This paper will review the field of mathematical modeling for metrology. It will consider light optical and electron optical microscopes. The physics and mathematics for these two types are very different. For light optics the two primary mathematical modeling techniques are eigenmode expansion methods which apply only to structures with a linear symmetry like lines and finite element methods which may be applied to general structures. For electron microscopes the principal technique is Monte Carlo trajectory analysis.
For light optical microscopes, models have been applied to reflective and transmitting systems, classical brightfield, confocal, and coherence probe instruments. For scanning electron microscopes, most of the focus has been on secondary electron detector systems and more recently backscattered electron systems.
This paper will also review applications of these models to the problem of acuracy and linearity. And will touch on the complex issue of the inverse scattering problem.