Page 253 - Electromagnetics

Materials are then classified as follows:

p in (r) = 0, lossless,
p in (r)> 0, lossy,
p in (r) ≥ 0, passive,
p in (r)< 0, active.
E
c
B
D
We see that if ξ = ξ , ξ H = ξ , and J = 0, then the material is lossless. This implies
i i i i
that (D,E) and (B,H) are exactly in phase and there is no conduction current. If the
material is isotropic, we may substitute from the constitutive relations (4.21)–(4.23) to
obtain
3 "
#
ˇ ω
2 | ˜σ| 2
µ
σ

p in (r) =− |E i | |˜ | sin(ξ ) − cos(ξ ) +| ˜µ||H i | sin(ξ ) . (4.148)
2 ˇ ω
i=1
The first two terms can be regarded as resulting from a single complex permittivity
(4.26). Then (4.148) simplifies to
3
ˇ ω
c 2 c 2 µ
p in (r) =− |˜ ||E i | sin(ξ ) +| ˜µ||H i | sin(ξ ) . (4.149)
2
i=1
Now we can see that a lossless medium, which requires (4.149) to vanish, has ξ c =
µ
ξ = 0 (or perhaps the unlikely condition that dissipative and active effects within the
µ
electric and magnetic terms exactly cancel). To have ξ = 0 we need B and H to be in

phase, hence we need ˜µ(r,ω) to be real. To have ξ c = 0 we need ξ = 0 (˜ (r,ω) real)
and ˜σ(r,ω) = 0 (or perhaps the unlikely condition that the active and dissipative effects
of the permittivity and conductivity exactly cancel).
µ
A lossy medium requires (4.149) to be positive. This occurs when ξ < 0 or ξ c < 0,
meaning that the imaginary part of the permeability or complex permittivity is negative.
The complex permittivity has a negative imaginary part if the imaginary part of ˜ is

negative or if the real part of ˜σ is positive. Physically, ξ < 0 means that ξ D <ξ E and
thus the phase of the response field D lags that of the excitation field E. This results
from a delay in the polarization alignment of the atoms, and leads to dissipation of power
within the material.
µ
An active medium requires (4.149) to be negative. This occurs when ξ > 0 or ξ c > 0,
meaning that the imaginary part of the permeability or complex permittivity is positive.
The complex permittivity has a positive imaginary part if the imaginary part of ˜ is
positive or if the real part of ˜σ is negative.
In summary, a passive isotropic medium is lossless when the permittivity and perme-
ability are real and when the conductivity is zero. A passive isotropic medium is lossy
when one or more of the following holds: the permittivity is complex with negative imag-
inary part, the permeability is complex with negative imaginary part, or the conductivity
has a positive real part. Finally, a complex permittivity or permeability with positive
imaginary part or a conductivity with negative real part indicates an active medium.
For anisotropic materials the interpretation of p in is not as simple. Here we find that
the permittivity or permeability dyadic may be complex, and yet the material may still
be lossless. To determine the condition for a lossless medium, let us recompute p in using
the constitutive relations (4.18)–(4.20). With these we have
∂D c ∂B
E E

3

E · + J + H · = ˇω |E i ||E j | −|˜ ij | sin( ˇωt + ξ + ξ ) cos( ˇωt + ξ ) +
i
j
ij
∂t ∂t
i, j=1
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