Effects of the curvature and third derivative of the field amplitude on the particle orbit. The field has (a) zero gradient and positive or negative curvature, (b) finite gradient and positive or negative curvature, and (c) zero gradient and positive or negative third derivative at . The blue line denotes the field pattern and the black solid line represents the particle orbit in the x-z plane. In the lower part, the unperturbed orbits are shown for comparison.
Transverse laser field profiles considered in Sec. IV A , i.e., Gaussian and super Gaussian , and B, i.e., and concave hollows.
Comparison between the numerical trajectories for the oscillation center of an electron obtained from the Lie perturbation method up to (I) the first and (II) third orders, and (III) the averaging method up to the third order with respect to ϵ. An electron interacts with a laser beam having an envelope , where and . The black solid line shows the electron trajectory calculated by the full-order equation of motion.
Numerical trajectories for the oscillation center of an electron interacting with a laser field having transverse envelopes (a) and (b) , where and . Blue and red lines show the oscillation center trajectories obtained by the ponderomotive formulae up to the first and third orders, respectively.
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