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
Now suppose that we have a point ato, at a distance s from the front surface of the glass, and another point O' at a distance s'inside the glass, and we desire to arrange the curved surface in such a manner that every ray from O which hits the surface, at any point P, will be bent so as to proceed toward the point o,. For that to be true, we have to shape the surface in such a way that the time it takes fo the light to go from O to P, that is, the distance oP divided by the speed of light (the speed here is unity), plus n. P, which is the time it takes to go from P to O is equal to a constant independent of the point P. This condition supplies us with an equation for determining the surface. The answer is that the surface is a very complicated fourth-degree curve, and the student may entertain himself by trying to calculate it by analytic geometry. It is simpler to try a special case that corre sponds to s-00, because then the curve is a second-degree curve and is more recognizable. It is interesting to compare this curve with the parabolic curve we found for a focusing mirror when the light is coming from infinit So the proper surface cannot easily be made--to focus the light from one point to another requires a rather complicated surface. It turns out in practice that we do not try to make such complicated surfaces ordinarily, but instead we make a compromise. Instead of trying to get all the rays to come to a focus, we arrange it so that only the rays fairly close to the axis 00 come to a focus. The farther ones may deviate if they want to, unfortunately, because the ideal surface is complicated, and we use instead a spherical surface with the right curvature at the axis. It is so much easier to fabricate a sphere than other surfaces that it is profitable for us to find out what happens to rays striking a spherical surface supposing that only the rays near the axis are going to be focused perfectly Those rays which are near the axis are sometimes called paraxial rays, and what we are analyzing are the conditions for the focusing of paraxial rays. We shall that troduced by the fact tha close to the axis he height P@ is h. For a moment, we imagine that the surface is a plane passing the through P. In that case, the time needed to go from o to P would exceed the time from 0 to 2, and also the time from P to o would exceed the time from g to o But that is why the glass must be curved, because the total excess time must be compensated by the delay in passing from V to g! Now the excess time along route OP is h/ 2s, and the excess time on the other route is nh2 /2s. This excess time, which must be matched by the delay in going along v@, differs from wha it would have been in a vacuum, because there is a medium present. In other yords, the time to go from V to e is not as if it were straight in the air, but it is slower by the factor n, so that the excess delay in this distance is then(n-1)vQ And now, how large is v@? If the point C is the center of the sphere and if its radius is R, we see by the same formula that the distance vg is equal to h2/2R. Therefore we discover that the law that connects the distances s and that gives us the radius of curvature R of the surface that we need t and s, (h27/2)+(mh2/2s)=(n-1)h2/2R (1/s)+(n/s)=(n-1)/R (273) If we have a position O and another position O', and want to focus light from O to o, then we can calculate the required radius of curvature R of the surface by this formula Now it turns out, interestingly, that the same lens, with the same curvature R, will focus for other distances, namely, for any pair of distances such that the sum of the two reciprocals, one multiplied by n, is a constant. Thus a given lens will(so long as we limit ourselves to paraxial rays)focus not only from O to out between an infinite number of other pairs of points, so long as those pairs of points bear the relationship that 1/s +n/s' is a constant, characteristic of the lens In particular, an interesting case is that in which s- oo. We can see from the formula that as one s increases, the other decreases. In other words, if point o
goes out, point O'comes in, and vice versa. As point O goes toward infinity, point O' keeps moving in until it reaches a certain distance, called the focal length inside the material. If parallel rays come in, they will meet the axis at a distance f. Likewise, we could imagine it the other way.( Remember the reciprocity rule if light will go from O to O, of course it will also go from O to O. )Therefore, if we had a light source inside the glass, we might want to know where the focus is In particular, if the light in the glass were at infinity(same problem)where wor it come to a focus outside? This distance is called f. Of course, we can also put it the other way. If we had a light source atf and the light went through the surface, then it would go out as a parallel beam. We can easily find out what fandf'are ∥=(n-1)/R f=Rn/(n-1), L/f=(n-1)/R f=R/(n-1) We see an interesting thing: if we divide each focal length by the corresponding dex of refraction we get the same result! This theorem, in fact, is general. It is true of any system of lenses, no matter how complicated, so it is interesting remember. We did not prove here that it is general-we merely noted it for a single are related in this way. Sometimes Eq.(27. 3)is written in the form t a syster surface, but it happens to be true in general that the two focal lengths of a system This is more useful than (27. 3) because we can measure more easily than we can measure the curvature and index of refraction of the lens if we are not interested in designing a lens or in knowing how it got that way, but simply lift it off a shelf, the interesting quantity is f, not the n and the l and the r Now an interesting situation occurs if s becomes less than f. what happens en? If s <f then (1/s)>(l/, and therefore s' is negative; our equation says that the light will focus only with a negative value of s, whatever that means It does mean something very interesting and very definite. It is still a useful formula 等转。k in other words, even when the numbers are negative. what it means is shown in o' Fig. 27-3. If we draw the rays which are diverging from O, they will be bent, it is rue, at the surface, and they will not come to a focus, because o is so close in that they are"beyond parallel. However, they diverge as if they had come from a point o outside the glass. This is an apparent image, sometimes called a virtual Fig. 27-3. A virtual image image. The image O' in Fig. 27-2 is called a real image. If the light really comes to a point, it is a real image. But if the light appears to be coming from a point, a fictitious point different from the original point, it is a virtual image. So when s' comes out negative, it means that o' is on the other side of the surface, and every- ling is all right Now consider the interesting case where R is equal to infinity; then we have (1/s)+(n/s)=0. In other words, s'=-ns, which means that if we look from a dense medium into a rare medium and see a point in the rare medium, it appears to be deeper by a factor n. Likewise, Ise the same equation backwards, so that if we look into a plane surface at an object that is at a certain distance inside the dense medium, it will appear as though the light is coming from not as far back(Fig. 27-4). When we look at the bottom of a swimming pool from above it does not look as deep as it really is, by a factor 3/4, which is the reciprocal of the Fig. 27-4. A plane surface re-images index of refraction of water the light from o to o We could go on, of course, to discuss the spherical mirror. But if one appreci ates the ideas involved, he should be able to work it out for himself. Therefore we leave it to the student to work out the formula for the spherical mirror, but we mention that it is well to adopt certain conventions concerning the distances involved (1)The object distance s is positive if the point O is to the left of the surface (2)The image distance s' is positive if the point o' is to the right of the surface (3)The radius of curvature of the surface is positive if the center is to the right of the surface
In Fig. 27-2, for example, s, s, and R are all positive; in Fig. 27-3, s and R are positive, but sis negative. If we had used a concave surface, our formula(27.3 would still give the correct result if we merely make r a negative quantity n working out the corresponding formula for a mirror, using the above conventions, you will find that if you put n =-I throughout the formula(27. 3) as though the material behind the mirror had an index-1), the right formula for a mirror results! Although the derivation of formula(27. 3)is simple and elegant, using least time, one can of course work out the same formula using Snell's law, remembering that the angles are so small that the sines of angles can be replaced by the angles themselves 27-3 The focal length of a lens Now we go on to consider another situation, a very practical one. Most of the lenses that we use have two surfaces, not just one. How does this affec matters? Suppose that we have two surfaces of different curvature, with glass filling the space between them(Fig. 27-5). We want to study the problem of focusing from a point O to an alternate point O'. How can we do that? The answer is this: First, use formula(27.3)for the first surface, forgetting about the second surface. This will tell us that the light which was diverging from O will appear to be converging or diverging, depending on the sign, from some other point, say O. Now we consider a new problem. We have a different surface. between glass and air, in which rays are converging toward a certain point o'. Where will they actually converge? We use the same formula again! We find that they con Fig. 27-5. Image formation by a verge at o. Thus, if necessary, we can go through 75 surfaces by just using the wo-surface lens same formula in succession, from one to the next There are some rather high-class formulas that would save us considerable energy in the few times in our lives that we might have to chase the light through five surfaces, but it is easier just to chase it through five surfaces when the problem arises than it is to memorize a lot of formulas, because it may be we will never have to chase it through any surfaces at all! any case, the principle is that when we go through one surface we find a new position, a new focal point, and then take that point as the starting point for the next surface, and so on. In order to actually do this, since on the second surface we are going from n to I rather than from 1 to n, and since in many systems there is more than one kind of glass, so that there are indices n1, n2, e really need o' a generalization of formula(27. 3) for a case where there are two different indices, nI and n2, rather than only n. Then it is not dificult to prove that the general form of (27. 3)is R Fig. 27-6. A thin lens with two posi- Particularly simple is the special case in which the two surfaces are very close tive rad together-so close that we may ignore small errors due to the thickness. If we draw the lens as shown in Fig. 27-6, we may ask this question: How must the lens be built so as to focus light from o to o? Suppose the light comes exactly to the edge of the lens, at point P. Then the excess time in going from o to Ois (n h 7 25)+(n h2/2s"), ignoring for a moment the presence of the thickness t of glass of index n2. Now, to make the time for the direct path equal to that fo the path OPO, we have to use a piece of glass whose thickness Tat the center is such that the delay introduced in going through this thickness is enough to compensate for the excess time above. Therefore the thickness of the lens at the center must be given by the relationship (n1h2/2)+(n1h2/2)=(m2-n1)7 (278) We can also express T in terms of the radii R, and r2 of the two surfaces. Paying attention to our convention (3), we thus find, for R1< R2(a convex lens), T=(h2/2R1)-(h2/2R2)
Therefore, we finally get (n1/s)+(n1/s)=(n2-m1)(1/R1-1/R2) (27.10) Now we note again that if one of the points is at infinity, the other will be at a point which we will call the focal lengths. The focal length is given by l=(n-1)(1/R1-1/R2), (27.11) where n= n2/nl Now, if we take the opposite case, where s goes to infinity, we see that s'is at the focal length,'. This time the focal lengths are equal.(This is another special case of the general rule that the ratio of the two focal lengths is the ratio of the indices of refraction in the two media in which the rays focus. In this particular ptical system, the initial and final indices are the same, so the two focal lengths are equal Forgetting for a moment about the actual formula for the focal length, if we bought a lens that somebody designed with certain radii of curvature and a certain index, we could measure the focal length, say, by seeing where a point at infinity focuses. Once we had the focal length, it would be better to write our equation in terms of the focal length directly, and the formula then is (1/s)+(1/)=1/ (27.12) Now let us see how the formula works and what it implies in different circu stances. First, it implies that if s or s'is infinite the other one is f. That means that parallel light focuses at a distance and this in effect defines f. Another interesting thing it says is that both points move in the same direction. If one moves to the right, the other does also. Another thing it says is that s and s are equal if they are both equal to 2/. In other words, if we want a symmetrical situation, we find that they will both focus at a distance 2f. 27-4M So far we have discussed the focusing action only for points on the axis. Now let us discuss also the imaging of objects not exactly on the axis, but a little bit off, so that we can understand the properties of magnification, When we set up a lens so as to focus light from a small filament onto a"point"on a screen, we notice that on the screen we get a"picture"of the same filament, except of a larger or smaller size than the true filament. This must mean that the light comes to a focus from each point of the filament. In order to understand this a little better, let us analyze the thin lens system shown schematically in Fig. 27-7. We know the follow ing facts 1 )Any ray that comes in parallel on one side proceeds toward a certain par- Fig. 27-7. The geometry of imaging ticular point called the focus on the other side, at a distance from the lens. by a thin ens. 2)Any ray that arrives at the lens from the fo out This is all we need to establish formula(27. 12)by geometry, as follows: Suppose we have an object at some distance x from the focus; let the height of the object be y. Then we know that one of the rays, namely Pe, will be bent so as to pass through the focus r on the other side. now if the lens will focus point P at all, can find out where if we find out where just one other ray goes, because the new focus will be where the two intersect again. We need only use our ingenuity to find the exact direction of one other ray. But we remember that a parallel ray goes through the focus and vice versa: a ray which goes through the focus will come out parallel! So we draw ray PT through U. (It is true that the actual rays which are doing the focusing may be much more limited than the two we have drawn, but they are harder to figure, so we make believe that we can make this ray. Since it would come out parallel, we draw TS parallel to XW. The intersection S is the point we need. This will determine the correct place and the correct height