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State-of-the-art petawatt laser beams may be focused down to few-micron spot sizes and can produce violent electron acceleration as a result of the extremely intense and asymmetric fields. Classical fifth-order calculations in the diffraction angle show that electrons, injected sideways into the tightly focused laser beam, get captured and gain energy in the GeV regime. We point out the most favorable points of injection away from the focus, along with an efficient means of extracting the energetic electron with a static magnetic field.

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Relativistic harmonic generation by the scattering of very-high-intensity laser light from fast free electrons is investigated theoretically. A general solution for the trajectory of an electron, moving initially with an arbitrary velocity in a light pulse of arbitrary intensity and polarization, is presented. This solution generalizes the classical analysis of Eberly ͓Progress in Optics, edited by E. Wolf ͑North-Holland, Amsterdam, 1969͒, Vol. 7͔ and that of Sarachik and Schappert ͓Phys. Rev. D 1, 2738 ͑1970͔͒ for the trajectory of an electron initially at rest. The result is then applied to the case of effective harmonic generation in a monochromatic, circularly polarized field under three different initial conditions for the electron, namely, ͑a͒ electron initially at rest, ͑b͒ electron initially moving in the direction of light propagation ͑and opposite to it͒, and ͑c͒ electron initially crossing the radiation beam at right angles. Angular distributions of the harmonics generated by the scattering process are presented in terms of the power scattering cross section in each case. Effects of increasing the light intensity and/or the initial electron speed and/or direction on the angular distributions are discussed. It is found, among other results, that at laser intensities higher than, say 10 18 W/cm 2 , the low-order harmonics are suppressed while the higher-order harmonics are enhanced.