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 and random projection to solve inversion problem appears when retrieving electric field of 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 solver FROG specify necessary modifications similar integrals. Unlike other approaches transform work in time domain where can be discretized as overdetermined polynomial system evaluated through list autocorrelations. The solution curve is partially continues stochastic, consisting small linked path segments enables computation optimal solutions presents noise. Interestingly, this alternative method find real systems, which are notoriously difficult find. show how implement adaptive Tikhonov-type regularization smooth dealing noisy data, compare results test data least-squares propose L-curve fine-tune parameter.