Vignetting of light beams for objects placed at a finite distance from an optical system

The paper analyses the vignetting phenomenon for optical systems that have the objects placed at finite distances from them, e.g. microscopes. Three possible definitions, only two of them existing in the literature, are used and discussed in the paper. The characteristic circles and zones in the object plane are ascertained with regard to the vignetting, that is, with regard to the stop of the incident beams produced by the various diaphragms of the optical system, i.e. by the entrance pupil and by the entrance stop. These zones are mathematically analyzed and a geometric vignetting coefficient, with two expressions, a linear, classical, easy-to-use one, and a non-linear, more precise one, that we propose, are obtained. We demonstrate that the latter is better one from the point of view of the precision of the mathematical model that characterizes the physical phenomenon, while using it proves not to be essentially more difficult than the previous, less precise one.