338
J. E. Vázquez-Lozano and A. Martínez
several expressions for the chirality flux density wherein the fields B and H are
used interchangeably, yet leading to same results. However, this does not occur in
dispersive media, and special care should be taken in dealing with EM fields either in
free space (E and H), or within a medium (D and B). Be that as it may, for symmetry
reasons, one may heuristically assume that the chirality flux density actually reads
as
F ≡
1
2
E × (∇ × H) − H × (∇ × E)
.
(13.39)
This authorative choice of the definition is in fact that used in [80], and coincides
with that originally introduced by Tang and Cohen [31] for EM fields in free space.
Following a similar procedure as for the EM energy conservation law, one can readily
calculate the divergence of the chirality flux density with the aid of the Maxwell’s
equations:
∇ · F =
1
2
H · ∇ × (∇ × E) − E · ∇ × (∇ × H)
= −
1
2
H · ∂ t (∇ × B) + E · ∂ t (∇ × D) + E · (∇ × J )
.
(13.40)
To find out the conserved quantity and its accompanying source-like term, one should
compare the latter expression with the mathematical structure of the continuity equation as given in (13.22), and identify the total time derivative operator. To this aim,
it follows that
E · ∂ t (∇ × D) = ∂ t [E · (∇ × D)] − ∂ t E · (∇ × D) ;
(13.41a)
H · ∂ t (∇ × B) = ∂ t
H · (∇ × B)
− ∂ t H · (∇ × B) .
(13.41b)
In this manner, (13.40) can be recast as a true continuity equation, i.e.,
∇ · F + ∂ t C = S,
(13.42)
where the optical chirality density and the source-like terms are defined as
C ≡
1
2
E · (∇ × D) + H · (∇ × B)
,
(13.43)
S ≡
1
2
∂ t E · (∇ × D) + ∂ t H · (∇ × B) − E · (∇ × J )
.
(13.44)
It is worth pointing out that the above expressions represent the most general result
for the optical chirality conservation law, without any restrictions on the nature of
the medium. Nonetheless, they differ considerably from the previously established
for EM fields in free space [17, 31–33, 44, 77, 80]:
∇ · F vacuum + ∂ t C vacuum = S vacuum ,
(13.45)
J. E. Vázquez-Lozano and A. Martínez
several expressions for the chirality flux density wherein the fields B and H are
used interchangeably, yet leading to same results. However, this does not occur in
dispersive media, and special care should be taken in dealing with EM fields either in
free space (E and H), or within a medium (D and B). Be that as it may, for symmetry
reasons, one may heuristically assume that the chirality flux density actually reads
as
F ≡
1
2
E × (∇ × H) − H × (∇ × E)
.
(13.39)
This authorative choice of the definition is in fact that used in [80], and coincides
with that originally introduced by Tang and Cohen [31] for EM fields in free space.
Following a similar procedure as for the EM energy conservation law, one can readily
calculate the divergence of the chirality flux density with the aid of the Maxwell’s
equations:
∇ · F =
1
2
H · ∇ × (∇ × E) − E · ∇ × (∇ × H)
= −
1
2
H · ∂ t (∇ × B) + E · ∂ t (∇ × D) + E · (∇ × J )
.
(13.40)
To find out the conserved quantity and its accompanying source-like term, one should
compare the latter expression with the mathematical structure of the continuity equation as given in (13.22), and identify the total time derivative operator. To this aim,
it follows that
E · ∂ t (∇ × D) = ∂ t [E · (∇ × D)] − ∂ t E · (∇ × D) ;
(13.41a)
H · ∂ t (∇ × B) = ∂ t
H · (∇ × B)
− ∂ t H · (∇ × B) .
(13.41b)
In this manner, (13.40) can be recast as a true continuity equation, i.e.,
∇ · F + ∂ t C = S,
(13.42)
where the optical chirality density and the source-like terms are defined as
C ≡
1
2
E · (∇ × D) + H · (∇ × B)
,
(13.43)
S ≡
1
2
∂ t E · (∇ × D) + ∂ t H · (∇ × B) − E · (∇ × J )
.
(13.44)
It is worth pointing out that the above expressions represent the most general result
for the optical chirality conservation law, without any restrictions on the nature of
the medium. Nonetheless, they differ considerably from the previously established
for EM fields in free space [17, 31–33, 44, 77, 80]:
∇ · F vacuum + ∂ t C vacuum = S vacuum ,
(13.45)