Biomedical Engineering Reference
In-Depth Information
Equation (2.2) thus reads
P
(
A
)
·
(
G
(
A
,
B
)
·
P
(
B
))
=
P
(
B
)
·
(
G
(
B
,
A
)
·
P
(
A
)) (2.3)
or equivalently in tensorial notation
i
,
j
(
G
ij
(
A
,
B
)
−
G
ji
(
B
,
A
))
P
i
(
A
)
P
j
(
B
)
=
=
1, 2, 3). This relation is valid for
every point dipoles in
A
and
B
and implies consequently
G
ij
(
A
,
B
)
0(
i
,
j
G
ji
(
B
,
A
). (2.4)
Actually, this relation constitutes a Maxwellian formulation of the
principle of light path reversal used in optical geometry. In the
next step of the proof, we consider
A
and
B
located in
z
=±∞
;
the fields can be then considered asymptotically as plane waves
and in the paraxial approximation
G
ij
(
−∞
,
+∞
) identifies with the
Jones matrix
J
ij
. We immediately see that the matrix
G
ij
(
+∞
,
−∞
)
identifies with the reciprocal matrix
J
rec
=
ij
. In other words, from the
point of view of Jones formalism, the principle of reciprocity states
J
xx
J
yx
J
xy
J
yy
.
J
rec
:
J
T
=
=
(2.5)
In this context, it is relevant to point out the similarity between
the reasoning given here for establishing the reciprocity theorem
and the one used in textbooks and articles [1, 2, 39] for establishing
the symmetry of the permittivity tensor
i
,
j
(
i
,
j
=
1, 2, 3) in solids.
Inparticular,bytakingintoaccountspatialnon-locality,itispossible
to obtaina version of the reciprocity theorem that reads
i
,
j
(
ω
,
−
k
)
=
j
,
i
(
ω
,
k
) (2.6)
where
k
is the wavevector of the monochromatic plane wave. The
analogy with Eq. (2.5) is complete if we choose the wave vector
along the
z
axis and if
i
,
j
corresponds to either
x
or
y
. Because of
thesesimilarities,manyreasoningdonefortheJonesmatrixthrough
thischaptercouldbeeasilyrestatedfortheelectricpermittivity
or
magnetic permeability
μ
tensors.
2.2.2
Rotation of the Optical Medium and Reciprocity:
Conserving the Handedness of the Reference Frame
Using the previous formalism, the reciprocity principle gives us a
univocal way to calculate the transmitted light beam propagating in
