where ,~, denotes proportional to. Equation (1.1.24) is the result of the Born approximation. in .NET

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where ,~, denotes proportional to. Equation (1.1.24) is the result of the Born approximation.
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1 ELECTROMAGNETIC SCATTERING BY SINGLE PARTICLE
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Absorption Cross Section
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The particle can also absorb energy from the incoming electromagnetic wave. Let
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Ep(f) = E~Cr) + iE~Cr)
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(1.1.25)
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r "1- 2 lv dr Ep(r) Eint(r) 1
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(1.1.26)
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where Eint(r) denotes internal field which is the electric field inside the particle, dr = dx dy dz, and the integration in (1.1.26) is over the threedimensional volume of the particle. The absorption cross section a a is defined by (1.1.27) The total cross section of the particle is (1.1.28) and the albedo of the particle is (1.1.29) Thus 0 ~ w~ 1. The albedo is a measure of the fraction of scattering cross section in the total cross section.
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1.2 Scattering Amplitude Matrix
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We next generalize the concept of scattering amplitudes to include polarization effects. For the incident wave, the electric field E i is perpendicular to the direction of propagation ki. There are two linearly independent vectors that are perpendicular to ki . Let us call them ai and bi. Then
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E i = ( ai E ai + bi E bi ) eik::;: '
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(1.1.30)
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where ki kk i . The directions of ki , ai, and b are such that they are i orthonormal unit vectors following the right-hand rule. Similarly for the scattered wave, let ks , as, and bs form an orthonormal system. Then (1.1.31)
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1.2 Scattering Amplitude Matrix
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Figure 1.1.3 Geometry for defining the orthonormal unit system based on scattering plane. The scattering plane contains ki and ks. The angle between ki and ks is e.
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The scattered field components, E as and Ebs' are linearly related to E ai and Ebi' The relation can be conveniently represented by a 2 x 2 scattering amplitude matrix
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Eas] [Ebs
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[faa(~s, ~i)
fba(k s , ki )
(1.1.32)
Orthonormal Unit Systems for Polarization Description
There are two common choices of the orthonormal unit systems (ai, bi, ki ) and (as, bs ' ks ) that describe scattering by a particle.
A. System Based on Scattering Plane
Let the angle between ki and ks be e (Fig. 1.1.3). The plane containing the incident direction ki and the scattered direction ks is known as the scattering plane. Let
(1.1.33)
be the unit vectors that are perpendicular to this plane and let
= Is = Iks
ks X ki
ki I
(1.1.34)
Then by orthonormality, (1.1.35)
1 ELECTROMAGNETIC SCATTERING BY SINGLE PARTICLE
(1.1.36)
In this 1-2 system, the scattering amplitude matrix obeys the relation
ElS] [E2s
[ill 121
(1.1.37)
The advantage of this system is that the scattering amplitudes can take simple forms for particles with symmetry. The disadvantage for this system is that the directions of ii and 2i depend on the scattered direction. For example, let E i = xeikz The incident wave is propagating in the direction ki = Z and with x polarization. If ks = y, then L = x and 2i = Y and the incident wave is polarized. However, if ks = x, then = -fJ and 2i = x and the incident wave is 2i polarized. The inconvenience is that E i is x polarized and propagation in z direction, and yet it is L or 2i polarized depending on whether the scattered direction ks is fJ or x. Some useful relations for this orthonormal system are
ks . ki = cose 2s . 2i = ks . ki = cos e
ks . 2i = ks . ki
(1.1.38) (1.1.39)
Ii = ks
ki . ii = Iks
ki I = sin e
(1.1.40)
B. Vertical and Horizontal Polarization
In many problems there is a preferred direction, for example the vertical direction that is labeled z. In geophysical probing and earth remote sensing problems, that will be the vertical axis which is perpendicular to the surface of the earth. Then we can have vertical polarization Vi and horizontal polarization hi that form an orthonormal system with ki. We choose
= hi =
, Iz x kil
z X ki
(1.1.41)
that is perpendicular to both
z and ki . Then
= Vi = hi
(1.1.42)
Other names for vertical polarization are TM polarization, parallel polarization, and p polarization. Other names for horizontal polarization are TE polarization, perpendicular polarization, and s polarization. If ki is charac-
2 Rayleigh Scattering
~"'=:-+-----+-~
in spherical coordinates.
terized by the angles Oi and <Pi in spherical coordinates (Fig. 1.1.4), then
ki =
sin Oi cos <PiX + sin Oi sin <Pry + cos OiZ (1.1.43)
Vi = cos Oi cos <PiX + cos Oi sin <pdl - sin 0iZ
Similarly, if ks is described by the angles Os and <Ps, then (Fig. 1.1.4)
ks =
sin Os cos <Psx + sin Os sin <Psy + cos Osz (1.1.44)