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

TL;DR

This paper introduces a novel numerical algorithm combining Newton's method, homotopy continuation, and random projection to accurately retrieve ultrashort laser pulse electric fields from 2D intensity measurements, even with noisy data.

Contribution

The authors develop a new approach transforming the integral inversion problem into a polynomial system solved in the time domain, enabling real solutions and noise robustness.

Findings

Effective in retrieving laser pulses from FROG measurements

Handles noisy data with adaptive Tikhonov regularization

Provides a novel method for solving difficult polynomial systems

Abstract

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.