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


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.

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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.

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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


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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.)).

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Authors and Affiliations



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.

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Correspondence to Frank Wise.

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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.

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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).

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