The most important thing, however, is thatthis formula (31.17)for Ea looks very much like the expression for Eathat we got in Eq. (31.8)by saying that the original wave was delayed in passing through a material withan index of refraction n . The two expressions will, in fact, beidentical if
然而，最重要的事情则是，下面两个表达式，看上去，非常像，一、这个关于Ea的公式（31.17）；二、在方程（31.8）中，通过说原始的波，在通过一个折射率为 n的材料时，被延迟了，而得到的一个关于Ea的表达式。事实上，如果
(31.18)
那么，这两个表达式，完全同等。
Notice that both sides are proportionalto Δz , since η , which is the number of atoms per unitarea, is equal to NΔz , where N is the number ofatoms per unit volume of the plate. Substituting NΔz for ηand cancelling the Δz , we get our main result, a formula for the indexof refraction in terms of the properties of the atoms of the material—and ofthe frequency of the light:
注意，两边都正比于Δz，由于η等于NΔz，这里，η是每单位面积中原子数目，N是板子中每单位体积的原子数目。用NΔz替换η，并消去Δz，我们就得到了我们的主要结果，一个关于折射率的公式，它是用材料中原子的属性、和光的频率的属性，来表达的。
(31.19)
This equation gives the “explanation” ofthe index of refraction that we wished to obtain.
对于折射率，我们希望得到一个“解释”，这个方程，给出了。

31–3Dispersion 31-3 散开
Notice that in the above process we haveobtained something very interesting. For we have not only a number for theindex of refraction which can be computed from the basic atomic quantities, butwe have also learned how the index of refraction should vary with the frequency ωof the light. This is something we would never understand from the simplestatement that “light travels slower in a transparent material.” We still havethe problem, of course, of knowing how many atoms per unit volume there are,and what is their natural frequency ω0 . We do not knowthis just yet, because it is different for every different material, and wecannot get a general theory of that now. Formulation of a general theory of theproperties of different substances—their natural frequencies, and so on—ispossible only with quantum atomic mechanics. Also, different materials havedifferent properties and different indexes, so we cannot expect, anyway, to geta general formula for the index which will apply to all substances.
要注意，在上面的过程中，我们已经得到了某些非常有趣的事情。因为，对于折射率，我们不仅得到了一个数，它可以从基本的原子的量出发，计算出，而且，我们也学习了，折射率是如何随着光的频率ω而变化的。有一个简单的声明：“光在透明材料中，传播地慢”，从它出发，我们永远也理解不了上面所说。当然，我们仍有：‘去了解每单位体积中有多少原子’这种问题，及它们的自然频率ω0是什么。这一点，我们仍不知道，因为，它对每种材料都不同，现在，我们还得不到一个关于它的普遍理论。对不同物品的属性--其频率等，要列出一个普遍理论的公式，只有用量子原子力学，才可能。另外，不同的材料，有不同的属性、及不同的折射率，所以，无论如何，我们不能期待，为折射率得到一个普遍公式，可应用于所有物品的。

However, we shall discuss the formula wehave obtained, in various possible circumstances. First of all, for mostordinary gases (for instance, for air, most colorless gases, hydrogen, helium,and so on) the natural frequencies of the electron oscillators correspond toultraviolet light. These frequencies are higher than the frequencies of visiblelight, that is, ω0 is much larger than ω of visiblelight, and to a first approximation, we can disregard ω2 incomparison with ω20 . Then we find that the indexis nearly constant. So for a gas, the index is nearly constant. This is alsotrue for most other transparent substances, like glass. If we look at ourexpression a little more closely, however, we notice that as ωrises, taking a little bit more away from the denominator, the index alsorises. So n rises slowly with frequency. The index is higher for bluelight than for red light. That is the reason why a prism bends the light morein the blue than in the red.
然而，在各种可能的情形中，我们地得到些公式，我们将讨论之。首先，对于大多数普通气体。（例如对于空气，大多数无色气体，氢、氦等）电子振荡器的自然频率，相当于紫外线的。这些频率，比可见光的频率要高，也就是说，ω0要远大于可见光的ω，对于一个一级近似，相比与ω20，我们可以忽略ω2。因此，我们发现，折射率接近于常数。于是，对于一种气体，折射率接近于常数。对于大多数透明的具体材料--比如玻璃--来说，这一点，也为真。然而，如果我们更仔细地查看我们的表达式，就会注意到，随着欧米伽的增加，即从分母中去掉一点，折射率就会增加。所以，n随着频率，缓慢增长。蓝光的折射率，要比红光的高。这就是为什么，一个棱镜，把蓝光折弯地多点，红光少点。

The phenomenon that the index depends uponthe frequency is called the phenomenon of dispersion, because it is thebasis of the fact that light is “dispersed” by a prism into a spectrum. Theequation for the index of refraction as a function of frequency is called a dispersionequation. So we have obtained a dispersion equation. (In the past few years“dispersion equations” have been finding a new use in the theory of elementaryparticles.)
折射率依赖于频率这一现象，被称为散开现象，因为，光被棱镜散开，形成光谱，它是基础。折射率的方程，作为频率的函数，被称为散开方程。于是，我们就得到了一个散开方程。（在过去的几年中，在基本粒子的理论中，散开方程又找到了一个新的应用。）

Our dispersion equation suggests otherinteresting effects. If we have a natural frequency ω0which lies in the visible region, or if we measure the index of refraction of amaterial like glass in the ultraviolet, where ω gets near ω0, we see that at frequencies very close to the natural frequency the index canget enormously large, because the denominator can go to zero. Next, supposethat ω is greater than ω0 . This would occur, forexample, if we take a material like glass, say, and shine x-ray radiation onit. In fact, since many materials which are opaque to visible light, likegraphite for instance, are transparent to x-rays, we can also talk about theindex of refraction of carbon for x-rays. All the natural frequencies of thecarbon atoms would be much lower than the frequency we are using in the x-rays,since x-ray radiation has a very high frequency. The index of refraction isthat given by our dispersion equation if we set ω0 equal tozero (we neglect ω20 in comparison with ω2).
我们的散开方程，提示了，还有其他有趣的效果。如果我们有一个自然的频率ω0，它处于可见光范围内，或者，如果我们在紫外线中，测量一个材料、比如玻璃的折射率，这里ω接近于ω0。我们看到，当频率非常接近于自然频率时，折射率可以变得非常巨大，因为分母趋于零。其次，假设ω大于ω0。这是可能发生的，例如，如果我们取一个材料、比如玻璃，用x射线辐射，照射它。事实上，由于很多材料，对可见光，是不透明的，比如石墨，但对x射线，是透明的，所以，我们也可以讨论，碳原子对x射线的折射率。所有碳原子的自然频率，比我们在x射线中所用的频率，都要低，由于x射线辐射，有一个非常高的频率。如果我们设ω0等于零，（与ω2平方相比，我们忽略ω20），那么，折射率就是通过我们的散开方程所给予的。

A similar situation would occur if we beamradiowaves (or light) on a gas of free electrons. In the upper atmosphereelectrons are liberated from their atoms by ultraviolet light from the sun andthey sit up there as free electrons. For free electrons ω0=0(there is no elastic restoring force). Setting ω0=0 in ourdispersion equation yields the correct formula for the index of refraction forradiowaves in the stratosphere, where N is now to represent the densityof free electrons (number per unit volume) in the stratosphere. But let us lookagain at the equation, if we beam x-rays on matter, or radiowaves (or anyelectric waves) on free electrons the term (ω20−ω2)becomes negative, and we obtain the result that n is less thanone. That means that the effective speed of the waves in the substance is fasterthan c ! Can that be correct?
如果我们用无线电波（或光）照射自由电子的气体，类似的情况，也会发生。在上层大气中，由于来自太阳的紫外线的作用，电子会从原子中，被释放出来，在那里，形成自由电子。对于自由电子，ω0=0（没有弹性恢复力）。在我们的散开方程中，设ω0=0，可以为平流层中的无线电波，产生正确的折射率的公式，现在N将代表平流层中的自由电子的密度（每单位体积的数目）。但是，让我们再看这个方程，如果我们用x射线照射物质，或用无线电波（或任何其他电波）照射自由电子，(ω20−ω2)就会变成负的。我们得到的结果就是：n小于1。这就意味着，波的有效速度，在这个具体材料中，比光速快。这能是正确的吗？

It is correct. In spite of the fact that it is saidthat you cannot send signals any faster than the speed of light, it isnevertheless true that the index of refraction of materials at a particularfrequency can be either greater or less than 1 . This just means that the phaseshift which is produced by the scattered light can be either positive ornegative. It can be shown, however, that the speed at which you can send a signalis not determined by the index at one frequency, but depends on what the indexis at many frequencies. What the index tells us is the speed at whichthe nodes (or crests) of the wave travel. The node of a wave isnot a signal by itself. In a perfect wave, which has no modulations of anykind, i.e., which is a steady oscillation, you cannot really say when it“starts,” so you cannot use it for a timing signal. 它是正确的。虽然有这种说法，即你不可能发送任何比光速快的信号，尽管这是一个事实，但在一个具体频率上，材料的折射率，还是可以大于1或小于1的，这也为真。这仅意味着，由光的散射所产生的相位偏移，即可为正，也可为负。然而，可以指出，你发射一个信号所用的速度，并不是被某一个频率的折射率所规定，而是依赖于很多频率的折射率。折射率告诉我们的，是波节、或波峰的传播速度。波节本身，并不是一个信号。在一个完美的播中，没有任何的调节，亦即，它是一个稳定的震荡器，所以，在其中，你不能真正地说：它是什么时候“开始”的，所以，你不能用它作为一个计时信号。

In order to send a signal you haveto change the wave somehow, make a notch in it, make it a little bit fatter orthinner. That means that you have to have more than one frequency in the wave,and it can be shown that the speed at which signals travel is notdependent upon the index alone, but upon the way that the index changes withthe frequency. This subject we must also delay (until Chapter 48).Then we will calculate for you the actual speed of signals through sucha piece of glass, and you will see that it will not be faster than the speed oflight, although the nodes, which are mathematical points, do travel faster thanthe speed of light.
为了发送一个信号，你必须以某种方式，改变波，在里面做个记号，让它稍微变胖点儿、或变瘦点儿。这就意味着，你必须让波中的频率，多于一个，可以指出，信号传播的速度，并不只依赖于折射率，而是依赖于折射率随着频率的变化。这个话题，我们也要推后（到第48章）。那时，我们将为你计算：信号通过这样一片玻璃的实际速度，你将会看到，它并不比光速快，虽然节点—它是数学上的点--，传播的比光速快。

How does the charge happen to be going inthe opposite direction? It certainly does not start off in the oppositedirection when the field is first turned on. When the motion first starts thereis a transient, which settles down after awhile, and only then is thephase of the oscillation of the charge opposite to the driving field. And it isthen that the phase of the transmitted field can appear to be advancedwith respect to the source wave. It is this advance in phase which ismeant when we say that the “phase velocity” or velocity of the nodes is greaterthan c . In Fig. 31–4 wegive a schematic idea of how the waves might look for a case where the wave issuddenly turned on (to make a signal). You will see from the diagram that the signal(i.e., the start of the wave) is not earlier for the wave whichends up with an advance in phase.
电荷是如何往相反的方向走的呢？当场最初被加上时，它当然不会往相反的方向走。当运动刚开始时，有那么一个瞬间，只有此时，电荷振荡的相位，才是相反于驱动场的；过一会儿，这个瞬间就过去了。且只有此时，传送场的相位，就源的波而言，可以表现为：被提前了。当我们说，“相位矢速”或节点矢速，大于c时，我们所说的，就是这个相位的提前。在图31-4中，我们提供了一个图示概念：对于一种案例，当波突然被加上时（以做出一个信号），波看上去，会是什么样子。从这个图，你将会看到，对于相位上有一个提前的波来说，信号（亦即，波的开始）并不更早。

Fig. 31–4.Wave “signals.” 图31-4 波的“信号”。

Let us now look again at our dispersionequation. We should remark that our analysis of the refractive index gives aresult that is somewhat simpler than you would actually find in nature. To becompletely accurate we must add some refinements. First, we should expect thatour model of the atomic oscillator should have some damping force (otherwiseonce started it would oscillate forever, and we do not expect that to happen).We have worked out before (Eq. 23.8) the motion of a damped oscillator and the resultis that the denominator in Eq. (31.16),and therefore in (31.19),is changed from (ω20−ω2)to (ω20−ω2+iγω) ,where γ isthe damping coefficient.
现在，让我们再看看我们的散开方程。我们应该说明，我们对折射率的分析，给出了一个结果，它比你在自然中实际上所能发现的，在某种意义上，要简单一些。要想完全准确，必须加一些提炼。首先，我们应该期待，我们的原子震荡器的模式，应该有一些阻尼力（否则，一旦开始，就会永远震荡下去，这并非我们所期待）。之前，我们已经得出了阻尼振荡的运动（方程23.8），结果就是，方程（31.16）中的分母，从而，方程（31.19）中的分母，从(ω20−ω2)变为 (ω20−ω2+iγω)，这里γ是阻尼系数。

You will note that so long as ωis not too close to one of the resonant frequencies, the slope of the curve ispositive. Such a positive slope is called “normal” dispersion (because it isclearly the most common occurrence). Very near the resonant frequencies,however, there is a small range of ω ’s for which the slope isnegative. Such a negative slope is often referred to as “anomalous” (meaningabnormal) dispersion, because it seemed unusual when it was first observed,long before anyone even knew there were such things as electrons. From ourpoint of view both slopes are quite “normal”!
你将会注意到，只要ω不是非常接近这些频率中的一个，那么，曲线的斜率，就是正的。这种正的斜率，被称为“正常的”散开！（因为，很清楚，这是最经常发生的）。然而，在非常接近共振频率的地方，有一个小范围的ω，对于它们，斜率为负。这样一种负的斜率，通常被称为“异常的”（意思是不正常的）散开，因为，初次发现它时，似乎是不正常的，很久之后，人们才知道，还有电子这种东西。从我们的观点，两种斜率，都相当“正常”。

又被吃楼了，微信也有：
https://mp.weixin.qq.com/s/VGJNTzdzUBTURoP4jhZHdg

31–4Absorption 31-4 吸收
Perhaps you have noticed something a littlestrange about the last form (Eq. 31.20)we obtained for our dispersion equation. Because of the term iγ we put in to take accountof damping, the index of refraction is now a complex number! What does thatmean? By working out what the real and imaginary parts of n are wecould write
对于我们散开方程，我们得到的最后形式是（方程31.20），或许你已经注意到，此形式中，有某个东西，略感奇怪。因为我们所放入的项 iγ，是用来考虑阻尼的，折射率现在是一个复数！其意为何？通过得出n的实部和虚部，我们可以写：
n=n′−in′′,(31.21)
where n′ and n′′ arereal numbers. (We use the minus sign in front of the in′′ because then n′′will turn out to be a positive number, as you can show for yourself.)
这里n′ 和n′′是实数。（我们在in′′之前，放了一个负号，因为n′′将是一个正数，这一点，你可以自己指出。）

We can see what such a complex index meanswhen there is only one resonant frequency by going back to Eq. (31.6),which is the equation of the wave after it goes through a plate of materialwith an index n . If we put our complex n into thisequation, and do some rearranging, we get
我们可以看到就是，通过回到方程（31.6），即当只有一个共振频率时，这样一个复数指数，所说的是什么；方程（31.6），是波通过一个指数为n的材料板之后的方程。如果我们把我们的复数n，放进方程，重新整理，就得到：
(31.22)
The last factors, marked B in Eq. (31.22),are just the form we had before, and again describe a wave whose phase has beendelayed by the angle ω(n′−1)Δz/c in traversingthe material. The first term (A) is new and is an exponential factor witha real exponent, because there were two i ’s that cancelled.Also, the exponent is negative, so the factor is a real number less than one.It describes a decrease in the magnitude of the field and, as we shouldexpect, by an amount which is more the larger Δz is. 最后一个因子，即在方程（31.11）中被标为B的项，正是我们曾拥有过的形式，它再次描述了，波穿过材料后，相位被延迟了一个角度ω(n′−1)Δz/c。第一项(A)，是一个新项，是一个指数型的因子，有一实数指数，因为，两个 i抵消了。另外，指数是负的，所以，此因子是一个小于1的实数。它描述了场的大小的减少，正如我们所期待，减少的量，比Δz大。{？}As the wave goes through the material, it is weakened. The materialis “absorbing” part of the wave. The wave comes out the other side with lessenergy. We should not be surprised at this, because the damping we put in forthe oscillators is indeed a friction force and must be expected to cause a lossof energy. We see that the imaginary part n′′ of a complex index ofrefraction represents an absorption (or “attenuation”) of the wave. In fact, n′′is sometimes referred to as the “absorption index.”
波穿过材料后，被变弱了。材料吸收了波的一部分。波从另一侧出来时，少了一些能量。对此，我们不应感到奇怪，因为，我们为了振荡器而放进去的阻尼项，实际上是一个摩擦力，可以期待，它会引起一些能量损失。我们看到，一个复数的折射率，其虚部n′′，代表着波的吸收（或“attenuation”）。事实上，n′′有时也被引用为“吸收指数”。

We may also point out that an imaginarypart to the index n corresponds to bending the arrow Eain Fig. 31–3toward the origin. It is clear why the transmitted field is then decreased.
我们还可以指出，指数n的虚部，相当于把图31-3中的箭头Ea，向起点弯折了。透射出的场，为什么会减弱，就很清楚了。