1. Real part of refractive index ( m r ): How matter slows down the light: where c is speed of light Question 3: Into which direction does the Scattered radiation go? In visible light, for water: m r ≈ 1.33 for glass: m r ≈ 1.5 m r is not related to mass density m r influences new direction because of refraction Answer depends on composition, size (relative to ), and shape of scatterers Influence of composition described through real part of refractive index

2. Scattering angle (  ): Why is angle sufficient (and no need for  0 and  a ): Spheres: Non-spherical particles (aerosols, ice crystals): Exception (may try experiment):   x y z How to calculate  ? 1: 2: Scattering angle  00 aa 

3. Equation of radiative transfer:  00 aa 11 ss cc bb  00 aa 11 ss Definition of P: Scattering phase function (P)

4. Phase function plots

5. Best approach depends on size parameter If x < 1 (that is, r << ): Rayleigh scattering Assumes: each atom scatters independently as a dipole If 1 < x < 1000: - Spherical particles: Mie scattering - Rotational symmetrical particles: T-matrix - Irregular shapes: Finite-Difference Time Domain method (FDTD) If x > 1000: geometric optics (Complication: diffraction) Approaches for calculating particle scattering properties

6. Idea of polarization, sources of polarization Two components of variations in electric field Dipole scattering depends on angle between E-variations and plane of scattering (specified by incoming and outgoing directions): Perpendicular component: P (  ) = 1 Parallel component: P (  )  cos 2 (  ) Overall: Clear-sky polarization Multiple scattering reduces polarization (e.g., clouds) Rayleigh phase function

7. Theory developed around 1890-1900 by Mie, Debye, Lorenz Theory based on Maxwell equations: PDEs for EM field solved in polar coordinates (r, ,  ), by using series expansions (sin & cos, Bessel functions, and Legendre polynomials for the 3 coordinates) Where a n and b n are Mie coefficients that depend on x and m Empirical formulas: pages 78-79 of Thomas & Stamnes book Publicly available codes, even on-line calculator at http://omlc.ogi.edu/calc/mie_calc.html http://omlc.ogi.edu/calc/mie_calc.html Problem for large x: long series is needed (n should go up to ~100 for cloud drops, much more for rain) Mie phase function Geometric optics for x > 1000

8. If x > 1000, diffraction is not too important (what examples?) Snell’s laws (1625): Critical angle:  t =90° (sin(  t )=1), If  is greater than critical angle: internal bouncing For light coming out of water, critical angle is about 50°. Nice online demonstrationNice online demonstration (http://www.physics.northwestern.edu/ugrad/vpl/optics/snell.html) Scattering by large particles—geometric optics (Figure uses a different notation, n instead of m r )

9. Rainbow Isaac Newton (1700s) When and where is it easiest to see rainbow?

10. Alexander’s band Secondary Rainbow

11. Goal: describe phase function ( P ) using few parameters so that it can be handled easily in equation of radiative transfer P l is l t h order Legendre polynomial (function for any x between –1 & 1) ; ; ;  l is case specific Legendre coefficient, given by (why?) ;, called asymmetry factor g = 0for isotropic scattering g = 1 for completely forward scattering g = -1for completely backward scattering Guess g using plot of phase functions (visible wavelengths): Rayleigh scattering: Aerosols: g ≈ 0.75 Cloud droplets: g ≈ ? Simple approximation that uses only three terms: Henyey-Greenstein phase fn. ; Legendre expansion

12. Sample Mie phase functions Figure from a book Why no ripples? Why no polarization? corona aureole glory

14. Scattering coefficient and phase function: Bulk parameters: Mean radius: Total cross-sectional area: Total volume: Liquid water content: (  is density of water) For a single size (r), Effective radius: When LWC integrated in entire column to give Liquid water path: Application in remote sensing: Combining droplets of various sizes related to LWC related to 

15. Non-spherical particles T-matrix method: Rotational symmetrical particles: Series expansion uses spherical Henkel and Bessel functions, etc. Free public codes (FORTRAN) available, fast FDTD method: irregular particles (e.g., ice crystals, aerosol) Finite difference time domain Computationally expensive Codes available (commercial too)

16. Sample ice crystal phase functions 22° and 46° halos