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

Abstract

We present an algorithm based on numerical techniques that have become standard for solving large-scale nonlinear integral equations: Newton's method, homotopy continuation and the multilevel method, to solve the inversion problem that appears when retrieving the electric field of an ultrashort laser pulse from 2d experimental data, the trace, via FROG or d-scan. Here we apply the solver to SHG-FROG and specify the necessary modifications for similar integrals. The electric field is discretised with a piecewise constant function on a uniform grid and the integral equation coarse-grained on pixelised grids. Every pixel surface average corresponds to a single equation of a 4th order polynomial system. The system is over-determined and reduced to square form through random matrices, similar to other approaches in stochastic optimisation and subsequently solved with Newton's method and homotopy continuation with altering random matrices along real continuation paths towards an optimal solution. Adaptive Tikhonovtype regularization enables smoothing 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 regularisation parameter. The algorithm paves the way for Quasi-Newton methods and is applicable to determine real solutions of other polynomial systems.