134
7 Modeling of the Optical Characteristics Fibrillar Structure
Expressions of the form (7.7)−(7.8) can be written as
U
j
V
j
=
U
j
inc (R j p )
V
j
inc (R j p )
+
U
j
scat (R j p )
V
j
scat (R j p )
+
N
k = j
U
k
scat (R kp )
V
k
scat (R kp )
.
(7.9)
The potential of the incident field for the case p- and s- polarization is the following
form
U
j
inc (R j p ) = e
−ikz sin θ e
−ikR jp , V
j
inc (R j p ) = 0,
(7.10)
V
j
inc (R j p ) = e
−ikz sin θ e
−ikR jp , U
j
inc (R j p ) = 0.
(7.11)
Using the expansion of a plane wave by a cylindrical wave functions and substituting
(7.12) for (7.10)−(7.11), we obtain
e
ikρ sin θ
=
∞
n=∞
(−i)
n J n (kρ)e
inθ
,
(7.12)
U
j
inc (R j p ) = e
−ikz sin θ
∞
n=∞
(−i)
n J n (kR j p cos θ n o )e
inϕ e
inγ jp , V
j
inc (R j p ) = 0,
(7.13)
V
j
inc (R j p ) = e
−ikz sin θ
∞
n=∞
(−i)
n J n (kR j p cos θ n o )e
inϕ e
inγ jp , U
j
inc (R j p ) = 0,
(7.14)
γ j p is angle to jth cylinder (see Fig. 7.1).
Combining the expressions (7.13)–(7.14) we obtain
Fig. 7.1 Geometric
illustration of scattering by
cylinders