A General Method for the Determination of Matrix Coefficients for High Order Optical System Modelling

摘要

The non-linear transformations incurred by the rays in an optical system canbe suitably described by matrices to any desired order of approximation. Insystems composed of uniform refractive index elements, each individual rayrefraction or translation has an associated matrix and a succession oftransformations correspond to the product of the respective matrices. Thispaper describes a general method to find the matrix coefficients fortranslation and surface refraction irrespective of the surface shape or theorder of approximation. The choice of coordinates is unusual as the orientationof the ray is characterised by the direction cosines, rather than slopes; thisis shown to greatly simplify and generalise coefficient calculation. Twoexamples are shown in order to demonstrate the power of the method: The firstis the determination of seventh order coefficients for spherical surfaces andthe second is the determination of third order coefficients for a toroidalsurface.