Abstract

As thermo-optic locking is widely used to establish a stable frequency detuning between an external laser and a high Q microcavity, it is important to understand how this method affects microcavity temperature and frequency fluctuations. A theoretical analysis of the laser-microcavity frequency fluctuations is presented and used to find the spectral dependence of the suppression of laser-microcavity, relative frequency noise caused by thermo-optic locking. The response function is that of a high-pass filter with a bandwidth and low-frequency suppression that increase with input power. The results are verified using an external-cavity diode laser and a silica disk resonator. The locking of relative frequency fluctuations causes temperature fluctuations within the microcavity that transfer pump frequency noise onto the microcavity modes over the thermal locking bandwidth. This transfer is verified experimentally. These results are important to investigations of noise properties in many nonlinear microcavity experiments in which low-frequency, optical-pump frequency noise must be considered.

© 2014 Optical Society of America

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  1. K. J. Vahala, “Optical microcavities,” Nature 424, 839–846 (2003).
    [CrossRef] [PubMed]
  2. A. B. Matkso and V. S. Ilchenko, “Optical resonators with whispering-gallery modes-part I: basics,” IEEE J. Quantum Electron. 12, 3–14 (2006).
    [CrossRef]
  3. M. L. Gorodetsky, A. A. Savchenkov, and V. S. Ilchenko, “Ultimate Q of optical microsphere resonators,” Opt. Lett. 21, 453–455 (1996).
    [CrossRef] [PubMed]
  4. D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
    [CrossRef] [PubMed]
  5. H. Lee, T. Chen, J. Li, K. Yang, S. Jeon, O. Painter, and K. J. Vahala, “Chemically etched ultrahigh-Q wedge-resonator on a silicon chip,” Nat. Photonics 6, 369–373 (2012).
    [CrossRef]
  6. V. S. Ilchenko and M. L. Gorodetskii, “Thermal nonlinear effects in optical whispering gallery microresonators,” Laser Phys. 2, 1004 (1992).
  7. 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, 459–465 (2005).
    [CrossRef]
  8. T. Carmon, L. Yang, and K. J. Vahala, ”Dynamical thermal behavior and thermal self-stability of microcavities,” Opt. Express 12, 4742–4750 (2004).
    [CrossRef] [PubMed]
  9. T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, “Analysis of Radiation-Pressure Induced Mechanical Oscillation of an Optical Microcavity,” Phys. Rev. Lett. 95, 033901 (2005).
    [CrossRef] [PubMed]
  10. T. J. Kippenberg and K. J. Vahala, “Cavity Optomechanics: Back-Action at the Mesoscale,” Science 321, 1172–1176 (2008).
    [CrossRef] [PubMed]
  11. M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram-and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459, 550–555 (2009).
    [CrossRef] [PubMed]
  12. Q. Lin, J. Rosenberg, X. Jiang, K. J. Vahala, and O. Painter, “Mechanical oscillation and cooling actuated by the optical gradient force,” Phys. Rev. Lett. 103, 103601 (2009).
    [CrossRef] [PubMed]
  13. P. DelHaye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450, 1214–1217 (2007).
    [CrossRef]
  14. T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-Based Optical Frequency Combs,” Science 322, 555–559 (2011).
    [CrossRef]
  15. J. Li, H. Lee, T. Chen, and K. J. Vahala, “Low-Pump-Power, Low-Phase-Noise, and Microwave to Millimeter-Wave Repetition Rate Operation in Microcombs,” Phys. Rev. Lett. 109, 233901 (2012).
    [CrossRef]
  16. S. B. Papp and S. A. Diddams, “Spectral and temporal characterization of a fused-quartzmicroresonator optical frequency comb,” Phys. Rev. A 84, 053833 (2011).
    [CrossRef]
  17. T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8, 145–152 (2014).
    [CrossRef]
  18. J. Li, H. Lee, T. Chen, and K. J. Vahala, “Characterization of a high coherence, Brillouin microcavity laser on silicon,” Opt. Express 20, 20170–20180 (2012).
    [CrossRef] [PubMed]
  19. J. Li, H. Lee, and K. J. Vahala, “Microwave synthesizer using an on-chip Brillouin oscillator,” Nat. Commun. 4, 2097 (2013).
    [PubMed]
  20. R. Drever, J. L. Hall, F. Kowalski, J. Hough, G. Ford, A. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
    [CrossRef]

2014 (1)

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8, 145–152 (2014).
[CrossRef]

2013 (1)

J. Li, H. Lee, and K. J. Vahala, “Microwave synthesizer using an on-chip Brillouin oscillator,” Nat. Commun. 4, 2097 (2013).
[PubMed]

2012 (3)

J. Li, H. Lee, T. Chen, and K. J. Vahala, “Characterization of a high coherence, Brillouin microcavity laser on silicon,” Opt. Express 20, 20170–20180 (2012).
[CrossRef] [PubMed]

J. Li, H. Lee, T. Chen, and K. J. Vahala, “Low-Pump-Power, Low-Phase-Noise, and Microwave to Millimeter-Wave Repetition Rate Operation in Microcombs,” Phys. Rev. Lett. 109, 233901 (2012).
[CrossRef]

H. Lee, T. Chen, J. Li, K. Yang, S. Jeon, O. Painter, and K. J. Vahala, “Chemically etched ultrahigh-Q wedge-resonator on a silicon chip,” Nat. Photonics 6, 369–373 (2012).
[CrossRef]

2011 (2)

S. B. Papp and S. A. Diddams, “Spectral and temporal characterization of a fused-quartzmicroresonator optical frequency comb,” Phys. Rev. A 84, 053833 (2011).
[CrossRef]

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-Based Optical Frequency Combs,” Science 322, 555–559 (2011).
[CrossRef]

2009 (2)

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram-and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459, 550–555 (2009).
[CrossRef] [PubMed]

Q. Lin, J. Rosenberg, X. Jiang, K. J. Vahala, and O. Painter, “Mechanical oscillation and cooling actuated by the optical gradient force,” Phys. Rev. Lett. 103, 103601 (2009).
[CrossRef] [PubMed]

2008 (1)

T. J. Kippenberg and K. J. Vahala, “Cavity Optomechanics: Back-Action at the Mesoscale,” Science 321, 1172–1176 (2008).
[CrossRef] [PubMed]

2007 (1)

P. DelHaye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450, 1214–1217 (2007).
[CrossRef]

2006 (1)

A. B. Matkso and V. S. Ilchenko, “Optical resonators with whispering-gallery modes-part I: basics,” IEEE J. Quantum Electron. 12, 3–14 (2006).
[CrossRef]

2005 (2)

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, 459–465 (2005).
[CrossRef]

T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, “Analysis of Radiation-Pressure Induced Mechanical Oscillation of an Optical Microcavity,” Phys. Rev. Lett. 95, 033901 (2005).
[CrossRef] [PubMed]

2004 (1)

2003 (2)

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

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

1996 (1)

1992 (1)

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

1983 (1)

R. Drever, J. L. Hall, F. Kowalski, J. Hough, G. Ford, A. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Arcizet, O.

P. DelHaye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450, 1214–1217 (2007).
[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, 925–928 (2003).
[CrossRef] [PubMed]

Brasch, V.

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8, 145–152 (2014).
[CrossRef]

Camacho, R.

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram-and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459, 550–555 (2009).
[CrossRef] [PubMed]

Carmon, T.

T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, “Analysis of Radiation-Pressure Induced Mechanical Oscillation of an Optical Microcavity,” Phys. Rev. Lett. 95, 033901 (2005).
[CrossRef] [PubMed]

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

Chan, J.

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram-and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459, 550–555 (2009).
[CrossRef] [PubMed]

Chen, T.

H. Lee, T. Chen, J. Li, K. Yang, S. Jeon, O. Painter, and K. J. Vahala, “Chemically etched ultrahigh-Q wedge-resonator on a silicon chip,” Nat. Photonics 6, 369–373 (2012).
[CrossRef]

J. Li, H. Lee, T. Chen, and K. J. Vahala, “Low-Pump-Power, Low-Phase-Noise, and Microwave to Millimeter-Wave Repetition Rate Operation in Microcombs,” Phys. Rev. Lett. 109, 233901 (2012).
[CrossRef]

J. Li, H. Lee, T. Chen, and K. J. Vahala, “Characterization of a high coherence, Brillouin microcavity laser on silicon,” Opt. Express 20, 20170–20180 (2012).
[CrossRef] [PubMed]

DelHaye, P.

P. DelHaye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450, 1214–1217 (2007).
[CrossRef]

Diddams, S. A.

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-Based Optical Frequency Combs,” Science 322, 555–559 (2011).
[CrossRef]

S. B. Papp and S. A. Diddams, “Spectral and temporal characterization of a fused-quartzmicroresonator optical frequency comb,” Phys. Rev. A 84, 053833 (2011).
[CrossRef]

Drever, R.

R. Drever, J. L. Hall, F. Kowalski, J. Hough, G. Ford, A. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Eichenfield, M.

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram-and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459, 550–555 (2009).
[CrossRef] [PubMed]

Fomin, A. E.

Ford, G.

R. Drever, J. L. Hall, F. Kowalski, J. Hough, G. Ford, A. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Gorodetskii, M. L.

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

Gorodetsky, M. L.

Grudinin, I. S.

Hall, J. L.

R. Drever, J. L. Hall, F. Kowalski, J. Hough, G. Ford, A. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Herr, T.

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8, 145–152 (2014).
[CrossRef]

Holzwarth, R.

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-Based Optical Frequency Combs,” Science 322, 555–559 (2011).
[CrossRef]

P. DelHaye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450, 1214–1217 (2007).
[CrossRef]

Hough, J.

R. Drever, J. L. Hall, F. Kowalski, J. Hough, G. Ford, A. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Ilchenko, V. S.

A. B. Matkso and V. S. Ilchenko, “Optical resonators with whispering-gallery modes-part I: basics,” IEEE J. Quantum Electron. 12, 3–14 (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, 459–465 (2005).
[CrossRef]

M. L. Gorodetsky, A. A. Savchenkov, and V. S. Ilchenko, “Ultimate Q of optical microsphere resonators,” Opt. Lett. 21, 453–455 (1996).
[CrossRef] [PubMed]

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

Jeon, S.

H. Lee, T. Chen, J. Li, K. Yang, S. Jeon, O. Painter, and K. J. Vahala, “Chemically etched ultrahigh-Q wedge-resonator on a silicon chip,” Nat. Photonics 6, 369–373 (2012).
[CrossRef]

Jiang, X.

Q. Lin, J. Rosenberg, X. Jiang, K. J. Vahala, and O. Painter, “Mechanical oscillation and cooling actuated by the optical gradient force,” Phys. Rev. Lett. 103, 103601 (2009).
[CrossRef] [PubMed]

Jost, J. D.

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8, 145–152 (2014).
[CrossRef]

Kippenberg, T.

P. DelHaye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450, 1214–1217 (2007).
[CrossRef]

Kippenberg, T. J.

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8, 145–152 (2014).
[CrossRef]

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-Based Optical Frequency Combs,” Science 322, 555–559 (2011).
[CrossRef]

T. J. Kippenberg and K. J. Vahala, “Cavity Optomechanics: Back-Action at the Mesoscale,” Science 321, 1172–1176 (2008).
[CrossRef] [PubMed]

T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, “Analysis of Radiation-Pressure Induced Mechanical Oscillation of an Optical Microcavity,” Phys. Rev. Lett. 95, 033901 (2005).
[CrossRef] [PubMed]

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

Kondratiev, N. M.

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8, 145–152 (2014).
[CrossRef]

Kowalski, F.

R. Drever, J. L. Hall, F. Kowalski, J. Hough, G. Ford, A. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Lee, H.

J. Li, H. Lee, and K. J. Vahala, “Microwave synthesizer using an on-chip Brillouin oscillator,” Nat. Commun. 4, 2097 (2013).
[PubMed]

J. Li, H. Lee, T. Chen, and K. J. Vahala, “Characterization of a high coherence, Brillouin microcavity laser on silicon,” Opt. Express 20, 20170–20180 (2012).
[CrossRef] [PubMed]

J. Li, H. Lee, T. Chen, and K. J. Vahala, “Low-Pump-Power, Low-Phase-Noise, and Microwave to Millimeter-Wave Repetition Rate Operation in Microcombs,” Phys. Rev. Lett. 109, 233901 (2012).
[CrossRef]

H. Lee, T. Chen, J. Li, K. Yang, S. Jeon, O. Painter, and K. J. Vahala, “Chemically etched ultrahigh-Q wedge-resonator on a silicon chip,” Nat. Photonics 6, 369–373 (2012).
[CrossRef]

Li, J.

J. Li, H. Lee, and K. J. Vahala, “Microwave synthesizer using an on-chip Brillouin oscillator,” Nat. Commun. 4, 2097 (2013).
[PubMed]

H. Lee, T. Chen, J. Li, K. Yang, S. Jeon, O. Painter, and K. J. Vahala, “Chemically etched ultrahigh-Q wedge-resonator on a silicon chip,” Nat. Photonics 6, 369–373 (2012).
[CrossRef]

J. Li, H. Lee, T. Chen, and K. J. Vahala, “Characterization of a high coherence, Brillouin microcavity laser on silicon,” Opt. Express 20, 20170–20180 (2012).
[CrossRef] [PubMed]

J. Li, H. Lee, T. Chen, and K. J. Vahala, “Low-Pump-Power, Low-Phase-Noise, and Microwave to Millimeter-Wave Repetition Rate Operation in Microcombs,” Phys. Rev. Lett. 109, 233901 (2012).
[CrossRef]

Lin, Q.

Q. Lin, J. Rosenberg, X. Jiang, K. J. Vahala, and O. Painter, “Mechanical oscillation and cooling actuated by the optical gradient force,” Phys. Rev. Lett. 103, 103601 (2009).
[CrossRef] [PubMed]

Matkso, A. B.

A. B. Matkso and V. S. Ilchenko, “Optical resonators with whispering-gallery modes-part I: basics,” IEEE J. Quantum Electron. 12, 3–14 (2006).
[CrossRef]

Munley, A.

R. Drever, J. L. Hall, F. Kowalski, J. Hough, G. Ford, A. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Painter, O.

H. Lee, T. Chen, J. Li, K. Yang, S. Jeon, O. Painter, and K. J. Vahala, “Chemically etched ultrahigh-Q wedge-resonator on a silicon chip,” Nat. Photonics 6, 369–373 (2012).
[CrossRef]

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram-and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459, 550–555 (2009).
[CrossRef] [PubMed]

Q. Lin, J. Rosenberg, X. Jiang, K. J. Vahala, and O. Painter, “Mechanical oscillation and cooling actuated by the optical gradient force,” Phys. Rev. Lett. 103, 103601 (2009).
[CrossRef] [PubMed]

Papp, S. B.

S. B. Papp and S. A. Diddams, “Spectral and temporal characterization of a fused-quartzmicroresonator optical frequency comb,” Phys. Rev. A 84, 053833 (2011).
[CrossRef]

Rokhsari, H.

T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, “Analysis of Radiation-Pressure Induced Mechanical Oscillation of an Optical Microcavity,” Phys. Rev. Lett. 95, 033901 (2005).
[CrossRef] [PubMed]

Rosenberg, J.

Q. Lin, J. Rosenberg, X. Jiang, K. J. Vahala, and O. Painter, “Mechanical oscillation and cooling actuated by the optical gradient force,” Phys. Rev. Lett. 103, 103601 (2009).
[CrossRef] [PubMed]

Savchenkov, A. A.

Scherer, A.

T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, “Analysis of Radiation-Pressure Induced Mechanical Oscillation of an Optical Microcavity,” Phys. Rev. Lett. 95, 033901 (2005).
[CrossRef] [PubMed]

Schliesser, A.

P. DelHaye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450, 1214–1217 (2007).
[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, 925–928 (2003).
[CrossRef] [PubMed]

Vahala, K. J.

J. Li, H. Lee, and K. J. Vahala, “Microwave synthesizer using an on-chip Brillouin oscillator,” Nat. Commun. 4, 2097 (2013).
[PubMed]

J. Li, H. Lee, T. Chen, and K. J. Vahala, “Characterization of a high coherence, Brillouin microcavity laser on silicon,” Opt. Express 20, 20170–20180 (2012).
[CrossRef] [PubMed]

J. Li, H. Lee, T. Chen, and K. J. Vahala, “Low-Pump-Power, Low-Phase-Noise, and Microwave to Millimeter-Wave Repetition Rate Operation in Microcombs,” Phys. Rev. Lett. 109, 233901 (2012).
[CrossRef]

H. Lee, T. Chen, J. Li, K. Yang, S. Jeon, O. Painter, and K. J. Vahala, “Chemically etched ultrahigh-Q wedge-resonator on a silicon chip,” Nat. Photonics 6, 369–373 (2012).
[CrossRef]

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram-and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459, 550–555 (2009).
[CrossRef] [PubMed]

Q. Lin, J. Rosenberg, X. Jiang, K. J. Vahala, and O. Painter, “Mechanical oscillation and cooling actuated by the optical gradient force,” Phys. Rev. Lett. 103, 103601 (2009).
[CrossRef] [PubMed]

T. J. Kippenberg and K. J. Vahala, “Cavity Optomechanics: Back-Action at the Mesoscale,” Science 321, 1172–1176 (2008).
[CrossRef] [PubMed]

T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, “Analysis of Radiation-Pressure Induced Mechanical Oscillation of an Optical Microcavity,” Phys. Rev. Lett. 95, 033901 (2005).
[CrossRef] [PubMed]

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

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

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

Wang, C. Y.

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8, 145–152 (2014).
[CrossRef]

Ward, H.

R. Drever, J. L. Hall, F. Kowalski, J. Hough, G. Ford, A. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Wilken, T.

P. DelHaye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450, 1214–1217 (2007).
[CrossRef]

Yang, K.

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Appl. Phys. B (1)

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Nat. Commun. (1)

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[PubMed]

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T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8, 145–152 (2014).
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H. Lee, T. Chen, J. Li, K. Yang, S. Jeon, O. Painter, and K. J. Vahala, “Chemically etched ultrahigh-Q wedge-resonator on a silicon chip,” Nat. Photonics 6, 369–373 (2012).
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Nature (4)

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram-and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459, 550–555 (2009).
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K. J. Vahala, “Optical microcavities,” Nature 424, 839–846 (2003).
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J. Li, H. Lee, T. Chen, and K. J. Vahala, “Low-Pump-Power, Low-Phase-Noise, and Microwave to Millimeter-Wave Repetition Rate Operation in Microcombs,” Phys. Rev. Lett. 109, 233901 (2012).
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Figures (4)

Fig. 1
Fig. 1

Plot of thermo-optic lock, laser-microcavity, relative-frequency-noise transfer function (R(Ω)) and the cavity-frequency-noise transfer function (K(Ω)) for several input laser power levels at a warm-cavity detuning of ΔogthΔTo =κ/2. Parameters used in these plots are γth/2π = 100 Hz, and α/2π = 10 kHz/mW.

Fig. 2
Fig. 2

(a) The laser-microcavity relative-frequency-noise spectra are measured for a 2 mm disk resonator with loaded linewidth 4.10 MHz under a series of input power levels: 25 μW (red curve), 100 μW (green curve), 500 μW (cyan curve) and 2 mW (blue curve). For comparison, the free-running laser frequency noise is also shown (black curve), measured using a Mach-Zehnder interferometer [5]. The magenta curve is the background noise when the laser is off-resonance and contains the laser relative intensity noise (RIN) and shot noise normalized by the collected power and transfer function H. Background noise (magenta curve) rises above 4 MHz is a result of the shot noise divided by the cavity transfer function. Note that the spurious noise at low frequencies (e.g. at 10 Hz) is believed to result from corresponding peaks in the frequency noise (e.g. mechanical noise) of the pump laser. (b) TOL laser-microcavity relative-frequency-noise transfer function, R, is plotted at a series of input power levels given in (a). The black-dashed curves are calculations based on Eq. (16) using the following fitting parameters γth/2π = 128 Hz and α/2π = 13.5 kHz/mW. TOL-induced laser-microcavity relative-frequency-noise suppression increases with the input power as does the TOL bandwidth.

Fig. 3
Fig. 3

Experimental layout is given for the pump-probe experiment to measure the cavity frequency noise induced by thermo-optic locking. Laser 1 is an external cavity diode laser, which is thermo-optically locked to a microcavity mode at the half-detuning point. Laser 2 is a narrow linewidth, tunable fiber laser, which is locked to a different cavity mode using Pound-Drever-Hall (PDH) technique. The fiber laser is also coupled to a Mach-Zehnder interferometer (MZI) for laser frequency noise measurement. AOM: acousto-optic modulator; ATT: tunable optical attenuator; PM: phase modulator; OBPF: optical band pass filter; PD: photodetector.

Fig. 4
Fig. 4

(a) Shown in the figure are the measured PDH-locked fiber laser (FL) frequency noise spectrum (green curve) and the predicted frequency noise spectrum of the thermally-slaved cavity pumped by an ECDL with 25 μW power (brown curve). Also plotted for comparison are the PDH-locked fiber laser frequency noise without ECDL thermo-optic locking (black) and the free-running fiber laser frequency noise (red). (b) The measured PDH-locked fiber laser frequency noise spectrum (green curve) and the predicted micro-cavity frequency noise spectrum induced by the thermo-optically locked ECDL with 2 mW power (brown curve) are shown.

Equations (21)

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a ˙ = [ i Δ i g th Δ T κ 2 ] a + κ ex s
Δ T ˙ = γ th Δ T + Γ | a | 2
a = a o + a 1 ( t )
Δ T = Δ T o + Δ T 1 ( t )
Δ = Δ 0 + Δ 1 ( t )
a o = κ ex s / ( i [ Δ o g th Δ T o ] + κ 2 )
Δ T o = Γ | a o | 2 γ th = Γ γ th κ ex | s | 2 [ Δ o g th Δ T o ] 2 + κ 2 / 4
a ˙ 1 = [ i Δ o i g th Δ o κ 2 ] a 1 + i [ Δ 1 g th Δ T 1 ] a o
Δ T ˙ 1 = γ th Δ T 1 + Γ ( a o a 1 * + a o * a 1 )
a 1 = i [ Δ 1 g th Δ T 1 ] a o i [ Δ o g th Δ T o ] + κ 2
Δ T ˙ 1 = [ γ th 2 Γ g th | a o | 2 ( Δ o g th Δ T o ) ( Δ o g th Δ T o ) 2 + κ 2 4 ] Δ T 1 2 Γ | a o | 2 ( Δ o g th Δ T o ) ( Δ o g th Δ T o ) 2 + κ 2 4 Δ 1
Δ T ˜ 1 ( Ω ) = 2 Γ | a o | 2 ( Δ o g th Δ T o ) ( Δ o g th Δ T o ) 2 + κ 2 4 γ th 2 Γ g th | a o | 2 ( Δ o g th Δ T o ) ( Δ o g th Δ T o ) 2 + κ 2 4 + i Ω Δ ˜ 1 ( Ω )
Δ ˜ 1 lc ( Ω ) = g th Δ T ˜ 1 ( Ω ) + Δ ˜ 1 ( Ω )
S Δ ω lc ( Ω ) = γ th 2 + Ω 2 [ γ th + α P in ] 2 + Ω 2 S Δ ω ( Ω )
α = d n d T ω o n Γ 2 κ ex ( Δ o g th Δ T o ) [ ( Δ o g th Δ T o ) 2 + κ 2 4 ] 2
R ( Ω ) = γ th 2 + Ω 2 [ γ th + α P in ] 2 + Ω 2
S Δ T ( Ω ) = [ α P in ] 2 / g th 2 [ γ th + α P in ] 2 + Ω 2 S Δ ω ( Ω )
S Δ ω cav ( Ω ) = g th 2 S Δ T ( Ω ) = [ α P in ] 2 [ γ th + α P in ] 2 + Ω 2 S Δ ω ( Ω )
K ( Ω ) = [ α P in ] 2 [ γ th + α P in ] 2 + Ω 2
S P ( Ω ) = P in 2 H ( Ω ) S Δ ω lc ( Ω )
H ( Ω ) = κ ex 2 [ Δ 2 + κ 2 / 4 ] 2 4 Δ 2 ( κ i 2 + Ω 2 ) [ ( Δ + Ω ) 2 + κ 2 / 4 ] [ ( Δ Ω ) 2 + κ 2 / 4 ]

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