Far field's starting distance

The far field of an antenna is generally considered to be the region where the outgoing wavefront is planar and the antenna radiation pattern has a polar variation and is independent of the distance from the antenna. Hence, to generate a locally plane wave in the far field the radial component of the electric field must be negligible compared to the transverse component. Also, the ratio of the electric and the magnetic far fields should equal the intrinsic impedance of the medium. These two requirements must hold in all angular directions from the antenna. The radial and the transverse components of the fields are space dependent so to determine the starting distance of the far field we need to examine the simultaneous satisfaction of these two properties for all θ and φ angular directions, where θ is the angle measured from z-axis and φ is the angle measured from the x-axis. The objective of this paper can be summarized in three points: First, this paper intends to illustrate that 2D² / λ formula, where D is the maximum dimension of the antenna and λ is the operating wavelength, is not universally valid, it is only valid for antennas where D " λ. Second, this paper intends to compute a more specific constraint so instead of D >> λ we compute a threshold for D after which the 2D² / λ formula applies. Third, this paper intends to properly interpret D in the formula 2D² / λ when the antenna is operating over an imperfect ground plane. In this paper, we do not use 2D² / λ for antennas operating over an imperfect ground instead we use a formula which depends on the transmitting and receiving antenna's heights over the air-Earth interface.