Variational regularization of complex deautoconvolution and phase retrieval in ultrashort laser pulse characterization

TL;DR

This paper develops a variational regularization framework for solving complex autoconvolution and phase retrieval problems in ultrashort laser pulse characterization, demonstrating stable numerical solutions with synthetic and real data.

Contribution

It introduces a variational regularization approach for complex autoconvolution and phase retrieval problems, including new numerical procedures and well-posedness analysis.

Findings

Numerical methods using NURBS and TIGRA effectively solve the regularized equations.

The approach is capable of handling noisy, full data in complex deautoconvolution.

Limitations arise due to measurement deficits in experimental data.

Abstract

The SD-SPIDER method for the characterization of ultrashort laser pulses requires the solution of a nonlinear integral equation of autoconvolution type with a device-based kernel function. Taking into account the analytical background of a variational regularization approach for solving the corresponding ill-posed operator equation formulated in complex-valued $L^2$-spaces over finite real intervals, we suggest and evaluate numerical procedures using NURBS and the TIGRA method for calculating the regularized solutions in a stable manner. In this context, besides the complex deautoconvolution problem with noisy but full data, a phase retrieval problem is introduced which adapts to the experimental state of the art in laser optics. For the treatment of this problem facet, which is formulated as a tensor product operator equation, we derive well-posedness of variational regularization methods. Case studies with synthetic and real optical data show the capability of the implemented approach as well as its limitation due to measurement deficits.