Real part of refractive index (mr):How matter slows down the light: where c is speed of light

cmedium cvacuum

mr

Question 3: Into which direction does theScattered radiation go?

In visible light, for water: mr ≈ 1.33

for glass: mr ≈ 1.5

mr is not related to mass densitymr 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

cos x x y y z z

cos cos cos sin sin cos

How to calculate ?

1:

2:

Scattering angle

0

a

Equation of radiative transfer:

dI

ds I

0

a

1

s c

b

I1 I0

s

Ia Ib I c ...

s I0

0

a

1

s

Ia

s I0 a

P

d 4

Definition of P:

Normalization: average value = 1:

P 4 d 1

4 d P

0

0

2

sin dd 4

Scattering phase function (P)

Phase function plots

Best approach depends on size parameter

• If x < 1 (that is, r << ): Rayleigh scatteringAssumes: 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 fieldDipole scattering depends on angle between E-variations and plane of scattering(specified by incoming and outgoing directions):

Perpendicular component: P() = 1Parallel component: P() cos2()

Overall:

Clear-sky polarization Multiple scattering reduces polarization(e.g., clouds)

Rayleigh phase function

P 3

41 cos2

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 an and bn 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

Problem for large x: long series is needed (n should go up to ~100 for cloud drops, much more for rain)

Mie phase function

Qs 2

x 22n 1

n1

an

2 bn

2

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 demonstration (http://www.physics.northwestern.edu/ugrad/vpl/optics/snell.html)

Scattering by large particles—geometric optics

1,out 1,in

sin1

sin2

c i

c2

mr,2

mr,1

(Figure uses a different notation, n instead of mr)

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

Pl is lth order Legendre polynomial(function for any x between –1 & 1)

; ; ;

l is case specific Legendre coefficient, given by

(why?) ; , called asymmetry factor

g = 0 for isotropic scatteringg = 1 for completely forward scatteringg = -1 for completely backward scattering

Guess g using plot of phase functions (visible wavelengths):

Rayleigh scattering:

Aerosols: g ≈ 0.75Cloud droplets: g ≈ ?

Simple approximation that uses only three terms: Henyey-Greenstein phase fn.

; ;

Legendre expansion

P cos 2l1 l0

2N 1

l Pl cos

Pl x 1

2l l!

d l

dx lx2 1 l

P0 x 1

P1 x x

P2 x 1

23x2 1

P3 x 1

25x3 3x

l 1

2P cos

1

1

Pl cos d cos

0 1

1 1

2P cos

1

1

cosd cos g

P 3

41 cos2

g 1 1

2P cos

1

1

cosd cos 1

2P

0

180

cosd ?

P 1 g2

1 g2 2gcos

0 1

1 g

2 g2

Sample Mie phase functions

Figure from a book

Why no ripples?

Why no polarization?

corona

aureoleglory

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

Q r 0

n r r 2 dr

P P ,r Q r 0

n r r 2 dr

r r n r

0

dr

n r 0

dr

A r 2 n r 0

dr

V 4

3r 3 n r

0

dr

LWC V

V

A

4

3r

V

A re or r

r 3 n r 0

dr

r 2 n r 0

dr

3

2

LWC

r

3

2

LWP

r

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 domainComputationally expensiveCodes available (commercial too)

Sample ice crystal phase functions

22° and 46° halos