Abstract

We study fundamental timing jitter in repetition rate of a mode locked Kerr frequency comb generated in an externally pumped nonlinear ring resonator. We show that the increase in the integrated power of the comb harmonics, and the corresponding decrease of the duration of the associated pulse, results in the increase of low frequency noise, and a decrease in high frequency noise.

© 2013 OSA

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  1. P. Del-Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature450, 1214–1217 (2007).
    [CrossRef]
  2. P. Del-Haye, O. Arcizet, A. Schliesser, R. Holzwarth, and T. J. Kippenberg, “Full stabilization of a microresonator-based optical frequency comb,” Phys. Rev. Lett.101, 053903 (2008).
    [CrossRef]
  3. A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, I. Solomatine, D. Seidel, and L. Maleki, “Tunable optical frequency comb with a crystalline whispering gallery mode resonator,” Phys. Rev. Lett.101, 093902 (2008).
    [CrossRef] [PubMed]
  4. I. S. Grudinin, N. Yu, and L. Maleki, “Generation of optical frequency combs with a CaF2resonator,” Opt. Lett.34, 878–880 (2009).
    [CrossRef] [PubMed]
  5. L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. E. Little, and D. J. Moss, “CMOS-compatible integrated optical hyper-parametric oscillator,” Nat. Photonics4, 41–45 (2010).
    [CrossRef]
  6. F. Ferdous, H. X. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, “Spectral line-by-line pulse shaping of on-chip microresonator frequency combs,” Nat. Photonics5, 770–776 (2011).
    [CrossRef]
  7. M. A. Foster, J. S. Levy, O. Kuzucu, K. Saha, M. Lipson, and A. L. Gaeta, “Silicon-based monolithic optical frequency comb source,” Opt. Express19, 14233–14239 (2011).
    [CrossRef] [PubMed]
  8. S. B. Papp and S. A. Diddams, “Spectral and temporal characterization of a fused quartz microresonator optical frequency comb,” Phys. Rev. A84, 053833 (2011).
    [CrossRef]
  9. P. Del-Haye, T. Herr, E. Gavartin, M. L. Gorodetsky, R. Holzwarth, and T. J. Kippenberg, “Octave spanning tunable frequency comb from a microresonator,” Phys. Rev. Lett.107, 063901 (2011).
    [CrossRef]
  10. W. Liang, A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, D. Seidel, and L. Maleki, “Generation of near-infrared frequency combs from a MgF2whispering gallery mode resonator,” Opt. Lett.36, 2290–2292 (2011).
    [CrossRef] [PubMed]
  11. A. A. Savchenkov, A. B. Matsko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Kerr combs with selectable central frequency,” Nat. Photonics5, 293–296 (2011).
    [CrossRef]
  12. Y. Okawachi, K. Saha, J. S. Levy, Y. H. Wen, M. Lipson, and A. L. Gaeta, “Octave spanning frequency comb generation in a silicon nitride chip,” Opt. Lett.36, 3398–3400 (2011).
    [CrossRef] [PubMed]
  13. I. S. Grudinin, L. Baumgartel, and N. Yu, “Frequency comb from a microresonator with engineered spectrum,” Opt. Express20, 6604–6609 (2012).
    [CrossRef] [PubMed]
  14. A. R. Johnson, Y. Okawachi, J. S. Levy, J. Cardenas, K. Saha, M. Lipson, and A. L. Gaeta, “Chip-based frequency combs with sub-100 GHz repetition rates,” Opt. Lett.37, 875–877 (2012).
    [CrossRef] [PubMed]
  15. F. Ferdous, H. X. Miao, P. H. Wang, D. E. Leaird, K. Srinivasan, L. Chen, V. Aksyuk, and A. M. Weiner, “Probing coherence in microcavity frequency combs via optical pulse shaping,” Opt. Express20, 21033–21043 (2012).
    [CrossRef] [PubMed]
  16. 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]
  17. A. A. Savchenkov, A. B. Matsko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Transient regime of Kerr-frequency-comb formation,” Phys. Rev. A86, 013838 (2012).
    [CrossRef]
  18. A. A. Savchenkov, A. B. Matsko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Kerr frequency comb generation in overmoded resonators,” Opt. Express20, 27290–27298 (2012).
    [CrossRef] [PubMed]
  19. P. H. Wang, F. Ferdous, H. X. Miao, J. Wang, D. E. Leaird, K. Srinivasan, L. Chen, V. Aksyuk, and A. M. Weiner, “Observation of correlation between route to formation, coherence, noise, and communication performance of Kerr combs,” Opt. Express20, 29284–29295 (2012).
    [CrossRef]
  20. S. B. Papp, P. Del’Haye, and S. A. Diddams, “Mechanical control of a microrod-resonator optical frequency comb,” Phys. Rev. X3, 031003 (2013).
    [CrossRef]
  21. K. Saha, Y. Okawachi, B. Shim, J. S. Levy, R. Salem, A. R. Johnson, M. A. Foster, M. R. E. Lamont, M. Lipson, and A. L. Gaeta, “Modelocking and femtosecond pulse generation in chip-based frequency combs,” Opt. Express21, 1335–1343 (2013).
    [CrossRef] [PubMed]
  22. T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Mode-locking in an optical microresonator via soliton formation,” arXiv:1211.0733v2 (2013).
  23. S. B. Papp, P. Del’Haye, and S. A. Diddams, “Parametric seeding of a microresonator optical frequency comb,” Opt. Express21, 17615–17624 (2013).
    [CrossRef] [PubMed]
  24. P. Del’Haye, S. B. Papp, and S. A. Diddams, “Self-injection locking and phase-locked states in microresonator-based optical frequency combs,” arXiv:1307.4091 (2013).
  25. T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science332, 555–559 (2011).
    [CrossRef] [PubMed]
  26. M. Nakazawa, K. Suzuki, and H. A. Haus, “The modulational instability laser-Part I: Experiment,” IEEE J. Quantum Electron.25, 2036–2044 (1989).
    [CrossRef]
  27. M. Haelterman, S. Trillo, and S. Wabnitz, “Additive-modulation-instability ring laser in the normal dispersion regime of a fiber,” Opt. Lett.17, 745–747 (1992).
    [CrossRef] [PubMed]
  28. S. Coen and M. Haelterman, “Modulational instability induced by cavity boundary conditions in a normally dispersive optical fiber,” Phys. Rev. Lett.79, 4139–4142 (1997).
    [CrossRef]
  29. S. Coen and M. Haelterman, “Continuous-wave ultrahigh-repetition-rate pulse-train generation through modulational instability in a passive fiber cavity,” Opt. Lett.26, 39–41 (2001).
    [CrossRef]
  30. D. K. Serkland and P. Kumar, “Tunable fiber-optic parametric oscillator,” Opt. Lett.24, 92–94 (1999).
    [CrossRef]
  31. L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passive optical systems,” Phys. Rev. Lett.58, 2209–2211 (1987).
    [CrossRef] [PubMed]
  32. Y. K. Chembo, D. V. Strekalov, and N. Yu, “Spectrum and dynamics of optical frequency combs generated with monolithic whispering gallery mode resonators,” Phys. Rev. Lett.104, 103902 (2010).
    [CrossRef] [PubMed]
  33. Y. K. Chembo and N. Yu, “On the generation of octave-spanning optical frequency combs using monolithic whispering-gallery-mode microresonators,” Opt. Lett.35, 2696–2698 (2010).
    [CrossRef] [PubMed]
  34. Y. K. Chembo and N. Yu, “Modal expansion approach to optical-frequency-comb generation with monolithic whispering-gallery-mode resonators,” Phys. Rev. A82, 033801 (2010).
    [CrossRef]
  35. A. B. Matsko, A. A. Savchenkov, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Mode-locked Kerr frequency combs,” Opt. Lett.36, 2845–2847 (2011).
    [CrossRef] [PubMed]
  36. A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, D. Seidel, and L. Maleki, “Hard and soft excitation regimes of Kerr frequency combs,” Phys. Rev. A85, 023830 (2012).
    [CrossRef]
  37. A. B. Matsko, A. A. Savchenkov, and L. Maleki, “Normal GVD Kerr frequency comb,” Opt. Lett.37, 43–45 (2012).
    [CrossRef] [PubMed]
  38. A. B. Matsko, A. A. Savchenkov, and L. Maleki, “On excitation of breather solitons in an optical microresonator,” Opt. Lett.37, 4856–4858 (2012).
    [CrossRef] [PubMed]
  39. T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang, E. Gavartin, R. Holzwarth, M. L. Gorodetsky, and T. J. Kippenberg, “Universal formation dynamics and noise of Kerr frequency combs in microresonators,” Nat. Photonics6, 480–487 (2012).
    [CrossRef]
  40. S. Coen, H. G. Randle, T. Sylvestre, and M. Erkintalo, “Modeling of octave-spanning Kerr frequency combs using a generalized mean-field Lugiato-Lefever model,” Opt. Lett.38, 37–39 (2013).
    [CrossRef] [PubMed]
  41. S. Coen and M. Erkintalo, “Universal scaling laws of Kerr frequency combs,” Opt. Lett.38, 1790–1792 (2013).
    [CrossRef] [PubMed]
  42. A. B. Matsko, W. Liang, A. A. Savchenkov, and L. Maleki, “Chaotic dynamics of frequency combs generated with continuously pumped nonlinear microresonators,” Opt. Lett.38, 525–527 (2013).
    [CrossRef] [PubMed]
  43. T. Hansson, D. Modotto, and S. Wabnitz, “Dynamics of the modulational instability in microresonator frequency combs,” Phys. Rev. A88, 023819 (2013).
    [CrossRef]
  44. C. Godey, I. Balakireva, A. Coillet, and Y. K. Chembo, “Stability analysis of the Lugiato-Lefever model for Kerr optical frequency combs. Part I: Case of normal dispersion,” arXiv:1308.2539 (2013).
  45. I. Balakireva, A. Coillet, C. Godey, and Y. K. Chembo, “Stability analysis of the Lugiato-Lefever model for Kerr optical frequency combs. Part II: Case of anomalous dispersion,” arXiv:1308.2542 (2013).
  46. M. Lamont, Y. Okawachi, and A. L. Gaeta, “Route to stabilized ultrabroadband microresonator-based frequency combs,” arXiv:1305.4921 (2013).
  47. S. Wabnitz, “Suppression of interactions in a phase-locked soliton optical memory,” Opt. Lett.18, 601–603 (1993).
    [CrossRef] [PubMed]
  48. I. V. Barashenkov and Y. S. Smirnov, “Existence and stability chart for the ac-driven, damped nonlinear Schrödinger solitons,” Phys. Rev. E54, 5707–5725 (1996).
    [CrossRef]
  49. J.-M. Ghidaglia, “Finite dimensional behavior for weakly damped driven Schrödinger equation,” Ann. Inst. Henri Poincare5, 365–405 (1988).
  50. N. I. Karachalios and N. M. Stavrakakis, “Global attractor for the weakly damped driven Schrodinger equation in H2(R),” Nonlinear Diff. Eq. Appl.9, 347–360 (2002).
    [CrossRef]
  51. C. Zhu, “Attractor of the nonlinear Schrodinger equation,” Commun. Math. Anal.4, 67–75 (2008).
  52. K. J. Blow and N. J. Doran, “Global and local chaos in the pumped nonlinear Schrödinger equation,” Phys. Rev. Lett.52, 526–529 (1984)
  53. Q.-H. Park and H. J. Shin, “Parametric control of soliton light traffic by cw traffic light,” Phys. Rev. Lett.82, 4432–4435 (1999).
    [CrossRef]
  54. S. Li, L. Li, Z. Li, and G. Zhou, “Properties of soliton solutions on a cw background in optical fibers with higher-order effects,” J. Opt. Soc. Am. B21, 2089–2094 (2004).
    [CrossRef]
  55. A. B. Matsko, A. A. Savchenkov, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Whispering gallery mode oscillators and optical comb generators,” in Proceedings of 7th Symposium on Frequency Standards and Metrology, L. Maleki, ed., (World Scientific, 2009), pp. 539–558.
  56. A. A. Savchenkov, A. B. Matsko, D. Strekalov, M. Mohageg, V. S. Ilchenko, and L. Maleki, “Low threshold optical oscillations in a whispering gallery mode CaF2resonator,” Phys. Rev. Lett.93, 243905 (2004).
    [CrossRef]
  57. A. A. Savchenkov, E. Rubiola, A. B. Matsko, V. S. Ilchenko, and L. Maleki, “Phase noise of whispering gallery photonic hyper-parametric microwave oscillators,” Opt. Express16, 4130–4144 (2008).
    [CrossRef] [PubMed]
  58. H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron.29, 983–996 (1993).
    [CrossRef]
  59. A. Hasegawa, “Soliton-based optical communications: An overwiew,” IEEE J. Sel. Top. Quantum Electron.6, 1161–1172 (2000).
    [CrossRef]
  60. A. Coillet, I. Balakireva, R. Henriet, K. Saleh, L. Larger, J. M. Dudley, C. R. Menyuk, and Y. K. Chembo, “Azimuthal Turing patterns, bright and dark cavity solitons in Kerr combs generated with whispering-gallery-mode resonators,” IEEE Photonics J.5, 6100409 (2013).
    [CrossRef]
  61. A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Optical hyperparametric oscillations in a whispering-gallery-mode resonator: Threshold and phase diffusion,” Phys. Rev. A71, 033804 (2005).
    [CrossRef]

2013 (11)

S. B. Papp, P. Del’Haye, and S. A. Diddams, “Mechanical control of a microrod-resonator optical frequency comb,” Phys. Rev. X3, 031003 (2013).
[CrossRef]

T. Hansson, D. Modotto, and S. Wabnitz, “Dynamics of the modulational instability in microresonator frequency combs,” Phys. Rev. A88, 023819 (2013).
[CrossRef]

C. Godey, I. Balakireva, A. Coillet, and Y. K. Chembo, “Stability analysis of the Lugiato-Lefever model for Kerr optical frequency combs. Part I: Case of normal dispersion,” arXiv:1308.2539 (2013).

I. Balakireva, A. Coillet, C. Godey, and Y. K. Chembo, “Stability analysis of the Lugiato-Lefever model for Kerr optical frequency combs. Part II: Case of anomalous dispersion,” arXiv:1308.2542 (2013).

M. Lamont, Y. Okawachi, and A. L. Gaeta, “Route to stabilized ultrabroadband microresonator-based frequency combs,” arXiv:1305.4921 (2013).

A. Coillet, I. Balakireva, R. Henriet, K. Saleh, L. Larger, J. M. Dudley, C. R. Menyuk, and Y. K. Chembo, “Azimuthal Turing patterns, bright and dark cavity solitons in Kerr combs generated with whispering-gallery-mode resonators,” IEEE Photonics J.5, 6100409 (2013).
[CrossRef]

S. Coen, H. G. Randle, T. Sylvestre, and M. Erkintalo, “Modeling of octave-spanning Kerr frequency combs using a generalized mean-field Lugiato-Lefever model,” Opt. Lett.38, 37–39 (2013).
[CrossRef] [PubMed]

K. Saha, Y. Okawachi, B. Shim, J. S. Levy, R. Salem, A. R. Johnson, M. A. Foster, M. R. E. Lamont, M. Lipson, and A. L. Gaeta, “Modelocking and femtosecond pulse generation in chip-based frequency combs,” Opt. Express21, 1335–1343 (2013).
[CrossRef] [PubMed]

A. B. Matsko, W. Liang, A. A. Savchenkov, and L. Maleki, “Chaotic dynamics of frequency combs generated with continuously pumped nonlinear microresonators,” Opt. Lett.38, 525–527 (2013).
[CrossRef] [PubMed]

S. Coen and M. Erkintalo, “Universal scaling laws of Kerr frequency combs,” Opt. Lett.38, 1790–1792 (2013).
[CrossRef] [PubMed]

S. B. Papp, P. Del’Haye, and S. A. Diddams, “Parametric seeding of a microresonator optical frequency comb,” Opt. Express21, 17615–17624 (2013).
[CrossRef] [PubMed]

2012 (11)

A. B. Matsko, A. A. Savchenkov, and L. Maleki, “Normal GVD Kerr frequency comb,” Opt. Lett.37, 43–45 (2012).
[CrossRef] [PubMed]

A. R. Johnson, Y. Okawachi, J. S. Levy, J. Cardenas, K. Saha, M. Lipson, and A. L. Gaeta, “Chip-based frequency combs with sub-100 GHz repetition rates,” Opt. Lett.37, 875–877 (2012).
[CrossRef] [PubMed]

I. S. Grudinin, L. Baumgartel, and N. Yu, “Frequency comb from a microresonator with engineered spectrum,” Opt. Express20, 6604–6609 (2012).
[CrossRef] [PubMed]

F. Ferdous, H. X. Miao, P. H. Wang, D. E. Leaird, K. Srinivasan, L. Chen, V. Aksyuk, and A. M. Weiner, “Probing coherence in microcavity frequency combs via optical pulse shaping,” Opt. Express20, 21033–21043 (2012).
[CrossRef] [PubMed]

A. A. Savchenkov, A. B. Matsko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Kerr frequency comb generation in overmoded resonators,” Opt. Express20, 27290–27298 (2012).
[CrossRef] [PubMed]

A. B. Matsko, A. A. Savchenkov, and L. Maleki, “On excitation of breather solitons in an optical microresonator,” Opt. Lett.37, 4856–4858 (2012).
[CrossRef] [PubMed]

P. H. Wang, F. Ferdous, H. X. Miao, J. Wang, D. E. Leaird, K. Srinivasan, L. Chen, V. Aksyuk, and A. M. Weiner, “Observation of correlation between route to formation, coherence, noise, and communication performance of Kerr combs,” Opt. Express20, 29284–29295 (2012).
[CrossRef]

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, D. Seidel, and L. Maleki, “Hard and soft excitation regimes of Kerr frequency combs,” Phys. Rev. A85, 023830 (2012).
[CrossRef]

T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang, E. Gavartin, R. Holzwarth, M. L. Gorodetsky, and T. J. Kippenberg, “Universal formation dynamics and noise of Kerr frequency combs in microresonators,” Nat. Photonics6, 480–487 (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]

A. A. Savchenkov, A. B. Matsko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Transient regime of Kerr-frequency-comb formation,” Phys. Rev. A86, 013838 (2012).
[CrossRef]

2011 (9)

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science332, 555–559 (2011).
[CrossRef] [PubMed]

F. Ferdous, H. X. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, “Spectral line-by-line pulse shaping of on-chip microresonator frequency combs,” Nat. Photonics5, 770–776 (2011).
[CrossRef]

S. B. Papp and S. A. Diddams, “Spectral and temporal characterization of a fused quartz microresonator optical frequency comb,” Phys. Rev. A84, 053833 (2011).
[CrossRef]

P. Del-Haye, T. Herr, E. Gavartin, M. L. Gorodetsky, R. Holzwarth, and T. J. Kippenberg, “Octave spanning tunable frequency comb from a microresonator,” Phys. Rev. Lett.107, 063901 (2011).
[CrossRef]

A. A. Savchenkov, A. B. Matsko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Kerr combs with selectable central frequency,” Nat. Photonics5, 293–296 (2011).
[CrossRef]

W. Liang, A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, D. Seidel, and L. Maleki, “Generation of near-infrared frequency combs from a MgF2whispering gallery mode resonator,” Opt. Lett.36, 2290–2292 (2011).
[CrossRef] [PubMed]

M. A. Foster, J. S. Levy, O. Kuzucu, K. Saha, M. Lipson, and A. L. Gaeta, “Silicon-based monolithic optical frequency comb source,” Opt. Express19, 14233–14239 (2011).
[CrossRef] [PubMed]

A. B. Matsko, A. A. Savchenkov, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Mode-locked Kerr frequency combs,” Opt. Lett.36, 2845–2847 (2011).
[CrossRef] [PubMed]

Y. Okawachi, K. Saha, J. S. Levy, Y. H. Wen, M. Lipson, and A. L. Gaeta, “Octave spanning frequency comb generation in a silicon nitride chip,” Opt. Lett.36, 3398–3400 (2011).
[CrossRef] [PubMed]

2010 (4)

Y. K. Chembo and N. Yu, “On the generation of octave-spanning optical frequency combs using monolithic whispering-gallery-mode microresonators,” Opt. Lett.35, 2696–2698 (2010).
[CrossRef] [PubMed]

L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. E. Little, and D. J. Moss, “CMOS-compatible integrated optical hyper-parametric oscillator,” Nat. Photonics4, 41–45 (2010).
[CrossRef]

Y. K. Chembo, D. V. Strekalov, and N. Yu, “Spectrum and dynamics of optical frequency combs generated with monolithic whispering gallery mode resonators,” Phys. Rev. Lett.104, 103902 (2010).
[CrossRef] [PubMed]

Y. K. Chembo and N. Yu, “Modal expansion approach to optical-frequency-comb generation with monolithic whispering-gallery-mode resonators,” Phys. Rev. A82, 033801 (2010).
[CrossRef]

2009 (1)

2008 (4)

A. A. Savchenkov, E. Rubiola, A. B. Matsko, V. S. Ilchenko, and L. Maleki, “Phase noise of whispering gallery photonic hyper-parametric microwave oscillators,” Opt. Express16, 4130–4144 (2008).
[CrossRef] [PubMed]

P. Del-Haye, O. Arcizet, A. Schliesser, R. Holzwarth, and T. J. Kippenberg, “Full stabilization of a microresonator-based optical frequency comb,” Phys. Rev. Lett.101, 053903 (2008).
[CrossRef]

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, I. Solomatine, D. Seidel, and L. Maleki, “Tunable optical frequency comb with a crystalline whispering gallery mode resonator,” Phys. Rev. Lett.101, 093902 (2008).
[CrossRef] [PubMed]

C. Zhu, “Attractor of the nonlinear Schrodinger equation,” Commun. Math. Anal.4, 67–75 (2008).

2007 (1)

P. Del-Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature450, 1214–1217 (2007).
[CrossRef]

2005 (1)

A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Optical hyperparametric oscillations in a whispering-gallery-mode resonator: Threshold and phase diffusion,” Phys. Rev. A71, 033804 (2005).
[CrossRef]

2004 (2)

A. A. Savchenkov, A. B. Matsko, D. Strekalov, M. Mohageg, V. S. Ilchenko, and L. Maleki, “Low threshold optical oscillations in a whispering gallery mode CaF2resonator,” Phys. Rev. Lett.93, 243905 (2004).
[CrossRef]

S. Li, L. Li, Z. Li, and G. Zhou, “Properties of soliton solutions on a cw background in optical fibers with higher-order effects,” J. Opt. Soc. Am. B21, 2089–2094 (2004).
[CrossRef]

2002 (1)

N. I. Karachalios and N. M. Stavrakakis, “Global attractor for the weakly damped driven Schrodinger equation in H2(R),” Nonlinear Diff. Eq. Appl.9, 347–360 (2002).
[CrossRef]

2001 (1)

2000 (1)

A. Hasegawa, “Soliton-based optical communications: An overwiew,” IEEE J. Sel. Top. Quantum Electron.6, 1161–1172 (2000).
[CrossRef]

1999 (2)

Q.-H. Park and H. J. Shin, “Parametric control of soliton light traffic by cw traffic light,” Phys. Rev. Lett.82, 4432–4435 (1999).
[CrossRef]

D. K. Serkland and P. Kumar, “Tunable fiber-optic parametric oscillator,” Opt. Lett.24, 92–94 (1999).
[CrossRef]

1997 (1)

S. Coen and M. Haelterman, “Modulational instability induced by cavity boundary conditions in a normally dispersive optical fiber,” Phys. Rev. Lett.79, 4139–4142 (1997).
[CrossRef]

1996 (1)

I. V. Barashenkov and Y. S. Smirnov, “Existence and stability chart for the ac-driven, damped nonlinear Schrödinger solitons,” Phys. Rev. E54, 5707–5725 (1996).
[CrossRef]

1993 (2)

H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron.29, 983–996 (1993).
[CrossRef]

S. Wabnitz, “Suppression of interactions in a phase-locked soliton optical memory,” Opt. Lett.18, 601–603 (1993).
[CrossRef] [PubMed]

1992 (1)

1989 (1)

M. Nakazawa, K. Suzuki, and H. A. Haus, “The modulational instability laser-Part I: Experiment,” IEEE J. Quantum Electron.25, 2036–2044 (1989).
[CrossRef]

1988 (1)

J.-M. Ghidaglia, “Finite dimensional behavior for weakly damped driven Schrödinger equation,” Ann. Inst. Henri Poincare5, 365–405 (1988).

1987 (1)

L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passive optical systems,” Phys. Rev. Lett.58, 2209–2211 (1987).
[CrossRef] [PubMed]

1984 (1)

K. J. Blow and N. J. Doran, “Global and local chaos in the pumped nonlinear Schrödinger equation,” Phys. Rev. Lett.52, 526–529 (1984)

Aksyuk, V.

Arcizet, O.

P. Del-Haye, O. Arcizet, A. Schliesser, R. Holzwarth, and T. J. Kippenberg, “Full stabilization of a microresonator-based optical frequency comb,” Phys. Rev. Lett.101, 053903 (2008).
[CrossRef]

P. Del-Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature450, 1214–1217 (2007).
[CrossRef]

Balakireva, I.

C. Godey, I. Balakireva, A. Coillet, and Y. K. Chembo, “Stability analysis of the Lugiato-Lefever model for Kerr optical frequency combs. Part I: Case of normal dispersion,” arXiv:1308.2539 (2013).

I. Balakireva, A. Coillet, C. Godey, and Y. K. Chembo, “Stability analysis of the Lugiato-Lefever model for Kerr optical frequency combs. Part II: Case of anomalous dispersion,” arXiv:1308.2542 (2013).

A. Coillet, I. Balakireva, R. Henriet, K. Saleh, L. Larger, J. M. Dudley, C. R. Menyuk, and Y. K. Chembo, “Azimuthal Turing patterns, bright and dark cavity solitons in Kerr combs generated with whispering-gallery-mode resonators,” IEEE Photonics J.5, 6100409 (2013).
[CrossRef]

Barashenkov, I. V.

I. V. Barashenkov and Y. S. Smirnov, “Existence and stability chart for the ac-driven, damped nonlinear Schrödinger solitons,” Phys. Rev. E54, 5707–5725 (1996).
[CrossRef]

Baumgartel, L.

Blow, K. J.

K. J. Blow and N. J. Doran, “Global and local chaos in the pumped nonlinear Schrödinger equation,” Phys. Rev. Lett.52, 526–529 (1984)

Brasch, V.

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Mode-locking in an optical microresonator via soliton formation,” arXiv:1211.0733v2 (2013).

Cardenas, J.

Chembo, Y. K.

A. Coillet, I. Balakireva, R. Henriet, K. Saleh, L. Larger, J. M. Dudley, C. R. Menyuk, and Y. K. Chembo, “Azimuthal Turing patterns, bright and dark cavity solitons in Kerr combs generated with whispering-gallery-mode resonators,” IEEE Photonics J.5, 6100409 (2013).
[CrossRef]

I. Balakireva, A. Coillet, C. Godey, and Y. K. Chembo, “Stability analysis of the Lugiato-Lefever model for Kerr optical frequency combs. Part II: Case of anomalous dispersion,” arXiv:1308.2542 (2013).

C. Godey, I. Balakireva, A. Coillet, and Y. K. Chembo, “Stability analysis of the Lugiato-Lefever model for Kerr optical frequency combs. Part I: Case of normal dispersion,” arXiv:1308.2539 (2013).

Y. K. Chembo and N. Yu, “On the generation of octave-spanning optical frequency combs using monolithic whispering-gallery-mode microresonators,” Opt. Lett.35, 2696–2698 (2010).
[CrossRef] [PubMed]

Y. K. Chembo, D. V. Strekalov, and N. Yu, “Spectrum and dynamics of optical frequency combs generated with monolithic whispering gallery mode resonators,” Phys. Rev. Lett.104, 103902 (2010).
[CrossRef] [PubMed]

Y. K. Chembo and N. Yu, “Modal expansion approach to optical-frequency-comb generation with monolithic whispering-gallery-mode resonators,” Phys. Rev. A82, 033801 (2010).
[CrossRef]

Chen, L.

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,” Phys. Rev. Lett.109, 233901 (2012).
[CrossRef]

Chu, S.

L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. E. Little, and D. J. Moss, “CMOS-compatible integrated optical hyper-parametric oscillator,” Nat. Photonics4, 41–45 (2010).
[CrossRef]

Coen, S.

Coillet, A.

A. Coillet, I. Balakireva, R. Henriet, K. Saleh, L. Larger, J. M. Dudley, C. R. Menyuk, and Y. K. Chembo, “Azimuthal Turing patterns, bright and dark cavity solitons in Kerr combs generated with whispering-gallery-mode resonators,” IEEE Photonics J.5, 6100409 (2013).
[CrossRef]

C. Godey, I. Balakireva, A. Coillet, and Y. K. Chembo, “Stability analysis of the Lugiato-Lefever model for Kerr optical frequency combs. Part I: Case of normal dispersion,” arXiv:1308.2539 (2013).

I. Balakireva, A. Coillet, C. Godey, and Y. K. Chembo, “Stability analysis of the Lugiato-Lefever model for Kerr optical frequency combs. Part II: Case of anomalous dispersion,” arXiv:1308.2542 (2013).

Del’Haye, P.

S. B. Papp, P. Del’Haye, and S. A. Diddams, “Mechanical control of a microrod-resonator optical frequency comb,” Phys. Rev. X3, 031003 (2013).
[CrossRef]

S. B. Papp, P. Del’Haye, and S. A. Diddams, “Parametric seeding of a microresonator optical frequency comb,” Opt. Express21, 17615–17624 (2013).
[CrossRef] [PubMed]

P. Del’Haye, S. B. Papp, and S. A. Diddams, “Self-injection locking and phase-locked states in microresonator-based optical frequency combs,” arXiv:1307.4091 (2013).

Del-Haye, P.

P. Del-Haye, T. Herr, E. Gavartin, M. L. Gorodetsky, R. Holzwarth, and T. J. Kippenberg, “Octave spanning tunable frequency comb from a microresonator,” Phys. Rev. Lett.107, 063901 (2011).
[CrossRef]

P. Del-Haye, O. Arcizet, A. Schliesser, R. Holzwarth, and T. J. Kippenberg, “Full stabilization of a microresonator-based optical frequency comb,” Phys. Rev. Lett.101, 053903 (2008).
[CrossRef]

P. Del-Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature450, 1214–1217 (2007).
[CrossRef]

Diddams, S. A.

S. B. Papp, P. Del’Haye, and S. A. Diddams, “Mechanical control of a microrod-resonator optical frequency comb,” Phys. Rev. X3, 031003 (2013).
[CrossRef]

S. B. Papp, P. Del’Haye, and S. A. Diddams, “Parametric seeding of a microresonator optical frequency comb,” Opt. Express21, 17615–17624 (2013).
[CrossRef] [PubMed]

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science332, 555–559 (2011).
[CrossRef] [PubMed]

S. B. Papp and S. A. Diddams, “Spectral and temporal characterization of a fused quartz microresonator optical frequency comb,” Phys. Rev. A84, 053833 (2011).
[CrossRef]

P. Del’Haye, S. B. Papp, and S. A. Diddams, “Self-injection locking and phase-locked states in microresonator-based optical frequency combs,” arXiv:1307.4091 (2013).

Doran, N. J.

K. J. Blow and N. J. Doran, “Global and local chaos in the pumped nonlinear Schrödinger equation,” Phys. Rev. Lett.52, 526–529 (1984)

Duchesne, D.

L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. E. Little, and D. J. Moss, “CMOS-compatible integrated optical hyper-parametric oscillator,” Nat. Photonics4, 41–45 (2010).
[CrossRef]

Dudley, J. M.

A. Coillet, I. Balakireva, R. Henriet, K. Saleh, L. Larger, J. M. Dudley, C. R. Menyuk, and Y. K. Chembo, “Azimuthal Turing patterns, bright and dark cavity solitons in Kerr combs generated with whispering-gallery-mode resonators,” IEEE Photonics J.5, 6100409 (2013).
[CrossRef]

Erkintalo, M.

Ferdous, F.

Ferrera, M.

L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. E. Little, and D. J. Moss, “CMOS-compatible integrated optical hyper-parametric oscillator,” Nat. Photonics4, 41–45 (2010).
[CrossRef]

Foster, M. A.

Gaeta, A. L.

Gavartin, E.

T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang, E. Gavartin, R. Holzwarth, M. L. Gorodetsky, and T. J. Kippenberg, “Universal formation dynamics and noise of Kerr frequency combs in microresonators,” Nat. Photonics6, 480–487 (2012).
[CrossRef]

P. Del-Haye, T. Herr, E. Gavartin, M. L. Gorodetsky, R. Holzwarth, and T. J. Kippenberg, “Octave spanning tunable frequency comb from a microresonator,” Phys. Rev. Lett.107, 063901 (2011).
[CrossRef]

Ghidaglia, J.-M.

J.-M. Ghidaglia, “Finite dimensional behavior for weakly damped driven Schrödinger equation,” Ann. Inst. Henri Poincare5, 365–405 (1988).

Godey, C.

I. Balakireva, A. Coillet, C. Godey, and Y. K. Chembo, “Stability analysis of the Lugiato-Lefever model for Kerr optical frequency combs. Part II: Case of anomalous dispersion,” arXiv:1308.2542 (2013).

C. Godey, I. Balakireva, A. Coillet, and Y. K. Chembo, “Stability analysis of the Lugiato-Lefever model for Kerr optical frequency combs. Part I: Case of normal dispersion,” arXiv:1308.2539 (2013).

Gorodetsky, M. L.

T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang, E. Gavartin, R. Holzwarth, M. L. Gorodetsky, and T. J. Kippenberg, “Universal formation dynamics and noise of Kerr frequency combs in microresonators,” Nat. Photonics6, 480–487 (2012).
[CrossRef]

P. Del-Haye, T. Herr, E. Gavartin, M. L. Gorodetsky, R. Holzwarth, and T. J. Kippenberg, “Octave spanning tunable frequency comb from a microresonator,” Phys. Rev. Lett.107, 063901 (2011).
[CrossRef]

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Mode-locking in an optical microresonator via soliton formation,” arXiv:1211.0733v2 (2013).

Grudinin, I. S.

Haelterman, M.

Hansson, T.

T. Hansson, D. Modotto, and S. Wabnitz, “Dynamics of the modulational instability in microresonator frequency combs,” Phys. Rev. A88, 023819 (2013).
[CrossRef]

Hartinger, K.

T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang, E. Gavartin, R. Holzwarth, M. L. Gorodetsky, and T. J. Kippenberg, “Universal formation dynamics and noise of Kerr frequency combs in microresonators,” Nat. Photonics6, 480–487 (2012).
[CrossRef]

Hasegawa, A.

A. Hasegawa, “Soliton-based optical communications: An overwiew,” IEEE J. Sel. Top. Quantum Electron.6, 1161–1172 (2000).
[CrossRef]

Haus, H. A.

H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron.29, 983–996 (1993).
[CrossRef]

M. Nakazawa, K. Suzuki, and H. A. Haus, “The modulational instability laser-Part I: Experiment,” IEEE J. Quantum Electron.25, 2036–2044 (1989).
[CrossRef]

Henriet, R.

A. Coillet, I. Balakireva, R. Henriet, K. Saleh, L. Larger, J. M. Dudley, C. R. Menyuk, and Y. K. Chembo, “Azimuthal Turing patterns, bright and dark cavity solitons in Kerr combs generated with whispering-gallery-mode resonators,” IEEE Photonics J.5, 6100409 (2013).
[CrossRef]

Herr, T.

T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang, E. Gavartin, R. Holzwarth, M. L. Gorodetsky, and T. J. Kippenberg, “Universal formation dynamics and noise of Kerr frequency combs in microresonators,” Nat. Photonics6, 480–487 (2012).
[CrossRef]

P. Del-Haye, T. Herr, E. Gavartin, M. L. Gorodetsky, R. Holzwarth, and T. J. Kippenberg, “Octave spanning tunable frequency comb from a microresonator,” Phys. Rev. Lett.107, 063901 (2011).
[CrossRef]

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Mode-locking in an optical microresonator via soliton formation,” arXiv:1211.0733v2 (2013).

Holzwarth, R.

T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang, E. Gavartin, R. Holzwarth, M. L. Gorodetsky, and T. J. Kippenberg, “Universal formation dynamics and noise of Kerr frequency combs in microresonators,” Nat. Photonics6, 480–487 (2012).
[CrossRef]

P. Del-Haye, T. Herr, E. Gavartin, M. L. Gorodetsky, R. Holzwarth, and T. J. Kippenberg, “Octave spanning tunable frequency comb from a microresonator,” Phys. Rev. Lett.107, 063901 (2011).
[CrossRef]

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science332, 555–559 (2011).
[CrossRef] [PubMed]

P. Del-Haye, O. Arcizet, A. Schliesser, R. Holzwarth, and T. J. Kippenberg, “Full stabilization of a microresonator-based optical frequency comb,” Phys. Rev. Lett.101, 053903 (2008).
[CrossRef]

P. Del-Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature450, 1214–1217 (2007).
[CrossRef]

Ilchenko, V. S.

A. A. Savchenkov, A. B. Matsko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Kerr frequency comb generation in overmoded resonators,” Opt. Express20, 27290–27298 (2012).
[CrossRef] [PubMed]

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, D. Seidel, and L. Maleki, “Hard and soft excitation regimes of Kerr frequency combs,” Phys. Rev. A85, 023830 (2012).
[CrossRef]

A. A. Savchenkov, A. B. Matsko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Transient regime of Kerr-frequency-comb formation,” Phys. Rev. A86, 013838 (2012).
[CrossRef]

W. Liang, A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, D. Seidel, and L. Maleki, “Generation of near-infrared frequency combs from a MgF2whispering gallery mode resonator,” Opt. Lett.36, 2290–2292 (2011).
[CrossRef] [PubMed]

A. B. Matsko, A. A. Savchenkov, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Mode-locked Kerr frequency combs,” Opt. Lett.36, 2845–2847 (2011).
[CrossRef] [PubMed]

A. A. Savchenkov, A. B. Matsko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Kerr combs with selectable central frequency,” Nat. Photonics5, 293–296 (2011).
[CrossRef]

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, I. Solomatine, D. Seidel, and L. Maleki, “Tunable optical frequency comb with a crystalline whispering gallery mode resonator,” Phys. Rev. Lett.101, 093902 (2008).
[CrossRef] [PubMed]

A. A. Savchenkov, E. Rubiola, A. B. Matsko, V. S. Ilchenko, and L. Maleki, “Phase noise of whispering gallery photonic hyper-parametric microwave oscillators,” Opt. Express16, 4130–4144 (2008).
[CrossRef] [PubMed]

A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Optical hyperparametric oscillations in a whispering-gallery-mode resonator: Threshold and phase diffusion,” Phys. Rev. A71, 033804 (2005).
[CrossRef]

A. A. Savchenkov, A. B. Matsko, D. Strekalov, M. Mohageg, V. S. Ilchenko, and L. Maleki, “Low threshold optical oscillations in a whispering gallery mode CaF2resonator,” Phys. Rev. Lett.93, 243905 (2004).
[CrossRef]

A. B. Matsko, A. A. Savchenkov, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Whispering gallery mode oscillators and optical comb generators,” in Proceedings of 7th Symposium on Frequency Standards and Metrology, L. Maleki, ed., (World Scientific, 2009), pp. 539–558.

Johnson, A. R.

Jost, J. D.

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Mode-locking in an optical microresonator via soliton formation,” arXiv:1211.0733v2 (2013).

Karachalios, N. I.

N. I. Karachalios and N. M. Stavrakakis, “Global attractor for the weakly damped driven Schrodinger equation in H2(R),” Nonlinear Diff. Eq. Appl.9, 347–360 (2002).
[CrossRef]

Kippenberg, T. J.

T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang, E. Gavartin, R. Holzwarth, M. L. Gorodetsky, and T. J. Kippenberg, “Universal formation dynamics and noise of Kerr frequency combs in microresonators,” Nat. Photonics6, 480–487 (2012).
[CrossRef]

P. Del-Haye, T. Herr, E. Gavartin, M. L. Gorodetsky, R. Holzwarth, and T. J. Kippenberg, “Octave spanning tunable frequency comb from a microresonator,” Phys. Rev. Lett.107, 063901 (2011).
[CrossRef]

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science332, 555–559 (2011).
[CrossRef] [PubMed]

P. Del-Haye, O. Arcizet, A. Schliesser, R. Holzwarth, and T. J. Kippenberg, “Full stabilization of a microresonator-based optical frequency comb,” Phys. Rev. Lett.101, 053903 (2008).
[CrossRef]

P. Del-Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature450, 1214–1217 (2007).
[CrossRef]

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Mode-locking in an optical microresonator via soliton formation,” arXiv:1211.0733v2 (2013).

Kondratiev, N. M.

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Mode-locking in an optical microresonator via soliton formation,” arXiv:1211.0733v2 (2013).

Kumar, P.

Kuzucu, O.

Lamont, M.

M. Lamont, Y. Okawachi, and A. L. Gaeta, “Route to stabilized ultrabroadband microresonator-based frequency combs,” arXiv:1305.4921 (2013).

Lamont, M. R. E.

Larger, L.

A. Coillet, I. Balakireva, R. Henriet, K. Saleh, L. Larger, J. M. Dudley, C. R. Menyuk, and Y. K. Chembo, “Azimuthal Turing patterns, bright and dark cavity solitons in Kerr combs generated with whispering-gallery-mode resonators,” IEEE Photonics J.5, 6100409 (2013).
[CrossRef]

Leaird, D. E.

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,” Phys. Rev. Lett.109, 233901 (2012).
[CrossRef]

Lefever, R.

L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passive optical systems,” Phys. Rev. Lett.58, 2209–2211 (1987).
[CrossRef] [PubMed]

Levy, J. S.

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,” Phys. Rev. Lett.109, 233901 (2012).
[CrossRef]

Li, L.

Li, S.

Li, Z.

Liang, W.

A. B. Matsko, W. Liang, A. A. Savchenkov, and L. Maleki, “Chaotic dynamics of frequency combs generated with continuously pumped nonlinear microresonators,” Opt. Lett.38, 525–527 (2013).
[CrossRef] [PubMed]

A. A. Savchenkov, A. B. Matsko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Transient regime of Kerr-frequency-comb formation,” Phys. Rev. A86, 013838 (2012).
[CrossRef]

A. A. Savchenkov, A. B. Matsko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Kerr frequency comb generation in overmoded resonators,” Opt. Express20, 27290–27298 (2012).
[CrossRef] [PubMed]

A. A. Savchenkov, A. B. Matsko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Kerr combs with selectable central frequency,” Nat. Photonics5, 293–296 (2011).
[CrossRef]

A. B. Matsko, A. A. Savchenkov, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Mode-locked Kerr frequency combs,” Opt. Lett.36, 2845–2847 (2011).
[CrossRef] [PubMed]

W. Liang, A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, D. Seidel, and L. Maleki, “Generation of near-infrared frequency combs from a MgF2whispering gallery mode resonator,” Opt. Lett.36, 2290–2292 (2011).
[CrossRef] [PubMed]

A. B. Matsko, A. A. Savchenkov, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Whispering gallery mode oscillators and optical comb generators,” in Proceedings of 7th Symposium on Frequency Standards and Metrology, L. Maleki, ed., (World Scientific, 2009), pp. 539–558.

Lipson, M.

Little, B. E.

L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. E. Little, and D. J. Moss, “CMOS-compatible integrated optical hyper-parametric oscillator,” Nat. Photonics4, 41–45 (2010).
[CrossRef]

Lugiato, L. A.

L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passive optical systems,” Phys. Rev. Lett.58, 2209–2211 (1987).
[CrossRef] [PubMed]

Maleki, L.

A. B. Matsko, W. Liang, A. A. Savchenkov, and L. Maleki, “Chaotic dynamics of frequency combs generated with continuously pumped nonlinear microresonators,” Opt. Lett.38, 525–527 (2013).
[CrossRef] [PubMed]

A. A. Savchenkov, A. B. Matsko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Transient regime of Kerr-frequency-comb formation,” Phys. Rev. A86, 013838 (2012).
[CrossRef]

A. B. Matsko, A. A. Savchenkov, and L. Maleki, “On excitation of breather solitons in an optical microresonator,” Opt. Lett.37, 4856–4858 (2012).
[CrossRef] [PubMed]

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, D. Seidel, and L. Maleki, “Hard and soft excitation regimes of Kerr frequency combs,” Phys. Rev. A85, 023830 (2012).
[CrossRef]

A. B. Matsko, A. A. Savchenkov, and L. Maleki, “Normal GVD Kerr frequency comb,” Opt. Lett.37, 43–45 (2012).
[CrossRef] [PubMed]

A. A. Savchenkov, A. B. Matsko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Kerr frequency comb generation in overmoded resonators,” Opt. Express20, 27290–27298 (2012).
[CrossRef] [PubMed]

A. A. Savchenkov, A. B. Matsko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Kerr combs with selectable central frequency,” Nat. Photonics5, 293–296 (2011).
[CrossRef]

A. B. Matsko, A. A. Savchenkov, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Mode-locked Kerr frequency combs,” Opt. Lett.36, 2845–2847 (2011).
[CrossRef] [PubMed]

W. Liang, A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, D. Seidel, and L. Maleki, “Generation of near-infrared frequency combs from a MgF2whispering gallery mode resonator,” Opt. Lett.36, 2290–2292 (2011).
[CrossRef] [PubMed]

I. S. Grudinin, N. Yu, and L. Maleki, “Generation of optical frequency combs with a CaF2resonator,” Opt. Lett.34, 878–880 (2009).
[CrossRef] [PubMed]

A. A. Savchenkov, E. Rubiola, A. B. Matsko, V. S. Ilchenko, and L. Maleki, “Phase noise of whispering gallery photonic hyper-parametric microwave oscillators,” Opt. Express16, 4130–4144 (2008).
[CrossRef] [PubMed]

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, I. Solomatine, D. Seidel, and L. Maleki, “Tunable optical frequency comb with a crystalline whispering gallery mode resonator,” Phys. Rev. Lett.101, 093902 (2008).
[CrossRef] [PubMed]

A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Optical hyperparametric oscillations in a whispering-gallery-mode resonator: Threshold and phase diffusion,” Phys. Rev. A71, 033804 (2005).
[CrossRef]

A. A. Savchenkov, A. B. Matsko, D. Strekalov, M. Mohageg, V. S. Ilchenko, and L. Maleki, “Low threshold optical oscillations in a whispering gallery mode CaF2resonator,” Phys. Rev. Lett.93, 243905 (2004).
[CrossRef]

A. B. Matsko, A. A. Savchenkov, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Whispering gallery mode oscillators and optical comb generators,” in Proceedings of 7th Symposium on Frequency Standards and Metrology, L. Maleki, ed., (World Scientific, 2009), pp. 539–558.

Matsko, A. B.

A. B. Matsko, W. Liang, A. A. Savchenkov, and L. Maleki, “Chaotic dynamics of frequency combs generated with continuously pumped nonlinear microresonators,” Opt. Lett.38, 525–527 (2013).
[CrossRef] [PubMed]

A. A. Savchenkov, A. B. Matsko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Transient regime of Kerr-frequency-comb formation,” Phys. Rev. A86, 013838 (2012).
[CrossRef]

A. B. Matsko, A. A. Savchenkov, and L. Maleki, “On excitation of breather solitons in an optical microresonator,” Opt. Lett.37, 4856–4858 (2012).
[CrossRef] [PubMed]

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, D. Seidel, and L. Maleki, “Hard and soft excitation regimes of Kerr frequency combs,” Phys. Rev. A85, 023830 (2012).
[CrossRef]

A. B. Matsko, A. A. Savchenkov, and L. Maleki, “Normal GVD Kerr frequency comb,” Opt. Lett.37, 43–45 (2012).
[CrossRef] [PubMed]

A. A. Savchenkov, A. B. Matsko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Kerr frequency comb generation in overmoded resonators,” Opt. Express20, 27290–27298 (2012).
[CrossRef] [PubMed]

A. A. Savchenkov, A. B. Matsko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Kerr combs with selectable central frequency,” Nat. Photonics5, 293–296 (2011).
[CrossRef]

W. Liang, A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, D. Seidel, and L. Maleki, “Generation of near-infrared frequency combs from a MgF2whispering gallery mode resonator,” Opt. Lett.36, 2290–2292 (2011).
[CrossRef] [PubMed]

A. B. Matsko, A. A. Savchenkov, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Mode-locked Kerr frequency combs,” Opt. Lett.36, 2845–2847 (2011).
[CrossRef] [PubMed]

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, I. Solomatine, D. Seidel, and L. Maleki, “Tunable optical frequency comb with a crystalline whispering gallery mode resonator,” Phys. Rev. Lett.101, 093902 (2008).
[CrossRef] [PubMed]

A. A. Savchenkov, E. Rubiola, A. B. Matsko, V. S. Ilchenko, and L. Maleki, “Phase noise of whispering gallery photonic hyper-parametric microwave oscillators,” Opt. Express16, 4130–4144 (2008).
[CrossRef] [PubMed]

A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Optical hyperparametric oscillations in a whispering-gallery-mode resonator: Threshold and phase diffusion,” Phys. Rev. A71, 033804 (2005).
[CrossRef]

A. A. Savchenkov, A. B. Matsko, D. Strekalov, M. Mohageg, V. S. Ilchenko, and L. Maleki, “Low threshold optical oscillations in a whispering gallery mode CaF2resonator,” Phys. Rev. Lett.93, 243905 (2004).
[CrossRef]

A. B. Matsko, A. A. Savchenkov, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Whispering gallery mode oscillators and optical comb generators,” in Proceedings of 7th Symposium on Frequency Standards and Metrology, L. Maleki, ed., (World Scientific, 2009), pp. 539–558.

Mecozzi, A.

H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron.29, 983–996 (1993).
[CrossRef]

Menyuk, C. R.

A. Coillet, I. Balakireva, R. Henriet, K. Saleh, L. Larger, J. M. Dudley, C. R. Menyuk, and Y. K. Chembo, “Azimuthal Turing patterns, bright and dark cavity solitons in Kerr combs generated with whispering-gallery-mode resonators,” IEEE Photonics J.5, 6100409 (2013).
[CrossRef]

Miao, H. X.

Modotto, D.

T. Hansson, D. Modotto, and S. Wabnitz, “Dynamics of the modulational instability in microresonator frequency combs,” Phys. Rev. A88, 023819 (2013).
[CrossRef]

Mohageg, M.

A. A. Savchenkov, A. B. Matsko, D. Strekalov, M. Mohageg, V. S. Ilchenko, and L. Maleki, “Low threshold optical oscillations in a whispering gallery mode CaF2resonator,” Phys. Rev. Lett.93, 243905 (2004).
[CrossRef]

Morandotti, R.

L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. E. Little, and D. J. Moss, “CMOS-compatible integrated optical hyper-parametric oscillator,” Nat. Photonics4, 41–45 (2010).
[CrossRef]

Moss, D. J.

L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. E. Little, and D. J. Moss, “CMOS-compatible integrated optical hyper-parametric oscillator,” Nat. Photonics4, 41–45 (2010).
[CrossRef]

Nakazawa, M.

M. Nakazawa, K. Suzuki, and H. A. Haus, “The modulational instability laser-Part I: Experiment,” IEEE J. Quantum Electron.25, 2036–2044 (1989).
[CrossRef]

Okawachi, Y.

Papp, S. B.

S. B. Papp, P. Del’Haye, and S. A. Diddams, “Mechanical control of a microrod-resonator optical frequency comb,” Phys. Rev. X3, 031003 (2013).
[CrossRef]

S. B. Papp, P. Del’Haye, and S. A. Diddams, “Parametric seeding of a microresonator optical frequency comb,” Opt. Express21, 17615–17624 (2013).
[CrossRef] [PubMed]

S. B. Papp and S. A. Diddams, “Spectral and temporal characterization of a fused quartz microresonator optical frequency comb,” Phys. Rev. A84, 053833 (2011).
[CrossRef]

P. Del’Haye, S. B. Papp, and S. A. Diddams, “Self-injection locking and phase-locked states in microresonator-based optical frequency combs,” arXiv:1307.4091 (2013).

Park, Q.-H.

Q.-H. Park and H. J. Shin, “Parametric control of soliton light traffic by cw traffic light,” Phys. Rev. Lett.82, 4432–4435 (1999).
[CrossRef]

Randle, H. G.

Razzari, L.

L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. E. Little, and D. J. Moss, “CMOS-compatible integrated optical hyper-parametric oscillator,” Nat. Photonics4, 41–45 (2010).
[CrossRef]

Riemensberger, J.

T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang, E. Gavartin, R. Holzwarth, M. L. Gorodetsky, and T. J. Kippenberg, “Universal formation dynamics and noise of Kerr frequency combs in microresonators,” Nat. Photonics6, 480–487 (2012).
[CrossRef]

Rubiola, E.

Saha, K.

Saleh, K.

A. Coillet, I. Balakireva, R. Henriet, K. Saleh, L. Larger, J. M. Dudley, C. R. Menyuk, and Y. K. Chembo, “Azimuthal Turing patterns, bright and dark cavity solitons in Kerr combs generated with whispering-gallery-mode resonators,” IEEE Photonics J.5, 6100409 (2013).
[CrossRef]

Salem, R.

Savchenkov, A. A.

A. B. Matsko, W. Liang, A. A. Savchenkov, and L. Maleki, “Chaotic dynamics of frequency combs generated with continuously pumped nonlinear microresonators,” Opt. Lett.38, 525–527 (2013).
[CrossRef] [PubMed]

A. A. Savchenkov, A. B. Matsko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Transient regime of Kerr-frequency-comb formation,” Phys. Rev. A86, 013838 (2012).
[CrossRef]

A. B. Matsko, A. A. Savchenkov, and L. Maleki, “On excitation of breather solitons in an optical microresonator,” Opt. Lett.37, 4856–4858 (2012).
[CrossRef] [PubMed]

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, D. Seidel, and L. Maleki, “Hard and soft excitation regimes of Kerr frequency combs,” Phys. Rev. A85, 023830 (2012).
[CrossRef]

A. B. Matsko, A. A. Savchenkov, and L. Maleki, “Normal GVD Kerr frequency comb,” Opt. Lett.37, 43–45 (2012).
[CrossRef] [PubMed]

A. A. Savchenkov, A. B. Matsko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Kerr frequency comb generation in overmoded resonators,” Opt. Express20, 27290–27298 (2012).
[CrossRef] [PubMed]

A. A. Savchenkov, A. B. Matsko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Kerr combs with selectable central frequency,” Nat. Photonics5, 293–296 (2011).
[CrossRef]

W. Liang, A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, D. Seidel, and L. Maleki, “Generation of near-infrared frequency combs from a MgF2whispering gallery mode resonator,” Opt. Lett.36, 2290–2292 (2011).
[CrossRef] [PubMed]

A. B. Matsko, A. A. Savchenkov, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Mode-locked Kerr frequency combs,” Opt. Lett.36, 2845–2847 (2011).
[CrossRef] [PubMed]

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, I. Solomatine, D. Seidel, and L. Maleki, “Tunable optical frequency comb with a crystalline whispering gallery mode resonator,” Phys. Rev. Lett.101, 093902 (2008).
[CrossRef] [PubMed]

A. A. Savchenkov, E. Rubiola, A. B. Matsko, V. S. Ilchenko, and L. Maleki, “Phase noise of whispering gallery photonic hyper-parametric microwave oscillators,” Opt. Express16, 4130–4144 (2008).
[CrossRef] [PubMed]

A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Optical hyperparametric oscillations in a whispering-gallery-mode resonator: Threshold and phase diffusion,” Phys. Rev. A71, 033804 (2005).
[CrossRef]

A. A. Savchenkov, A. B. Matsko, D. Strekalov, M. Mohageg, V. S. Ilchenko, and L. Maleki, “Low threshold optical oscillations in a whispering gallery mode CaF2resonator,” Phys. Rev. Lett.93, 243905 (2004).
[CrossRef]

A. B. Matsko, A. A. Savchenkov, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Whispering gallery mode oscillators and optical comb generators,” in Proceedings of 7th Symposium on Frequency Standards and Metrology, L. Maleki, ed., (World Scientific, 2009), pp. 539–558.

Schliesser, A.

P. Del-Haye, O. Arcizet, A. Schliesser, R. Holzwarth, and T. J. Kippenberg, “Full stabilization of a microresonator-based optical frequency comb,” Phys. Rev. Lett.101, 053903 (2008).
[CrossRef]

P. Del-Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature450, 1214–1217 (2007).
[CrossRef]

Seidel, D.

A. A. Savchenkov, A. B. Matsko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Kerr frequency comb generation in overmoded resonators,” Opt. Express20, 27290–27298 (2012).
[CrossRef] [PubMed]

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, D. Seidel, and L. Maleki, “Hard and soft excitation regimes of Kerr frequency combs,” Phys. Rev. A85, 023830 (2012).
[CrossRef]

A. A. Savchenkov, A. B. Matsko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Transient regime of Kerr-frequency-comb formation,” Phys. Rev. A86, 013838 (2012).
[CrossRef]

W. Liang, A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, D. Seidel, and L. Maleki, “Generation of near-infrared frequency combs from a MgF2whispering gallery mode resonator,” Opt. Lett.36, 2290–2292 (2011).
[CrossRef] [PubMed]

A. B. Matsko, A. A. Savchenkov, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Mode-locked Kerr frequency combs,” Opt. Lett.36, 2845–2847 (2011).
[CrossRef] [PubMed]

A. A. Savchenkov, A. B. Matsko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Kerr combs with selectable central frequency,” Nat. Photonics5, 293–296 (2011).
[CrossRef]

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, I. Solomatine, D. Seidel, and L. Maleki, “Tunable optical frequency comb with a crystalline whispering gallery mode resonator,” Phys. Rev. Lett.101, 093902 (2008).
[CrossRef] [PubMed]

A. B. Matsko, A. A. Savchenkov, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Whispering gallery mode oscillators and optical comb generators,” in Proceedings of 7th Symposium on Frequency Standards and Metrology, L. Maleki, ed., (World Scientific, 2009), pp. 539–558.

Serkland, D. K.

Shim, B.

Shin, H. J.

Q.-H. Park and H. J. Shin, “Parametric control of soliton light traffic by cw traffic light,” Phys. Rev. Lett.82, 4432–4435 (1999).
[CrossRef]

Smirnov, Y. S.

I. V. Barashenkov and Y. S. Smirnov, “Existence and stability chart for the ac-driven, damped nonlinear Schrödinger solitons,” Phys. Rev. E54, 5707–5725 (1996).
[CrossRef]

Solomatine, I.

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, I. Solomatine, D. Seidel, and L. Maleki, “Tunable optical frequency comb with a crystalline whispering gallery mode resonator,” Phys. Rev. Lett.101, 093902 (2008).
[CrossRef] [PubMed]

Srinivasan, K.

Stavrakakis, N. M.

N. I. Karachalios and N. M. Stavrakakis, “Global attractor for the weakly damped driven Schrodinger equation in H2(R),” Nonlinear Diff. Eq. Appl.9, 347–360 (2002).
[CrossRef]

Strekalov, D.

A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Optical hyperparametric oscillations in a whispering-gallery-mode resonator: Threshold and phase diffusion,” Phys. Rev. A71, 033804 (2005).
[CrossRef]

A. A. Savchenkov, A. B. Matsko, D. Strekalov, M. Mohageg, V. S. Ilchenko, and L. Maleki, “Low threshold optical oscillations in a whispering gallery mode CaF2resonator,” Phys. Rev. Lett.93, 243905 (2004).
[CrossRef]

Strekalov, D. V.

Y. K. Chembo, D. V. Strekalov, and N. Yu, “Spectrum and dynamics of optical frequency combs generated with monolithic whispering gallery mode resonators,” Phys. Rev. Lett.104, 103902 (2010).
[CrossRef] [PubMed]

Suzuki, K.

M. Nakazawa, K. Suzuki, and H. A. Haus, “The modulational instability laser-Part I: Experiment,” IEEE J. Quantum Electron.25, 2036–2044 (1989).
[CrossRef]

Sylvestre, T.

Trillo, S.

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,” Phys. Rev. Lett.109, 233901 (2012).
[CrossRef]

Varghese, L. T.

F. Ferdous, H. X. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, “Spectral line-by-line pulse shaping of on-chip microresonator frequency combs,” Nat. Photonics5, 770–776 (2011).
[CrossRef]

Wabnitz, S.

Wang, C. Y.

T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang, E. Gavartin, R. Holzwarth, M. L. Gorodetsky, and T. J. Kippenberg, “Universal formation dynamics and noise of Kerr frequency combs in microresonators,” Nat. Photonics6, 480–487 (2012).
[CrossRef]

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Mode-locking in an optical microresonator via soliton formation,” arXiv:1211.0733v2 (2013).

Wang, J.

P. H. Wang, F. Ferdous, H. X. Miao, J. Wang, D. E. Leaird, K. Srinivasan, L. Chen, V. Aksyuk, and A. M. Weiner, “Observation of correlation between route to formation, coherence, noise, and communication performance of Kerr combs,” Opt. Express20, 29284–29295 (2012).
[CrossRef]

F. Ferdous, H. X. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, “Spectral line-by-line pulse shaping of on-chip microresonator frequency combs,” Nat. Photonics5, 770–776 (2011).
[CrossRef]

Wang, P. H.

Weiner, A. M.

Wen, Y. H.

Wilken, T.

P. Del-Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature450, 1214–1217 (2007).
[CrossRef]

Yu, N.

I. S. Grudinin, L. Baumgartel, and N. Yu, “Frequency comb from a microresonator with engineered spectrum,” Opt. Express20, 6604–6609 (2012).
[CrossRef] [PubMed]

Y. K. Chembo, D. V. Strekalov, and N. Yu, “Spectrum and dynamics of optical frequency combs generated with monolithic whispering gallery mode resonators,” Phys. Rev. Lett.104, 103902 (2010).
[CrossRef] [PubMed]

Y. K. Chembo and N. Yu, “Modal expansion approach to optical-frequency-comb generation with monolithic whispering-gallery-mode resonators,” Phys. Rev. A82, 033801 (2010).
[CrossRef]

Y. K. Chembo and N. Yu, “On the generation of octave-spanning optical frequency combs using monolithic whispering-gallery-mode microresonators,” Opt. Lett.35, 2696–2698 (2010).
[CrossRef] [PubMed]

I. S. Grudinin, N. Yu, and L. Maleki, “Generation of optical frequency combs with a CaF2resonator,” Opt. Lett.34, 878–880 (2009).
[CrossRef] [PubMed]

Zhou, G.

Zhu, C.

C. Zhu, “Attractor of the nonlinear Schrodinger equation,” Commun. Math. Anal.4, 67–75 (2008).

Ann. Inst. Henri Poincare (1)

J.-M. Ghidaglia, “Finite dimensional behavior for weakly damped driven Schrödinger equation,” Ann. Inst. Henri Poincare5, 365–405 (1988).

Commun. Math. Anal. (1)

C. Zhu, “Attractor of the nonlinear Schrodinger equation,” Commun. Math. Anal.4, 67–75 (2008).

IEEE J. Quantum Electron. (2)

H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron.29, 983–996 (1993).
[CrossRef]

M. Nakazawa, K. Suzuki, and H. A. Haus, “The modulational instability laser-Part I: Experiment,” IEEE J. Quantum Electron.25, 2036–2044 (1989).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

A. Hasegawa, “Soliton-based optical communications: An overwiew,” IEEE J. Sel. Top. Quantum Electron.6, 1161–1172 (2000).
[CrossRef]

IEEE Photonics J. (1)

A. Coillet, I. Balakireva, R. Henriet, K. Saleh, L. Larger, J. M. Dudley, C. R. Menyuk, and Y. K. Chembo, “Azimuthal Turing patterns, bright and dark cavity solitons in Kerr combs generated with whispering-gallery-mode resonators,” IEEE Photonics J.5, 6100409 (2013).
[CrossRef]

J. Opt. Soc. Am. B (1)

Nat. Photonics (4)

T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang, E. Gavartin, R. Holzwarth, M. L. Gorodetsky, and T. J. Kippenberg, “Universal formation dynamics and noise of Kerr frequency combs in microresonators,” Nat. Photonics6, 480–487 (2012).
[CrossRef]

L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. E. Little, and D. J. Moss, “CMOS-compatible integrated optical hyper-parametric oscillator,” Nat. Photonics4, 41–45 (2010).
[CrossRef]

F. Ferdous, H. X. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, “Spectral line-by-line pulse shaping of on-chip microresonator frequency combs,” Nat. Photonics5, 770–776 (2011).
[CrossRef]

A. A. Savchenkov, A. B. Matsko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Kerr combs with selectable central frequency,” Nat. Photonics5, 293–296 (2011).
[CrossRef]

Nature (1)

P. Del-Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature450, 1214–1217 (2007).
[CrossRef]

Nonlinear Diff. Eq. Appl. (1)

N. I. Karachalios and N. M. Stavrakakis, “Global attractor for the weakly damped driven Schrodinger equation in H2(R),” Nonlinear Diff. Eq. Appl.9, 347–360 (2002).
[CrossRef]

Opt. Express (8)

A. A. Savchenkov, E. Rubiola, A. B. Matsko, V. S. Ilchenko, and L. Maleki, “Phase noise of whispering gallery photonic hyper-parametric microwave oscillators,” Opt. Express16, 4130–4144 (2008).
[CrossRef] [PubMed]

M. A. Foster, J. S. Levy, O. Kuzucu, K. Saha, M. Lipson, and A. L. Gaeta, “Silicon-based monolithic optical frequency comb source,” Opt. Express19, 14233–14239 (2011).
[CrossRef] [PubMed]

I. S. Grudinin, L. Baumgartel, and N. Yu, “Frequency comb from a microresonator with engineered spectrum,” Opt. Express20, 6604–6609 (2012).
[CrossRef] [PubMed]

F. Ferdous, H. X. Miao, P. H. Wang, D. E. Leaird, K. Srinivasan, L. Chen, V. Aksyuk, and A. M. Weiner, “Probing coherence in microcavity frequency combs via optical pulse shaping,” Opt. Express20, 21033–21043 (2012).
[CrossRef] [PubMed]

A. A. Savchenkov, A. B. Matsko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Kerr frequency comb generation in overmoded resonators,” Opt. Express20, 27290–27298 (2012).
[CrossRef] [PubMed]

P. H. Wang, F. Ferdous, H. X. Miao, J. Wang, D. E. Leaird, K. Srinivasan, L. Chen, V. Aksyuk, and A. M. Weiner, “Observation of correlation between route to formation, coherence, noise, and communication performance of Kerr combs,” Opt. Express20, 29284–29295 (2012).
[CrossRef]

K. Saha, Y. Okawachi, B. Shim, J. S. Levy, R. Salem, A. R. Johnson, M. A. Foster, M. R. E. Lamont, M. Lipson, and A. L. Gaeta, “Modelocking and femtosecond pulse generation in chip-based frequency combs,” Opt. Express21, 1335–1343 (2013).
[CrossRef] [PubMed]

S. B. Papp, P. Del’Haye, and S. A. Diddams, “Parametric seeding of a microresonator optical frequency comb,” Opt. Express21, 17615–17624 (2013).
[CrossRef] [PubMed]

Opt. Lett. (15)

A. B. Matsko, W. Liang, A. A. Savchenkov, and L. Maleki, “Chaotic dynamics of frequency combs generated with continuously pumped nonlinear microresonators,” Opt. Lett.38, 525–527 (2013).
[CrossRef] [PubMed]

S. Coen and M. Erkintalo, “Universal scaling laws of Kerr frequency combs,” Opt. Lett.38, 1790–1792 (2013).
[CrossRef] [PubMed]

S. Coen, H. G. Randle, T. Sylvestre, and M. Erkintalo, “Modeling of octave-spanning Kerr frequency combs using a generalized mean-field Lugiato-Lefever model,” Opt. Lett.38, 37–39 (2013).
[CrossRef] [PubMed]

A. B. Matsko, A. A. Savchenkov, and L. Maleki, “On excitation of breather solitons in an optical microresonator,” Opt. Lett.37, 4856–4858 (2012).
[CrossRef] [PubMed]

A. B. Matsko, A. A. Savchenkov, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Mode-locked Kerr frequency combs,” Opt. Lett.36, 2845–2847 (2011).
[CrossRef] [PubMed]

Y. Okawachi, K. Saha, J. S. Levy, Y. H. Wen, M. Lipson, and A. L. Gaeta, “Octave spanning frequency comb generation in a silicon nitride chip,” Opt. Lett.36, 3398–3400 (2011).
[CrossRef] [PubMed]

A. B. Matsko, A. A. Savchenkov, and L. Maleki, “Normal GVD Kerr frequency comb,” Opt. Lett.37, 43–45 (2012).
[CrossRef] [PubMed]

A. R. Johnson, Y. Okawachi, J. S. Levy, J. Cardenas, K. Saha, M. Lipson, and A. L. Gaeta, “Chip-based frequency combs with sub-100 GHz repetition rates,” Opt. Lett.37, 875–877 (2012).
[CrossRef] [PubMed]

I. S. Grudinin, N. Yu, and L. Maleki, “Generation of optical frequency combs with a CaF2resonator,” Opt. Lett.34, 878–880 (2009).
[CrossRef] [PubMed]

Y. K. Chembo and N. Yu, “On the generation of octave-spanning optical frequency combs using monolithic whispering-gallery-mode microresonators,” Opt. Lett.35, 2696–2698 (2010).
[CrossRef] [PubMed]

W. Liang, A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, D. Seidel, and L. Maleki, “Generation of near-infrared frequency combs from a MgF2whispering gallery mode resonator,” Opt. Lett.36, 2290–2292 (2011).
[CrossRef] [PubMed]

M. Haelterman, S. Trillo, and S. Wabnitz, “Additive-modulation-instability ring laser in the normal dispersion regime of a fiber,” Opt. Lett.17, 745–747 (1992).
[CrossRef] [PubMed]

S. Wabnitz, “Suppression of interactions in a phase-locked soliton optical memory,” Opt. Lett.18, 601–603 (1993).
[CrossRef] [PubMed]

D. K. Serkland and P. Kumar, “Tunable fiber-optic parametric oscillator,” Opt. Lett.24, 92–94 (1999).
[CrossRef]

S. Coen and M. Haelterman, “Continuous-wave ultrahigh-repetition-rate pulse-train generation through modulational instability in a passive fiber cavity,” Opt. Lett.26, 39–41 (2001).
[CrossRef]

Phys. Rev. A (6)

A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Optical hyperparametric oscillations in a whispering-gallery-mode resonator: Threshold and phase diffusion,” Phys. Rev. A71, 033804 (2005).
[CrossRef]

T. Hansson, D. Modotto, and S. Wabnitz, “Dynamics of the modulational instability in microresonator frequency combs,” Phys. Rev. A88, 023819 (2013).
[CrossRef]

Y. K. Chembo and N. Yu, “Modal expansion approach to optical-frequency-comb generation with monolithic whispering-gallery-mode resonators,” Phys. Rev. A82, 033801 (2010).
[CrossRef]

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, D. Seidel, and L. Maleki, “Hard and soft excitation regimes of Kerr frequency combs,” Phys. Rev. A85, 023830 (2012).
[CrossRef]

S. B. Papp and S. A. Diddams, “Spectral and temporal characterization of a fused quartz microresonator optical frequency comb,” Phys. Rev. A84, 053833 (2011).
[CrossRef]

A. A. Savchenkov, A. B. Matsko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Transient regime of Kerr-frequency-comb formation,” Phys. Rev. A86, 013838 (2012).
[CrossRef]

Phys. Rev. E (1)

I. V. Barashenkov and Y. S. Smirnov, “Existence and stability chart for the ac-driven, damped nonlinear Schrödinger solitons,” Phys. Rev. E54, 5707–5725 (1996).
[CrossRef]

Phys. Rev. Lett. (10)

K. J. Blow and N. J. Doran, “Global and local chaos in the pumped nonlinear Schrödinger equation,” Phys. Rev. Lett.52, 526–529 (1984)

Q.-H. Park and H. J. Shin, “Parametric control of soliton light traffic by cw traffic light,” Phys. Rev. Lett.82, 4432–4435 (1999).
[CrossRef]

S. Coen and M. Haelterman, “Modulational instability induced by cavity boundary conditions in a normally dispersive optical fiber,” Phys. Rev. Lett.79, 4139–4142 (1997).
[CrossRef]

L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passive optical systems,” Phys. Rev. Lett.58, 2209–2211 (1987).
[CrossRef] [PubMed]

Y. K. Chembo, D. V. Strekalov, and N. Yu, “Spectrum and dynamics of optical frequency combs generated with monolithic whispering gallery mode resonators,” Phys. Rev. Lett.104, 103902 (2010).
[CrossRef] [PubMed]

P. Del-Haye, T. Herr, E. Gavartin, M. L. Gorodetsky, R. Holzwarth, and T. J. Kippenberg, “Octave spanning tunable frequency comb from a microresonator,” Phys. Rev. Lett.107, 063901 (2011).
[CrossRef]

P. Del-Haye, O. Arcizet, A. Schliesser, R. Holzwarth, and T. J. Kippenberg, “Full stabilization of a microresonator-based optical frequency comb,” Phys. Rev. Lett.101, 053903 (2008).
[CrossRef]

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, I. Solomatine, D. Seidel, and L. Maleki, “Tunable optical frequency comb with a crystalline whispering gallery mode resonator,” Phys. Rev. Lett.101, 093902 (2008).
[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]

A. A. Savchenkov, A. B. Matsko, D. Strekalov, M. Mohageg, V. S. Ilchenko, and L. Maleki, “Low threshold optical oscillations in a whispering gallery mode CaF2resonator,” Phys. Rev. Lett.93, 243905 (2004).
[CrossRef]

Phys. Rev. X (1)

S. B. Papp, P. Del’Haye, and S. A. Diddams, “Mechanical control of a microrod-resonator optical frequency comb,” Phys. Rev. X3, 031003 (2013).
[CrossRef]

Science (1)

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science332, 555–559 (2011).
[CrossRef] [PubMed]

Other (6)

C. Godey, I. Balakireva, A. Coillet, and Y. K. Chembo, “Stability analysis of the Lugiato-Lefever model for Kerr optical frequency combs. Part I: Case of normal dispersion,” arXiv:1308.2539 (2013).

I. Balakireva, A. Coillet, C. Godey, and Y. K. Chembo, “Stability analysis of the Lugiato-Lefever model for Kerr optical frequency combs. Part II: Case of anomalous dispersion,” arXiv:1308.2542 (2013).

M. Lamont, Y. Okawachi, and A. L. Gaeta, “Route to stabilized ultrabroadband microresonator-based frequency combs,” arXiv:1305.4921 (2013).

A. B. Matsko, A. A. Savchenkov, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Whispering gallery mode oscillators and optical comb generators,” in Proceedings of 7th Symposium on Frequency Standards and Metrology, L. Maleki, ed., (World Scientific, 2009), pp. 539–558.

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Mode-locking in an optical microresonator via soliton formation,” arXiv:1211.0733v2 (2013).

P. Del’Haye, S. B. Papp, and S. A. Diddams, “Self-injection locking and phase-locked states in microresonator-based optical frequency combs,” arXiv:1307.4091 (2013).

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Equations (77)

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L ϕ = π 2 P t h P in ( D ) g γ 2 [ 1 + π 2 96 P th ( D ) P in γ 2 π 2 f 2 + 1 24 ( 1 + π 2 f 2 γ 2 ) 1 γ 2 π 2 f 2 P in ( D ) P th ] ,
D = 2 ω 0 ω + ω γ ,
g = n 2 n 0 h ¯ ω 0 2 c 𝒱 n 0
T R A T + i 2 β 2 Σ 2 A t 2 i γ Σ | A | 2 A = ( α Σ + T c 2 + i δ 0 ) A + i T c P in e i ϕ in ,
A out = P in e i ϕ in + i T c A ,
A out 1 = P in e i ϕ in + i T c 1 A ,
A out 2 = i T c 2 A .
A ( T , t ) = A c + A p ( T , t ) , A c = P c e i ϕ c , A p ( T , t ) = [ P p 2 ] 1 / 2 [ sech ( t ξ τ ) ] 1 + i q e i Ω ( t ξ ) + i ϕ p ,
P c ( α Σ + T c 2 + i δ 0 ) i ξ D C e i ϕ c = i T c P in e i ( ϕ in ϕ c ) ,
ξ D C = γ Σ P c e i ϕ c [ P c + P p τ T R ( 2 + e 2 i ( ϕ c ϕ p ) + π ( P p + 8 P c ) 4 2 P p P c e i ( ϕ c ϕ p ) + π P c 2 P p e i ( ϕ c ϕ p ) ) ]
P c ( α Σ + T c 2 + i δ 0 i γ Σ P c ) = i T c P in e i ( ϕ in ϕ c ) .
ϕ c ϕ in α + T c / 2 δ 0 ,
P c T c P in δ 0 2 ( 1 + 2 T c γ Σ P in δ 0 3 ) .
= T R 2 ( A p * A p T A p A p * T ) i 2 ( β 2 Σ | A p t | 2 + γ Σ | A p | 4 )
δ δ A * = A * T ( A * / T ) t ( A * / t ) = T R A p T + i 2 β 2 Σ 2 A p t 2 i γ Σ | A p | 2 A p = 0 .
L = d t ,
A p T = A p P p P p T + A p ϕ p ϕ p T + A p ξ ξ T + A p Ω Ω T + A p q q T + A p τ τ T .
L = i β 2 Σ P p 6 τ ( 1 + q 2 + 3 Ω 2 τ 2 ) i 6 γ Σ P p 2 τ + i 2 P p T R [ q τ T + 2 τ ( q T ( ln ( 2 ) 1 ) Ω ξ T + ϕ p T ) ] .
d d T ( L r ˙ j ) L r j = 0 ,
R = [ α Σ + T c 2 + i δ 0 ] A p + i [ γ Σ ( | A c + A p | 2 ( A c + A p ) | A p | 2 A p ) ξ D C ]
d d T ( L r ˙ j ) L r j = ( R * A p r j R A p * r j ) d t ,
T R d E d T = E [ T c + 2 α Σ + π 2 2 γ Σ P c sin ( ϕ c ϕ p ) ( P p P c + 8 2 π cos ( ϕ c ϕ p ) ) ] ,
T R d ξ d T = β 2 Σ Ω ,
T R d Ω d T = T R E d E d T Ω ( T c + 2 α Σ ) Ω ,
T R d q d T = T R E d E d T q + 2 β 2 Σ 3 τ 2 ( 1 + q 2 + 3 Ω 2 τ 2 ) + 1 3 γ Σ P p + ( T c + 2 α Σ ) q + π 2 2 γ Σ P c [ P p P c cos ( ϕ c ϕ p ) + π 2 τ T R ( 8 + 4 cos 2 ( ϕ c ϕ p ) + P p P c ) ] ,
T R τ d τ d T = T R 2 E d E d T τ 2 + 6 π 2 β 2 Σ q τ 2 2 [ T c + 2 α Σ + 3 2 ( π 2 8 ) 4 π γ Σ P c sin ( ϕ c ϕ p ) ( P p P c + 16 π 3 2 ( π 2 8 ) cos ( ϕ c ϕ p ) ) ] ,
T R d ϕ p d T = T R d q d T ( 1 ln 2 ) + T R d ξ d T Ω T R 2 τ d τ d T q + β 2 Σ 6 τ 2 ( 1 + q 2 + 3 Ω 2 τ 2 ) + 1 3 γ Σ P p δ 0 + γ Σ P c ( 2 + cos 2 ( ϕ c ϕ p ) + 3 π 4 2 P p P c cos ( ϕ c ϕ p ) ) .
τ 2 = 2 β 2 Σ γ Σ P p ,
ϕ p = 1 4 T T R γ Σ P p .
2 β 2 Σ 3 τ 2 + 1 3 γ Σ P p 0 ,
β 2 Σ 6 τ 2 + 1 3 γ Σ P p δ 0 0 ,
P p = 4 δ 0 γ Σ .
α Σ + T c 2 + π 4 2 γ Σ P c sin ( ϕ c ϕ p ) ( P p P c + 8 2 π cos ( ϕ c ϕ p ) ) = 0 ,
sin ( ϕ c ϕ p ) = 8 δ 0 ( α Σ + T c / 2 ) 2 π 2 T c γ Σ P in .
δ 0 π 2 T c γ Σ P in 8 ( α Σ + T c / 2 ) 2 ,
P out P in [ 1 π 2 sin 2 ( ϕ p ϕ c ) sech ( t τ ) ] 2 ,
P out 2 P in = T c 2 δ 0 2 + π 2 4 sin 4 ( ϕ p ϕ c ) sech 2 ( t τ ) .
g = h ¯ ω 0 γ Σ T R 2
γ Σ P c α Σ + T c / 2 = g N p h γ ,
γ c = T c 2 T R , γ 0 = α Σ T R , γ = γ c + γ 0 , Δ 0 = δ 0 T R , D = β 2 Σ ω R 2 c γ n 0 .
i Ψ τ 0 + 1 2 2 Ψ θ 2 + | Ψ | 2 Ψ = ( i + ζ 0 ) Ψ + if ,
τ 0 = T ( α Σ + T c / 2 ) T R , ζ 0 = δ 0 α Σ + T c / 2 , f = T c γ Σ P in ( α Σ + T c / 2 ) 3 / 2 , θ = t / ( β 2 Σ ) α Σ + T c / 2 , Ψ = A γ Σ α Σ + T c / 2 ( i e i ϕ in ) .
T R A T + i 2 β 2 Σ 2 A t 2 i γ Σ | A | 2 A = ( α Σ + T c 2 + i δ 0 ) A + i T c P in e i ϕ in + T R F ( t , T ) .
F ( t , T ) F ( t , T ) = D F F + δ ( t t ) δ ( T T ) ,
F ( t , T ) F ( t , T ) = D F + F δ ( t t ) δ ( T T ) ,
D F F + = h ¯ ω 0 T R [ 2 α Σ + T c ] ,
D F + F = 0 .
d ξ d T = β 2 Σ Ω T R + F ξ ,
d Ω d T = T c + 2 α σ T R Ω + F Ω .
F ξ ( T ) = 1 E t [ A * F ( t , T ) + F ( t , T ) A ] d t ,
F Ω ( T ) = i E [ A * t F ( t , T ) F ( t , T ) A t ] d t ,
[ A * F ( t , T ) t + A * t F ( t , T ) ] d t = A * F ( t , T ) | = 0 .
F ξ ( T ) F ξ ( T ) = D ξ ξ δ ( T T ) = π 2 τ 2 12 E D F F + δ ( T T ) ,
F Ω ( T ) F Ω ( T ) = D Ω Ω δ ( T T ) = 1 3 E τ 2 D F F + δ ( T T ) ,
F ξ ( T ) F Ω ( T ) = D ξ Ω δ ( T T ) = i 2 E D F F + δ ( T T ) ,
F Ω ( T ) F ξ ( T ) = D Ω ξ δ ( T T ) = i 2 E D F F + δ ( T T ) .
ξ ( ω ) = i ω [ β 2 Σ T R F Ω ( ω ) i ω + ( T c + 2 α ) / T R + F ξ ( ω ) ] .
ξ ( T ) = ξ ( ω ) e i ω T d ω 2 π ,
ξ ( ω ) = ξ ( T ) e i ω T d T .
t m ( ω ) = ξ ( ω ) + F m w ( ω ) ,
F m w ( T ) F m w ( T ) = D m w δ ( T T ) ,
D m w = 1 ω R 2 2 q ρ P ¯ D + F k B Θ P m w | D ,
P m w | D = ρ 2 T c 2 | T R / 2 T R / 2 | A ( T , t ) | 2 exp [ i 2 π ( t 1 t ) T R ] d t 1 T R | 2 ρ 2 T c 2 P a v e 2
P ¯ D = T c P ave
S t m ( ω ) = t m ( T ) t m ( T + T ) e i ω T d T
S t m ( ω ) 1 ω 2 [ D ξ ξ + ( β 2 Σ T R ) 2 D Ω Ω ω 2 + [ ( T c + 2 α Σ ) / T R ] 2 β 2 Σ T R D ξ Ω i ω + ( T c + 2 α Σ ) / T R β 2 Σ T R D Ω ξ i ω + ( T c + 2 α Σ ) / T R ] + D m w .
S ϕ = π 2 ω 2 h ¯ ω 0 P ave ( α Σ + T c / 2 ) ω R 2 τ 2 6 T R 2 [ 1 + ( 2 β 2 Σ π τ 2 T R ) 2 1 ω 2 + 4 ( α Σ + T c / 2 ) 2 / T R 2 ] + 2 q ρ P ¯ D + F k B Θ P m w | D
D m w = 1 ω R 2 2 h ¯ ω 0 η T c P ave = 2 π ω R 2 γ Δ 0 ( D ) g η γ 2 ,
S ϕ = 2 π γ Δ 0 ( D ) g η γ 2 [ 1 + 1 96 γ ( D ) Δ 0 η γ 2 f 2 + 1 24 ( 1 + π 2 f 2 γ 2 ) 1 η γ 2 π 2 f 2 Δ 0 ( D ) γ ]
S ϕ | f 0 = g f 2 1 12 2 π ( ( D ) Δ 0 γ ) 1 / 2
S ϕ min | f 0 g f 2 3 1 / 4 12 2 π D
σ min 2 g 1 / 2 ( D ) 1 / 4 5 π ν R F 1 τ ,
σ 2 = 2 0 S ϕ f 2 ν R F 2 sin 4 ( π f τ ) ( π f τ ) 2 d f ,
σ = S ϕ f 2 2 ν R F 2 τ ,
σ minhp g 1 / 2 2 2 π ν R F 1 τ .
P t h ( α Σ + T c / 2 ) 3 T c γ Σ ,
Δ 0 γ = π 2 8 sin 2 ( ϕ c ϕ p ) P i n P t h P i n P t h .

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