336
J. E. Vázquez-Lozano and A. Martínez
for its part, have a double dependency on both ω 0 and ω
0 . This distinction between
ω
0 and ω 0 is actually artificial, and it is intended to simplify the calculations. So, in
order to deal with this inconsistency, we make the following algebraic manipulation:
W
e
=
ε 0
4
ω
0 Re
ε(ω
0 )
− ω 0 Re[ε(ω 0 )]
ω
0 − ω 0
|E 0 |
2
.
(13.36)
In this way, ω
0 and ω 0 are treated on equal footing, thus showing much more clearly
a singular behavior when ω
0 = ω 0 . To overcome this issue one should take the limit
ω
0 → ω 0 , transforming the above expression into a derivative with respect to ω. By
performing the same calculation for the magnetic contribution one obtains that
W
e
≡ ≡W
e
T =
ε 0 ˜
ε eff (ω)
4
|E 0 |
2
ω→ω 0
;
(13.37a)
W
m
≡ ≡W
m
T =
μ 0 ˜
μ eff (ω)
4
|H 0 |
2
ω→ω 0
,
(13.37b)
where ˜
ε eff and ˜
μ eff are the real-valued effective material parameters (i.e., the electric
permittivity and magnetic permeability), which are defined as
˜
ε eff (ω) ≡
d
ωε
dω
,
˜
μ eff (ω) ≡
d
ωμ
dω
,
(13.38)
with ε
≡ Re[ε(ω)] and μ
≡ Re[μ(ω)].
By comparing (13.37a) and (13.37b) with (13.28a) and (13.28b), respectively, one
can observe that the differences among them are found in the dispersion-modified
material parameters. After some lengthy but straightforward calculations it can be
shown that ˜
ε eff (ω) = ε eff (ω) and ˜
μ eff (ω) = μ eff (ω) if and only if (ω
2
+ ω
2
n )γ
2
n = 0,
and (ω
2
+ ˜
ω
2
n ) ˜
γ
2
n = 0. Namely, if γ n = 0 and/or ω = ±iω n , and ˜
γ n = 0 and/or
ω = ±i ˜
ω n , for all n. Since ω must be real (otherwise one would deal with, a
priori, unphysical imaginary frequencies [64]), the only possible solution is that
γ n = ˜
γ n = 0, i.e., a lossless media. Furthermore, (13.37a) and (13.37b) equal (13.9)
if and only if ε = ε
, μ = μ
, and they do not depend on frequency (which is certainly
redundant on account of the Kramers-Kronig relations, since non-dispersive necessarily implies lossless, and vice versa). Hence, we can state that the sum of (13.28a)
and (13.28b), provide ultimately the most general definition of the EM energy density in dispersive and lossy media, since it allows us to recover the more particular
expressions successively, just by relaxing further assumptions.
J. E. Vázquez-Lozano and A. Martínez
for its part, have a double dependency on both ω 0 and ω
0 . This distinction between
ω
0 and ω 0 is actually artificial, and it is intended to simplify the calculations. So, in
order to deal with this inconsistency, we make the following algebraic manipulation:
W
e
=
ε 0
4
ω
0 Re
ε(ω
0 )
− ω 0 Re[ε(ω 0 )]
ω
0 − ω 0
|E 0 |
2
.
(13.36)
In this way, ω
0 and ω 0 are treated on equal footing, thus showing much more clearly
a singular behavior when ω
0 = ω 0 . To overcome this issue one should take the limit
ω
0 → ω 0 , transforming the above expression into a derivative with respect to ω. By
performing the same calculation for the magnetic contribution one obtains that
W
e
≡ ≡W
e
T =
ε 0 ˜
ε eff (ω)
4
|E 0 |
2
ω→ω 0
;
(13.37a)
W
m
≡ ≡W
m
T =
μ 0 ˜
μ eff (ω)
4
|H 0 |
2
ω→ω 0
,
(13.37b)
where ˜
ε eff and ˜
μ eff are the real-valued effective material parameters (i.e., the electric
permittivity and magnetic permeability), which are defined as
˜
ε eff (ω) ≡
d
ωε
dω
,
˜
μ eff (ω) ≡
d
ωμ
dω
,
(13.38)
with ε
≡ Re[ε(ω)] and μ
≡ Re[μ(ω)].
By comparing (13.37a) and (13.37b) with (13.28a) and (13.28b), respectively, one
can observe that the differences among them are found in the dispersion-modified
material parameters. After some lengthy but straightforward calculations it can be
shown that ˜
ε eff (ω) = ε eff (ω) and ˜
μ eff (ω) = μ eff (ω) if and only if (ω
2
+ ω
2
n )γ
2
n = 0,
and (ω
2
+ ˜
ω
2
n ) ˜
γ
2
n = 0. Namely, if γ n = 0 and/or ω = ±iω n , and ˜
γ n = 0 and/or
ω = ±i ˜
ω n , for all n. Since ω must be real (otherwise one would deal with, a
priori, unphysical imaginary frequencies [64]), the only possible solution is that
γ n = ˜
γ n = 0, i.e., a lossless media. Furthermore, (13.37a) and (13.37b) equal (13.9)
if and only if ε = ε
, μ = μ
, and they do not depend on frequency (which is certainly
redundant on account of the Kramers-Kronig relations, since non-dispersive necessarily implies lossless, and vice versa). Hence, we can state that the sum of (13.28a)
and (13.28b), provide ultimately the most general definition of the EM energy density in dispersive and lossy media, since it allows us to recover the more particular
expressions successively, just by relaxing further assumptions.