Page 217 - Electromagnetics
P. 217
We have obtained the second form of each of these expressions using the property (4.3)
for the transform of a real function, and by using the change of variables ω =−ω.
Multiplying the two forms of the expressions and adding half of each, we find that
1 ∞ dω ∞
∂w e dω ˜ ∗ ˜ ˜ ˜ ∗ − j(ω −ω)t
= jωE (ω ) · D(ω) − jω E(ω) · D (ω ) e . (4.57)
∂t 2 −∞ 2π −∞ 2π
Now let us consider a dispersive isotropic medium described by the constitutive rela-
˜ ˜
˜
˜
tions D = ˜ E, B = ˜µH. Since the imaginary parts of ˜ and ˜µ are associated with power
dissipation in the medium, we shall approximate ˜ and ˜µ as purely real. Then (4.57)
becomes
∂w e 1 ∞ dω ∞ dω ˜ ∗ ˜ − j(ω −ω)t
= E (ω ) · E(ω) jω˜ (ω) − jω ˜ (ω ) e .
∂t 2 −∞ 2π −∞ 2π
Substitution from (4.55) now gives
1 ∞ dω ∞
∂w e dω
= jω˜ (ω) − jω ˜ (ω ) ·
∂t 8 −∞ 2π −∞ 2π
ˇ ˇ ∗ ˜ ˜ ˇ ˇ ∗ ˜ ˜
· E · E F(ω − ω 0 )F(ω − ω 0 ) + E · E F(ω + ω 0 )F(ω + ω 0 )+
ˇ ∗ ˇ ∗ ˜
˜
ˇ
ˇ ˜
˜
+ E · EF(ω − ω 0 )F(ω + ω 0 ) + E · E F(ω + ω 0 )F(ω − ω 0 ) e − j(ω −ω)t .
˜
Let ω →−ω wherever the term F(ω + ω 0 ) appears, and ω →−ω wherever the term
˜
˜
˜
F(ω + ω 0 ) appears. Since F(−ω) = F(ω) and ˜ (−ω) = ˜ (ω), we find that
1 ∞ dω ∞ dω
∂w e ˜ ˜
= F(ω − ω 0 )F(ω − ω 0 ) ·
∂t 8 −∞ 2π −∞ 2π
j(ω−ω )t j(ω −ω)t
ˇ
ˇ
ˇ ∗
ˇ ∗
· E · E [ jω˜ (ω) − jω ˜ (ω )]e + E · E [ jω ˜ (ω ) − jω˜ (ω)]e +
ˇ
ˇ
ˇ ∗ ˇ ∗
+ E · E[ jω˜ (ω) + jω ˜ (ω )]e j(ω+ω )t + E · E [− jω˜ (ω) − jω ˜ (ω )]e − j(ω+ω )t .
(4.58)
For small α the spectra are concentrated near ω = ω 0 or ω = ω 0 . For terms involving
the difference in the permittivities we can expand g(ω) = ω˜ (ω) in a Taylor series about
ω 0 to obtain the approximation
ω˜ (ω) ≈ ω 0 ˜ (ω 0 ) + (ω − ω 0 )g (ω 0 )
where
∂[ω˜ (ω)]
g (ω 0 ) = .
∂ω
ω=ω 0
This is not required for terms involving a sum of permittivities since these will not tend
to cancel. For such terms we merely substitute ω = ω 0 or ω = ω 0 . With these (4.58)
becomes
∂w e 1 ∞ dω ∞ dω ˜ ˜
= F(ω − ω 0 )F(ω − ω 0 ) ·
∂t 8 −∞ 2π −∞ 2π
j(ω−ω )t j(ω −ω)t
ˇ
ˇ
ˇ ∗
ˇ ∗
· E · E g (ω 0 )[ j(ω − ω )]e + E · E g (ω 0 )[ j(ω − ω)]e +
ˇ
ˇ
ˇ ∗ ˇ ∗
+ E · E˜ (ω 0 )[ j(ω + ω )]e j(ω+ω )t + E · E ˜ (ω 0 )[− j(ω + ω )]e − j(ω+ω )t .
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