Modal Analysis Investigation of Mechanical Kerr Frequency Combs

  • Samer HouriEmail author
  • Daiki Hatanaka
  • Yaroslav M. Blanter
  • Hiroshi Yamaguchi
Conference paper
Part of the Springer Proceedings in Physics book series (SPPHY, volume 228)


The aim of this work is to theoretically investigate the possibility of Kerr frequency combs in mechanical systems. In particular, whether microelectromechanical devices (MEMS) can be used to generate frequency combs in a manner that is analogous to the optical frequency combs generated in optical microresonators with Kerr-type nonlinearity. The analysis assumes a beam-like micromechanical structure, and starting from the Euler-Bernoulli beam equation derives the necessary conditions in parameter space for the comb generation. The chapter equally presents potential means for the physical implementation of mechanical Kerr combs.


MEMS Frequency combs Kerr nonlinearity 



This work is partly supported by a MEXT Grant-in-Aid for Scientific Research on Innovative Areas “Science of hybrid quantum systems” (Grant No. JP15H05869 and JP15K21727).


  1. 1.
    N. Akhmediev, J.M. Dudley, D.R. Solli, S.K. Turitsyn, Recent progress in investigating optical rogue waves. J. Opt. 15(6), 060201 (2013)ADSGoogle Scholar
  2. 2.
    C. Bao, Y. Xuan, D.E. Leaird, S. Wabnitz, M. Qi, A.M. Weiner, Spatial mode-interaction induced single soliton generation in microresonators. Optica 4(9), 1011–1015 (2017)Google Scholar
  3. 3.
    R.D. Blevins, R. Plunkett, Formulas for natural frequency and mode shape. J. Appl. Mech. 47, 461 (1980)ADSGoogle Scholar
  4. 4.
    L.S. Cao, D.X. Qi, R.W. Peng, M. Wang, P. Schmelcher, Phononic frequency combs through nonlinear resonances. Phys. Rev. Lett. 112(7), 075505 (2014)ADSGoogle Scholar
  5. 5.
    J. Cha, C. Daraio, Electrical tuning of elastic wave propagation in nanomechanical lattices at MHz frequencies. Nat. Nanotechnol. 13(11), 1016 (2018)ADSGoogle Scholar
  6. 6.
    Y.K. Chembo, N. Yu, Modal expansion approach to optical-frequency-comb generation with monolithic whispering-gallery-mode resonators. Phys. Rev. A 82(3), 033801 (2010)ADSGoogle Scholar
  7. 7.
    Y.K. Chembo, C.R. Menyuk, Spatiotemporal Lugiato-Lefever formalism for Kerr-comb generation in whispering-gallery-mode resonators. Phys. Rev. A 87(5), 053852 (2013)ADSGoogle Scholar
  8. 8.
    Chia, C.Y., 1980. Nonlinear analysis of plates. McGraw-Hill International Book CompanyGoogle Scholar
  9. 9.
    N.A. Cleland, Foundations of Nanomechanics: From Solid-State Theory to Device Applications (Springer, New York, 2003), pp. 312–319Google Scholar
  10. 10.
    A. Coillet, Y.K. Chembo, Routes to spatiotemporal chaos in Kerr optical frequency combs. Chaos: Interdiscip. J. Nonlinear Sci. 24(1), 013113 (2014)MathSciNetzbMATHGoogle Scholar
  11. 11.
    A. Coillet, J. Dudley, G. Genty, L. Larger, Y.K. Chembo, Optical rogue waves in whispering-gallery-mode resonators. Phys. Rev. A 89(1), 013835 (2014)ADSGoogle Scholar
  12. 12.
    J. Crawford, S. Atluri, Non-linear vibrations of a flat plate with initial stresses. J. Sound Vib. 43(1), 117–129 (1975)ADSzbMATHGoogle Scholar
  13. 13.
    S.T. Cundiff, J. Ye, J.L. Hall, Optical frequency synthesis based on mode-locked lasers. Rev. Sci. Instrum. 72(10), 3749–3771 (2001)ADSGoogle Scholar
  14. 14.
    D.A. Czaplewski, C. Chen, D. Lopez, O. Shoshani, A.M. Eriksson, S. Strachan, S.W. Shaw, Bifurcation generated mechanical frequency comb. Phys. Rev. Lett. 121(24), 244302 (2018)ADSGoogle Scholar
  15. 15.
    S.A. Diddams, D.J. Jones, J. Ye, S.T. Cundiff, J.L. Hall, J.K. Ranka, R.S. Windeler, R. Holzwarth, T. Udem, T.W. Hänsch, Direct link between microwave and optical frequencies with a 300 THz femtosecond laser comb. Phys. Rev. Lett. 84(22), 5102 (2000)ADSGoogle Scholar
  16. 16.
    P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, T.J. Kippenberg, Optical frequency comb generation from a monolithic microresonator. Nature 450(7173), 1214 (2007)ADSGoogle Scholar
  17. 17.
    J.N. Eckstein, A.I. Ferguson, T.W. Hänsch, High-resolution two-photon spectroscopy with picosecond light pulses. Phys. Rev. Lett. 40(13), 847 (1978)ADSGoogle Scholar
  18. 18.
    A. Erbe, H. Krömmer, A. Kraus, R.H. Blick, G. Corso, K. Richter, Mechanical mixing in nonlinear nanomechanical resonators. Appl. Phys. Lett. 77(19), 3102–3104 (2000)ADSGoogle Scholar
  19. 19.
    A. Ganesan, C. Do, A. Seshia, Phononic frequency comb via intrinsic three-wave mixing. Phys. Rev. Lett. 118(3), 033903 (2017)ADSGoogle Scholar
  20. 20.
    A. Ganesan, C. Do, A. Seshia, Phononic frequency comb via three-mode parametric resonance. Appl. Phys. Lett. 112(2), 021906 (2018)ADSGoogle Scholar
  21. 21.
    C. Godey, I.V. Balakireva, A. Coillet, Y.K. Chembo, Stability analysis of the spatiotemporal Lugiato-Lefever model for Kerr optical frequency combs in the anomalous and normal dispersion regimes. Phys. Rev. A 89(6), 063814 (2014)ADSGoogle Scholar
  22. 22.
    P. Grelu, N. Akhmediev, Dissipative solitons for mode-locked lasers. Nat. Photonics 6(2), 84 (2012)ADSGoogle Scholar
  23. 23.
    D.S. Greywall, B. Yurke, P.A. Busch, A.N. Pargellis, R.L. Willett, Evading amplifier noise in nonlinear oscillators. Phys. Rev. Lett. 72(19), 2992 (1994)ADSGoogle Scholar
  24. 24.
    J.L. Hall, Nobel lecture: defining and measuring optical frequencies. Rev. Mod. Phys. 78(4), 1279 (2006)ADSGoogle Scholar
  25. 25.
    D. Hatanaka, I. Mahboob, K. Onomitsu, H. Yamaguchi, Phonon waveguides for electromechanical circuits. Nat. Nanotechnol. 9(7), 520 (2014)ADSGoogle Scholar
  26. 26.
    D. Hatanaka, A. Dodel, I. Mahboob, K. Onomitsu, H. Yamaguchi, Phonon propagation dynamics in band-engineered one-dimensional phononic crystal waveguides. New J. Phys. 17(11), 113032 (2015)ADSGoogle Scholar
  27. 27.
    D. Hatanaka, T. Darras, I. Mahboob, K. Onomitsu, H. Yamaguchi, Broadband reconfigurable logic gates in phonon waveguides. Sci. Rep. 7(1), 12745 (2017)ADSGoogle Scholar
  28. 28.
    D. Hatanaka, A. Bachtold, H. Yamaguchi, Electrostatically induced phononic crystal. Phys. Rev. Appl. 11(2), 024024 (2019)ADSGoogle Scholar
  29. 29.
    H. Haus, Parameter ranges for CW passive mode locking. IEEE J. Quantum Electron. 12(3), 169–176 (1976)ADSGoogle Scholar
  30. 30.
    T. Herr, V. Brasch, J.D. Jost, C.Y. Wang, N.M. Kondratiev, M.L. Gorodetsky, T.J. Kippenberg, Temporal solitons in optical microresonators. Nat. Photonics 8(2), 145 (2014)ADSGoogle Scholar
  31. 31.
    S. Houri, U. Bhaskar, B. Gallacher, L. Francis, T. Pardoen, J.P. Raskin, Dynamic analysis of multi-beam MEMS structures for the extraction of the stress-strain response of thin films. Exp. Mech. 53(3), 441–453 (2013)Google Scholar
  32. 32.
    S. Houri, S.J. Cartamil-Bueno, M. Poot, P.G. Steeneken, H.S.J. van der Zant, W.J. Venstra, Direct and parametric synchronization of a graphene self-oscillator. Appl. Phys. Lett. 110(7), 073103 (2017)ADSGoogle Scholar
  33. 33.
    S. Houri, R. Ohta, M. Asano, Y.M. Blanter, H. Yamaguchi, Pulse-width modulated oscillations in a nonlinear resonator under two-tone driving as a means for MEMS sensor readout. Jpn. J. Appl. Phys., 58(SB), SBBI05 (2019)Google Scholar
  34. 34.
    S.Houri, D. Hatanaka, M. Asano, R. Ohta, H. Yamaguchi, Limit cycles and bifurcations in a nonlinear MEMS resonator with a 1: 3 internal resonance. Appl. Phys. Lett., 114(10), 103103 (2019)ADSGoogle Scholar
  35. 35.
    N. Jaber, A. Ramini, M.I. Younis, Multifrequency excitation of a clamped–clamped microbeam: analytical and experimental investigation. Microsystems & Nanoeng. 2, 16002 (2016)Google Scholar
  36. 36.
    R.B. Karabalin, M.C. Cross, M.L. Roukes, Nonlinear dynamics and chaos in two coupled nanomechanical resonators. Phys. Rev. B 79(16), 165309 (2009)ADSGoogle Scholar
  37. 37.
    T.J. Kippenberg, S.M. Spillane, K.J. Vahala, Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity. Phys. Rev. Lett. 93(8), 083904 (2004)ADSGoogle Scholar
  38. 38.
    T.J. Kippenberg, R. Holzwarth, S.A. Diddams, Microresonator-based optical frequency combs. Science 332(6029), 555–559 (2011)ADSGoogle Scholar
  39. 39.
    M. Kurosu, D. Hatanaka, K. Onomitsu, H. Yamaguchi, On-chip temporal focusing of elastic waves in a phononic crystal waveguide. Nat. Commun. 9(1), 1331 (2018)ADSGoogle Scholar
  40. 40.
    J. Laconte, D. Flandre, J.P. Raskin, Micromachined Thin-Film Sensors for SOI-CMOS Co-integration (Springer Science & Business Media, 2006)Google Scholar
  41. 41.
    J.S. Levy, A. Gondarenko, M.A. Foster, A.C. Turner-Foster, A.L. Gaeta, M. Lipson, CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects. Nat. Photonics 4(1), 37 (2010)ADSGoogle Scholar
  42. 42.
    R. Lifshitz, M.C. Cross, Nonlinear dynamics of nanomechanical and micromechanical resonators. Rev. Nonlinear Dyn. Complex. 1, 1–52 (2008)zbMATHGoogle Scholar
  43. 43.
    L.A. Lugiato, R. Lefever, Spatial dissipative structures in passive optical systems. Phys. Rev. Lett. 58(21), 2209 (1987)ADSGoogle Scholar
  44. 44.
    K.J. Lulla, R.B. Cousins, A. Venkatesan, M.J. Patton, A.D. Armour, C.J. Mellor, J.R. Owers-Bradley, Nonlinear modal coupling in a high-stress doubly-clamped nanomechanical resonator. New J. Phys. 14(11), 113040 (2012)ADSGoogle Scholar
  45. 45.
    L.-S. Ma, Z. Bi, A. Bartels, L. Robertsson, M. Zucco, R.S. Windeler, G. Wilpers, C. Oates, L. Hollberg, S.A. Diddams, Optical frequency synthesis and comparison with uncertainty at the 10–19 level. Science 303(5665), 1843–1845 (2004)ADSGoogle Scholar
  46. 46.
    M.J. Madou, Manufacturing Techniques for Microfabrication and Nanotechnology, vol. 2 (CRC press, 2011)Google Scholar
  47. 47.
    I. Mahboob, Q. Wilmart, K. Nishiguchi, A. Fujiwara, H. Yamaguchi, Tuneable electromechanical comb generation. Appl. Phys. Lett. 100(11), 113109 (2012)ADSGoogle Scholar
  48. 48.
    I. Mahboob, R. Dupuy, K. Nishiguchi, A. Fujiwara, H. Yamaguchi, Hopf and period-doubling bifurcations in an electromechanical resonator. Appl. Phys. Lett. 109(7), 073101 (2016)ADSGoogle Scholar
  49. 49.
    P. Marin-Palomo, J.N. Kemal, M. Karpov, A. Kordts, J. Pfeifle, M.H. Pfeiffer, P. Trocha et al., Microresonator-based solitons for massively parallel coherent optical communications. Nature 546(7657), 274 (2017)ADSGoogle Scholar
  50. 50.
    M.H. Matheny, L.G. Villanueva, R.B. Karabalin, J.E. Sader, M.L. Roukes, Nonlinear mode-coupling in nanomechanical systems. Nano Lett., 13(4), 1622–1626 (2013)ADSGoogle Scholar
  51. 51.
    A.B. Matsko, A.A. Savchenkov, D. Strekalov, V.S. Ilchenko, L. Maleki, Optical hyperparametric oscillations in a whispering-gallery-mode resonator: threshold and phase diffusion. Phys. Rev. A 71(3), 033804 (2005)ADSGoogle Scholar
  52. 52.
    A.B. Matsko, W. Liang, A.A. Savchenkov, L. Maleki, Chaotic dynamics of frequency combs generated with continuously pumped nonlinear microresonators. Opt. Lett. 38(4), 525–527 (2013)ADSGoogle Scholar
  53. 53.
    A.H. Nayfeh, D.T. Mook, Nonlinear Oscillations (Willey, New York, 1979)zbMATHGoogle Scholar
  54. 54.
    A.H. Nayfeh, R.A. Ibrahim, Nonlinear interactions: analytical, computational, and experimental methods. Appl. Mech. Rev., 54, B60 (2001)Google Scholar
  55. 55.
    E. Obrzud, S. Lecomte, T. Herr, Temporal solitons in microresonators driven by optical pulses. Nat. Photonics 11(9), 600 (2017)Google Scholar
  56. 56.
    K. Panajotov, M.G. Clerc, M. Tlidi, Spatiotemporal chaos and two-dimensional dissipative rogue waves in Lugiato-Lefever model. Eur. Phys. J. D 71(7), 176 (2017)ADSGoogle Scholar
  57. 57.
    M. Sathyamoorthy, Nonlinear Analysis of Structures (1997) (CRC Press 2017)Google Scholar
  58. 58.
    M.J. Seitner, M. Abdi, A. Ridolfo, M.J. Hartmann, E.M. Weig, Parametric oscillation, frequency mixing, and injection locking of strongly coupled nanomechanical resonator modes. Phys. Rev. Lett. 118(25), 254301 (2017)ADSGoogle Scholar
  59. 59.
    D.E. Spence, P.N. Kean, W. Sibbett, 60-fsec pulse generation from a self-mode-locked Ti: sapphire laser. Opt. Lett. 16(1), 42–44 (1991)ADSGoogle Scholar
  60. 60.
    D.T. Spencer, T. Drake, T.C. Briles, J. Stone, L.C. Sinclair, C. Fredrick, Q. Li, et al., An optical-frequency synthesizer using integrated photonics. Nature 557(7703), 81–85 (2018)ADSGoogle Scholar
  61. 61.
    B. Stern, X. Ji, Y. Okawachi, A.L. Gaeta, M. Lipson, Battery-operated integrated frequency comb generator. Nature 562(7727), 401 (2018)ADSGoogle Scholar
  62. 62.
    H.A. Tilmans, M. Elwenspoek, J.H. Fluitman, Micro resonant force gauges. Sens. Actuators, A 30(1–2), 35–53 (1992)Google Scholar
  63. 63.
    T. Udem, J. Reichert, R. Holzwarth, T.W. Hänsch, Absolute optical frequency measurement of the cesium D 1 line with a mode-locked laser. Phys. Rev. Lett. 82(18), 3568 (1999)ADSGoogle Scholar
  64. 64.
    S.S. Verbridge, J.M. Parpia, R.B. Reichenbach, L.M. Bellan, H.G. Craighead, High quality factor resonance at room temperature with nanostrings under high tensile stress. J. Appl. Phys. 99(12), 124304 (2006)ADSGoogle Scholar
  65. 65.
    H.J.R. Westra, M. Poot, H.S.J. Van Der Zant, W.J. Venstra, Nonlinear modal interactions in clamped-clamped mechanical resonators. Phys. Rev. Lett. 105(11), 117205 (2010)ADSGoogle Scholar
  66. 66.
    H. Yamaguchi, I. Mahboob, Parametric mode mixing in asymmetric doubly clamped beam resonators. New J. Phys. 15(1), 015023 (2013)ADSMathSciNetGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  • Samer Houri
    • 1
    Email author
  • Daiki Hatanaka
    • 1
  • Yaroslav M. Blanter
    • 2
  • Hiroshi Yamaguchi
    • 1
  1. 1.NTT Basic Research Laboratories, NTT CorporationKanagawaJapan
  2. 2.Kavli Institute of Nanoscience, Delft University of TechnologyDelftThe Netherlands

Personalised recommendations