Biomedical Engineering Reference
In-Depth Information
λ
±
=
λ
±
, but the
The eigenvalues are the same as for
J
2D
, that is,
eigenvectors are now
±
√
β
|
√
γ
|
L
R
|±
enant
=
.
(2.55)
|
β
|+|
γ
|
Importantly her,e
J
flip
J
enant
and therefore by applying Eq. (2.48)
=
|
=|±
on the initial states
,weget
|
out
=−
(
J
flip
)
2
ˆ
in
2
x
|
in
=∓
λ
±
|±
enant
(2.56)
whereweusedtherelations
ˆ
,
ˆ
,and
ˆ
x
|± =
±|±
enant
.Likewedidforopticalactivity,wecangobacktotheinitial
coordinate system
x
,
y
,
z
and the final states read now
ˆ
x
|
=|
x
|
=|
L
R
R
L
2
(2.57)
As before that again illustrates the effectiveness of the reciprocity
principle. Furthermore, as the transformation is not unitary, we
could notobtain such a result using
J
inv
as defined by Eq. (2.15).
x
|
out
=−
λ
±
|±
2.4 Discussion and Examples
As we mentioned in the Introduction, it is very interesting to
observe that the property concerning the change of twist for
genuine 2D chiral systems when watched from two different sides
stirred a considerable debate in the recent year in the context of
metamaterials.
To understand this in more detail, we remind that partly
boosted by practical motivations, such as the quest of negative
refractive lenses [20] or the possibility to obtain giant optical
activity for applications in optoelectronics, there is currently a
renewedinterest[20-30]intheopticalactivityinartificialphotonic
media with planar chiral structures. It was shown for instance
that planar gammadionic structures, which have by definition no
axis of reflection but a four-fold rotational invariance [21, 23], can
generate optical activity with giant gyrotropic factors [24, 28-30].
Importantly, and in contrast to the usual three-dimensional (3D)
chiral medium (like quartz and its helicoidal structure [3, 17]),
planar chiral structures change their observed handedness when
the direction of light is reversed through the system [21, 40]. This
