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 Nature and 54 other Nature Portfolio journals
Get Nature+, our best-value online-access subscription
$29.99 per month
cancel any time
Subscribe to this journal
Receive 12 print issues and online access
$209.00 per year
only $17.42 per issue
Rent or buy this article
Get just this article for as long as you need it
Prices may be subject to local taxes which are calculated during checkout
Segev, M., Silberberg, Y. & Christodoulides, D. N. Anderson localization of light. Nat. Photon. 7, 197–204 (2013).
El-Ganainy, R. et al. Non-Hermitian physics and PT symmetry. Nat. Phys. 14, 11–19 (2018).
Ozawa, T. et al. Topological photonics. Rev. Mod. Phys. 91, 015006 (2019).
Haus, H. A. Mode-locking of lasers. IEEE J. Sel. Top. Quantum Electron. 6, 1173–1185 (2000).
Kippenberg, T. J., Gaeta, A. L., Lipson, M. & Gorodetsky, M. L. Dissipative Kerr solitons in optical microresonators. Science 361, eaan8083 (2018).
Fleck, J. A. Jr & Kidder, R. E. Coupled-mode laser oscillation. J. Appl. Phys. 35, 2825–2831 (1964).
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).
Baldeck, P. L., Raccah, F. & Alfano, R. R. Observation of self-focusing in optical fibers with picosecond pulses. Opt. Lett. 12, 588–589 (1987).
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).
Ishimaru, A. Diffusion of light in turbid material. Appl. Opt. 28, 2210–2215 (1989).
Anderson, P. W. Absence of diffusion in certain random lattices. Phys. Rev. 109, 1492–1505 (1958).
John, S. Electromagnetic absorption in a disordered medium near a photon mobility edge. Phys. Rev. Lett. 53, 2169–2172 (1984).
Wiersma, D. S., Bartolini, P., Lagendijk, A. & Righini, R. Localization of light in a disordered medium. Nature 390, 671–673 (1997).
De Raedt, H., Lagendijk, A. D. & de Vries, P. Transverse localization of light. Phys. Rev. Lett. 62, 47–50 (1989).
Abdullaev, S. S. & Abdullaev, F. K. On propagation of light in fiber bundles with random parameters. Radiofizika 23, 766–767 (1980).
Christodoulides, D. N., Lederer, F. & Silberberg, Y. Discretizing light behaviour in linear and nonlinear waveguide lattices. Nature 424, 817–823 (2003).
Pertsch, T. et al. Nonlinearity and disorder in fiber arrays. Phys. Rev. Lett. 93, 053901 (2004).
Karbasi, S. et al. Observation of transverse Anderson localization in an optical fiber. Opt. Lett. 37, 2304–2306 (2012).
Schwartz, T., Bartal, G., Fishman, S. & Segev, M. Transport and Anderson localization in disordered two-dimensional photonic lattices. Nature 446, 52–55 (2007).
Lahini, Y. et al. Anderson localization and nonlinearity in one-dimensional disordered photonic lattices. Phys. Rev. Lett. 100, 013906 (2008).
Bromberg, Y., Lahini, Y., Morandotti, R. & Silberberg, Y. Quantum and classical correlations in waveguide lattices. Phys. Rev. Lett. 102, 253904 (2009).
Lahini, Y. et al. Observation of a localization transition in quasiperiodic photonic lattices. Phys. Rev. Lett. 103, 013901 (2009).
Christodoulides, D. N. & Joseph, R. I. Discrete self-focusing in nonlinear arrays of coupled waveguides. Opt. Lett. 13, 794–796 (1988).
Leonetti, M., Karbasi, S., Mafi, A. & Conti, C. Experimental observation of disorder induced self-focusing in optical fibers. Appl. Phys. Lett. 105, 171102 (2014).
Fishman, S., Krivolapov, Y. & Soffer, A. The nonlinear Schrodinger equation with a random potential: results and puzzles. Nonlinearity 25, R53 (2012).
Fan, S. & Kahn, J. M. Principal modes in multimode waveguides. Opt. Lett. 30, 135–137 (2005).
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).
Ambichl, P. et al. Super-and anti-principal-modes in multimode waveguides. Phys. Rev. 7, 041053 (2017).
Matthes, M. W., Bromberg, Y., de Rosny, J. & Popoff, S. M. Learning and avoiding disorder in multimode fibers. Phys. Rev. 11, 021060 (2021).
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).
Bertolotti, J. & Katz, O. Imaging in complex media. Nat. Phys. https://doi.org/10.1038/s41567-022-01723-8 (2022).
Poletti, F. & Horak, P. Description of ultrashort pulse propagation in multimode optical fibers. J. Opt. Soc. Am. B 25, 1645–1654 (2008).
Mafi, A. Pulse propagation in a short nonlinear graded-index multimode optical fiber. J. Lightwave Technol. 30, 2803–2811 (2012).
Antonelli, C., Shtaif, M. & Mecozzi, A. Modeling of nonlinear propagation in space-division multiplexed fiber-optic transmission. J. Lightwave Technol. 34, 36–54 (2015).
Krupa, K. et al. Multimode nonlinear fiber optics, a spatiotemporal avenue. APL Photonics 4, 110901 (2019).
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).
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).
Wright, L. G. Spatiotemporal Nonlinear Optics in Multimode Fibers. PhD thesis, Cornell Univ. (2018).
Kartashov, Y. V. et al. Frontiers in multidimensional self-trapping of nonlinear fields and matter. Nat. Rev. Phys. 1, 185–197 (2019).
Krupa, K. et al. Spatial beam self-cleaning in multimode fibres. Nat. Photon. 11, 237–U299 (2017).
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).
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).
Agrawal, G. P. Nonlinear Fiber Optics 5th edn (Academic Press, 2012).
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).
Wright, L. G. et al. Self-organized instability in graded-index multimode fibres. Nat. Photon. 10, 771–776 (2016).
Podivilov, E. V. et al. Hydrodynamic 2D turbulence and spatial beam condensation in multimode optical fibers. Phys. Rev. Lett. 122, 103902 (2019).
Pourbeyram, H. et al. Direct observations of thermalization to a Rayleigh–Jeans distribution in multimode optical fibers. Nat. Phys 18, 685–690 (2022).
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).
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).
Laegsgaard, J. Spatial beam cleanup by pure Kerr processes in multimode fibers. Opt. Lett. 43, 2700–2703 (2018).
Wu, F. O., Hassan, A. U. & Christodoulides, D. N. Thermodynamic theory of highly multimoded nonlinear optical systems. Nat. Photon. 13, 776–782 (2019).
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).
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).
Mangini, F. et al. Statistical mechanics of beam self-cleaning in GRIN multimode optical fibers. Opt. Express 30, 10850–10865 (2022).
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).
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).
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).
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).
Nakazawa, M., Yamada, E., Kubota, H. & Suzuki, K. 10 Gbit/s soliton data-transmission over one million kilometers. Electron. Lett. 27, 1270–1272 (1991).
Skryabin, D. V. & Gorbach, A. V. Colloquium: looking at a soliton through the prism of optical supercontinuum. Rev. Mod. Phys. 82, 1287–1299 (2010).
Chen, Z. G., Segev, M. & Christodoulides, D. N. Optical spatial solitons: historical overview and recent advances. Rep. Prog. Phys. 75, 086401 (2012).
Renninger, W. H. & Wise, F. W. Optical solitons in graded-index multimode fibres. Nat. Commun. 4, 1719 (2013).
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).
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).
Wright, L. G., Christodoulides, D. N. & Wise, F. W. Controllable spatiotemporal nonlinear effects in multimode fibres. Nat. Photon. 9, 306–310 (2015).
Zitelli, M. et al. High-energy soliton fission dynamics in multimode GRIN fiber. Opt. Express 28, 20473–20488 (2020).
Wright, L. G., Renninger, W. H., Christodoulides, D. N. & Wise, F. W. Spatiotemporal dynamics of multimode optical solitons. Opt. Express 23, 3492–3506 (2015).
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).
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).
Hasegawa, A. Self-confinement of multimode optical pulse in a glass fiber. Opt. Lett. 5, 416–417 (1980).
Crosignani, B. & Di Porto, P. Soliton propagation in multimode optical fibers. Opt. Lett. 6, 329–330 (1981).
Zitelli, M., Ferraro, M., Mangini, F. & Wabnitz, S. Single-mode spatiotemporal soliton attractor in multimode GRIN fibers. Photon. Res. 9, 741–748 (2021).
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).
Barland, S. et al. Cavity solitons as pixels in semiconductor microcavities. Nature 419, 699–702 (2002).
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).
Tanguy, Y., Ackemann, T., Firth, W. J. & Jager, R. Realization of a semiconductor-based cavity soliton laser. Phys. Rev. Lett. 100, 13907 (2008).
Gustave, F. et al. Observation of mode-locked spatial laser solitons. Phys. Rev. Lett. 118, 044102 (2017).
Renninger, W. H., Chong, A. & Wise, F. W. Dissipative solitons in normal-dispersion fiber lasers. Phys. Rev. A 77, 023814 (2008).
Wright, L. G., Christodoulides, D. N. & Wise, F. W. Spatiotemporal mode-locking in multimode fiber lasers. Science 358, 94–97 (2017).
Wright, L. G. et al. Mechanisms of spatiotemporal mode-locking. Nat. Phys. 16, 565–570 (2020).
Ding, Y. H. et al. Spatiotemporal mode-locking in lasers with large modal dispersion. Phys. Rev. Lett. 126, 93901 (2021).
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).
Tegin, U., Kakkava, E., Rahmani, B., Psaltis, D. & Moser, C. Spatiotemporal self-similar fiber laser. Optica 6, 1412–1415 (2019).
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).
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).
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).
Longhi, S. Modulational instability and space time dynamics in nonlinear parabolic-index optical fibers. Opt. Lett. 28, 2363–2365 (2003).
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).
Guasoni, M. Generalized modulational instability in multimode fibers: wideband multimode parametric amplification. Phys. Rev. A 92, 033849 (2015).
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).
Demas, J. et al. Intermodal nonlinear mixing with Bessel beams in optical fiber. Optica 2, 14–17 (2015).
Dupiol, R. et al. Intermodal modulational instability in graded-index multimode optical fibers. Opt. Lett. 42, 3419–3422 (2017).
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).
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).
Cheng, J. et al. Intermodal Cerenkov radiation in a higher-order-mode fiber. Opt. Lett. 37, 4410–4412 (2012).
Lpken, N. M. et al. Numerical and experimental demonstration of intermodal dispersive wave generation. Laser Photon. Rev. 15, 2100125 (2021).
Eftekhar, M. A. et al. Versatile supercontinuum generation in parabolic multimode optical fibers. Opt. Express 25, 9078–9087 (2017).
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).
Florentin, R. et al. Shaping the light amplified in a multimode fiber. Light Sci. Appl. 6, e16208 (2017).
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).
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).
Tegin, U., Yldrm, M., Ouz, l, Moser, C. & Psaltis, D. Scalable optical learning operator. Nat. Comput. Sci. 1, 542–549 (2021).
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).
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).
Silberberg, Y. Collapse of optical pulses. Opt. Lett. 15, 1282–1284 (1990).
Kibler, B. & Bejot, P. Discretized conical waves in multimode optical fibers. Phys. Rev. Lett. 126, 23902 (2021).
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).
Raghavan, S. & Agrawal, G. P. Spatiotemporal solitons in inhomogeneous nonlinear media. Opt. Commun. 180, 377–382 (2000).
Mayteevarunyoo, T., Malomed, B. A. & Skryabin, D. V. Spatiotemporal solitons in dispersion-managed multimode fibers. J. Opt. 23, 015501 (2021).
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).
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).
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).
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).
Hanna, M. et al. Nonlinear optics in multipass cells. Laser Photon. Rev. 15, 2100220 (2021).
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).
Mecozzi, A., Antonelli, C. & Shtaif, M. Nonlinear propagation in multi-mode fibers in the strong coupling regime. Opt. Express 20, 11673–11678 (2012).
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).
Mecozzi, A., Antonelli, C. & Shtaif, M. Coupled Manakov equations in multimode fibers with strongly coupled groups of modes. Opt. Express 20, 23436–23441 (2012).
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).
Patsyk, A., Sivan, U., Segev, M. & Bandres, M. A. Observation of branched flow of light. Nature 583, 60–65 (2020).
Eslami, Z. et al. Two octave supercontinuum generation in a non-silica graded-index multimode fiber. Nat. Commun. 13, 2126 (2022).
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).
Eftekhar, M. A. et al. Accelerated nonlinear interactions in graded-index multimode fibers. Nat. Commun. 10, 1638 (2019).
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).
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).
Piccoli, R. et al. Intense few-cycle visible pulses directly generated via nonlinear fibre mode mixing. Nat. Photon. 15, 884–889 (2021).
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).
Safaei, R. et al. High-energy multidimensional solitary states in hollow-core fibres. Nat. Photon. 14, 733–739 (2020).
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).
Gao, X. H. et al. Ionization-assisted spatiotemporal localization in gas-filled capillaries. Opt. Lett. 43, 3112–3115 (2018).
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).
Tradonsky, C. et al. High-resolution digital spatial control of a highly multimode laser. Optica 8, 880–884 (2021).
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).
Guo, H. et al. Intermode breather solitons in optical microresonators. Phys. Rev. 7, 041055 (2017).
Ji, X. et al. Exploiting ultralow loss multimode waveguides for broadband frequency combs. Laser Photon. Rev. 15, 6–11 (2021).
Molesky, S. et al. Inverse design in nanophotonics. Nat. Photon. 12, 659–670 (2018).
Frostig, H. et al. Focusing light by wavefront shaping through disorder and nonlinearity. Optica 4, 1073–1079 (2017).
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).
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).
Guenard, R. et al. Kerr self-cleaning of pulsed beam in an ytterbium doped multimode fiber. Opt. Express 25, 4783–4792 (2017).
Jankowski, M. et al. Temporal simultons in optical parametric oscillators. Phys. Rev. Lett. 120, 053904 (2018).
Bai, N. et al. Mode-division multiplexed transmission with inline few-mode fiber amplifier. Opt. Express 20, 2668–2680 (2012).
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).
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).
Dorrer, C. Spatiotemporal metrology of broadband optical pulses. IEEE J. Sel. Top. Quantum Electron. 25, 3100216 (2019).
Jolly, S. W., Gobert, O. & Quere, F. Spatio-temporal characterization of ultrashort laser beams: a tutorial. J. Opt. 22, 103501 (2020).
Leventoux, Y. et al. 3D time-domain beam mapping for studying nonlinear dynamics in multimode optical fibers. Opt. Lett. 46, 66–69 (2021).
Guo, Y. K. et al. Real-time multispeckle spectral-temporal measurement unveils the complexity of spatiotemporal solitons. Nat. Commun. 12, 67 (2021).
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).
Esmaeelpour, M. et al. Power fluctuations of intermodal four-wave mixing in few-mode fibers. J. Lightwave Technol. 35, 2429–2435 (2017).
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).
Shapira, O., Abouraddy, A. F., Joannopoulos, J. D. & Fink, Y. Complete modal decomposition for optical waveguides. Phys. Rev. Lett. 94, 143902 (2005).
An, Y. et al. Learning to decompose the modes in few-mode fibers with deep convolutional neural network. Opt. Express 27, 10127–10137 (2019).
Manuylovich, E. S., Dvoyrin, V. V. & Turitsyn, S. K. Fast mode decomposition in few-mode fibers. Nat. Commun. 11, 5507 (2020).
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).
Pierangeli, D. et al. Observation of replica symmetry breaking in disordered nonlinear wave propagation. Nat. Commun. 8, 1501 (2017).
Ramos, A., Fernandez-Alcazar, L., Kottos, T. & Shapiro, B. Optical phase transitions in photonic networks: a spin-system formulation. Phys. Rev. 10, 031024 (2020).
Angelani, L., Conti, C., Ruocco, G. & Zamponi, F. Glassy behavior of light. Phys. Rev. Lett. 96, 065702 (2006).
Conti, C. & Leuzzi, L. Complexity of waves in nonlinear disordered media. Phys. Rev. B 83, 134204 (2011).
Carleo, G. & Troyer, M. Solving the quantum many-body problem with artificial neural networks. Science 355, 602–605 (2017).
Borhani, N., Kakkava, E., Moser, C. & Psaltis, D. Learning to see through multimode fibers. Optica 5, 960–966 (2018).
Caramazza, P., Moran, O., Murray-Smith, R. & Faccio, D. Transmission of natural scene images through a multimode fibre. Nat. Commun. 10, 2029 (2019).
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).
Tegin, U. et al. Controlling spatiotemporal nonlinearities in multimode fibers with deep neural networks. APL Photonics 5, 030804 (2020).
Karniadakis, G. E. et al. Physics-informed machine learning. Nat. Rev. Phys. 3, 422–440 (2021).
Mounaix, M. et al. Time reversed optical waves by arbitrary vector spatiotemporal field generation. Nat. Commun. 11, 5813 (2020).
Baudin, K. et al. Classical Rayleigh-Jeans condensation of light waves: observation and thermodynamic characterization. Phys. Rev. Lett. 125, 244101 (2020).
Haus, H. A. & Kogelnik, H. Electromagnetic momentum and momentum flow in dielectric waveguides. J. Opt. Soc. Am. 66, 320–327 (1976).
Pathria, R. K. & Beale, P. D. Statistical Mechanics (Academic Press, 2011).
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).
Picozzi, A. et al. Optical wave turbulence: towards a unified nonequilibrium thermodynamic formulation of statistical nonlinear optics. Phys. Rep. 542, 1–132 (2014).
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).
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).
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 information
Nature Physics thanks Stefan Wabnitz and the other, anonymous, reviewer(s) 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.
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.
About this article
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
This article is cited by
Shaping the propagation of light in complex media
Nature Physics (2022)