Symplecticity and relationships among the fundamental properties in linear optics

W F Harris

doi:10.4102/aveh.v69i1.119

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)

Highlights

Symplecticity is of profound significance to modern science and to optometry in particular
There are several good sources that deal with symplecticity[1,2,3,4] they tend to be mathematically sophisticated and not readily accessible for most people working in visual optics
As for symplectic matrices one can define an augmented Hamiltonian matrix to be a Hamiltonian matrix with an additional right-hand column and an additional bottom row

Summary

Introduction

Symplecticity is of profound significance to modern science and to optometry in particular. - WF Harris Symplecticity and relationships among the fundamental properties in linear optics where I is an identity matrix, B is the inverse of. Proofs are given elsewhere2-4.) This means a symplectic matrix S is never singular (that is, det S ≠ 0 ) and its inverse S−1 always exists Premutiplying both sides of Equation 17 by S−T and postmultiplying by S−1 one obtains. The sequence of submatrices on the left-hand sides of Equations 40 to 43 the following may be helpful: beginning diagonally across in from the submatrix in question one writes the submatrices in cyclical order going clockwise for the block-diagonal matrices and anticlockwise for the others There is another class of matrices which is important in modern science and which bears a surprising relationship to symplectic matrices: it is the class of Hamiltonian matrices.

T M O P

Concluding remarks