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We present an algorithm based on numerical techniques that have becomestandard for solving nonlinear integral equations: Newton's method, homotopycontinuation, the multilevel method and random projection to solve theinversion problem that appears when retrieving the electric field of anultrashort laser pulse from a 2-dimensional intensity map measured withFrequency-resolved optical gating (FROG), dispersion-scan or amplitude-swingexperiments. Here we apply the solver to FROG and specify the necessarymodifications for similar integrals. Unlike other approaches we transform theintegral and work in time-domain where the integral can be discretised as anover-determined polynomial system and evaluated through list auto-correlations.The solution curve is partially continues and partially stochastic, consistingof small linked path segments and enables the computation of optimal solutionsin the presents of noise. Interestingly, this is a novel method to find realsolutions of polynomial systems which are notoriously difficult to find. Weshow how to implement adaptive Tikhonov-type regularization to smooth thesolution when dealing with noisy data, we compare the results for noisy testdata with a least-squares solver and propose the L-curve method to fine-tunethe regularization parameter.

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