1.5 Structure-Dependent Refractive Index
13
Fig. 1.7 Schematic illustration of metal-dielectric-layer-based electric metamaterial, illuminated
by parallel and perpendicular polarization
1
⊥
=
E ⊥
D ⊥
(1.47)
=
f m E m
D ⊥
+
f d E d
D ⊥
(1.48)
=
f m
m
+
f d
d
(1.49)
On the contrary, for parallel polarization, the electric field is continuous across
the interface, i.e., E m = E d = E || , whereas the displacement vector is discontinuous.
Hence, in this case, the effective displacement vector needs to be taken as a weighted
mean, given by
D || = f m D m + f d D d
(1.50)
Therefore, the effective permittivity is
|| =
D ||
E ||
(1.51)
=
f m D m
E ||
+
f d D d
E ||
(1.52)
= f m m + f d d
(1.53)
The effective permittivity as a function of wavelength, for the perpendicular and
parallel polarization, has been shown in Figs. 1.8 and 1.9. Looking at these figures,
it is clear that the effective permittivity, hence the effective refractive index, can be
controlled at will by controlling the structural parameters.