Dipole Radiation Effected by the Plane Earth

Method of operators is utilized effectively to simplify the rigorous integral expression for the Hertz vector, as well as the relation between Hermite polynomials and their integral expressions under the condition that k R ≫1, where R is the distance from the dipole to the point of observation, and k is the wave number for the air. The obtained formulas for the Hertz vector of the radiation field hold without restrictions on the earth's refractive index and heights of the dipole or the point of observation, except the assumption that k R ≫1, for all its simplicity.