Conditions in linear optics for sharp and undistorted retinal images, including Le Grand's conditions for distant objects

- Conditions are derived for sharp retinal images of distant objects.
- Conditions are presented in particular for sharp undistorted images.
- The condition for unrotated images is also presented.
- The results hold for an eye and for an eye plus optical instrument.
- The conditions are on the transference of linear optics.

In 1945 Yves Le Grand published conditions, now largely forgotten, on the 4 × 4 matrix of an astigmatic eye for the eye to be emmetropic and an additional condition for retinal images to be undistorted. The conditions also applied to the combination of eye and the lens used to compensate for the refractive error. The conditions were presented with almost no justification. The purpose of this paper is to use linear optics to derive such conditions. It turns out that Le Grand's conditions are correct for sharp images but his condition such that the images are undistorted prove to be neither necessary nor sufficient in general although they are necessary but not sufficient in most situations of interest in optometry and vision science. A numerical example treats a model eye which satisfies Le Grand's condition of no distortion and yet forms elliptical and noncircular images of distant circles on the retina. The conditions for distant object are generalized to include the case of objects at finite distances, a case not examined by Le Grand.