Scattering Fundamentals - Introduction to Imaging from Scattered Fields - page 37

The scattering cross section is defined as σ = N s / N i where N s is the number
of scattered photons/area and N i the number of incident photons per unit area.
There are several different classifications of scattering regimes, which we will
briefly discuss next.

3.3.2 Rayleigh Scattering
Rayleigh scattering occurs when the particles are small, on the scale of λ. The
mean scattered power from N Rayleigh scatterers in the direction θ (Figure
3.2) is proportional to ωα θπε (i.e., proportional to ω 4 , and so

42 2

E sin/

2

32

2

rc

3

0

the scattered power is proportional to 1/λ 4 ).
It is understood that the primary model for describing the field from “small”
scattering structures is that of a dipole field (recall Sihvola's dipolarizability
(Sihvola, 2007)). For a dipole, the field can be written as follows:

1 3 ( rprrp

r

⋅ − + )

2

rp

2

−⋅ ( rpr )

�

E = ( 1 + ikr ) k

2

e

− ik r

(3.26)

0

4 πε

0

2

r

3

0

w here p is the dipole moment. For a magnetic dipole, we can just replace p by
m , the magnetic moment and replace ε 0 by μ 0 . Dropping the terms that fall off
fast with r and rewriting this in polar coordinates, we can simplify this to give:

2
4

p cos θ

E = ( 1 + ik r ) e

− ik r

(3.27)

0

r 0

πε r

3

0

This is the same as the expression obtained previously when calculating
the net phase retardation from a sheet of dipoles.
Other important definitions discussed previously can be extended by sub-
stituting σ in the previous scattering equations with C for cross sections.
Using this substitution, the scattering cross section is now defined as

1
2

∫

C = F (, ) θφ Ω d (3.28)

sc

k

where dΩ = sinθ dθ dϕ, F is dimensionless, and F / k 2 is the area. Similarly, we
can define C abs as the absorption cross section. As previously shown, we can
now define the extinction cross section as

CCC
ext

=+

(3.29)

sc abs

θ

Incident field direction

Figure 3.2 Rayleigh scattering definition of variables.