Biomedical Engineering Reference
In-Depth Information
Finally, we point out that all genuine 2D chiral systems share
another important property, namely a breaking of time invariance
or reversal [42, 43, 47, 48]. To understand this peculiarity, we go
back to our discussio
n
of time reversal and consider the condition
thattheJonesmatrix
J
2
D
defi
n
edbyEq.(2.33)shouldfulfillinorder
to be unitary, that is,
J
−
1
J
2
D
. From Eq. (2.41), this leads to the
=
2
D
relation:
α
−
β
−
γα
α
∗
γ
∗
β
∗
α
∗
.
1
=
(2.44)
α
2
−
βγ
This implies
α
α
−
βγ
=
α
∗
,
β
−
βγ
=−
γ
∗
and
γ
−
βγ
=−
β
∗
.Bytaking
2
α
2
α
2
2
the norms of each terms we deduce
|
α
−
βγ
|=
1and
|
β
|=|
γ
|
.
Thislastequalitycontradictsthedefinitionof
J
2
D
andconsequently
suchaplanarchiralJonesmatrixcannotbeunitary.Thisimpliesthat
J
2
D
=
J
−
1,
∗
2
D
and that therefore
J
rec
2
D
is different from
J
inv
2
D
.
In other words, a 2D chiral system provides a perfect illustration
that time-reversal is necessarily different from reciprocity, that is,
path reversal. As time-reversal is a key property of fundamental
physical laws at the
microscopic
level, the only solution is to assume
that this breaking of time-reversal at the level of 2D chiral objects
is associated with
macroscopic
irreversibility. Indeed, the imaginary
part of the permittivity, for example, is connected to losses and
dissipation into the environment (seen as a thermal bath) and
the condition for its positivity implies a strong irreversibility in
the propagation. Similarly here, 2D optical chirality means that
some sources of irreversibility must be present in order to prohibit
unitarity of the Jones matrix. This is an interesting example where
two fundamental aspects of nature, namely chirality and time
irreversibility (intrinsically linked to the entropic time arrow,) are
intimately connected.
2.3.4
Generalization
ThemostgeneralJonesmatrix
J
characterizingachiralmediumcan
bewritten:
(
J
ll
+
J
rr
)
/
2
10
0
−
1
,
(
J
ll
−
J
rr
)
2
J
lr
J
chiral
=
+
J
rl
(
J
ll
+
J
rr
)
/
2
(2.45)
