Customizing supercontinuum generation via on-chip adaptive temporal pulse-splitting

Modern optical systems increasingly rely on complex physical processes that require accessible control to meet target performance characteristics. In particular, advanced light sources, sought for, for example, imaging and metrology, are based on nonlinear optical dynamics whose output properties must often finely match application requirements. However, in these systems, the availability of control parameters (e.g., the optical field shape, as well as propagation medium properties) and the means to adjust them in a versatile manner are usually limited. Moreover, numerically finding the optimal parameter set for such complex dynamics is typically computationally intractable. Here, we use an actively controlled photonic chip to prepare and manipulate patterns of femtosecond optical pulses that give access to an enhanced parameter space in the framework of supercontinuum generation. Taking advantage of machine learning concepts, we exploit this tunable access and experimentally demonstrate the customization of nonlinear interactions for tailoring supercontinuum properties.


Supplementary Discussion
Numerical simulation of supercontinuum generation dynamics for single pulse evolution: In Supplementary Figure 1, we provide an overview of the initial conditions and evolution dynamics of a single pulse propagating into our 1 km-length highly nonlinear fibre (HNLF) for given experimental parameters. The initial pulse properties, measured experimentally at the HNLF input 1 , are summarized in Supplementary Fig. 1a-d. Here, the pulse had an initial average power of ~15 mW and was propagated through the chip in the absence of pulse-splitting (i.e. taking the shortest possible path in the sample waveguide). It was then characterized after undergoing amplification in the erbium-doped fibre amplifier (EDFA), in order to reach an average power of up to 50 mW (corresponding to a peak power of ~ 1 kW). The pulse evolution for supercontinuum generation, numerically obtained assuming a 5 mW average input power, is shown in Supplementary Fig. 1e From Supplementary Fig. 1, one can clearly observe that there is good qualitative agreement with the experimental results reported in Figure 3 of the main manuscript. As we increase the power, more solitons (and corresponding dispersive waves) are radiated to contribute to the spectral broadening during propagation 3-5 -Note that in our case, spectral broadening induced by soliton self-frequency shift is limited by the significantly high losses above 1900 nm, which cannot be neglected for such a long fibre. Interestingly, although a careful characterization of the experimental parameters has been conducted, it is worth mentioning that some quantitative discrepancies with Figure 3a are observed in our simulation results. This can be attributed to uncertainty in both the fibre and input pulse parameters (e.g. due to an imperfect reconstruction of the input pulse caused by a limited spectral resolution of our frequency-resolved optical gating traces), as well as ambiguity in the type and magnitude of noise (and distortions) induced by the EDFA for different amplification levels. Additional contributions, such as polarization-mode dispersion and/or nonlinear effects (e.g. cross-phase modulation) due to polarization cross-talk in the integrated waveguides, can also be responsible for discrepancies. The fact that such numerical simulation approaches have been shown to usually provide trustworthy agreement with experimental results 2 is here a key point to highlight the importance of the results reported here: In many cases, several experimental parameters may not be perfectly known, and they are thus difficult to assess a priori for modelling a target application. Our approach, enabling the efficient tuning of multiple variables in the parameter space, hence represents a powerful alternative for the optimization of supercontinua -as well as similar complex dynamics observed in a variety of ultrafast nonlinear systems. Initial input spectra for different pulse patterns: For completeness, we also provide the spectra measured in our experimental setup at the input of the HNLF. The spectra, shown in Supplementary Figure 3 after pulse splitting and amplification, are provided for selected pattern configurations, also depicted in Figure 2 of the main manuscript (in the temporal domain). Starting from the very typical and smooth spectrum, the pulse-splitter leads to coherent temporal modulation, which yields interference in the spectrum. In particular, depending on the integrated pulse-splitter configuration, the otherwise smooth initial spectral envelope (black line) presents a modulation based on the number and separation of the generated pulses. This modulation is relatively clean when considering only the generation of two pulses (see, e.g., the plain red line, for 2 pulses separated by 1 ps) and typically exhibits a contrast above 10 dB (in agreement with the measured extinction ratio of the splitter). As expected, a more complex interference pattern can be observed in the spectrum when considering more pulses (see e.g. dashed blue line, for 8 pulses separated by 2 ps) and therefore, multiple modulation frequencies between the pulses of adjacent patterns.
Genetic algorithm -Optimization approach, results and processing: The evolutionary (genetic) algorithm settings were chosen to provide a suitable supercontinuum in a reasonable time frame. In particular, we observed that even when fully adjusting 5 interferometers' settings in the sample (i.e. changing the pulse path from 0 to 100 %), the predominant spectral modifications were stable and reproducible after only a few hundred ms settling time (including the response time of the electronic driver and spectrometer). In this 'worst-case scenario' of supercontinuum tuning, we found that a perfectly stable and reproducible spectrum was typically obtained after 3 s, when slight thermal crosstalk between adjacent waveguide heaters reached equilibrium.
Based on these observations, we performed several rounds of supercontinuum optimization and found that consistent results could be obtained considering a settling time of 500 ms. Supplementary Figure 4 illustrates typical evolutions of the genetic algorithm (GA) iterative process for different parameters and target wavelengths. In the first example (i.e. with a target wavelength set to 1900 nm), one can see that similar optimization results are obtained regardless of the number of interferometers used. This is due to the fact that, at such a wavelength, a soliton with a minimal power needs to be ejected during HNLF propagation of the input pulses. In this case, the potential spectral optimization is therefore strongly limited (~25%) by the overall power budget of the initial pulse patterns, as well as from the increasing fibre attenuation at wavelengths approaching 2000 nm. In the second example, however, the spectral intensity enhancement obtained at 1600 nm is already significant (~ 370 %) and reached after only a few generations, even when just 16 pulses are used for optimization. At this wavelength, we can observe that an approximately two-fold spectral intensity enhancement (~ 750 %) can be readily obtained using an optimization based on 32 pulses. This confirms that spectral components with a wavelength closer to the optical pump require much less energy to be produced from the initial pump pulse. Increasing the number of pulses enables to drastically enhance the power spectral density at this wavelength via controlled soliton ejection from each pulse (i.e. in a way similar to using a pump laser with increased repetition rate).
For this wavelength, we also performed a GA optimization using longer settling times between various iterations of the integrated splitter thermal adjustments. In Supplementary Fig. 4, we also provide an example of such an optimization procedure with a 3 s settling time, clearly illustrating that the previous optimization results are consistent (~ 350 % for 16 pulses). Note that comparable results were also observed for other wavelength optimization procedures and/or variable settling times (in the range 100 ms -20 s).
Finally, we mention in the main text that further tunability in the supercontinuum could be obtained from multiple pulse seeding compared to the single pulse case. Besides the ability to control both temporal and spectral location of the radiated structures (see Fig. 5 of the main manuscript), an example of our approach's versatility is illustrated in the dual-wavelength optimization shown in Supplementary   Figure 5. Indeed, in this figure, we only present the optimized supercontinuum, where the two target wavelengths possess similar intensities (i.e. for which the intensity at one wavelength is no more than twice the intensity at the other wavelength). Yet, the optimization of multiple objectives clearly reveals the potential of our pulse splitting approach, allowing to further control the intensity at each wavelength of the supercontinuum. This aspect is illustrated in Supplementary Fig. 5a, where we show the optimization results in the form of a Pareto front [9][10][11] . In this case, the weight of each optimization criterion (i.e. the spectral intensity for the two wavelengths) can be post-selected to adjust the respective intensity at each optimized wavelength. Typical spectra obtained for dual-wavelength optimization with various relative spectral intensities are reported in Supplementary Fig. 5b.
Once again, one can readily observe that the control of the spectral intensity at both wavelengths of interest is not easily addressed when considering only a single input pulse with adjustable power (as seen by the highly skewed distribution of black dots in Supplementary Fig. 5). On the other hand, a clear intensity enhancement, along with a selective control of the power distribution between the two wavelengths of interest, can be readily achieved using our optimization technique. Such results further highlight the versatility of the proposed pulse splitting scheme as an effective tool for the control of supercontinuum generation and complex (nonlinear) optical processes.