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.
Your institute does not have access to this article
Open Access articles citing this article.
Nature Communications Open Access 29 July 2022
Subscribe to Nature+
Get immediate online access to the entire Nature family of 50+ journals
Subscribe to Journal
Get full journal access for 1 year
only $8.25 per issue
All prices are NET prices.
VAT will be added later in the checkout.
Tax calculation will be finalised during checkout.
Get time limited or full article access on ReadCube.
All prices are NET prices.
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.
Zhao, L.-D. et al. Ultralow thermal conductivity and high thermoelectric figure of merit in SnSe crystals. Nature 508, 373–377 (2014).
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).
Fermi, E., Pasta, P., Ulam, S. & Tsingou, M. Studies of the Nonlinear Problems (Los Alamos Scientific Lab., 1955).
Infeld, E. & Rowlands, G. Nonlinear Waves, Solitons and Chaos 2nd edn (Cambridge Univ. Press, 2000).
Kippenberg, T. J. & Vahala, K. J. Cavity optomechanics: back-action at the mesoscale. Science 321, 1172–1176 (2008).
Razzari, L. et al. CMOS-compatible integrated optical hyper-parametric oscillator. Nat. Photon. 4, 41–45 (2010).
Richardson, D., Fini, J. & Nelson, L. Space-division multiplexing in optical fibres. Nat. Photon. 7, 354–362 (2013).
Wright, L. G., Christodoulides, D. N. & Wise, F. W. Spatiotemporal mode-locking in multimode fiber lasers. Science 358, 94–97 (2017).
Wright, L. G., Christodoulides, D. N. & Wise, F. W. Controllable spatiotemporal nonlinear effects in multimode fibres. Nat. Photon. 9, 306–310 (2015).
Krupa, K. et al. Spatial beam self-cleaning in multimode fibres. Nat. Photon. 11, 237–241 (2017).
Longhi, S. Modulational instability and space time dynamics in nonlinear parabolic-index optical fibers. Opt. Lett. 28, 2363–2365 (2003).
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).
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).
Fusaro, A. et al. Dramatic acceleration of wave condensation mediated by disorder in multimode fibers. Phys. Rev. Lett. 122, 123902 (2019).
Podivilov, E. V. et al. Hydrodynamic 2D turbulence and spatial beam condensation in multimode optical fibers. Phys. Rev. Lett. 122, 103902 (2019).
Eftekhar, M. A. et al. Accelerated nonlinear interactions in graded-index multimode fibers. Nat. Commun. 10, 1638 (2019).
Lopez-Galmiche, G. et al. Visible supercontinuum generation in a graded index multimode fiber pumped at 1064 nm. Opt. Lett. 41, 2553–2556 (2016).
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).
Agrawal, G. P. Nonlinear Fiber Optics 5th edn (Springer, 2000).
Zakharov, V. E., L’vov, V. S. & Falkovich, G. Kolmogorov Spectra of Turbulence I: Wave Turbulence 1st edn (Springer Science & Business Media, 2012).
Picozzi, A. et al. Optical wave turbulence: towards a unified nonequilibrium thermodynamic formulation of statistical nonlinear optics. Phys. Rep. 542, 1–132 (2014).
Turitsyna, E. G. et al. The laminar–turbulent transition in a fibre laser. Nat. Photon. 7, 783–786 (2013).
Churkin, D. V. et al. Wave kinetics of random fibre lasers. Nat. Commun. 6, 6214 (2015).
Sun, C. et al. Observation of the kinetic condensation of classical waves. Nat. Phys. 8, 470–474 (2012).
Weill, R., Fischer, B. & Gat, O. Light-mode condensation in actively-mode-locked lasers. Phys. Rev. Lett. 104, 173901 (2010).
Wu, F. O., Hassan, A. U. & Christodoulides, D. N. Thermodynamic theory of highly multimoded nonlinear optical systems. Nat. Photon. 13, 776–782 (2019).
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).
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).
Haus, H. A. & Kogelnik, H. Electromagnetic momentum and momentum flow in dielectric waveguides. J. Opt. Soc. Am. 66, 320–327 (1976).
Landau, L. D. & Lifshitz, E. M. Statistical Physics: Volume 5 3rd edn, Vol. 5 (Butterworth-Heinemann, 1980).
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).
Baudin, K. et al. Classical Rayleigh-Jeans condensation of light waves: observation and thermodynamic characterization. Phys. Rev. Lett. 125, 244101 (2020).
Kim, M. Principles and techniques of digital holographic microscopy. SPIE Rev. 1, 018005 (2010).
Pathria, R. K. & Beale, P. D. Statistical Mechanics (Academic Press, 2011).
Siegman, A. New Developments in Laser Resonators Vol. 1224 (SPIE, 1990).
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).
Rokitski, R. & Fainman, S. Propagation of ultrashort pulses in multimode fiber in space and time. Opt. Express 11, 1497–1502 (2003).
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).
Guang, Z., Rhodes, M. & Trebino, R. J. Measuring spatiotemporal ultrafast field structures of pulses from multimode optical fibers. Appl. Opt. 56, 3319–3324 (2017).
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).
Shapira, O., Abouraddy, A. F., Joannopoulos, J. D. & Fink, Y. Complete modal decomposition for optical waveguides. Phys. Rev. Lett. 94, 143902 (2005).
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).
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).
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).
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.)).
The authors declare no competing interests.
Peer review information
Nature Physics thanks the anonymous reviewers 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.
About this article
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
Nature Communications (2022)
Nature Physics (2022)