Inverse Scattering Fundamentals - Introduction to Imaging from Scattered Fields - page 45

where r inc is the unit vector that specifies the direction of the incident field
An expression for the total field Ψ( r ) in terms of the coordinate system r can be
obtained by looking at the interaction of the incident field, Ψ inc ( r ), and the tar-
get V ( r ). This relationship can be expressed as an inhomogeneous Helmholtz
equation (Avish and Slaney, 1988) as follows:

( ∇+

2

k

2

) Ψ () r =− k

2

V () () r Ψ r (4.5)

The total field in Equation 4.5 can now be expressed generally as the sum of
the incident field and the scattered field as follows:

ΨΨ Ψ () r =

inc ()

r + () r (4.6)

s

where Ψ s is the scattered field All fields in Equation 4.6 are expressed in
terms of the position as defined by the point r . Now the total field, Ψ( r ), can be
expressed in terms of an inhomogeneous Fredholm integral equation of the
first kind (Morse and Feshbach, 1953) as follows:

kV

· ·

Ψ ( rr , ) = Ψ () r −

2

( r rr rr ′ ) Ψ ( ′ , ) G ( , ′ ) d r ′

(4.7)

inc inc inc 0

D

where G 0 ( r , r ′) is (free space) Green's function which is the solution of the sca-
lar Helmholtz equation (Equation 4.4), and it satisfies the differential equation
(Morse and Feshbach, 1953)

( ∇+ =−

2

kG rr rr

2

) ( , ′ ) δ ( ′ ) (4.8)

0

Green's function basically gives the field amplitude at any point r , generated
by any given point source located at r ′. Since the modeling space in this case
is homogenous and rotationally symmetrical, Green's function can be solved
in spherical coordinates with r ′ at the origin as follows:

( ∇+

2

kG

2

) () r =− =− δ () r δ δδ ()()() xyz (4.9)

0

For homogenous, spherically symmetrical partial differential equations
(PDEs), the solution for Green's function in free space can be written as

Ce
r

ikr

Ue
r

− ikr

(4.10)

G

0 ( r =

+

Utilizing the radiation boundary condition, which is simply that the
sources cannot be present at infinity, translates to this: that only the outgoing
wave solution(s) exist(s) and therefore U must be equal to 0 ( U = 0). This sim-
plifies Equation 4.10 as follows:

C e ikr

(4.11)

G

0 ( r =

r