Electromagnetic Waves - Introduction to Imaging from Scattered Fields - page 30

= ∑

ikxkykzt ( ++− ω )

Er () E (, ) kke

x y z

(2.91)

0 xy

kk ,

xy

where

k = ω 2 / c

2

− kk

2

−

2

(2.92)

z 0 x y

However, k z takes only real values if

ωπ
λ

2

2

=>+ kk

x

2 2

(2.93)

y

c

2

0

Those waves or spatial frequencies with imaginary propagation constants
carry the highest resolution information about the scattering object, but decay
exponentially away from the object. Near-field measurements, that is, gather-
ing data within a distance of ~λ from the surface of the scatterer, can capture
some of this information. More typically, the scattered field is measured many
wavelengths from the scatterer, and only the propagating scattered waves con-
tribute to the signal. This limits the resolution available about the scatter-
ing target to ~λ/2. As will be discussed in later chapters, many methods to
improve the resolution of an image exist and most implicitly involve using
some kind of cost function to extract a single solution from a space of solu-
tions which minimizes (or maximizes) some meaningful quantity such as the
total energy or entropy of a data consistent (or almost consistent) image. The
incorporation of prior knowledge (when available) about the anticipated shape
or structure of the scattering object can greatly help here, but this is frequently
only known in special cases such as medical or nondestructive testing appli-
cations of inverse scattering.
Propagation from the near to the far field is easily done by using a plane
wave expansion of the propagating wave and recognizing that propagation
in a homogeneous medium is a convolution which can be executed by taking
the Fourier transform of the wave in one plane, applying a phase shift to each
plane wave component corresponding to the distance the wave is to be propa-
gated, and then inverse transforming the results (see Appendix C).