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A solution to the Klein Gordon equation for a laser-driven electron is constructed from a superposition of Volkov states. The time-and space-dependent three-dimensional superposition integral can be evaluated analytically for an initial Gaussian momentum distribution when the expression for relativistic energy is expanded in a Taylor series over the scaled initial momenta. The solution preserves many complicated wave-packet dynamics in a strong field, including so-called wave-packet shearing and the formation of multiple peaks when the wave packet spreads to the scale of the driving-field wavelength. The range of applicability of the solution applies to much of the parameter space accessible by current intense ultra-short laser systems.

Electron wave-packet dynamics in a relativistic electromagnetic field: 3-D analytical approximation