Art wrote:
"The three phases of radiation are the current application and the
electron emission, formation of the near field and finally formation of
the far field."

The questioner asked: "If I want to compute the voltage induced on the
antenna by the field, do I add the two E field components or do I
compute the norm of the vector on the RHS of eqn (II) ?"

Art replied:
"Gain can mean many things."

Terman defines on page 870 of his 1955 opus:
"The extent of such concentration relative to that of some standard
antenna, termed the directive gain, is defined quantitatively as the
ratio of power that must be radiated by the comparison antenna to
develop a particular field strength in the direction of maximum
radiation to the power that must be radiated by the directional antenna
system to obtain the same field strength in the same direction."

Kraus shows how to handle arrays of point sources. All the math is
included.

Kraus wrote on page 12 of the 3rd edition of "Antennas":
"Antennas convert electrons to photons or vice versa."

If the questioner draws his information from Terman and Kraus, he won`t
err.

Best regards, Richard Harrison, KB5WZI