2013/01/25 by Daniel Gerth, Bernd Hofmann, Gerth, Daniel +7 · 1 citation
Engineering · Physics and Astronomy · #Advanced X-ray Imaging Techniques #FOS: Physical sciences #Laser-Matter Interactions and Applications #Laser-Plasma Interactions and Diagnostics #Mathematical Physics (math-ph) #Optics (physics.optics) #Photoacoustic and Ultrasonic Imaging #Thermography and Photoacoustic Techniques
paper · pdf · doi:10.48550/arxiv.1301.6061
openalex publication_date 2013/01/25 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
An ill-posed inverse problem of autoconvolution type is investigated. This\ninverse problem occurs in nonlinear optics in the context of ultrashort laser\npulse characterization. The novelty of the mathematical model consists in a\nphysically required extension of the deautoconvolution problem beyond the\nclassical case usually discussed in literature: (i) For measurements of\nultrashort laser pulses with the self-diffraction SPIDER method, a stable\napproximate solution of an autocovolution equation with a complex-valued kernel\nfunction is needed. (ii) The considered scenario requires complex functions\nboth, in the solution and the rhs of the integral equation. Since, however,\nnoisy data are available not only for amplitude and phase functions of the rhs,\nbut also for the amplitude of the solution, the stable approximate\nreconstruction of the associated smooth phase function represents the main goal\nof the paper. An iterative regularization approach is described that is\nspecifically adapted to the physical situation in pulse characterization, using\na non-standard stopping rule for the iteration process of computing regularized\nsolutions. Our approach is illustrated by several case studies for synthetic\nnoisy data and physically realistic complex-valued kernel functions. Based on\nan example with focus on amplitude perturbations, we show that the\nautoconvolution equation is locally ill-posed everywhere. To date, the\nanalytical treatment of the impact of noisy data on phase perturbations remains\nan open question. However, we show its influence with the help of numerical\nexperiments. Moreover, we formulate assertions on the non-uniqueness of the\ncomplex-valued autoconvolution problem, at least for the simplified case of a\nconstant kernel. The presented results and figures associated with case studies\nillustrate the ill-posedness phenomena also for the case of non-trivial complex\nkernel functions.\n