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Week 6 Notes

A Mie Scattering Code and Lab

Download MieLab.zip from course web page

Files:

1. lab exercise file: mielab.pdf

2. Fortran source for Mie code: miegamma.f

3. Fortran source for index of refraction code: waterindex.f

4. IDL phase function plotting file: plotmie.pro

5. IDL procedure to read phase function file: readphase.pro

6. A Matlab mie code: matlab mie.pdf

7. An idl source code: IDL mie_single.pro

You will need to know the gamma distribution:

)exp()1(

)(

1

brrNb

rn

where rc = /b is the modal radius and controls the width.

Moments: )1(

)1()(

0

kNbdrrnr

kk

Effective radius:

Effective variance:

Liquid water content:

breff 3

31 effv

1233

4LWC 3

Nb

2

More on particle scattering: other methods

Anomalous Diffraction Theory (ADT)

Simple scattering theory - explains main Qext oscillations.

ADT applies to limits: x >> 1 and m 1 << 1 so no little refraction or reflection.

Phase difference between incident and directly transmitted waves: = 2x(m-1).

ADT integrates sum of incident and transmitted E field in forward direction

Uses optical theorem )(04

2 S

kCext

to obtain extinction cross section, and bulk

absorption coefficient to obtain absorption cross section.

Oscillations in Qext due to constructive and destructive interference of diffracted and

transmitted waves.

For non-absorbing spheres ADT gives:

cos14

sin4

22

extQ

First maximum at ~ 4.1; subsequent maxima at = 2(n +3/4)

3

Extinction curves computed from Lorenz-Mie theory for m=1.5, 1.33, 0.93, 0.8. The

abscissa is = 2x(m-1) and is common to the upper two Mie curves as well as to the

bottom anomalous diffraction theory (ADT) curves [van del Hulst, 1957; Stephens, 1994]

Angular dependence form diffraction

ADT provides extinction and absorption, but not phase function.

In geometric optics limit (x >> 1) light may be treated as rays, except for Fraunhofer

diffraction around a particle.

Babinet’s principle - diffraction pattern is the same from an aperture as for opaque

particle of same size.

For sphere (circular aperture) the diffraction pattern is:

2

1

4

22

0

sin

sin2

4)(

x

xJx

Rk

II

where x is the size parameter, wavenumber k = 2/, and J1 is a Bessel function.

Diffraction peak in phase function (for geometric optics limit): the angular width of the

forward scattering peak (defined at half the maximum, at p(0°)) occurs at Θ ≈1/x. The

height at p(0°) x2.

4

First zero at x sin = 3.83, max at x sin = 5.14.

Some non-spherical computational methods

• Ray Tracing

• Discrete Dipole Approximation

• T–Matrix Method

Ray Tracing

In the geometric optics limit ray optics provide an accurate scattering method for non-

spherical particles.

Ray optics consists of two parts:

1) Diffraction theory for the forward scattering peak,

2) Ray tracing using Fresnel reflection and transmission formulas.

Many rays incident on particle are followed to determine what fraction scatter in each

direction or are absorbed.

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Left: schematic representation of the components of the phase function for

randomly oriented hexagonal ice crystals. Right: geometrical reflection and

refraction by hexagonal crystals.

Fresnel Reflection and Transmission

Plane wave incident on planar dielectric surface. Solve for reflection and transmission

using boundary conditions from Maxwell’s equations (tangential components of fields

are continuous).

Snel’s Law: ti m sinsin

Fresnel formula for polarized reflection amplitude coefficients:

ii

ii

m

mr

22

22

sincos

sincos

ii

ii

mm

mmr

cossin

cossin

222

222

||

Reflection and transmission coefficients for irradiance:

RTRTrRrR 1 and 1 ; and ||||

22

||||

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Note that and

Using Snel’s law, , we derive the Brewster angle where :

Another way of thinking about Brewster:

• Consider that induced electric dipoles in a material will oscillate in

the direction of the polarization of the transmitted light.

• Dipoles do not radiate energy in the direction in which they oscillate.

• If the direction of the transmitted light is perpendicular to the

direction of specular reflectance the dipoles will not create any

reflected light.

• This occurs at r + t = /2 = i + t

• Same condition as above; using Sne’s law,

• For water in air, m = 1.33, B = 53

Ray tracing compares well with Mie theory for large size parameters:

1

1)(0 )(0

2

||

m

mRR 2 when 0 || tiR

ri

ti m sinsin 0 || R

mi tan

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Yang, P., Liou, K., Wyser, K., and Mitchell, D.: Parameterization of the scattering

and absorption properties of individual ice crystals, J. Geophys. Res., 105, 4699–

4718, 2000.

Discrete Dipole Approximation

• A numerical method for scattering from any shape particle that is not too large (x

< 5). See review paper by Draine and Flatau, J. Opt. Soc. Am. A, 11, 1491.

– Particle is divided into dipoles of size d small compared to wavelength.

– The field at one dipole is determined by all other dipoles.

– For a particular incident E field, a linear system of equations may be

solved for the dipole moments.

– The far field scattering properties are calculated from the dipole moments.

• Finite-Difference Time Domain (FDTD) is a competing method that is also used

for small non-spherical particles.

• In both DDA and FDTD the number of dipoles or elements is proportional to the

particle volume quickly becomes computationally prohibitive.

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T–Matrix Method

• Expand incident field into vector spherical harmonics (similar to Mie theory).

• Resulting equations of the expansion coefficients of the incident and the scattered

electric fields are linear.

• The matrix realizing the transformation of the expansion coefficients from the

incident to the scattered fields is called the T- matrix. (A kind of a Müller matrix).

• Can be used for any scattering property of nonspherical particles

• Highly accurate and fast, public codes available, size parameter may exceed 100

• The T-matrix is independent of the incident and scattered fields. It only depends

on the shape of the particle, the size parameter, the refractive index, and the

orientation of the particle.

• It needs to be simulated only once and is then applicable for any direction of

incidence and scattering angle.

• Large computation time

Geometric Optics approximation for single scattering albedo in the limit of weak

absorption

Consider a sphere of radius a. A ray entering the sphere at an angle of

i to the normal to the surface traverses a path =2rcosr before

exiting or undergoing an internal reflection. i .and i are related

through Snel’s law: msini = sinr where m is the refractive index. Let

t be the fraction of incident radiation penetrating the sphere and r be

the fraction of internally incident radiation internally reflected. The

fractional absorption over a distance is )exp(1 where is the

absorption coefficient of water. If reflected, it traverses another

distance , and so on. Therefore, the total fraction of the radiation

incident from this direction that is absorbed is:

)exp(1

)exp(1...)exp()exp()exp(1)exp(1

...)exp(1)exp()exp(1)exp()exp(1

32

2

r

trrrt

trtrt

For weak absorption << 1 and from Fresnel equations t = 1 – r. Substituting, fractional

absorption becomes

.cos21)exp(1

)exp(1rr

r

t

r

t

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To get the absorption cross section we must integrate all incident angles over the

hemisphere:

law). sSnel' (using 113

4

sincossin

14sincoscos2

23

223

2

0

2

23

2

0

2

0

2

mmr

dm

rddrrC iii

iiirabs

113

21 2

322

0

mmrC

C

sca

abs using the geometric optics approximation

.2 2rCsca For m = 1.33, we get r84.01 0

Problem:

A corona is a colored ring of light around the moon or Sun when viewed through a thin

cloud. The outer part of the disk often has a brownish-red tinge. If the diameter of the

disk is 10 times that of the moon, what is the approximate radius of particles in the cloud?

Check out http://www.atoptics.co.uk/droplets/corim1.htm for images of the corona.

Recall the scattering diagram for a circular disk.

The forward lobe is contained within xsin 3. Recall, this is geometric optics so x >> 1

and therefore, sin << 1 so sin sin and so the edge of the primary lobe is at 3/x

or 170/x. If the diffraction ring is ten times the diameter of the moon, then

5.22

5.010

so 68

5.2

170

x and

2

68r . Using 0.65 m for red light,

r 7 m. This is appropriate for liquid water droplet clouds.

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Legendre Polynomial Expansion of Phase Functions

Real phase functions are usually expressed as infinite series of orthogonal functions,

usually Legendre polynomials (Pl): )(cos)(cos0

l

ll Pp

P0(x)= 1

P1(x)= x

P2(x) = (3x2 - 1)/2

P3(x) = (5x3 - 3x)/2

.

.

.

Legendre polynomials are orthogonal:

mn ,12

2

,0)()(

1

1

n

mndxxPxP mn

Legendre coefficients can be obtained from phase function:

xcosd)x(cosP)x(cospl

ll

1

12

12

Recall:

2

0

1

1

)(coscos)(cos4

1cos ddpg

Show that 3

1g

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Term project: Monte Carlo Radiative Transfer

Three parts:

1. Getting started: reproduce probability distributions for mean-free path

(optical depth) and H-G phase function

2. 2-stream MC simulation

3. Multi-stream simulation

1. Getting started

a. Reproduce figures 6.4 and 6.5 (p() and p(), respectively).

b. Show plots for three values of N

c. Define some test of convergence (for example, rms difference < )

2. Two-Stream Simulation

a. Isolated slab of thickness (no surface or surface completely absorbing)

b. Compute reflectance, R, transmittance, T, absorptance, A

• R is fraction of incident photons that are reflected

• T is fraction of incident photons that are transmitted

• A is fraction of incident photons that are absorbed

c. Compare results with 2-stream solutions

• A photon travels an optical path before something happens to it:

– Track position inside medium

• i > , photon transmitted; increment number of photons

transmitted, nt – If it is first “event” (i=1 or 0, depending on how counter is

initialized), increment number of photons directly transmitted,

nt,dir.

– If scattered one or more times, increment number of photons

diffusely transmitted, nt,dif.

• i < 0, photon reflected; increment number of photons reflected,

nr

– Probability it is scattered is the single-scattering albedo ; probability it is

absorbed is 1 − .

– To determine if a photon is scattered or absorbed, compute a random

number. If it is less than or equal to , the photon is scattered.

– If greater than , the photon is absorbed; increment number of photons

absorbed, na

• If a photon is scattered, the direction of its next path (forward or back, increment

or decrement ) is determined by a probability distribution for scattering

directions (phase function):

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– Probability of forward scattering, pf( =1) = (1+g)/2

– Probability of back scattering, pb( = 1) = (1g)/2

– To determine if a photon is scattered forward or backward, compute a

random number. If it is less than or equal to pf, the photon is scattered

forward.

– If it is greater than pf, the photon is scattered backward (reverse sign of ).

2-stream MC summary:

Input: , g, , N (number of photons)

Output: R(= nr/N), T(= nt/N), Tdir(= nt,dir/N), Tdif(= nt,dif/N), A(= na/N)

Test for number of photons for convergence

Compare results against 2-stream solutions, without and with absorption.