Recent Developments in Inverse Problems - Abstract
Steinmeyer, Günter
Femtosecond and attosecond laser pulses constitute the shortest controllable events in nature, which enables experimental studies of the physical dynamics on time frames down to the duration of a single cycle of the optical field. Despite the enabling character of such pulses, their shortness also gives rise to an inherent problem. As there simply are no faster phenomena available, one can simply not gate the short laser pulses with any such faster phenomenon. The pragmatic solution to this dilemma has been gating the pulse with itself, a method which is called optical autocorrelation. While it is possible to obtain some information on the shortness of the pulse, it is fundamentally impossible to unambiguously reconstruct the intensity envelope of the pulse. Several methods have been invented to overcome this fundamental deficiency. One class of these methods has been termed tomographic pulse characterization, measuring, e.g., spectrally resolved autocorelations. While it is not possible to determine the intensity envelope from such measurements in an analytic way, there exists a one-to-one correspondence between pulse shape and measured trace. Consequently, retrieving the pulse shape from tomographic measurements (aka as FROG) is an inverse problem. Substantial effort has been invested in order to prevent stagnation in reconstruction of the pulse shape an to accelerate numerical solution of this problem. Here I will discuss two fairly novel scenarios, which target current developments in ultrafast laser optics. One of these scenarios is the trend towards ever shorter pulses, approaching the duration of a single cycle of the carrier field. Once pulse durations approach the latter duration, nearly all methods start to fail, and no reliable reconstruction can be done anymore. In collaboration with the group at TU Chemnitz, we have recently devised a method to further push limitation to well beyond the single cycle. Numerical reconstruction of the pulse shape is fairly involved and requites regularization strategies. As an outlook, we also envision characterization of coherence problems in ultrafast optics, which arise when unstable pulse trains are to be characterized. This situation leads to a seemingly unsolvable pulse characterization problem, i.e., to measure the average pulse shape as well as to measure the pulse-to-pulse variation of pulse shapes. Here we devise new methods to retrieve characteristic functions of both phenomena at the same time. Retrieval strategies heavily rely on regularization. We are confident that this joint effort of optical physics together with applied mathematics will be able to shed new light on this admittedly dark spot of ultrashort pulse characterization. vspace1cm noindentbf References: beginenumerate item R. Trebino, “Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses: The Measurement of Ultrashort Laser Pulses” Springer, Heidelberg (2000). item D. Gerth, B. Hofmann, S. Birkholz, S. Koke, and G. Steinmeyer, “Regularization of an autoconvolution problem in ultrashort laser pulse characterization,” Inverse Problems in Science and Engineering bf 22, 245--266 (2014). item S. Birkholz, G. Steinmeyer, S. Koke, D. Gerth, S. Bürger, and B. Hofmann, “Phase retrieval via regularization in self-diffraction-based spectral interferometry,” J. Opt. Soc. Am. B bf 32, 983--992 (2015). item M. Rhodes, G. Steinmeyer, J. Ratner, and R. Trebino, “Pulse-shape instabilities and their measurement,“ Laser Photonics Reviews bf 7, 557-?565 (2013). endenumerate