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 fo r off-axis points and that accounted f o r by the cosinefourth and vignetting effects. A closer examination of the image format ion process fo r real lenses revealed that even an the absence of vignetting a point light source does not uniformly illuminate the aperture, a n effect known as pupil aberration. For example, we found the variation f o r a 16" lens to be as large as 31% f o r a field angle of 10". I n this paper, we critically evaluate the roles of cosine-fourth and vignetting effects and demonstrate the significance of the pupil aberration o n the fall-off in irradiance away f rom image center. The pupil aberration effect strongly depends o n 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 a n d vignetting effects, that mus t be taken in to account in applications that rely heavily o n photometric variation such as shape f rom shading and mosaicing.