Symplecticity and relationships among the fundamental properties in linear optics

Abstract

Because of symplecticity the four   fundamental first-order optical properties of an optical system are not independent.  Relationships among them reduce the number of degrees of freedom of a system’s   transference from 16 to 10.  There are many such relationships, they are not easy to remember, they take many forms and they are often needed in derivations.  The purpose of this paper is to provide in one place a comprehensive collection of those that have proved useful in linear optics generally and in the context of the eye particularly.  The paper also offers aids to memorizing some of the results, derives most of them and along the way introducesthe basic notions underlying symplecticity.  The relationship to another important class of matrices, the Hamiltonian matrices, is discussed together with their potential role in statistical analysis of the eye.  Augmented symplectic matrices are also defined and their relationship to augmented Hamiltonian matrices described.  An appendix gives numerical examples of symplectic and Hamiltonian matrices and shows how they may be recognized and constructed.  (S Afr Optom 2010 69(1) 3-13)