Page 215 - Electromagnetics

4.5.2 Energy stored in a dispersive material
In the previous section we were able to isolate the dissipative effects for a dispersive
material under special circumstances. It is not generally possible, however, to isolate
a term describing the stored energy. The Kronig–Kramers relations imply that if the
constitutive parameters of a material are frequency-dependent, they must have both real
and imaginary parts; such a material, if isotropic, must be lossy. So dispersive materials
are generally lossy and must have both dissipative and energy-storage characteristics.
However, many materials have frequency ranges called transparency ranges over which

˜ c and ˜µ are small compared to ˜ and ˜µ . If we restrict our interest to these ranges,
c
we may approximate the material as lossless and compute a stored energy. An important
special case involves a monochromatic field oscillating at a frequency within this range.
To study the energy stored by a monochromatic field in a dispersive material we
must consider the transient period during which energy accumulates in the fields. The
assumption of a purely sinusoidal field variation would not include the effects described
by the temporal constitutive relations (2.29)–(2.31), which show that as the field builds
the energy must be added with a time lag. Instead we shall assume fields with the
temporal variation
3

E
E(r, t) = f (t) ˆ i i |E i (r)| cos[ω 0 t + ξ (r)] (4.51)
i
i=1
where f (t) is an appropriate function describing the build-up of the sinusoidal field. To
compute the stored energy of a sinusoidal wave we must parameterize f (t) so that we
may drive it to unity as a limiting case of the parameter. A simple choice is

π ω 2
2 2
−α t
˜
f (t) = e ↔ F(ω) = e − 4α 2 . (4.52)
α 2
Note that since f (t) approaches unity as α → 0, we have the generalized Fourier trans-
form relation
˜
lim F(ω) = 2πδ(ω). (4.53)
α→0
Substituting (4.51) into the Fourier transform formula (4.1) we find that
3
3
1
E 1
E
˜ ˆ jξ (r) ˜ ˆ − jξ (r) ˜
E(r,ω) = i i |E i (r)|e i F(ω − ω 0 ) + i i |E i (r)|e i F(ω + ω 0 ).
2 2
i=1 i=1
We can simplify this by defining
3
E
ˇ
ˆ jξ (r)
E(r) = i i |E i (r)|e i (4.54)
i=1
as the phasor vector field to obtain
˜ 1 ˇ ˜ ˇ ∗ ˜
E(r,ω) = E(r)F(ω − ω 0 ) + E (r)F(ω + ω 0 ) . (4.55)
2
We shall discuss the phasor concept in detail in § 4.7.
˜
The field E(r, t) is shown in Figure 4.2 as a function of t, while E(r, ω) is shown in
Figure 4.2 as a function of ω. As α becomes small the spectrum of E(r, t) concentrates
around ω =±ω 0 . We assume the material is transparent for all values α of interest so

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