Image Processing Reference

In-Depth Information

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

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