The propagation of ultrashort pulses in optical fibre plays a central role in the development of light sources and photonic technologies, with applications from fundamental studies of light–matter interactions to high-resolution imaging and remote sensing. However, short pulse dynamics are highly nonlinear, and optimizing pulse propagation for application purposes requires extensive and computationally demanding numerical simulations. This creates a severe bottleneck in designing and optimizing experiments in real time. Here, we present a solution to this problem using a recurrent neural network to model and predict complex nonlinear propagation in optical fibre, solely from the input pulse intensity profile. We highlight particular examples in pulse compression and ultra-broadband supercontinuum generation, and compare neural network predictions with experimental data. We also show how the approach can be generalized to model other propagation scenarios for a wider range of input conditions and fibre systems, including multimode propagation. These results open up novel perspectives in the modelling of nonlinear systems, for the development of future photonic technologies and more generally in physics for studies in Bose–Einstein condensates, plasma physics and hydrodynamics.
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L.S. acknowledges support from the Faculty of Engineering and Natural Sciences Graduate School of Tampere University. J.M.D. acknowledges support from the French Agence Nationale de la Recherche (ANR-20-CE30-0004, ANR-15-IDEX-0003 and ANR-17-EURE-0002). G.G. acknowledges support from the Academy of Finland (298463, 318082 and Flagship PREIN 320165). We also thank D. Brunner for useful discussions.
The authors declare no competing interests.
Peer review information Nature Machine Intelligence 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.
Extended Data Fig. 1 Comparison of spectral intensity evolution modelling by the RNN when trained using data input in linear and logarithmic (dB) representation.
The dashed red line shows the simulated spectra of a higher-order soliton (N = 6) at selected distances along the fibre similar to those plotted in Fig. 5. The solid blue lines shows the RNN prediction when trained directly with spectral data input in dB. The solid green line shows the RNN prediction (plotted in dB) when trained with spectral data input in linear scale.
Extended Data Fig. 2 Generalization of RNN applicability to modelling NLSE dynamics using normalized training simulations.
Training data were generated using the normalized form of the NLSE (see Methods). The results are plotted in dimensional units. (a) and (b) shows the temporal and spectral evolution corresponding to the initial stage of higher-order soliton compression and breakup with soliton number N = 4, pulse duration (FWHM) of 300 fs and peak power of 157 W. (c) and (d) shows the temporal and spectral evolution corresponding to the initial stage of higher-order soliton compression and breakup with soliton number N = 7, pulse duration (FWHM) of 3 fs and peak power of 4.8 W. In each panel, we show the evolution map directly obtained from the numerical NLSE simulations and that obtained from the RNN network model. The RNN was trained with 950 normalized NLSE realizations where the soliton number was randomly varied in the range 1 to 7. The r.m.s. error (see Eq. (4) in Methods) computed over 50 test realizations was R = 0.152 and R = 0.077 for the temporal and spectral intensities, respectively.
(a) and (b) shows temporal evolution for input pulse parameter similar to those in Extended Data Fig. 2(a) and (c), but when a linear chirp (parabolic phase) of C = − 5 and C = 5, respectively, was added to the input pulse (see Methods). The left and middle panels show temporal evolution from NLSE simulations and prediction by the RNN, respectively, and the right panel shows the comparison between these two at selected distances. The network was trained with 5900 NLSE realizations where the soliton number and chirp parameter were varied randomly in the range 1 to 7 and -8 to 8, respectively. The r.m.s. error computed over 100 test realizations is R = 0.156.
Training data were generated using the normalized form of the GNLSE including Raman effect, self-steepening and third-order dispersion (see Methods). The results are plotted in dimensional units. The network was trained using 11800 normalized GNLSE realizations where the soliton number, normalized third-order dispersion parameter, and pulse duration were respectively randomly varied in the range 2 to 8, 1 to 9 and 30 and 130 fs (FWHM). (a) shows the results for a transform-limited N = 4 input pulse centered at 830 nm with 7.6 kW peak power and 40 fs duration. Corresponding fibre parameters are γ = 0.1 W−1m−1, β2 = − 8 × 10−27 s2m−1 and β3 = 9 × 10−41 s3m−1. (b) shows the results for a transform-limited N = 7 input pulse with 2.9 kW peak power and 120 fs duration. Corresponding fibre parameters are γ = 0.0184 W−1m−1, β2 = − 5.1 × 10−27 s2m−1 and β3 = 4.3 × 10−41 s3m−1. (c) shows the results for a transform-limited N = 4.5 input pulse with 3.0 kW peak power and 60 fs duration. Corresponding fibre parameters are γ = 0.01 W−1m−1, β2 = − 1.7 × 10−27 s2m−1 and β3 = 6.5 × 10−42 s3m−1. In each panel, we show the evolution map directly obtained from the numerical GNLSE simulations and that obtained from the RNN network model. The r.m.s. error computed over 200 test realizations is R = 0.092.
The left and right panels shows how ± 10 and ± 20 % relative random multiplicative intensity noise added to the examples shown in Extended Data Fig. 2(a) and (c) affect the RNN predictions (see also Methods). The r.m.s. error computed over 50 test realizations was R = 0.200 and R = 0.271 for the ± 10 and ± 20 % cases, respectively (R = 0.152 for noise-free data). Although we do note a residual shift in the point of maximum compression, the dynamics of the higher-order soliton compression are overall well reproduced even under noisy conditions.
RNN modelling of nonlinear propagation of an ultrashort pulse of 150 fs duration (FWHM) and 2.5 MW peak power with 1500 nm center wavelength in a 50 μm radius multimode silica fibre. The network was trained with 950 realizations where the energy distribution between the different modes at the fibre input was varied. (a-d) show the spectral evolution integrated over all the modes for input energy distributions as indicated. In each panel, we show the evolution map directly obtained from the numerical multimode GNLSE simulations and that obtained from the RNN network model. The r.m.s. error computed over 50 test realizations is R = 0.104.
(a) Simulation time to compute evolution maps of supercontinuum similar to those presented in Fig. 6 using the GNLSE and RNN, as a function of the number of points Np in the temporal (or spectral) grid and number of computed maps as indicated in the parentheses. (b) Simulation time to compute evolution maps of multimode simulations similar to those shown in Extended Data Fig. 6 using the multimode GNLSE and RNN, as a function of the number of realizations for a constant number of modes and grid points.
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Salmela, L., Tsipinakis, N., Foi, A. et al. Predicting ultrafast nonlinear dynamics in fibre optics with a recurrent neural network. Nat Mach Intell 3, 344–354 (2021). https://doi.org/10.1038/s42256-021-00297-z
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