13 Theoretical Generalization of the Optical Chirality to Arbitrary Optical Media
333
the electric and the magnetic contributions, it can be demonstrated that (13.21a) and
(13.21b) can be recast as
E · ∂ t D = ∂ t
ε 0
2
E
2
+
N −1
n=0
1
2ε 0 f n ω 2
p
(∂ t P n )
2
+ ω
2
n P n
2
+
N −1
n=0
γ n
ε 0 f n ω 2
p
(∂ t P n )
2
;
(13.23a)
H · ∂ t B = ∂ t
μ 0
2
H
2
+
N −1
n=0
μ 0
2 ˜
f n ˜
ω 2
n
(∂ t M n )
2
+ ˜
ω
2
n (M n )
2
+
N −1
n=0
μ 0 ˜
γ n
˜
f n ˜
ω 2
n
(∂ t M n )
2
.
(13.23b)
Equations (13.23a) and (13.23b) result in a total time derivative plus a residual term,
i.e.,
E · ∂ t D = ∂ t
W
e
vac + W
e
med
+ L
e
;
(13.24a)
H · ∂ t B = ∂ t
W
m
vac + W
m
med
+ L
m
.
(13.24b)
Hence, from (13.23a) and (13.23b) it is easy to see that the electric and magnetic
contributions to the energy density are given by
W
e
= W
e
vac + W
e
med =
ε 0
2
E
2
+
N −1
n=0
1
2ε 0 f n ω 2
p
(∂ t P n )
2
+ ω
2
n P n
2
; (13.25a)
W
m
= W
m
vac + W
m
med =
μ 0
2
H
2
+
N −1
n=0
μ 0
2 ˜
f n ˜
ω 2
n
(∂ t M n )
2
+ ˜
ω
2
n (M n )
2
,
(13.25b)
where the contributions related to the vacuum and the medium, are, in turn, separated
from each other. Furthermore, the terms accounting for the power loss densities are
L
e
=
N −1
n=0
γ n
ε 0 f n ω 2
p
(∂ t P n )
2
;
(13.26a)
L
m
=
N −1
n=0
μ 0 ˜
γ n
˜
f n ˜
ω 2
n
(∂ t M n )
2
;
(13.26b)
L
c
= −J · E.
(13.26c)
Putting it all together, the EM energy conservation law finally reads as