Introduction

When the construction details of a lens are known, it is possible to avoid all theory and to determine its aberrations numerically by using Snell's law and geometry. This process is called ray tracing. The procedure is straightforward. Select a ray coming from a chosen object point. Follow its path as it traverses the lens, and find its intersection point with the image plane. Do this for several other rays coming from the same object point. Ideally all the intersection points should coincide and form a perfect image point, but this is hardly ever the case. Usually the intersection points fall in slightly different spots, and the information obtained about their spread must be used to predict how well the lens will perform when it is put to its intended use.

Ray tracing consists of two steps applied alternately, translation and refraction. Given a ray specified by a point and a direction, (1) its intersection point with the next lens surface must be determined, and (2) this point found, Snell's law must be used to find the direction of the ray after refraction. These two steps are repeated until the image plane is reached. We used this process in chapter 10, but there we restricted ourselves to the small angle approximation. This led to simple linear relations. Now we have to use the exact equations, which are unpleasantly complicated.

Prior to the introduction of mechanical desk calculators in the 1930s and 1940s, logarithms were used to carry out these calculations.