Scattering Fundamentals - Introduction to Imaging from Scattered Fields - page 33

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