13 Theoretical Generalization of the Optical Chirality to Arbitrary Optical Media
341
well with experiments [72–74], and thus, can be regarded as absolutely general for
characterizing the optical response of whatever material system, for any frequency
and bandwidth.
13.4.1 Optical Chirality Density in Dispersive and Lossless
Media: Brillouin’s Approach
By regarding the original definition of the optical chirality density in the timedependent representation, it is easy to show that, for monochromatic electric and magnetic fields in free space, E(r, t) = Re[E(r)e
−iωt
] and H(r, t) = Re[H(r)e
−iωt
], the
time-averaged optical chirality density is given by [31, 32]:
C vacuum ≡ ≡C vacuum T =
ω
2c 2 Im
E · H
∗
.
(13.52)
From this, it is straightforward to verify that the maximum value of C is achieved
when assuming freely propagating EM plane waves with circular polarization (CP),
i.e., corresponding to the eigenstates associated to the optical chirality [44]:
C
(±)CP
vacuum = ±
ω
2c 2
1
Z 0
|E|
2
,
(13.53)
where Z 0 ≡
√
μ 0 /ε 0 is the vacuum impedance, and the signs + and − correspond
to the left- and right-handed circular polarizations.
Now, the aim of this section is finding out a closed expression generalizing the
above to dispersive media in the lossless case. For simplicity, hereinafter it will be
assumed a linear, homogeneous and isotropic medium such that D = ε 0 ε(ω)E and
B = μ 0 μ(ω)H. Then, starting from the continuity equation as given in (13.40), and
taking into account the above discussion, let us expand the right-hand side by using
the Fourier integrals:
1
2
E · ∂ t (∇ × D) =
1
2
+∞
−∞
E(ω
)e
−iω
t dω
· ∂ t
∇ ×
+∞
−∞
D(ω)e
−iωt dω
=
−i
2
+∞
−∞
+∞
−∞
ωE(ω
) · [∇ × D(ω)] e
−i(ω
+ω)t dω
dω;
(13.54a)
1
2
H · ∂ t (∇ × B) =
1
2
+∞
−∞
H(ω
)e
−iω
t dω
· ∂ t
∇ ×
+∞
−∞
B(ω)e
−iωt dω
=
−i
2
+∞
−∞
+∞
−∞
ωH(ω
) · [∇ × B(ω)] e
−i(ω
+ω)t dω
dω.
(13.54b)