Optical matrix applied to the centered systems

Geometrical optics is the first approach to a model describing the trajectory of light energy, more commonly known as a "light ray". These rays are used to construct, with the help of drawings, the final image of an object through optical systems (diopter, mirror, lens). When an optical system incorporates a large number of optical elements and becomes more complex, the construction of rays more or less distant from the optical axis can be tedious, if not impossible. If these rays remain close to the center of the optical systems, the use of so-called "conjugation" formulas can help calculate two essential parameters: the position and size of the image in relation to its object. Here again, however, calculations can be time-consuming, cumbersome and prone to error. Both approaches are therefore inappropriate.

The matrix formalism applied to geometrical optics, more commonly known as matrix optics, is much simpler to use. This mathematical tool provides easy access to the same properties, especially as it can be used on a simple calculator. As a preliminary to an in-depth study, it offers the advantage of a quick and inexpensive first approach using equivalent optical systems. Although this approach is deduced directly from the theory of electromagnetism, it is applied within the framework of an approximation called paraxial optics, based on the geometry of light rays evolving at the centers of optical systems. It has its own limitations, which can be overcome by more sophisticated computational means using ray tracing (or "throwing") software, for example. Nevertheless, this matrix formalism remains highly effective for guiding the initial design of optical systems, as it fits in perfectly with the miniaturization of optical devices.

This article begins by presenting the background to matrix optics, and its principle of calculation based on two-dimensional matrices. These matrices are then calculated and associated with each basic optical element, such as the propagation of a ray, the crossing of a diopter or the reflection on a mirror. Based on an inventory of these associated matrices, concrete examples are given of systems made up of two diopters (plano-convex, biconvex lenses) to determine focal length, for example. A second part then shows the method of calculation based on matrices of equivalent systems (matrices related to conjugate, principal and focal points), facilitating a preliminary study of even more complex systems, and allowing us to set out, in a final part, a method of global calculation on more complex centered optical systems.

A glossary and a table of symbols used are presented at the end of the article.