If this book were to follow historical order, the present chapter should have preceded the previous one, since lenses and mirrors were known and studied long before wave theory was understood. However, once we have grasped the elements of wave theory, it is much easier to appreciate the strengths and limitations of geometrical optics, so logically it is quite appropriate to put this chapter here. Essentially, geometrical optics, which considers light waves as rays that propagate along straight lines in uniform media and are related by Snell's law (§2.6.2 and §5.4) at interfaces, has a relationship to wave optics similar to that of classical mechanics to quantum mechanics. For geometrical optics to be strictly true, it is important that the sizes of the elements we are dealing with be large compared with the wavelength λ. Under these conditions we can neglect diffraction, which otherwise prevents the exact simultaneous specification of the positions and directions of rays on which geometrical optics is based.
Analytical solutions of problems in geometrical optics are rare, but fortunately there are approximations, in particular the Gaussian or paraxial approximation, which work quite well under most conditions and will be the basis of the discussion in this chapter. Exact solutions can be found using specialized computer programs, which will not be discussed here. However, from the practical point of view, geometrical optics answers most questions about optical instruments extremely well and in a much simpler way than wave theory could do.