Light Scattering: Theory and Applications

1 Fundamental Equation of Radiative Transfer

),,(),,(),,( φμτφμττ

φμτμ JII−=

dd (1)

∫ ∫−

−=1

1

2

0

'')',',()',',(4

),,(π

μφφμτφφμμπωφμτ ddIPJ (2)

2 Phase Matrix P and Scattering Matrix F

• Phase matrix usually known with reference to scattering plane: scattering

matrix F

• Many different planes of scattering in multiple scattering problems

• Necessary to choose one common plane of reference for Stokes parameters

• Normally use local meridian plane, specified by local normal and direction of

emergence

• P related to F through a coordinate transformation (from the scattering plane

to the local meridian plane)

• F, in general, a function of all the angles

• Special cases:

o randomly oriented particles, each with a plane of symmetry

o randomly oriented asymmetric particles and their mirror images in equal

numbers

o Rayleigh and Mie scattering

• For special cases, F is a function only of scattering angle Θ and has the form

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

− 4434

3433

2212

1211

0000

0000

FFFF

FFFF

• Isotropic Rayleigh scattering:

)cos1(43 2

2211 Θ+== FF

Θ== cos23

4433 FF

Θ−== 22112 sin

43FF

• Mie scattering:

44332211 , FFFF ==

3 Scattering Behavior in Different Regimes [Fig. 5, Hansen and Travis]

• Rayleigh scattering

o strong positive linear polarization ( 1121 / FF− ), with a maximum at 90°

scattering angle

• Geometric optics

o small scattering angles: phase function large and linear polarization small

because diffracted light is predominant

o other than diffraction, most of the light scattered into forward hemisphere

due to rays passing through particle with two refractions => negative

linear polarization (Fresnel’s equations)

o positive linear polarization at ~ 80-120° scattering angle: reflection from

outside of particles

o positive polarization maximum at ~ 150° scattering angle (primary

rainbow): internal reflection – scattering angle has a maximum as a

function of the incident angle on the particle; weaker feature at ~ 120°

scattering angle (secondary rainbow): two internal reflections – scattering

angle has a minimum as a function of the incident angle on the particle

o maximum in backscattering direction (glory): incident edge rays

• Transition region in between

o linear polarization complicated function of size parameter

o as size parameter decreases, degree to which paths of separate light rays

can be localized decreases

o secondary rainbow, with a more detailed ray path, lost from polarization

before primary rainbow

o primary rainbow becomes blurred with decreasing size parameter

4 Effect of Nonsphericity [Figure 1, Mishchenko et al.]

Define )(/)( 1111 sphereFspheroidF≡ρ . There are five distinct regions:

• Nearly direct forward scattering

o ρ ~ 1

o region least sensitive to particle sphericity because diffraction dominates

and is determined by the average area of the particle geometrical cross

section

• ~10 to 30° scattering angle

o ρ > 1

o ratio increases with increasing aspect ratio ε

• ~30 to 90° scattering angle

o ρ < 1

o region becomes narrower with increasing ε

o ratio decreases with increasing ε

• ~90 to 150° scattering angle

o ρ >> 1

o strongly enhanced side scattering as opposed to deep and wide side

scattering minimum in spherical particles

o region becomes wider with increasing ε

• ~150 to 180° scattering angle

o ρ << 1

o strong rainbow and glory features suppressed by nonsphericity

In general, the polarization generated by spheroids is more neutral than that for

spheres and shows less variability with size parameter and scattering angle

[Figure 10.6, Mishchenko et al.].

• >~60° scattering angle

o degree of linear polarization p is strongly ε-dependent

o deviation from spherical behavior becomes more pronounced with

increasing ε

o Lorenz-Mie theory inappropriate for nonspherical particles in this region

• <~60° scattering angle

o p weakly dependent on particle shape

• ~120° scattering angle

o most prominent polarization feature for spheroids: bridge of positive

polarization, which separates two regions of negative or neutral

polarization at small and large scattering angles

o bridge absent in spherical particles

5 Global Climatology of Aerosol Types [Figure 2]

• Seven basic aerosol types: sulfate(land/water), seasalt, carbonaceous, black

carbon, mineral dust (accumulated/coarse)

• Each mixing group is a combination of 4 aerosol components

• Lognormal distribution

6 Phase Function for Kahn Mixing Group Types

7 Linear Polarization for Kahn Mixing Group Types

8 Radiative Effect I: Weighting Functions in 0.76 µm O2 A Band

9 Radiative Effect II: Weighting Functions in 1.61 µm CO2 Band

1.590 1.595 1.600 1.605 1.610 1.615 1.620

-0.4

-0.2

0.0

0.2

0.4 Type 1a 1b 1c 2a 2b 3a 3b 4a 4b 4c 5a 5b 5c

Nor

mal

ized

Jac

obia

n

Wavelength [μm]

0.758 0.760 0.762 0.764 0.766 0.768 0.770 0.772-10

-8

-6

-4

-2

0

2

4

6

8N

orm

aliz

ed J

acob

ian

Wavelength [μm]

Type 1a 1b 1c 2a 2b 3a 3b 4a 4b 4c 5a 5b 5c

10 Radiative Effect III: Weighting Functions in 2.06 µm CO2 Band

11 Reducing Mixing Group Types Using SSA and Extinction Behavior

0.755 0.785 1.56 1.65 2.03 2.09

0.80

0.85

0.90

0.95

1.00

0.755 0.785 1.56 1.65 2.03 2.090.80

0.85

0.90

0.95

1.00

0.755 0.785 1.56 1.65 2.03 2.090.80

0.85

0.90

0.95

1.00

0.755 0.785 1.56 1.65 2.03 2.090.80

0.85

0.90

0.95

1.00

0.755 0.785 1.56 1.65 2.03 2.090.80

0.85

0.90

0.95

1.00

Type 5b

Wavelength [μm]

Sing

le S

catte

ring

Albe

do Group 4 Type 3b

Group 5

Type 2b 4b 4c

Sing

le S

catte

ring

Albe

do

Group 2

Type 3a 5a 5c

Group 3

Type 1a 1b 1c 2a 4a

Sing

le S

catte

ring

Albe

do

Group 1

2.04 2.05 2.06 2.07 2.08

-0.10

-0.05

0.00

0.05

0.10

Nor

mal

ized

Jac

obia

n Type

1a 1b 1c 2a 2b 3a 3b 4a 4b 4c 5a 5b 5c

Wavelength [μm]

0.755 0.785 1.56 1.65 2.03 2.090.10.20.30.40.50.60.70.80.91.01.1

0.755 0.785 1.56 1.65 2.03 2.090.10.20.30.40.50.60.70.80.91.01.1

0.755 0.785 1.56 1.65 2.03 2.090.10.20.30.40.50.60.70.80.91.01.1

0.755 0.785 1.56 1.65 2.03 2.090.10.20.30.40.50.60.70.80.91.01.1

0.755 0.785 1.56 1.65 2.03 2.090.10.20.30.40.50.60.70.80.91.01.1

Nor

mal

ized

Ext

inct

ion

Coe

ffici

ent

Type 5b

Wavelength [μm]

Group 4

Type 3b

Group 5

Type 2b 4b 4c

Nor

mal

ized

Ext

inct

ion

Coe

ffici

ent Group 2

Type 3a 5a 5c

Group 3 N

orm

aliz

edE

xtin

ctio

n C

oeffi

cien

t

Type 1a 1b 1c 2a 4a

Group 1

Figure 2. Global Climatology of Aerosols (Kahn et al., 2001)