Page 235 - Electromagnetics

The imaginary part of the permittivity peaks near the resonant frequency, dropping
off monotonically in each direction away from the peak. The width of the curve is an
important parameter that we can most easily determine by approximating the behavior

of ˜ near ω 0 . Letting ¯ω = (ω 0 − ω)/ω 0 and using
2
2
2
ω − ω = (ω 0 − ω)(ω 0 + ω) ≈ 2ω ¯ω,
0
0
we get
¯
1 2

˜ (ω) ≈− 0 ¯ω p .
2
¯ 2
2 ( ¯ω) +
This approximation has a maximum value of
1 2 1

˜ max = ˜ (ω 0 ) =− 0 ¯ω p
¯
2
¯
located at ω = ω 0 , and has half-amplitude points located at ¯ω =± . Thus the width
of the resonance curve is
W = 2 .
Note that for a material characterized by a low-loss resonance ( ω 0 ), the location of

˜ max can be approximated as

ωmax = ω 0 1 − 2 /ω 0 ≈ ω 0 −
while ˜ is located at
min

ω min = ω 0 1 + 2 /ω 0 ≈ ω 0 + .
The region of anomalous dispersion thus lies between the half amplitude points of ˜ :

ω 0 − < ω < ω 0 + .
As → 0 the resonance curve becomes narrower and taller. Thus, a material charac-
terized by a very low-loss resonance may be modeled very simply using ˜ = Aδ(ω − ω 0 ),

where A is a constant to be determined. We can find A by applying the Kronig–Kramers
formula (4.37):
∞
2 d 2 ω 0

˜ (ω) − 0 =− P.V. Aδ( − ω 0 ) =− A .
2
2
π − ω 2 π ω − ω 2
0
0
Since the material approaches the lossless case, this expression should match the Sellmeier
equation (4.105):
2 ω 0 ω 2 p
− A = 0 ,
2
2
π ω − ω 2 ω − ω 2
0 0
2
giving A =−π 0 ω /2ω 0 . Hence the permittivity of a material characterized by a low-loss
p
resonance may be approximated as
2 2
ω π ω
c p p
˜ (ω) = 0 1 + 2 − j 0 δ(ω − ω 0 ).
ω − ω 2 2 ω 0
0
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