Scattering Fundamentals - Introduction to Imaging from Scattered Fields - page 35

3.2 weAk SCAtteReRS
Note that for all dielectric media for which ε r > 1 it is apparent that the local
field is greater than the external applied field This sometimes causes confu-
sion because we usually think of the dielectric constant of a material as pro-
viding a screening of the applied field, thereby making the local field smaller.
However, this ignores the polarization of the material itself, and inside the
sphere, we observe the external field plus the field due to the medium's polar-
ization. It will be explained later how, in the frequency domain, the real and
imaginary parts of the complex permittivity are locked together through dis-
persion relations (Kramer-Kronig dispersion relations or in the mathematics
community, Hilbert transforms). This is very fundamental, being based on the
causality principle, and very important since it allows one to compute the real
part from the imaginary part and vice versa. It will be shown that controlling
spectral absorption profiles directly determines spectral refractive index pro-
files, and we usually are trying hard to manage the absorption and dispersion
in the materials. Another remark worth noting is that while we have used the
polarizability α e here, it could represent a polarizability arising from a num-
ber of different physical mechanisms including electronic, ionic, permanent
dipole rotational, or displacement effects.
We will assume for the sake of simplicity that our scattering objects will
not have a permanent dipole moment. There are models to deal with this case
and incorporate their thermally randomized alignment known as the Debye
model, but even this fails for simple units like H 2 O because of the complexity
of local interactions. The Onsager model attempts to create a cavity around a
single unit, and the Kirkwood model creates a small cluster around the unit.
Both may be important to help advance models for strongly resonant atoms or
structures. It is very interesting to note also that in practice when it comes to
coupling effects between neighboring units, usually less than 10 neighbors are
important and sometimes only immediate nearest neighbors are important.
3.3 SCAtteRIng FRoM CoMpACt StRuCtuReS
Most objects to be imaged by electromagnetic waves are 3-D in nature, and
there is to date no well-developed diffraction or scattering theory for such
objects. Solutions of the scattering problem based on Maxwell's equations
have been obtained for some idealized objects such as spheres or cylinders
(Barber and Yeh, 1975; Logan, 1965). However, even in these cases, the solu-
tions are generally expressed as infinite series of special functions, which
makes the interpretation of the solutions difficult As a result of this, many
approximate methods have been proposed which do allow the scattered field
to be expressed analytically or which are convenient for numerical experi-
ments (Keller, 1961).
Inverse scattering methods for studying the detailed local structure
inside penetrable compact objects are the focus of much of this text. There
are two approaches to determining the internal structure of such objects
from scattered and transmitted field data. The first relies on a geometrical
optics description, which requires the assumption that the wavelength used
is considerably smaller than the structural detail of interest. This assump-
tion is adequate at x-ray wavelengths when used in computerized tomography