See our recent paper on the spatio-temporal control of the energy of a laser pulse using non-linear medium. The idea is that the nonlinear index of refraction of the material can be interpreted as an intensity-dependent focal length for a laser pulse. The task is then to “design” a laser pulse power spectrum based on the location where the pulse is to be focused. This is a wicked cool idea and it’s an iteration on the flying focus that circumvents the need for complicated fabrication of novel optics by leveraging natural nonlinearity of material. See the paper for detailed calculations and explanations:

Tanner Simpson, D. Ramsey, P. Franke, N. Vafaei-Najafabadi, D. Turnbull, D. H. Froula, and J. P. Palastro, Nonlinear Spatiotemporal Control of Laser Intensity, Optics Express, 28, 38516-38526 (2020) https://doi.org/10.1364/OE.411011; arXiv:2009.11783 [physics.optics]