Incidentally, you should notice that thereis another effect caused by the motion of the charges in the plate. Thesecharges will also radiate waves back toward the source S . Thisbackward-going field is the light we see reflected from the surfaces oftransparent materials. It does not come from just the surface. The backwardradiation comes from everywhere in the interior, but it turns out that thetotal effect is equivalent to a reflection from the surfaces. These reflectioneffects are beyond our approximation at the moment because we shall be limitedto a calculation for a material with an index so close to 1 that verylittle light is reflected.
顺便说说，我们应该注意，平板中电荷的运动，还会引起另外一个效果。这些电荷，也会辐射出波，返回到源S。这个往回走的场，就是我们看到的：从透明材料表面所反射的光。它并不是只来自于表面。往回的辐射，来自内部的每一个地方。但结果则是，总的效果，等价于表面的反射。这些反射效果，已经超出了我们现在的近似。因为，我们的计算，有一个限制条件，就是材料的折射率，非常接近于1，以至于被反射的光很少。

Fig. 31–2.Relation between refraction andvelocity change. 图31-2 折射与矢速变化的关系
Beforewe proceed with our study of how the index of refraction comes about, we shouldunderstand that all that is required to understand refraction is to understandwhy the apparent wave velocity is different in different materials. The bendingof light rays comes about just because the effective speed of the wavesis different in the materials. To remind you how that comes about we have drawnin Fig. 31–2several successive crests of an electric wave which arrives from a vacuum ontothe surface of a block of glass. The arrow perpendicular to the wave crestsindicates the direction of travel of the wave. Now all oscillations in the wavemust have the same frequency. (We have seen that driven oscillationshave the same frequency as the driving source.) This means, also, that the wavecrests for the waves on both sides of the surface must have the same spacingalong the surface because they must travel together, so that a chargesitting at the boundary will feel only one frequency. 折射率是如何发生的，这是一个问题，在我们继续研究它之前，应该明白，要理解折射，本质上，就是理解：为什么在不同的材料中，显现出来的波的矢速，是不同的。光线之所以折弯，正是因为，在不同的材料中，波的有效速度，是不同的。为了让大家记住，这是如何发生的，我们在图31-2中，画了几个相继电波的波峰。这些电波，是从真空中，来到一块玻璃的表面。垂直于波峰的箭头，表示了波传播的方向。现在，波中的所有震荡，都应该有同样的频率。（我们曾看到，被驱动的震荡，有同样的频率，作为驱动源。）这同样也意味着，对于表面两侧的波峰来说，它们沿着表面，应该有相同的空间。因为，它们应该一起传播，所以，一个位于边界处的电荷，将只会感觉到同一个频率。

Fig. 31–2.Fig. 31–2.Relation between refraction andvelocity change. 图31-2 折射与矢速变化的关系

The shortest distance between crestsof the wave, however, is the wavelength which is the velocity divided by thefrequency. On the vacuum side it is λ0=2πc/ω, and on the other side it is λ=2πv/ω or 2πc/ωn, if v=c/n is the velocity of the wave. From the figure wecan see that the only way for the waves to “fit” properly at the boundary isfor the waves in the material to be travelling at a different angle withrespect to the surface. From the geometry of the figure you can see that for a“fit” we must have λ0/sinθ0=λ/sinθ, or sinθ0/sinθ=n , which is Snell’s law. Weshall, for the rest of our discussion, consider only why light has an effectivespeed of c/n in material of index n , and nolonger worry, in this chapter, about the bending of the light direction.
然而，波峰之间的最短距离，就是波长，即速度除以频率。在真空一侧，它是λ0=2πc/ω，在另外一侧，它是λ=2πv/ω ，或者，如果v=c/n是波的矢速，则λ= 2πc/ωn。从图中，我们可以看到，在边界处，唯一能够让波适当地“合身”的方式，就是让材料中的波，以一个不同的角度传播，此角度，是就表面而言的。从图的几何形状，你可以看到，对于一个“合身”的波来说，我们应该有，λ0/sinθ0=λ/sinθ，或sinθ0/sinθ=n，这就是斯涅耳规律。在余下的讨论中，我们将只考虑，为什么在折射率为n的材料中，光的有效速度为c/n，并且，在本章中，将不会继续担忧：光的方向的弯折。

We go back now to the situation shown inFig. 31–1.We see that what we have to do is to calculate the field produced at Pby all the oscillating charges in the glass plate. We shall call this part ofthe field Ea , and it is just the sum written as the second term inEq. (31.2).When we add it to the term Es , due to the source, we will have thetotal field at P .
我们现在，回到图31-1所示的情况中。我们看到，我们要做的，就是计算:由玻璃中的所有振荡电荷，在 P，所产生的场。我们将把这部分场，称为Ea，在方程（31.2）中，这个和，是第二项。当我们把它加给可归于源的项 Es时，我们就有了P点的总场。

This is probably the most complicated thingthat we are going to do this year, but it is complicated only in that there aremany pieces that have to be put together; each piece, however, is very simple.Unlike other derivations where we say, “Forget the derivation, just look at theanswer!,” in this case we do not need the answer so much as the derivation. Inother words, the thing to understand now is the physical machinery for theproduction of the index.
在我们今年将要做的事情中，这件事情，大概是最复杂的了，但是，其复杂性，也仅在于，我们要把很多部分，拼在一起，而每一部分，都很简单。在其他的推导中，我们可能会说：“忘掉推导，只要看答案就行！”但是，在这里不会。在这个案例中，我们更看重推导，而不是答案。换句话说，我们现在要理解的事情，就是折射率产生的物理机制。

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。{？}