Assumptions and formalisms

Mie Scattering

3.2 Assumptions and formalisms

When an arbitrary electromagnetic wave propagating in a medium encounters a particle with dielectric properties differing from the medium, the incident energy is scattered in various directions. Part of the incident energy is also absorbed by the particle itself.

Thus both absorption and scattering remove energy from the incident electromagnetic wave. This attenuation is also called extinction and these terms are related as follows

Extinction

=

scattering+absorbtion . (3.1)

Throughout this study, independent, single scattering is assumed. This is the standard assumption and proven to be a suitable approximation for the evaluation of rain attenuation in coherent communication systems (e.g. Watson 1976; Oguchi 1981, 1983;

Crane 1983; Matzler 2002,2003).

The assumptions are as follows

1. The scattered wave and the incident wave have the same frequency. Effects such as the Raman effect and quantum transitions can be ignored, thus scattering only changes the direction of propagation and not the wavelength.

2. Independent scattering is assumed. Particles are sufficiently far from each other such that each particle can form its own scattering pattern undisturbed by the presence of other particles in the medium.

3. Multiple scattering will be neglected. It is assumed that each particle is exposed only to the radiation of the incident electromagnetic wave and not the scattered radiation from other particles. Hence for a medium containing N identical particles the total absorption, scattering and extinction due to the medium is the

N times that for a single particle.

The geometry and formula for the scattering of electromagnetic wave by spheres are available in a variety of texts (e.g. Stratton 1941; van de Hulst 1957; Ishimaru 1978;

Bohren and Huffman 1983; Sadiku 2001). Unfortunately, the notations and definitions in these texts are not always uniform and depend upon the approach of the author. The definitions, geometry and formula adopted in this study are established in the forthcoming sections.

3.2.1 Scattering Amplitudes and Cross Sections

Consider a linearly polarised electromagnetic plane wave propagating in a medium with dielectric constant ^£0 and permeability 110' Let Eⁱ denote the electric field given by

(3.2)

where Eo is the amplitude of the electric field,

E;

is the unit vector in the direction of its polarization and

i

_l is the unit vector in the direction of wave propagation (also called the forward direction). A e^iW( time convention is assumed and is suppressed throughout this study. The wave number k_o=OJ~Jio£o

= 21l/Au

and

Au

is the wavelength in the medium. For a unit-amplitude electric field Eo equals 1 volt mete{'.

Figure 3-1 illustrates the geometry when such an electromagnetic wave falls incident upon an arbitrary particle with dielectric constant £ and permeability Jio The origin has been chosen within the particle. The scattered field at any observation point r, with distance r from the origin, behaves as a spherical wave if r> >D^2/ A, where D is the dimension of the particle. This is called the far-field region.

z

y

~

^E

^,

^{j" .j K"',1}

^if.

^{' "}

x

Figure 3-1 : Geometry of the incident and scattered fields for an arbitrary scatterer.

The electric field of the scattered wave in the far-field region, is denoted by E^S and may be given by

(3.3)

where

i

₂ is the unit vector directed from the origin to the observation point rand

t(i

2

.i

t ) is the vector-scattering amplitude. The scattering amplitude

t(i

2

.i.)

provides a complete description of the amplitude, phase and polarization of the scattered wave in direction

i

₂ when a unit-amplitude plane wave propagating in the direction

i.

falls incident upon the particle.

t (i

2

.i.)

depends on the polarization of the incident wave, the shape, size, orientation and dielectric properties of the particle and the direction K₂•

It is convenient to define the differential cross section Cd as follows

(3.4)

If the scattered power flux density as observed in the direction K₂ was extended uniformly over I steradian of solid angle about

i

2, then the differential cross section is the cross section of a particle that would provide the equivalent amount of scattering.

Cd thus varies with

i

2 • Integrating Cd over the entire 4n steradians of solid angle yields the total scattered power around the particle. The cross section of a particle that will produce this amount of scattering is called the scattering cross section C_sca and is given by

Csca

= f

^Cd

⁽ⁱ

²

^.i

^{t ) dq}

4JT

= fl t (i

²

.i

^t

t

^d

^q,

4JT

(3.5)

where dq

=

sinBsdBsdrPs is the differential solid angle. Similarly, the absorption cross section ^Cabs can be defined as the cross section of particle that corresponds to the total power absorbed by the particle. According to (3.1) the extinction cross section also called the total cross section is then given by

(3.6)

The single-scattering albedo of the particle Wo is the ratio of the scattering cross section to the total cross section and given by

W - Csca^{0 - - -} Cext

(3.7)

Other useful cross sections include the bistatic radar cross section Cbiand the back- scattering cross section Cb'Each are related to Cd as follows

Cbi (K2.K.)=4Jr'Cd(K².K.), and Cb (K2,Kt )

=

4Jr' Cd (-Kt.K.).

3.2.2 General properties of cross sections

(3.8)

It is sometimes convenient to relate the various cross sections to the geometric cross section C_g of the particle. For a sphere with radius a, the geometric cross section C_g

=

^Jra2• The ratio between each cross section and the geometric cross section IS

referred to as efficiency and they are defined as follows

Qe.tl

=

Cexl/Cg ,

Qsca

=

Csca/Cg ,

Qabs

=

Cabs/Cg , and (3.9)

For D«)., Rayleigh scattering theory is relevant and Qsca ex(Dj).rand Qabs ex(Dj).). When the size of the particle D»)., optical scattering theory applies.

The extinction efficiency Qexl approaches 2 and the absorption efficiency Qabs approaches a constant somewhat less than I (Ishimaru 1978). These properties are proven for hydrometeors inthe next section.

Another important relation is given by the forward scattering theorem or extinction theorem which states that the extinction cross section C_ex1 is related to the imaginary part of the scattering amplitude in the forward direction

f( ^X

2

'X

_I⁾ as follows

(3.10)

where Im denotes the imaginary component.