Abstract
Optical frequency-comb sources, which emit perfectly periodic and coherent waveforms of light1, have recently rapidly progressed towards chip-scale integrated solutions. Among them, two classes are particularly significant—semiconductor Fabry–Perót lasers2,3,4,5,6 and passive ring Kerr microresonators7,8,9. Here we merge the two technologies in a ring semiconductor laser10,11 and demonstrate a paradigm for the formation of free-running solitons, called Nozaki–Bekki solitons. These dissipative waveforms emerge in a family of travelling localized dark pulses, known within the complex Ginzburg–Landau equation12,13,14. We show that Nozaki–Bekki solitons are structurally stable in a ring laser and form spontaneously with tuning of the laser bias, eliminating the need for an external optical pump. By combining conclusive experimental findings and a complementary elaborate theoretical model, we reveal the salient characteristics of these solitons and provide guidelines for their generation. Beyond the fundamental soliton circulating inside the ring laser, we demonstrate multisoliton states as well, verifying their localized nature and offering an insight into formation of soliton crystals15. Our results consolidate a monolithic electrically driven platform for direct soliton generation and open the door for a research field at the junction of laser multimode dynamics and Kerr parametric processes.
This is a preview of subscription content, access via your institution
Access options
Access Nature and 54 other Nature Portfolio journals
Get Nature+, our best-value online-access subscription
$29.99 / 30 days
cancel any time
Subscribe to this journal
Receive 51 print issues and online access
$199.00 per year
only $3.90 per issue
Buy this article
- Purchase on Springer Link
- Instant access to full article PDF
Prices may be subject to local taxes which are calculated during checkout
Similar content being viewed by others
Data availability
Source data are provided with this paper. Additional data that support the findings of this study are available from the corresponding authors upon reasonable request.
Code availability
Information on the code developed to simulate the QCL dynamics and its results are available from the corresponding authors upon reasonable request.
Change history
30 January 2024
In the version of the article initially published, the name of a peer reviewer, Wenle Weng, was mispelled in the reviewer acknowledgements, and has now been amended in the HTML and PDF versions of the article.
References
Udem, T., Holzwarth, R. & Hänsch, T. W. Optical frequency metrology. Nature 416, 233 (2002).
Hugi, A., Villares, G., Blaser, S., Liu, H. C. & Faist, J. Mid-infrared frequency comb based on a quantum cascade laser. Nature 492, 229 (2012).
Täschler, P. et al. Femtosecond pulses from a mid-infrared quantum cascade laser. Nat. Photon. 15, 919 (2021).
Hillbrand, J., Andrews, A. M., Detz, H., Strasser, G. & Schwarz, B. Coherent injection locking of quantum cascade laser frequency combs. Nat. Photon. 13, 101 (2019).
Villares, G. et al. On-chip dual-comb based on quantum cascade laser frequency combs. Appl. Phys. Lett. 107, 251104 (2015).
Opačak, N. & Schwarz, B. Theory of frequency-modulated combs in lasers with spatial hole burning, dispersion, and Kerr nonlinearity. Phys. Rev. Lett. https://doi.org/10.1103/physrevlett.123.243902 (2019).
Herr, T. et al. Temporal solitons in optical microresonators. Nat. Photon. 8, 145 (2013).
Guo, H. et al. Universal dynamics and deterministic switching of dissipative Kerr solitons in optical microresonators. Nat. Phys. 13, 94 (2016).
Xue, X. et al. Mode-locked dark pulse Kerr combs in normal-dispersion microresonators. Nat. Photon. 9, 594 (2015).
Piccardo, M. et al. Frequency combs induced by phase turbulence. Nature 582, 360 (2020).
Meng, B. et al. Mid-infrared frequency comb from a ring quantum cascade laser. Optica 7, 162 (2020).
Aranson, I. S. & Kramer, L. The world of the complex Ginzburg–Landau equation. Rev. Mod. Phys. 74, 99 (2002).
Bekki, N. & Nozaki, K. Formations of spatial patterns and holes in the generalized Ginzburg–Landau equation. Phys. Lett. A 110, 133 (1985).
Lega, J. Traveling hole solutions of the complex Ginzburg–Landau equation: a review. Physica D 152–153, 269 (2001).
Karpov, M. et al. Dynamics of soliton crystals in optical microresonators. Nat. Phys. 15, 1071 (2019).
Akhmediev, N. & Ankiewicz, A. (eds) Dissipative Solitons: From Optics to Biology and Medicine (Springer, 2008).
Grelu, P. & Akhmediev, N. Dissipative solitons for mode-locked lasers. Nat. Photon. 6, 84 (2012).
Englebert, N., Arabí, C. M., Parra-Rivas, P., Gorza, S.-P. & Leo, F. Temporal solitons in a coherently driven active resonator. Nat. Photon. 15, 536 (2021).
Leo, F. et al. Temporal cavity solitons in one-dimensional Kerr media as bits in an all-optical buffer. Nat. Photon. 4, 471 (2010).
Rowley, M. et al. Self-emergence of robust solitons in a microcavity. Nature 608, 303 (2022).
Zhang, S .et al. Dark-bright soliton bound states in a microresonator. Phys. Rev. Lett. https://doi.org/10.1103/physrevlett.128.033901 (2022).
Marin-Palomo, P. et al. Microresonator-based solitons for massively parallel coherent optical communications. Nature 546, 274 (2017).
Riemensberger, J. et al. Massively parallel coherent laser ranging using a soliton microcomb. Nature 581, 164 (2020).
Suh, M.-G., Yang, Q.-F., Yang, K. Y., Yi, X. & Vahala, K. J. Microresonator soliton dual-comb spectroscopy. Science 354, 600 (2016).
Spencer, D. T. et al. An optical-frequency synthesizer using integrated photonics. Nature 557, 81 (2018).
Yao, Y., Hoffman, A. J. & Gmachl, C. F. Mid-infrared quantum cascade lasers. Nat. Photon. 6, 432 (2012).
Williams, B. S. Terahertz quantum-cascade lasers. Nat. Photon. 1, 517 (2007).
Opačak, N., Cin, S. D., Hillbrand, J. & Schwarz, B. Frequency comb generation by Bloch gain induced giant Kerr nonlinearity. Phys. Rev. Lett. https://doi.org/10.1103/physrevlett.127.093902 (2021).
Friedli, P. et al. Four-wave mixing in a quantum cascade laser amplifier. Appl. Phys. Lett. 102, 222104 (2013).
Gaeta, A. L., Lipson, M. & Kippenberg, T. J. Photonic-chip-based frequency combs. Nat. Photon. 13, 158 (2019).
Jaidl, M. et al. Comb operation in terahertz quantum cascade ring lasers. Optica 8, 780 (2021).
Paolo Micheletti et al. Terahertz optical solitons from dispersion-compensated antenna-coupled planarized ring quantum cascade lasers. Sci. Adv. 9, eadf9426 (2023).
Meng, B. et al. Dissipative Kerr solitons in semiconductor ring lasers. Nat. Photon. 16, 142 (2021).
Columbo, L. et al. Unifying frequency combs in active and passive cavities: temporal solitons in externally driven ring lasers. Phys. Rev. Lett. https://doi.org/10.1103/physrevlett.126.173903 (2021).
Prati, F. et al. Soliton dynamics of ring quantum cascade lasers with injected signal. Nanophotonics 10, 195 (2020).
Henry, C. Theory of the linewidth of semiconductor lasers. IEEE J. Quantum Electron. 18, 259 (1982).
Opačak, N. et al. Spectrally resolved linewidth enhancement factor of a semiconductor frequency comb. Optica 8, 1227 (2021).
Efremidis, N., Hizanidis, K., Nistazakis, H. E., Frantzeskakis, D. J. & Malomed, B. A. Stabilization of dark solitons in the cubic Ginzburg–Landau equation. Phys. Rev. E 62, 7410 (2000).
Perraud, J.-J. et al. One-dimensional “spirals”: novel asynchronous chemical wave sources. Phys. Rev. Lett. 71, 1272 (1993).
Burguete, J., Chaté, H., Daviaud, F. & Mukolobwiez, N. Bekki–Nozaki amplitude holes in hydrothermal nonlinear waves. Phys. Rev. Lett. 82, 3252 (1999).
Slepneva, S. et al. Convective Nozaki–Bekki holes in a long cavity OCT laser. Opt. Express 27, 16395 (2019).
Gowda, U. et al. Turbulent coherent structures in a long cavity semiconductor laser near the lasing threshold. Opt. Lett. 45, 4903 (2020).
Popp, S., Stiller, O., Aranson, I., Weber, A. & Kramer, L. Localized hole solutions and spatiotemporal chaos in the 1D complex Ginzburg–Landau equation. Phys. Rev. Lett. 70, 3880 (1993).
Popp, S., Stiller, O., Aranson, I. & Kramer, L. Hole solutions in the 1D complex Ginzburg–Landau equation. Physica D 84, 398 (1995).
Kazakov, D. et al. Active mid-infrared ring resonators. Nat. Commun. https://doi.org/10.1038/s41467-023-44628-7 (2024).
Burghoff, D. et al. Evaluating the coherence and time-domain profile of quantum cascade laser frequency combs. Opt. Express 23, 1190 (2015).
Jang, J. K., Erkintalo, M., Murdoch, S. G. & Coen, S. Observation of dispersive wave emission by temporal cavity solitons. Opt. Lett. 39, 5503 (2014).
Anderson, M. H. et al. Zero dispersion Kerr solitons in optical microresonators. Nat. Commun. https://doi.org/10.1038/s41467-022-31916-x (2022).
Obrzud, E., Lecomte, S. & Herr, T. Temporal solitons in microresonators driven by optical pulses. Nat. Photon. 11, 600 (2017).
Liu, D., Zhang, L., Tan, Y. & Dai, D. High-order adiabatic elliptical-microring filter with an ultra-large free-spectral-range. J. Lightw. Technol. 39, 5910 (2021).
Mansuripur, T. S. et al. Single-mode instability in standing-wave lasers: the quantum cascade laser as a self-pumped parametric oscillator. Phys. Rev. https://doi.org/10.1103/physreva.94.063807 (2016).
White, A. D. et al. Integrated passive nonlinear optical isolators. Nat. Photon. 17, 143–149 (2022).
Xiang, C. et al. 3D integration enables ultralow-noise isolator-free lasers in silicon photonics. Nature 620, 78–85 (2023).
Villares, G., Hugi, A., Blaser, S. & Faist, J. Dual-comb spectroscopy based on quantum-cascade-laser frequency combs. Nat. Commun. https://doi.org/10.1038/ncomms6192 (2014).
Acknowledgements
This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 Research and Innovation programme (grant agreement number 853014) and from the National Science Foundation under grant number ECCS-2221715. T.P.L. thanks the support of the Department of Defense (DoD) through the National Defense Science and Engineering Graduate (NDSEG) Fellowship Program.
Author information
Authors and Affiliations
Contributions
N.O., D.K., F. Pilat, T.P.L. and B.S. carried out the experiments and analysed the data. M.B. fabricated the device. N.O. performed the master-equation simulations. L.L.C., M.B. and F. Prati did the CGLE simulations and contributed to the analysis of the experimental results and their interpretation in the framework of the CGLE theory. N.O. prepared the paper with input from all co-authors. N.O., T.P.L., D.K. and F. Pilat wrote sections of the Supplementary Information. B.S., M.P. and F.C. supervised the project. All authors contributed to the discussion of the results.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing interests
Peer review
Peer review information
Nature thanks Wenle Weng and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.
Additional information
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 Experimental evidence suggesting an NB soliton crystal.
Intensity spectrum of a probable fifth harmonic frequency comb, where the intermode spacing equals 5 FSRs. The soliton crystal regime is suggested by the smooth bell-shaped envelope of the spectrum. In the Fig. 3 of the main manuscript, we have shown an experimental and theoretical characterization of a multisoliton state comprised of two co-propagating NB solitons in a single roundtrip. A special case of multisolitonic states, where all of the solitons within one roundtrip are equidistant, is called a soliton crystal15. In the frequency domain, these waveforms correspond to a harmonic frequency comb whose spacing between adjacent comb modes is equal to an integer multiple of the FSR: N × FSR, where N is the number of solitons in the soliton crystal. The coherence of the state is suggested by the high suppression ratio of the fundamental modes that fall beneath the noise floor, leaving only the harmonic equidistant modes. Furthermore, the modes form a smooth bell-shaped spectral envelope that indicates the soliton nature of the state. The high frequency of the intermode beatnote (around 68 GHz) lies well above the cutoff frequency of our optical detector, thus prohibiting SWIFTS characterization to truly assess the coherence of the state. This begs for the future use of another coherent technique to study soliton crystal dynamics in active ring resonators.
Extended Data Fig. 2 Shifting of the soliton spectral envelope relative to the primary mode.
Experimental characterization of two NB solitons where the tuning of the bias current results in a shift of the spectral soliton envelope from the red to the blue side of the primary mode (a and b respectively). The shift of the soliton spectral envelope happens as the currents of the ring and the waveguide are changed. The main reason for this likely lies in the large change of the total GVD, as discussed in the main manuscript, and recently observed experimentally in passive microresonators21. Although the soliton envelope may be positioned differently relative to the primary mode, the expected two π jumps of the intermode phases around the primary mode are still present – indicating that this is indeed a salient feature of NB solitons. The temporal profile of the phase exhibits the familiar 2π ramp within the width of the soliton. We can observe that the direction of the ramp depends on whether the soliton spectral envelope is on the red or on the blue side relative to the primary mode. In a hypothetical state where the soliton envelope would be perfectly symmetric relative to the primary mode (if the soliton spectral center of mass coincides with the position of the primary mode), the 2π phase ramp would comprise two separate π ramps with an opposite direction. However, this state does not represent a stable fixed point, and is likely never to occur experimentally.
Extended Data Fig. 3 Laser operation under delayed optical feedback.
a, Fabry-Perot QCL spectrogram of the intermode beat note under delayed feedback induced by placing a mirror at the laser output. The feedback intensity is varied by rotating a polarizer placed between the laser facet and the mirror. The frequency axis of the spectrogram is centered at 5.833 GHz. b, Same measurement as in a, performed on a ring QCL generating a unidirectional Nozaki-Bekki soliton. The frequency axis of the spectrogram is centered at 18.623 GHz. In both measurements resolution bandwidth of the RF spectrum analyzer is 16 kHz.
Supplementary information
Supplementary Information
This file contains Supplementary Information, including Supplementary Figs. 1–8, Table 1 and additional references.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Opačak, N., Kazakov, D., Columbo, L.L. et al. Nozaki–Bekki solitons in semiconductor lasers. Nature 625, 685–690 (2024). https://doi.org/10.1038/s41586-023-06915-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1038/s41586-023-06915-7
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.
