Biomedical Engineering Reference
In-Depth Information
reflection axis parallel to
x
.Through
R
x
, the field vectors transform
as
E
x
ˆ
·
x
−
1
.
Fundamentally, it is interesting to note that a symmetry
operation, which is defined geometrically (and independently from
any set of physical laws), manifests itself specifically according to
a particular physical environment or context. From the point of
view of Maxwell's equations, this symmetry operation is indeed
implemented at the level of the susceptibility of the medium
interaction with light, that is, at the level ofthe Jones matrices.
Actually,thereciprocaltransformationdefinedaboveconstitutes
a rigorous and operational optical definition of the medium flipping
x
E
, so that the Jones matrix becomes
x
·
J
T
=
J
flip
:
=
x
·
J
rec
·
x
−
1
with
J
rec
=
J
T
.
(2.10)
It will be also convenient in the following to express the electric
field in the left (L) and rig
ht
(R) circul
ar
ly polarized light basis
defined by
L
,
R
=
(
x
±
i
y
)
/
√
2. We write
J
the Jones matrix in such
basis, and wehave
J
ll
J
lr
J
rl
J
rr
=
UJU
−
1
,
J
=
(2.11)
defines the unitary matrix associated with
this vectorbasis transformation.
1
−
i
1
i
1
√
2
where
U
=
With these definitions, Eq. (2.10) reads in the RCP and LCP basis
J
ll
J
rl
J
lr
J
rr
J
flip
J
T
=
=
(2.12)
J
rr
J
lr
J
rl
J
ll
=
J
T
as the transposition
T
and the
transformation
U
are not commutativeoper
at
ions).
To summarize, we showed that
J
flip
and
J
flip
define the genuine
representation of reciprocity and path reversal in a coordinate
systemhavingthesamehandednessastheoriginalone(i.e.,asseen
from the other side of the object). This corresponds to the axes
transformation
x
(remark that
J
T
=
x
,
y
y
,and
z
=
=−
=−
z
and it implies an
exchangein the role of LCP and RCP.
