Regularization of an autoconvolution problem in ultrashort laser pulse characterization

Abstract

An ill-posed inverse problem of autoconvolution type is investigated. This inverse problem occurs in non-linear optics in the context of ultrashort laser pulse characterization. The novelty of the mathematical model consists in a physically required extension of the deautoconvolution problem beyond the classical case usually discussed in literature: (i) For measurements of ultrashort laser pulses with the self-diffraction SPIDER method, a stable approximate solution of an autocovolution equation with a complex-valued kernel function is needed. (ii) The considered scenario requires complex functions both, in the solution as well as in the right-hand side of the integral equation. Since, however, noisy data are available not only for amplitude and phase functions of the right-hand side, but also for the amplitude of the solution, the stable approximate reconstruction of the associated smooth phase function represents the main goal of the paper. An iterative regularization approach will be described that is specifically adapted to the physical situation in pulse characterization, using a non-standard stopping rule for the iteration process of computing regularized solutions. The opportunities and limitations of regularized solutions obtained by our approach are illustrated by means of several case studies for synthetic noisy data and physically realistic complex-valued kernel functions. Based on an example with focus on amplitude perturbations, we show that the autoconvolution equation is locally ill-posed everywhere. To date, the analytical treatment of the impact of noisy data on phase perturbations remains an open question. However, we show its influence with the help of numerical experiments. Moreover, we formulate assertions on the non-uniqueness of the complex-valued autoconvolution problem, at least for the simplified case of a constant kernel. The presented results and figures associated with case studies illustrate the ill-posedness phenomena also for the case of non-trivial complex kernel functions.

Regularization Methods for Ill-Posed Problems

“…Take into account L For the SPIDER technology, in particular, the phase function is to be determined from noisy observations of the complex function y, whereas information about the amplitude function can be verified by alternative measurements. In [47] some mathematical studies and a regularization approach for this specific problem have been presented, and further analytic investigations for the specific case of a constant kernel k can be found in [24].…”

“…In a second variant of this example with applications in laser optics, the full data case is exploited, but for complex functions and with an additional kernel. This variety considers a generalized autoconvolution equation motivated by problems of ultrashort laser pulse characterization arising in the context of the self-diffraction SPIDER method, and the reader is referred to the recent paper [47] for physical details and the experimental setting of this problem. In this variant, the focus is on a kernel-based, complex-valued, and full data analog to (58).…”

About a deficit in low-order convergence rates on the example of autoconvolution

“…Hence, the problem is locally ill-posed at any point x † ∈ L 2 C (0, 1). It was formulated in [13] as an open question whether the deautocovolution process remains always instable if only phase perturbations occur. This question is motivated by the laser pulse problem, where a complex-valued measuring tool based kernel function k(s, t) is added to the integral equation (3.1), but the amplitude function A = |x † | as part of the solution x † (t) = A(t) e iφ † (t) , 0 ≤ t ≤ 1, can be measured and only the phase function φ † is to be determined from observed y ∈ L 2 C (0, 2).…”

“…The question of ill-posedness must be reset in the case of the complex-valued autoconvolution equations motivated by an application from laser optics (cf. [13]). We will show in Section 3 that both locally well-posed and ill-posed situations occur for such complex-valued problems with full data in dependence of the domain D(F ) under consideration.…”

“…The complex-valued and full data analog to equation (2.1) was motivated by problems of ultrashort laser pulse characterization arising in the context of the self-diffraction SPIDER method, and we refer to [13,28] for physical details. Taking into account L Finally, we note that well-posedness situations for the real autoconvolution operator…”