Light Scattering: Theory and Applications 1 Fundamental Equation of Radiative Transfer ) , , ( ) , , ( ) , , ( φ μ τ φ μ τ τ φ μ τ μ J I I − = d d (1) ∫∫ − − = 1 1 2 0 ' ' ) ' , ' , ( ) ' , ' , ( 4 ) , , ( π μ φ φ μ τ φ φ μ μ π ω φ μ τ d d I P J (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
12
Embed
Light Scattering: Theory and Applicationsvijay/pdf/Light Scattering.pdf · Light Scattering: Theory and Applications 1 Fundamental Equation of Radiative Transfer ( , , ) ( , , ) (
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
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