Image Processing Reference

In-Depth Information

We can now define an “efficiency” factor as

C

G

Q

=

ext

(3.30)

ext

where
G
is the geometrical cross section.

Similarly, we can now define the corresponding efficiency factors
Q
sc
,
Q
abs

and
Q
ext
(
Q
ext
=
Q
sc
+
Q
abs
). Extinction from a single particle,
Q
ext
, can be very

large, especially at resonance. Defining
N
as the number of particles per unit

volume, we can express the relationship

Q

=

Q

(3.31)

ext

ext single particle

(

)

3.3.3 Mie Scattering

One very specialized and important scenario of the inverse scattering prob-

lem that will be looked at in detail later in this topic involves a target that is

an isotropic, homogeneous, dielectric sphere in 3-D or a disk in 2-D. This type

of problem is called a Lorenz-Mie scattering type, which was first published

in 1908. The basic setup with variable definitions is shown in Figure 3.3. In

general, Maxwell's equations are solved in spherical coordinates utilizing the

separation of variables method. The problem is solved for the case when the

field is determined at a distance that is much larger than the wavelength,

otherwise known as the far-field condition or zone. In the far field, the solu-

tion can be expressed in terms of two scattering functions as follows (van de

Hulst, 1957):

∞

∑

n

nn

21

1

+

+

S

()

Θ

=

)
[

a

π

(

cos

Θ

)

+

b

τ

(

cos

Θ

)]

(3.32)

1

(

nn

nn

n

=

1

∞

∑

n

nn

21

1

+

+

[

]

S

()

Θ

=

b

π

(

cos

Θ

)

+

a

τ

(

cos

Θ

)

(3.33)

2

(

)

nn

nn

n

=

1

X

S
scat

m
o

Φ

Θ

E

Z

S
inc

B

m

Y

Figure 3.3
Typical setup and coordinate geometry with variable definitions for the Lorenz-Mie scattering

problem.

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