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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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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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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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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
q
e|| sca
analyzer I sca
analyzer ||
I || sca
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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 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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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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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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II. 1. THE DIRECT PROBLEM
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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II. 1. THE DIRECT PROBLEM
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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II.3 LIGHT SCATTERING - FUNDAMENTALS -
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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II.3 LIGHT SCATTERING - FUNDAMENTALS -
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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II.3 LIGHT SCATTERING - FUNDAMENTALS -
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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II.3 LIGHT SCATTERING - FUNDAMENTALS -
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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II.3 LIGHT SCATTERING - FUNDAMENTALS -
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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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
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
III.1 LIGHT SCATTERING BY A SMALL PARTICLE- RAYLEIGH THEORY -
m
Einc
3 volume material electric field
III 1
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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 t incEp
polarizability
III.1 LIGHT SCATTERING BY A SMALL PARTICLE- RAYLEIGH THEORY -
m
Einc
2
13
2
2
m
mv
III 1
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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 t incEp
polarizability
III.1 LIGHT SCATTERING BY A SMALL PARTICLE- RAYLEIGH THEORY -
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:
III.1 LIGHT SCATTERING BY A SMALL PARTICLE- RAYLEIGH THEORY -
)(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:
III.1 LIGHT SCATTERING BY A SMALL PARTICLE- RAYLEIGH THEORY -
)(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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III.1 LIGHT SCATTERING BY A SMALL PARTICLE- RAYLEIGH THEORY -
)(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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III.1 LIGHT SCATTERING BY A SMALL PARTICLE- RAYLEIGH THEORY -
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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III.1 LIGHT SCATTERING BY A SMALL PARTICLE- RAYLEIGH THEORY -
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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III.1 LIGHT SCATTERING BY A SMALL PARTICLE- RAYLEIGH THEORY -
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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III.1 LIGHT SCATTERING BY A SMALL PARTICLE- RAYLEIGH THEORY -
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
consequences of the wavelength dependence of the scattered intensity in the Rayleigh theory
(provided m does not depend on the wavelength)
III 1
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III.1 LIGHT SCATTERING BY A SMALL PARTICLE- RAYLEIGH THEORY -
4
1
l sca I
consequences of the wavelength dependence of the scattered intensity in the Rayleigh theory
but why blue and not violet ?...
III 1
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III.1 LIGHT SCATTERING BY A SMALL PARTICLE- RAYLEIGH THEORY -
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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III.1 LIGHT SCATTERING BY A SMALL PARTICLE- RAYLEIGH THEORY -
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
SUMMARY OF THE SECTION:
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SUMMARY OF THE SECTION:
“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:
III 2 RAYLEIGH THEORY
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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)
III 2 RAYLEIGH THEORY
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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
III 2 RAYLEIGH THEORY
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III.2 RAYLEIGH THEORY- APPLICATIONS -
any very small particles
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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
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
IV 1 LIGHT SCATTERING BY A SPHERE
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IV.1 LIGHT SCATTERING BY A SPHERE- MIE THEORY -
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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IV.1 LIGHT SCATTERING BY A SPHERE- MIE THEORY -
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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IV.1 LIGHT SCATTERING BY A SPHERE- MIE THEORY -
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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IV.1 LIGHT SCATTERING BY A SPHERE- MIE THEORY -
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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IV.1 LIGHT SCATTERING BY A SPHERE- MIE THEORY -
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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IV.1 LIGHT SCATTERING BY A SPHERE- MIE THEORY -
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
nennonn sph d c E NME
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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IV.1 LIGHT SCATTERING BY A SPHERE- MIE THEORY -
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
nennonn sph d c E NME
the first term (n 1) is called the dipolar term, the other terms (n 2) are the multipolar terms
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IV.1 LIGHT SCATTERING BY A SPHERE- MIE THEORY -
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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IV.1 LIGHT SCATTERING BY A SPHERE- MIE THEORY -
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
SUMMARY OF THE SECTION:
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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
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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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- 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)