Solving non-linear integral equations for laser pulse retrieval with Newton's method

We present an algorithm based on numerical techniques that have become standard for solving nonlinear integral equations: Newton's method, homotopy continuation, the multilevel method and random projection to solve the inversion problem that appears when retrieving the electric field of an ultrashort laser pulse from a 2-dimensional intensity map measured with Frequency-resolved optical gating (FROG), dispersion-scan or amplitude-swing experiments. Here we apply the solver to FROG and specify the necessary modifications for similar integrals. Unlike other approaches we transform the integral and work in time-domain where the integral can be discretised as an over-determined polynomial system and evaluated through list auto-correlations. The solution curve is partially continues and partially stochastic, consisting of small linked path segments and enables the computation of optimal solutions in the presents of noise. Interestingly, this is a novel method to find real solutions of polynomial systems which are notoriously difficult to find. We show how to implement adaptive Tikhonov-type regularization to smooth the solution when dealing with noisy data, we compare the results for noisy test data with a least-squares solver and propose the L-curve method to fine-tune the regularization parameter.