Longitudinal fields in cylindrical and spherical modes

The structure and properties of the longitudinal field in electromagnetic waves have been studied in the case of the superposition of plane waves, in cylindrical and spherical modes. It has been ascertained that the phase velocity of the longitudinal field is directed along the field vector. Another feature is the pi/2 phase shift of the longitudinal electric field relative to the magnetic field preventing the energy transport by this field. The analytical method for the theoretical description of the radially and azimuthally polarized beams is presented. It does not have any inherent contradictions. The longitudinal field is formed with the radial component of the wavevectors directed to the axis of the beam. Coherent superposition of two equal contradirectional radially or azimuthally polarized beams gives a special point in the waist. All tree field components including the longitudinal field equal zero at this point. In spherical high order modes the radius of the 'dark area' in the centre of the sphere, where the field is close to zero, increases with the mode number.