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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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: https://figshare.com/articles/Codes_for_Benjamin-Feir_space_parameters_and_dynamic_QCL_gratings/11967552/1; https://figshare.com/articles/Plot_code_and_datasets_for_phase_turbulence_space-time_plots/11967756/1; https://figshare.com/articles/Plot_code_and_datasets_for_defect_engineered_laser_space-time_plots/11967828/1. Information on the code developed to simulate the QCL dynamics and its results are available from the corresponding authors upon reasonable request.
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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.
The authors declare no competing interests.
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.
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Extended data figures and tables
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.
a–e, 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.
a–c, 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. d–f, 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).
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.
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.
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.
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). https://doi.org/10.1038/s41586-020-2386-6
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