Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Article
  • Published:

Photonic snake states in two-dimensional frequency combs

Nature Photonics volume 17, pages 767–774 (2023)Cite this article

Subjects

Abstract

Taming the instabilities inherent to many nonlinear optical phenomena is of paramount importance for modern photonics. In particular, the so-called snake instability is universally known to severely distort localized wave stripes, leading to the occurrence of transient, short-lived dynamical states that eventually decay. This phenomenon is ubiquitous in nonlinear science—from river meandering to superfluids—and so far it apparently remains uncontrollable; however, here we show that optical snake instabilities can be harnessed by a process that leads to the formation of stationary and robust two-dimensional zigzag states. We find that such a new type of nonlinear waves exists in the hyperbolic regime of cylindrical microresonators, and that it naturally corresponds to two-dimensional frequency combs featuring spectral heterogeneity and intrinsic synchronization. We uncover the conditions of the existence of such spatiotemporal photonic snakes and confirm their remarkable robustness against perturbations. Our findings represent a new paradigm for frequency comb generation, thus opening the door to a whole range of applications in communications, metrology and spectroscopy.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Cylindrical microresonator and multi-stable resonances.
Fig. 2: Photonic snake families bifurcating from the single stripe background.
Fig. 3: Onset of snakes.
Fig. 4: Coupled photonic snakes.
Fig. 5: Two-dimensional heterogeneous combs.
Fig. 6: Effect of perturbations.

Similar content being viewed by others

Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding author on reasonable request.

Code availability

The analysis codes will be made available on reasonable request.

References

  1. Zakharov, V. E. & Rubenchik, A. M. Instability of waveguides and solitons in nonlinear media. Sov. Phys. JETP 38, 494–500 (1974).

    ADS  Google Scholar 

  2. Kuznetsov, E. A., Rubenchik, A. M. & Zakharov, V. E. Soliton stability in plasmas and hydrodynamics. Phys. Rep. 142, 103–65 (1986).

    Article  ADS  MathSciNet  Google Scholar 

  3. Kivshar, Y. S. & Pelinovsky, D. E. Self-focusing and transverse instabilities of solitary waves. Phys. Rep. 331, 117–195 (2000).

    Article  ADS  MathSciNet  Google Scholar 

  4. Constantin, J. A., Dunne, T., Ahmed, J., Legleiter, C. & Lazarus, E. D. Sediment supply as a driver of river meandering and floodplain evolution in the Amazon Basin. Nat. Geosci. 7, 899–903 (2014).

    Article  ADS  Google Scholar 

  5. Brown, E. D. et al. Observations of a nonlinear solitary wave packet in the Kelvin wake of a ship. J. Fluid Mech. 204, 263–293 (1989).

    Article  ADS  Google Scholar 

  6. Yefsah, T. et al. Heavy solitons in a fermionic superfluid. Nature 499, 426–430 (2013).

    Article  ADS  Google Scholar 

  7. Cetoli, A., Brand, J., Scott, R. G., Dalfovo, F. & Pitaevskii, L. P. Snake instability of dark solitons in fermionic superfluids. Phys. Rev. A 88, 043639 (2013).

    Article  ADS  Google Scholar 

  8. Claude, F. et al. Taming the snake instabilities in a polariton superfluid. Optica 7, 1660–1665 (2020).

    Article  ADS  Google Scholar 

  9. Anderson, B. P. et al. Watching dark solitons decay into vortex rings in a Bose–Einstein condensate. Phys. Rev. Lett. 86, 2926–2929 (2001).

    Article  ADS  Google Scholar 

  10. Luengviriya, C. et al. Scroll wave instabilities in an excitable chemical medium. Phys. Rev. Lett. 100, 148302 (2008).

    Article  ADS  Google Scholar 

  11. Mamaev, A. V., Saffman, M. & Zozulya, A. A. Propagation of dark stripe beams in nonlinear media: snake instability and creation of optical vortices. Phys. Rev. Lett. 76, 2262–2265 (1996).

    Article  ADS  Google Scholar 

  12. Tikhonenko, V., Christou, J., Luther-Davies, B. & Kivshar, Y. S. Observation of vortex solitons created by the instability of dark soliton stripes. Opt. Lett. 21, 1129–1131 (1996).

    Article  ADS  Google Scholar 

  13. Gorza, S.-P., Roig, N., Emplit, P. & Haelterman, M. Snake instability of a spatiotemporal bright soliton stripe. Phys. Rev. Lett. 92, 084101 (2004).

    Article  ADS  Google Scholar 

  14. Gorza, S.-P., Deconinck, B., Emplit, P., Trogdon, T. & Haelterman, M. Experimental demonstration of the oscillatory snake instability of the bright soliton of the (2 + 1)D hyperbolic nonlinear Schrödinger equation. Phys. Rev. Lett. 106, 094101 (2011).

    Article  ADS  Google Scholar 

  15. Gorza, S.-P., Emplit, P. & Haelterman, M. Observation of the snake instability of a spatially extended temporal bright soliton. Opt. Lett. 31, 1280–1282 (2006).

    Article  ADS  Google Scholar 

  16. Brasch, V. et al. Photonic chip-based optical frequency comb using soliton Cherenkov radiation. Science 351, 357–360 (2016).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Yi, X. et al. Single-mode dispersive waves and soliton microcomb dynamics. Nat. Commun. 8, 14869 (2017).

    Article  ADS  Google Scholar 

  18. Karpov, M. et al. Raman self-frequency shift of dissipative Kerr solitons in an optical microresonator. Phys. Rev. Lett. 116, 103902 (2016).

    Article  ADS  Google Scholar 

  19. Yang, Q.-F., Yi, X., Yang, K. Y. & Vahala, K. Stokes solitons in optical microcavities. Nat. Phys. 13, 53–57 (2017).

    Article  Google Scholar 

  20. Yu, M. et al. Raman lasing and soliton mode-locking in lithium niobate microresonators. Light: Sci. Appl. 9, 9 (2020).

    Article  ADS  Google Scholar 

  21. Stone, J. R. et al. Thermal and nonlinear dissipative-soliton dynamics in Kerr-microresonator frequency combs. Phys. Rev. Lett. 121, 063902 (2018).

    Article  ADS  Google Scholar 

  22. Obrzud, E., Lacomte, S. & Herr, T. Temporal solitons in microresonators driven by optical pulses. Nat. Photon. 11, 600–607 (2017).

    Article  Google Scholar 

  23. Pasquazi, A. et al. Micro-combs: a novel generation of optical sources. Phys. Rep. 729, 1–81 (2017).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. Cundiff, S. T. & Ye, J. Colloquium: Femtosecond optical frequency combs. Rev. Mod. Phys. 75, 325–342 (2003).

    Article  ADS  Google Scholar 

  25. Yang, Q. F., Yi, X., Yang, K. Y. & Vahala, K. Counter-propagating solitons in microresonators. Nat. Photon. 11, 560–564 (2017).

    Article  Google Scholar 

  26. Weng, W., Bouchand, R., Lucas, E. & Kippenberg, T. J. Polychromatic Cherenkov radiation induced group velocity symmetry breaking in counterpropagating dissipative Kerr solitons. Phys. Rev. Lett. 123, 253902 (2019).

    Article  ADS  Google Scholar 

  27. Tikan, A. et al. Emergent nonlinear phenomena in a driven dissipative photonic dimer. Nat. Phys. 17, 604–610 (2021).

    Article  Google Scholar 

  28. Jang, J. K. et al. Observation of Arnold Tongues in Coupled Soliton Kerr Frequency Combs. Phys. Rev. Lett. 123, 153901 (2019).

    Article  ADS  Google Scholar 

  29. Lucas, E. et al. Spatial multiplexing of soliton microcombs. Nat. Photon. 12, 699–705 (2018).

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  31. Conti, C. et al. Nonlinear electromagnetic X waves. Phys. Rev. Lett. 90, 170406 (2003).

    Article  ADS  Google Scholar 

  32. Di Trapani, P. et al. Spontaneously generated X-shaped light bullets. Phys. Rev. Lett. 91, 093904 (2003).

    Article  ADS  Google Scholar 

  33. Cisneros-Ake, L. A., Carretero-González, R., Kevrekidis, P. B. & Malomed, B. A. Dynamics and stabilization of bright soliton stripes in the hyperbolic-dispersion nonlinear Schrödinger equation. Commun. Nonlinear Sci. Numer. Simul. 74, 268–281 (2019).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  34. Barashenkov, I. & Smirnov, Y. S. Existence and stability chart for the ac-driven, damped nonlinear Schrödinger solitons. Phys. Rev. E 54, 5707–5725 (1996).

    Article  ADS  Google Scholar 

  35. Mc Laughlin, D. W., Moloney, J. V. & Newell, A. C. Solitary Waves as Fixed Points of Infinite-Dimensional Maps in an Optical Bistable Ring Cavity. Phys. Rev. Lett. 51, 75–78 (1983).

    Article  ADS  Google Scholar 

  36. Pelinovsky, D. E., Kivshar, Y. S. & Afanasjev, V. V. Internal modes of envelope solitons. Physica D 116, 121–142 (1998).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  37. Skryabin, D. V. & Firth, W. J. Modulational instability of solitary waves in non-degenerate three wave mixing: the role of phase symmetries. Phys. Rev. Lett. 81, 3379–3382 (1998).

    Article  ADS  Google Scholar 

  38. Brazhnyi, V. A. & Pérez-García, V. M. Stable multidimensional soliton stripes in two-component Bose-Einstein condensates. Chaos Solit. Fractals 44, 381–389 (2011).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  39. Parra-Rivas, P., Knobloch, E., Gomila, D. & Gelens, L. Dark solitons in the Lugiato-Lefever equation with normal dispersion. Phys. Rev. A 93, 063839 (2016).

    Article  ADS  Google Scholar 

  40. Godey, C., Balakireva, I. V., Coillet, A. & Chembo, Y. K. Stability analysis of the spatiotemporal Lugiato-Lefever model for Kerr optical frequency combs in the anomalous and normal dispersion regimes. Phys. Rev. A, 89, 063814 (2014).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  42. Demchenko, Y. A. & Gorodetsky, M. L. Analytical estimayes of eigenfrequencies, dispersion, and field distribution in whispering gallery resonators. J. Opt. Soc. Am. B 30, 3056–3063 (2013).

    Article  ADS  Google Scholar 

  43. Milián, C., Kartashov, Y. V., Skryabin, D. V. & Torner, L. Clusters of cavity solitons bounded by conical radiation. Phys. Rev. Lett. 121, 103903 (2018).

    Article  ADS  Google Scholar 

  44. Hamidfar, T. et al. Localisation of light in an optical microcapillary induced by a droplet. Optica 5, 382–388 (2018).

    Article  ADS  Google Scholar 

  45. Ilchenko, V. S. & Gorodetskii, M. L. Thermal nonlinear effects in optical whispering gallery microresonators. Laser Phys. 2, 1004–1009 (1992).

    Google Scholar 

  46. Leshem, A., Qi, Z., Carruthers, T. F., Menyuk, C. R. & Gat, O. Thermal instabilities, frequency-comb formation, and temporal oscillations in Kerr microresonators. Phys. Rev. A 103, 013512 (2021).

    Article  ADS  MathSciNet  Google Scholar 

  47. Savchenkov, A. A. et al. Kerr frequency comb generation in overmoded resonators. Opt. Express 20, 27290–27298 (2012).

    Article  ADS  Google Scholar 

  48. Xue, X. et al. Mode-locked dark pulse Kerr combs in normal-dispersion microresonators. Nat. Photon. 9, 594–600 (2015).

    Article  ADS  Google Scholar 

  49. Jang, J. K. et al. Dynamics of mode-coupling-induced microresonator frequency combs in normal dispersion. Opt. Express 24, 28794–28803 (2016).

    Article  ADS  Google Scholar 

  50. Nazemosadat, E. et al. Switching dynamics of dark-pulse Kerr frequency comb states in optical microresonators. Phys. Rev. A 103, 013513 (2021).

    Article  ADS  Google Scholar 

  51. Xue, X. et al. Normal-dispersion microcombs enabled by controllable mode interactions. Laser Photon. Rev. 9, L23–L28 (2015).

    Article  Google Scholar 

  52. Kim, B. Y. et al. Turn-key, high-efciency Kerr comb source. Opt. Lett. 44, 4475–4478 (2019).

    Article  ADS  Google Scholar 

  53. Jin, W. et al. Hertz-linewidth semiconductor lasers using CMOS-ready ultra-high-Q microresonators. Nat. Photon. 15, 346–353 (2021).

    Article  ADS  Google Scholar 

  54. Shen, B. et al. Integrated turnkey soliton microcombs. Nature 582, 365–369 (2020).

    Article  ADS  Google Scholar 

  55. Rowley, M. et al. Self-emergence of robust solitons in a microcavity. Nature 608, 303–309 (2022).

    Article  ADS  Google Scholar 

  56. Nie, M. et al. Dissipative soliton generation and real-time dynamics in microresonator-filtered fiber lasers. Light Sci. Appl. 11, 296 (2022).

    Article  ADS  Google Scholar 

  57. Weng, W. et al. Heteronuclear soliton molecules in optical microresonators. Nat. Commun. 11, 2402 (2020).

    Article  ADS  Google Scholar 

  58. Jang, J. K. et al. Synchronization of coupled optical microresonators. Nat. Photon. 12, 688–693 (2018).

    Article  ADS  Google Scholar 

  59. Cheo, P. K. & Wagner, R. Infrared Electrooptic Waveguides. IEEE J. Quantum Electron. 13, 159–164 (1977).

    Article  ADS  Google Scholar 

  60. Deuflhard, P. Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms Vol. 35 (Springer, 2004).

Download references

Acknowledgements

J.A.C. and C.M. acknowledge support from the Spanish government via grant PID2021-124618NB-C21, which was funded by MCIN/AEI/10.13039/501100011033 and ‘ERDF: a way of making Europe’ of the European Union. C.M. acknowledges support from Generalitat Valenciana PROMETEO/2021/082. P.F.d.C. acknowledges partial support from the Spanish government via project PID2021-128676OB-I00 (MICINN). L.T. acknowledges support by CEX2019-000910-S (MCIN/AEI/10.13039/501100011033), F. Cellex, F. M. Puig and Generalitat de Catalunya (CERCA). Y.V.K.’s academic research has been supported by the research project FFUU-2021-0003 of the Institute of Spectroscopy of the Russian Academy of Sciences.

Author information

Authors and Affiliations

  1. Institut Universitari de Matemàtica Pura i Aplicada, Universitat Politècnica de València, València, Spain

    Salim B. Ivars, P. Fernández de Córdoba, J. Alberto Conejero & Carles Milián

  2. Departament de Física, Universitat Politècnica de Catalunya, Barcelona, Spain

    Salim B. Ivars

  3. Institute of Spectroscopy, Russian Academy of Sciences, Moscow, Russia

    Yaroslav V. Kartashov

  4. ICFO–Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, Barcelona, Spain

    Lluis Torner

  5. Universitat Politècnica de Catalunya, Barcelona, Spain

    Lluis Torner

Authors
  1. Salim B. Ivars
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yaroslav V. Kartashov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. P. Fernández de Córdoba
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. J. Alberto Conejero
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Lluis Torner
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Carles Milián
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

S.B.I. and C.M. performed numerical simulations. C.M. conceived the project. All of the authors substantially contributed to this work, discussed the results, and contributed to the preparation of the manuscript.

Corresponding author

Correspondence to Carles Milián.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature Photonics thanks Stefan Wabnitz and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Additional information

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

Supplementary information

Supplementary Information

Supplementary Figs. 1–7, Sections 1–8 and 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.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ivars, S.B., Kartashov, Y.V., de Córdoba, P.F. et al. Photonic snake states in two-dimensional frequency combs. Nat. Photon. 17, 767–774 (2023). https://doi.org/10.1038/s41566-023-01220-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41566-023-01220-1

This article is cited by

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing