Page 233 - Electromagnetics

is the resonance frequency of the dipole moments. We see that this frequency is reduced
from the resonance frequency of the electron oscillation because of the polarization of
the surrounding medium.
We can now obtain a dispersion equation for the electrical susceptibility by taking the
Fourier transform of (4.102). We have

Nq 2
˜
2 ˜ 2 ˜ e ˜
−ω P + jω2 P + ω P = E.
0
m e
Thus we obtain the dispersion relation
P ˜ ω 2 p
˜ χ e (ω) = = 2
˜
2
0 E ω − ω + jω2
0
where ω p is the plasma frequency (4.74). Since ˜ r (ω) = 1 + ˜χ e (ω) we also have
ω 2 p
˜ (ω) = 0 + 0 2 . (4.103)
2
ω − ω + jω2
0
If more than one type of oscillating moment contributes to the permittivity, we may
extend (4.103) to
ω 2
pi

˜ (ω) = 0 + 0 (4.104)
2 2
ω − ω + jω2 i
i i
2
where ω pi = N i q / 0 m i is the plasma frequency of the ith resonance component, and
e
ω i and i are the oscillation frequency and damping coefficient, respectively, of this
component. This expression is the dispersion formula for classical physics, so called
because it neglects quantum effects. When losses are negligible, (4.104) reduces to the
Sellmeier equation
ω 2
pi

˜ (ω) = 0 + 0 . (4.105)
2
ω − ω 2
i i
Let us now study the frequency behavior of the dispersion relation (4.104). Splitting
the permittivity into real and imaginary parts we have
2
ω − ω 2

2 i
˜ (ω) − 0 = 0 ω ,
pi 2 2 2 2 2
i
i [ω − ω ] + 4ω i

2ω i
2
˜ (ω) =− 0 ω .
pi 2 2 2 2 2
i
i [ω − ω ] + 4ω i
As ω → 0 the permittivity reduces to
2

ω
pi
= 0 1 + ,
ω 2
i i
which is the static permittivity of the material. As ω →∞ the permittivity behaves as

! 2 ! 2
ω 2
pi
i pi i ω i

˜ (ω) → 0 1 − , ˜ (ω) →− 0 .
ω 2 ω 3
This high frequency behavior is identical to that of a plasma as described by (4.76).
© 2001 by CRC Press LLC