Biomedical Engineering Reference
In-Depth Information
respectively, written in cartesian and circular bases with
ϑ
=
0
=
x
. Through such
ϑ
symmetry operation, the Jones matrix
transforms as
J
=
ϑ
J
ˆ
ˆ
−
1
ϑ
in the cartesian basis and as
J
rr
J
rl
e
−
i
2
ϑ
J
lr
e
+
i
2
ϑ
.
J
=
ϑ
·
J
·
ϑ
−
1
=
(2.17)
J
ll
in the circular basis.
With Kelvin's definition, a system will be optically non-chiral
if, and only if, it is invariant under
ϑ
, meaning that
J
=
J
or equivalently that the operators, respectively, associated with
the Jones matrix and the mirror-symmetry matrix c
o
mm
ut
e as
[
J
,
ˆ
ϑ
J
=
0. The invariance condition
J
=
J
mirror
enforces two constraints on the Jones matrix coe
cients, namely
that
J
ˆ
ˆ
ϑ
]
=
ϑ
−
J
ll
=
J
rr
and
J
rl
=
J
lr
e
2
i
ϑ
.
(2.18)
This implies that the Jones matrix associated with a non-chiral
optical system has the followinggeneral form
A
+
B
cos
ϑ
B
sin
ϑ
J
mirror
=
(2.19)
B
sin
ϑ
A
−
B
cos
ϑ
By contrapositiveof conditions (2.18), we seethat
Theorem:
Optical chiralityis possible if, and only if,
J
ll
=
J
rr
OR
|
J
lr
| =|
J
rl
|
,
OR beingthe logical disjunction.
This constitutes a theorem equivalent to Kelvin's statement that
an optically chiral system has no mirror symmetry, with
J
=
J
or, equivalently, with non-commuting operators, respectively,
associated with the Jones matrix and the mirror-symmetry matrix
as
[
J
,
ˆ
J
ˆ
ϑ
J
=
0forany
ϑ
.
ˆ
ϑ
]
=
ϑ
−
(2.20)
Such a Jones matrix can bewritten in thefollowing form:
(
J
ll
+
J
rr
)
/
2
10
0
−
1
. (2.21)
(
J
ll
−
J
rr
)
2
J
lr
J
=
+
J
rl
(
J
ll
+
J
rr
)
/
2
