Biomedical Engineering Reference
In-Depth Information
produced by the electric field, as the current runs along the spiral
axis only by passing through the spiral rings. As a result, one will
have
m
z
=
α
HH
H
z
+
α
HE
E
z
, (4.9)
where
α
HH
istheregularmagneticpolarizabilityofthespiraland
α
HE
is magnetoelectric cross-polarizability of the particle. Moreover, the
spiral is characterized by some other (transverse relatively to the
spiral axis) components of electric and magnetic dipole moments,
which also have a chiral origin. The above arguments become even
moreillustrativewhenappliedtothesecondpairofelementsshown
in Fig. 4.2b.
It is very important that the electromagnetic cross-polariza-
bilities are of inductive nature and are related to time dependence
of the incident fields. Thus, for the harmonic time dependence of
the electromagnetic fields, exp(
−
i
ω
t
), the electromagnetic cross-
polarizabilities can be presented in the form
α
EH
=
i
β
,
α
HE
=
−
i
β
where
β
is a real number, positive for left-handed spirals, and
negative for right-handed spirals. So, the longitudinal structure of
the spiral dipole moments can be rewritten as:
d
z
=
α
EE
E
z
+
i
β
H
z
,
m
z
=−
i
β
E
z
+
α
HH
H
z
. (4.10)
The properties of spirals (helices) were considered in more details
in ([4, 33, 53, 71, 79]). In particular, Jaggard, Mickelson, Papas [33]
have shown that for chiral “meta-atom” shown in Fig. 4.2B, the
expressions forcross-polarizabilities can bepresented in the form:
k
π
R
2
=
α
EE
k
π
R
2
L
β
=
α
HH
,
(4.11)
L
where
L
is the total height of the element in Fig. 4.2b and
R
is the
radius of its ring part and
K
is the wavenumber.
After the polarization properties of a single chiral nanoparticle
have been determined, the question of optical properties of
metamaterialsmadeofchiralmeta-atomsmaybeconsidered.Chiral
medium can be either an ordered structure in the form of a spatial
lattice of chiral meta-atoms or a chaotic mixture of chiral elements.
Asanexcitingexampleofaregularchiralstructure,theplanarchiral
metamaterial made from 3D gold nanohelices is shown in Fig. 4.3
([23]). This structure was recently fabricated by laser “writing” of
