Solving nonlinear 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 two-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 discretized as an overdetermined polynomial system and evaluated through list autocorrelations. 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 an alternative 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, and 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.