We see an interesting thing: if we divideeach focal length by the corresponding index of refraction we get the sameresult! This theorem, in fact, is general. It is true of any system of lenses,no matter how complicated, so it is interesting to remember. We did not provehere that it is general—we merely noted it for a single surface, but it happensto be true in general that the two focal lengths of a system are related inthis way. Sometimes Eq. (27.3)is written in the form
我们看到一个有趣的事情：如果我们把每个焦距，都除以相应的折射指数，那么，我们就会得到：同样的结果！事实上，这个定理，是普遍的。对于任何棱镜系统，它都成立，不论该系统多么复杂，所以，它很容易记。我们这里不去证明，其普遍性--我们只是对于单独的界面，指出这一点，但是，一个系统的两个焦距，以这种方式相关，碰巧普遍为真。有时，方程（27.3），可写为这种形式：
1/s+n/s′=1/f. (27.6)
This is more useful than (27.3)because we can measure f more easily than we can measure the curvatureand index of refraction of the lens: if we are not interested in designing alens or in knowing how it got that way, but simply lift it off a shelf, theinteresting quantity is f , not the n and the 1and the R !
这比（27.3）更有用，因为，测量一个系统的f，比测量其曲率和折射指数，更容易些：如果我们对于设计透镜、或者知道它是如何变成这样的，并不感兴趣，而只是从架子上，拿起一个透镜，感兴趣的量是f , 而不是 n、1、和 R !{？}

Now an interesting situation occurs if sbecomes less than f . What happens then? If s<f , then(1/s)>(1/f) , and therefore s′ is negative; ourequation says that the light will focus only with a negative value of s′, whatever that means! It does mean something very interesting and verydefinite. It is still a useful formula, in other words, even when the numbersare negative. What it means is shown in Fig. 27–3. Ifwe draw the rays which are diverging from O , they will be bent, it istrue, at the surface, and they will not come to a focus, because O is soclose in that they are “beyond parallel.” 现在，如果s变得小于f，那么，一个有趣的情况，就出现了。出现了什么呢？如果s<f ，那么(1/s)>(1/f)，因此，s′就是负的；我们的方程说，光只会在一个负的s′处聚焦，它究竟意味着什么呢！它确实意味着，某个非常有趣的、也非常确定的事情。它仍是一个有用的公式，换句话说，即便数字是负的。它所意味的，如图27-3所示。如果我们要画，从O散射出去的光线，那么，它们在界面上会被弯曲，这是真的，但它们不会来到焦点，因为太近了，它们已经超越了“平行”。However, they diverge as if they had come from a point O′outside the glass. This is an apparent image, sometimes called a virtualimage. The image O′ in Fig. 27–2 iscalled a real image. If the light really comes to a point, it is a realimage. But if the light appears to be coming from a point, afictitious point different from the original point, it is a virtual image. Sowhen s′ comes out negative, it means that O′ is on the other sideof the surface, and everything is all right.
然而，它们好像是从玻璃外面的点O′，散射出来。这幅图像，显然如此，有时，它被称为虚拟图像。图27-2中的图像O′，被称为真实图像。如果光真地来到了一个点，那么，它就是一个真实图像。但是，如果光只是显得是从一个点来的，这是一个虚构的点，与始发点有区别，那么，它就是一个虚拟图像。所以，当s′变负时，就意味着，O′是在界面的另一边，所有的事情，就都正确了。

Fig. 27–3.A virtual image. 图27-3 一个虚拟图像。

We could go on, of course, to discuss thespherical mirror. But if one appreciates the ideas involved, he should be ableto work it out for himself. Therefore we leave it to the student to work outthe formula for the spherical mirror, but we mention that it is well to adoptcertain conventions concerning the distances involved:
1. The object distance s is positive if the point Ois 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 isto the right of the surface.
In Fig. 27–2, forexample, s , s′ , and R are all positive; inFig. 27–3,s and R are positive, but s′ is negative. If we hadused a concave surface, our formula (27.3)would still give the correct result if we merely make R a negativequantity.
当然，我们可以继续讨论球形镜子。但是，如果有人同意上面所牵扯到的想法，那么，他自己应该就能做。因此，我们将把球面镜子的公式，留给学生去做，但是，我们要提一下，考虑到所牵扯的距离，我们最好采用一些约定：
1、如果点O，在界面的左边，那么，对象的距离s，就是正的。
2、如果点O′，在界面的右边，那么，图像的距离s′，就是正的。
3、如果中心，在界面的右边，那么，面的曲率半径，就是正的。
例如，在图27-2中，s , s′ , 和 R，都为正；在图27-3中，s和 R为正，但 s′ 为负。我们如果用一个凹界面，那么，如果我们只要使得R为一个负的量的话，则我们的公式（27.3），就会给出正确的结果。

27–3The focal length of a lens 27-3 透镜的焦距
Now we go on to consider another situation,a very practical one. Most of the lenses that we use have two surfaces, notjust one. How does this affect matters? Suppose that we have two surfaces ofdifferent curvature, with glass filling the space between them (Fig. 27–5). Wewant to study the problem of focusing from a point O to analternate 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. 现在，我们接着考虑另外一种情况，它很有实际意义。我们使用的大多数透镜，都有两个表面，而不是一个。这会有什么影响呢？假设我们有两个表面，其曲率不同，玻璃填充其间，如图（27-5）。我们想研究，从点O聚焦到可替换的点O′这一问题。我们该怎么做呢？答案就是这样：首先，对于第一个表面，使用公式（27.3），先忘掉第二个表面。This will tellus that the light which was diverging from O will appear to beconverging 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 andair, in which rays are converging toward a certain point O′ . Wherewill they actually converge? We use the same formula again! We find that theyconverge at O′′ . Thus, if necessary, we can go through 75 surfaces by just using the same formula insuccession, from one to the next!
这会告诉我们，从O散射出的光，可能表现为，是从另外一个点，比如说O′，聚集的或发散的，这依赖于符号{？}。现在，我们考虑一个新的问题。在玻璃和空气之间，我们有一个不同的表面，在其中，光线聚集到某确定点O′。它们实际上会聚集在哪里呢？我们再次使用同样的公式！我们发现，会聚焦在O′′。这样，如果必要的话，通过从一个表面到另一个，相继地使用同样的公式，我们就可以经过75表面。{？75，意义不明}

Fig. 27–5.Image formation by a two-surfacelens. 图27-5 两个表面透镜的图像信息。

There are some rather high-class formulasthat would save us considerable energy in the few times in our lives that wemight have to chase the light through five surfaces, but it is easier just tochase it through five surfaces when the problem arises than it is to memorize alot of formulas, because it may be we will never have to chase it through anysurfaces at all!
在我们的人生中，可能只有很少的机会，需要去追逐光通过五个表面，对此，有一些非常高级的公式，可以让我们节省心力，但是，当问题产生时，再去追逐光通过五个表面，会比记住那一堆公式，更容易一些，因为，追踪光通过表面这种事，我们可能永远也不会去做！{kidding}

In any case, the principle is that when wego through one surface we find a new position, a new focal point, and then takethat point as the starting point for the next surface, and so on. In order toactually do this, since on the second surface we are going from nto 1 rather than from 1 to n , and since in many systems thereis more than one kind of glass, so that there are indices n1, n2 , …, we really need a generalization of formula (27.3)for a case where there are two different indices, n1and n2 , rather than only n . Then it is notdifficult to prove that the general form of (27.3)is
适用于任何情况的原理就是，当我们通过一个表面，我们找到一个新的位置，一个新的聚焦点，然后，对于下一个表面，我们把此点，当作起点，如此等等。由于在第二个表面，我们是从n到1，而不是从1到n，且由于，在很多系统中，玻璃种类，多于一种，于是，就有指数n1 , n2, …, 为了实际上做到这一点，对于有两种指数的情况，即有n1和n2 ,而不是只有n，我们确实需要把公式（27.3），进行普遍化。因此，证明下式，是（27.3）的普遍形式，并不困难：
(n1/s)+(n2/s′)=(n2−n1)/R. (27.7)

Now, if we take the opposite case, where sgoes to infinity, we see that s′ is at the focal length f′ .This time the focal lengths are equal. (This is another special case of thegeneral rule that the ratio of the two focal lengths is the ratio of theindices of refraction in the two media in which the rays focus. In thisparticular optical system, the initial and final indices are the same, so thetwo focal lengths are equal.)
现在，如果我们取相反的情况，s走向无穷，我们看到s′，就在焦距f′处。这次，焦距是相等的。（这是普遍规则的另外一种特殊情况，光线在两个媒介中聚焦，这两个媒介的折射指数的比率，就是两个焦距的比率。在这个具体的光学系统中，最初和最终的指数，是一样的，所以，两个焦距，是相等的）。

Forgetting for a moment about the actualformula for the focal length, if we bought a lens that somebody designed withcertain radii of curvature and a certain index, we could measure the focallength, say, by seeing where a point at infinity focuses. Once we had the focallength, it would be better to write our equation in terms of the focal lengthdirectly, and the formula then is
关于焦距的实际公式，暂时忘记了，比如，某人设计了一个透镜，有确定的曲率半径和指数，如果我们买了它，那么，我们可以通过看聚焦点有多远，而测量焦距。一旦我们有了焦距，那么，直接用聚焦来写我们的方程，会更好：
(1/s)+(1/s′)=1/f. (27.12)

This is all we need to establishformula (27.12)by geometry, as follows: Suppose we have an object at some distance xfrom the focus; let the height of the object be y . Then we know thatone of the rays, namely PQ , will be bent so as to pass through thefocus R on the other side. Now if the lens will focus point Pat all, we 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 useour ingenuity to find the exact direction of one other ray. But weremember that a parallel ray goes through the focus and vice versa: aray which goes through the focus will come out parallel! 要通过几何，来建立公式（27.12），如下就是我们所需的一切：假设我们有一个对象，距离焦点为x，高度为y。因此，我们就知道，光线之一，即PQ，会被这样折弯，即它将通过另一侧的焦点R。现在，如果透镜，将把P点的光，完全聚焦，那么，如果我们能够找出，另外一条光线怎么走，那么，我们就可以找出聚焦在何处，因为，新的焦点，应为这两条线的交点。我们只需使用我们的聪明才智，找出另外一条光线的准确方向。我们记得，平行光线，会过焦点，及反过来：过焦点的光，出来时，会变平行。So we draw ray PT through U . (It is true thatthe actual rays which are doing the focusing may be much more limited than thetwo we have drawn, but they are harder to figure, so we make believe that wecan make this ray.) 所以，我们过U画光线PT。（确实，实际上做这个聚焦行动的光线，比我们所画的这两条，要更受限制{？}，但是，要把它们想清楚，会更困难，所以，我们就相信，我们可以做出这条光线。）Since it would come out parallel, we draw TS parallelto XW . The intersection S is the point we need. Thiswill determine the correct place and the correct height. Let us call theheight y′ and the distance from the focus, x′ . Now we mayderive a lens formula. Using the similar triangles PVU and TXU, we find
由于光线出来，会变平行，所以，我们画TS，平行于XW。交点S，就是我们所需。这将会规定：正确的位置、和正确的高度。我们称高度为 y′，到焦点的距离为x′。现在，我们就可以导出一个透镜公式。利用相似三角形PVU和 TXU，我们发现：
y′/f=y/x. (27.13)
Similarly, from triangles SWR and QXR, we get
类似，从三角形SWR和 QXR，我们得到：
y′/x′=y/f. (27.14)
Solving each for y′/y , wefind that
为两式求y′/y，我们发现：
y′/y=x′/f=f/x. (27.15)

Equation (27.15)is the famous lens formula; in it is everything we need to know about lenses:It tells us the magnification, y′/y , in terms of the distancesand the focal lengths. It also connects the two distances x and x′with f :
方程（27.15），就是著名的透镜公式；关于透镜，我们所需要知道的一切，里面都有了：它用距离和焦距，告诉了我们放大率，y′/y。它也把两个距离x和 x′，与f联系了起来：
xx′=f2, (27.16)
which is a much neater form to work withthan Eq. (27.12).We leave it to the student to demonstrate that if we call s=x+fand s′=x′+f , Eq. (27.12)is the same as Eq. (27.16).
上式使用起来，比方程（27.12），更为简明。如果我们设s=x+f ，及s′=x′+f，那么，方程（27.12）与方程（27.16），就是一样的；这一点，我们留给学生去演证。

27–5Compound lenses 27-5 复合透镜
Without actually deriving it, we shallbriefly describe the general result when we have a number of lenses. If we havea system of several lenses, how can we possibly analyze it? That is easy. Westart with some object and calculate where its image is for the first lens,using formula (27.16)or (27.12)or any other equivalent formula, or by drawing diagrams. So we find an image.Then we treat this image as the source for the next lens, and use the secondlens with whatever its focal length is to again find an image. We simply chasethe thing through the succession of lenses. That is all there is to it. Itinvolves nothing new in principle, so we shall not go into it. 当我们有若干透镜时，这里，我们不去实际地推导结果，我们将只简单地描述一下一般结果。如果我们有一个系统，由若干透镜组成，我们该如何分析它呢？这很容易。我们从某个对象开始，然后，利用公式（27.16）、或（27.12）、或任何其他等价的公式、或通过画图，来计算出，它的对于第一个透镜的图像，该在何处。于是，我们就找到了第一个图像。然后，我们把这个图像，当做下一个透镜的源，然后，利用下一个透镜，不论其焦距为何，再次去找到另外一个图像。我们只是按照透镜的序列，追逐图像。这就是所有的工作。它不包含任何新的原理，所以，我们不去讨论它。However, there is a very interesting net result of the effects ofany sequence of lenses on light that starts and ends up in the same medium, sayair. Any optical instrument—a telescope or a microscope with any number oflenses and mirrors—has the following property: There exist two planes, called theprincipal planes of the system (these planes are often fairly close tothe first surface of the first lens and the last surface of the last lens),which have the following properties: 然而，对于任何透镜序列，其对光的影响，有一个净结果，非常有趣，那就是，它开始于、也终结于同一个媒介，比如说空气。任何光学设备--一个望远镜、或一个具有任意数量透镜和镜子的显微镜--，都具有如下属性：存在两个平面，被称为主平面（这些平面，通常会非常接近于第一个透镜的第一个表面，和最后一个透镜的最后一个表面），它有如下属性：(1) If light comes into the system parallel from the firstside, it comes out at a certain focus, at a distance from the secondprincipal plane equal to the focal length, just as though the system were athin lens situated at this plane. (2) If parallel light comes in the otherway, it comes to a focus at the same distance f from the firstprincipal plane, again as if a thin lens where situated there. (See Fig. 27–8.)
（1）如果光从第一主平面的一侧，平行地来到系统，那么，它出来时，会来到某一确定的焦点，该点到第二主平面的距离，等于焦距，就好像这个系统，是一个薄的透镜一样，处于这个平面处。（2）如果平行光，从另外一个方向来，那么，它会来到一个焦点，该点到第一主平面的距离为同样的f，再次好像，是有一个薄的透镜，处于那里一样。（见图27-8。）

Fig. 27–8.Illustration of the principalplanes of an optical system. 图 27-8 一个光学系统的主平面的示意图

Of course, if we measure the distances xand x′ , and y and y′ as before, theformula (27.16)that we have written for the thin lens is absolutely general, provided that wemeasure the focal length from the principal planes and not from the center ofthe lens. It so happens that for a thin lens the principal planes arecoincident. It is just as though we could take a thin lens, slice it down themiddle, and separate it, and not notice that it was separated. Every ray thatcomes in pops out immediately on the other side of the second plane from thesame point as it went into the first plane! The principal planes and the focallength may be found either by experiment or by calculation, and then the wholeset of properties of the optical system are described. It is very interestingthat the result is not complicated when we are all finished with such a big,complicated optical system.
当然，如果我们像以前一样，测量距离x和x′ ,及 y和y′，那么，对于薄透镜，我们曾写过的公式（27.16），就是绝对普遍的，这里假设了，我们测量焦距，是从主平面开始，而不是从透镜的中心。对于薄透镜，主平面是一致的，就是如此{？}。就好像，我们可以拿一个薄透镜，从中间把它划开，分开，但却并未注意到，它被分开了。每条来到它的光线，都立即从第二个平面的另一侧的同一个点，被弹出，这一点，与光线进入第一个平面的那个点，就好像是一样的。主平面和焦距，既可以通过实验来发现，也可以通过计算来发现，因此，这个光学系统的整个属性集合，就已经被描述。有一点，很有趣，那就是，此光学系统，大且复杂，但是，当我们完成了对它的分析之后，结果则并不复杂。

27–6Aberrations 27-6 偏差
Before we get too excited about howmarvelous lenses are, we must hasten to add that there are also seriouslimitations, because of the fact that we have limited ourselves, strictlyspeaking, to paraxial rays, the rays near the axis. A real lens having a finitesize will, in general, exhibit aberrations. For example, a ray that ison the axis, of course, goes through the focus; a ray that is very close to theaxis will still come to the focus very well. But as we go farther out, the raybegins to deviate from the focus, perhaps by falling short, and a ray strikingnear the top edge comes down and misses the focus by quite a wide margin. 透镜真是不可思议，在我们对此过于激动之前，我们应该抓紧补充一点，即还是有一些限制的，因为，事实上，或严格地说，我们是把我们限制在近轴光线上，即接近于轴的光线上。一个真实的透镜，尺寸有限，一般来说，会表现出偏差。例如，轴上的光线，当然就会通过焦点，而一个离轴很近的光线，来到焦点也没问题。但是，当我们走得更远时，光线就开始偏离焦点，或许变短，且打在边缘顶端的光线，会错过焦点，且离的很远。So, instead of getting a point image, we get a smear. This effect iscalled spherical aberration, because it is a property of the sphericalsurfaces we use in place of the right shape. This could be remedied, for anyspecific object distance, by re-forming the shape of the lens surface, orperhaps by using several lenses arranged so that the aberrations of theindividual lenses tend to cancel each other. 这样，我们得到的，就不是一个点的图像，而是一团模糊。这个效果，被称为球面偏差，因为，它是球形表面的属性，我们用此球面，来代替正确的形状{？}。这个偏差，可以修正，对任何具体对象的距离，通过调整透镜的表面形状，或者，通过合理安排几个透镜，这样，单个透镜的偏差，就会被相互抵消。

Lenses have another fault: light ofdifferent colors has different speeds, or different indices of refraction, inthe glass, and therefore the focal length of a given lens is different fordifferent colors. So if we image a white spot, the image will have colors,because when we focus for the red, the blue is out of focus, or vice versa.This property is called chromatic aberration.
透镜还有另外一个缺陷：不同颜色的光，在玻璃中，速度不同，折射指数不同，因此，给定透镜，对于不同的颜色，其焦距不同。所以，如果我们要给一个白色的点成像，那么，这个图像，将会是彩色的，因为，当我们红色聚焦时，则蓝色就不会在焦点上，反之亦然。这个属性，被称为色差。

There are still other faults. If the objectis off the axis, then the focus really isn’t perfect any more, when it gets farenough off the axis. The easiest way to verify this is to focus a lens and thentilt it so that the rays are coming in at a large angle from the axis. Then theimage that is formed will usually be quite crude, and there may be no placewhere it focuses well. There are thus several kinds of errors in lenses thatthe optical designer tries to remedy by using many lenses to compensate eachother’s errors.
还有其它的缺陷。如果对象不在轴上，那么，当它离轴足够远时，焦点就不再是完美的。要演证这一点，最容易的方式，就是先让一个透镜聚焦，然后，倾斜透镜，这样，到来的光线，就与轴，成一个大角。因此，所形成的图像，将会非常粗糙，图像不知道在什么地方，才能聚好焦。在透镜中，这种错误，有若干个，光学设计者们，尝试通过使用多个透镜，相互之间，进行补偿，来纠正这种错误。