To see where we are going, let us firstfind out what the “correction field” Ea would have to beif the total field at P is going to look like radiation from thesource that is slowed down while passing through the thin plate. If the platehad no effect on it, the field of a wave travelling to the right (along the z-axis) would be
Es=E0cosω(t−z/c) (31.3)
or, using the exponential notation,
Es=E0eiω(t−z/c). (31.4)
为了看清楚，我们将要往哪里走，让们首先找出：如果在P的总场，看上去像是来自源的辐射，且此辐射，在通过薄板的时候会变慢，那么，“修正场”Ea，会是什么。如果板子对它没有影响，那么，一个波的场，在向右（沿着z轴）传播时，将会是：
Es=E0cosω(t−z/c) (31.3)
或者，使用指数表达法就是：
Es=E0eiω(t−z/c). (31.4)

Now what would happen if the wave travelledmore slowly in going through the plate? Let us call the thickness of theplate Δz . If the plate were not there the wave would travel thedistance Δz in the time Δz/c . But if it appearsto travel at the speed c/n then it should take the longertime nΔz/c or the additional time Δt=(n−1)Δz/c. After that it would continue to travel at the speed c again. Wecan take into account the extra delay in getting through the plate byreplacing t in Eq. (31.4)by (t−Δt) or by [t−(n−1)Δz/c]. So the wave after insertion of the plate should be written
现在，如果波在通过平板时，走的更慢，那么，会发生什么呢？让我们设板的厚度为 Δz。如果板子不在那儿，那么，波通过此距离 Δz，用时为Δz/c。但是，如果它要以速度 c/n传播，那么，就应该去花更长的时间 nΔz/c，或者，附加的时间就是 Δt=(n−1)Δz/c。在此之后，它又继续以速度c传播。通过平板时，有额外的延迟，在方程（31.4）中，我们可以用 (t−Δt) 或 [t−(n−1)Δz/c]，来替换 t，从而，就可把此延迟，考虑进来。于是，在插入平板之后，波就应被写为：
(31.5)
We can also write this equation as
我们也可把这个方程写为：
(31.6)
which says that the wave after the plate isobtained from the wave which could exist without the plate, i.e., from Es, by multiplying by the factor e−iω(n−1)Δz/c. Now we know that multiplying an oscillating function like eiωtby a factor eiθ just says that we change the phaseof the oscillation by the angle θ , which is, of course, what theextra delay in passing through the thickness Δz has done. It hasretarded the phase by the amount ω(n−1)Δz/c(retarded, because of the minus sign in the exponent).
它说的就是，板子后面的波，可以通过没有板子时的波，即 Es，乘以因子e−iω(n−1)Δz/c，来得到。现在，我们知道，用一个因子 eiθ，乘以一个振荡函数，比如eiωt，所说的正是：我们通过角度 θ，改变了振荡的相位，而这{?θ}当然就是:通过厚度Δz所带来的额外延迟，所做放入事。它把相位，迟滞了ω(n−1)Δz/c（被迟滞，因为指数中的负号）。

What we have been doing is easilyvisualized if we look at the complex number diagram in Fig. 31–3. Wefirst draw the number Es (we chose some values for zand t so that Es comes out horizontal, but thisis not necessary). The delay due to slowing down in the plate would delay thephase of this number, that is, it would rotate Esthrough a negative angle. But this is equivalent to adding the smallvector Ea at roughly right angles to Es. But that is just what the factor −i means in the second term ofEq. (31.8).It says that if Es is real, then Ea is negativeimaginary or that, in general, Es and Eamake a right angle.
如果我们在图31-3中，看一个复数图，那么，我们一直所做的事情，就可以可视化。首先，我们画一个数Es（我们为z和t，选某些值，以让Es水平地出来，但这并非必需）。板子中的减速所引起的延迟，将会延迟这个数的相位，也就是说，它会让Es转一个负的角度。但是，这等价于在大约垂直于 Es的方向，增加一个小的矢量Ea。但是，它所表示的，正是方程（31.8）中的第二项的意思。它说，如果Es是实的，那么，Ea就是负的虚的，或者，一般来说，Es和Ea，成一个直角。

Fig. 31–3.Diagram for the transmitted waveat a particular t and z . 图31-3 在具体的t和z上，所传输的波的图形。

31–2The field due to the material 31-2 可归于材料的场
We now have to ask: Is the field Eaobtained in the second term of Eq. (31.8)the kind we would expect from oscillating charges in the plate? If we can showthat it is, we will then have calculated what the index n shouldbe! [Since n is the only nonfundamental number in Eq. (31.8).]We turn now to calculating what field Ea the charges inthe material will produce. (To help you keep track of the many symbols we haveused up to now, and will be using in the rest of our calculation, we have putthem all together in Table 31–1.)
我们现在要问：从方程（31.8）的第二项所得到的场Ea，就是我们期待的、从板子中震荡着电荷中所得到的场吗？如果我们可以指出，它就是，那么，我们将因此去计算，折射率n应该是什么。[由于在方程（31.8）中，n是唯一的非基础性的数。]现在，我们要转向计算，材料中的电荷将会产生的场Ea（到目前为止，我们已经用了很多符号，在下面的计算中，也会用到一些，为了帮助大家跟踪这些符号，我们把它们全放在表31-1中。）

Table 31–1Symbols used in the calculations 表31-1 计算中用到的符号
Es= field from the source 来自源的场
Ea= field produced by charges in the plate 板子中的电荷所产生的场
Δz= thickness of the plate 板子的厚度
z= perpendicular distance from the plate 到板子的垂直距离
n= index of refraction 折射率
ω= frequency (angular) of the radiation 辐射的频率（角度的）
N= number of charges per unit volume in the plate 板子中每单位体积中的电荷数目
η= number of charges per unit area of the plate 板子上每单位面积的电荷数目
qe= charge on an electron 一个电子的电量
m= mass of an electron 一个电子的质量
ω0= resonant frequency of an electron bound in an atom 原子中绑定的电子的共振频率

If the source S (of Fig. 31–1) isfar off to the left, then the field Es will have the samephase everywhere on the plate, so we can write that in the neighborhood of theplate
如果（图31-1中的）源S，在左边很远处。那么，在板子上任何地方，场Es的相位，都将一样，于是，我们可以把板子附近的场，写作：
Es=E0eiω(t−z/c). (31.9)
Right at the plate, where z=0 , wewill have
在板子的右边，z=0处，我们有：
Es=E0eiωt (atthe plate 在板子上). (31.10)

Each of the electrons in the atoms of theplate will feel this electric field and will be driven up and down (we assumethe direction of E0 is vertical) by the electricforce qE . To find what motion we expect for the electrons, we willassume that the atoms are little oscillators, that is, that the electrons arefastened elastically to the atoms, which means that if a force is applied to anelectron its displacement from its normal position will be proportional to theforce.
板子上的原子中的每一个电子，都将会感觉到这个电场，也会被电力 qE驱动着，上下运动（我们假定，E0的方向，是垂直的。）电子是如何运动的，我们有个期待，要找这个运动，我们将假定，原子就是小的振荡器，也就是说，电子被弹性地扣紧在原子上，这就意味着，如果一个力，被应用到一个电子上，那么，电子到通常位置的位移，将正比于此力。

You may think that this is a funny model ofan atom if you have heard about electrons whirling around in orbits. But thatis just an oversimplified picture. The correct picture of an atom, which isgiven by the theory of wave mechanics, says that, so far as problemsinvolving light are concerned, the electrons behave as though they wereheld by springs. So we shall suppose that the electrons have a linear restoringforce which, together with their mass m , makes them behave likelittle oscillators, with a resonant frequency ω0 . Wehave already studied such oscillators, and we know that the equation of theirmotion is written this way:
如果你听说过电子在轨道中，做旋转运动，那么，你可能会想，这个原子模式，非常有趣。但是，那只是一个超级简化的图像。原子的正确图像，是通过波动力学的理论，给出的，该理论说，只要牵扯到光的问题，被考虑到，那么，电子的表现，就好像它们是被绳子拴着一样。所以，我们将假设，电子有一个线性的恢复力，它与质量m一起，使得它们的表现，像一个小的震荡器，且有一个共振频率ω0。这种振荡器，我们已经研究过，我们知道，其运动方程，是这样写的：
(31.11)
where F is the driving force.
这里F是驱动力。

Now we are ready to find the field Eathat these atoms produce at the point P , because we have alreadyworked out (at the end of Chapter 30)what field is produced by a sheet of charges that all move together. Referringback to Eq. (30.19), we see that the field Eaat P is just a negative constant times the velocity of the chargesretarded in time by the amount z/c . Differentiating xin Eq. (31.16)to get the velocity, and sticking in the retardation [or just putting x0from (31.15)into (30.18)] yields
现在，我们已经准备好了，去找出这些原子在的点P所产生的场Ea，因为，（在30章的最后）我们已经得出了，一个薄板上的电子所产生的场，这些电子，都在一起运动。回头参考方程（31.19），我们看到，在点P的场Ea，正是一个负的常数，乘以某电荷的速度，该电荷，在时间上，被迟滞了一个量 z/c。在方程（30.16）中，对x求微分，可以得到矢速，加上迟滞[或者，只是把（31.15）中的x0带入（30.18）]就会产生：
(31.17)
Just as we expected, the driven motion ofthe electrons produced an extra wave which travels to the right (that is whatthe factor eiω(t−z/c)says), and the amplitude of this wave is proportional to the number of atomsper unit area in the plate (the factor η ) and also proportional tothe strength of the source field (the factor E0 ). Thenthere are some factors which depend on the atomic properties (qe, m , and ω0 ), as we should expect.
正如我们所期待，被驱动的电子的运动，产生了另外一个波，它向右边传播（这就是因子eiω(t−z/c)所说），这个波的振幅，正比于板子中每单位面积中原子的数目（即因子η），也正比于源场的强度（即因子E0）。因此，就有一些因子，依赖于原子的属性(qe, m , and ω0 )，我们应如此期待。{？}

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？