The limits of Gaussian optics

The initial layout of an optical instrument is usually made using simple Gaussian optics, where all apertures and pupils are infinitesimal and all angles are considered small, so that the approximations sin θ = tan θ = θ and cos θ = 1 are adequate.

Geometrical optics, however, is a non-linear discipline and ray-tracing, using the proper values of sines and cosines, quickly reveals that rays do not arrive in the expected places to make sharp images. The quality of an image can be tested by modifying the Gaussian optics to the approximations that sin θ = θ − θ3/6, cos θ = 1 and tan θ = θ + θ3/3 and adjustments can then be made to the positions, curvatures and refractive indices of the various elements of a system to produce better images at the focal surface. Except in rare instances, perfection is not to be expected, and the art of the optical designer is to adjust the values to produce an acceptable result. The problem confronting the designer is that the adjustments all interfere with each other, making the problem miserably complicated.

The problem was first analysed by Ludwig von Seidel of Munich (1821–1896), who codified the five so-called Seidel aberrations of a monochromatic system and derived formulae for computing them.