Mie Scattering Summary

7/25/2019 Mie Scattering Summary

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Week 8: October 15-19

Lorenz-Mie Scattering

Topics:

1. Lorenz-Mie theory2. Scattering amplitudes/phase matrix

3. Mie scattering results: extinction, absorption, phase function vs. x

4. Anomalous Diffraction Theory

5. Index of refraction for water/ice/aerosols

6. Mie scattering results for size distributions

Reading: Liou 3.3.2& ; Thomas 9.3

Lorenz-Mie Theory

Mie scattering is a solution method for light scattering from spheres.Applicable for any size parameter, but Mie regime 0.1< x


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Fundamental extinction formula (optical theorem):

ext=Cext=4

k2Re[S1,2(0

)]

Extinction cross section is related to scattering in forward direction.

Phase Matrix

The phase matrix is the phase function with polarization.

For randomly oriented particles it is

IscaQscaUsca

Vsca

= sca4R2

P11 P12 0 0P12 P22 0 0

0 0 P33 P34

0 0 P34 P44

I0Q0U0

V0

Each element depends on scattering angle (1/R2 is from solid angle).For spheresP22=P11and P44=P33.The off diagonal terms are usually small for Mie scattering, so polarization does

not affect intensity (then need onlyP11 forI).Intensity component of phase matrix

P11() = 4

k2sca

|S1|2 + |S2|

2

2

Mie Scattering Amplitudes

Mie theory scattering amplitudes

S1() =n=1

2n + 1

n(n + 1)[ann(cos ) +bnn(cos )]

S2() =

n=1

2n + 1

n(n + 1)

[bnn(cos ) + ann(cos )]

The complex Mie coefficientsanandbnare obtained from matching the boundary

conditions at the surface of the sphere. They are expressed in terms of spherical

Bessel functions evaluated atx andmx.

The Mie angular functions are

n(cos) = 1

sinP1n(cos ) n(cos ) =

d

dP1n(cos)

2


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P1n are associate Legendre functions.The number of terms needed and amount of angular structure is proportional to

size parameterx.

Spherical Bessel functiuons of the first (a) and second (b) kind. [Bohren and Huffman, 1993; Fig.

4.2]

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Polar plots of the first five Mie angular functionsnand n. Both functions are plotted to the same

scale. [Bohren and Huffman, 1993; Fig. 4.3]

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Mie Efficiency Factors

The Mie efficiency factors are derived from the scattering amplitudes.

Extinction efficiency:

Qext= 2x2

n=1

(2n + 1)Re(an+ bn)

Scattering efficiency:

Qsca= 2

x2

n=1

(2n + 1)(|an|2 + |bn|

2)

Asymmetry parameter:

Qscag=

4

x2

n

n(n + 2)

n+ 1 Re(ana

n+1+ bnb

n+1) +

2n + 1

n(n + 1)Re(anb

n)

Mie Code Algorithm

How a Mie code works:

1. Compute an and bn for n = 1 . . . N from size parameter x and index ofrefraction m (uses recursion relations for the spherical Bessel functions).

N x + 4x1/3 + 2.

2. ComputeQext,Qsca, andg fromanandbn.

3. (optional) Compute S1() and S2() at desired scattering angles from anand bn and n() and n() (n and n from recursion). Compute phasematrix elementsP11, P12, P33, P34fromS1,S2.

4. Integrate numerically over a size distribution n(r) to get volume extinction, single scattering albedo, and phase functionP().

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Mie Scattering Results

Lorenz-Mie theory applies to spheres of all size parametersx.

Extinction efficiency vs size parameter (no absorption):

1) Small in Rayleigh limitQext x

4

2) LargestQextwhen particle and wavelength have similar size.

3)Qext 2in geometric limit (x ).4) Oscillations from interference of transmitted and diffracted waves.

5) Ripple structure from surface waves - resonance effects

Period inx of interference oscillation depends onm.

Absorption reduces interference oscillations and kills resonance ripples.

Scattering and absorption efficiency vs size parameter with absorbingm:as x , Qsca 1, Qabs 1; entering rays are absorbed inside particle.Smaller imaginary part ofm requires larger particle to fully absorb internal rays.

Phase functions: Forward peak height increases dramatically with x.

For single particles - number of oscillations in P()increases withx.

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0 5 10 15 20 25 30 35 40Size parameter x

0.00.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

Qext

Mie results: Extinction Efficiency

m=1.55

m=1.33

0 10 20 30 40 50 60 70 80Size parameter x

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Qext

m=1.33-0.10i

m=1.33-0.03i

m=1.33-0.01i

m=1.33-0.00i

Extinction efficiency vs. size parameter. Top panel shows effect of real part of index of refraction,

while bottom panel shows effect of imaginary part.

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0 10 20 30 40 50 60 70 80Size parameter x

0.00.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Qsca

Mie results: Scattering Efficiency

m=1.33-0.1i

m=1.33-0.03im=1.33-0.01i

m=1.33-0.003i

0 10 20 30 40 50 60 70 80Size parameter x

0.0

0.5

1.0

1.5

2.0

2.53.0

3.5

4.0

Qabs

Mie results: Absorption Efficiency

m=1.33-0.1i

m=1.33-0.03i

m=1.33-0.01i

m=1.33-0.003i

Scattering and absorption efficiencies vs. size parameter for varying amounts of absorption.

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0 20 40 60 80 100 120 140 160 180Scattering angle

0

2

4

6

8

10

Phase

function

Mie results: Phase function (m=1.33)

x=3x=1

x=0.1


10-2

10-1

100

101

102

103

Phase

function

Single particle results

x=30

x=10

x=3

x=0.1

Phase functions for single nonabsorbing spheres of increasing size parameter x: linear scale (top),

log scale (bottom).

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Extinction Paradox and Geometric Optics Limit

Geometric optics limit isx .

Asx extinction efficiency isQext= 2.

Extinction cross section istwiceparticle area!Oner2 from blockage by particle, secondr2 from diffraction.Light diffracted by particle edge is scattered by small angles.

Solution to paradox: need to be in far field (xr 1) to see diffraction.

If optical path in particle4r/ 1, all light entering is absorbed:Cabs=r

2 Csca=r2 = 0.5

If4r/ 1, all light entering particle is transmitted:Cabs= 0 Csca= 2r

2 = 1

Optical depth is proportional to second moment of size distribution:

=

02r2n(r)dr dz =

3

2

LWP

lreforx

where LWP is liquid water path and reis effective radius.

Anomalous Diffraction Theory (ADT)

Simple scattering theory - explains main MieQext(x)oscillations.

ADT applies to limits:x 1so treat waves as rays,m 1 1so no refraction or reflection.

But phase lag in particle is significant = 2x(m 1).

ADT integrates sum of incident and transmitted Efield (in = 0direction) forall rays through particle. Then uses optical theorem (Cext = (4/k2)Re[S(0)])to obtain extinction cross section, and bulk absorption coefficient (4Im[m]/) toobtain absorption cross section.

Oscillations inQext due to constructive and destructive interference of diffractedand transmitted waves.

For non-absorbing spheres ADT gives

Qext= 2 4

sin +

4

2(1 cos )

First maximum at 4.1. Asymptotically, maxes at = 2(n + 3/4).

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ADT provides extinction and absorption, but not phase function.

ADT can be used on any convex shaped particle. But not that accurate for realistic

index of refraction.

Extinction curves computed from Lorenz-Mie theory for m=1.5,1.33,0.93,0.8. The abscissa is

= 2x(m 1) and is common to the upper two Lorenz-Mie curves as well as to the bottom

anomalous diffraction theory (ADT) curves [van del Hulst, 1957; Stephens, 1994]

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Single Particle Mie Scattering Summary

x 0 0.2< x 0) 0 reaches maximum 1/2g 0 increases constant (.71)

0 10 20 30 40 50 60 70 80 90 100Size parameter x

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Resu

lts

Mie results: =3.9 m m=1.357-.0038i (water)

g

Qext

0 1 2 3 4 5 6 7 8 9 10Size parameter x

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

g

Qext

Extinction efficiency (Qext), single scattering albedo (), and asymmetry parameter (g) as a func-

tion of size parameter for a slightly absorbing index of refraction.

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Index of Refraction of Water and Ice

See graphs for complex index of refractionm= n iDebye and Lorentz models used to understand index (see Stephens).

Microwave: water - very highn and , ice -n= 1.78, low

Index is temperature dependent for water but not much for ice.

Thermal IR: high (highest at 3m); wiggle inn with each peak in.

Visible/near IR: mostly constantn,= 0in visible and increases with wavelength in near IR.

0 20 40 60 80 100 120 140 160 180 200Frequency (GHz)

10-4

10

-3

10-2

10-1

100

101

ImaginaryPart

1

2

3

4

5

6

7

8

910

RealPart

Index of Refraction for Water and Ice

Ice T=-20 C

Water T=0 C

Water T=20 C

Real and imaginary part of the index of refraction of water and ice in the microwave.

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0 500 1000 1500 2000 2500 3000 3500 4000

Wavenumber (cm-1

)

10-3

10-2

10-1

100

Imagin

aryPart

1.0

1.2

1.4

1.6

1.8

2.0

RealPart


IceWater

Real and imaginary part of the index of refraction of water and ice in the infrared.

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0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0Wavelength ( m)

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

ImaginaryPart

1.0

1.1

1.2

1.3

1.4

1.51.6

1.7

RealPart


Ice

Water

Real and imaginary part of the index of refraction of water and ice in the visible and near infrared.

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Index of refraction of aerosol materials: mostly nonabsorbing in shortwave,

except for soot (carbon) and dust (e.g. hematite) aerosols. Sulfates and quartz

absorb in 8-12m region.

Imaginary part of index of refraction as a function of wavelength for some common aerosol mate-

rials. [Bohren and Huffman; Fig. 5.16]

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Mie Scattering Results for Distributions

Real distributions of particles smooth out oscillations in extinction efficiency Qext(x)and phase functionP().

The extinction efficiency as a function of the effective size parameter xe= 2re/for gamma size

distributions of various effective variances b. Mie theory with indexm = 1.33 was used. [after

Hansen and Travis, 1974; Stephens, Fig. 5.16]

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For cloud droplets at solar wavelengths: still have forward diffraction peak (width

1/x), rainbow near = 140, and glory at = 180.


10-2

10-1

100

101

102

103

104

Phase

function

Mie Phase Function for Distributions

reff=20 m

reff=10 m

reff=5 m

Water droplets =1.65 m

The phase function for gamma distributions (= 7) of water droplets for three different effective

radii.

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Mie scattering results vs. wavelength for cloud droplets:

Extinction: constant in visible and near IR; decreases in far IR.

Single scattering albedo: 1 for

Mie Scattering Summary

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