Inverse Scattering Fundamentals - Introduction to Imaging from Scattered Fields - page 44

Rx ( r , φ s )

y

r - r ′

r

x

r ′
V ( r )= ε ( r )-1

Incident plane wave

Tx ( r , φ inc )

Figure 4.1 A typical 2-D inverse scattering experimental setup. Transmitter Tx transmits the incident quasi-
monochromatic plane wave to the scattering object V ( r ). The receivers Rx are located all around the target which
collects the scattered field data after the interaction of the incident wave with the scattering object.

ε 0 is usually the permittivity of free space. This target or scattering object has
a permittivity of ε( r ), which is related to the target by the equation

(4.1)

V () r = ε () r − 1

r

V ( r ) is an expression that describes the fluctuations of the permittivity rela-
tive to free space in terms of the coordinate system described in Figure 4.1
where r = ( x , y ). Examining this relationship outside of the target boundary
yields the following:

V () r = ε () r − 1 = −=−= ε 1 11 0

(4.2)

r r

0

This demonstrates that V ( r ) is zero at all points outside of the target bound-
ary which has a compact support domain of D , which implies that the Fourier
transform of V ( r ) is an entire function which is completely determined by its
exact values on some region in Fourier or k -space from which it could, in prin-
ciple, be determined everywhere using analytic continuation (Ritter, 2012).
The incident plane wave (in the absence of the target) is governed by the scalar
homogeneous Helmholtz equation (Avish and Slaney, 1988) as given here:

(4.3)

( ∇+

2

k Ψ inc

2

) () r = 0

where k is the wave number as defined by k = 2π/λ. The solution for the inci-
dent field, Ψ inc , can be written in terms of the standard exponential form of a
plane wave

(4.4)

Ψ inc () r = e ik ·

r − r

inc