Biomedical Engineering Reference
In-Depth Information
(with
δ
=
0,) which actually induces the chiral behavior. In the
cartesian basis, wecan equivalentlywrite
A
0
i
, (2.29)
+
B
cos
ϑ
B
sin
ϑ
γ
J
o.a.
=
+
B
sin
ϑ
A
−
B
cos
ϑ
−
i
γ
0
J
lr
e
i
ϑ
and
with
A
=
(
J
ll
+
/
=
γ
=
(
J
rr
−
/
2. The
presenceoftheantisymmetricalpartisthesignatureofchiralityand
the coe
cient
γ
=
0 is called (natural) gyromagnetic factor.
Animportantparticular caseconcernsJonesmatrices, whichare
invariant through a rotation by an angle
ϑ
around the
Z
axis. Such a
rotation is defined by the matrix
R
ϑ
with
J
rr
)
2,
B
J
ll
)
cos
ϑ
sin
ϑ
−
sin
ϑ
cos
ϑ
,
R
ϑ
=
e
+
i
ϑ
0
0
e
−
i
ϑ
.
R
ϑ
=
(2.30)
Theinv
a
riancebyrotationimpliesthatthematrix
J
R
=
R
ϑ
·
J
·
R
ϑ
−
1
equals
J
, that is:
J
ll
0
0
J
rr
.
J
rotation axis
=
(2.31)
If
J
ll
=
J
rr
then Eq. (2.31) is clearly a particular case of Eq. (2.27)
that actually describes optical activity in isotropic media, such as
quartz crystals or molecular solutions, and corresponds to the
circular birefringence (and dichroism) introduced by Fresnel in
1825. It is interesting to observe that this is also the matrix which
is associated with the gammadions artificial structure considered
in [21-24] and which have a four-fold rotational invariance
around
Z
.
Following its definition, the Jones enantiomorphic matrix as-
A
0
0
B
writes as
J
enant
B
0
0
A
.
=
ϑ
=
sociated with
J
o.a.
o.a.
(
)
Because of rotational invariance,
J
enant
o.a.
(
ϑ
) is independent of
ϑ
.
These two enantiomorphic matrices are associated with opposite
optical rotatory powers.
Consider for example the Jones matrix associated with optical
activity in an isotropic medium, such as a random distribution of
helices for
ex
ample.
Fr
om Eqs. (2.10, 2.22, 2.31) it is immediately
seen that
J
flip
=
J
. This invariance means that an observer
