Biomedical Engineering Reference
In-Depth Information
Figure 2.1
Thetwodifferentreciprocalhistoriesforalightbeamthrougha
medium.Theinitialpathgoingfromlefttorightisrepresented bytheJones
matrix,
J
whilethereversedpaththatgoesfromrighttoleftcorrespondsto
the matrix
J
rec
. (See Eqs. (2.5)and (2.10)).
plane wave propagating along the
−
z
direction
E
(in)
impinging
from the other side of the system and which after transmission
gives the output state
E
(out)
←
. Here, by definition,
J
rec
is the reciprocal Jones operator associated with
J
. Using Maxwell
equations, one can easily show that
J
rec
J
rec
E
(in)
=
←
←
J
T
,where
T
denotes the
transposition in the cartesian basis
x
,
y
. The proof reads as follow:
first, from Maxwell equations, one deduces [1] the reciprocity
theorem of Lorentz, which states that if in a passive and linear
environment,weconsidertwospacepoints
A
and
B
,thenthevector
field
E
(
B
)in
B
produced by a (harmonic) point-like dipole source
P
(
A
) located in
A
is linked to the field
E
(
A
)in
A
produced by a
second point-likedipole
P
(
B
) located in
B
through the formula
=
E
(
A
)
P
(
B
)
P
(
A
)
·
=
·
E
(
B
)
(2.2)
(the time harmonic dependency
e
−
i
ω
t
has been dropped every-
where).
Now, the electric field produced in
M
by a point-like dipole
located in
M
is written
E
(
M
)
P
(
M
)where
G
(
M
,
M
)
denotes the dyadic Green's function for this environment [46].
G
(
M
,
M
)
=
·
