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:

Direct observations of thermalization to a Rayleigh–Jeans distribution in multimode optical fibres

Abstract

Nonlinear multimode optical systems support a host of intriguing effects that are impossible in single-mode settings. Although nonlinearity can provide a rich environment where the chaotic power exchange among thousands of modes can lead to novel behaviours, understanding and harnessing these processes to our advantage is challenging. Over the years, statistical models have been developed to macroscopically describe the response of these complex systems. One of the cornerstones of these theoretical formalisms is the prediction of a photon–photon-mediated thermalization process that leads to a Rayleigh–Jeans distribution of mode occupations. Here we report the use of mode-resolved measurement techniques to directly observe the thermalization to a Rayleigh–Jeans power distribution in a multimode optical fibre. We experimentally demonstrate that the underlying system Hamiltonian remains invariant during propagation, whereas power equipartition takes place among degenerate groups of modes—all in full accordance with theoretical predictions. Our results may pave the way towards a new generation of high-power optical sources whose brightness and modal content can be controlled using principles from thermodynamics and statistical mechanics.

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

Access options

Rent or buy this article

Prices vary by article type

from$1.95

to$39.95

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

Fig. 1: Process of optical thermalization and experimental setup used to observe RJ distribution.
Fig. 2: Experimental observation of optical thermalization to an RJ distribution.
Fig. 3: RJ ensemble produced by speckled input fields.
Fig. 4: Irreversibility of optical thermalization.
Fig. 5: Numerical simulations of the optical thermalization process leading to an RJ distribution.

Similar content being viewed by others

Data availability

Source data are available for this paper. Raw data for Figs. 2b, 3e, 4 and 5 are included. All other data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

Code availability

The principal components of the codes used in the manuscript have been made publicly available along with extensive documentation at https://github.com/WiseLabAEP/GMMNLSE-Solver-FINAL.

References

  1. Zhao, L.-D. et al. Ultralow thermal conductivity and high thermoelectric figure of merit in SnSe crystals. Nature 508, 373–377 (2014).

    Article  Google Scholar 

  2. Yildirim, T. et al. Giant anharmonicity and nonlinear electron-phonon coupling in MgB2: a combined first-principles calculation and neutron scattering study. Phys. Rev. Lett. 87, 037001 (2001).

    Article  ADS  Google Scholar 

  3. Fermi, E., Pasta, P., Ulam, S. & Tsingou, M. Studies of the Nonlinear Problems (Los Alamos Scientific Lab., 1955).

  4. Infeld, E. & Rowlands, G. Nonlinear Waves, Solitons and Chaos 2nd edn (Cambridge Univ. Press, 2000).

  5. Kippenberg, T. J. & Vahala, K. J. Cavity optomechanics: back-action at the mesoscale. Science 321, 1172–1176 (2008).

    Article  Google Scholar 

  6. Razzari, L. et al. CMOS-compatible integrated optical hyper-parametric oscillator. Nat. Photon. 4, 41–45 (2010).

    Article  Google Scholar 

  7. Richardson, D., Fini, J. & Nelson, L. Space-division multiplexing in optical fibres. Nat. Photon. 7, 354–362 (2013).

    Article  Google Scholar 

  8. Wright, L. G., Christodoulides, D. N. & Wise, F. W. Spatiotemporal mode-locking in multimode fiber lasers. Science 358, 94–97 (2017).

    Article  Google Scholar 

  9. Wright, L. G., Christodoulides, D. N. & Wise, F. W. Controllable spatiotemporal nonlinear effects in multimode fibres. Nat. Photon. 9, 306–310 (2015).

    Article  Google Scholar 

  10. Krupa, K. et al. Spatial beam self-cleaning in multimode fibres. Nat. Photon. 11, 237–241 (2017).

    Article  Google Scholar 

  11. Longhi, S. Modulational instability and space time dynamics in nonlinear parabolic-index optical fibers. Opt. Lett. 28, 2363–2365 (2003).

    Article  Google Scholar 

  12. Krupa, K. et al. Observation of geometric parametric instability induced by the periodic spatial self-imaging of multimode waves. Phys. Rev. Lett. 116, 183901 (2016).

    Article  ADS  Google Scholar 

  13. Pourbeyram, H., Agrawal, G. P. & Mafi, A. Stimulated Raman scattering cascade spanning the wavelength range of 523 to 1750 nm using a graded-index multimode optical fiber. Appl. Phys. Lett. 102, 201107 (2013).

    Article  ADS  Google Scholar 

  14. Fusaro, A. et al. Dramatic acceleration of wave condensation mediated by disorder in multimode fibers. Phys. Rev. Lett. 122, 123902 (2019).

    Article  ADS  Google Scholar 

  15. Podivilov, E. V. et al. Hydrodynamic 2D turbulence and spatial beam condensation in multimode optical fibers. Phys. Rev. Lett. 122, 103902 (2019).

    Article  ADS  Google Scholar 

  16. Eftekhar, M. A. et al. Accelerated nonlinear interactions in graded-index multimode fibers. Nat. Commun. 10, 1638 (2019).

    Article  ADS  Google Scholar 

  17. Lopez-Galmiche, G. et al. Visible supercontinuum generation in a graded index multimode fiber pumped at 1064 nm. Opt. Lett. 41, 2553–2556 (2016).

    Google Scholar 

  18. Krupa, K. et al. Spatiotemporal characterization of supercontinuum extending from the visible to the mid-infrared in a multimode graded-index optical fiber. Opt. Lett. 41, 5785–5788 (2016).

    Article  Google Scholar 

  19. Agrawal, G. P. Nonlinear Fiber Optics 5th edn (Springer, 2000).

  20. Zakharov, V. E., L’vov, V. S. & Falkovich, G. Kolmogorov Spectra of Turbulence I: Wave Turbulence 1st edn (Springer Science & Business Media, 2012).

  21. Picozzi, A. et al. Optical wave turbulence: towards a unified nonequilibrium thermodynamic formulation of statistical nonlinear optics. Phys. Rep. 542, 1–132 (2014).

    Article  MathSciNet  Google Scholar 

  22. Turitsyna, E. G. et al. The laminar–turbulent transition in a fibre laser. Nat. Photon. 7, 783–786 (2013).

    Article  Google Scholar 

  23. Churkin, D. V. et al. Wave kinetics of random fibre lasers. Nat. Commun. 6, 6214 (2015).

    Article  ADS  Google Scholar 

  24. Sun, C. et al. Observation of the kinetic condensation of classical waves. Nat. Phys. 8, 470–474 (2012).

    Google Scholar 

  25. Weill, R., Fischer, B. & Gat, O. Light-mode condensation in actively-mode-locked lasers. Phys. Rev. Lett. 104, 173901 (2010).

    Article  ADS  Google Scholar 

  26. Wu, F. O., Hassan, A. U. & Christodoulides, D. N. Thermodynamic theory of highly multimoded nonlinear optical systems. Nat. Photon. 13, 776–782 (2019).

    Google Scholar 

  27. Makris, K. G., Wu, F. O., Jung, P. S. & Christodoulides, D. N. Statistical mechanics of weakly nonlinear optical multimode gases. Opt. Lett. 45, 1651–1654 (2020).

    Article  Google Scholar 

  28. Ramos, A., Fernández-Alcázar, L., Kottos, T. & Shapiro, B. Optical phase transitions in photonic networks: a spin-system formulation. Phys. Rev. X 10, 031024 (2020).

    Google Scholar 

  29. Haus, H. A. & Kogelnik, H. Electromagnetic momentum and momentum flow in dielectric waveguides. J. Opt. Soc. Am. 66, 320–327 (1976).

    Article  ADS  MathSciNet  Google Scholar 

  30. Landau, L. D. & Lifshitz, E. M. Statistical Physics: Volume 5 3rd edn, Vol. 5 (Butterworth-Heinemann, 1980).

  31. Aschieri, P., Garnier, J., Michel, C., Doya, V. & Picozzi, A. Condensation and thermalization of classsical optical waves in a waveguide. Phys. Rev. A 83, 033838 (2011).

    Article  ADS  Google Scholar 

  32. Baudin, K. et al. Classical Rayleigh-Jeans condensation of light waves: observation and thermodynamic characterization. Phys. Rev. Lett. 125, 244101 (2020).

    Article  ADS  Google Scholar 

  33. Kim, M. Principles and techniques of digital holographic microscopy. SPIE Rev. 1, 018005 (2010).

    Google Scholar 

  34. Pathria, R. K. & Beale, P. D. Statistical Mechanics (Academic Press, 2011).

  35. Siegman, A. New Developments in Laser Resonators Vol. 1224 (SPIE, 1990).

  36. Nicholson, J., Yablon, A., Ramachandran, S. & Ghalmi, S. Spatially and spectrally resolved imaging of modal content in large-mode-area fibers. Opt. Express 16, 7233–7243 (2008).

    Article  Google Scholar 

  37. Rokitski, R. & Fainman, S. Propagation of ultrashort pulses in multimode fiber in space and time. Opt. Express 11, 1497–1502 (2003).

    Article  Google Scholar 

  38. Guang, Z., Rhodes, M., Davis, M. & Trebino, R. J. Complete characterization of a spatiotemporally complex pulse by an improved single-frame pulse-measurement technique. J. Opt. Soc. Am. B 11, 2736–2743 (2014).

    Google Scholar 

  39. Guang, Z., Rhodes, M. & Trebino, R. J. Measuring spatiotemporal ultrafast field structures of pulses from multimode optical fibers. Appl. Opt. 56, 3319–3324 (2017).

    Google Scholar 

  40. Pariente, G., Gallet, V., Borot, A., Gobert, O. & Quéré, F. Space–time characterization of ultra-intense femtosecond laser beams. Nat. Photon. 10, 547–553 (2016).

    Article  Google Scholar 

  41. Shapira, O., Abouraddy, A. F., Joannopoulos, J. D. & Fink, Y. Complete modal decomposition for optical waveguides. Phys. Rev. Lett. 94, 143902 (2005).

    Article  ADS  Google Scholar 

  42. Lü, H., Zhou, P., Wang, X. & Jiang, J. Fast and accurate modal decomposition of multimode fiber based on stochastic parallel gradient descent algorithm. Appl. Opt. 52, 2905–2908 (2013).

    Google Scholar 

  43. Paurisse, M., Lévèque, L., Hanna, M., Druon, F. & Georges, P. Complete measurement of fiber modal content by wavefront analysis. Opt. Express 20, 4074–4084 (2012).

    Article  Google Scholar 

  44. Wright, L. G. et al. Multimode nonlinear fiber optics: massively parallel numerical solver, tutorial and outlook. IEEE J. Sel. Topics Quantum Electron. 24, 1–16 (2018).

    Article  Google Scholar 

Download references

Acknowledgements

This effort was sponsored, in part, by the Department of the Navy, Office of Naval Research, under award nos. N00014-20-1-2789 (P.S., N.B., F.W., F.O.W. and D.N.C.) and N00014-18-1-2347 (D.N.C.). Portions of the work were sponsored by the National Science Foundation award nos. ECCS-1912742 (H.P. and F.W.) and EECS-1711230 (F.O.W. and D.N.C.), the Army Research Office (award no. W911NF1710481 (D.N.C.)), the Simons Foundation (733682 (D.N.C.)) and the BSF (2016381 (D.N.C.)).

Author information

Authors and Affiliations

Authors

Contributions

H.P., P.S. and N.B. conducted the experiments and performed the data analysis. L.W. developed the laser and P.S. developed the spatiotemporal measurement instrument. H.P. and F.O.W. performed the numerical simulations. F.O.W. and D.N.C. developed the theory. L.W., H.P., F.O.W., N.B. and F.W. conceived the experiments. All the authors contributed to the writing and editing of the manuscript.

Corresponding author

Correspondence to Frank Wise.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature Physics thanks the anonymous reviewers 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 Sections 1–21, Appendix and Figs. 1–20.

Source data

Source Data Fig. 2b

Source data for the graph of distribution of modal occupancies.

Source Data Fig. 3e

Source data for the graph of distribution of modal occupancies.

Source Data Fig. 4

Source data for the three plots of data points.

Source Data Fig. 5

Source data for the graphs of modal occupancies.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pourbeyram, H., Sidorenko, P., Wu, F.O. et al. Direct observations of thermalization to a Rayleigh–Jeans distribution in multimode optical fibres. Nat. Phys. 18, 685–690 (2022). https://doi.org/10.1038/s41567-022-01579-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41567-022-01579-y

This article is cited by

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