Scattering from a particle can be described by dividing the particles volume into many small volume

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Scattering from a particle can be described by
dividing the particles volume into many small
volume elements.
volume element
scatterer
These volume elements will act as the source of
the scattered wave. This is analogous to the idea
of Huygens point sources except more general

• account for the polarization of the wave
• account for variations in composition throughout
  the particle
• consider arbitrary, complicated particle shapes
• develop an intuitive description of scattering
  that can be made exact

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Polarization
Consider how one of the particles volume
elements responds to the wave incident on the
particle
sub-volume





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p

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the polarization of the volume element gives it a
dipole moment, p
volume element in the absence of the incident
field
the electric field polarizes the volume element
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The incident waves electric field oscillates in
time
This will cause a volume elements dipole moment
to vary in time too
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Dipole radiation
When the dipole moment oscillates with the
(incident) field driving it, it produces an
electromagnetic wave of its own with electric
field,
so the volume element radiates
The wave travels radially outward.
The wavelength of the radiation is the same
as the incident light.
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Coupling
Nearby volume elements will respond to the
incident field and the field radiated by other
volume elements
One elements radiation influences the
others, and vice versa
The elements are coupled together.
The the simple, diffraction / Fourier transform
description scattering neglects this.
We can find how the dipoles of a particle respond
to the incident wave and couple to each other by
solving a large system of linear algebraic
equations.
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An example of the effect of coupling A sphere
look inside of a sphere in the x-z plane
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What coupling does
the effects of coupling
propagation
polarization
Coupling changes the direction magnitude of the
dipole moments from what they would be if they
had responded to the incident wave only.
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The DDA
Once the coupling between all the dipoles is
know, the wave scattered by the particle is found
by adding-up the waves that each dipole radiates
out to the observation point
observation point
This description of scattering is the Discrete
Dipole Approximation, (DDA)
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the DDA can be made more rigorous
Maxwells equations
electric field inside of the particle this
is where the dipoles enter
wave equation
Volume Integral Equation (VIE)
Dividing the particle into its volume elements
discretizes the VIE which generates a system of
algebraic equations that can be solved to yield
the dipole coupling.
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The fineness of the division of the particles
volume is the major approximation in DDA its
error reduces with smaller subdivision of the
particle (but at the cost of more computation).
If the incident field is substituted for the
field inside of the particle,
This corresponds to neglecting the coupling
between the volume elements. The VIE then
reduces to
which is the Fourier transform of the particles
volume as we saw before
simple Fourier-transform description the
no-coupling limit of scattering
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What is this good for ?
the DDA is well suited to model scattering from
complex, particles since there are no
restrictions on the particle shape composition
atmospheric particles water, ice, carbon soot,
dust
biological particle
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Some applications
a typical scattering arrangement
horizontal scattering plane
observation point detector CCD, PM, eye
but, the intensity is not the only
measurable quantity of interest look at the
scattered waves polarization
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Review The polarization state
For a typical incident laser wave, the most
general polarization that a scattered wave can
have at a detector far from the particle is
elliptical. Elliptical polarization can be
though of as the combination of two
linearly polarized waves that are out of phase
with each other
electric field vector traces an ellipse in time
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• the polarization state
• is described by three
• measurable quantities
• ellipticity
• orientation
• sense or rotation (handedness)

particle
each of these quantities can be measured
using combinations of polarizers and wave plates
in front of the detector
show the polarization state on a large
observation sphere surrounding the particle
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A spherical particle kR4, m1.25
scattered waves polarization state varies with
direction!
observation sphere
incident wave is linearly polarized
polarization state is the same as the incident
wave for directions contained in the plane of
reflection symmetry
black circular polarization white linear
polarization
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An ice crystal kRve3.6, m1.33
symmetric orientation
polarization state still preserved in planes of
reflection symmetry
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Same ice crystal kRve3.6, m1.33
symmetry broken
polarization state is no longer preserved!
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Explanation for relation between symmetry and the
polarization state
The symmetry of the particle shape the
position of the observation point causes
components of the particles dipole fields to
cancel, producing a linearly polarized scattered
wave.
Reflection Symmetry of a Spheres Internal Field
and its Consequences on Scattering A
Microphysical Approach, M. J. Berg, C. M.
Sorensen and A. Chakrabarti, JOSA A, (submitted
July, 2007, in review).
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An application
The polarization state of the backscattered wave
can indicate the presence of nonspherical
particles, e.g. a cloud full of ice or of water
drops.
ice, no symmetry
water only sphere - symmetry