lecture on rayleigh scatterring

8/18/2019 Lecture on Rayleigh Scatterring

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Module A:electromagnetic theory

of l ight scatter ing

LECTURE 1 :

small particles and spheres


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OUTLINE - LECTURE 1 -

0. prerequisite – what you should know about the electromagnetic waves –

0.1 electromagnetic field and Maxwell equations

0.2 plane transverse em wave, shape and polarization

0.3 the oscillating electric dipole

I. concept – what is light scattering? –

I.1 light scattering geometry

I.2 fundamentals

I.3 types of light – types of particles

I.4 types of scattering I.5 main parameters governing light scattering

II. context – why light scattering? –

II.1 the direct problem

II.2 the inverse problem

II.3 light scattering – fundamentals –

III. case of the small particle – Rayleigh scattering –

III.1 Rayleigh theory

III.2 instructions of use for the Rayleigh theory

IV. case of the sphere – the Mie solution –

IV.1 Mie theory

IV.2 instructions of use for the Mie solution IV.3 Mie theory numerical codes


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0. PREREQUISITE

what you should know about the em waves


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0.1 EM FIELD AND MAXWELL EQUATIONS- THE UNCHARGED, NON-MAGNETIC, ISOTROPIC CASE -

em field (E , B) is a couple of complex-valued vectors, solution anywhere anytime ofthe Maxwell equations:

;

for isotropic dielectric material, e r ( = relative electric permittivity) is a scalar depending

on the material. The refractive index m is such that: m2 = e r

for this kind of material, only three components of the vector field are independent

(for example : vector E)

at boundary between two dielectric uncharged non-magnetic materials:

the normal components of D and of B are continuous through the interface

the tangential components of E and of H are continuous through the interface

ED r oe e HB o

0 D 0 B

t

BE

t

D

H

no charge

no current

non-magnetic


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0.2 PLANE TRANSVERSE EM WAVE – SHAPE AND POLARIZATION -

em monochromatic plane wave of frequency w and wavenumber k , propagating alongthe z -axis and linearly polarized along the x-axis, is:

which is solution of the Helmoltz equation:

k = m(w /c) (dispersion relation )

k = 2p/l (wavelength def ini tion )

any em wave can be expanded in a series of em monochromatic wave, since the

Maxwell equations are linear ( superposition theorem and Fourier theorem)

x

z k t

oe E eE w i

0EE 2

222

cm

w


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0.3 THE OSCILLATING ELECTRIC DIPOLE – RADIATED ELECTRIC FIELD -

an oscillating electric dipole of dipole moment p radiates electric field of the form(spherical coordinates) :

Every piece of volume v of

a dielectric material submitted to external

electric field, becomes an electric dipole of moment :

kr r k r

ek r r r r

kr

o

dip

i1)(3)(

4 22

i2

ppeeepeEpe

Electric dipole radiation. The dipole lies in the plane of the drawing, point vertically upward and oscillates.

Colour indicates the strength of the field travelloing outward (Wikipedia)

radiative static induction

ext vEp

polarizability per unit of volume = 3 (m21)/(m22)


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I.2 fundamentals











IV.1 Mie theory

IV.2 instructions of use for the Mie solution IV.3 Mie theory numerical codes


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the source of light is not seen directly

sunlight is scattered by small dust particles

I. 1. DEFINITION


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I. 1. LIGHT SCATTERING GEOMETRY

q


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I. 1. LIGHT SCATTERING GEOMETRY

q


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wave scattering = redirection of radiation out of the incident direction of propagation

scatter ing resul ts from light-matter interaction

(e.g. interaction with particles)

reflection, refraction, diffraction, are forms of wave scattering

I. 2. LIGHT SCATTERING - FUNDAMENTALS -

reflection nebula IC 2631 MPG/ESO


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frequency = actual speed of light/wavelengthI. 3. TYPES OF “LIGHT”...


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Aitken particles

gas molecules

viruses

tobacco smoke

bacteria

ice crystal

fog

mist

rain

objects visible by human eye

drizzle

pollen

snowflake

° PM2.5°

PM10

l i g h t v i s i b l e b y h u m a n e y eI. 3. TYPES OF “LIGHT” AND OF “PARTICLES”

Diesel smokesnowflake

ice crystal

pollen

Diesel smoke

bacteria

smoke

viruses


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I. 4. TYPES OF SCATTERING (1)

elastic scatter ing wavelength of scattered light

= wavelength of incident light

inelastic scatter ing wavelength of scattered light

wavelength of incident light


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I. 4. TYPES OF SCATTERING (1)

elastic scatter ing wavelength of scattered light

= wavelength of incident light

inelastic scatter ing wavelength of scattered light

wavelength of incident light


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I. 5. MAIN PARAMETERS GOVERNING LIGHTSCATTERING

optical size parameter : x p L /l

contrast : difference of refractive index m m 0

m 0

wavelength :m

incident wave

particle shape

particle material

sphere of radius a L = diameter = 2a

cube of side a L = a

…


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I. 5. MAIN OBSERVABLES FOR LIGHT SCATTERING

angular distr ibution of the scattered intensity :

I sca(q , f ) = ( I sca I // sca)

l inear polari zation: P = ( I sca I // sca)/( I sca I // sca)

scatter ing, extinction cross sections : C sca , C ext

einc

e||ince

sca

x

z

y

qf

e|| sca


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scatter ing, extinction cross sections : C sca , C ext

einc

e||ince

sca

x

z

y

q

e|| sca

analyzer I sca

analyzer ||

I || sca


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scatter ing, extinction cross sections : C sca , C ext x

y

Note : Im{m} = 0 absorption = 0

scattering

scattering scattering

scattering

absorption

loss of intensity due to

scatter ing and absorption

I inc I trans

I trans = I inc eC

extn l

Beer-Lambert law

n


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SUMMARY OF THE SECTION:

“GENERALITIES ABOUT LIGHT SCATTERING”

light scattering = em wave/particle interaction

two fundamental non-dimensional parameters:

the optical size parameter x p L/l

the contrast mm0


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0.2 plane transverse em wave, shape and polarization 0.3 the oscillating electric dipole



I.2 fundamentals











IV.1 Mie theory

IV.2 instructions of use for the Mie solution

IV.3 Mie theory numerical codes


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II. 1. THE DIRECT PROBLEM

you know the incident em wave

you know the particle shape and material

m 0

goal: controlling the scattered wave

(intensity, polarization)

incident wave

wavelength :m


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1) write the Maxwell equations for all the em waves

2) write the boundary conditions on the particle

m 0

3) solve the vector linear equations…

incident wave

wavelength :m


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1) the homogeneous sphere (G. Mie, 1905)

3) the inf ini te cylinder (J.R. Wait, 1955)

the mathematical problem is well-posed and

it is possible to obtain the exact solution in a few cases:

4) the aggregate of homogeneous spheres (F. Borghese, 1979)

2) the coated sphere (A. Aden and M. Kerker, 1951)

core

coating


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II. 2. THE INVERSE PROBLEM

you know the incident em wave

you know the scattered em wave

m 0

m

goal: characterizing the scattering particles

incident wave

wavelength :


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II. 2. THE INVERSE PROBLEM

the mathematical problem is ill-posed and

there is no general way to characterize the particles

though it may be the most important problem…

…only : guess-fit-errors

pollution : which particles? remote objects: which structure? medical: which hazard?


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II.3 LIGHT SCATTERING - FUNDAMENTALS -

incident field = em transverse plane wave

x-linearly polarized em plane wave


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einc

e||inc

e||sca

esca

incincincincinc E E eeE

sca sca sca sca sca E E eeE

= plane wave

= transverse

wave


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einc

e||inc

e||sca

esca

incincincincinc E E eeE

sca sca sca sca sca E E eeE

inc

inc z r k

sca

sca

E

E

S S

S S

kr

e

E

E

14

32)(i

i

amplitude-scatteringmatrix elements


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inc

inc z r k

sca

sca

E

E

S S

S S

kr

e

E

E

14

32)(i

i

I sca= I inc

2

4

2

3

2

2

2

12

1S S S S

= angular distribution of the scattered intensity

= degree of linear polarization of the scattered light

…

to know the amplitude-scattering coefficients

to know everything about the far-field scattered light

P =

2

4

2

3

2

2

2

1 S S S S 2

4

2

3

2

2

2

1 S S S S


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inc

inc z r k

sca

sca

E

E

S S

S S

kr

e

E

E

14

32)(i

i

…

to know the amplitude-scattering coefficients

to know everything about the far-field scattered light

C sca =

f q q f q d d I sca

sin),(

I inc I trans

I trans = I inc eC

ext N l

)0()0(Re2

212

q q p

S S k

C ext

Beer-Lambert law

N


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when several scattering particles are illuminated at the same time, two cases may appear:

• either the particle positions are correlated one each other (they form a rigid aggregate)

fields are additive (coherence)

• or the particle positions are uncorrelated (they all move independently)

intensities are additive (no coherence)

i

sca sca sca i E i E I 22

)()(

i

sca

i

sca sca i E i E I

22

)()(


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0.2 plane transverse em wave, shape and polarization 0.3 the oscillating electric dipole



I.2 fundamentals


I.4 types of scattering

I.5 main parameters governing light scattering









IV.1 Mie theory



III 1


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III.1 LIGHT SCATTERING BY A SPHERE- RAYLEIGH THEORY -

Step 1 : replace the particle by em dipole

Step 2 : the Maxwell equations give the polarizability of the dipole

Step 3 : the scattered field is the resulting dipolar field

d e r i v a t i o n i n a n u t s h e l l

III 1


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III.1 LIGHT SCATTERING BY A SMALL PARTICLE- RAYLEIGH THEORY -

incident wave

scattering particle

radius


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particle of volume v, with refractive index m embedded in a medium of relative permittivity

m =1 and a uniform electric field Einc

solution of the Maxwell equations is dipole radiation from the dipole moment:

t m

mat incEp

2

14

2

23

p


m

Einc

3 volume material electric field

III 1


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t t incEp

polarizability


m

Einc

2

13

2

2

m

mv

III 1


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t t incEp

polarizability


m

Einc

2

13

2

2

m

mv

small particle = electric dipole

III 1


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an oscillating electric dipole radiates the far-field electric field:


)(4

3i

incr r

kr

sca k kr e EeeE

p vector form

erincident wave

III 1


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an oscillating electric dipole radiates the electric field:


)(4

3i

incr r

kr

sca k kr e EeeE

p vector form

erincident wave

exercise : write the formula in the matrix form

inc

inc z r k

sca

sca

E

E

S S

S S

kr

e

E

E

14

32)(i

i

III 1


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)(4

3i

incr r

kr

sca

k

kr

e

EeeE p vector form

e z ’ er

incident wave

exercise : write the formula in the matrix form

e xe y

e z

e x’

e y’ e y

Einc

E//inc

E// sca

E scaq

z k

o

inc e

E i

0

0

E

q

q

cos

0

sin

r e;

q

q

q

sin

0

cos

cos)(oincr r

E E ee

e x’

inc

inc z r k

sca

sca

E

E k

kr

e

E

E

10

0cos

4

ii

3)(i q

p

z = 0

matrix

form

III 1


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inc

inc z r k

sca

sca

E

E k

kr

e

E

E

10

0cos

4

ii

3)(i q

p

particle volume particle material

angular dependence

III 1


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the scattered intensity at 90° is ½ the forward intensity

q 2cos1 sca I

|| || x

z

z

y

q

.

. .

.

y

graphic representations of the angular distribution of the scattered intensity in the Rayleigh theory

inc

inc z r k

sca

sca

E

E k

kr

e

E

E

10

0cos

4

ii

3)(i q

p

242322212

1/ S S S S I I inc sca and

III 1


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I sca(l 450 nm) = 10 I sca(l 800 nm)

small particles scatter much more the small than the large wavelengths

consequences of the wavelength dependence of the scattered intensity in the Rayleigh theory

inc

inc z r k

sca

sca

E

E k

kr

e

E

E

10

0cos

4

ii

3)(i q

p

242322212

1/ S S S S I I inc sca and

4

1

l sca I

III 1


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4

1

l sca I

explains the color blue of the sky

( scattering by molecules 0.3 nm)

and the reddening of forward transmissionscattered blue

not-scattered red


(provided m does not depend on the wavelength)

III 1


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4

1

l sca I


but why blue and not violet ?...

III 1


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4

1

l sca I

but why blue and not violet?...

atmosphere optical absorption

Rayleigh scattering

blue sky spectrum (white curve)

III 1 LIGHT SCATTERING BY A SMALL PARTICLE


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degree of polarization of the scattered intensity in the Rayleigh theory

q

q 2

2

cos1

sin

P

inc

inc z r k

sca

sca

E

E k

kr

e

E

E

10

0cos

4

ii

3)(i q

p

and P =

24

23

22

21 S S S S

2

4

2

3

2

2

2

1 S S S S

q 90°

q 0°

q 180°

Paraselene, 2007, Nikon D80, Sigma lens 10-20 mm, polarizing filter. unprocessed image

backscattering

forward

linearly polarized

partially polarized

unpolarized

q



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“RAYLEIGH THEORY”

when particle is small enough, it can be represented by an

em dipole

the dipole may be anisotropic if the real particle is notspherical

the scattered field is the em dipolar field

III 2 RAYLEIGH THEORY


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III.2 RAYLEIGH THEORYREQUIREMENTS

1) refractive index, shape and volume of the particle

1) table relating the effective (anisotropic) refractive index to the shape

of the particle

INPUT:

REQUIREMENT:



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III.2 RAYLEIGH THEORYCONDITIONS OF USE

1) x < 0.3

2) a less restrictive condition is sometimes used : |m |x < 1, though it is verysimilar

CONDITIONS :

that is very small particles : L < 0.1 l ex. : L < 50 nm for l 500 nm(or : N < 150000 Fe atoms)



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III.2 RAYLEIGH THEORYPROS & CONS

PROS :

• valid for any finite particle shape

• fully analytical, then easy to handle

CONS :

• only for very small particles



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III.2 RAYLEIGH THEORY- APPLICATIONS -

any very small particles


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I.2 fundamentals


I.4 types of scattering

I.5 main parameters governing light scattering









IV.1 Mie theory



IV 1 LIGHT SCATTERING BY A SPHERE


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IV.1 LIGHT SCATTERING BY A SPHERE- MIE THEORY -

as plane waves form natural basis for a medium invariant by translation,

vector spherical harmonics form natural basis for the spherical symmetry

Step 1 : search for scalar functions y solutions of the scalar Helmoltz equation

under the form y = f(r ) g(q ) h(f ), and which are finite everywhere

Step 2 : for each scalar function y , define the two vector functions :

then: M and N are both solutions of the correspondingvector Helmoltz equation

Step 3 : expand the incident em plane wave Einc and the em scattered wave E sca as sums of thevector spherical wave functions M and N

Step 4 : write the boundary conditions for the em fields at the surface of the sphere to find

equations relating the various coefficients in the expansions of the waves inM and N

Step 5 : solve the equations

022 y y k

y rM MN k

1

d e r i v a t i o n i n a n u t s h e l l



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the incident plane wave writes :

where the vector spherical wave functions M(R), N(R) can be written in terms

of associated Legendre and Bessel functions of the reduced distance r = k r :

with the angular functions :

1

)R (

1

)R (

1

cosi in

nenon x

kr

oinc E e E NMeEq

r r p

r y 2/12

nn J

r y

r p r

y p q q d

d nn nnnr nnne

1sincossincos)1( 2

)R (

1 eeeN

nnnnoy

r p q

1sincos

)R (

1 eeM

q q p sin/cos1nn P

q q d dP nn /cos1

)1(

12i

nn

n E E n

on



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examples of Legendre and Bessel functions :

p 1 1 1 = cos q y 1 = (sin r – r cos r )/ r

p 2 3 cos q 2 = 3(2cos

2 q 1) y 2 = ((3- r 2

)sin r 3 r cos r )/ r 2

…

r r p

r y 2/12

nn J

r y

r p r

y p q q d

d nn nnnr nnne

1sincossincos)1( 2

)R (

1 eeeN

nnnnoy

r p q

1sincos

)R (

1 eeM

q q p sin/cos1nn P

q q d dP nn /cos1



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the wave inside the spherical particle writes generally :

with the same vector spherical wave functions M(R), N(R) as for the incident wave

(because the field has to be finite at the origin)

1

)R (

1

)R (

1 in

nennonn sph d c E NME



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the scattered wave writes generally :

where the vector spherical wave functions M and N are the same as M(R), N(R),

replacing the regular y n by the general x n (because the field has not to be finite at the origin):

in which :

1

11 in

nennonn sca ab E NME

q q p sin/cos1nn P

q q d dP nn /cos1 r r r

p r x 2/12/1 i

2 nnn Y J

r x

r p r

x p q q d

d nn nnnr nnne

1sincossincos)1( 21 eeeN

nnnnox

r p q

1sincos1 eeM



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note that only the radial component of the corresponding vector spherical

functions are different :

is the regular part of Mo1n

r r r p r x 2/12/1 i2 nnn Y J

nnnnox

r p q

1sincos1 eeM

nnnnoy

r p q

1sincos

)R (

1 eeM

r r p

r y 2/12

nn J

)R (1noM



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we then write the em boundary conditions, namely:

• the tangential components of the electric field inside the particle just beneath the surface must be equal to the tangential components of the

sum of the incident field and the scattered field just above the surface

• the same for the components of H (that is essentially )E

1

)R (

1

)R (

1 in


1

1

)R (

11

)R (

1 in

nennenonnon scainc ab E NNMMEE

and using the orthogonality of the vector spherical wave functions (angular part),

one eventually finds four linear equations in the four unknownsan, bn, cn, d n

at r a

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

k

k

k

k

A

d

c

b

a

k

k

k

k

d

c

b

a

A

,4

,3

,2

,1

1

,4

,3

,2

,1



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the coefficients an, bn allow to compute straightforwardly the scattered em field

anywhere outside the sphere

and the coefficients cn, d n can be used to compute the em field inside the sphere

1

11 in

nennonn sca ab E NME

1

)R (

1

)R (

1 in


the first term (n 1) is called the dipolar term, the other terms (n 2) are the multipolar terms



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inc

inc z r k

sca

sca

E

E

S

S

kr

e

E

E

1

2)(i

0

0i

in particular, the amplitude-scattering matrix writes simply :

from which all the optical scattered quantities can be deduced…

1

1 )()1(

12

n

nnnn bann

nS p

1

2 )(

)1(

12

n

nnnn ba

nn

nS p



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the coefficients an, bn write :

11 )()1(

12

nnnnn bann

n

S p

1

2 )()1(

12

n

nnnn bann

nS p

)(')()(')(

)(')()(')(

mx xm xmx

mx xm xmx

b nnnn

nnnn

n y x x y

y y y y

)(')()(')(

)(')()(')(

mx x xmxm

mx x xmxma

nnnn

nnnnn

y x x y

y y y y

xY x J x x nnn 2/12/1 i2 p

x

x J x xnn 2/1

2

p y



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“MIE THEORY”

the known incident field, the unknown inner field and the

unknown scattered field are expanded on the basis of the

vector spherical functions attached to the spherical particle

the coefficients of the expansion are given by the em

boundary conditions on the surface of the sphere

IV.2 MIE THEORY


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IV.2 MIE THEORYREQUIREMENTS

1) refractive index and radius of the sphere

1) numerical code

INPUT:

REQUIREMENT:

IV.2 MIE THEORY


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IV.2 MIE THEORYCONDITIONS OF USE

1) restricted to homogeneous simple shape with specific symmetries

(sphere, spheroid, infinite cylinder)

CONDITIONS :

IV.2 MIE THEORY


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IV.2 MIE THEORYPROS & CONS

PROS :

• very precise (exact) and stable

• extended to multi-layers

CONS :

• only for a few perfect homogeneous shapes

IV.2 MIE THEORY


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IV.2 MIE THEORY- APPLICATIONS -

perfect sphere

coated sphere

IV.2 LIGHT SCATTERING BY A SPHERE


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IV.2 LIGHT SCATTERING BY A SPHERE- THE TYPES OF THEORIES -

Mie solution is the complete solution

(any wavelength, refractive index

and particle radius)

…though approximate theories are

useful for simple formula at hand, better understanding, etc



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IV.3 LIGHT SCATTERING BY A SPHERE- THE TYPES OF THEORIES -

Rayleigh scattering

.

Geometric scattering

.



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V.- THE TYPES OF THEORIES -

absorbance is measured by the

extinction efficiency :

1

22

2 )12(

2

n

nnext ban

k C

p



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- RAYLEIGH FROM MIE THEORY -

inc

inc z r k

sca

sca

E

E

S

S

kr

e

E

E

1

2)(i

0

0i

if truncated to the first term (|m |x


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- MIE THEORY -

numerics

most popular free code in FORTRAN from Bohren and Huffman : BHMIE.f

for example here https://code.google.com/archive/p/scatterlib/



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- MIE THEORY -

optical efficiencies

S 11 is the scattered intensity (unpolarized/unpolarized)

POL is the degree of polarization of the scattered light

S 33 = Re(S 1S 2*+S 3S 4*)

S 34 = Re(S 2S 1*+S 4S 3*)

easy to modify the code to obtain the desired quantities

numerics

most popular free code in FORTRAN from Bohren and Huffman : BHMIE.f



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R. Botet 04/2016

- MIE THEORY -

numerics

there are also a number of alternative free code …

a list http://diogenes.iwt.uni-bremen.de/vt/laser/wriedt/Mie_Type_Codes/body_mie_type_codes.html

Summary of the Lecture 1

http://diogenes.iwt.uni-bremen.de/vt/laser/wriedt/Mie_Type_Codes/body_mie_type_codes.htmlhttp://diogenes.iwt.uni-bremen.de/vt/laser/wriedt/Mie_Type_Codes/body_mie_type_codes.htmlhttp://diogenes.iwt.uni-bremen.de/vt/laser/wriedt/Mie_Type_Codes/body_mie_type_codes.htmlhttp://diogenes.iwt.uni-bremen.de/vt/laser/wriedt/Mie_Type_Codes/body_mie_type_codes.html


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Summary of the Lecture 1

very small particles Rayleigh scattering

approximation L 0

analytical

any shape

spherical particles Mie theory

exact

computer code

homogeneous spheres (coated, infinite cylinders, spheroids)

lecture on rayleigh scatterring

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