Backward canonical energy flow in the near field of three non-paraxial 2D beams

In this paper, several 2D non-paraxial light fields near the initial plane (sinc beam, Gaussian beam, and Hankel beam) are considered. These TE-polarized beams are shown theoretically and numerically to have a reverse canonical energy flux in the initial plane. The negative value of the longitudinal projection of the canonical flux vector means that the longitudinal derivative of the phase is negative, and hence, the local wave vector is directed against the positive direction of the optical axis. The reverse flux occurs in the presence of evanescent waves. Although running along the initial plane, evanescent waves themselves do not transfer energy along the optical axis perpendicular to the initial plane; they participate in the formation of the reverse flux. Near the initial plane, the amplitude of the light field consists of a linear combination of amplitudes of the propagating and evanescent waves. Therefore, evanescent waves participate in the formation of the phase function of the complex amplitude of the entire field and affect the sign of the longitudinal phase derivative. For a sinc beam, the backflow occurs in the first intensity side lobes, where the standing evanescent wave has odd antinodes.