Distortions in focusing laser pulses due to spatio-temporal couplings – An analytic description

In ultra-short laser pulses, small changes in dispersion properties before the final focusing mirror can lead to severe pulse distortions around the focus and therefore to very different pulse properties at the point of laser-matter interaction yielding unexpected interaction results. The mapping between far and near-field laser properties intricately depends on the spatial and angular dispersion properties as well as the focal geometry. For a focusing Gaussian laser pulse subject to angular, spatial, and group delay dispersion, we derive analytical expressions for its pulse-front tilt, duration, and width from a fully analytic expression for its electric field in time-space domain. This expression is not only valid in and near the focus but along the entire propagation distance from the focusing mirror to the focus. Together with expressions relating angular, spatial, and group delay dispersion before focusing at an off-axis parabola to the respective values in the pulse’s focus, these formulas are used to show in example setups that pulse-front tilts of lasers with small initial dispersion can become several ten degrees large in the vicinity of the focus while being small directly in the focus. The formulas derived here provide the analytical foundation for observations previously made in numerical experiments. By numerically simulating Gaussian pulse propagation and measuring properties of the pulse at distances several Rayleigh lengths off the focus we verified the analytic expressions.