13 Theoretical Generalization of the Optical Chirality to Arbitrary Optical Media
339
where F vacuum ≡ F , but C vacuum ≡
ε 0 E · (∇ × E) + μ 0 H · (∇ × H)
/2, i.e., that
given in (13.1), and S vacuum ≡ − [J · (∇ × E) + E · (∇ × J )] /2 being, respectively, the chirality flux density, the optical chirality density, and the source-like
term in free space. Curiously, albeit (13.45) was initially posed for optical fields in
the vacuum, it has been widely used for investigating chiroptical effects occurring in
material systems, including metals (or plasmonic structures) as well as metamaterials. However, it can be demonstrated that the general result given in (13.42) reduces
to (13.45) only for linear and lossless media, i.e., when D = ε 0 E and B = μ 0 H,
and assuming ε = μ = 1:
C =
1
2
E · (∇ × D) + H · (∇ × B)
=
1
2
ε 0 E · (∇ × E) + μ 0 H · (∇ × H)
= C vacuum ,
(13.46)
and
S =
1
2
∂ t E · (∇ × D) + ∂ t H · (∇ × B) − E · (∇ × J )
=
1
2
ε 0 ∂ t E · (−μ 0 ∂ t H) + μ 0 ∂ t H · (ε 0 ∂ t E + J ) − E · (∇ × J )
=
1
2
μ 0 ∂ t H · J − E · (∇ × J )
= −
1
2
[J · (∇ × E) + E · (∇ × J )] = S vacuum ,
(13.47)
where μ 0 ∂ t H = − (∇ × E). As one can see at a glance, the essential discrepancy
arises on account of the dispersion-related terms. This becomes much more evident
by rewriting the above expressions in terms of those for vacuum. Indeed, since
E · (∇ × D) = D · (∇ × E) + ∇ · (P × E) ;
(13.48a)
H · (∇ × B) = B · (∇ × H) + μ 0 ∇ · (M × H) ,
(13.48b)
the optical chirality density as given in (13.43) can be generally expressed as
C =
1
2
D · (∇ × E) + B · (∇ × H) + ∇ ·
P × E + μ 0 (M × H)
=
1
2
{ε 0 E · (∇ × E) + E · (∇ × P) + μ 0 H · (∇ × H) + μ 0 H · (∇ × M)}
= C vacuum + C medium = C vacuum + C
e
medium + C
m
medium ,
(13.49)
where the free-space contribution C vacuum , i.e., the original expression as given in
(13.1), has been separated from the one accounting for the medium contribution,
C medium ≡
E · (∇ × P) + μ 0 H · (∇ × M)
/2. Similarly, the source-like term can
also be recast as follows