Biomedical Engineering Reference
In-Depth Information
sign of the wave vectors in the propagative sector corresponding
to a reversal of propagation direction for every plane-wave of the
modal expansion. Going back to the Jones matrix formalism, we can
transform the relation
E
(out)
J
E
(in)
J
−
1,
∗
E
(out),
∗
into
E
(in),
∗
.
Now, from the previous discussion concerning time reversibility,
the complex conjugated input field
E
(in),
∗
=
=
→
→
→
→
at
z
=−∞
corresponds
→
to the time-reversed output
E
(out)
computed at
z
=−∞
,
←
whereas the complex conjugated output field
E
(out),
∗
=+∞
at
z
→
corresponds to the time-reversed output
E
(in)
=+
computed at
z
←
∞
. The Jones matrix associated with time-reversal is therefore
J
yy
.
J
xy
−
J
yx
J
xx
−
1
J
xx
J
yy
−
J
xy
J
yx
J
inv
:
=
J
−
1,
∗
=
(2.15)
As it is clear from its definition,
J
inv
is in general different from
J
rec
, exemplifying the importance of losses and dissipation in the
relationbetweentimereversibilityandreciprocityinoptics.Thetwo
operators are indeed identical if, and only if,
J
is unitary, that is,
J
−
1
J
†
, meaning that an optical system through which energy
isconservedandwhichissimultaneouslyreciprocalwillbetheonly
optical system to be time-reversal invariant. This reveals the non-
equivalence between time reversibility and reciprocity. The latter is
more general: reciprocity can hold for systems in which irreversible
processes take place, as a fundamental consequence of Onsager's
principle of microscopic reversibility [50]. In the context of planar
chirality, this subtle link plays a fundamental role, as it will be
discussed in Section 2.3.3.
=
2.3 Optical Chirality
2.3.1
Chiral Jones Matrix
Following the operational definition of Lord Kelvin, the study
of chirality demands to characterize the optical behavior of the
considered system through a planar mirror symmetry
ϑ
.By
ϑ/
definition, an in-plane symmetry axis making an angle
2 with
respecttothe
x
-directionisassociatedwithtransformationmatrices
cos
ϑ
,
ϑ
=
0
e
−
i
ϑ
e
+
i
ϑ
0
,
sin
ϑ
ϑ
=
(2.16)
sin
ϑ
−
cos
ϑ
