Controlling spatiotemporal nonlinearities in multimode fibers with deep neural networks

Multimode fibers (MMFs) have found applications in several fields in the last few decades mainly in telecommunication and imaging.^1,2 In recent years, spatiotemporal nonlinearities in MMFs have also become the subject of strong interest in various fundamental and applied areas from single-pass propagation to spatiotemporally mode-locked laser cavities.^3–5 In single-pass nonlinear propagation studies, numerous interesting phenomena,^6–15 including spatiotemporal instability, self-beam cleaning, and supercontinuum generation, were reported with graded-index multimode fibers (GRIN MMFs). Although the importance of the excitation condition is mentioned, these studies were reported with only a Gaussian beam profile.

In the linear regime, the spatial control of light propagation in MMFs via wavefront shaping with a spatial light modulator is now well understood. Over the last few years, several methods, including iterative, phase conjugation, and transmission matrix, were successfully applied to demonstrate fluorescence confocal, two photon, Raman, CARS imaging and material processing through a MMF.^16–18 Recently, machine learning tools have been proposed to simplify the calibration and improve the robustness of the system in the absence of optical nonlinearities.¹⁹ Deep neural networks (DNNs) proved useful for classification/reconstruction of the information sent to km-long MMFs solely from the intensity measurement at the output of the fibers.^20,21 For the nonlinear propagation regime, adaptive algorithms have been brought forth and been shown to be successful in harnessing the entangled spatiotemporal nonlinearities such as Kerr beam self-cleaning of low-order modes²² and optimization of the intensity of targeted spectral peaks generated by Raman scattering or four-wave mixing.²³ 

In this article, we report the results of our studies on the effect of the initial spatial excitation condition of a GRIN MMF on the output spectrum by employing machine learning. Specifically, we achieved control over multimodal nonlinear frequency conversion dynamics and demonstrated that spatiotemporal nonlinear pulse propagation can be learned by DNNs. Once trained, the DNNs can predict the spatial beam shape for the input pump pulses to produce a desired spectral shape within the limitations of the triggered nonlinear effects at the end of the GRIN MMF. In particular, we showed that cascaded stimulated Raman scattering (SRS) based broadening of the spectrum and supercontinuum generation, two highly nonlinear phenomena, can be experimentally controlled for the first time in the literature with machine learning tools.

Numerical calculations are performed to determine the suitable approach to define the preliminary excitation patterns applied on the GRIN MMF in the experimental studies. We numerically investigate the effect of excitation to nonlinear pulse propagation by changing the initial energy splitting ratio between the simulated modes (LP₀₁, LP_11a, LP_11b, LP_21a, LP_21b, LP₀₂, LP_31a, LP_31b, LP_12a, and LP_12b). To investigate the nonlinear effects in a 1 m GRIN MMF, pump pulses with 2 ps duration and 500 kW peak power were used in the numerical studies. The multimode nonlinear Schrödinger equation with a Raman scattering term [Eq. (1)] is numerically solved.²⁴ For numerical integration with high accuracy, we used a fourth-order Runge–Kutta in the interaction picture method.²⁵ We used an integration step of 5 µm and a time resolution of 2.4 fs with 20 ps time window width. Although the GRIN MMF with 62.5 µm core diameter supports more than 250 modes, to reduce the computational time, we considered the 10 linearly polarized modes in our numerical studies, which required 14 days of computation.

Here, $ηplmn$ is the nonlinear coupling coefficient, $fR$ is the fractional contribution of the Raman effect, $hR$ is the delayed Raman response function, and $δβ0(p)$ (⁠$δβ1(p)$⁠) is the difference between first (second) Taylor expansion coefficient of the propagation constant for the corresponding and the fundamental mode. Relative index difference between fiber core and clad is assumed to be 0.01. Figure 1 demonstrates the variations in the nonlinear pulse propagation with the different initial excitation condition. As shown in Fig. 1(a), spectral broadening can be achieved by favoring the lower order modes (LP₀₁, LP_11a, and LP_11b). For the same pump pulse parameters, equal excitation of all the modes interestingly resulted in broader spectra [see Fig. 1(b)]. Since more energy couples to higher order modes, spectral formations around 1.5 µm occur in a shorter propagation distance. When most of the energy was coupled to higher-order modes, the optical spectrum evolved to a smoother supercontinuum formation, as presented in Fig. 1(c).

Our simulations revealed significant spectral differences entirely due to the initial power distribution between the fiber modes. Similar behavior was numerically reported in the literature for multimode holey fibers.²⁶ The propagation differences observed in Fig. 1 can be understood by studying the nonlinear coupling term used in the numerical calculations. According to the nonlinear coupling between the modes determined by the overlap integrals, some of the intermodal processes are favored by different modal symmetry classes and particular modes act as a pump for these nonlinear effects. In our calculations, we found that by initially favoring high order modes in spatiotemporal nonlinear propagation, a medium to generate a broad output spectrum can be achieved since excitation of the higher-order modes creates an environment that encourages the nonlinear intermodal interactions. In the literature, the importance of the beam size on the fiber facet to initiate multimode propagation is emphasized in experimental studies related to self-beam cleaning and spatiotemporal instability.^6,12

The experimental setup is shown in Fig. 2. We launched 10 ps pulses centered around 1030 nm with adjustable peak power into a 20 m GRIN MMF with 62.5 µm core diameter and 0.275 NA (Thorlabs, GIF625). The fiber is coiled with 25 cm diameter and rests on the optical table without additional cooling. A phase-only spatial light modulator (SLM), an 8f imaging system, and a quarter-wave plate are placed before the GRIN MMF. Here, we study two particular phenomena by adjusting the peak power of the pump pulses to either 85 kW or 150 kW. In the first case, spectral broadening induced by cascaded SRS is observed, while in the second case, supercontinuum generation was recorded at the output of the GRIN. Cascaded SRS based spectral broadening in GRIN MMFs has been extensively studied in the past, and spatiotemporal pulse propagation is the leading mechanism for the cascaded Raman Stokes generation. In GRIN MMFs, Raman Stokes peaks are reported with different beam shapes, which lead to different propagation paths for each Raman peak. By compensating the chromatic dispersion difference, the multimode propagation enhances the cross-phase modulation between the Raman peaks, hence triggering the generation of a supercontinuum formation for higher peak powers after reaching the zero-dispersion wavelength (ZDW).^13,15

To experimentally study the effect of the spatial profile of the excitation condition on the nonlinear pulse propagation, a set of beam profiles containing 3000 samples are calculated by superposing pre-defined base patterns with random non-repeating amplitudes from 0 to 1 and fixed sum for each candidate beam shape. Guided by simulations, a mixture of the five lowest order LG_4X modes are selected as the base patterns to create beam profiles of the pump pulses. To minimize the power level changes due to beam shaping, the number of coefficients is intentionally limited as five and the LG₄₀ mode is added to each calculated beam shape as a background. Here, we would like to emphasize that the selected LG modes are calculated for the free-space propagation, and the energy distribution of these patterns cannot be directly related to the energy distribution of the modes supported by the fibers. The complex amplitude modulation method described in Ref. 27 is applied to calculate the required phase patterns to generate the designed beam profiles with the phase-only SLM.²⁷ The achieved beam profiles are ∼80× demagnified with the imaging system to excite the GRIN MMF.

Pulses with 85 kW and 150 kW peak powers and different beam shapes impinge on the GRIN MMF facet, while the output spectra are being recorded for each applied pattern. For both cases of interest, strong variations at the output spectra are observed. For 150 kW pump peak power, as the spectrum approaches the ZDW, the driving nonlinear effect changes and instead of cascaded SRS, modulation instability based spectral broadening occurs. Therefore, we focused on the wavelength range above the ZDW (1350 nm–1700 nm).^13,15 The extreme cases recorded in the datasets are presented in Fig. 3.

For both peak power levels, the machine learning approach is employed to analyze the experimentally collected data. In both cases, the same network architecture employed is comprised of four hidden layers, as demonstrated in Fig. 4. The measured spectra are fed to the network as inputs, and for each spectrum, the coefficients to generate the corresponding beam profiles are the output variables of the network. DNNs can learn the spatiotemporal nonlinear pulse propagation inside the tested GRIN MMF by adjusting their weights to learn the relation between the generated spectra and the excitation condition, as described by the coefficients used to determine the input beam shape. The training and validation results of the DNNs are described in Note 1 of the supplementary material.

To experimentally investigate the performance of the trained DNN, the control on the output spectra is tested with the schematic presented in Fig. 5. For this, a collection of synthetic spectral shapes is generated via summations of Gaussian distributions with different amplitudes and widths. It should be noted that the synthetic spectra must still lie within the limits of the recorded spectral dataset. These target spectra are then fed to the DNN to predict the required beam shapes of the target pump pulses. For each designed target spectra, the DNN predicts the coefficients of the LG4x patterns, and from these coefficients, the required input beam shapes are synthesized. Here, we rely on the ability of the neural network to generalize since the user-defined spectra are not from the test or the training dataset. For the 85 kW pump peak power level, summations of different Gaussian functions centered around the Raman Stokes peaks of the silica medium are used to design the targeted spectra. Results of the tests for controlling cascaded SRS generation are presented in Fig. 6. The measured spectra corresponding to the DNN-predicted beam shapes are well matched with the target spectra [Figs. 6(a)–6(c)]. For the 150 kW pump peak power level, spectral shapes are designed based on the trailing edge of a Gaussian function centered around 1350 nm since the targeted wavelength range corresponds to the trailing edge of the supercontinuum spectra. The same procedure explained for the 85 kW pump peak power level is applied to control the supercontinuum generation. Targeted and experimentally measured spectra are presented in Figs. 7(a)–7(c), and the experimentally measured spectra are in good agreement with the targeted spectra. DNN predicted the beam profiles of the pump pulses, which are shown in insets of Figs. 7(d)–7(f).

The main result of this article is that machine learning tools can master the underlying basis of the spatiotemporal nonlinear propagation in MMFs, which have been considered chaotic and complex over the years. Here, DNNs are employed to learn the relation between the initial modal energy distribution of the fiber and nonlinear frequency conversion. Trained with experimental data, our DNNs are shown to be a powerful tool to harness the nonlinear dynamics of the MMF within the nonlinear dynamics defined by the system.

Due to non-one-to-one relation between the initial modal energy distribution of the fiber and nonlinear frequency conversion dynamics, the experimental controlling efficiency of the DNNs is a significant topic. For both peak power levels (85 kW and 150 kW), more than 50% efficiency is achieved in the control experiments. By introducing artificial noise to the designed spectra, the experimental estimation efficiencies of DNNs are improved and reached 80%. We believe that this improvement is due to the noisy nature of the experimentally collected dataset, but it needs to be further investigated in future work. We observed that DNNs learned the behavior of initially favoring high order modes to achieve a broader output spectrum, which is also presented in our numerical simulations. For both of the peak power levels, to generate the targeted spectra, which requires more nonlinear conversions, DNNs increased the coefficients of the high order base patterns [Fig. 6 (inset) and Fig. 7 (inset)].

In conclusion, we showed that spatiotemporal nonlinear pulse propagation can be learned and controlled by machine learning tools. We demonstrated that spectral broadening based on both cascaded SRS and supercontinuum generation can be altered by tuning the initial modal energy distribution of the fiber through shaping the beam profile of the pump pulse by implementing experimentally trained DNNs. The machine learning approach reported here can be employed to understand and tune other nonlinear effects and relations. Our results present a novel path toward automated tunable fiber-based broadband sources.

See the supplementary material for further details regarding the training of neural networks and output beam profile measurements.

The authors would like to thank D. Loterie and P. Hadikhani for fruitful discussions regarding the use of beam shaping and machine learning tools.