Margenberg, Nils

The subject of this thesis is the efficient simulation of nonlinear optical phenomena, especially frequency mixing processes. Unlike linear optics, where the response of the material to light remains constant regardless of the light's intensity, in nonlinear optics, this response varies as a function of the light's intensity. Nonlinear optical phenomena have paved the way for significant scientific innovations, e.g. the development of novel optical sources, and have become central to a wide range of technological applications. This work is located in the emerging field of scientific machine learning and contributes to the efficient simulation of nonlinear optical phenomena by combining the rigor of finite element methods with the empirical strength of deep neural networks. Specifically, we focus on frequency mixing processes in the context of THz generation in periodically poled nonlinear crystals through Quasi-Phase-Matching. We develop a mathematical model and numerical simulation techniques in the time domain to simulate complex nonlinear optical processes without the usual slowly varying envelope approximation. The accuracy is illustrated through numerical simulations and comparisons to experimental data. The simulations elucidate the THz generation in periodically poled Lithium Niobate (PPLN), including optical harmonic generation. This thesis introduces a novel approach for solving optimal Dirichlet boundary control problems of nonlinear optics using deep learning. For computing high-resolution approximations of the solution to the nonlinear wave model, we propose higher order space-time finite element methods combined with collocation techniques, ensuring C^l-regularity in time of the global discrete solution. The simulation data trains solution operators that effectively leverage the higher regularity of the training data. The operators, represented by Fourier Neural Operators, are used as the forward solver in the optimal Dirichlet boundary control problem. This approach demonstrates an innovative integration of traditional numerical methods with data-driven techniques. The proposed algorithm is implemented and tested on high-performance computing platforms, with a focus on efficiency and scalability. We showcase its effectiveness on the problem of generating Terahertz radiation in PPLN, optimizing the parametrization of the optical input pulse to maximize the yield of 0.3THz-frequency radiation. By exploiting the periodic layering of the crystal, the networks are trained to learn propagation through one period of the layers. The recursive application of the network onto itself yields an approximation to the full problem. The results indicate a significant speedup compared to classical methods and, when compared to experimental data, demonstrate the potential to revolutionize optimization in nonlinear optics. The proposed approach of data-driven acceleration contributes to a paradigm shift towards more efficient and innovative problem-solving methods in the realm of nonlinear optics and beyond. In addition, we review a promising approach to showing the well-posedness of the nonlinear Maxwell equation and show how it can be applied in the settings we consider throughout this thesis. To this end, the nonlinear wave equations are rewritten as a first-order system in space and time. The well-posedness is ensured by a combination of the abstract theory for evolutionary problems by R. Picard and fixed point arguments.