Biomedical Engineering Reference
In-Depth Information
this chapter. First, we consider the case ofoptical activity where
α
0
0
β
=
J
flip
.
=
J
o.a.
(2.49)
The two eigenstates are
|
L
and
|
R
corresponding to eigenvalues
α
,respectively.However,
J
flip
J
J
enant
and although
and
β
=
=
the two eigenstates of
J
enant
are still
|
L
and
|
R
the eigenvalues
β
α
are now exchanged, that is,
, respectively. Therefore, a direct
application of Eq. (2.48) on any vector
and
|
=
|
+
|
in
a
L
b
R
leads to
J J
enant
ˆ
|
out
=−
x
|
=−
αβ
|
+
|
in
(
a
R
b
L
).
(2.50)
If we are only interested into the field expression in the original
coordinate system, wecan alternatively rewrite
ˆ
x
|
out
=−
αβ
(
a
|
L
+
b
|
R
)
=−
αβ
|
in
. (2.51)
This is a direct formulation of the fact that path reversal should
lead us here back to the initial state
|
in
as expected. It illustrates
the impact of reciprocity on propagation and shows that a 3D chiral
mediumdoesnotactasanopticalisolator.Wealsopointoutthatfor
a
loss-l
ess
ideal medium represented by a unitary rotation matrix
J
o.a.
=
R
ϑ
with eigenvalues
e
±
i
ϑ
, we have exactly
ˆ
x
|
out
=−|
in
,
which, up to the minus sign coming from the mirror reflection, is a
perfectillustrationoftimereversalandsymmetryfornaturaloptical
activity.
We now consider 2D planar chirality. The eigenstates and
eingenvalues of the chiral Jones matrix
αβ
γα
J
2D
=
(2.52)
defined by
J
2D
|±
|±
=
λ
±
|±
are by definition states
.After
straightforward calculations, we obtain
λ
±
=
α
±
(
βγ
),
(2.53)
and
√
β
|
±
√
γ
|
L
R
|± =
.
(2.54)
|
β
|+|
γ
|
Using similar method
s,
we c
an
easily find eigenstates and values of
thereciprocalmatrix
J
flip
J
enant
2D
suchas
J
flip
|±
enant
=
λ
±
|±
enant
.
=
2D
2D
