title: Optical magnification matrix: near objects and strongly oblique incidence.
abstract: OBJECTIVE
  In every day spectacle use relevant parts of a spectacle lens involve large gaze angles and lie outside the paraxial region. The purpose here is to derive a general equation for the local 2 x 2 magnification matrix of an arbitrary K-surface optical instrument in front of an eye for any given gaze direction. The general case comprises a finite object distance as well as astigmatic surfaces and strongly oblique incidence angles of the involved rays.
  
  
  METHODS
  A 2 x 2 stepalong method for the contribution factors to the magnification matrix is developed that allows to treat oblique rays, which propagate through a system possessing tilted refracting interfaces. Extensive use is made of the wave-tracing method to describe propagating wavefronts by 2 x 2 vergence matrices for the corresponding general oblique case.
  
  
  RESULTS
  An analytical 2 x 2 matrix as a function of the ray geometry and the dioptric power matrices of the interfaces in the intersection points is given explicitly. The expression is an intuitive product representation in which all the usual factors occur, which also in the paraxial case would correspond to the gaps between two subsequent surfaces. Additionally, however, the result contains now exact matrix-valued correction factors, which take the oblique incidences at each interface into account, and which would not be present in the paraxial case.
  
  
  CONCLUSIONS
  The result is a generalization of any of the formerly known results for the magnification matrix in the paraxial case. The treatment is complete in the sense that it covers any situation that can be encountered when light propagates through an optical system with a finite number of refracting surfaces to an eye. In particular, the oblique case is treated exactly even if the chief ray lies beyond the domain in which the transference theory of linear optics could be applied.
authors.affiliation: Research and Development, Rodenstock GmbH, München, Federal Republic of Germany. wolfgang.becken@rodenstock.de
authors.forename: Wolfgang
authors.initials: W
authors.lastname: Becken
authors.forename: Helmut
authors.initials: H
authors.lastname: Altheimer
authors.forename: Gregor
authors.initials: G
authors.lastname: Esser
authors.forename: Werner
authors.initials: W
authors.lastname: Mueller
authors.forename: Dietmar
authors.initials: D
authors.lastname: Uttenweiler
mesh_terms: D005139:Eyeglasses; D006801:Humans; D008433:Mathematics; D055095:Optics and Photonics; D012029:Refraction, Ocular / Q000502:physiology*