Light Scattering predictions.
G. Grehan
L. Méès, S. Saengkaew, S. Meunier-Guttin-Cluzel
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1

Rainbow: Far field scattering
 Fluorescence : Internal field
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2
Theories
• Airy theory (1838): A scalar solution. Could be applied only close
of rainbow
• Lorenz-Mie theory (1890-1908): rigorous solution of Maxwell
equations. All the scattering effects are merged.
Extension to multilayered spheres.
• Debye theory (1909): post processing of Lorenz-Mie. The different
scattering effects could be separated.
• Nussenzveig theory (1969) : is “analytical integration” of Debye
series, leading to a generalization of Airy. It is clean to have
a larger domain of application than Airy
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One particle
Rainbow
Airy, Lorenz-Mie,
Debye, Nussenzveig
Fluorescence
Lorenz-Mie, Debye
Internal field
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A cloud (section)
Global
Multiple scattering
4
List of program
Internal field and homogeneous sphere: INTGLMT
Internal fields+near field : NEARINT
1or 2 beam(s) impinging on a sphere, internal field :
3D2F (3 dimensions)
2D2F (2dimensions)
DEBYE internal field : INTDEBYE
Far field and homogeneous sphere: DIFFGLMT
Far field and multilayered sphere : MCDIFF
DEBYE Far field : DIFFDEBYE
Far field for pulses
: PULSEDIFF
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Rainbow far the rainbow angle
according with Nussenzveig
Impact
parameter
III
I
II
IV
V
Geometrical
rainbow ray
impact
parameter
Scattering angle
Geometrical
optics angle
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Comparison of Lorenz-Mie, Debye, Nussenzveig and
Airy predictions for one particle
Fig 4. Scattering diagram around first rainbow simulated by Lorenz-Mie,
Debye, Airy and Nussenzveig theories (d=95.5 µm, m=1.33-0.0i, =500)
1.6
Mie Mie
Debye,Debye,
p=2 p=2
Nussenzveig
Airy
1.4
Normed intensity
1.2
1.0
0.8
0.6
0.4
0.2
0.0
136
138
140
142
144
146
148
150
Scattering angle
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Rainbow far the rainbow angle
according with Nussenzveig
1.2
Nussenzveig, Eqs (19) and (20)
Debye, p = 2
Normed Intensity
1.0
0.8
0.6
0.4
0.2
0.0
130
135
140
145
150
155
160
Scattering angle
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Z is the argument of Airy function
Intensity Ratio Debye/Nussenzveig)
1.6
10 micron
20 micron
50 micron
100 micron
1.4
1.2
1/ 3
 11 
  2 / 3  
Z  

R
2
 h 

y = -0.2523x + 1.2807
y = -0.1642x + 1.1722
y = -0.0946x + 1.0982
1
0.6
hm 
Comparision scattering diagrame between Debye & Nussenzveig without and with coef160°
For water (m=1.333)
0.4
0.2
1.2
0
2
4
Normalized scattering Intensity
-2
,
Y= -0.2523Z+1.2807
Y= -0.1642Z+1.1722
Y= -0.0946Z+1.0982
Y= -0.0593Z+1.0639
D<=15
15>D<=35
35>D<=75
75>D<=150
y = -0.0593x + 1.0639
0.8

6
8
10
1.0Z
Debye 10
20 micron
Original20
Nus20
micron
Nus160°
Debye
20
micron
micron
Original50
Debye
Nus20
micron
micron
Col 1 vs Debye_100
Nus160° 10 micron
Nus160° 20 micron
Nus160° 50 micron
Nus160° 100 micron
Original_Nus 10 micron
Original Nus20 micron
Original_Nus50 micron
Original_Nus100 micron
.8
.6
.4
.2
0.0
125
130
135
140
145 2004
150
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September
Scattering Angle
155
160
165
9
Comparison of Lorenz-Mie, Debye and Nussenzveig
predictions for one particle
Fig 6 Comparison scattering diagram between Lorenz-Mie,
Debye (p=0+2) and Nussenzveig (p=0+2)
Normed scattered intensity
1.6
1.4
Nussenzveig, p = 0 and 2
Debye p = 0 and 2
Lorenz-Mie
1.2
1.0
0.8
0.6
0.4
0.2
0.0
135
140
145
150
155
160
Scattering angle
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Comparison of Lorenz-Mie, Debye and Nussenzveig
predictions for cloud of particle
Global rainbow
rainbow distribution
distributionsimulated
simulatedby
by
Fig 7. Global
Lorenz-Mie, Nussenzveig Theory
Theory
Lorenze-Mie,
(mean diameter
diameter == 50
50 micron,
micron,rms
rms==200)
200)
(mean
1.2
1.2
Lorenz-Mie
Lorenz-Mie
NussenzveigOriginal
by add coefficient
Nussenzveig
Nussenzveig Original
Normalized
Normalized Intensity
Intensity
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0
125
125
130
130
135
135
140
140
145
145
150
150
155
155
160
160
165
165
Angle
Angle
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Effect of an imagining part of the refractive index
nk=0.0005
nk=0.001
nk=0.0025
nk=0.005
1.0
Normed intensity
0.8
0.6
0.4
0.2
maximum
Refractive
index
nk=0.0005
139.60
1.330
nk=0.001
139.63
1.3299
nk=0.025
139.58
1.3298
nk=0.005
/////// ///////
0.0
120
125
130
135
140
145
150
Scattering angle
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Refractive index at center is nc
Refractive index at surface is ns
The law is :
 bx
e 1
n  nc  (ns  nc) b
e 1
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Global rainbow for a particle with gradient. The refracive index at center
is equal to 1.33 and at surface to 1.36.
Behaviour of the radial rafractive index
1.365
1.2
b = -2
b=2
b = -6
b=6
1.0
1.355
Normed intensity
Real part of refractive index
1.360
1.350
1.345
1.340
1.335
0.2
0.4
0.6
0.8
1.0
0.4
0.0
1.325
0.0
0.6
0.2
b = -2
b=2
b = -6
b=6
1.330
0.8
1.2
120
130
140
150
160
Scattering angle
Normad radius
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b=2
1.3259
b=-2
1.3459
b=6
1.3198
b=-6
1.3578
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2D model
Two steps:
•Excitation by the laser
•Collection in a given solid angle of the fluorescence
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2D model : Excitation map
Internal intensity (in log-scale) created by a beam with a beam waist diameter equal to 20 µm, and a
wavelength equal to 0.6 µm. The particle is a water droplet with a diameter equal to 100 µm and a
complex refractive index equal to 1.33 – 0.0 i. The parameter is the impact location of the beam: (a)
 = 50 µm (on the edge of the droplet), (b)  = 30 µm and (c)  = 0 µm (on the symmetry axis of the
droplet).
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2D model : Detection map
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2D model : Answer
Map of fluorescence emission. The particle is a water droplet of 100 µm on which impinges a laser
beam with a diameter equal to 20 µm, and for an impact location equal to 50 µm (Fig. 2a). The
parameter is the location of the collecting lens: (a) 0°, (b) 90° and (c) 180°.
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2D model
60 µm
50 µm
40 µm
30 µm
20 µm
10 µm
0 µm
-10 µm
-20 µm
-30 µm
-40 µm
-50 µm
-60 µm
Logarithm scale
2
20
1
15
10
2
10
0.1
0.01
3
0
20
15
10
5
1
1
5
3
60 µm
50 µm
40 µm
30 µm
20 µm
10 µm
0 µm
-10 µm
-20 µm
-30 µm
-40 µm
-50 µm
-60 µm
Logarithm scale
0
0
0
5
10
15
10
20
1
0.1
0.01
0.01
0.1
1
10
0.01
5
6
6
0.1
10
1
15
4
4
10
20
5
5
Diagram of fluorescence for a water droplet of 100 µm. The parameter is the impact
location  which runs from 60 µm to –60 µm by steps of 10 µm. The left figure is in
linear scale while rigth figure is in logarithm scale.
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3D model : Excitation map
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