Optimized on-demand ultrafast and broadband light sources using machine learning – OPTIMAL

A. Numerical and Theoretical Methods

1. For our theoretical analysis and numerical simulations, we developed realistic models of waveguides and laser systems using generalized propagation equations. Single-pass simulations were used to build databases linking input and output characteristics, taking into account both spectral phase and amplitude as well as technical and quantum noise. After an extensive parameter exploration, we targeted initial conditions that optimized broadband spectra in the 1500–1600 nm range, which guided the experimental design.

2. Two data-driven approaches were also studied: SINDy (Sparse Identification of Nonlinear Dynamics) and Dominant Balance. The SINDy method, applied to the case of four-wave mixing in optical fibers, successfully retrieved the governing differential equations from dynamic data. Its application to a more complex experimental system revealed how non-conservative losses disrupted the ideal dynamics. The Dominant Balance approach, in turn, allowed the automatic identification of dominant terms in a numerical model by combining a spatio-temporal dynamic map with statistical clustering (Gaussian Mixture Model).

3. Coding was performed in MATLAB for direct nonlinear Schrödinger equation modeling, and in Python (open-source code on GitHub) for the SINDy and Dominant Balance methods.

B. Experimental Work

Our experimental methodology builds on many years of joint experience in nonlinear fiber optics around the 1550 nm telecommunications band. This expertise leverages the availability of high-quality sources and components for manipulating and detecting picosecond and femtosecond pulses. Specific experimental setups employed tailored approaches, summarized below:

- Development of a new experimental setup based on sideband truncation and feedback to study the ideal dynamics of four-wave mixing.

- Development of an ultrasensitive real-time spectral measurement method.

- Demonstration of active control of supercontinuum generation using a genetic algorithm, in both single-pass fiber and fiber-laser configurations.

- Demonstration of active control of supercontinuum generation applied to multiphoton microscopy.

- Demonstration of a neural network capable of interpreting noise-driven incoherent nonlinear dynamics in fiber propagation (selected by Optica as a 2025 research highlight).

OPTIMAL has led to a number of major advances, both in terms of theoretical/numerical objectives and experimental goals. In particular:

Theoretical and Numerical Studies:

-Data-driven discovery of the underlying physical model governing four-wave mixing in optical fibers, using a noise-adapted variant of the SINDy (Sparse Identification of Nonlinear Dynamics) method.

-Automation of physical intuition for nonlinear propagation in optical fibers using the Dominant Balance method, which combines sparse regression and combinatorial approaches (selected by *Optica* as a 2024 research highlight).

-Emulation of nonlinear propagation equations using neural networks, particularly via an iterative feedforward approach.

Experimental Results:

-Development of a new experimental setup based on sideband truncation and feedback to study the ideal dynamics of four-wave mixing.

-Development of an ultrasensitive real-time spectral measurement method.

-Demonstration of active control of supercontinuum generation using a genetic algorithm, both in single-pass fiber and fiber-laser configurations.

-Demonstration of active control of supercontinuum generation applied to multiphoton microscopy.

-Demonstration of the use of a neural network to interpret noise-driven incoherent nonlinear dynamics during fiber propagation (selected by Optica as a 2025 Research Highlight).

Summary of Publications:

The work carried out within the OPTIMAL project has resulted in significant scientific output:

45 publications in peer-reviewed journals

99 contributions to international peer-reviewed conferences (including many invited conferences)

Publications resulting from OPTIMAL have already been cited 250 times in scientific literature.

The outstanding focus result of OPTIMAL is the development of actively-controlled smart broadband optical sources using evolutionary algorithms. These sources are designed by optimising initial conditions injected into optical fibre in the case of supercontinuum generation, or by optimising the intracavity saturable absorber properties in the case of laser operation. The results in OPTIMAL have attracted broad interest, and future work will continue to further optimise these methods. In addition, the process of supercontinuum generation itself is being further studied to exploit its nonlinear transformation properties as a computational resource.

OPTIMAL aims to develop advanced ultrafast and broadband light sources by bringing the power of machine learning into the mainstream of laser design and ultrafast nonlinear optics. Specific objectives include: (i) designing new approaches to generate broadband light with tailored spectral and temporal properties using machine-learning to optimize initial conditions for nonlinear propagation in optical fibre waveguides; (ii) developing new ultrafast sources based on the Mamyshev regenerative oscillation concept using machine-learning to optimize intra-cavity control of amplitude and phase; (iii) combining the development of these novel sources with fundamentally-oriented studies of the underlying nonlinear propagation dynamics, building on analogies such as cavity hydrodynamics and turbulence, and using machine learning to analyse and interpret experimental data.