1. Scattering by particles
PHY Lecture 6
Scattering by particles

2. Outline Refractive index Mie scattering – “white clouds”
an exact solution for homogeneous spherical particles
Geometric scattering – refractive effects: rainbows, halos..
Real atmospheric particles
Scattering effects due to shape and variations in composition

3. Refractive index Liou: see Appendix D, p529-532
Last time: Rayleigh scattering – what is the origin of the real and imaginery components of refractive index? – Fundamentally it is the dispersion relationship for modes of oscillation in the Lorenz atom:
Liou: see Appendix D, p
Refractive index,m, by definition is the ratio of speed of light in a medium to that in a vacuum (m2=eo)
We have an expression for refractive index in terms of “polarizability” – how do we relate this to EM frequency…
Definition of polarizability is the separation of charges in the dipole induced by the electric field..
Relationship of polarizability to frequency is found by solving the equation for displacement r generated by the Lorenz force on a charged particle.
The solution is in terms of a resonant frequency of oscillation of the dipole, and dispersion of wave frequencies induced by the medium about this frequency
The half width of the natural broadening depends on the damping an=g/4p and line strength S is pNe2/mec . Thus the absorption coeffiecient, k, is 4pnomi/c.

4. Plots of refractive index components

5. Mie scattering Hard part
An exact scattering solution for homogeneous spherical particles.
Derivation not too difficult but very long…
won’t go into in detail here..just main points:
Based on wave equation in spherical polar co-ordinates, origin centre of particle….Most of the mathematical complexity in this theory is due to expressing a plane wave as an expansion in spherical polar functions
Scalar solutions to the wave equation are
Related to vector solutions by
Series expansion for E and H of form
Legendre Bessel Neumann
Hard part

6. Mie scattering fields
Consider particle sphere radus r, refractive index m, surrounded by vacuum, m=1
To find the coefficients defining the scattered wave, use boundary condition at surface of sphere
Incident field
Scattered field
Giving coefficients:
Field expressed by incoming wave, Bessel function Y (k1r)
Field expressed by superposition of incoming and outgoing waves

7. Angular dependence
Define functions
…from Bohren & Huffmann, 1986

8. Scattering matrix form
Define scattering functions:
Then by considering parallel and perpendicular components of the field
Scattering matrix

9. Extinction efficiency
Find extinction cross section by superimposing incident and scattered fields in the forward direction S1(q=0) and integrating over the sphere
Extinction efficiency
Approximations to Mie theory are based on a power series expansion of the Bessel functions
where
Rayleigh term

10. Extinction efficiency for a sphere

11. Geometric optics A glory
A glory

12. Geometric optics
In the geometric limit light can be thought of as a collection of individual rays.
This approximation is increasingly bad as the size parameter gets smaller, where phase effects become important and the effect of a wavefront is smeared out over the particle
To express the scattered wave field in the geometric scattering limit we must superimpose the fields due to effects of diffraction, reflection and refraction governed by fixed phase relations
Far field= Fraunhofer diffraction:
Bessel function solution:

13. Geometric optics Reflection and refraction:
Responsible for rainbows and glories:
Deviation due to multiple reflection and
refraction
p is number of internal reflections
Differenciating Snell’s law,
we get a minimum which governs the
angle the ray exits:

14. Geometric optics

15. Comparison of geometric and Mie scattering
Geometric optics is poor approximation for small x, asymptotic improvement for large x:
Computation of exact solutions for spheres,
spheroids and cyllinders possible from
Mie approach
(any shape where boundary can be expressed
on a surface of the co-ordinate system)
And for coated inhomogenous
particles with spherical symmetry..
BUT..
Computationally expensive:
a rough rule of thumb is that x
terms must be retained in Bessel expansion,
so for a raindrop – implies 12,000

16. Real particles

17. Real particles: thought expt
Smoothing out of features as size parameter decreases…also observable with “geometric effects”..
Inner rainbow observable only for raindrops above a certain size parameter….with small raindrops, it is smeared out and disappears

18. Real particles: refractive index
Smoothing out of features as size parameter decreases…also observable with “geometric effects”:
Inner rainbow observable only for raindrops above a certain size parameter….with small raindrops, it is smeared out and disappears

19. Equation of radiative transfer through a scattering layer
1 Attenuation by extinction
2. Single scattering
3. Multiple scattering
4. Emission
Coefficients:

20. RTE scattering parameters
To account for shape variation and inhomogenuity in real particles, we introduce two parameters to characterize particles:
Single scattering albedo, w = bs/be
Asymmetry parameter, g = the “average” scattering angle
From expansion of the phase function

21. Summary
We have looked at Mie scattering, geometric scattering and the refractive index of real particles
…and related this to the radiative transfer equation
Next time we are going to look at thermal radiation…(Liou, chapter 4)
..radiative transfer models & computational techniques (practical 1)
..then look at the radiative transfer problem through a cloudy atmosphere