Brightness distribution in the frame area
Brightness of the image produced by a lens is not even - it decreases toward the frame edges. Further from the image center (corresponding to the lens optical axis) - higher drop of the brightness. Brightness distributions is determined as relation of the brightness at the frame edge to brightness at the center, and it is measured in percents. Usually brightness drops faster in wide-angle lenses, Formula based purely on the geometrical factors is following:
E'ω = E'0•cos4ω
where ω - angle between the lens optical axis and main beam coming to the point where brightness drop is measured; E'0 - brightness in the image center; E'ω - brightness in the measured point (at the frame edge, for example).
Moreover, brightness at the edges drops due to vignetting of the slanting light beams by the lens housing or by diaphragm.
In the result, brightness drop at the frame edge of wide-angle lenses may reach significant values. For example, 16mm OKC1-16-1 purposed for 16x22mm frame size has brightness at the frame edge only 14% compared to the brightness in the center. For 18mm OKS1-18-1 lens this value equals to 30%.
When super wide angle lenses are designed, special measures should be applied to reduce brightness drop. Method of such calculations was introduced by professor M.M.Rusinov. His 28mm "Kinorussar-10A" lens purposed to shoot wide-screen movies (52.5x23mm frame size) gives 52% edge brightness.
Thanks to features of the human eye, we don't notice brightness drop at the large screen edges unless it is really significant. But in some cases a DP should take the brightness drop into consideration (especially during shots on color film with low sensitivity range, or when wide angle lens is used), and compensate it with extra light if possible to get more evenly illuminated negative image density.