Biomedical Engineering Reference
In-Depth Information
2.2.3
Time-Reversal versus Reciprocity
We point out that the reciprocity relations (2.10) and (2.12) should
notbeconfusedwiththetime-reversaltransform.Time-reversalisa
fundamental symmetry that dictates the invariance of physical laws
between exchange of past and future. In classical mechanics, this
symmetry corresponds to a system described by its position and
momentum (
q
,
p
), which equations of motion are invariant through
the transformation (
q
,
p
,
t
)
→
(
q
,
−
p
,
−
t
). Although an isolated
mechanical system is time-reversal, a real system is obviously
always coupled with its environment, resulting in friction that
immediately breaks the time-reversal symmetry.
In the presence of electromagnetic fields
E
(
x
,
t
),
B
(
x
,
t
), this
time-reversal invariance is preserved if in addition to the variables
(
q
,
p
), one also transforms the field into
E
(
x
,
t
)
t
),
B
(
x
,
t
)
=−
B
(
x
,
−
t
). In order that Maxwell's equation be automatically
satisfied by the new solutions
E
,
B
, the electric current and charge
distributions are changed accordingly into
J
(
x
,
t
)
=−
J
(
x
,
−
t
),
ρ
(
x
,
t
)
=
ρ
(
x
,
−
t
). In the monochromatic regime where fields,
electric currents, and charges are conveniently described by their
time-Fourier transforms at the positive pulsation
ω
, the time-
reversal operation reads:
E
ω
(
x
)
=
E
ω
(
x
),
B
ω
(
x
)
=−
B
ω
(
x
)
J
ω
(
x
)
=−
J
ω
(
x
),
ρ
ω
(
x
)
=
ρ
ω
(
x
).
=
−
E
(
x
,
(2.13)
The time dependence is restored after integration over the
pulsation Spectrum, for example,
E
(
x
,
t
)
=
+∞
0
ω
E
ω
(
x
)
e
−
i
ω
t
d
+
+∞
0
ω
E
ω
(
x
)
e
+
i
ω
t
etc.
For optical situations in which a mo
dal expansi
on into plane
waves of vector
k
d
(
c
)
2
=
k
2
is considered
(
ω
beingthecomplex-valueddielectricpermittivityofthemedium),
time-reversal implies new modal components such as
k
x
x
+
k
y
y
,
k
z
=
ω
−
ω
=
ω
,
k
z
=
k
z
,
E
ω
,
±
(
k
)
=
E
ω
,
∓
(
−
k
),
B
ω
,
±
(
k
)
=−
B
ω
,
∓
(
−
k
),
(2.14)
d
2
k
e
i
kx
[
E
ω
,
+
(
k
)
e
ik
z
z
+
E
ω
,
−
(
k
)
e
−
ik
z
z
]etc.In
particular, if the medium is lossless, that is, Im
ag
[
where
E
ω
(
x
)
=
ω
]
=
0, the time-
|≤
ω
√
ω
/
|
reversal operation dictates (i.e., for
k
c
) a change in the
