Paraxial approximation

Figure 13 shows that the ray tranversal aberration increases with the height of incidence h of I on the diopter. Let us seek the limit A' of the intersection of the refracted ray when h leads to 0. When I leads towards S, the relationship (7) becomes :

The study of aberrations shows that the distance gap dy' between the refracted ray and A' on a plane going through A' and perpendicular to the axis is approximately proportional to h 3 .

For small values of h, dy' is very short, there is an approximate stigmatism.

In this case, rays incidences i and i' on the surface of the diopter are low, sinus and radian angles are very close, the relationship becoming : n.i = n'.i'

An incidence of 5° for which we have [ i - sin(i)]/i ≈ 1,2.10-3 is a good limit for this approximation. It is thus said that A' is the image of A in the paraxial approximation and is also called Gaussian approximation. In paraxial approximation, all diopters, therefore all optical systems, are stigmatic.