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several resonant frequencies. 'Therefore jWy is not really the number of electrons having the
frequency œ4, but is replaced instead by ) ƒ„, where is the number of atoms per unit volume
and ƒ„ (called the oscillator strength) is a factor that tells how strongly the atom exhibits each
of its resonant frequencies œy;.
--- Trang 548 ---
“normail” dispersion (because it is clearly the most common occurrence). Very
near the resonant frequencies, however, there is a small range oŸ œ”s for which
the slope is negative. Such a negative slope is often referred to as “anomalous”
(meaning abnormal) dispersion, because i% seemed unusual when it was first
observed, long before anyone even knew there were such things as electrons.
ttrom our point of view both slopes are quite “normail”!
31-4 Absorption
Perhaps you have noticed something a little strange about the last form
(Eq. 31.20) we obtained for our dispersion equation. Because of the term 2y
we put in to take account of damping, the Index of refraction 1s now a complez
nưmber! What does that mean? By working out what the real and imaginary
parts of m are we could write
tr — TỦ — ?nẺ, (31.21)
where ø and ø” are real numbers. (We use the minus sign in front of the ¿nw”
because then ø” will turn out to be a positive number, as you can show Íor
yourself.)
W©e can see what such a complex index means by going back to Eq. (31.6),
which is the equation of the wave after it goes through a plate of material with
an index nø. IÝ we put our complex ø into this equation, and do some rearranging,
W© getE
Eatey plate — «n7 Az/c e— 1(m°—1) ^z/e macf2Œ—=2/5) : (31.22)
"¬——ễ_—————
The last factors, marked B in Eq. (31.22), are just the form we had before, and
again describe a wave whose phase has been delayed by the angle /j(m' — 1) Az/c
in traversing the material. 'The first term (A) is new and is an exponential
factor with a reøl exponent, because there were ©wo ?)s that cancelled. Also, the
exponent is negative, so the factor is a real number less than one. It describes a
đecrease 1n the magnitude of the field and, as we should expect, by an amount
which is more the larger Az is. As the wave goes through the material, it is
weakened. 'Phe material is “absorbing” part of the wave. 'Phe wave comes out
the other side with less energy. We should not be surprised at this, because
the damping we put in for the oscillators is indeed a friction force and must
--- Trang 549 ---
be expected to cause a loss of energy. We see that the imaginary part øé of a
complex index of refraction represents an absorption (or “attenuation”) of the
wave. In fact, ?ø7 is sometimes referred to as the “absorption Index.”
We may also point out that an imaginary part to the index ø corresponds
to bending the arrow „in Fig. 3Í-3 toward the origin. It is clear why the
transmitted fñeld is then decreased.
Normally, for instance as in glass, the absorption of light is very smaill.
This is to be expected from our Eq. (31.20), because the imaginary part oŸ the
denominator, 2œ, is much smaller than the term (œ£ — œ2). But ïf the light
frequency œ is very close to œ„ then the resonance term (œÿ — @”) can become
small compared with 7+„œ and the index becomes almost completely imaginary.
The absorption of the light becomes the dominant efect. It is just this efect
that gives the dark lines in the spectrum of light which we receive from the sun.
The light from the solar surface has passed through the sun's atmosphere (as
well as the earth's), and the light has been strongly absorbed at the resonant
frequencies of the atoms in the solar atmosphere.
The observation of such spectral lines in the sunlight allows us to tell the
resonant frequencies of the atoms and hence the chemical composition of the
sun's atmosphere. The same kind of observations tell us about the materials in
the stars. From such measurements we know that the chemical elements in the
sun and in the stars are the same as those we ñnd on the earth.
31-5 The energy carried by an electric wave
W©e have seen that the imaginary part of the index means absorption. We
shall now use this knowledge to fnd out how much energy is carried by a light
wave. We have given earlier an argument that the energy carried by light is
proportional to #2, the tỉme average of the square of the electrie feld in the
wave. The decrease in # due to absorption must mean a loss of energy, which
would go into some friction of the electrons and, we might guess, would end up
as heat in the material.
Tf we consider the light arriving on a unit area, say one square centimeter,
of our plate in Eig. 31-1, then we can write the following energy equation (ïŸ we
assume that energy is conserved, as we đo!):
lnergy in per sec = energy out per sec -- work done per sec. (31.23)
--- Trang 550 ---
For the fñrst term we can write œ22, where œ is the as yet unknown constant
of proportionality which relates the average value of #2 to the energy being
carried. For the second term we must include the part from the radiating
atoms oŸ the material, so we should use œ(#; + !4)2, or (evaluating the square)
a(EJ + 2E,l2„ + E2).
All of our calculations have been made for a thin layer of material whose
index is not too far from 1, so that #„ would always be much less than #2; (Just
to make the calculations easier). In keeping with our approximations, we should,
therefore, leave out the term E2, because it is much smaller than #;„. You
may say: “Then you should leave out #,„ also, because #£ is much smaller
than F2” It is true that ¿„ is much smaller than #2, but we must keep
2; F„ or our approximation will be the one that would apply if we neglected the
presence of the material completely! One way of checking that our calculations
are consistent is to see that we always keep terms which are proportional to Ñ Az,
the area density of atoms in the material, but we leave out terms which are
proportional to (W Az)2 or any higher power of Az. Ours is what should be
called a “low-density approximation.”
In the same spirit, we might remark that our energy equation has neglected
the energy in the reflected wave. But that is OK because this term, Èoo, is
proportional to (N A2), since the amplitude of the reflected wave is proportional
to N Az.
Eor the last term in Eq. (31.23) we wish to compute the rate at which the
incoming wave is doing work on the electrons. We know that work is Íorce tỉmes
distance, so the røứe of doïing work (also called power) is the force times the
velocity. It is really #!- ø, but we do not need to worry about the dot produect
when the velocity and force are along the same direction as they are here (except
for a possible minus sign). So for each atom we take ge2;ò for the average rate
of doing work. Since there are W Az atoms in a unit area, the last term in