92
5 Study of the Optical Characteristics of a Biotissue …
G(r, r
) =
1
4π
e
−ik R
R
, R = |r − r
|.
Taking into account expression (5.4), we obtain
E(r) = E inc (r) +
S
E re f (r
)
∂G(r, r
)
∂n
−
∂ E re f (r
)
∂n
G(r, r
)
d S
(5.5)
S
E inc (r
)
∂G(r, r
)
∂n
−
∂ E inc (r
)
∂n
G(r, r
)
d S = 0.
(5.6)
Expression (5.6) implies that all sources of the field lie within the surface. Subtracting expression (5.6) from (5.5), we arrive to the formula
E(r) = E inc (r) +
S
(E inc (r
) − E re f (r
))
∂G(r, r
)
∂n
−
−
∂ E inc (r
)
∂n
−
∂ E re f (r
)
∂n
G(r, r
)d S.
(5.7)
Substituting the value of the field and its derivative into this expression, we obtain
E(r) = E inc (r) +
S
V
E inc (r
)
∂G(r, r
)
∂n
−
∂ E inc (r
)
∂n
G(r, r
)
d S.
(5.8)
We will henceforth consider only the scattered field defined as
E scat (r) =
S
V
E inc (r
)
∂G(r, r
)
∂n
−
∂ E inc (r
)
∂n
G(r, r
)
d S.
(5.9)
Substituting into this formula the approximate expression for the derivative of
e
ik R
/R with respect to n and the corresponding approximate expression for e
ik R
/R,
which are defined as [9]
∂G
∂n
∼ =
1
4π
e
−ikr
r
∂e
i(k,r)
∂n
=
1
4π
e
−ikr
r
i(n, k)e
i(k,r)
,
∂ E inc
∂n
∼ =
1
4π
e
−ikr
r
∂e
−i(k 1 ,r)
∂n
=
=
1
4π
e
−ikr
r
i(n, k 1 )e
−i(k 1 ,r)
,
e
ik R
R
∼ =
1
4π
e
ikr
r
e
i(k,r)
,
(5.10)
we obtain
E scat (r ) = i
e
−ikr
4πr
q
S
V
ne
i(q,r)
d S,
(5.11)