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To be published in Applied Optics:
Droplet characteristics measurement in Fourier Interferometry Imaging (FII)
and behavior at rainbow angle
Paul Briard, Sawitree Saengkaew, Xuecheng Wu, Siegfried
Authors:  
Meunier-Guttin-Cluzel, Linghong Chen, Kefa Cen, and Gerard Grehan
Accepted:   16 October 2012
Posted:   16 October 2012
Doc. ID:   174337
Title:

 

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Droplet characteristics measurement in Fourier
Interferometry Imaging (FII) and behavior at rainbow angle
Paul Briard,1 Sawitree Saengkaew,1 Xuecheng Wu,2 Siegfried Meunier-Guttin-Cluzel,1
Linghong Chen,2 Kefa Cen,2 and Gérard Gréhan,1,*
1

UMR 6614/CORIA, CNRS/Université et INSA de Rouen, LABEX EMC3,

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Département ‘Optique & Laser’, BP 8, 76800 Saint Etienne du Rouvray, France
2

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State Key Laboratory of Clean Energy Utilization, Institute for Thermal Power Engineering,
Zhejiang University, 38#, Zheda Road, Hangzhou, 31002
*

Corresponding author: grehan@coria.fr

This paper presents the possibility of measuring the 3D relative locations and diameters of

a set of spherical particles and discusses the behavior of the light recorded around the
rainbow angle, an essential step toward refractive index measurements. When a set of

particles is illuminated by a pulsed incident wave, the particles act as spherical light wave
sources. When the pulse duration is short enough to fix the particles location (typically

about 10 ns), interference fringes between these different spherical waves can be recorded.
The Fourier Transform of the fringes divides the complex fringe systems into a series of
spots, with each spot characterizing the interference between a pair of particles. The
analyses of these spots (in position and shape) potentially allow the measurement of
particle characteristics (3D relative position, particle diameter and particle refractive index
value).

1

OCIS codes: 090.0090, 070.0070, 120.0120, 290.4020, 290.5820.

1. Introduction
The study of disperse two-phase flows requires one to understand and quantify physical
phenomena such as the evaporation or combustion of a particle cloud. In such a cloud, the
evaporation of a droplet is influenced by its neighborhood: the vapor from the other droplets alter

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the evaporation process and drag forces [1]. Many parameters must be measured in order to

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quantify the interactions between the particles, including the distances between particles, their

sizes, velocities, and their representative refractive indices. Many optical characterization
techniques exist. In all optical techniques, particles are illuminated and scatter the light toward

one or many detectors. The recorded signal characterizes the interaction between the light and
the particles. The analysis of this signal allows one to measure the parameters of the particles.

Optical techniques can be grouped into families. Techniques such as phase Doppler

anemometry [2] or standard rainbow refractometry [3] measure parameters for a single drop.

Others techniques such as global rainbow refractometry [4], diffractometry, extinction etc.
directly measure the size distribution for large numbers of particles.

The Fourier Interferometry Imaging (FII) method is used for the simultaneous
measurement of the 3D relative locations of particles from the interference fringes created by
illuminating a section of a cloud [5]. The first aim of the method was the measurement of the 3D
relative locations of particles, similarly to the digital holography method [6], but with the same
accuracy in each of the 3 spatial directions. This paper presents an extension of the FII method to
the measurement of diameters, including a discussion on light characteristics around the rainbow

2

angle, a step toward refractive indices measurements. This approach was validated using
numerical simulations and the Lorenz-Mie theory. Multiple scattering is neglected.
That paper is organized as follows. In Section 2, the coordinate systems to be used are
described and the principle of the FII method is recalled. Section 3 presents the influences of
droplet sizes and refractive indices on the Fourier spectrum. In Section 4, the field of particles
(3D relative locations and diameters) is reconstructed by using the geometric optics from fringe
systems simulated by the Lorenz-Mie theory. Section 5 concerns the composite rainbow which is

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very sensitive to the refractive index and the diameters of particle pairs. Section 6 concludes the
discussion.

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2. Principle of Fourier Interferometry Imaging

In the Fourier Interferometry Imaging approach, as in classical holography, some particles are

illuminated by a pulsed laser beam (typical pulse duration is 10 ns) giving rise, in the far field, to
a response analogous to that of a set of spherical light wave sources. The resulting interference
fringe field is recorded by a CCD camera. The analysis of the field of fringes permits the
measurement of particle characteristics. The scheme of such a setup is shown in Fig. 1.

Fig. 1. Fourier Interferometry Imaging set up and the two coordinates systems which are used.

3

In Fig. 1, two systems of coordinates are shown. The 3D locations of the particles are
represented in the system {OXYZ } and the position of a pixel in the surface of the detector is
represented in the system {O 'ηξ } . The center of the detector O ' lies in the XOZ plane. M is a
point on the surface of the detector. The location of M on the surface of the CCD camera is

(ηM , ξM ) .

The distance from the camera to the measurement volume is large compared to

distances between particles, sizes of the particles and the dimensions of the detector (far field
approximation). The distance OO ' is R0 . To the 2D coordinate system {O 'ηξ } , the 2D Fourier

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Transform associates another 2D coordinate system,

{f

f f } , where fη and fξ are the angular

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0 η ξ

frequencies (number of fringes by degree) and f 0 corresponds to the zero frequency component
(DC component).

The OZ and OX axes are the axis of propagation and the axis of polarization of the

incident beam, respectively. The axis OY is parallel to the axis O ' ξ . The average scattering

angle θ0 is the angle between the line OO ' and the axis of propagation OZ . In order to avoid
Gabor Holography with interferences between the light scattered by the particles and the incident

beam, angle θ0 cannot be equal to 0° or 180°. The scattering angle θ is the angle between an axis

Oi Zi parallel to OZ but including the center Oi of the i-particle and the direction Oi M . The
angles θmin and θ max are the extreme values of θ and characterize the edges of the camera. The
angle θ max − θ min is named the collecting angle of the detector. The dimension of the detector in
direction η is approximately equal to R0 θ max − θ min (far field approximation). In this paper, the
detector has the same dimensions and resolution in the directions η and ξ . The collection angle

4

θmax − θmin does not exceed 20°. The particles are spherical, homogenous and isotropic. They
have a radius Ri and a real refractive index value Ni . Here, i is an arbitrary integer to identify
the particles and Ntot is the total number of illuminated particles. The incident beam is a
monochromatic plane wave characterized by pulsation ω and wavelength λ . The illuminated
particles create interference fringes recorded by the detector. Fig. 2 shows an example of a
numerical simulation of fringes carried out using the Lorenz-Mie theory for 3 water droplets.
The recording angle is 20  5°.

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Fig. 2. Numerical simulation of interferences fringes created by 3 water droplets (forward
scattering off-axis, θ = 20 ± 5° ).

The code based on the Lorenz-Mie theory used for numerical simulations is described in

reference [7]. The code calculates the magnetic and electric fields scattered by each particle, the
total electric and magnetic fields, the associated Poynting Vector and the intensity of the light.
Multiple scattering is not considered. Particles scatter the light independently of each other.
The total field on a pixel (ηM , ξ M ) is a complex vector corresponding to the sum of the fields
scattered by each particle. The expression of light intensity on a pixel (ηM , ξ M ) can be written
as:
5

Ntot

Ntot Ntot

i =1

k =1 l = k +1

I =  Ii + 

2

I k I l cos (φl − φk )

(1)

where Ii , I k and Il are the light intensity recorded from a pixel (ηM , ξ M ) when particles i , k or l
are alone. The terms Ii correspond to the interferences fringes with low spatial frequency
associated with the interference between the different contributions to the scattering by one
particle (for example, refracted and reflected light for forward scattering).
The term 2 [ I k I l ] cos (φl − φk ) corresponds to the interferences between the light
1/ 2

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scattered by each pair {k , l} of particles, associated with fringes of a higher frequency. The aim

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of the FII technique is to measure the characteristics of particles from the terms
2 [ I k I l ] cos (φl − φk ) . The interference fringes system is a very complex 2D signal because of
1/ 2

the multitude of interference terms. In addition, the 2D signal can be altered by the Moiré effect.
For these reasons, the interference fringe system will be analyzed in the associated 2D Fourier
space. The 2D Fourier transform permits the analysis of the different interferential systems to be
performed separately. For the following, the detector surface is named “physical space”.

The 2D Fourier Transform of a 2D matrix is a matrix of complex numbers which are

characterized by a magnitude and phase. Figure 3 displays 2D Fourier maps associated with the

fringe system of figure 2. Fig. 3.a shows the magnitude of the Fourier spectrum of the
interferences fringe transformation displayed in Fig. 2. In order to reduce the effects of the
truncation of the physical space [8] (detector dimensions are finite), which appear as gray bands
in Fig. 3.a, a Blackman-Harris apodization function is applied to the physical space before the
Fourier transform to obtain Fig. 3.b. The grayscale in the magnitude Fourier spectrum is

6

presented on a logarithmic scale. Figures 3.c and 3.d present the phase spectrum associated with
the magnitude spectrum of Figs. 3.a and 3.b, respectively.

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Fig. 3. Magnitude Fourier spectrum (a and b) and phase Fourier spectrum (c and d) of
interferences fringes created by 3 particles (Fig. 2) without using (a and c) and after using (b and
d) the Blackman-Harris apodization function.

The magnitude Fourier spectrum is a 2D real matrix where many spots, associated with

fringes in physical space, are observed. The spot in the center of the spectrum corresponds to the
fringes with low spatial frequency. It is the Fourier transform of the sum of the terms Ii . It

contains all of the information about diameters and refractive indices of particles. The spots
outside the center of the Fourier magnitude spectrum correspond to the fringes with high

frequency. For each pair of particles, there are 2 symmetrical, relative to the center of the
spectrum, spots. The Fourier transform corresponds to the term 2 [ I k I l ] cos (φl − φk ) . In Fig. 3,
1/ 2

there are 3 pairs of particles, 3 associated spots and 3 symmetrical spots. These spots contain the
information about each pair of particles.
The spots in the associated Fourier space code the information of interest (particle 3D
location, particle size, particle refractive index value). To extract this information, several
processing strategies are possible. In this paper, two different approaches will be used:
7



Directly analyze the spot shape in the associated Fourier space.



Carry out the processing in the physical space of the detector for a limited number of
particles by computing the inverse Fourier transform of a limited number of spots (1 spot
in this paper).
The shape of the spots is mainly associated with the Fourier transform of the term

2 [ I k I l ] . The analysis of the shape to measure diameters with is the subject of Section 3. The
1/ 2

reconstruction of the particle field (diameters and 3D relative locations) is discussed in Section 4.

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The analysis of the physical space after filtering in the Fourier space is developed in Section 5.

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3. Shape of spots in magnitude Fourier spectrum

If θ max − θ min is small enough, the spots have an axis of symmetry, parallel to the direction fη ,

associated to η . The spots contain many peaks, with the same ordinate fξ . It is possible to
process a profile of the spot along the axis of symmetry. With this simplification, the study of the

shape of a spot is a 1D problem. The spot shape has been studied for the off-axis forward
scattering. A profile of a spot (red line Fig. 3.b.) for the forward scattering is shown in Fig. 4.

8

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Fig. 4. Profile of a spot (red line Fig. 3.b.) for off-axis forward scattering.

In the case of forward scattering off-axis, the geometric optics provides a sufficient

model to partially describe the shape of the spots. In geometric optics (Fig. 5), light waves are

scattered by a pair of particles by reflection (p = 0 order), refraction (p = 1 order) with, according
to van de Hulst’s notation [9], p, the number of chords inside the particle. In order to identify

them, the order of the rays scattered by particle “1” are named order “ p1 ” and the orders of the
rays scattered by particle “2” are named order “ p2 ”.

Fig. 5. Reflection and refraction of light rays by a pair of particles and associated reference rays.

9

The main peaks at the ends of the spot (peaks number 1 and 3 in Fig. 4) correspond to
interference between waves refracted by a particle and waves reflected by the other particle of
the pair ( p1 = 1 and p2 = 0 or p1 = 0 and p2 = 1 ).
The main peak near the center of the spot (peak number 2 in Fig. 4), according to the
angle θ0 , corresponds to the light waves refracted by the particles, or the waves reflected by the
particles : p1 = 1 and p2 = 1 if θ0 < 70° , p1 = 0 and p2 = 0 if θ0 > 70° . The physical meaning is,
from Fig. 5, that the optical path difference is the same between ray pair ( p1 = 1 and p2 = 1 ) and

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pair ( p1 = 0 and p2 = 0 ), while the path difference is smaller between pair ( p1 = 1 and p2 = 0 ),

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and larger between pair ( p1 = 0 and p2 = 1 ). The optical path difference of two rays scattered by

the pair of particles can be expressed as:

δ p − δ p = (δ p − δ ref ,1 ) − (δ p − δ ref ,2 ) + (δ ref ,1 − δ ref ,2 )
1

2

1

(2)

2

Where δ p1 is the optical path for a ray scattered by particle “1” and δ p2 is the optical

path for a ray scattered by particle “2”. The reference rays are “fictive” rays which lead from the
centers of the particles directly to a point M on the detector. δ ref ,1 and δ ref ,2 are the optical paths
for reference rays associated to particles “1” and “2”. The optical path difference (δ ref ,1 − δ ref ,2 )

has been calculated in reference [5]. The differences



p1

− δ ref ,1

)



and

p2

− δ ref ,2

)

are

respectively the optical path difference between p1 and p2 order rays scattered by the particles

(

)

(

“1” and “2” and the associated reference rays. The path differences δ p1 − δ ref ,1 and δ p2 − δ ref ,2

)

have been calculated according to Glantschnig and Chen [10]. From the optical path difference
between light rays scattered by the two particles, it is possible to calculate the location of the

10

main peaks of the spot. The peak locations fη p =1, p =1 and fη p =0 , p =0 correspond to interferences
1

2

1

2

between waves refracted or reflected by the two particles respectively and are expressed in
equation (3).



 fη p1=1, p2 =1 
 π θ max − θ min

 ≅ ±
180λ
 fη p1=0, p2 =0 








θ 
θ 
R1 N1 sin  0 
R2 N 2 sin  0 



2
2







θ
θ
 0
 
2
1 + N 2 2 − 2 N 2 cos  0   + fη 
 1 + N1 − 2 N1 cos  2 
 
 2 





θ

( R2 − R1 ) cos  0 



2



(3)

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In equation (3), fη is the location of the spot associated with the difference path of the

reference rays (δ ref ,1 − δ ref ,2 ) . It corresponds to an average spatial frequency in physical space
and depends on the 3D relative locations of particles which are described in Section 4.

From Glantschnig and Chen [10], the proportionality relation between the distance of the

main peaks Δfη1 and Δfη2 (Fig. 4) and the radius of the particles can be demonstrated, leading to:

Δfη1 ≅

Δfη2 ≅

π θ max − θ min
180λ

π θ max − θ min
180λ

θ 
N1 sin  0 
2

(4)

θ 
N 2 sin  0 
2

(5)

θ 
+ cos  0  R1
θ
2
1 + N12 − 2 N1 cos 0
2

θ 
+ cos  0  R2
2
θ 
1 + N 2 2 − 2 N 2 cos  0 
2

11

In order to validate the expressions of Δfη1 and Δfη2 , numerical simulations using the
Lorenz-Mie theory were carried out. Interference fringes created by a pair of particles are
simulated and their magnitude Fourier spectrum is calculated using a Fast Fourier Transform
algorithm in FORTRAN [8]. Fig. 6 presents distances between the main peaks of spots simulated
by the Lorenz-Mie theory versus the sizes of particles for different scattering angles θ0 . They are
compared to the calculations using geometrical optics (equations (4) and (5)).

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Fig. 6. Δfη1 versus particle radius. The studied parameter is the scattering angle θ0 . The points
correspond to the Lorenz-Mie computation and dashed lines correspond to geometrical optics
computations (equations (4) and (5)).

From the measurement of the distances Δfη1 and Δfη2 between the main peaks, the

measurement of the diameters of each particle is possible. The location fη can be measured if

diameters are known (equation (3)). Only the main peak locations fη p =1, p =1 or
1

2

fη p =0 , p =0
1

2

corresponding to interferences between refracted waves scattered by particle pairs, or reflected
waves scattered by particle pairs, are necessary.
The next section will exemplify the use of equations (3), (4), (5) and (6) to reconstruct a
particle field from the Fourier spectrum.

12

4. Particle field reconstruction from magnitude Fourier spectrum analysis.
The locations ( fη , fξ ) of the spots depend on the phase difference (φl − φk ) which is connected
to the 3D relative locations of the pair of particles. From reference [5], equation (6) gives the
relation between the 2D location ( fη , fξ ) of a spot in Fourier space and the 3D coordinates of a
pair of particles ( xk , yk , zk ) and ( xl , yl , zl ) , with θ max − θ min expressed in degrees.

 fη 
π θmax − θmin − ( xl − xk ) cosθ0 + ( zl − zk ) sin θ0 
f =±


180λ
yl − yk


 ξ

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(6)

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Equation (6) shows that the measurements of ( xl − xk ) and ( zl − zk ) from spot locations

in Fourier space ( fη , fξ ) requires one to record the fringe system at two different angles θ0 .

To validate the method, interference fringes created by 6 water droplets (refractive index value
equal to 1.3333) with random diameters and 3D relative locations were simulated using the

Lorenz-Mie theory. The particles are in a cubic volume equal to 6003 µm3. The wavelength λ of
the incident beam is equal to 0.532 µm.

Two detectors record the fringe systems at two angles θ0 equal to 20° and 90°. The

collection angle θ max − θ min is equal to 10° for both detectors. The number of pixels for the two

detectors is equal to 512x512. The distances R0 are equal to 1 meter. Fig. 7 shows the simulated
interferences fringes, and the associated Fourier magnitude spectrum for each detector.

13

a

b

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c

d

Fig. 7. Interferences fringes created by 6 water droplets with two forward scattering angles (a
and b) and their magnitude Fourier spectrums (c and d). The scattering angles θ0 are equal to 20°
(a and c) and 90° (b and d).

Because of the symmetrical properties of Fourier space, only half of the magnitude

spectrums needs to be analyzed. The identification of the sets of spots corresponding to a set of
interactions between one particle and the other particles is described in paper [5]. The set of

spots corresponding to interactions between one particle and the remaining particles are
identified. The locations of the peaks of each spot in Fourier magnitude spectrums have been
measured with the software ImageJ. For a given pair of particles, four peak coordinates are
measured: three from the spot identifying the pair at 20° and one from the spot corresponding to
, fη202 , fη20
, with the following inequalities:
the same pair at 90°. These coordinates are noted fη20
1
3

14

and fη902 is the main peak of the corresponding spot at 90°. From the difference
< fη202 < fη20
fη20
1
3
- fη20
, Δfη1 is measured. Then applying equation (4), the radius R1 can be deduced if the
fη20
2
1
- fη202 , Δfη2 is measured, and
value of the refractive index is known. From the difference fη20
3
applying equation (9), the radius R2 is deduced. Using equation (3), with fη202 , R1 and R2
known, the value of fη20 is obtained (using the same procedure with fη902 , fη90 can be obtained).
Finally, fξ20 , fη20 , fξ90 and fη90 are injected into the system of equations (6). The solution of this

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system of equations gives access to the coordinates x1 , y1 , z1 , x2 , y2 and z2 for the two particles

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under investigation.

The measurement of each parameter (3D relative coordinates and diameters) can be made

(N-1) times where N is the number of particles in the control volume: in this case, one particle
creates five pairs with the other particles, and its size can be measured five times in a magnitude

spectrum. The initial particle field and the reconstructed particle field are shown in table 1 and in

Fig. 8. The 3D relative locations and diameters have been measured with an error close to a

micrometer (relatively to particle 6). The measured diameters in table 1 are an average of five
measurements.

15

Original data of particle field
Particle
number

1
2
3
4
5
6

Relative
coordinate
x(µm)

Relative
coordinate
y(µm)

Relative
coordinate
z(µm)

-89
-39
-254
-392
192
-361
128
333
-328
106
-94
-381
-309
-101
-128
0
0
0
Reconstructed particle field by FII
-90
-40
-254
-394
192
-359
128
335
-329
106
-94
-383
-310
-100
-129
0
0
0

Diameters
(µm)

36.04
43.99
27.82
67.34
45.62
49.54

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1
2
3
4
5
6

36.19
43.79
29.35
66.99
46.20
49.09

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Table 1. Diameters and 3D relative locations of initial and reconstructed particle fields.

16

6
0

3

Z (µm)

-100

5
1 4

-200

2

-300

200
100

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-400


m
)

0
-100
-200
-300

300

X

-500

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200

100

Y (µm
)

0

-400

-100

-200

-500

Fig. 8. Initial (red) and reconstructed (blue) particle field.

The possibility to reconstruct the particle field using optical geometry has thus been

shown. The sizes and the 3D relative locations are measured with micrometric accuracy.

5. The composite rainbow

For one spherical particle, the light scattering properties at the rainbow angle are very

sensitive to the refractive index. In this section, the particularities of light scattered around the
rainbow angle by a pair of particles are studied. Fig. 9 shows the interference fringe system
created by 3 water droplets, and their associated magnitude and phase Fourier spectra close to the
rainbow angles of the droplets (in Fig. 9, the scattering angle θ0 is equal to 140°, with a
collecting angle equal to 10°). For individual spherical water droplets, the first rainbow angle is
located at about 138°, in agreement with geometric optics. In the associated Fourier space, a pair

17

of particles (one spot) is selected. The selection is carried out by applying the same rectangular
mask on both the magnitude and phase 2D spectra (see Figure 10). Then the inverse Fourier
transform is applied. The result, a matrix of complex numbers, is the dual contribution of a
unique pair of particles, approximately proportional to the contribution [ I k I l ] exp [ ± j (φl − φk ) ]
1/2

which has a modulus approximately proportional to [ I k I l ]

1/2

(see equation (1)) as exemplified by

Fig. 11. Our interest is focused on the distribution of intensity versus the scattering angle. We
will call the intensity distribution along the horizontal symmetry axis of Fig. 11 versus the

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scattering angle, the composite rainbow.

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Fig. 9. Simulation using the Lorenz-Mie theory of interferences fringes created by 3 water
droplets close to their rainbow angle (a), associated magnitude (b) and phase (c) Fourier
spectrums.

18

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Filtered Fourier space (magnitude spectrum: a, phase spectrum: b) using a
Fig. 10.
rectangular filter. The phases of the number complexes equal to zero are not defined. They are
represented in gray in Fig 9.b.

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Fig. 11.

Composite rainbow associated with a pair of particles and calculated from the
inversion of Filtered Fourier space in Fig. 10.

To prove the validity of this approach, a series of profiles extracted by inverse FFT from
a filtered FFT map have been compared with profiles directly computed in the framework of the
Lorenz-Mie theory by multiplying the profile of the individual particles creating the pair.

5.1. Identical particles

19

In the case of particles with the same diameters and refractive indices, we have [ I k I l ]

1/2

≅I

where I is the intensity scattered by one of the droplets of the pair. Fig. 12 shows an example of
the composite rainbow and standard rainbow created by two identical particles.

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Composite rainbow (green dots) and standard rainbow (red line) created by 2
Fig. 12.
pairs of identical particles (refractive index value equals to 1.33, diameter equal to 130 µm).

Such rainbows can be inversed by using a procedure developed for standard refractometry [3].

5.2. Particles have the same refractive index and different diameters

Figure 13 shows an example of a composite rainbow and two standard rainbows created by two
particles of different sizes but with the same refractive index.

20

Fig. 13.

Composite rainbow (green dots), associated standard rainbows (black line: I1 and

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blue line: I 2 ) and [ I1 I 2 ] function created by two particles with the same refractive index
(refractive index equal to 1.33) and different diameters (130 and 100 µm). The composite
1/2
rainbow has been normalized by the maximum of [ I1 I 2 ] .
1/2

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The composite rainbow has the same shape as the standard rainbows: a low spatial

frequency structure with a main bow and the supernumerary bows, and a ripple structure. The

rising parts of the main bows of the two standard rainbows and the composite rainbow have the

same location. The falling part of the composite rainbow and the falling part of the rainbow
created by the biggest particle are at the same location.

5.3. Particles have different refractive indices

Fig. 14 shows the composite rainbow and associated standard rainbows with different
configurations where the particle pairs have different refractive indexes.

21

3e-9

3e-9

I1 (N1=1,32 D1=100 µm)

I1 (N1=1,32 D1=130 µm)
I2 (N1=1,33 D2=130 µm)

3e-9

I2 (N2=1,33 D2=130 µm)

3e-9

1/2

[I1I2]

[I1I2]1/2
Composite rainbow

Composite rainbow

Intensity

Intensity

2e-9

2e-9

2e-9

2e-9

1e-9

1e-9

5e-10

5e-10

0

0

136

138

140

142

144

136

Scattering angle (°)

4e-9

138

140

142

I1 (N1=1,36 D1=130 µm)

3e-9

I2 (N1=1,33 D2=130 µm)

I2 (N1=1,36 D2=100 µm)
I2 (N1=1,33 D2=130 µm)

[I1I2]1/2
Composite rainbow

Composite rainbow

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Intensity

Intensity

[I1I2]1/2

3e-9

3e-9

2e-9

144

Scattering angle (°)

2e-9

2e-9

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1e-9

1e-9

5e-10

0

0

136

138

140

142

136

144

Scattering angle (°)

138

140

142

144

Scattering angle (°)

Composite rainbows (red lines) and associated standard rainbows (blue and black
Fig. 14.
line) created by two particles with different refractive indices: 1.32 and 1.33; 1.33 and 1.36. The
pair of particles has the same diameter equal to 130 µm and diameters equal to 130 and 100 µm.
The scattering angle is equal to 140±5°.

Figure 14 shows the agreement between the composite rainbow extracted from a spot in

the Fourier space and the product of the intensity scattered by the two particles in the pair. This
is a proof of the validity of the proposed approach, based on the exploitation of the term [ I k I l ]

1/2

in equation (1).

6. Conclusion
We have presented an extension of the FII technique, initially developed for 3D relative position
measurements, to the measurement of spherical particle diameters. The sizes and 3D relative
locations of particles are measured from the magnitude Fourier space associated with the
interference fringes system. Furthermore, the analysis of the FII signal around the rainbow angle

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has been carried out, a necessary step for refractive index measurement. The light scattered
around the rainbow angle characterizes a composite rainbow. This composite rainbow
corresponds to the product of the intensity scattered by a pair of particles. The composite
rainbow is obtained by computing the inverse Fourier transform of an individual spot. The next
step is to use this composite rainbow to measure the refractive index of each particle in the
control volume.
Acknowledgments:

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The authors gratefully acknowledge financial support from La region Haute-Normandie to grant
P. Briard, the Chinese Program of Introducing Talents of Discipline to University (B08026), the

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National Natural Science Foundation of China (NSFC) projects (grants 5080606) and the
National Basic Research Program of China (grants 2009CB219802).

References

1. V. Devarakonda and A. K. Ray, “Effect of inter-particle interactions on evaporation of
droplets in a linear array,” J. Aerosol Sci. 34, 837–857 (2003).

2. W. D. Bachalo and M. J. Houser, “Phase/Doppler spray analyzer for simultaneous
measurements of drop size and velocity distributions,” Opt. Eng. 23, 583–590 (1984).
3. S. Saengkaew, T. Charinpanikul, C. Laurent, Y. Biscos, G. Lavergne, G. Gouesbet, and G.
Gréhan, “Processing of individual rainbow signals,” Exp. Fluids 48, 111–119 (2010).

4. J. P. A. J. Van Beeck, L. Zimmer, and M. L. Riethmuller, “Global rainbow thermometry for
mean temperature and size measurement of spray droplets,” Part. Part. Syst. Charact. 118,
196–204 (2001).

23

5. P. Briard, S. Saengkaew, X. C. Wu, S. Meunier-Guttin-Cluzel, L. H. Chen, K. F. Cen, and G.
Gréhan, “Measurements of 3D relative locations of particles by Fourier Interferometry
Imaging (FII),” Opt. Exp. 19, 13, 12700–12718 (2011).
6. E. Darakis, T. Khanam, and A. Rajendran, “Microparticle characterization using digital
holography,” Chem. Eng. Sci. 65, 1037–1044 (2010).
7. X. Wu, S. Meunier-Guttin-Cluzel, Y. Wu, S. Saengkaew, D. Lebrun, M. Brunel, L. Chen, S.
Coetmellec, K. Cen, and G. Gréhan, “Holography and micro-holography of particle fields: A

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numerical standard,” Opt. Commun. 285, 3013–3020 (2012).

8. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The

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Art of Scientific Computing (Cambridge University Press, 1986).

9. H. C. van de Hulst, Light Scattering by Small Particles (Dover Publications Inc, 1981)

10. W. J. Glantschnig and S.-H. Chen, “Light scattering from water droplets in the geometrical
optics approximation,” Appl. Optics 20, 2499–2509 (1981).

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Figure 1 :

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Figure 2

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Figure 3a

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Figure 3b

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Figure 3c

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Figure 3d

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Figure 4

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Figure 5

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Figure 6

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Figure 7a

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Figure 7b

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Figure 7c

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Figure 7d

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Figure 8

6
0

3

Z (µm)

-100

5
1 4

-200

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2

-300

200
100

-400

0
-100
-200
-300


m
)

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300

X

-500

200

100

Y (µm
)

0

-400

-100

38

-200

-500

Figure 9a

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Figure 9b

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Figure 9c

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Figure 10a

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Figure 10b

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Figure 11

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Figure 12

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Figure 13

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Figure 14a

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Figure 14b

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