22 Geometrical optics 27-1 Introduction In this chapter we shall discuss some elementary applications of the ideas of 27-1 Introduction he previous chapter to a number of practical devices, using the approximation 27-2 The focal length of a spherical called geometrical optics. This is a most useful approximation in the practical surface design nany optical systems and instruments. Geometrical optics is either very simple or else it is very complicated. By that we mean that we can either study 27-3 The focal length of a lens it only superficially, so that we can design instruments roughly, using rules that are so simple that we hardly need deal with them here at all, since they are practi- cally of high school level, or else, if we want to know about the small errors in 27-5 Compound lenses lenses and similar details, the subject gets so complicated that it is too advanced 27-6 aberrations to discuss here! If one has an actual, detailed problem in lens design, including analysis of aberrations, then he is advised to read about the subject or else simply 27-7 Resolving power to trace the rays through the various surfaces(which is what the book tells how to do), using the law of refraction from one side to the other, and to find out where they come out and see if they form a satisfactory image People have said that this is too tedious, but today, with computing machines, it is the right way to do it One can set up the problem and make the calculation for one ray after another very easily. So the subject is really ultimately quite simple, and involves no new principles. Furthermore, it turns out that the rules of either elementary or advanced tics are seldom characteristic of other fields, so that there is no special reason to follow the subject very far, with one im The most advanced and abstract theory of geometrical optics was worked out by Hamilton, and it turns out that this has very important applications in mechanics. It is actually even more important in mechanics than it is in optics, and so we leave Hamilton's theory for the subject of advanced analytical mechanics which is studied in the senior yea graduate school. So, appreciating that geometrical optics contributes very little, except for its own sake, we now go on to discuss the elementary properties of simple optical systems on the basis of the principles outlined in the last chapter Figure 27-1 In order to go on, we must have one geometrical formula, which is the follow ing: if we have a triangle with a small altitude h and a long base d, then the diagonal s(we are going to need it to find the difference in time between two different routes) is longer than the base(Fig. 27-1). How much longer? The difference A=s-d can be found in a number of ways. One way is this. We see that s2-d2=hi or(s-d(s+ d)=h2. But s-d= 4, and s+ d w 2s. Thus This is all the geometry we need to discuss the formation of images by curved surfaces 27-2 The focal length of a spherical surface The first and simplest situation to discuss is a single refracting surface, sep arating two media with different indices of refraction( Fig. 27-2). We leave the case of arbitrary indices of refraction to the student, because ideas are always the acting surface most important thing, not the specific situation, and the problem is easy enough to do in So we shall suppose that, on the left, the speed is 1 right it is 1/n, where n is the index of refraction. The light travels more slowly in the glass by a factor n 27-1