect and forces the resonances to appear linear with both red and blue slopes stable.

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References

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  1. K. J. Vahala, “Optical microcavities,” Nature 424(6950), 839–846 (2003).
    [CrossRef] [PubMed]
  2. S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A 71(1), 013817 (2005).
    [CrossRef]
  3. T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature 443(7112), 671–674 (2006).
    [CrossRef] [PubMed]
  4. T. J. Kippenberg and K. J. Vahala, “Cavity opto-mechanics,” Opt. Express 15(25), 17172–17205 (2007).
    [CrossRef] [PubMed]
  5. V. S. Ilchenko and A. B. Matsko, “Optical resonators with whispering-gallery modes-part II: Applications,” IEEE J. Sel. Top. Quantum Electron. 12(1), 15–32 (2006).
    [CrossRef]
  6. S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature 415(6872), 621–623 (2002).
    [CrossRef] [PubMed]
  7. T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity,” Phys. Rev. Lett. 93(8), 083904 (2004).
    [CrossRef] [PubMed]
  8. F. Vollmer and S. Arnold, “Whispering-gallery-mode biosensing: label-free detection down to single molecules,” Nat. Methods 5(7), 591–596 (2008).
    [CrossRef] [PubMed]
  9. V. S. Ilchenko and M. L. Gorodetskii, “Thermal nonlinear effects in optical whispering gallery microresonators,” Laser Phys. 2, 1004–1009 (1992).
  10. T. Carmon, L. Yang, and K. J. Vahala, “Dynamical thermal behavior and thermal self-stability of microcavities,” Opt. Express 12(20), 4742–4750 (2004).
    [CrossRef] [PubMed]
  11. L. He, Y.-F. Xiao, C. Dong, J. Zhu, V. Gaddam, and L. Yang, “Compensation of thermal refraction effect in high-Q toroidal microresonator by polydimethylsiloxane coating,” Appl. Phys. Lett. 93(20), 201102 (2008).
    [CrossRef]
  12. H. S. Choi and A. M. Armani, “Thermal nonlinear effects in hybrid optical microresonators,” Appl. Phys. Lett. 97(22), 223306 (2010).
    [CrossRef]
  13. I. S. Grudinin and K. J. Vahala, “Thermal instability of a compound resonator,” Opt. Express 17(16), 14088–14097 (2009).
    [CrossRef] [PubMed]
  14. 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]
  15. A. C. Liu, M. J. F. Digonnet, and G. S. Kino, “Measurement of the dc Kerr and electrostrictive phase modulation in silica,” J. Opt. Soc. Am. B 18(2), 187–194 (2001).
    [CrossRef]
  16. M. Cai, O. Painter, and K. J. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system,” Phys. Rev. Lett. 85(1), 74–77 (2000).
    [CrossRef] [PubMed]
  17. S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91(4), 043902 (2003).
    [CrossRef] [PubMed]

2010

H. S. Choi and A. M. Armani, “Thermal nonlinear effects in hybrid optical microresonators,” Appl. Phys. Lett. 97(22), 223306 (2010).
[CrossRef]

2009

2008

L. He, Y.-F. Xiao, C. Dong, J. Zhu, V. Gaddam, and L. Yang, “Compensation of thermal refraction effect in high-Q toroidal microresonator by polydimethylsiloxane coating,” Appl. Phys. Lett. 93(20), 201102 (2008).
[CrossRef]

F. Vollmer and S. Arnold, “Whispering-gallery-mode biosensing: label-free detection down to single molecules,” Nat. Methods 5(7), 591–596 (2008).
[CrossRef] [PubMed]

2007

2006

V. S. Ilchenko and A. B. Matsko, “Optical resonators with whispering-gallery modes-part II: Applications,” IEEE J. Sel. Top. Quantum Electron. 12(1), 15–32 (2006).
[CrossRef]

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature 443(7112), 671–674 (2006).
[CrossRef] [PubMed]

2005

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A 71(1), 013817 (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

T. Carmon, L. Yang, and K. J. Vahala, “Dynamical thermal behavior and thermal self-stability of microcavities,” Opt. Express 12(20), 4742–4750 (2004).
[CrossRef] [PubMed]

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity,” Phys. Rev. Lett. 93(8), 083904 (2004).
[CrossRef] [PubMed]

2003

K. J. Vahala, “Optical microcavities,” Nature 424(6950), 839–846 (2003).
[CrossRef] [PubMed]

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91(4), 043902 (2003).
[CrossRef] [PubMed]

2002

S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature 415(6872), 621–623 (2002).
[CrossRef] [PubMed]

2001

2000

M. Cai, O. Painter, and K. J. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system,” Phys. Rev. Lett. 85(1), 74–77 (2000).
[CrossRef] [PubMed]

1992

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

Aoki, T.

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature 443(7112), 671–674 (2006).
[CrossRef] [PubMed]

Armani, A. M.

H. S. Choi and A. M. Armani, “Thermal nonlinear effects in hybrid optical microresonators,” Appl. Phys. Lett. 97(22), 223306 (2010).
[CrossRef]

Arnold, S.

F. Vollmer and S. Arnold, “Whispering-gallery-mode biosensing: label-free detection down to single molecules,” Nat. Methods 5(7), 591–596 (2008).
[CrossRef] [PubMed]

Bowen, W. P.

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature 443(7112), 671–674 (2006).
[CrossRef] [PubMed]

Cai, M.

M. Cai, O. Painter, and K. J. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system,” Phys. Rev. Lett. 85(1), 74–77 (2000).
[CrossRef] [PubMed]

Carmon, T.

Choi, H. S.

H. S. Choi and A. M. Armani, “Thermal nonlinear effects in hybrid optical microresonators,” Appl. Phys. Lett. 97(22), 223306 (2010).
[CrossRef]

Dayan, B.

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature 443(7112), 671–674 (2006).
[CrossRef] [PubMed]

Digonnet, M. J. F.

Dong, C.

L. He, Y.-F. Xiao, C. Dong, J. Zhu, V. Gaddam, and L. Yang, “Compensation of thermal refraction effect in high-Q toroidal microresonator by polydimethylsiloxane coating,” Appl. Phys. Lett. 93(20), 201102 (2008).
[CrossRef]

Fomin, A. E.

Gaddam, V.

L. He, Y.-F. Xiao, C. Dong, J. Zhu, V. Gaddam, and L. Yang, “Compensation of thermal refraction effect in high-Q toroidal microresonator by polydimethylsiloxane coating,” Appl. Phys. Lett. 93(20), 201102 (2008).
[CrossRef]

Goh, K. W.

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A 71(1), 013817 (2005).
[CrossRef]

Gorodetskii, M. L.

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

Gorodetsky, M. L.

Grudinin, I. S.

He, L.

L. He, Y.-F. Xiao, C. Dong, J. Zhu, V. Gaddam, and L. Yang, “Compensation of thermal refraction effect in high-Q toroidal microresonator by polydimethylsiloxane coating,” Appl. Phys. Lett. 93(20), 201102 (2008).
[CrossRef]

Ilchenko, V. S.

V. S. Ilchenko and A. B. Matsko, “Optical resonators with whispering-gallery modes-part II: Applications,” IEEE J. Sel. Top. Quantum Electron. 12(1), 15–32 (2006).
[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]

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

Kimble, H. J.

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature 443(7112), 671–674 (2006).
[CrossRef] [PubMed]

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A 71(1), 013817 (2005).
[CrossRef]

Kino, G. S.

Kippenberg, T. J.

T. J. Kippenberg and K. J. Vahala, “Cavity opto-mechanics,” Opt. Express 15(25), 17172–17205 (2007).
[CrossRef] [PubMed]

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature 443(7112), 671–674 (2006).
[CrossRef] [PubMed]

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A 71(1), 013817 (2005).
[CrossRef]

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity,” Phys. Rev. Lett. 93(8), 083904 (2004).
[CrossRef] [PubMed]

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91(4), 043902 (2003).
[CrossRef] [PubMed]

S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature 415(6872), 621–623 (2002).
[CrossRef] [PubMed]

Liu, A. C.

Matsko, A. B.

V. S. Ilchenko and A. B. Matsko, “Optical resonators with whispering-gallery modes-part II: Applications,” IEEE J. Sel. Top. Quantum Electron. 12(1), 15–32 (2006).
[CrossRef]

Painter, O.

M. Cai, O. Painter, and K. J. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system,” Phys. Rev. Lett. 85(1), 74–77 (2000).
[CrossRef] [PubMed]

Painter, O. J.

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91(4), 043902 (2003).
[CrossRef] [PubMed]

Parkins, A. S.

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature 443(7112), 671–674 (2006).
[CrossRef] [PubMed]

Spillane, S. M.

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A 71(1), 013817 (2005).
[CrossRef]

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity,” Phys. Rev. Lett. 93(8), 083904 (2004).
[CrossRef] [PubMed]

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91(4), 043902 (2003).
[CrossRef] [PubMed]

S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature 415(6872), 621–623 (2002).
[CrossRef] [PubMed]

Vahala, K. J.

I. S. Grudinin and K. J. Vahala, “Thermal instability of a compound resonator,” Opt. Express 17(16), 14088–14097 (2009).
[CrossRef] [PubMed]

T. J. Kippenberg and K. J. Vahala, “Cavity opto-mechanics,” Opt. Express 15(25), 17172–17205 (2007).
[CrossRef] [PubMed]

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature 443(7112), 671–674 (2006).
[CrossRef] [PubMed]

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A 71(1), 013817 (2005).
[CrossRef]

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity,” Phys. Rev. Lett. 93(8), 083904 (2004).
[CrossRef] [PubMed]

T. Carmon, L. Yang, and K. J. Vahala, “Dynamical thermal behavior and thermal self-stability of microcavities,” Opt. Express 12(20), 4742–4750 (2004).
[CrossRef] [PubMed]

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91(4), 043902 (2003).
[CrossRef] [PubMed]

K. J. Vahala, “Optical microcavities,” Nature 424(6950), 839–846 (2003).
[CrossRef] [PubMed]

S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature 415(6872), 621–623 (2002).
[CrossRef] [PubMed]

M. Cai, O. Painter, and K. J. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system,” Phys. Rev. Lett. 85(1), 74–77 (2000).
[CrossRef] [PubMed]

Vollmer, F.

F. Vollmer and S. Arnold, “Whispering-gallery-mode biosensing: label-free detection down to single molecules,” Nat. Methods 5(7), 591–596 (2008).
[CrossRef] [PubMed]

Wilcut, E.

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature 443(7112), 671–674 (2006).
[CrossRef] [PubMed]

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A 71(1), 013817 (2005).
[CrossRef]

Xiao, Y.-F.

L. He, Y.-F. Xiao, C. Dong, J. Zhu, V. Gaddam, and L. Yang, “Compensation of thermal refraction effect in high-Q toroidal microresonator by polydimethylsiloxane coating,” Appl. Phys. Lett. 93(20), 201102 (2008).
[CrossRef]

Yang, L.

L. He, Y.-F. Xiao, C. Dong, J. Zhu, V. Gaddam, and L. Yang, “Compensation of thermal refraction effect in high-Q toroidal microresonator by polydimethylsiloxane coating,” Appl. Phys. Lett. 93(20), 201102 (2008).
[CrossRef]

T. Carmon, L. Yang, and K. J. Vahala, “Dynamical thermal behavior and thermal self-stability of microcavities,” Opt. Express 12(20), 4742–4750 (2004).
[CrossRef] [PubMed]

Zhu, J.

L. He, Y.-F. Xiao, C. Dong, J. Zhu, V. Gaddam, and L. Yang, “Compensation of thermal refraction effect in high-Q toroidal microresonator by polydimethylsiloxane coating,” Appl. Phys. Lett. 93(20), 201102 (2008).
[CrossRef]

Appl. Phys. Lett.

L. He, Y.-F. Xiao, C. Dong, J. Zhu, V. Gaddam, and L. Yang, “Compensation of thermal refraction effect in high-Q toroidal microresonator by polydimethylsiloxane coating,” Appl. Phys. Lett. 93(20), 201102 (2008).
[CrossRef]

H. S. Choi and A. M. Armani, “Thermal nonlinear effects in hybrid optical microresonators,” Appl. Phys. Lett. 97(22), 223306 (2010).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

V. S. Ilchenko and A. B. Matsko, “Optical resonators with whispering-gallery modes-part II: Applications,” IEEE J. Sel. Top. Quantum Electron. 12(1), 15–32 (2006).
[CrossRef]

J. Opt. Soc. Am. B

Laser Phys.

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

Nat. Methods

F. Vollmer and S. Arnold, “Whispering-gallery-mode biosensing: label-free detection down to single molecules,” Nat. Methods 5(7), 591–596 (2008).
[CrossRef] [PubMed]

Nature

S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature 415(6872), 621–623 (2002).
[CrossRef] [PubMed]

K. J. Vahala, “Optical microcavities,” Nature 424(6950), 839–846 (2003).
[CrossRef] [PubMed]

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature 443(7112), 671–674 (2006).
[CrossRef] [PubMed]

Opt. Express

Phys. Rev. A

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A 71(1), 013817 (2005).
[CrossRef]

Phys. Rev. Lett.

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity,” Phys. Rev. Lett. 93(8), 083904 (2004).
[CrossRef] [PubMed]

M. Cai, O. Painter, and K. J. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system,” Phys. Rev. Lett. 85(1), 74–77 (2000).
[CrossRef] [PubMed]

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91(4), 043902 (2003).
[CrossRef] [PubMed]

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Figures (4)

Fig. 1
Fig. 1

Schematic of the experimental setup.

Fig. 2
Fig. 2

A) experimental frequency scans of a WGM resonance at 1450 nm without (red) and with stabilization (blue). B) solutions of the system (4). WGM resonance with thermal effect (red dots show unstable solutions). Solutions for the same mode when the stabilizing laser is activated constitute a nearly Lorentzian lineshape. Note the frequency shift of the stabilized resonance caused by heating from the stabilizing WGM.

Fig. 3
Fig. 3

Optical cooling and amplification of a 70 MHz mechanical mode of the optical resonator. Broadening and narrowing of the mechanical spectrum occurs due to optomechanical cooling and amplification in the unresolved sideband regime. Colored lines represent Lorentzian fits of the experimentally recorded spectra. Inset shows an optical photograph of the resonator near the fiber coupler.

Fig. 4
Fig. 4

Left panel: Phase diagram of temperature change of the cavity upon application of a blue detuned pump. There is always a single, stable steady-state solution. Right panel: Phase diagram of temperature change of the cavity upon using a red detuned pump. The thermal bistability may arise at sufficient pump power. The excitation of a stabilizing mode can compensate the nonlinearity via a saddle-node bifurcation.

Tables (1)

Tables Icon

Table 1 Parameters and variables used in computation of resonance curves in Fig. 2

Equations (13)

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

Δ ω ω = ( 1 n n T + α T ) Δ T = β Δ T
E ( r , t ) = a ( t ) E 0 ( r ) e i ω t ,
| E 0 | 2 d r = 1.
{ a ˙ + a ( γ a + i [ ω 1 ω a ( 1 μ | a | 2 β θ ) ] ) = i F 1 b ˙ + b ( γ b + i [ ω 2 ω b ( 1 μ | b | 2 β θ ) ] ) = i F 2 θ ˙ + δ θ θ = ν ( | a | 2 + | b | 2 ) .
{ a ˙ 1 = a 1 γ a + a 2 ( ω 1 ω a ( 1 μ | a | 2 β θ ) ) a ˙ 2 = a 2 γ a a 1 ( ω 1 ω a ( 1 μ | a | 2 β θ ) ) + F 1 b ˙ 1 = b 1 γ b + b 2 ( ω 2 ω b ( 1 μ | b | 2 β θ ) ) b ˙ 2 = b 2 γ b b 1 ( ω 2 ω b ( 1 μ | b | 2 β θ ) ) + F 2 θ ˙ = ν ( a 2 + b 2 ) δ θ θ
{ y ˙ 1 = y 1 + y 2 Δ ω a / γ a + y 2 y 1 2 + y 2 3 + y 2 y 5 y ˙ 2 = y 2 y 1 Δ ω a / γ a y 1 3 y 1 y 2 2 y 1 y 5 + F 1 μ ω a / γ a 3 y ˙ 3 = y 3 γ b / γ a + y 4 Δ ω b / γ a + y 4 y 5 ω b / ω a + y 4 y 3 2 + y 4 3 y ˙ 4 = y 4 γ b / γ a y 3 Δ ω b / γ a y 3 y 5 ω b / ω a y 3 3 y 3 y 4 2 + F 2 μ ω b / γ a 3 y ˙ 5 = y 5 δ θ / γ a + ( y 1 2 + y 2 2 ) ν β / ( μ γ a ) + ( y 3 2 + y 4 2 ) ν β ω a / ( μ γ a ω b ) .
P c i r c = ( y 1 2 + y 2 2 ) a c 2 n c V a ( 2 π ) 2 R × 10 7 [ W a t t ] .
C θ ˙ = q ˙ i n q ˙ o u t = W h 1 [ ( ω 2 ω b , 0 ( 1 β θ ) ) / γ b ] 2 + 1 K θ
C K { [ ω 2 ω b , 0 ( 1 β θ ) ] 2 + γ b 2 } θ ˙ = f ( θ , ω 2 , W h ) = β 2 ω b , 0 2 θ 3 + 2 β ω b , 0 ( ω b , 0 ω 2 ) θ 2 [ ( ω 2 ω b , 0 ) 2 + γ b 2 ] θ + γ b 2 W h K
C θ ˙ = W h 1 [ ( ω 2 ω b ) / γ b ] 2 + 1 K θ + W s , c ( θ ) = W h , c ( θ ) K θ + W s , c ( θ ) .
C δ T ˙ = W h , c θ δ T K δ T + W s , c θ δ T = W h , c θ δ T ( K + ω a β W s , c ω a ) δ T
| W h , c θ | < ( K + ω a β W s , c ω a ) ω b β | W h , c ω b | < ( K + ω a β | W s , c ω a | )
3 3 4 W h Q b < 3 3 4 W s Q a + K β

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