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enormously large, because the denominator can go to zero. Next, suppose that œ
1s preater than œọ. This would occur, for example, If we take a material like glass,
say, and shine x-ray radiation on it. In fact, since many materials which are
opaque to visible light, like graphite for instance, are transparent to x-rays, we
can also talk about the index of refraction of carbon for x-rays. All the natural
frequencies of the carbon atoms would be much lower than the frequency we are
using in the x-rays, since x-ray radiation has a very high fÍrequency. 'Phe index of
refraction is that given by our dispersion equation iŸ we set œ«ọ equal to zero (we
neglect œđ in comparison with œŸ).
A similar situation would occur if we beam radiowaves (or lighÈ) on a gas
of free electrons. In the upper atmosphere electrons are liberated from their
atoms by ultraviolet light rom the sun and they sỉt up there as free electrons.
Eor free electrons œo = 0 (there is no elastic restoring force). Setting «go =0 in
our dispersion equation yields the correct formula for the index of refraction for
radiowaves In the stratosphere, where / is now to represent the density of free
electrons (number per unit volume) in the stratosphere. But let us look again at
the equation, iŸ we beam x-rays on matter, or radiowaves (or any electric waves)
on free electrons the term (œä — œ2) becomes ø„ega7e, and we obtain the result
that tò is less than one. That means that the efective speed of the waves in the
substanee is ƒøsfer than cl Can that be correct?
Tt is correct. In spite of the fact that it is said that you cannot send signals any
faster than the speed of light, it is nevertheless true that the index of refraction of
materials at a particular Írequency can be either greater or less than 1. Thịis just
means that the phase shøff which is produced by the scattered light can be either
posifive or negative. It can be shown, however, that the speed at which you can
send a signadl is not determined by the index at one frequency, but depends on
what the index is at mømy frequencies. What the index tells us is the speed at
which the œodes (or crests) of the wave travel. The node of a wave is not a signal
by itself. In a perfect wave, which has no modulations of any kind, i.e., which is
a steady oscillation, you cannot really say when it “starts,” so you cannot use
1t for a timing signal. In order to send a siøgna/ you have to change the wave
somehow, make a notch in it, make it a little bit fatter or thinner. 'Phat means
that you have to have more than one frequenecy in the wave, and it can be shown
that the speed at which s¿ønais travel is not dependent upon the index alone,
but upon the way that the index changes with the frequency. 'Phis subject we
must also delay (until Chapter 48). Then we will calculate for you the acbual
speed of s7ønals through such a piece of glass, and you will see that ít will not
--- Trang 546 ---
be faster than the speed of light, although the nodes, which are mathematical
points, do travel faster than the speed of light.
Just to give a slipht hint as to how that happens, you will note that the real
dificulty has to do with the fact that the responses of the charges are opposite
to the field, i.e., the sign has gotten reversed. 'Thus in our expression Íor
(Eq. 31.16) the displacement of the charge is in the direction opposite to the
driving feld, because (œ — œ2) is negative for small œọ. The formula says that
when the electric fñeld is pulling in one direction, the charge is moving in the
opposite direction.
How does the charge happen to be going in the opposite direction? lt certainly
does not start of in the opposite direction when the fñeld is frst turned on. When
the motion first starts there is a transient, which settles down after awhile, and
only hen 1s the phase of the oscillation of the charge opposite to the driving
field. And it is then that the phase of the transmitted field can appear to be
aduanccd with respect to the source wave. l§ is this œduance ín phase which 1s
meant when we say that the “phase velocity” or velocity of the nodes is greater
than c. In Fig. 31-4 we give a schematic idea of how the waves might look for
a case where the wave is suddenly turned on (to make a signal). You will see
from the diagram that the signal (i.e., the sfar£ of the wave) is not earlier Íor
the wave which ends up with an advance in phase.
Let us now look again at our dispersion equation. We should remark that
our analysis of the refractive Index gives a result that is somewhat simpler than
(a) E /Strt |
Wave with no HA
material I I | I t
(b) (| pc «4
Transmitted wave t
withn>1 ⁄ I t
delay of phase
I I 1
Transmitted wave
with n< 1 ' ' h ! t
advance of phase
Fig. 31-4. Wave “signals.”
--- Trang 547 ---
you would actually ñnd in nature. To be completely accurate we must add some
refinements. First, we should expect that our model of the atomic oscillator
should have some damping force (otherwise once started it would oscillate forever,
and we do not expectE that to happen). We have worked out before (Eq. 23.8)
the motion of a damped oscillator and the result is that the denominator in
E4. (31.16), and therefore in (31.19), is changed from (uậT— œ2) to (uẩ —œ 2+7),
where + is the damping coeficient.
W©e need a second modification to take into account the fact that there are
several resonant frequencies for a particular kind of atom. Ït is easy to ñx up
our dispersion equation by imagining that there are several different kinds of
oscillators, but that each oscillator acts separately, and so we simply add the
contributions of all the oscillators. Let us say that there are j„ electrons per
unit of volume, whose natural frequeney is ¿„ and whose damping factOr is +.
We would then have for our dispersion equation
đề Ahụ
n=1+s —ÐÖ ` —s..—- (31.20)
2corn T § — Ố † 7k9
W©e have, finally, a complete expression which describes the index of refraction
that is observed for many substances.* 'The index described by this formula varies
with frequency roughly like the curve shown in Pig. 3Í-5.
You will note that so long as œ is not too close to one of the resonant
frequencies, the slope of the curve is positive. Such a positive slope is called
1 “-‡-~ “-‡->
0 ŒỊ (2 FC. 1)
Fig. 31-5. The index of refraction as a function of frequency.
* Actually, although in quantum mechanics Eq. (31.20) is still valid, its interpretation is
somewhat diferent. In quantum mechanics even an atom with one electron, like hydrogen, has