The image of a luminous point in a diopter

Let us consider a spherical diopter separating two mediums of indexes n and n', defined by its curvature center as C, its vertex as S, its curvature ray .

All lengths and angles are orientated in accordance with the trigonometry convention

A point A is situated on the object space on line SC. The ray arising from A through S is perpendicular to the diopter, it is not deviated. Another ray arising from A going through any point I from the diopter is subject to refraction, the ray arising cuts SC at a point A'.

Let us look for the position of A'. According to figure 12 :

i is the incidence angle of the ray on the dioptic.

i' is the refraction angle and, after (5), n.sin(i) = n'.sin(i').

An usual formula in the triangle CAI gives :

One can also write :

Therefore :

Conjugate stigmatism would mean that A' does not depend on the position of I. It is thus necessary that CA' remains fixed, similarly for the IA/IA' ratio. This is obtained only in a particular position of A and is not achieved in general cases.

Figure 13 concretely shows an example of ray tracing in a diopter, the aberation here is substantial. This tracing is obained by means of the free software Oslo-Edu1 which can be dowloaded from this address : http://www.lambdares.com/downloads/index.phtml#osloedu