Antenna [aperture] illumination efficiency is a measure of the extent to which an antenna or array is uniformly excited or illuminated.
It is typical for an antenna [aperture] or array to be intentionally under-illuminated or under-excited in order to mitigate sidelobes and reduce antenna temperature.
[1] Antenna [aperture] illumination efficiency is defined as "The ratio, usually expressed in percent, of the maximum directivity of an antenna [aperture] to its standard directivity."
Standard [reference] directivity is defined as "The maximum directivity from a planar aperture of area A, or from a line source of length L, when excited with a uniform-amplitude, equiphase distribution.
"[1] Key to understanding these definitions is that "maximum" directivity refers to the direction of maximum radiation intensity, i.e., the main lobe.
Therefore, illumination efficiency is not a function of angle with respect to the antenna [aperture], but rather is a constant of the aperture for all aspect angles.
However, one can infer that, if an antenna [aperture] were excited [illuminated] uniformly with no phase difference (equiphase) over the entire aperture, then the illumination efficiency would be equal to unity.
It is very typical for an antenna [aperture] to be intentionally under-excited [illuminated] with a "taper" in order to reduce radiation pattern sidelobes and antenna temperature.
In such a design, the maximum directivity is reduced because the full aperture is not being used to the full extent possible, and the illumination efficiency will be less than unity.
"[1] and under effective area of an antenna, IEEE states, "The effective area of an antenna in a given direction is equal to the square of the operating wavelength times its gain in that direction divided by 4π."
Gain is also defined to be less than directivity by the radiation efficiency,