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

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 

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)

Phase function plots

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

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

Theory developed around 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 of Thomas & Stamnes book Publicly available codes, even on-line calculator at 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

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 ( Scattering by large particles—geometric optics (Figure uses a different notation, n instead of m r )

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

Alexander’s band Secondary Rainbow

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

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

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 

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)

Sample ice crystal phase functions 22° and 46° halos