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The SD-SPIDER method for the characterization of ultrashort laser pulsesrequires the solution of a nonlinear integral equation of autoconvolution typewith a device-based kernel function. Taking into account the analyticalbackground of a variational regularization approach for solving thecorresponding ill-posed operator equation formulated in complex-valued$L^2$-spaces over finite real intervals, we suggest and evaluate numericalprocedures using NURBS and the TIGRA method for calculating the regularizedsolutions in a stable manner. In this context, besides the complexdeautoconvolution problem with noisy but full data, a phase retrieval problemis introduced which adapts to the experimental state of the art in laseroptics. For the treatment of this problem facet, which is formulated as atensor product operator equation, we derive well-posedness of variationalregularization methods. Case studies with synthetic and real optical data showthe capability of the implemented approach as well as its limitation due tomeasurement deficits.

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