Far-field distance of antenna and a wave phase In a following reference [David M. Pozar 2012, 4ed, Microwave Engineering, p 661], it is told that the far field distance of a relatively large antenna is given from the formula
\begin{equation}
R_{ff}=\frac{2D^2}{\lambda}
\end{equation}
where D is a maximum dimension of an antenna, $\lambda$ is a wavelength. And it is told that:"This result is derived from the condition that the actual spherical wave front radiated by
the antenna departs less than $\pi/8 =22.5^\circ$ from a true plane wave front over the maximum
extent of the antenna". Does this mean that we have to compare the "far-field" component of electric field with a "near-field" component and at phase $\pi/8$ they are equal or something else? I think that I miss the point.
 A: The condition on the far field in the free space is obtained from wave equations. First of all, the equation on the retarded potentials is
\begin{equation} \label{eq:dAlembert}
 \Box{\vec{A}=-\mu_0 \vec{j}}
\end{equation}
where the temporal part of of potential is $e^{j \omega t}$, so we can get the following equation assuming $k = \omega /c$
\begin{equation} \label{eq:Helmholtz}
 \Delta\vec{A}+  k^{2}\vec{A} = -\mu_0\vec{j}
\end{equation}
Next we solve this equation assuming Green function formalism.
\begin{equation}
 G(\vec{r}) = \frac {-e^{\pm ikr}}{4\pi r}
\end{equation}
After a long calculation and using Laurenz jauge, we can achieve precise equations on electric and magnetic field. For exemple, for the electric field we have
\begin{equation}
 \vec{H} = -\frac{1} {4 \pi} 
 \int \ e^{ik|\vec{r}-\vec{r}\,'|} \frac {1} {|\vec{r}-\vec{r}\,'|^2}  (-\frac{\vec{r}-\vec{r}\,'} {|\vec{r}-\vec{r}\,'|}  + ik (\vec{r}-\vec{r}\,') ) \times \vec{j}(\vec{r}\,') \, d^3\vec{r}\,'
\end{equation}
We are interested in the term $ e^{ik|\vec{r}-\vec{r}\,'|} $, where the Taylor series of the following function is
\begin{equation} \label{Eq:Abs_r-r'}
 |\vec{r}-\vec{r}\,'| = \sqrt{(\vec{r}-\vec{r}\,')\cdot(\vec{r}-\vec{r}\,')}\approx r (1-\vec{e}_r \vec{r}\,'/r +\frac{{\vec{r}\,'}^2} {2 r^2})
\end{equation}
In order to decrease the last term in the previous equation we have the far field criteria:
\begin{equation}
  \frac{{r\,'}^2} {2 r^2} < \lambda/16
 \end{equation}
or equally, $$phase = k\frac{{r\,'}^2} {2 r^2}  < \frac{\pi}{8}$$.
It is well explained in [p83, A.B. Smolders, H.J. Visser, U. Johannsen, Modern Antennas and Microwave Circuits, September 2020, Eindhoven University of Technology].
