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.

  • Review Article
  • Published:

Physics of highly multimode nonlinear optical systems

An Author Correction to this article was published on 14 February 2023

This article has been updated

Abstract

Linear multimode optical systems have enabled clean experimental observations and the applications of numerous phenomena that continually extend the boundaries of wave physics. The infrastructure that has enabled these studies facilitates the study of an even richer world of nonlinear multimode optical systems. Multimode nonlinear optical physics is full of emergent phenomena, including robust spatial attractors, multimode wave instabilities, and conservative and dissipative multimode solitons. Many of these effects push the limits of existing theoretical techniques, demanding new insights and approaches that could emerge from other fields, such as statistical mechanics, physics-informed machine learning, network science and beyond. Here we provide an overview of recent investigations of wave propagation in highly multimode nonlinear systems, principally multimode fibre waveguides and laser cavities. These systems, with their multifaceted control, low cost, scalability and ultrahigh bandwidth, are ideal physical platforms for exploring—and ultimately applying—high-dimensional nonlinear physics, from orderly but elusive objects like spatiotemporal solitons to dynamical complexity itself, both near and far from equilibrium.

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

Access options

Buy this article

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

Fig. 1: Beam self-cleaning in GRIN multimode fibre.
Fig. 2: Multimode solitons and spatiotemporal dynamics in multimode waveguides and cavities.
Fig. 3: Spatiotemporal instabilities of nonlinear waves in multimode waveguides.
Fig. 4: Future directions for MMNLO systems.

Similar content being viewed by others

Change history

References

  1. Segev, M., Silberberg, Y. & Christodoulides, D. N. Anderson localization of light. Nat. Photon. 7, 197–204 (2013).

    Article  ADS  Google Scholar 

  2. El-Ganainy, R. et al. Non-Hermitian physics and PT symmetry. Nat. Phys. 14, 11–19 (2018).

    Article  Google Scholar 

  3. Ozawa, T. et al. Topological photonics. Rev. Mod. Phys. 91, 015006 (2019).

    Article  ADS  MathSciNet  Google Scholar 

  4. Haus, H. A. Mode-locking of lasers. IEEE J. Sel. Top. Quantum Electron. 6, 1173–1185 (2000).

    Article  ADS  Google Scholar 

  5. Kippenberg, T. J., Gaeta, A. L., Lipson, M. & Gorodetsky, M. L. Dissipative Kerr solitons in optical microresonators. Science 361, eaan8083 (2018).

    Article  Google Scholar 

  6. Fleck, J. A. Jr & Kidder, R. E. Coupled-mode laser oscillation. J. Appl. Phys. 35, 2825–2831 (1964).

    Article  ADS  Google Scholar 

  7. Stolen, R. H., Bjorkholm, J. E. & Ashkin, A. Phase-matched 3-wave mixing in silica fiber optical-waveguides. Appl. Phys. Lett. 24, 308–310 (1974).

    Article  ADS  Google Scholar 

  8. Baldeck, P. L., Raccah, F. & Alfano, R. R. Observation of self-focusing in optical fibers with picosecond pulses. Opt. Lett. 12, 588–589 (1987).

    Article  ADS  Google Scholar 

  9. Grudinin, A. B., Dianov, E. M., Korbkin, D. V., Prokhorov, A. M. & Khaǐdarov, D. V. Nonlinear mode coupling in multimode optical fibers; excitation of femtosecond-range stimulated-Raman-scattering solitons. Sov. J. Exp. Theor. Phys. Lett. 47, 356 (1988).

    ADS  Google Scholar 

  10. Ishimaru, A. Diffusion of light in turbid material. Appl. Opt. 28, 2210–2215 (1989).

    Article  ADS  Google Scholar 

  11. Anderson, P. W. Absence of diffusion in certain random lattices. Phys. Rev. 109, 1492–1505 (1958).

    Article  ADS  Google Scholar 

  12. John, S. Electromagnetic absorption in a disordered medium near a photon mobility edge. Phys. Rev. Lett. 53, 2169–2172 (1984).

    Article  ADS  Google Scholar 

  13. Wiersma, D. S., Bartolini, P., Lagendijk, A. & Righini, R. Localization of light in a disordered medium. Nature 390, 671–673 (1997).

    Article  ADS  Google Scholar 

  14. De Raedt, H., Lagendijk, A. D. & de Vries, P. Transverse localization of light. Phys. Rev. Lett. 62, 47–50 (1989).

    Article  ADS  Google Scholar 

  15. Abdullaev, S. S. & Abdullaev, F. K. On propagation of light in fiber bundles with random parameters. Radiofizika 23, 766–767 (1980).

    Google Scholar 

  16. Christodoulides, D. N., Lederer, F. & Silberberg, Y. Discretizing light behaviour in linear and nonlinear waveguide lattices. Nature 424, 817–823 (2003).

    Article  ADS  Google Scholar 

  17. Pertsch, T. et al. Nonlinearity and disorder in fiber arrays. Phys. Rev. Lett. 93, 053901 (2004).

    Article  ADS  Google Scholar 

  18. Karbasi, S. et al. Observation of transverse Anderson localization in an optical fiber. Opt. Lett. 37, 2304–2306 (2012).

    Article  ADS  Google Scholar 

  19. Schwartz, T., Bartal, G., Fishman, S. & Segev, M. Transport and Anderson localization in disordered two-dimensional photonic lattices. Nature 446, 52–55 (2007).

    Article  ADS  Google Scholar 

  20. Lahini, Y. et al. Anderson localization and nonlinearity in one-dimensional disordered photonic lattices. Phys. Rev. Lett. 100, 013906 (2008).

    Article  ADS  Google Scholar 

  21. Bromberg, Y., Lahini, Y., Morandotti, R. & Silberberg, Y. Quantum and classical correlations in waveguide lattices. Phys. Rev. Lett. 102, 253904 (2009).

    Article  ADS  Google Scholar 

  22. Lahini, Y. et al. Observation of a localization transition in quasiperiodic photonic lattices. Phys. Rev. Lett. 103, 013901 (2009).

    Article  ADS  Google Scholar 

  23. Christodoulides, D. N. & Joseph, R. I. Discrete self-focusing in nonlinear arrays of coupled waveguides. Opt. Lett. 13, 794–796 (1988).

    Article  ADS  Google Scholar 

  24. Leonetti, M., Karbasi, S., Mafi, A. & Conti, C. Experimental observation of disorder induced self-focusing in optical fibers. Appl. Phys. Lett. 105, 171102 (2014).

    Article  ADS  Google Scholar 

  25. Fishman, S., Krivolapov, Y. & Soffer, A. The nonlinear Schrodinger equation with a random potential: results and puzzles. Nonlinearity 25, R53 (2012).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  26. Fan, S. & Kahn, J. M. Principal modes in multimode waveguides. Opt. Lett. 30, 135–137 (2005).

    Article  ADS  Google Scholar 

  27. Carpenter, J., Eggleton, B. J. & Schroder, J. Observation of Eisenbud–Wigner–Smith states as principal modes in multimode fibre. Nat. Photon. 9, 751–757 (2015).

    Article  ADS  Google Scholar 

  28. Ambichl, P. et al. Super-and anti-principal-modes in multimode waveguides. Phys. Rev. 7, 041053 (2017).

    Article  Google Scholar 

  29. Matthes, M. W., Bromberg, Y., de Rosny, J. & Popoff, S. M. Learning and avoiding disorder in multimode fibers. Phys. Rev. 11, 021060 (2021).

    Article  Google Scholar 

  30. Cao, H., Mosk, A. P. & Rotter, S. Shaping the propagation of light in complex media. Nat. Phys. https://doi.org/10.1038/s41567-022-01677-x (2022).

  31. Bertolotti, J. & Katz, O. Imaging in complex media. Nat. Phys. https://doi.org/10.1038/s41567-022-01723-8 (2022).

  32. Poletti, F. & Horak, P. Description of ultrashort pulse propagation in multimode optical fibers. J. Opt. Soc. Am. B 25, 1645–1654 (2008).

    Article  ADS  Google Scholar 

  33. Mafi, A. Pulse propagation in a short nonlinear graded-index multimode optical fiber. J. Lightwave Technol. 30, 2803–2811 (2012).

    Article  ADS  Google Scholar 

  34. Antonelli, C., Shtaif, M. & Mecozzi, A. Modeling of nonlinear propagation in space-division multiplexed fiber-optic transmission. J. Lightwave Technol. 34, 36–54 (2015).

    Article  ADS  Google Scholar 

  35. Krupa, K. et al. Multimode nonlinear fiber optics, a spatiotemporal avenue. APL Photonics 4, 110901 (2019).

    Article  ADS  Google Scholar 

  36. Agrawal, G. P. Invite paper: self-imaging in multimode graded-index fibers and its impact on the nonlinear phenomena. Opt. Fiber Technol. 50, 309–316 (2019).

    Article  ADS  Google Scholar 

  37. Wright, L. G. et al. Multimode nonlinear fiber optics: massively parallel numerical solver, tutorial and outlook. IEEE J. Sel. Top. Quantum Electron. 24, 1–16 (2017).

    Article  Google Scholar 

  38. Wright, L. G. Spatiotemporal Nonlinear Optics in Multimode Fibers. PhD thesis, Cornell Univ. (2018).

  39. Kartashov, Y. V. et al. Frontiers in multidimensional self-trapping of nonlinear fields and matter. Nat. Rev. Phys. 1, 185–197 (2019).

    Article  Google Scholar 

  40. Krupa, K. et al. Spatial beam self-cleaning in multimode fibres. Nat. Photon. 11, 237–U299 (2017).

    Article  ADS  Google Scholar 

  41. Lopez-Galmiche, G. et al. Visible supercontinuum generation in a graded index multimode fiber pumped at 1,064 nm. Opt. Lett. 41, 2553–2556 (2016).

    Article  ADS  Google Scholar 

  42. Liu, Z., Wright, L. G., Christodoulides, D. N. & Wise, F. W. Kerr self-cleaning of femtosecond-pulsed beams in graded-index multimode fiber. Opt. Lett. 41, 3675–3678 (2016).

    Article  ADS  Google Scholar 

  43. Agrawal, G. P. Nonlinear Fiber Optics 5th edn (Academic Press, 2012).

  44. Terry, N. B., Alley, T. G. & Russell, T. H. An explanation of SRS beam cleanup in graded-index fibers and the absence of SRS beam cleanup in step-index fibers. Opt. Express 15, 17509–17519 (2007).

    Article  ADS  Google Scholar 

  45. Wright, L. G. et al. Self-organized instability in graded-index multimode fibres. Nat. Photon. 10, 771–776 (2016).

    Article  ADS  Google Scholar 

  46. 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 

  47. Pourbeyram, H. et al. Direct observations of thermalization to a Rayleigh–Jeans distribution in multimode optical fibers. Nat. Phys 18, 685–690 (2022).

    Article  Google Scholar 

  48. Fusaro, A., Garnier, J., Krupa, K., Millot, G. & Picozzi, A. Dramatic acceleration of wave condensation mediated by disorder in multimode fibers. Phys. Rev. Lett. 122, 123902 (2019).

    Article  ADS  Google Scholar 

  49. Sidelnikov, O. S., Podivilov, E. V., Fedoruk, M. P. & Wabnitz, S. Random mode coupling assists Kerr beam self-cleaning in a graded-index multimode optical fiber. Opt. Fiber Technol. 53, 101994 (2019).

    Article  Google Scholar 

  50. Laegsgaard, J. Spatial beam cleanup by pure Kerr processes in multimode fibers. Opt. Lett. 43, 2700–2703 (2018).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  52. Wu, F. O., Jung, P. S., Parto, M., Khajavikhan, M. & Christodoulides, D. N. Entropic thermodynamics of nonlinear photonic chain networks. Commun. Phys. 3, 216 (2020).

    Article  Google Scholar 

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

    Article  ADS  Google Scholar 

  54. Mangini, F. et al. Statistical mechanics of beam self-cleaning in GRIN multimode optical fibers. Opt. Express 30, 10850–10865 (2022).

    Article  ADS  Google Scholar 

  55. 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  ADS  Google Scholar 

  56. Wu, Y., Pourbeyram, H., Christodoulides, D. N. & Wise, F. W. Weak beam self-cleaning of femtosecond pulses in the anomalous dispersion regime. Opt. Lett. 46, 3312–3315 (2021).

    Article  ADS  Google Scholar 

  57. Kharenko, D. S. et al. Mode-resolved analysis of pump and Stokes beams in LD-pumped GRIN fiber Raman lasers. Opt. Lett. 47, 1222–1225 (2022).

    Article  ADS  Google Scholar 

  58. Tegin, U., Rahmani, B., Kakkava, E., Psaltis, D. & Moser, C. Single-mode output by controlling the spatiotemporal nonlinearities in mode-locked femtosecond multimode fiber lasers. Adv. Photon 2, 056005 (2020).

    Article  ADS  Google Scholar 

  59. Nakazawa, M., Yamada, E., Kubota, H. & Suzuki, K. 10 Gbit/s soliton data-transmission over one million kilometers. Electron. Lett. 27, 1270–1272 (1991).

    Article  ADS  Google Scholar 

  60. Skryabin, D. V. & Gorbach, A. V. Colloquium: looking at a soliton through the prism of optical supercontinuum. Rev. Mod. Phys. 82, 1287–1299 (2010).

    Article  ADS  Google Scholar 

  61. Chen, Z. G., Segev, M. & Christodoulides, D. N. Optical spatial solitons: historical overview and recent advances. Rep. Prog. Phys. 75, 086401 (2012).

    Article  ADS  Google Scholar 

  62. Renninger, W. H. & Wise, F. W. Optical solitons in graded-index multimode fibres. Nat. Commun. 4, 1719 (2013).

    Article  ADS  Google Scholar 

  63. Zhu, Z., Wright, L. G., Christodoulides, D. N. & Wise, F. W. Observation of multimode solitons in few-mode fiber. Opt. Lett. 41, 4819–4822 (2016).

    Article  ADS  Google Scholar 

  64. Wright, L. G., Wabnitz, S., Christodoulides, D. N. & Wise, F. W. Ultrabroadband dispersive radiation by spatiotemporal oscillation of multimode waves. Phys. Rev. Lett. 115, 223902 (2015).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  66. Zitelli, M. et al. High-energy soliton fission dynamics in multimode GRIN fiber. Opt. Express 28, 20473–20488 (2020).

    Article  ADS  Google Scholar 

  67. Wright, L. G., Renninger, W. H., Christodoulides, D. N. & Wise, F. W. Spatiotemporal dynamics of multimode optical solitons. Opt. Express 23, 3492–3506 (2015).

    Article  ADS  Google Scholar 

  68. Eftekhar, M. A., Lopez-Aviles, H., Wise, F. W., Amezcua-Correa, R. & Christodoulides, D. N. General theory and observation of Cherenkov radiation induced by multimode solitons. Commun. Phys. 4, 137 (2021).

    Article  Google Scholar 

  69. Zitelli, M., Mangini, F., Ferraro, M., Sidelnikov, O. & Wabnitz, S. Conditions for walk-off soliton generation in a multimode fiber. Commun. Phys. 4, 182 (2021).

    Article  Google Scholar 

  70. Hasegawa, A. Self-confinement of multimode optical pulse in a glass fiber. Opt. Lett. 5, 416–417 (1980).

    Article  ADS  Google Scholar 

  71. Crosignani, B. & Di Porto, P. Soliton propagation in multimode optical fibers. Opt. Lett. 6, 329–330 (1981).

    Article  ADS  Google Scholar 

  72. Zitelli, M., Ferraro, M., Mangini, F. & Wabnitz, S. Single-mode spatiotemporal soliton attractor in multimode GRIN fibers. Photon. Res. 9, 741–748 (2021).

    Article  Google Scholar 

  73. Maggipinto, T., Brambilla, M., Harkness, G. K. & Firth, W. J. Cavity solitons in semiconductor microresonators: existence, stability and dynamical properties. Phys. Rev. E 62, 8726–8739 (2000).

    Article  ADS  Google Scholar 

  74. Barland, S. et al. Cavity solitons as pixels in semiconductor microcavities. Nature 419, 699–702 (2002).

    Article  ADS  Google Scholar 

  75. Brambilla, M., Maggipinto, T., Patera, G. & Columbo, L. Cavity light bullets: three-dimensional localized structures in a nonlinear optical resonator. Phys. Rev. Lett. 93, 203901 (2004).

    Article  ADS  Google Scholar 

  76. Tanguy, Y., Ackemann, T., Firth, W. J. & Jager, R. Realization of a semiconductor-based cavity soliton laser. Phys. Rev. Lett. 100, 13907 (2008).

    Article  ADS  Google Scholar 

  77. Gustave, F. et al. Observation of mode-locked spatial laser solitons. Phys. Rev. Lett. 118, 044102 (2017).

    Article  ADS  Google Scholar 

  78. Renninger, W. H., Chong, A. & Wise, F. W. Dissipative solitons in normal-dispersion fiber lasers. Phys. Rev. A 77, 023814 (2008).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  80. Wright, L. G. et al. Mechanisms of spatiotemporal mode-locking. Nat. Phys. 16, 565–570 (2020).

    Article  Google Scholar 

  81. Ding, Y. H. et al. Spatiotemporal mode-locking in lasers with large modal dispersion. Phys. Rev. Lett. 126, 93901 (2021).

    Article  ADS  Google Scholar 

  82. Qin, H. Q., Xiao, X. S., Wang, P. & Yang, C. X. Observation of soliton molecules in a spatiotemporal mode-locked multimode fiber laser. Opt. Lett. 43, 1982–1985 (2018).

    Article  ADS  Google Scholar 

  83. Tegin, U., Kakkava, E., Rahmani, B., Psaltis, D. & Moser, C. Spatiotemporal self-similar fiber laser. Optica 6, 1412–1415 (2019).

    Article  ADS  Google Scholar 

  84. Liou, L. W., Cao, X. D., Mckinstrie, C. J. & Agrawal, G. P. Spatiotemporal instabilities in dispersive nonlinear media. Phys. Rev. A 46, 4202–4208 (1992).

    Article  ADS  Google Scholar 

  85. Hill, K. O., Johnson, D. C. & Kawasaki, B. S. Efficient conversion of light over a wide spectral range by 4-photon mixing in a multimode graded-index fiber. Appl. Opt. 20, 1075–1079 (1981).

    Article  ADS  Google Scholar 

  86. Nazemosadat, E., Pourbeyram, H. & Mafi, A. Phase matching for spontaneous frequency conversion via four-wave mixing in graded-index multimode optical fibers. J. Opt. Soc. Am. B 33, 144–150 (2016).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  88. 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  ADS  Google Scholar 

  89. Guasoni, M. Generalized modulational instability in multimode fibers: wideband multimode parametric amplification. Phys. Rev. A 92, 033849 (2015).

    Article  ADS  Google Scholar 

  90. Essiambre, R. J. et al. Experimental investigation of inter-modal four-wave mixing in few-mode fibers. IEEE Photon. Technol. Lett. 25, 539–542 (2013).

    Article  ADS  Google Scholar 

  91. Demas, J. et al. Intermodal nonlinear mixing with Bessel beams in optical fiber. Optica 2, 14–17 (2015).

    Article  ADS  Google Scholar 

  92. Dupiol, R. et al. Intermodal modulational instability in graded-index multimode optical fibers. Opt. Lett. 42, 3419–3422 (2017).

    Article  ADS  Google Scholar 

  93. 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 

  94. Conforti, M., Arabi, C. M., Mussot, A. & Kudlinski, A. Fast and accurate modeling of nonlinear pulse propagation in graded-index multimode fibers. Opt. Lett. 42, 4004–4007 (2017).

    Article  ADS  Google Scholar 

  95. Cheng, J. et al. Intermodal Cerenkov radiation in a higher-order-mode fiber. Opt. Lett. 37, 4410–4412 (2012).

    Article  ADS  Google Scholar 

  96. Lpken, N. M. et al. Numerical and experimental demonstration of intermodal dispersive wave generation. Laser Photon. Rev. 15, 2100125 (2021).

    Article  ADS  Google Scholar 

  97. Eftekhar, M. A. et al. Versatile supercontinuum generation in parabolic multimode optical fibers. Opt. Express 25, 9078–9087 (2017).

    Article  ADS  Google Scholar 

  98. Tzang, O., Caravaca-Aguirre, A. M., Wagner, K. & Piestun, R. Adaptive wavefront shaping for controlling nonlinear multimode interactions in optical fibres. Nat. Photon. 12, 368–374 (2018).

    Article  ADS  Google Scholar 

  99. Florentin, R. et al. Shaping the light amplified in a multimode fiber. Light Sci. Appl. 6, e16208 (2017).

    Article  Google Scholar 

  100. Deliancourt, E. et al. Wavefront shaping for optimized many-mode Kerr beam self-cleaning in graded-index multimode fiber. Opt. Express 27, 17311–17321 (2019).

    Article  ADS  Google Scholar 

  101. Wei, X. M., Jing, J. C., Shen, Y. C. & Wang, L. H. V. Harnessing a multi-dimensional fibre laser using genetic wavefront shaping. Light Sci. Appl. 9, 1–10 (2020).

    Article  Google Scholar 

  102. Tegin, U., Yldrm, M., Ouz, l, Moser, C. & Psaltis, D. Scalable optical learning operator. Nat. Comput. Sci. 1, 542–549 (2021).

    Article  Google Scholar 

  103. Dong, J., Rafayelyan, M., Krzakala, F. & Gigan, S. Optical reservoir computing using multiple light scattering for chaotic systems prediction. IEEE J. Sel. Top. Quantum Electron. 26, 1–12 (2020).

    Article  Google Scholar 

  104. Zhu, Z. et al. Mode-resolved control and measurement of nonlinear pulse propagation in multimode fibers. In Proc. 2018 Conference on Lasers and Electro-Optics (OSA) (OPG, 2018).

  105. Silberberg, Y. Collapse of optical pulses. Opt. Lett. 15, 1282–1284 (1990).

    Article  ADS  Google Scholar 

  106. Kibler, B. & Bejot, P. Discretized conical waves in multimode optical fibers. Phys. Rev. Lett. 126, 23902 (2021).

    Article  ADS  MathSciNet  Google Scholar 

  107. Yu, S.-S., Chien, C.-H., Lai, Y. & Wang, J. Spatio-temporal solitary pulses in graded-index materials with Kerr nonlinearity. Opt. Commun. 119, 167–170 (1995).

    Article  ADS  Google Scholar 

  108. Raghavan, S. & Agrawal, G. P. Spatiotemporal solitons in inhomogeneous nonlinear media. Opt. Commun. 180, 377–382 (2000).

    Article  ADS  Google Scholar 

  109. Mayteevarunyoo, T., Malomed, B. A. & Skryabin, D. V. Spatiotemporal solitons in dispersion-managed multimode fibers. J. Opt. 23, 015501 (2021).

    Article  ADS  Google Scholar 

  110. Chekhovskoy, I. S., Shtyrina, O. V., Wabnitz, S. & Fedoruk, M. P. Finding spatiotemporal light bullets in multicore and multimode fibers. Opt. Express 28, 7817–7828 (2020).

    Article  ADS  Google Scholar 

  111. Shtyrina, O. V., Fedoruk, M. P., Kivshar, Y. S. & Turitsyn, S. K. Coexistence of collapse and stable spatiotemporal solitons in multimode fibers. Phys. Rev. A 97, 013841 (2018).

    Article  ADS  Google Scholar 

  112. Kalashnikov, V. L. & Wabnitz, S. Distributed Kerr-lens mode locking based on spatiotemporal dissipative solitons in multimode fiber lasers. Phys. Rev. A 102, 023508 (2020).

    Article  ADS  Google Scholar 

  113. Mayteevarunyoo, T., Malomed, B. A. & Skryabin, D. V. Spatiotemporal dissipative solitons and vortices in a multi-transverse-mode fiber laser. Opt. Express 27, 37364–37373 (2019).

    Article  ADS  Google Scholar 

  114. Hanna, M. et al. Nonlinear optics in multipass cells. Laser Photon. Rev. 15, 2100220 (2021).

    Article  ADS  Google Scholar 

  115. Akhmediev, N., Soto-Crespo, J. M. & Grelu, P. Spatiotemporal optical solitons in nonlinear dissipative media: from stationary light bullets to pulsating complexes. Chaos 17, 037112 (2007).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  116. Mecozzi, A., Antonelli, C. & Shtaif, M. Nonlinear propagation in multi-mode fibers in the strong coupling regime. Opt. Express 20, 11673–11678 (2012).

    Article  ADS  Google Scholar 

  117. Mumtaz, S., Essiambre, R. J. & Agrawal, G. P. Nonlinear propagation in multimode and multicore fibers: generalization of the Manakov equations. J. Lightwave Technol. 31, 398–406 (2013).

    Article  ADS  Google Scholar 

  118. Mecozzi, A., Antonelli, C. & Shtaif, M. Coupled Manakov equations in multimode fibers with strongly coupled groups of modes. Opt. Express 20, 23436–23441 (2012).

    Article  ADS  Google Scholar 

  119. Buch, S., Mumtaz, S., Essiambre, R. J., Tulino, A. M. & Agrawal, G. P. Averaged nonlinear equations for multimode fibers valid in all regimes of random linear coupling. Opt. Fiber Technol. 48, 123–127 (2019).

    Article  ADS  Google Scholar 

  120. Patsyk, A., Sivan, U., Segev, M. & Bandres, M. A. Observation of branched flow of light. Nature 583, 60–65 (2020).

    Article  ADS  Google Scholar 

  121. Eslami, Z. et al. Two octave supercontinuum generation in a non-silica graded-index multimode fiber. Nat. Commun. 13, 2126 (2022).

    Article  ADS  Google Scholar 

  122. Sobon, G. et al. Infrared supercontinuum generation in soft-glass photonic crystal fibers pumped at 1,560 nm. Opt. Mater. Express 4, 7–15 (2014).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  124. Niang, A. et al. Spatial beam self-cleaning and supercontinuum generation with Yb-doped multimode graded-index fiber taper based on accelerating self-imaging and dissipative landscape. Opt. Express 27, 24018–24028 (2019).

    Article  ADS  Google Scholar 

  125. Petersen, C. R. et al. Mid-infrared supercontinuum covering the 1.4–13.3-μm molecular fingerprint region using ultra-high NA chalcogenide step-index fibre. Nat. Photon. 8, 830–834 (2014).

    Article  ADS  Google Scholar 

  126. Piccoli, R. et al. Intense few-cycle visible pulses directly generated via nonlinear fibre mode mixing. Nat. Photon. 15, 884–889 (2021).

    Article  ADS  Google Scholar 

  127. Tani, F., Travers, J. C. & Russell, P. S. Multimode ultrafast nonlinear optics in optical waveguides: numerical modeling and experiments in kagome photonic-crystal fiber. J. Opt. Soc. Am. B 31, 311–320 (2014).

    Article  ADS  Google Scholar 

  128. Safaei, R. et al. High-energy multidimensional solitary states in hollow-core fibres. Nat. Photon. 14, 733–739 (2020).

    Article  ADS  Google Scholar 

  129. Carpeggiani, P. A. et al. Extreme Raman red shift: ultrafast multimode nonlinear space-time dynamics, pulse compression and broadly tunable frequency conversion. Optica 7, 1349–1354 (2020).

    Article  ADS  Google Scholar 

  130. Gao, X. H. et al. Ionization-assisted spatiotemporal localization in gas-filled capillaries. Opt. Lett. 43, 3112–3115 (2018).

    Article  ADS  Google Scholar 

  131. Zhang, S. et al. Solitary beam propagation in periodic layered Kerr media enables high-efficiency pulse compression and mode self-cleaning. Light. Sci. Appl. 10, 53 (2021).

    Article  ADS  Google Scholar 

  132. Tradonsky, C. et al. High-resolution digital spatial control of a highly multimode laser. Optica 8, 880–884 (2021).

    Article  ADS  Google Scholar 

  133. Lupken, N. M. et al. Low-power broadband all-optical switching via intermodal cross-phase modulation in integrated optical waveguides. Opt. Lett. 43, 1631–1634 (2018).

    Article  ADS  Google Scholar 

  134. Guo, H. et al. Intermode breather solitons in optical microresonators. Phys. Rev. 7, 041055 (2017).

    Article  Google Scholar 

  135. Ji, X. et al. Exploiting ultralow loss multimode waveguides for broadband frequency combs. Laser Photon. Rev. 15, 6–11 (2021).

    Article  Google Scholar 

  136. Molesky, S. et al. Inverse design in nanophotonics. Nat. Photon. 12, 659–670 (2018).

    Article  ADS  Google Scholar 

  137. Frostig, H. et al. Focusing light by wavefront shaping through disorder and nonlinearity. Optica 4, 1073–1079 (2017).

    Article  ADS  Google Scholar 

  138. Ung, B. et al. Few-mode fiber with inverse-parabolic graded-index profile for transmission of OAM-carrying modes. Opt. Express 22, 18044–18055 (2014).

    Article  ADS  Google Scholar 

  139. Ma, Z., Kristensen, P. & Ramachandran, S. Light guidance based on topological confinement yielding fiber mode counts exceeding 50. In Proc. 2021 Conference on Lasers and Electro-Optics (CLEO) (eds Kang, J. et al.) (IEEE, 2021).

  140. Guenard, R. et al. Kerr self-cleaning of pulsed beam in an ytterbium doped multimode fiber. Opt. Express 25, 4783–4792 (2017).

    Article  ADS  Google Scholar 

  141. Jankowski, M. et al. Temporal simultons in optical parametric oscillators. Phys. Rev. Lett. 120, 053904 (2018).

    Article  ADS  Google Scholar 

  142. Bai, N. et al. Mode-division multiplexed transmission with inline few-mode fiber amplifier. Opt. Express 20, 2668–2680 (2012).

    Article  ADS  Google Scholar 

  143. Askarov, D. & Kahn, J. M. Design of transmission fibers and doped fiber amplifiers for mode-division multiplexing. IEEE Photon. Technol. Lett. 24, 1945–1948 (2012).

    Article  ADS  Google Scholar 

  144. Chen, H. et al. Integrated cladding-pumped multicore few-mode erbium-doped fibre amplifier for space-division-multiplexed communications. Nat. Photon. 10, 529–533 (2016).

    Article  ADS  Google Scholar 

  145. Dorrer, C. Spatiotemporal metrology of broadband optical pulses. IEEE J. Sel. Top. Quantum Electron. 25, 3100216 (2019).

    Article  Google Scholar 

  146. Jolly, S. W., Gobert, O. & Quere, F. Spatio-temporal characterization of ultrashort laser beams: a tutorial. J. Opt. 22, 103501 (2020).

    Article  ADS  Google Scholar 

  147. Leventoux, Y. et al. 3D time-domain beam mapping for studying nonlinear dynamics in multimode optical fibers. Opt. Lett. 46, 66–69 (2021).

    Article  ADS  Google Scholar 

  148. Guo, Y. K. et al. Real-time multispeckle spectral-temporal measurement unveils the complexity of spatiotemporal solitons. Nat. Commun. 12, 67 (2021).

    Article  ADS  Google Scholar 

  149. Dacha, S. K. & Murphy, T. E. Spatiotemporal characterization of nonlinear intermodal interference between selectively excited modes of a few-mode fiber. Optica 7, 1796–1803 (2020).

    Article  ADS  Google Scholar 

  150. Esmaeelpour, M. et al. Power fluctuations of intermodal four-wave mixing in few-mode fibers. J. Lightwave Technol. 35, 2429–2435 (2017).

    Article  ADS  Google Scholar 

  151. Zhu, P., Jafari, R., Jones, T. & Trebino, R. Complete measurement of spatiotemporally complex multi-spatial-mode ultrashort pulses from multimode optical fibers using delay-scanned wavelength-multiplexed holography. Opt. Express 25, 24015–24032 (2017).

    Article  ADS  Google Scholar 

  152. 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 

  153. An, Y. et al. Learning to decompose the modes in few-mode fibers with deep convolutional neural network. Opt. Express 27, 10127–10137 (2019).

    Article  ADS  Google Scholar 

  154. Manuylovich, E. S., Dvoyrin, V. V. & Turitsyn, S. K. Fast mode decomposition in few-mode fibers. Nat. Commun. 11, 5507 (2020).

    Article  ADS  Google Scholar 

  155. Fan, C. C. et al. Seeing the beam cleanup effect in a high-power graded-index-fiber Raman amplifier based on mode decomposition. Opt. Lett. 46, 4220–4223 (2021).

    Article  ADS  Google Scholar 

  156. Pierangeli, D. et al. Observation of replica symmetry breaking in disordered nonlinear wave propagation. Nat. Commun. 8, 1501 (2017).

    Article  ADS  Google Scholar 

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

    Article  Google Scholar 

  158. Angelani, L., Conti, C., Ruocco, G. & Zamponi, F. Glassy behavior of light. Phys. Rev. Lett. 96, 065702 (2006).

    Article  ADS  Google Scholar 

  159. Conti, C. & Leuzzi, L. Complexity of waves in nonlinear disordered media. Phys. Rev. B 83, 134204 (2011).

    Article  ADS  Google Scholar 

  160. Carleo, G. & Troyer, M. Solving the quantum many-body problem with artificial neural networks. Science 355, 602–605 (2017).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  161. Borhani, N., Kakkava, E., Moser, C. & Psaltis, D. Learning to see through multimode fibers. Optica 5, 960–966 (2018).

    Article  ADS  Google Scholar 

  162. Caramazza, P., Moran, O., Murray-Smith, R. & Faccio, D. Transmission of natural scene images through a multimode fibre. Nat. Commun. 10, 2029 (2019).

    Article  ADS  Google Scholar 

  163. Rahmani, B. et al. Actor neural networks for the robust control of partially measured nonlinear systems showcased for image propagation through diffuse media. Nat. Mach. Intell. 2, 403–410 (2020).

    Article  Google Scholar 

  164. Tegin, U. et al. Controlling spatiotemporal nonlinearities in multimode fibers with deep neural networks. APL Photonics 5, 030804 (2020).

    Article  ADS  Google Scholar 

  165. Karniadakis, G. E. et al. Physics-informed machine learning. Nat. Rev. Phys. 3, 422–440 (2021).

    Article  Google Scholar 

  166. Mounaix, M. et al. Time reversed optical waves by arbitrary vector spatiotemporal field generation. Nat. Commun. 11, 5813 (2020).

    Article  ADS  Google Scholar 

  167. 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 

  168. 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 

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

  170. Dyachenko, S., Newell, A. C., Pushkarev, A. & Zakharov, V. Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear Schrodinger equation. Phys. D 57, 96–160 (1992).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  172. Parto, M., Wu, F. O., Jung, P. S., Makris, K. & Christodoulides, D. N. Thermodynamic conditions governing the optical temperature and chemical potential in nonlinear highly multimoded photonic systems. Opt. Lett. 44, 3936–3939 (2019).

    Article  ADS  Google Scholar 

Download references

Acknowledgements

This effort was sponsored, in part, by the Department of the Navy, Office of Naval Research under ONR awards nos. N00014-20-1-2789 and N00014-18-1-2347. Portions of the work were sponsored by the National Science Foundation (ECCS-1912742 and EECS-1711230), the Army Research Office (award no. W911NF1710481), the Simons Foundation (733682) and the BSF (2016381).

Author information

Authors and Affiliations

Authors

Contributions

All authors contributed to the writing and editing of the manuscript.

Corresponding author

Correspondence to Frank W. Wise.

Ethics declarations

Competing interests

L.G.W. and F.W.W. hold patent number US 10,965,092 B2 on spatiotemporal mode-locking. The other authors declare no competing interests.

Peer review

Peer review information

Nature Physics 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.

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

Wright, L.G., Wu, F.O., Christodoulides, D.N. et al. Physics of highly multimode nonlinear optical systems. Nat. Phys. 18, 1018–1030 (2022). https://doi.org/10.1038/s41567-022-01691-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41567-022-01691-z

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