The fact that a focused light beam has a longitudinal component was shown for example by W. Carter: J. Opt. Soc. Am. 62
, 1195 (1972); Expression (1) was derived by Louisell using a perturbation analysis in Phys. of Quant. Elec.
Vol. III, ed. by S. Jacobs, M. Scully, M. Sargent, C. Cantrell, p. 378. See also M. Lax, W. Louisell, W. McKnight: Phys. Rev. A 41
, 3727 (1975). In their paper they concentrated on the importance of the longitudinal components of a laser beam interacting with a plasma to accelerate electrons. In the present paper we consider the case of fast electrons interacting with a laser pulse. Equation (12b) follows from the works of Carter and of Cicchitelli et al. by using the asymptotic expression for Bessel functions in the limit kz ≫ 1, which is the case in our problemGoogle Scholar