Abstract

We theoretically and experimentally investigate the chaotic regime of optical frequency combs generated in nonlinear ring microresonators pumped with continuous wave light. We show that the chaotic regime reveals itself by a flat top symmetric envelope of the frequency spectrum, when observed by means of an optical spectrum analyzer. The comb, demodulated on a fast photodiode, produces a noisy radio frequency signal with spectral width significantly exceeding the linear bandwidth of the microresonator mode. We discuss practical ways of excitation of a coherent frequency comb and avoiding the chaotic regime.

© 2013 Optical Society of America

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  1. P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, Nature 450, 1214 (2007).
    [CrossRef]
  2. A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, I. Solomatine, D. Seidel, and L. Maleki, Phys. Rev. Lett. 101, 093902 (2008).
    [CrossRef]
  3. T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, Science 332, 555 (2011).
    [CrossRef]
  4. T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang, E. Gavartin, R. Holzwarth, M. L. Gorodetsky, and T. J. Kippenberg, Nat. Photonics 6, 480 (2012).
    [CrossRef]
  5. K. J. Blow and N. J. Doran, Phys. Rev. Lett. 52, 526 (1984).
    [CrossRef]
  6. A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, D. Seidel, and L. Maleki, Phys. Rev. A 85, 023830 (2012).
    [CrossRef]
  7. S. B. Papp and S. A. Diddams, Phys. Rev. A 84, 053833 (2011).
    [CrossRef]
  8. F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, Nat. Photonics 5, 770 (2011).
    [CrossRef]
  9. F. Ferdous, H. Miao, P.-H. Wang, D. E. Leaird, K. Srinivasan, L. Chen, V. Aksyuk, and A. M. Weiner, Opt. Express 20, 21033 (2012).
    [CrossRef]
  10. 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,” arXiv:1208.5256v1 (2012).
  11. W. Liang, A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, D. Seidel, and L. Maleki, Opt. Lett. 36, 2290 (2011).
    [CrossRef]
  12. L. Maleki, V. S. Ilchenko, A. A. Savchenkov, and A. B. Matsko, in Practical Applications of Microresonators in Optics and Photonics, A. B. Matsko, ed. (CRC, 2009), Chap. 3.
  13. W. Liang, V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, D. Seidel, and L. Maleki, Opt. Lett. 35, 2822 (2010).
    [CrossRef]
  14. Y. K. Chembo and N. Yu, Phys. Rev. A 82, 033801 (2010).
    [CrossRef]
  15. A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, Phys. Rev. A 71, 033804 (2005).
    [CrossRef]
  16. T. Herr, V. Brasch, M. L. Gorodetsky, and T. J. Kippenberg, “Soliton mode-locking in optical microresonators,” arXiv:1211.0733 (2012).

2012

T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang, E. Gavartin, R. Holzwarth, M. L. Gorodetsky, and T. J. Kippenberg, Nat. Photonics 6, 480 (2012).
[CrossRef]

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, D. Seidel, and L. Maleki, Phys. Rev. A 85, 023830 (2012).
[CrossRef]

F. Ferdous, H. Miao, P.-H. Wang, D. E. Leaird, K. Srinivasan, L. Chen, V. Aksyuk, and A. M. Weiner, Opt. Express 20, 21033 (2012).
[CrossRef]

2011

W. Liang, A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, D. Seidel, and L. Maleki, Opt. Lett. 36, 2290 (2011).
[CrossRef]

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, Science 332, 555 (2011).
[CrossRef]

S. B. Papp and S. A. Diddams, Phys. Rev. A 84, 053833 (2011).
[CrossRef]

F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, Nat. Photonics 5, 770 (2011).
[CrossRef]

2010

2008

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, I. Solomatine, D. Seidel, and L. Maleki, Phys. Rev. Lett. 101, 093902 (2008).
[CrossRef]

2007

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, Nature 450, 1214 (2007).
[CrossRef]

2005

A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, Phys. Rev. A 71, 033804 (2005).
[CrossRef]

1984

K. J. Blow and N. J. Doran, Phys. Rev. Lett. 52, 526 (1984).
[CrossRef]

Aksyuk, V.

Arcizet, O.

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, Nature 450, 1214 (2007).
[CrossRef]

Blow, K. J.

K. J. Blow and N. J. Doran, Phys. Rev. Lett. 52, 526 (1984).
[CrossRef]

Brasch, V.

T. Herr, V. Brasch, M. L. Gorodetsky, and T. J. Kippenberg, “Soliton mode-locking in optical microresonators,” arXiv:1211.0733 (2012).

Chembo, Y. K.

Y. K. Chembo and N. Yu, Phys. Rev. A 82, 033801 (2010).
[CrossRef]

Chen, L.

F. Ferdous, H. Miao, P.-H. Wang, D. E. Leaird, K. Srinivasan, L. Chen, V. Aksyuk, and A. M. Weiner, Opt. Express 20, 21033 (2012).
[CrossRef]

F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, Nat. Photonics 5, 770 (2011).
[CrossRef]

Chen, T.

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,” arXiv:1208.5256v1 (2012).

Del’Haye, P.

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, Nature 450, 1214 (2007).
[CrossRef]

Diddams, S. A.

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, Science 332, 555 (2011).
[CrossRef]

S. B. Papp and S. A. Diddams, Phys. Rev. A 84, 053833 (2011).
[CrossRef]

Doran, N. J.

K. J. Blow and N. J. Doran, Phys. Rev. Lett. 52, 526 (1984).
[CrossRef]

Ferdous, F.

F. Ferdous, H. Miao, P.-H. Wang, D. E. Leaird, K. Srinivasan, L. Chen, V. Aksyuk, and A. M. Weiner, Opt. Express 20, 21033 (2012).
[CrossRef]

F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, Nat. Photonics 5, 770 (2011).
[CrossRef]

Gavartin, E.

T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang, E. Gavartin, R. Holzwarth, M. L. Gorodetsky, and T. J. Kippenberg, Nat. Photonics 6, 480 (2012).
[CrossRef]

Gorodetsky, M. L.

T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang, E. Gavartin, R. Holzwarth, M. L. Gorodetsky, and T. J. Kippenberg, Nat. Photonics 6, 480 (2012).
[CrossRef]

T. Herr, V. Brasch, M. L. Gorodetsky, and T. J. Kippenberg, “Soliton mode-locking in optical microresonators,” arXiv:1211.0733 (2012).

Hartinger, K.

T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang, E. Gavartin, R. Holzwarth, M. L. Gorodetsky, and T. J. Kippenberg, Nat. Photonics 6, 480 (2012).
[CrossRef]

Herr, T.

T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang, E. Gavartin, R. Holzwarth, M. L. Gorodetsky, and T. J. Kippenberg, Nat. Photonics 6, 480 (2012).
[CrossRef]

T. Herr, V. Brasch, M. L. Gorodetsky, and T. J. Kippenberg, “Soliton mode-locking in optical microresonators,” arXiv:1211.0733 (2012).

Holzwarth, R.

T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang, E. Gavartin, R. Holzwarth, M. L. Gorodetsky, and T. J. Kippenberg, Nat. Photonics 6, 480 (2012).
[CrossRef]

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, Science 332, 555 (2011).
[CrossRef]

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, Nature 450, 1214 (2007).
[CrossRef]

Ilchenko, V. S.

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, D. Seidel, and L. Maleki, Phys. Rev. A 85, 023830 (2012).
[CrossRef]

W. Liang, A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, D. Seidel, and L. Maleki, Opt. Lett. 36, 2290 (2011).
[CrossRef]

W. Liang, V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, D. Seidel, and L. Maleki, Opt. Lett. 35, 2822 (2010).
[CrossRef]

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, I. Solomatine, D. Seidel, and L. Maleki, Phys. Rev. Lett. 101, 093902 (2008).
[CrossRef]

A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, Phys. Rev. A 71, 033804 (2005).
[CrossRef]

L. Maleki, V. S. Ilchenko, A. A. Savchenkov, and A. B. Matsko, in Practical Applications of Microresonators in Optics and Photonics, A. B. Matsko, ed. (CRC, 2009), Chap. 3.

Kippenberg, T. J.

T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang, E. Gavartin, R. Holzwarth, M. L. Gorodetsky, and T. J. Kippenberg, Nat. Photonics 6, 480 (2012).
[CrossRef]

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, Science 332, 555 (2011).
[CrossRef]

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, Nature 450, 1214 (2007).
[CrossRef]

T. Herr, V. Brasch, M. L. Gorodetsky, and T. J. Kippenberg, “Soliton mode-locking in optical microresonators,” arXiv:1211.0733 (2012).

Leaird, D. E.

F. Ferdous, H. Miao, P.-H. Wang, D. E. Leaird, K. Srinivasan, L. Chen, V. Aksyuk, and A. M. Weiner, Opt. Express 20, 21033 (2012).
[CrossRef]

F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, Nat. Photonics 5, 770 (2011).
[CrossRef]

Lee, H.

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,” arXiv:1208.5256v1 (2012).

Li, J.

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,” arXiv:1208.5256v1 (2012).

Liang, W.

Maleki, L.

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, D. Seidel, and L. Maleki, Phys. Rev. A 85, 023830 (2012).
[CrossRef]

W. Liang, A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, D. Seidel, and L. Maleki, Opt. Lett. 36, 2290 (2011).
[CrossRef]

W. Liang, V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, D. Seidel, and L. Maleki, Opt. Lett. 35, 2822 (2010).
[CrossRef]

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, I. Solomatine, D. Seidel, and L. Maleki, Phys. Rev. Lett. 101, 093902 (2008).
[CrossRef]

A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, Phys. Rev. A 71, 033804 (2005).
[CrossRef]

L. Maleki, V. S. Ilchenko, A. A. Savchenkov, and A. B. Matsko, in Practical Applications of Microresonators in Optics and Photonics, A. B. Matsko, ed. (CRC, 2009), Chap. 3.

Matsko, A. B.

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, D. Seidel, and L. Maleki, Phys. Rev. A 85, 023830 (2012).
[CrossRef]

W. Liang, A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, D. Seidel, and L. Maleki, Opt. Lett. 36, 2290 (2011).
[CrossRef]

W. Liang, V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, D. Seidel, and L. Maleki, Opt. Lett. 35, 2822 (2010).
[CrossRef]

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, I. Solomatine, D. Seidel, and L. Maleki, Phys. Rev. Lett. 101, 093902 (2008).
[CrossRef]

A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, Phys. Rev. A 71, 033804 (2005).
[CrossRef]

L. Maleki, V. S. Ilchenko, A. A. Savchenkov, and A. B. Matsko, in Practical Applications of Microresonators in Optics and Photonics, A. B. Matsko, ed. (CRC, 2009), Chap. 3.

Miao, H.

F. Ferdous, H. Miao, P.-H. Wang, D. E. Leaird, K. Srinivasan, L. Chen, V. Aksyuk, and A. M. Weiner, Opt. Express 20, 21033 (2012).
[CrossRef]

F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, Nat. Photonics 5, 770 (2011).
[CrossRef]

Papp, S. B.

S. B. Papp and S. A. Diddams, Phys. Rev. A 84, 053833 (2011).
[CrossRef]

Riemensberger, J.

T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang, E. Gavartin, R. Holzwarth, M. L. Gorodetsky, and T. J. Kippenberg, Nat. Photonics 6, 480 (2012).
[CrossRef]

Savchenkov, A. A.

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, D. Seidel, and L. Maleki, Phys. Rev. A 85, 023830 (2012).
[CrossRef]

W. Liang, A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, D. Seidel, and L. Maleki, Opt. Lett. 36, 2290 (2011).
[CrossRef]

W. Liang, V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, D. Seidel, and L. Maleki, Opt. Lett. 35, 2822 (2010).
[CrossRef]

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, I. Solomatine, D. Seidel, and L. Maleki, Phys. Rev. Lett. 101, 093902 (2008).
[CrossRef]

A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, Phys. Rev. A 71, 033804 (2005).
[CrossRef]

L. Maleki, V. S. Ilchenko, A. A. Savchenkov, and A. B. Matsko, in Practical Applications of Microresonators in Optics and Photonics, A. B. Matsko, ed. (CRC, 2009), Chap. 3.

Schliesser, A.

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, Nature 450, 1214 (2007).
[CrossRef]

Seidel, D.

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, D. Seidel, and L. Maleki, Phys. Rev. A 85, 023830 (2012).
[CrossRef]

W. Liang, A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, D. Seidel, and L. Maleki, Opt. Lett. 36, 2290 (2011).
[CrossRef]

W. Liang, V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, D. Seidel, and L. Maleki, Opt. Lett. 35, 2822 (2010).
[CrossRef]

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, I. Solomatine, D. Seidel, and L. Maleki, Phys. Rev. Lett. 101, 093902 (2008).
[CrossRef]

Solomatine, I.

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, I. Solomatine, D. Seidel, and L. Maleki, Phys. Rev. Lett. 101, 093902 (2008).
[CrossRef]

Srinivasan, K.

F. Ferdous, H. Miao, P.-H. Wang, D. E. Leaird, K. Srinivasan, L. Chen, V. Aksyuk, and A. M. Weiner, Opt. Express 20, 21033 (2012).
[CrossRef]

F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, Nat. Photonics 5, 770 (2011).
[CrossRef]

Strekalov, D.

A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, Phys. Rev. A 71, 033804 (2005).
[CrossRef]

Vahala, K. J.

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,” arXiv:1208.5256v1 (2012).

Varghese, L. T.

F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, Nat. Photonics 5, 770 (2011).
[CrossRef]

Wang, C. Y.

T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang, E. Gavartin, R. Holzwarth, M. L. Gorodetsky, and T. J. Kippenberg, Nat. Photonics 6, 480 (2012).
[CrossRef]

Wang, J.

F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, Nat. Photonics 5, 770 (2011).
[CrossRef]

Wang, P.-H.

Weiner, A. M.

F. Ferdous, H. Miao, P.-H. Wang, D. E. Leaird, K. Srinivasan, L. Chen, V. Aksyuk, and A. M. Weiner, Opt. Express 20, 21033 (2012).
[CrossRef]

F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, Nat. Photonics 5, 770 (2011).
[CrossRef]

Wilken, T.

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, Nature 450, 1214 (2007).
[CrossRef]

Yu, N.

Y. K. Chembo and N. Yu, Phys. Rev. A 82, 033801 (2010).
[CrossRef]

Nat. Photonics

T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang, E. Gavartin, R. Holzwarth, M. L. Gorodetsky, and T. J. Kippenberg, Nat. Photonics 6, 480 (2012).
[CrossRef]

F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, Nat. Photonics 5, 770 (2011).
[CrossRef]

Nature

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, Nature 450, 1214 (2007).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A

Y. K. Chembo and N. Yu, Phys. Rev. A 82, 033801 (2010).
[CrossRef]

A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, Phys. Rev. A 71, 033804 (2005).
[CrossRef]

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, D. Seidel, and L. Maleki, Phys. Rev. A 85, 023830 (2012).
[CrossRef]

S. B. Papp and S. A. Diddams, Phys. Rev. A 84, 053833 (2011).
[CrossRef]

Phys. Rev. Lett.

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, I. Solomatine, D. Seidel, and L. Maleki, Phys. Rev. Lett. 101, 093902 (2008).
[CrossRef]

K. J. Blow and N. J. Doran, Phys. Rev. Lett. 52, 526 (1984).
[CrossRef]

Science

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, Science 332, 555 (2011).
[CrossRef]

Other

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,” arXiv:1208.5256v1 (2012).

L. Maleki, V. S. Ilchenko, A. A. Savchenkov, and A. B. Matsko, in Practical Applications of Microresonators in Optics and Photonics, A. B. Matsko, ed. (CRC, 2009), Chap. 3.

T. Herr, V. Brasch, M. L. Gorodetsky, and T. J. Kippenberg, “Soliton mode-locking in optical microresonators,” arXiv:1211.0733 (2012).

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Figures (4)

Fig. 1.
Fig. 1.

Optical spectrum of an incoherent Kerr comb [(a),(b)] and associated RF signals [(c),(d)] generated by the comb on a fast photodiode. (a) and (c) Represent the theoretical simulation results, and (b) and (d) are for the measurement results. To simulate the spectrum of the chaotic comb (a), we time averaged the chaotic comb spectra over tave=100/γ. This averaging corresponds to the averaging introduced by an OSA used to observe the comb spectrum in the experiment.

Fig. 2.
Fig. 2.

Dependence of the normalized field amplitude (A=b51, for the case described in the Letter) in the optically pumped mode on the frequency of the pump light. Solid blue lines show stable branches corresponding to generation of various Kerr combs (see Fig. 3); dots correspond to generation of chaotic combs or combs with time-depending envelopes. Plot (a) stands for the soft excitation regime [6]. The initial amplitude in this case is selected to be |bj|(g/γ)1/2=104, which approximately corresponds to the field in the mode containing 1/2 photon. Branches (1–7) are not accessible in this case. Plot (b) stands for hard excitation. It is assumed that A(g/γ)1/2=10, b50(g/γ)1/2=0.33, and b52(g/γ)1/2=0.33 to get the solutions. Stable solutions are realized for the particular nonzero initial condition and detuning selection. The comb regimes are unstable with respect to the initial conditions, though. In other words, jumping to a particular solution branch is not guaranteed if the initial conditions are slightly modified. The comb can also be excited from a nearly zero initial excitation of modes, if the pump frequency is shifted nonadiabatically (see Fig. 4).

Fig. 3.
Fig. 3.

Simulated spectra of the combs corresponding to stable solution branches shown in Fig. 2.

Fig. 4.
Fig. 4.

Density plot illustrating time dependence of amplitude of comb harmonics. This is an example of excitation of a regular frequency comb in an empty microresonator via nonadiabatic change of the oscillator parameters. The comb forms starting at |bj(t=0)|(g/γ)1/2=104, when the normalized frequency detuning ((ωω0)/γ) between the pump light and the pumped mode abruptly changes from 5 to 27 at t=1.4/γ. A chaotic frequency comb is generated if the detuning stays constant at the initial level (5). The abrupt change of the detuning results in generation of a regular broad coherent comb at branch (1) of Fig. 2. The transient process reveals breather formation in the resonator.

Equations (1)

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b^˙j=(γ+iωj)b^j+i[V^,b^j]+Feiωtδj0,j,

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