Image Processing Reference
In-Depth Information
Equations 3.42 and 3.43 are very useful in that they are easily implemented
and approximate the scattering cross section predicted by the Lorenz-Mie
theory to within 1% (Mohlenhoff et al. 2005; Walstra, 1964).
In a later chapter, the “
Q
” factor will be utilized in explaining a cyclic phe-
nomenon associated with the performance of reconstructed images obtained
from the Born approximation method. It will be shown that resonance, or lack
thereof, plays a vital role in the ability to successfully reconstruct an image
using the techniques described in this topic.
The important and valuable feature of Mie scattering is that one can exactly
know the resonances and all one needs for this is the index or (ε
r
, μ
r
) and
x
= 2π
a
/λ (
a
is sphere radius, λ is wavelength in the external medium). Mie
scattering is rigorous for spheres and other convex shapes. Only the index
m
and
a
/λ are more important. One can typically expect strong resonances,
especially for larger |
m
|.
If we recall that the polarizability of a scattering atom is expressed as
(3.44)
PE
=α
where
a
=∑ cos (where ω
n
is the Mie resonant frequency of the
n
th
mode). In addition, consideration should be given to other contributions to α:
a
wt
n
n
ααααα
=++++
e
(3.45)
a
d
s
where α
e
is the electronic component, α
a
is the atomic component, α
d
is the
orientational component, α
s
is the shape component, etc.
The next logical question to explore is whether we can relate the effective
refractive index of a material to its dipolar or Mie scattering. The scattered
field from one particle is
ee
ikr
−
ikr ikz
(3.46)
uS u
=
(, )
θφ
0
where
S
(θ, ϕ) is the individual particle scattering pattern and
u
0
is the incident
wave,
e
−
ikz
+
i
ω
t
. In the forward direction, the total field is
10
1
�
∑
uu
+
S
()
ikr
e
−
((
ik xy r
2
+
2
)
/)
2
(3.47)
0
2
π
�
u
1
−
NLS
()
0
(3.48)
0
k
2
which is the approximate expression for total field from integrating over a slab
of length L.
If we replace the medium by an equivalent medium having complex index
profile,
m
, and write
e
1
−
ikL m
(
−
1 , then we can write
)
−
ikLm
(
−
1
)
()
π
2
N
miS
(3.49)
=−
1
0
=−′
n n
k
3
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