Stigmatic optical systems

There would appear to be little disagreement on what constitutes an astigmatic system in the case of a thin lens: the cylinder is not zero. A spherical thin lens is stigmatic or not astigmatic. The issue is less clear in the case of a thick system. For example, is an eye stigmatic merely because its refraction is stigmatic (spherical)? In this article, a system is defined to be stigmatic if and only if, through the system, every point object maps to a point image. Every other system is astigmatic. Thus, a system is astigmatic if and only if there exists a point object for which the image is not a point. This article is restricted to linear optics. The optical character of a system is completely determined by the ray transference of the system. The objective here is to find those conditions on the transference for which the system is stigmatic or astigmatic. The result is that, for a stigmatic system, all the 2 x 2 submatrices are scalar multiples of a common orthogonal matrix. For a system to be stigmatic, it is not sufficient that its power be stigmatic. An eye may be astigmatic despite having a stigmatic refraction.