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In order to investigate ponderomotive force in the relativistic regime, the interaction of ultraintense laser pulses with free electrons in vacuum is studied both theoretically and numerically. Various expressions for the electromagnetic field of the laser in the case of a Gaussian transverse profile are given, which take into account corrections to the monochromatic paraxial approximation, and the effects of finite pulse duration. A detailed demonstration of relativistic ponderomotive force (RPF) is established which makes apparent the domain of validity of this concept. Computer simulations are carried out using a three-dimensional test-particle code. They show the importance of the correct description of the fields, and confirm the domain of validity of the RPF which is 1-vₙ/c1/kw₀, where vₙ is the component of the electron velocity parallel to the laser propagation direction, c is the velocity of light, k is the laser wave vector, and w₀ the beam waist at focus. Outside of this domain, the electron motion is more complicated, with a high sensitivity on the initial distance from the laser propagation axis and a relatively low energy gain.
Quesnel et al. (Tue,) studied this question.