Radiometric versus geometric, linear, and nonlinear vignetting coefficient.

We analyze the vignetting phenomenon both for optical systems with objects placed at finite distances and for systems with objects at infinity. Four of the possible definitions of the vignetting coefficient k, only two of them existing in the literature, are discussed. We propose two new definitions, i.e., a nonlinear geometric coefficient that is, in part, an analytical model of the vignetting characterization using optical software and a radiometric vignetting coefficient. The object space of each type of optical systems is studied first, defining its characteristic light circles and cones. Several simplifying assumptions are made for each of the two cases considered to derive analytical equations of the vignetting coefficient and thus to determine the best definition to be used. A geometric vignetting coefficient with two expressions, a linear classical and easy-to-use one and a nonlinear, that we propose for both types of systems is obtained. This nonlinear geometric vignetting coefficient proves to be more adequate in modeling the phenomenon, but it does not entirely fit the physical reality. We finally demonstrate that the radiometric vignetting coefficient we define and derive as a view factor for both types of optical systems is the most appropriate one. The half vignetting level, necessary in most optical design procedures to obtain a satisfactory illumination level in the image plane, is also discussed.