Electromagnetic Waves - Introduction to Imaging from Scattered Fields - page 25

2

We can now define the differential cross section as d σθϕ (, ) /d Ω≡ f ( ,). θ ϕ

This cross section is defined as σ = (power scattered/power incident). We also
note that

4 π

∫ ∫

2

∫

σ = d σ = f (, θϕ d Ω = f (, )(,) θϕ θϕ f d Ω = Imf (, ) (2.63)

*

00

K

that is, the imaginary part of the forward scattering amplitude measures the
loss of intensity that the incident beam suffers due to scattering. This is known
as the Optical Theorem.
In the scattering examples that follow, we usually also make the so-called
far-field approximation. We assume an incident plane wave of the form e iKz as

1
4

e
rr Ur

iK rr − ′

∫

ψ () r = e − ()() ψ r r

iKz

′ ′ d

3

′ (2.64)

π

− ′

and we assume that the “potential” U ( r ) has limited range such that the integral
is over some finite range of the variable r '. From this assumption, we can write

12 /

2 2

rr −=−= +−− ′ rr ′ r r ′ 2 rr ′

12 /

2 rr

r

− ′

= r 1 − �

�

2

rr −

= r 1 − + �

�

2

r

··

=−− r ( r rr ′ ) ′

Therefore

··

e
rr

iK rr − ′

ee
r

iKr − iKr ′ ( rr − ′ )

ee
r

iKr − iKr ′′

→ =

(2.65)

− ′

This now allows us to write

e

iKr

∫

ψ () r →− e eUr ()() ′ ψ r ′ d r ′

iKz − iK r ′′ 3

(2.66)

4 π r

r →∞

so that now the scattering amplitude becomes

1
4

∫

f () θ =− e Ur ()() ′ ψ r ′ d r ′ (2.67)

− iK r ′′ 3

π

This is simply the Fourier transform of the product U ψ and is an exact
representation in the far field, also referred to as the Fraunhofer approxima-
tion. U ψ is referred to as the object wave or as a secondary source since it is U