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Frequency combs induced by phase turbulence


Wave instability—the process that gives rise to turbulence in hydrodynamics1—represents the mechanism by which a small disturbance in a wave grows in amplitude owing to nonlinear interactions. In photonics, wave instabilities result in modulated light waveforms that can become periodic in the presence of coherent locking mechanisms. These periodic optical waveforms are known as optical frequency combs2,3,4. In ring microresonator combs5,6, an injected monochromatic wave becomes destabilized by the interplay between the resonator dispersion and the Kerr nonlinearity of the constituent crystal. By contrast, in ring lasers instabilities are considered to occur only under extreme pumping conditions7,8. Here we show that, despite this notion, semiconductor ring lasers with ultrafast gain recovery9,10 can enter frequency comb regimes at low pumping levels owing to phase turbulence11—an instability known to occur in hydrodynamics, superconductors and Bose–Einstein condensates. This instability arises from the phase–amplitude coupling of the laser field provided by linewidth enhancement12, which produces the needed interplay of dispersive and nonlinear effects. We formulate the instability condition in the framework of the Ginzburg–Landau formalism11. The localized structures that we observe share several properties with dissipative Kerr solitons, providing a first step towards connecting semiconductor ring lasers and microresonator frequency combs13.

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Fig. 1: Fabry–Pérot and ring frequency combs.
Fig. 2: Conditions for the phase instability in a monolithic ring laser.
Fig. 3: Defect-engineered ring frequency comb.

Data availability

Source data for Figs. 13 are provided with the paper. Additional data that support the findings of this study are available from the corresponding authors upon reasonable request.

Code availability

The codes used to plot the Benjamin–Feir space and related datasets, to calculate the Ginzburg–Landau cD and cNL parameters with error propagation, and to simulate the dynamic microwave gratings are available at:;; Information on the code developed to simulate the QCL dynamics and its results are available from the corresponding authors upon reasonable request.


  1. 1.

    Reynolds, O. XXIX. An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels. Philos. Trans. R. Soc. Lond. 174, 935–982 (1883).

    ADS  MATH  Google Scholar 

  2. 2.

    Hänsch, T. W. Nobel lecture: passion for precision. Rev. Mod. Phys. 78, 1297–1309 (2006).

    ADS  Article  Google Scholar 

  3. 3.

    Udem, T., Holzwarth, R. & Hänsch, T. W. Optical frequency metrology. Nature 416, 233–237 (2002).

    ADS  CAS  PubMed  PubMed Central  Article  Google Scholar 

  4. 4.

    Picqué, N. & Hänsch, T. W. Frequency comb spectroscopy. Nat. Photon. 13, 146–157 (2019).

    ADS  Article  Google Scholar 

  5. 5.

    Vahala, K. J. Optical microcavities. Nature 424, 839–846 (2003).

    ADS  CAS  PubMed  PubMed Central  Article  Google Scholar 

  6. 6.

    Gaeta, A. L., Lipson, M. & Kippenberg, T. J. Photonic-chip-based frequency combs. Nat. Photon. 13, 158–169 (2019).

    ADS  CAS  Article  Google Scholar 

  7. 7.

    Risken, H. & Nummedal, K. Self-pulsing in lasers. J. Appl. Phys. 39, 4662–4672 (1968).

    ADS  Article  Google Scholar 

  8. 8.

    Graham, R. & Haken, H. Quantum theory of light propagation in a fluctuating laser-active medium. Z. Phys. 213, 420–450 (1968).

    ADS  Article  Google Scholar 

  9. 9.

    Mujagić, E. et al. Grating-coupled surface emitting quantum cascade ring lasers. Appl. Phys. Lett. 93, 011108 (2008).

    ADS  Article  CAS  Google Scholar 

  10. 10.

    Hugi, A., Villares, G., Blaser, S., Liu, H. C. & Faist, J. Mid-infrared frequency comb based on a quantum cascade laser. Nature 492, 229–233 (2012).

    ADS  CAS  PubMed  PubMed Central  Article  Google Scholar 

  11. 11.

    Aranson, I. S. & Kramer, L. The world of the complex Ginzburg–Landau equation. Rev. Mod. Phys. 74, 99–143 (2002).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    Henry, C. Theory of the linewidth of semiconductor lasers. IEEE J. Quantum Electron. 18, 259–264 (1982).

    ADS  Article  Google Scholar 

  13. 13.

    Lugiato, L. A. & Lefever, R. Spatial dissipative structures in passive optical systems. Phys. Rev. Lett. 58, 2209–2211 (1987).

    ADS  CAS  PubMed  PubMed Central  Article  Google Scholar 

  14. 14.

    Kues, M. et al. Quantum optical microcombs. Nat. Photon. 13, 170–179 (2019).

    ADS  CAS  Article  Google Scholar 

  15. 15.

    Kippenberg, T. J., Holzwarth, R. & Diddams, S. A. Microresonator-based optical frequency combs. Science 332, 555–559 (2011).

    ADS  CAS  PubMed  Article  Google Scholar 

  16. 16.

    Herr, T. et al. Universal formation dynamics and noise of Kerr-frequency combs in microresonators. Nat. Photon. 6, 480–487 (2012).

    ADS  CAS  Article  Google Scholar 

  17. 17.

    Agrawal, G. P. Population pulsations and nondegenerate four-wave mixing in semiconductor lasers and amplifiers. J. Opt. Soc. Am. B 5, 147–159 (1988).

    ADS  CAS  Article  Google Scholar 

  18. 18.

    Faist, J. et al. Quantum cascade laser frequency combs. Nanophotonics 5, 272–291 (2016).

    Article  Google Scholar 

  19. 19.

    Piccardo, M. et al. The harmonic state of quantum cascade lasers: origin, control, and prospective applications. Opt. Express 26, 9464–9483 (2018).

    ADS  CAS  PubMed  PubMed Central  Article  Google Scholar 

  20. 20.

    Opačak, N. & Schwarz, B. Theory of frequency-modulated combs in lasers with spatial hole burning, dispersion, and Kerr nonlinearity. Phys. Rev. Lett. 123, 243902 (2019).

    ADS  PubMed  PubMed Central  Article  Google Scholar 

  21. 21.

    Matsumoto, N. & Kumabe, K. AlGaAs-GaAs semiconductor ring laser. Jpn. J. Appl. Phys. 16, 1395–1398 (1977).

    ADS  CAS  Article  Google Scholar 

  22. 22.

    Krauss, T., Laybourn, P. J. R. & Roberts, J. CW operation of semiconductor ring lasers. Electron. Lett. 26, 2095–2097 (1990).

    ADS  Article  Google Scholar 

  23. 23.

    Gelens, L. et al. Exploring multistability in semiconductor ring lasers: theory and experiment. Phys. Rev. Lett. 102, 193904 (2009).

    ADS  CAS  PubMed  PubMed Central  Article  Google Scholar 

  24. 24.

    Villares, G., Hugi, A., Blaser, S. & Faist, J. Dual-comb spectroscopy based on quantum-cascade-laser frequency combs. Nat. Commun. 5, 5192 (2014).

    ADS  CAS  PubMed  PubMed Central  Article  Google Scholar 

  25. 25.

    Consolino, L. et al. Fully phase-stabilized quantum cascade laser frequency comb. Nat. Commun. 10, 2938 (2019).

    ADS  PubMed  PubMed Central  Article  CAS  Google Scholar 

  26. 26.

    Piccardo, M. et al. Radio frequency transmitter based on a laser frequency comb. Proc. Natl Acad. Sci. USA 116, 9181–9185 (2019); correction 116 17598 (2019).

    ADS  CAS  PubMed  PubMed Central  Article  Google Scholar 

  27. 27.

    Hillman, L. W., Krasiński, J., Boyd, R. W. & Stroud, C. R. Observation of higher order dynamical states of a homogeneously broadened laser. Phys. Rev. Lett. 52, 1605–1608 (1984).

    ADS  CAS  Article  Google Scholar 

  28. 28.

    Staliunas, K. Laser Ginzburg–Landau equation and laser hydrodynamics. Phys. Rev. A 48, 1573–1581 (1993).

    ADS  CAS  PubMed  PubMed Central  Article  Google Scholar 

  29. 29.

    Gil, L. & Lippi, G. L. Phase instability in semiconductor lasers. Phys. Rev. Lett. 113, 213902 (2014).

    ADS  CAS  PubMed  PubMed Central  Article  Google Scholar 

  30. 30.

    Chate, H. Spatiotemporal intermittency regimes of the one-dimensional complex Ginzburg–Landau equation. Nonlinearity 7, 185–204 (1994).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  31. 31.

    Columbo, L. L., Barbieri, S., Sirtori, C. & Brambilla, M. Dynamics of a broad-band quantum cascade laser: from chaos to coherent dynamics and mode-locking. Opt. Express 26, 2829–2847 (2018).

    ADS  CAS  PubMed  PubMed Central  Article  Google Scholar 

  32. 32.

    Shraiman, B. et al. Spatiotemporal chaos in the one-dimensional complex Ginzburg–Landau equation. Physica D 57, 241–248 (1992).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  33. 33.

    Kippenberg, T. J., Gaeta, A. L., Lipson, M. & Gorodetsky, M. L. Dissipative Kerr solitons in optical microresonators. Science 361, eaan8083 (2018).

    PubMed  PubMed Central  Article  CAS  Google Scholar 

  34. 34.

    Herr, T. et al. Temporal solitons in optical microresonators. Nat. Photon. 8, 145–152 (2014).

    ADS  CAS  Article  Google Scholar 

  35. 35.

    Cole, D. C., Lamb, E. S., Del’Haye, P., Diddams, S. A. & Papp, S. B. Soliton crystals in Kerr resonators. Nat. Photon. 11, 671–676 (2017).

    ADS  CAS  Article  Google Scholar 

  36. 36.

    Karpov, M. et al. Dynamics of soliton crystals in optical microresonators. Nat. Phys. 15, 1071–1077 (2019).

    CAS  Article  Google Scholar 

  37. 37.

    Brusch, L., Zimmermann, M. G., van Hecke, M., Bär, M. & Torcini, A. Modulated amplitude waves and the transition from phase to defect chaos. Phys. Rev. Lett. 85, 86–89 (2000).

    ADS  CAS  PubMed  PubMed Central  Article  Google Scholar 

  38. 38.

    Lugiato, L. A., Oldano, C. & Narducci, L. M. Cooperative frequency locking and stationary spatial structures in lasers. J. Opt. Soc. Am. B 5, 879–888 (1988).

    ADS  CAS  Article  Google Scholar 

  39. 39.

    Kaige, W., Abraham, N. B. & Lugiato, L. A. Leading role of optical phase instabilities in the formation of certain laser transverse patterns. Phys. Rev. A 47, 1263–1273 (1993).

    ADS  Article  Google Scholar 

  40. 40.

    Del’Haye, P. et al. Optical frequency comb generation from a monolithic microresonator. Nature 450, 1214–1217 (2007).

    ADS  PubMed  PubMed Central  Article  CAS  Google Scholar 

  41. 41.

    Bao, H. et al. Laser cavity-soliton microcombs. Nat. Photon. 13, 384–389 (2019).

    ADS  CAS  Article  Google Scholar 

  42. 42.

    Wang, C. A. et al. MOVPE growth of LWIR AlInAs/GaInAs/InP quantum cascade lasers: impact of growth and material quality on laser performance. IEEE J. Sel. Top. Quantum Electron. 23, 1–13 (2017).

    ADS  Google Scholar 

  43. 43.

    Hofstetter, D. & Faist, J. Measurement of semiconductor laser gain and dispersion curves utilizing Fourier transforms of the emission spectra. IEEE Photonics Technol. Lett. 11, 1372–1374 (1999).

    ADS  Article  Google Scholar 

  44. 44.

    von Staden, J., Gensty, T., Elsäßer, W., Giuliani, G. & Mann, C. Measurements of the α factor of a distributed-feedback quantum cascade laser by an optical feedback self-mixing technique. Opt. Lett. 31, 2574–2576 (2006).

    ADS  Article  Google Scholar 

  45. 45.

    Jumpertz, L. et al. Measurements of the linewidth enhancement factor of mid-infrared quantum cascade lasers by different optical feedback techniques. AIP Adv. 6, 015212 (2016).

    ADS  Article  CAS  Google Scholar 

  46. 46.

    Kumazaki, N. et al. Spectral behavior of linewidth enhancement factor of a mid-infrared quantum cascade laser. Jpn. J. Appl. Phys. 47, 6320–6326 (2008).

    ADS  CAS  Article  Google Scholar 

  47. 47.

    Szedlak, R. et al. Ring quantum cascade lasers with twisted wavefronts. Sci. Rep. 8, 7998 (2018).

    ADS  PubMed  PubMed Central  Article  CAS  Google Scholar 

  48. 48.

    Malara, P. et al. External ring-cavity quantum cascade lasers. Appl. Phys. Lett. 102, 141105 (2013).

    ADS  Article  CAS  Google Scholar 

  49. 49.

    Wojcik, A. K. et al. Generation of picosecond pulses and frequency combs in actively mode locked external ring cavity quantum cascade lasers. Appl. Phys. Lett. 103, 231102 (2013).

    ADS  Article  CAS  Google Scholar 

  50. 50.

    Revin, D. G., Hemingway, M., Wang, Y., Cockburn, J. W. & Belyanin, A. Active mode locking of quantum cascade lasers in an external ring cavity. Nat. Commun. 7, 11440 (2016).

    ADS  CAS  PubMed  PubMed Central  Article  Google Scholar 

  51. 51.

    Faist, J. et al. Quantum cascade disk lasers. Appl. Phys. Lett. 69, 2456–2458 (1996).

    ADS  CAS  Article  Google Scholar 

  52. 52.

    Meng, B. et al. Mid-infrared frequency comb from a ring quantum cascade laser. Optica 7, 162–167 (2020).

    ADS  Article  Google Scholar 

  53. 53.

    Gmachl, C. et al. High-power directional emission from microlasers with chaotic resonators. Science 280, 1556–1564 (1998).

    ADS  CAS  PubMed  PubMed Central  Article  Google Scholar 

  54. 54.

    Wang, Q. J. et al. Whispering-gallery mode resonators for highly unidirectional laser action. Proc. Natl Acad. Sci. USA 107, 22407–22412 (2010).

    ADS  CAS  PubMed  PubMed Central  Article  Google Scholar 

  55. 55.

    Lončar, M. et al. Design and fabrication of photonic crystal quantum cascade lasers for optofluidics. Opt. Express 15, 4499–4514 (2007).

    ADS  PubMed  PubMed Central  Article  Google Scholar 

  56. 56.

    Piccardo, M. et al. Time-dependent population inversion gratings in laser frequency combs. Optica 5, 475–478 (2018).

    ADS  CAS  Article  Google Scholar 

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We acknowledge support from the National Science Foundation under award numbers ECCS-1614631 and CCSS-1807323. Any opinions, findings, conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. This work was performed in part at the Center for Nanoscale Systems (CNS), a member of the National Nanotechnology Coordinated Infrastructure Network (NNCI), which is supported by the National Science Foundation under NSF award number 1541959. B.S. was supported by the Austrian Science Fund (FWF) within the project NanoPlas. We gratefully acknowledge C. A. Wang, M. K. Connors and D. McNulty for providing the QCL material. We thank V. Ginis, T. S. Mansuripur and F. Grillot for discussions, G. Strasser for enabling the device fabrication, P. Chevalier for cleaving the devices and the D. Ham group for lending us microwave amplifiers. We acknowledge discussions with L. A. Lugiato on spatial patterns in lasers and Kerr microcombs.

Author information




M.P. initiated the project. B.S. and M.B. fabricated the devices. M.P, B.S., D.K., S.J. and J.H. carried out the experiments. M.P. derived the grating model. B.S. and N.O. derived the Ginzburg–Landau theory with suggestions from L.L.C. B.S., N.O. and Y.W. performed the laser space–time domain simulations. D.K., M.T., W.T.C. and A.Y.Z. performed the defect engineering based on simulations from M.P. and D.K. M.P. wrote most of the manuscript. M.P., N.O., D.K. and S.J. wrote sections of the Supplementary Information. A.B. and F.C. supervised the project. M.P., B.S., D.K., M.B., N.O., Y.W., S.J., L.L.C., A.B. and F.C. contributed to the analysis, discussion and writing of the paper.

Corresponding authors

Correspondence to Marco Piccardo or Federico Capasso.

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The authors declare no competing interests.

Additional information

Peer review information Nature thanks Roberto Morandotti, Johann Riemensberger and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Extended data figures and tables

Extended Data Fig. 1 Coherence and phase of the ring frequency combs.

a, Schematic of the Shifted Wave Interference Fourier Transform Spectroscopy (SWIFTS) setup. The inset shows a microscope image of the QCL ring. LNA, low-noise amplifier; FTIR, Fourier transform infrared spectrometer; LO, local oscillator; QWIP, quantum well infrared photodetector. b, Comparison of the simulations with the experimental results obtained by SWIFTS. The displayed simulation results show the spectral amplitudes (top), the intermodal difference phases (middle) and the corresponding time-domain signals (bottom). Different seeds for spontaneous emission noise were used in the two simulations. The experimental data show the spectrum (top), the measured intermodal difference phases (middle) and the SWIFTS amplitudes (bottom). The red crosses on top of the SWIFTS amplitudes are given by |An||An−1|, that is, the geometric average of adjacent modes of the intensity spectrum. The red crosses agree well with the expected values for full phase coherence.

Extended Data Fig. 2 Evolution of the experimental optical spectra with injected current.

ae, Spectral series corresponding to five distinct ring lasers. The multimode regime can switch on and off. The current density normalized to the lasing threshold is given to the right of each spectrum. Int., intensity.

Extended Data Fig. 3 Dynamic gratings in ring lasers.

ac, Beat patterns, calculated from the analytical model of a ring with a defect, that oscillate at the fundamental, second harmonic and third harmonic of the round-trip frequency frt. Patterns are shown both for the unwrapped angular coordinate (top) and as projected onto a two-dimensional ring (bottom). Here it is assumed that the counterpropagating optical beats have the same intensity. df, Different beat patterns calculated assuming various beat balance ratios rBB, that is, different relative intensities of the counterpropagating optical beats, as discussed in the text. Also shown are the electric fields of the clockwise (ECW) and counterclockwise (ECCW) waves (red curves). The wavenumber is small for visual representation. The black lines correspond to the envelope of the fields, from which the mean values E and modulation amplitudes ΔE are calculated. The three cases correspond to: unidirectional lasing, which gives a uniform beat power across the cavity (d); bidirectional lasing with counterpropagating optical beats that are not fully balanced, which gives a beat grating with limited fringe visibility (e); bidirectional lasing with fully balanced optical beats, which gives a dynamic grating with strongly suppressed nodes (f).

Extended Data Fig. 4 Spectral gaps.

Simulations of ring QCL states, showing spectral gaps reminiscent of multisoliton spectra in microresonators. A sech2 envelope is fitted to the dominant modes of the spectra. The simulations are carried out for slightly different initial conditions in terms of noise seed and GVD.

Extended Data Fig. 5 Central-mode suppression in ring spectra.

a, b, Experimental optical spectra of a ring frequency comb at two different pump currents, showing that the carrier (central mode) can become suppressed with respect to the first pair of sidebands. c, Pump-dependent evolution of the carrier and the first two pairs of sidebands. The colours of the series match the modes of the optical spectra.

Extended Data Table 1 Parameters used in the numerical simulations of QCLs

Supplementary information

Supplementary Information

The Supplementary Information contains 12 display items (Supplementary Figs. 1–11, Supplementary Table 1). These display items and the related text discuss: the Ginzburg-Landau theory and simulations; beat note measurements; dynamic gratings effects; linewidth enhancement factor measurements; rings fabrication.

Video 1

Space–time simulation showing the evolution of the intensity in the ring cavity over 600'000 roundtrips. It corresponds to the space-time plot of Fig. 2b of the text.

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Piccardo, M., Schwarz, B., Kazakov, D. et al. Frequency combs induced by phase turbulence. Nature 582, 360–364 (2020).

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