Image Processing Reference
InDepth Information
2. At high frequencies this simplifies to
�
ω
ω
2

‚
p
ε
=
ε
1
−
(3.16)
eff
0
2
We can now use this expression to define
Ne
m
2
ω
=
(3.17)
2
p
ε
0
where ω
p
is defined as the plasma frequency. It should now be noted
that when ω < ω
p
, then ε
eff
< 0 and when ω > ω
p
, then ε
eff
> 0.
It will be shown that when we develop a wave equation, that wave equation
still holds when we replace ε by this new quantity ε
eff
. However, when ε
eff
is
complex, then the wavenumber or propagation constant
k
is complex. If
k
has
a nonzero imaginary part, then there is no propagation and the wave is said to
be evanescent, which means it exponentially decays in the direction of propa
gation. We will see later in the context of imaging, that
k
becomes imaginary
when very high resolution information (i.e., high “spatial” frequency informa
tion) is imposed on an applied propagating field
3.1.4 Increasing
N
and local Fields
If polarizable regions in a material do not interact, as might be the case in a gas,
then we can express the macroscopic properties of a bulk material or object
we are illuminating in terms of the polarizability of each unit, for example,
atom or (nonpolar) molecule (Table 3.1). We expressed that
PNE
=α where α
was the polarizability of the “unit” in the material and
N
, the number density
of these polarizable units. From this we can write ε
and since
=+
1
N
α ε
/
r
0
=
2
, where
n
is the refractive index, it follows that
ε
r
n
=+
[
1
N
αε
/
]
12
/
~
1
+
N
αε
/
2
(3.18)
r
0
This last step is from truncating a Taylor series expansion, which is justi
fied for a “low concentration” of scattering units (such as a gas) and defines it
as a weak scatterer showing that the index is simply proportional to
N
.
As the density of the units increases, we cannot neglect the effects of the local
field, coupling, and multiple scatter
i
ng between neighboring units or inho
mogenieties. We should now write
PNENgE
loc ext
and ε
r
= 1 +
N
α
g
/ε
0
.
A very simple model to approximate these local field effects is to define a
=
α
=
α
table 3.1
Sample Material Properties
Form
Density
N
x
/
ε
0
ε
r
CS
2
Gas
0.00339
0.0029
1.0029
Liquid
1.293
1.11
2.76
O
2
Gas
0.00143
0.000523
1.000523
Liquid
1.19
0.435
1.509
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