On cosine-fourth and vignetting effects in real lenses

This paper has been prompted by observations of disparities between the observed fall-off in irradiance for off-axis points and that accounted for by the cosine-fourth and vignetting effects. A closer examination of the image formation process for real lenses revealed that even in the absence of vignetting a point light source does not uniformly illuminate the aperture, an effect known as pupil aberration. For example, we found the variation for a 16 mm lens to be as large as 31% for a field angle of 10°. In this paper, we critically evaluate the roles of cosine-fourth and vignetting effects and demonstrate the significance of the pupil aberration on the fall-off in irradiance away from image center. The pupil aberration effect strongly depends on the aperture size and shape and this dependence has been demonstrated through two sets of experiments with three real lenses. The effect of pupil aberration is thus a third important cause of fall in irradiance away from the image center in addition to the familiar cosine-fourth and vignetting effects, that must be taken into account in applications that rely heavily on photometric variation such as shape from shading and mosaicing