Overview
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ABSTRACT
The optical matrix offers a very efficient mathematical tool to find an image quickly through an optical system and discern its properties easily before further study is undertaken. This article describes how this approach is learnt from basic optical components (mirror, diopter, lens) and then gives more rapid methods using an equivalent matrix. Applied examples are given for each case to facilitate learning.
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Read the articleAUTHORS
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Christophe LABBÉ: Lecturer at the University of Caen - Normandie Univ, UNICAEN, IUT de Caen, Département Mesures Physiques, Caen, France - Normandie Univ, ENSICAEN, UNICAEN, CEA, CNRS, CIMAP, Caen, France
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Benoît PLANCOULAINE: Lecturer at the University of Caen - Lecturers at Caen Normandie University - Normandie Univ, UNICAEN, IUT de Caen, Département Mesures Physiques, Caen, France - Normandie Univ, UNICAEN, INSERM, BioTICLA, Caen, France
INTRODUCTION
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.
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KEYWORDS
optical matrix | lens | achromatic | optical systems | aspheric
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Optics and photonics
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Matrix optics for the analysis of centered systems
Bibliography
- (1) - Base de données des indices de réfraction : Mikhail Polyanskiy, - 2008-2016, http://refractive index.info/ .
- (2) - Tout Savoir sur les Lentilles Asphériques,...
Works
Optics Learning by Computing, with Examples Using Mathcad, MATLAB, Mathematica, and Maple, second edition, Karl Dieter Moller, ISBN: 9780387261683, 454 pp, (2007)
Eugene Hecht, Optics, 5th edition, Adelphi University, pages 261-266, 2015
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Course, with extensions on the telescope/microscope in matrix optics, 3x3 matrix theory for diffractive...
Scientific articles
Joseph (S.) Choi and John (C.) Howell, "Paraxial ray optics cloaking," Opt. Express 22, 29465-29478 (2014)
Gatinel (D.), Haouat (M.), Hoang-Xuan (T.), "Étude des paramètres permettant la description mathématique de l'asphéricite cornéenne", J FR. Ophthalmol; 25, 1, 81-90 (2002)
Halbach (K.) " Matrix representation of Gaussian...
University courses
– Geometrical matrix optics course, introduction to interference and diffraction phenomena: Yann Vaills, Université d'Orléans (2011)
Software tools (non-exhaustive list)
Ray tracing software :
The calculation principle of these programs is based on the intersection between the ray (characterized by the 2 components of its wave vector) and the nearest surface. A calculation is then performed to find the new wave vector using Descartes' laws after reflection and/or refraction on the surface, up to the next nearest surface encountered. Here are...
Websites
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Laser optics and Gaussian beams – Course, exercise and example applications, S. Forget (2015)
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Choosing the right telescope, how to get started in astronomy
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