Effective corneal patch of an astigmatic heterocentric eye

The pupil admits to the back of the eye only some of the light arriving from a point in space. As a result only a portion of the cornea is involved when an eye views the point; it is the effective corneal patch for that point. The location, size and shape of the patch are of interest for corneal refractive surgery inter alia. Previous studies have used geometrical optics and a simple model eye (a naked eye with a spherical, single-surface, centred cornea and a concentric circular pupil). Even for the simplest situations geometrical optics provides only implicit formulae which give little insight into relationships and require numerical solution.

To show how linear optics leads to explicit formulae that estimate the geometry of the effective corneal patch in a very wide range of situations. The eye is not restricted to a single refracting surface; the surfaces may be astigmatic and decentred or tilted and the pupil may be decentred and elliptical. The eye may contain implants and it may be looking through a spectacle lens or other optical instrument which may also contain astigmatic and decentred surfaces.

Linear optics is used to provide general formulae for the geometry of the corneal patch. An appendix illustrates application to some particular cases.

General formulae are obtained for the location and geometry of the effective corneal patch for object points that may be near or distant. Formulae are presented in particular for the special case of the naked eye and the case in which all surfaces are spherical and centred on a common axis. Numerical examples in the appendix allow comparison of results obtained via geometrical and linear optics.

In using linear optics one sacrifices some accuracy at increasing angles away from the longitudinal axis but there is considerable gain in the complexity and range of problems that can be tackled, and the explicit formula one obtains clearly exhibit relationships among parameters of clinical relevance.