Page 213 - Electromagnetics

we find that dissipation is associated with negative imaginary parts of the constitutive
parameters. To see this we write

1 ∞ jωt 1 ∞ jω t
˜
˜

E(r, t) = E(r,ω)e dω, D(r, t) = D(r,ω )e dω ,
2π 2π
−∞ −∞
and thus find
∂D 1 ∞ ∞
c ˜ c ˜ j(ω+ω )t

J · E + E · = 2 E(ω) · ˜ ¯ (ω ) · E(ω )e jω dω dω
∂t (2π)
−∞ −∞
c
where ˜ ¯ is the complex dyadic permittivity (4.24). Then
1 ∞ ∞ c
˜
˜
˜
˜

w Q = 2 E(ω) · ˜ ¯ (ω ) · E(ω ) + H(ω) · ˜ ¯µ(ω ) · H(ω ) ·
(2π)
−∞ −∞
∞

· e j(ω+ω )t dt jω dω dω . (4.42)
−∞
Using (A.4) and integrating over ω we obtain
1 ∞ c
˜
˜
˜
˜

w Q = E(−ω ) · ˜ ¯ (ω ) · E(ω ) + H(−ω ) · ˜ ¯µ(ω ) · H(ω ) jω dω . (4.43)
2π
−∞
Let us examine (4.43) more closely for the simple case of an isotropic material for
which
1 ∞
˜
˜
c

c
w Q = j ˜ (ω ) − ˜ (ω ) E(−ω ) · E(ω )+
2π
−∞
˜
˜

+ j ˜µ (ω ) − ˜µ (ω ) H(−ω ) · H(ω ) ω dω .
Using the frequency symmetry property for complex permittivity (4.17) (which also holds
for permeability), we find that for isotropic materials
c
c
c
c
˜ (r,ω) = ˜ (r, −ω), ˜ (r,ω) =−˜ (r, −ω), (4.44)

˜ µ (r,ω) = ˜µ (r, −ω), ˜ µ (r,ω) =− ˜µ (r, −ω). (4.45)
Thus, the products of ω and the real parts of the constitutive parameters are odd

functions, while for the imaginary parts these products are even. Since the dot products
of the vector fields are even functions, we find that the integrals of the terms containing
the real parts of the constitutive parameters vanish, leaving
1 ∞ c ˜ 2 ˜ 2
w Q = 2 −˜ |E| − ˜µ |H| ω dω. (4.46)
2π 0
Here we have used (4.3) in the form
˜
˜
˜ ∗
˜ ∗
E(r, −ω) = E (r,ω), H(r, −ω) = H (r,ω). (4.47)
Equation (4.46) leads us to associate the imaginary parts of the constitutive parameters
with dissipation. Moreover, a lossy isotropic material for which w Q > 0 must have at

least one of c and µ less than zero over some range of positive frequencies, while an
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