Page 214 - Electromagnetics

active isotropic medium must have at least one of these greater than zero. In general,
we speak of a lossy material as having negative imaginary constitutive parameters:

˜ c < 0, ˜ µ < 0, ω > 0. (4.48)
A lossless medium must have
˜ = ˜µ = ˜σ = 0

for all ω.
Things are not as simple in the more general anisotropic case. An integration of (4.42)

over ω instead of ω produces
1 ∞ c
˜
˜
˜
˜
w Q =− E(ω) · ˜ ¯ (−ω) · E(−ω) + H(ω) · ˜ ¯µ(−ω) · H(−ω) jω dω.
2π
−∞
Adding half of this expression to half of (4.43) and using (4.25), (4.17), and (4.47), we
obtain
1 ∞ c
˜
˜
˜
˜
∗
c∗
˜ ∗
˜ ∗
˜ ∗
˜ ∗
w Q = E · ˜ ¯ · E − E · ˜ ¯ · E + H · ˜ ¯µ · H − H · ˜ ¯µ · H jω dω.
4π
−∞
Finally, using the dyadic identity (A.76), we have
1 ∞ c
˜
˜
˜ ∗
˜ ∗
w Q = E · ˜ ¯ − ˜ ¯ c† · E + H · ˜ ¯µ − ˜ ¯µ † · H jω dω
4π
−∞
where the dagger (†) denotes the hermitian (conjugate-transpose) operation. The condi-
tion for a lossless anisotropic material is
c
c†
†
˜ ¯ = ˜ ¯ , ˜ ¯ µ = ˜ ¯µ , (4.49)
or
∗
∗
∗
˜ ij = ˜ , ˜ µ ij = ˜µ , ˜ σ ij = ˜σ . (4.50)
ji
ji
ji
These relationships imply that in the lossless case the diagonal entries of the constitutive
dyadics are purely real.
Equations (4.50) show that complex entries in a permittivity or permeability matrix
do not necessarily imply loss. For example, we will show in § 4.6.2 that an electron
plasma exposed to a z-directed dc magnetic field has a permittivity of the form
 
− jδ 0
[ ˜ ¯ ] =  jδ 0 
0 0 z
where , z , and δ are real functions of space and frequency. Since ˜ ¯ is hermitian it
describes a lossless plasma. Similarly, a gyrotropic medium such as a ferrite exposed to
a z-directed magnetic field has a permeability dyadic
 
µ − jκ 0
[ ˜ ¯µ] =  jκµ 0  ,
0 0 µ 0
which also describes a lossless material.
© 2001 by CRC Press LLC