This appendix gives a recipe for calculating the third order aberrations of a lens when its construction data are known. As an example we use the 4 diopter eye glass corrected for astigmatism described in section 30.6. This is a thin lens with a focal length of 250 mm, and the exit pupil 25 mm to the right of the lens.

The first step is to trace two paraxial rays through the system, as shown in table A2.1. The first ray, called the object ray, comes from the axial point of the object and must be chosen such that its direction in the image space is +1. For the eye glass used as an example the object is located at infinity, so the incident ray must be parallel to the axis, and 250 mm below it to obtain the direction +1 in the image space. The first two columns of numbers in table A2.1 show the data for this ray. The unit of length used is the dm (100 mm); this avoids very large and very small numbers. Lines 1, 2, 3, and 6 contain the radii, refractive indices, and surface spacings of the lens. The incident ray is specified by h = –2.5 dm on line 7 and ψ=0 on line and 10. Filling in lines 11 through 15 of the first column yields the direction of the ray after the first surface, and lines 8 and 9 provide the height of the ray at the next surface. These two numbers, 2.0951 for the new direction and –2.5 for the new height, are transferred to the next column, which is then completed to find the height and direction of the ray as it emerges from the second surface. In our case there are only two surfaces; if there are more surfaces the process is repeated till the last surface is reached. Line 16 must be completed as well.