1

EE 543Theory and Principles of

Remote Sensing

Scattering and Absorption of Waves by a Single Particle

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Theory of Radiative Transfer

We will be considering techniques to derive expressions for the apparent temperature, TAP of different scenes as shown below.

Atmosphere

Terrain

TA

TUP

TA

Terrain could be smooth, irregular, slab (such as layer of snow) over a surface.

STEP 1: Derive equation of radiative transferSTEP 2: Apply to different scenes

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Radiation and Matter

• Interaction between radiation and matter is described by two processes:– Extinction– Emission

• Usually we have both phenomenon simultaneously.

• Extinction: radiation in a medium is reduced in intensity (due to scattering and absorption)

• Emission: medium adds energy of its own (through scattering and self emission)

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Mediums of Interest

• The mediums of interest will typically consist of numerous types of single scatterers (rain, vegetation, atmosphere,etc.)

• First we will consider a single particle and examine its scattering and absorption characteristics.

• Then we will derive the transport equation for a collection of particles in a given volume.

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Scattering from a Single Particle

• When a single particle is illuminated by a wave, a part of the incident power is scattered out and another is absorbed by the particle.

• The characteristics of these two phenomena can be expressed assuming an incident plane wave.

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Particle Scattering Phenomena

Particle scattering is a complex topic—but we can simplify the view of particle scattering by visualizing the scattered radiation as composed of contributions of many waves generated by oscillating dipoles that make up the particle.

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Sub-Particle Scattering Effects

The radiation scattered by a particle and observed at P results from the superposition of all wavelets scattered by the subparticle regions (dipoles)—from Bohren and Huffman (1983).

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Scattering by a Single Dipole• Recall the dipole radiation pattern• Same amount of energy in the forward and backward direction• For particles that are very small compared to , forward and

backscattering values are same.• The incident energy will be redistributed uniformly in azimuth

plane less energy reaching the forward direction.

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Complex Phenomena

• Each dipole affects the other such that the resulting field emitted by an oscillating dipole has a contribution that is due to stimulation by the incident field and a contribution due to stimulation by the fields of neighboring dipoles.

• The charge distribution in the particle forms higher order poles than dipoles and these multipoles also contribute to the scattered radiation.

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Radiation from Multiple Dipoles – Understanding Larger Particles

Recall the N element array:• Directivity increases with N2.• Therefore the larger the particle, the more energy is scattered forward.• As N increases, there are more sidelobes, thus more nulls.• Therefore the larger the particle, the more involved the scattering

characteristic gets (more maxima and minima as varies)

0 1 2 3 N-1d d d

dcos

dcos

dcos

rsin

2

sin2

o

N

AF I

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Scattering from Large Particles

• Although the simple discussion of multiple dipoles given above ignores the complicating effects of dipole-dipole interactions, the broad behavior predicted carries over to the more complete calculation of particle scattering.

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Two Dipoles

d

P(x,y,z)

x

z

(1) (2)

r

r1

r2

dcos

Ei

Two isolated dipoles emit waves in all directions. At some point P far from the 'particle', these waves superimpose to create the scattered wave along the direction θ. These waves either constructively or destructively interfere depending on their relative phase difference ΔΦ, which is defined in terms of the extra distance the incident wave travels first to the second dipole (r) less the extra distance from the first dipole to P(r cos θ).

2 1

1 2

( )

( )

cos

(1 cos )

L d r r

d r r

d d

kd

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Excited by an incident wave, two dipoles scattering all directions. In the forward direction (= 0), the two waves are always exactly in phase regardless of the separation of dipoles.

Two Dipole Case

0(1 cos ) 0kd

180(1 cos ) 2kd kd

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Multiple DipolesThe greater the number of dipoles in the particle array, the more they collectively scatter toward the forward direction, For the example shown here, all dipoles lie on the same line, are separated by one wavelength, and interact with each other. The scattered intensity is obtained as an average over all orientations of the line of dipoles (from Bohren, 1988)

Forward scattering

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Particle Extinction

• Depletion of forward scattering radiation is known as extinction.

• This decrease occurs due to two main effects: scattering and absorption

• Total extinction = scattering + absorption• Measure of total extinction: Extinction

Coefficient or Cross Section• Measure of extinction due to scattering:

Particle Albedo (ratio)

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Scattering Cross Section Concept

• Consider radiation as a stream of photons that flow into a volume containing the scattering particles.

• Each particle within the volume blocks a certain amount of radiation resulting in a reduction of the amount of radiation directly transmitted through the volume.

• The reduction in the radiation as it passes through a volume of scatterers can be expressed in terms of a cross-sectional area ext , which is generally different from the geometric cross-sectional area of the particle.

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Cross Section (ext) and Efficiency (Qext) of Particle Extinction

• Particles 'block' a certain amount of radiation from penetrating through the slab. • The reduced transmission can be described in terms of a cross-sectional area

ext shown as the shaded area around the particle which is different than the

geometric cross section, g (area of the particle projected onto a plane perpendicular to the direction of incident wave)

While this view of extinction is a simple one to visualize, it is not entirely correct as extinction occurs through subtle interference effects.

2sphere

extext

g

extext

Q

Qr

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Absorption and Scattering

ext a s t

ext a sQ Q Q

When ext exceeds the value of the geometrical cross-sectional area of the particle, Qext > 1 and more radiation is attenuated by the particle than is actually intercepted by its physical cross-sectional area.

Since this extinction occurs by absorption or by scattering or by a combination of both, it follows that:

Qext depends on the refractive index of the material, the wavelength of radiation and the size of the particle.

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Qext vs Size for Spherical Water Droplets

Plot of Qext, calculated from Lorentz-Mie theory, as a function of the size parameter x for a water sphere illuminated by light of a wavelength of 500 nm.

Extinction is highly dependent on the size of the particle.

x = 2Dmax/

Qe

xt

blue = 450 nmred = 650 nm

D<nm

Blue is scattered

D =

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Qext vs Size

• The rapid rise in extinction as x increases (i.e., toward shorter wavelengths = higher frequencies) is a general characteristic of non-absorbing particles that are smaller than the incident wavelength.

• Thus, blue light is extinguished (scattered) more than red light, leaving the transmitted light reddened in comparison to the incident light.

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The Extinction Paradox

• The tendency for the extinction Qext either to oscillate around the value of 2 as x → ∞ or to converge to the value of 2.

• Intuition suggests that if we consider extinction as just the radiation that is blocked by the particle, then the extinction cross section is just the shadow projected by the very large particle. This geometrical view of extinction predicts that the limiting value of Qext is 1 and not 2.

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The Extinction Paradox (2)

• However, no matter how large the particle, in the vicinity of its edges rays do not behave the way simple geometrical arguments say they should.

• The energy removed from the forward direction can be thought of as being made up of a part that represents the amount blocked by the cross-sectional area of the particle and a part diffracted around the particle's edge.

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Fundamental Extinction Formula

jwt jkzoE e

Consider a component of a plane wave (with unit amplitude) incident on a particle

Suppose the scattered wave at O’ is a spherical wave (i.e. far field)

( )jwt jkr

s

eE S

jkr

Ref: Van de Hulst (1957, p.29)

Amplitude function; accounts for the fact that the intensity of the scattered radiation varies with scattering angle.

z

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Extinction Formula

• Extinction is defined at θ = 0 (forward direction) and ext can be derived in terms of S(θ)

• This relation is known as the fundamental extinction formula.

• In deriving this relation, consider points near θ = 0 provided x, y < < z.

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Forward Direction Approximation

For x, y < < z

Total field in the forward direction:

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Intensity in the Forward Direction

Total Intensity:

O is the integral of the first term and corresponds to the geometric area projected by the particle on a distant screen.

ext is the integral of the second term and can be interpreted as the amount of power

reduced by the presence of the particle (due to extinction).

O - ext

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Extinction Term

2

2

4gx

where

2

2

4Re 0

4Re 0

ext

ext

Sk

Q Sx

Extinction depends on particle size, and amplitude function along forward direction.

Using the method of stationary phase the integral of the second term is reduced to:

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Extinction Factors

• One may gain the false impression that extinction occurs through blocking an amount of incident em wave of magnitude ext. • In fact, extinction is a subtle interference phenomenon and not one of merely blocking as the extinction paradox reminds us. • In discussing this paradox, we learn that the particle not only 'blocks' an amount of incident energy that is defined by the geometric cross section, but also removes some of the energy of the original wave via interference.

•More to come on how to calculate S()…

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Terminology for Radiation/Scattering from a Particle• Scattering Amplitude and its relation to

amplitude function, S()• Differential Scattering Cross Section:

– Bistatic radar cross section– Backscattering cross section

• Scattering Cross Section• Absorption Cross Section• Total Cross Section (Extinction Cross Section)• Albedo• Phase Function

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Scattering AmplitudeConsider an arbitrary scatterer, and a linearly polarized

incident plane wave with unit amplitude:

2

ˆˆ( , ) ;

ikR

s

e DE f o i R

R

Imaginary, smallest sphere

D

i

Ei

oEs

R

The scatterer redistributes the incident electric field in space:

ˆ.ˆ( ) jki ri iE r ee

r

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Scattering Amplitude (2)

• f(o,i) is a vector which represents the amplitude, phase, and polarization of the scattered field in the far field in the direction of o when the particle is illuminated in the direction i with unit amplitude f(o,i) depends on four angles.

• Note that even if the incident wave is linearly polarized, the scattered wave in general is elliptically polarized.

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Scattering Amplitude(3)2

s iˆˆ ˆˆ( , ) ; , o: , , i: ,

ikR

s s i

e DE f o i R

R

Note: Comparison to slide #22 suggests: (ik)f(o,i) = S()

Es behaves like a spherical wave in the far field.

4 ˆ ˆIm ( , )ext fi ik

: polarization of the incident wave

Forward scattering Theorem:

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Scattering Cross Section Definitions: Power Relations

• Differential Scattering Cross-section

• Scattering Cross-section

• Absorption Cross-section

• Total Cross-section

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Differential Scattering Cross Section (m2/St)

• Consider the scattered power flux density Ss at a distance R from the particle along the direction, o caused by an incident power flux density Si.

2 2

ˆ ˆˆ ˆ, lim ,sd R

i

R So i f o i

S

2

*i

2

*s

1 ˆS ; 2 2

1ˆS ;

2 2

i

i i i i

o

s

s s s s

o

ES S E H i

ES S E H o

where

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Differential Scattering Cross Section

2

2

2,

ˆˆ, lim

4lim lim

4 4

s s sd R

si si

s

i s

s totals

R Ri i

R S A Po i

AS A SR

PS

PR SS S

s

i

o

R

As

Units: m2/St

Suppose the observed scattered power flux density in the direction o is extended uniformly over 1 Steradian of solid angle about o. Then the cross section of a

particle which would cause just this amount of scattering would be d. The differential cross section varies with o and i.

Total power if Ss was uniformly distributed over the sphereOr, equivalently:

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Bistatic Radar Cross Section

ˆ ˆˆ ˆ, 4 ,bi do i o i

Suppose the observed power flux density in the direction o is extended uniformly in all directions from the particle over the entire 4 steradians of solid angle. Then the cross section which would cause this would be 4 times d for the direction o.

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Backscattering Cross Section(Radar Cross Section)

ˆ ˆ ˆ ˆ4 , ,b d bii i i i

Suppose the observed power flux density in the direction opposite to the incident illumination (o = -i) is extended uniformly in all directions from the particle over the entire 4 steradians of solid angle. Then the cross section which would cause this would be 4 times d for the backscatter direction.

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Scattering Cross Section (m2)

• Consider the total observed scattered power at all angles surrounding the particle.

• The cross section of a particle which would produce this amount of scattering is called the scattering cross section, s.

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Scattering Cross Section (2)

4

22

4 4

ˆ ˆˆ( ) ( , )

1

s d s

ss s s

i i

s

i

i o i d

R Sd S R d

S S

PS

Unit: m2

Integrate over all scattered angles over a sphere surrounding the particle

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Absorption Cross Section (m2)

• Consider the total power absorbed by the particle.

• The cross section of a particle which would correspond to this much power is called the absorption cross section, a.

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Absorption Cross Section (2)

*

ˆ( )

1Re . ;

2

aa

i

aV

Pi

S

P J E dv J E

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Total Cross Section(Extinction Cross Section)

The sum of the scattering and the absorption cross sections is called the total cross section, t (or the extinction cross section)

t s a

tt s a

i

P P P

PS

Recall that these all depend on the incident energy direction.

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Size Dependence

• If the particle size is much larger than : total cross section is twice the geometric cross section. (Van de Hulst: Extinction paradox discussed earlier). Use GO approximation.

• If the size is much smaller than the scattering cross section is inversely proportional to 4 and proportional to the square of the particle volume. (Kerker). Use Rayleigh approximation.

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AlbedoThe ratio of the scattering cross section to the total

cross section is called the albedo of a particle.

s

t

a

Low Albedo: If a >> s a<<1

• Incident energy is mostly absorbed than scattered• Usually occurs for moderately lossy particles at large compared to D.• e.g. rain at microwave frequencies

High Albedo: If s >> a a>>1

• Incident energy is mostly scattered than absorbed• Usually occurs for large particle sized compared to • e.g. tree trunks at microwave frequencies

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Phase Functions

• A dimensionless quantity that relates the differential cross section to the total cross section.

• It represents the amount of scattered power and has no relation to the phase of the wave.

• It is commonly used in radiative transfer theory.• The name “phase function” has its origins in

astronomy, where it refers to lunar phases.

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Phase Functions: p and

ˆ ˆˆ ˆ, ,4

ˆ ˆˆ ˆ, ,4

td

sd

o i p o i

o i o i

Note that phase functions are dimensionless.

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Phase Function (2)

4t

4t

4t

i

ˆˆ( , )ˆˆ( , )

4ˆˆ( , )ˆˆ( , )

4

d

t

d

s

o ip o i

o io i

Analogy: directivity

p: The ratio of the differential scattering cross-section of a particle to an equivalent particle that would scatter the total extinction power uniformly.

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Phase Function Relations

ˆ ˆˆ

ˆ ˆ ˆˆ ˆ ˆ, , ,4 4

ˆ, ,

s

t

t sd

a

o i p o i

p o a

o

o i

i

i

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Phase Function Relations (2)

4

4

4

ˆˆ( , )

ˆˆ( ,

1 ˆˆ(

)

)4

4

,

s d s

ts

ss

t

a p o

o i d

i d

p o i d

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Phase Function Relations (3)

4

4

4

ˆˆ( , )

ˆˆ( , )

1 ˆˆ1 ( , )4

4

s d s

s

s

s

o i d

o i d

o i d

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Integral Representations of Scattering Amplitude and Absorption Cross Section

• If the shape of a particle is simple (e.g. sphere, cylinder), one can obtain exact expressions for cross sections and scattering amplitude.

• For instance, Mie solution is used for spheres.• However, in many practical cases, the shape of

a particle is arbitrary.• In such cases, general integral representations

of scattering amplitude can be used.

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Arbitrary Dielectric Body

' ''

0

( )( ) ( ) ( )p

r r r

rr r j r

( )p r

i

Consider a dielectric body whose relative dielectric constant is a function of position:

Assume that the dielectric body occupies a volume V, and is surrounded

by free space, 0. Also assume that the permeability of the body is 0.

0

V

o

ˆr Ro'r

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Maxwell’s Equations

0

( )

E jw H

H J jw r E

0

0

( ); (inside the particle)

; (outside the particle)r r r V

r

Maxwell’s equations for this medium are:

where J is the source for the incident field and is the position dependent permittivity given as follows:

(1)

(2)

(3)

( )r

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Maxwell’s Equations for the Incident and Total Fields

0

0

i i

i

E jw H

H J jw E

The total field which satisfies eqn 1 and 2 is the sum of the incident and scattered fields for the region outside the particle.

( ) ( ) ( )

( ) ( ) ( )

i s

i s

E r E r E r

H r H r H r

(4)

The incident field satisfies the Maxwell’s equations in the following form:

(5)

and exists independent of the particle.

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Maxwell’s Equations for the Scattered Field

0 0

0

0

0

( )

( )

s s

i s r

i i

s

i

s

s r

E jw H

H H J jw r E

E jw H

H jw r E

E jw H

J H

Using the linearity of Maxwell’s equations, the scattered field expressions can be derived as follows:

0 ijw E

(6)

(7)

(4) in (1)

(4) in (2)

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Equivalent Current for Scattered Field

0

0

0

0 0

0 0

0

0

0

0

( )

(

( )

( )

( ) 1

) 1

s r

r

r

r s

eq

i

i

s

s

s s

s eq s

eq r

H jw r E

jw r E

j

J H

jw

w jw r E

jw r E jw E

J jw E

E

J jw r E

j

E

w H

H J jw E

E E

From (5)

From (4)

(8)

(9)

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Equivalent Current Source

0s eq sH J jw E

Equivalent source inside the particle that generates the scattered fields.

0 ( ) 1 in V

0 outsider

eq

jw rJ

Note that:

V

o

'r

JEs

(10)

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Solution

Recall the use of vector potential A and scalar potential to solve for the radiated fields given a current source distribution in space.

= vector potential

= scalar potential

H A

E jw A

A(11)

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Solution (2)

within the source region

region far away from source

( ) ( ') '4

( )

1( )

1( )

1( )

jkR

V

eq

eA r J r dv

Rj

E r jw A Aw

H r Jjw

H rjw

H r A

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Solution for Scattered Fields

'

( ) ( ') '4

( ') ( , ') '

where ( , ') is the free space Green's function:

( , ')4 4 '

( ) ' 1 ' ( , ') '

jkR

eqV

eq oV

o

jk r rjkR

o

r oV

eA r J r dv

R

J r G r r dv

G r r

e eG r r

R r r

A r r E r G r r dv

Jeq

(12)

(13)

(14)

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Far Field Solution for the Es Field• In order to obtain the scattering amplitude, we

consider Es in the far field.

0

0

0 0

0

0 0

'

1( ) ( ) ( )

1( ); ( ) 0 for

1

1' 1 ' ( , ') '

1' 1 ' '

4 '

s s eq

s eq

r oV

jk r r

rV

E r H r J rjw

H r J r r Vjw

Ajw

r E r G r r dvjw

er E r dv

r r

(15)

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Far Field ApproximationsIn the far field,

2 2

2 2

'

' ' '

' 2 '

ˆ '

ˆ' 2 '

ˆ

ikR ikR

r r

r r r r r r

r r r r

r o r

e eik R

R R

rR r r r r o

riko

(16)

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Scattering Amplitude – Integral Form

2ˆ'

2ˆ'

ˆ ˆ ' ' 1 '4

ˆˆ,

ˆˆ ˆ ˆ, ' ' 1 '4

ikRjkr o

s rV

ikR

jkr or

V

e kE r o o E r r e dv

R

ef o i

Rk

f o i o o E r r e dv

This is an exact expression for the scattering amplitude f(o,i) in terms of the total electric field E(r) inside the particle.

E(r) is not known in general, therefore this eqn is not a complete description in terms of known quantities.

Using (16) in (15),

(17)

Dependent on incidence direction

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Approximate Calculations of f(o,i)

• Rayleigh Scattering: When the particle size is much smaller than the incident wavelength (kD <<1), the internal field can be calculated approximately for simple geometries.

• Because of the small size, the impinging field within or near the object can be treated as a static field; i.e. no time variation.

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Small Dielectric Sphere

2 0

Boundary Conditions:

is finite at the origin

is continous at r=a

ˆ is continous at r=a (no charges)

at r>>ao

D r

E E

r

Solve for Laplace’s equation:(Note that there are no charges)

a

ˆo oE zE

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Solution of Laplace’s Equation for Dielectric Sphere

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Electrostatic Solution for Dielectric Sphere Illuminated by a Plane Wave

3

2

3 3cos

2 2

1cos cos

2

3 3ˆ

2 2

o oinside

r r

routside o o

r

o oinside inside

r r

E Er z

aE r E

r

E EE z

(18)

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Scattering Amplitudes of a Dielectric Sphere (Rayleigh)

Using (18) in (17),

2ˆ'

2

ˆˆ ˆ ˆ, ' ' 1 '4

3 1ˆ ˆ ˆ

4 2

ˆ

jkr or

V

r

r

o o

kf o i o o E r r e dv

kV o o z

where

E E z

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General Form of Scattering Amplitudes for a Dielectric Sphere (Rayleigh)

2ˆ'

2

ˆˆ ˆ ˆ, ' ' 1 '4

3 1ˆ ˆ ˆ

4 2

ˆ

jkr or

V

r

r

o o

kf o i o o E r r e dv

kV o o q

where

E E q

o

q

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ ˆ ˆ sin

o o q o o q q o o

o o q q

o q o q

o o q q o q o

o o q q o q o

ˆ ˆq o

ˆ ˆ ˆq q o

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Differential Cross Section of a Dielectric Sphere (Rayleigh)

2

2

24

2 22

22

3 1ˆˆ ˆ ˆ ˆ,4 2

ˆ ˆˆ ˆ, ,

13 sin

24

ˆ ˆsin 1

r

r

d

r

r

kf o i V o o q

o i f o i

kV

o q

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Scattering Cross Section

24

2 22

4 4

24 2

2 32

0 0

3

0

25 6

4

13 sin

24

1 3 sin

24

where

4sin

3:

11283 2

rs d

r

r

r

rs

r

kd V d

kV d d

d

Thus

a

2

4s

V

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Out of the blue…• Why is the sky blue?• The blue of the sky (shorter wavelength)

scatters more than the red portion because of the -4 dependence;

4 4

, ,

blue red

blue red

s blue s red

This was a great puzzle in the last century, and was finally explained by Rayleigh. Rayleigh noted that the scatterers need not be water or ice as wa commonly believed at that time, but that the molecules of air itself can contribute to this scattering.

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Efficiency, Qs

2

248 1

3 2

s ss

g

r

r

Qa

ka

Note: This is valid for small ka (Rayleigh).

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Absorption Cross Section

• The absorption cross section for a dielectric body is the volume integral of the loss inside the particle.

2''

' ''

2''

2

''

12 ;

for an incident wave of unit amplitude;

1 12 2

' ' '

3;

2

o rloss V

a r r r

i i

oi

o o

a rV

r o o

r

w E dvP

jS S

S

k r E r dv

k V k w

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Efficiency, Qa

2

2

'' 3 42 3

a aa

g

r

r

Qa

ka

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Total Cross Section

• The total cross section is the sum of scattering and absorption cross sections.

• Note that we can not apply the forward scattering theorem, since this would give a total cross section of zero when .

4 ˆ ˆIm ( , )ext fi ik

2 3 1ˆˆ ˆ ˆ ˆ,4 2

r

r

kf o i V o o q

'' 0r

Real if r is real.

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Total Cross Section

• Rayleigh approximation does not hold for forward scattering theorem.

• In general summing the absorption and scattering cross sections for a particle for a given approximate value of the internal field gives a much better approximation to the total scattering cross section.

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Example

• Consider a plane wave with unit magnitude, polarized in the x direction and propagating in the z direction. This wave is incident upon a small dielectric sphere of radius a. Calculate the scattered field in the far field.

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Solution

ˆˆ ˆ,ˆ

ˆˆ ˆ,

ˆˆ,

ikr ikr

ikr

ikr

e eE E E f o i

r r

f o iE e

r

f o iE e

r

In the far zone of the sphere, the electric field components are given by:

2 3 1ˆˆ ˆ ˆ ˆ,4 2

r

r

kf o i V o o q

where

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E Component

2

2

ˆˆ ˆ, 3 1 ˆ ˆ ˆ ˆ4 2

where

3 1ˆˆ ˆ ˆ ˆ,4 2

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆˆ ˆUsing ; and

ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ cos cos

ˆ ˆsince 0;

ikrrikr

r

r

r

f o i k eE e V o o q

r r

kf o i V o o q

q x o r o o q q o q o

o o q x r x r

r

2

ˆ ˆ cos cos

Thus:

3 1cos cos

4 2

ikrr

r

x

k eE V

r

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E Component

2

2

ˆˆ,ˆ 3 1ˆ ˆ ˆˆ

4 2

ˆ ˆ ˆ ˆ ˆ ˆ ˆ cosˆ ˆ

ˆ ˆsince 0; sinˆ ˆThus:

3 1sin

4 2

ikrrikr

r

ikrr

r

f o i k eE e V o o q

r r

o o q x r x r

r x

k eE V

r

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Solution

2

ˆ ˆ

cos cos

sin

where

3 1

4 2

ikr

ikro

ikro

r

o

r

eE E E

rE E e

E E e

k VE

r

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Rayleigh-Debye Scattering (Born Approximation)

• Now consider the scattering characteristics of a particle whose relative dielectric constant r is close to unity.

• In this case, the field inside the particle may be approximated by the incident field.

ˆˆ iki riE r E r qe

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Scattering Amplitude – Born Approximation

2ˆ'

ˆ

'

2

ˆˆ ˆ ˆ, ' ' 1 '4

where

ˆ

1 ˆ ˆ' 1 '; ; 2sin2

ˆˆ ˆ ˆ ˆ,4

s

jkr or

V

jki r

jk r

s r s sV

s

kf o i o o E r r e dv

E r qe

S k r e dv k k i o kV

kf o i o o q VS k

1 1r kD

Using the integral equation for scattering amplitude

Define

Thus i

o

i-o

Largest dimension of the particle; i.e. diameter

Fourier Transform of ' 1r r

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Observations

• The scattering amplitude is proportional to the Fourier transform of evaluated at the wave number ks.

• In general if is concentrated in a region small compared with the wavelength, the cross section is spread out in ks and thus in angle , and the scattering is almost isotropic.

• If the particle size is large compared to a wavelength, the scattering is mostly concentrated in a small forward angular region, ~0.

' 1r r

' 1r r

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Rayleigh-Debye Scattering Cross Section

2

4 4

22

4

ˆˆ,

ˆ ˆ ˆ4

s d

s

d f o i d

ko o q VS k d

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Rayleigh-Debye Absorption Cross Section

2''

''

' ' '

'

a rV

rV

k r E r dv

k r

ˆˆ iki riE r E r qe

Where we use:

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Example: Homogenous Sphere• Let ks be along a unit vector direction denoted by is.• Let is be along the z’ direction. Because of spherical

symmetry we can assign this arbitrarily.

x’ y’

z’

r’is

2

'

2

2ˆ ' ' c2

0 0 0

ˆˆ ˆ ˆ ˆ,4

1 ˆ ˆ' 1 '; ; 2sin2

'

ˆˆ ˆ ˆ ˆ, 141

' ' sin ' ' ' '

s

s s

s

jk r

s r s sV

r r

r

ajki r ik r

V

kf o i o o q VS k

where

S k r e dv k k i o kV

r

kf o i o o q VF

F e dv d d r dr eV

os '

3 3

3sin cos

2 sin / 2

s s s

s

s

k a k a k ak a

k k

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Limitations of Rayleigh and Born

• The forward scattering theorem is applicable to any scatterer of arbitrary shape and size.

• However, for Rayleigh and Born approximations, the forward scattering theorem does not yield the total cross section.

• In fact, the scattering amplitude f(o,i) is real for a lossless particle in these approximations no loss as the way passes through the particle!

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Limitations of Rayleigh and Born

• The reason for this shortcoming is that the scattering amplitude f(o,i) is predominantly real, and this real part describes the general angular characteristics of scattering.

• The imaginary part is small, but this represents the total loss, which includes both the part scattered out and that absorbed by the object related to the extinction cross section.

• If we have a solution which is better than Rayleigh and Born, forward scattering theorem can be used to calculate extinction cross section.

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WKB Technique

• This better approximation may be provided by the WKB interior wave number technique.

• Note that Rayleigh is applicable to small particles; i.e. ka <<1

• Born is applicable when (r-1)kD << 1, where kD itself can be large, as long as the dielectric value of the particle is close to free space.

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WKB Technique

• WKB technique is valid when (r-1)kD >> 1

and r-1 < 1

• Therefore, it provides a solution for which Rayleigh and Born approximations are not applicable.

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WKB Technique

• In WKB approximation, the field inside the particle is approximated by a field which propagates with the propagation constant of the medium of the particle.

• It is also assumed that since r-1 < 1, the refraction angle is equal to the incident angle,

• And the wave propagates in the direcion of the incident wave.

• The transmission coefficient T at the surface of the particle is approximated by that for normal incidence.

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WKB Technique

1 1( )

1 2

ˆ

ˆ

: index of refraction

21

ikzi i

ikz ikn z z

i

r

E r E e q

E r TE e q z z z

n

Tn

For an incident field:

The internal field at point B is approximated by:z1 z2

A BC

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1 1

1

2ˆ'

( )

2

( 1) ( 1)

ˆˆ ˆ ˆ, ' ' 1 '4

ˆ ˆ;

ˆˆ ˆ ˆ ˆ ˆ ˆ, ,41

ˆ ˆ, 2 ' 1 ';

jkr or

V

ikz ikn z z ikzi i i

ik n z ik n z

rV

kf o i o o E r r e dv

E r TE e q E r E e q

kf o i o o q V S o z

S o z n r e dv nV

From

where

We obtain

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Mie Theory

• The exact solution of scattering of a plane wave by an isotropic, homogenous sphere was obtained by Mie in 1908.

• See Kerker, 1969, Chapter 3 for a detailed derivation or any other standard textbook on radiation and scattering.

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Mie Theory

1 2

11

21

11

sin , cos

(2 1)(cos ) (cos )

( 1)

(2 1)(cos ) (cos )

( 1)

(cos )(cos ) , (cos ) (cos )

sin

ikr ikr

n n n nn

n n n nn

nn n n

e eE i S E i S

kr krn

S a bn n

nS a b

n n

P dP

d

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References

• Microwave Remote Sensing, F. T. Ulaby, et.al. Addison-Wesley

• http://langley.atmos.colostate.edu/at622/notes/at622_sp06_sec14.pdf

• Van deHulst, Light Scattering by Small Particles (1957, p.29)

• Bohren and Huffman, Absorption and Scattering of Light by Small Particles.

• Ishimaru