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Measurements of 3D relative locations of
particles by Fourier Interferometry Imaging
(FII)
Paul Briard,1 Sawitree Saengkaew,1 Xuecheng Wu,2 Siegfried Meunier-Guttin-Cluzel,1
Linghong Chen,2 Kefa Cen,2 and Gérard Grehan1,*
1

UMR 6614/CORIA, CNRS/Université et INSA de Rouen, Departement ‘Optique & Laser’, BP 8, 76800 Saint Etienne
du Rouvray, France
2
State Key Laboratory of Clean Energy Utilization, Institute for Thermal Power Engineering, Zhejiang University,
38#, Zheda Road, Hangzhou, 310027, China
*grehan@coria.fr

Abstract: In a large number of physical systems formed of discrete
particles, a key parameter is the relative distance between the objects, as for
example in studies of spray evaporation or droplets micro-explosion. This
paper is devoted to the presentation of an approach where the relative 3D
location of particles in the control volume is accurately extracted from the
interference patterns recorded at two different angles. No reference beam is
used and only ten (2 + 8) 2D-FFT have to be computed.
©2011 Optical Society of America
OCIS codes: (290.4020) Mie scattering; (070.0070) Fourier optics and signal processing;
(120.0120) Instrumentation, measurement, and metrology.

References and links
1.
2.

3.
4.
5.
6.

V. Devarakonda and A. K. Ray, “Effect of inter-particle interactions on evaporation of droplets in a linear array,”
J. Aerosol Sci. 34(7), 837–857 (2003).
G. Castanet, P. Lavieille, F. Lemoine, M. Lebouché, A. Atthasit, Y. Biscos, and G. Lavergne, “Energetic budget
on an evaporating monodisperse droplet stream using combined optic methods evaluation of the convective heat
transfer,” Int. J. Heat Mass Transfer 45(25), 5053–5067 (2002).
G. E. Elsinga, F. Scarano, B. Wieneke, and B. W. Oudheusden, “Tomographic particle image velocimetry,” Exp.
Fluids 41(6), 933–947 (2006).
C. Haigermoser, F. Scarano, and M. Onorato, “Investigation of the flow in a circular cavity using stereo and
tomographic particle image velocimetry,” Exp. Fluids 46(3), 517–526 (2009).
S. L. Pu, D. Allano, B. Patte-Rouland, M. Malek, D. Lebrun, and K. F. Cen, “Particle field characterization by
digital in-line holography: 3D location and sizing,” Exp. Fluids 39(1), 1–9 (2005).
G. Pan and H. Meng, “Digital holography of particle fields: reconstruction by use of complex amplitude,” Appl.
Opt. 42(5), 827–833 (2003).

Introduction
In combustion studies, for example, the evaporation of one droplet depends on the position of
other droplets in its neighborhood. The vapor coming from the other droplets will change its
evaporation process and the drag forces [1,2]. Complementary, the characterization of the size
and relative velocities of the droplets created by the micro-explosion of a large emulsion
droplet is a key parameter to understand this kind of atomization. For these two cases, the
knowledge of the 3D relative locations of the droplets in the control volume is necessary to
understand the physics and validate numerical simulations.
To measure the 3D locations of droplets essentially two approaches are used up to now:
holography and Tomographic PIV Approach [3,4].
Nowadays, the most popular configurations used in holography of sprays are digital
holography configurations where the field of interference is recorded by a CCD camera and
the reconstruction of the particle field is numerically achieved [5]. In this digital holographic
approach the field of particles is lit by a laser beam, and the interferences between the light

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scattered by the droplets and a reference beam are recorded. In the particular case of Gabor
holography configuration, the reference beam is also the incident beam on the particle field. In
a second step, the field of particles is numerically reconstructed by computing the
electromagnetic field at different distances from the hologram in the framework of the
diffraction theory. The main limitation of this technique is the relatively long computing time
for the image reconstruction as a 2D-FFT must be computed for each reconstruction plane and
furthermore the error on the longitudinal position is much larger than for the transverse
locations [6].
Alternatively, The PIV technique initially developed to measure 2D velocity in a plane is
under extension to 3D-velocity measurements. Basically, PIV technique is based on the
measurement of the displacement of particle images during a short delay. Then in 3D-PIV, the
3D locations are extracted from intensity variation, assuming more or less than the diffusion
by the object is essentially isotropic. The tomographic PIV system, however, is very
complicated (normally with four individual cameras) and time consuming both in calibration
and reconstruction processes.
In this paper, a new method based on the direct analysis of the interference fringes created
by the scattering of a plane wave by a set of particles is proposed. The main advantages are:
an equivalent accuracy on the particle location in the three space directions, a relative
economy in the computation effort and a very reduced sensitivity to Moiré effect (the Moiré
effect is due to an under sampling of oscillating patterns, in classical holography the
oscillating patterns are created by interference between spherical waves and a plane wave),
but only the relative locations of the particles are obtained.
The paper is organized in 4 sections, including this introduction. The second section is
devoted to i) the introduction of the principle of the method ii) the presentation of a rigorous
numerical model of the fringes patterns iii) the deduction of an analytical expression of the
fringe spacing under a far-field approximation iv) the description of the 2D-FFT associated to
the fringe patterns. In the third section the analytical predictions are validated by processing
numerically simulated images. The main characteristics of the 2D-FFT associated to the fringe
systems are described, and the quantification of the inter-particle distance is introduced, then
some exampling results are given. The last section is a conclusion.

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Principle, tools and numerical analysis
Geometry and measurement principle

Fig. 1. Configuration under study.

Assume a cloud of spherical, homogeneous and isotropic particles on which impinges a
pulsed plane wave and a detector which is located in a direction (  0 ), at a distance R
relatively to a point O located in the cloud on the axis of the incident beam. The incident plane
wave is assumed to be propagating along the z axis towards the positive z. The coordinates of
each particle, relatively to the point O, are

xi , yi , zi  . On the surface of the detector,

assumed to be perfectly perpendicular to the direction O’O, the light scattered by the particles
interfere together, creating a complex fringes system. This fringes system codes the 3D
structure of the field of particles. Then the challenge is to extract the particle positions from
the fringes characteristics. It is the aim of that paper to expose a possible strategy to achieve
this task.
The next step is devoted to the introduction of the rigorous computation of fringe patterns
in the framework of the Lorenz-Mie theory. The second step is to analytically deduce a
relationship between the fringe steps and the distance between pair of particles. The third step
is to identify and quantify the fringe system created by each pair of particles, a task achieved
by using a 2D-FFT. The fourth step, starting from the identified fringe systems and the
relationship between fringe systems and particle distance, is to reconstruct clouds of particles,
creating equivalent fringe system.
Numerical computation of fringe patterns
The configuration under study is schematized in Fig. 1. A plane wave, called incident wave,
impinges on a cloud of N spherical particles. Each particle is characterized by its coordinates
xi, yi and zi relatively to a Cartesian coordinate system (OXYZ). A detector, assumed to be
represented by a section of a plane, is located in the angular direction (  0 ) at a distance R (R
= OO’) of the center of the Cartesian coordinate system. On this detector the interference
between the light scattered by the particles is recorded.

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Under the assumption that multiple scattering can be neglected (no light scattered by more
than a particle reaches the detector), this problem is N times the basic problem of the
scattering of a plane wave by a spherical particle which has been solved by Mie.
The field scattered by one particle
A Mie scatter center (a perfectly spherical, isotropic and homogeneous particle with a
diameter d and a complex refractive index m) is located at a point O j of a Cartesian
coordinate system Oxyz. The incident wave propagates from the negative z to the positive z. In
the non-absorbing medium surrounding the particle, the scattered electromagnetic field
s, j

components (noted Vk

where V stands for E or H, s stands for scattered, j is the particle

number and k stands for the coordinate r, θ or φ), in a direction

 j ,  j and at a distance rj

from the particle center are given by:
s, j

Er



 E0 cos  j  i
n 1

E

 1

n

2n  1
n ( n  1)

  

''
1
an  n ( krj )   n krj Pn cos  j





(1)

 ann'  krj  n  cos  j  

E0
n 2n  1


cos  j  i n 1  1
krj
n(n  1)  ibnn (krj ) n  cos  j 
n 1



(2)

 ann'  krj   n  cos  j  

 E0
n 2n  1
n 1



sin  j  i  1
krj
n(n  1)  ibnn (krj ) n  cos  j 
n 1



(3)

Es , j 

s, j

n 1

1/2


 
n 2n  1
H rs , j  E0   sin  j  i n 1  1
bn n'' ( krj )  n  krj Pn1  cos  j  (4)

n
(
n

1)
n 1
 

1/2

s, j



H

s, j

H

E0   



  sin  j n1 i
krj   
E
 0
krj

n 1

 1

n

  



'
2n  1   ann krj  n cos  j 



n ( n  1)  ib  ( kr )  cos   
j 
 n n j n

1/2
 iann'  krj  n  cos  j 

 
n 2n  1
n 1


  cos  j  i  1
n(n  1)  bnn (krj ) n  cos  j  
n 1




Where E 0 is the amplitude of the incident plane wave and the Legendre functions

(5)

(6)

 n and  n

are defined by:

 n  cos    Pn  cos   / sin 

(7)

 n (cos  )  dPn (cos  ) / d

(8)

1

1

Pn (cos  )   sin 
1

dPn (cos  )
d cos 

(9)

where the Pn (cos  ) are the classical Legendre polynomials.
The functions

n  kr 

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are given by:

Received 25 Feb 2011; revised 31 May 2011; accepted 1 Jun 2011; published 16 Jun 2011

20 June 2011 / Vol. 19, No. 13 / OPTICS EXPRESS 12703

n (kr)   n (kr)  i n (kr)

(10)

Where

  kr 

 2 

1/2

 n (kr )  

J n 1/2 (kr )

(11)

1/2

  kr 
 n ( kr )  ( 1) 
 J  n1/2 (kr )
 2 
n

(12)

Where the J n 1/2 (kr ) are the classical half-order Bessel functions, and k is the wavenumber 2 /  .
The scattering coefficients read

 n ( ) 'n (  )  m 'n ( ) n (  )
n ( ) 'n (  )  m 'n ( ) n (  )

(13)

m n   'n     'n   n   
mn   'n      'n   n   

(14)

an 
bn 

Where    d /  is the size parameter of the particle under study and   m . The prime
indicates the derivative of the function with respect to the argument for the value of the
argument indicated between parentheses.
The total field from several particles
Assuming that multiple scattering can be neglected, the total field at the running point P
located on the detector surface will be the sum of all the scattered fields. Then the six
components of the total field are given by:
N

Vwt  Yws, j

(15)

j 1

where N is the total number of particles in the control volume. V stands for E or H and where
w stands for x, y or z, in the Cartesian system associated to the particles field, Y stands for the
component E or H of the scattered field in the same Cartesian system.
When the total field is known, the intensities (and the direction of propagation) are given
by the Poynting’s vector:
1
Re  E t . H t * 
(16)
2 
The intensity recorded by one pixel is directly proportional to the flux of S through its
surface that is to say to the component of S perpendicular to the detector.
A code has been written to compute interference systems for an arbitrary number of
particles according with the theoretical background introduced above. The main
characteristics of this code are:
S

• The properties (diameter and complex refractive index) as well as the 3D location of
each particle can be freely defined
• The detector can be located arbitrarily. Forward as well as rainbow or backward
detection can be accurately simulated.

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• The distance between the detector and the cloud of particles can be defined arbitrarily
(near-field configurations as well as far-field configurations can be simulated)
Simplified analytical relationship
Let us start with two particles, called G1 and G2, located in the neighborhood of the point O
and a point detector M located in or very close to the plane XOZ and at a large distance from
the point O (OM>>>λ and OM>>max(OG1,OG2)), where λ is the wavelength of the incident
beam. The incident laser beam propagates from the negative towards the positive z. The
particle centers G1 and G2 have coordinates {x1 , y1 , z1} and

x2 , y2 , z2 , respectively. A first

assumption is that each particle can be viewed as the center of a scattered spherical wave,
described by the following relations:

E0
exp  j ( kr1 t 1 )
r1

E1 
E2 

E0

exp

(17)

 j ( kr2 t  2 )

(18)

r2

Where k is the wave number ( k  2
initial phase, that is to say that


 ) where λ is the wavelength, 1 and  2 are the
2  1  k ( z2  z1 ) and r1 and r2 are the distance from

particles G1 and G2 to point M, given by:

r1 

 xM  x1    yM  y1    zM  z1 

r2 

 x M  x2    y M  y 2    z M  z 2 

2

2

2

2

2

2

(19)
(20)

By using the Poynting theorem, the intensity at point M is:



E02
IM 
1  2 cos k  r2  r1   2  1
r1r2



Using the fact that point M is assumed in or close of plane xOz, we define a distance

 MG 
i

 xM  xi    zM  zi 
2

2

(21)

 MG by
i

, then Eq. (19) and Eq. (20) can be rewritten as:
1/2


  yM  y1 2  
2
r1    MG1 1 

2
 MG

1

 

2
r2    MG
2

2
As  MG
i

 yM  yi 

2

(22)

1/2

  y M  y2  2  
1 

2
 MG

 
2

(23)

, the Fresnel approximation can be used, and Eq. (22) and Eq. (23) read

now as:

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r1   MG1 

( yM  y1 )2
2  MG1

(24)

r2   MG2 

( yM  y2 )2
2  MG2

(25)

Starting from expressions (24) and (25), the difference r2  r1 can be written as depending on
two terms A and B, with r2  r1  A  B and:
A   MG2   MG1

B

 MG

1

 y M  y2 

2

(26)

  MG

2

 yM  y1 

2

2  MG  MG
1

(27)

2

These expressions can be developed, using again the Fresnel approximation and the fact that
 MG1   MG2  RM where RM is the distance from the detector center to the coordinate center.
The next step is to relate the coordinates of the detector point M expressed in the 3D-space
xM , yM , zM to its 2D-surface coordinates M , M in the coordinate system linked to the
surface of the detector. Without loss of generality, we assume: i) that the direction ξ is parallel
to the plane x0z, ii) that the detector center is also in the plane x0z, at the distance RM from
point O and iii) that the detector center position is defined by its angular position  0 .
Then, Eq. (21) is written in this new coordinate system, using all the previously introduced
approximations, and leading to:


  M
 1  2cos  k 
r1r2 
  RM
2

I , 

E0



M
 [   x2  x1  cos  0   z2  z1  sin  0 ]  k R  y 2  y1   

M

(28)

This equation can be analyzed as follows:
• k

M
RM

 y2  y1  shows

that the fringe spacing (or angular frequency) along the

direction ξ depends only on the distance between the pair of particles in the y
direction.
• while

 
k  M  [  x2  x1  cos 0   z2  z1  sin 0 ] shows that the fringe spacing
 RM 

along the direction η depends on the distance between the pair of particles along the x
and z direction multiplied by cos θ0 (sin θ0, respectively).
Associated 2D-FFT
As classically, an efficient way to extract frequencies from a signal is Fast Fourier Transform
(FFT). Then we will carry out the 2D-FFT of the recorded interference patterns. With such a
procedure, each pair of particles will create a fringe system which will appear as a pair of

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spots in the 2D-FFT associated plane (one spot will be associated to positive frequency and
the other to negative frequency).
Figure 2 displays four images which correspond to:
a) The scattering pattern by one particle
b) The scattering pattern by two particles
c) The 2D-FFT associated to the scattering pattern by one particle
d) The 2D-FFT associated to the scattering pattern by two particles

Fig. 2. Scattering patterns from one and two particles and associated 2D-FFT maps, the
detector is a square of 512 x 512 pixels, located at 1 meter form the coordinate system center
with a collecting angle  0 equal to 35° ± 5°. The incident wavelength is equal to 0.532 µm. a:
the scattering from one particle (d = 30 µm, m = 1.5-0.0i, (x = 0, y = 0, z = 0)); b: the scattering
from two particles (the first one and a second identical to the first but located at (x = 150, y =
150, z = 150); c: the 2D-FFT with a Gaussian windowing associated to Fig. 2(a); d: The 2DFFT with Gaussian windowing associated to Fig. 2(b).

From Fig. 2 the following points must be noted:
• In Fig. 2(a), the scattered field from one particle is not due to a single spherical wave,
but to several spherical waves. Each of these spherical waves corresponds to a kind

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of interaction between the particle and the plane wave (here, in forward essentially to
the externally reflected light (p = 0) and the light refracted twice (p = 1)). The
interference between these waves creates the fringes visible in Fig. 2(a).
• When two particles are in the control volume, the light scattered by each particle
interferes, creating a new fringe system of higher frequency, which is well visible in
Fig. 2(b).
Then, in the associated 2D-FFT space, to one pair of particles corresponds one pair of
complex spots. The structure of each spot depends on the size, refractive index, scattering
angle, and relative position of the two particles. Such a complex spot is displayed in Fig. 3.
Nevertheless, in that paper, this complex structure will not be studied. Only the mean η
coordinate of each complex spot, as defined in Fig. 3, will be used.

Fig. 3. Detail of a complex spot in the 2D-FFT space. The value of coordinate α used is the
mathematical average of the coordinates α of the elementary spots.

Next, a numerical study of the fringe frequency is carried out. For this study, the collecting
angle θ0 varies from 10° to 370° by steps of 10° (but the position θ 0 = 180° is not studied).
The distance between a pair of particles has been varied, for each scattering angle, along the
direction x (respectively z) from 100 to 400 µm in 4 steps. For each x (z) distance, the fringe
pattern has been computed with the Mie theory and then the 2D-FFT. From the 2D-FFT, the
spot location have been measured and the coefficient k/RM sinθ0 (k/RM cosθ0) deduced. The
results are plotted in Fig. 4. In agreement with Eq. (28), the dependence of η as (x2-x1) cosθ0
and (z2-z1) sinθ0 is obtained. This result proves that, when the detector is far enough from the
control volume, the analysis of the fringe frequency can be carried out by using formula (28),
i.e., the assumptions used to obtain Eq. (28) are satisfied.

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value of coefficient (k sin 0 /RM ) and (k cos0 /RM )

measured behavior for x
measured behaviour for z
computed behaviour for x
computed behaviour for z

0.4
0.3
0.2
0.1
0.0
-0.1
-0.2
-0.3
-0.4
0

100

200

300

collecting angle

Fig. 4. Comparison between numerical and analytical values of Eq. (28) coefficients.

As the analyze of the fringe patterns will be carried out in associated 2D-FFT plane, we
propose to call this technique Fourier Interferometric Imaging (FII). The challenge is now to
process interferometric images created by several particles with the tools introduced in this
section. It is the aim of the next section.
Image Processing
In this section, a possible processing strategy to extract the relative position between several
particles is introduced. Starting from an example, the different steps of the processing are
introduced one by one.
First a remark: as η depends on both x and z coordinates, at least two cameras will be
used, collecting the light in two different directions θ 0, in order to extract the three particle
relative coordinates ( x , y , z ) .

0 

40° and  0  90° for a cloud of six
particles. These test images have been computed for the diameter and the absolute coordinates
given in Table 1. The incident wavelength is equal to 0.532 µm, the distance between the
detector center and the coordinate center is equal to 1 m and the aperture is assumed to be
equal to 10° and the CCD camera to have 512 x 512 pixels. In all cases the refractive index is
equal to 1.5-0.0i. From such images, it is difficult to easily know how many particles are at
the origin of the image. Figure 6 displays the 2D-FFT images associated with the images of
Fig. 5. The 2D-FFT images are characterized by spots symmetric to the image center. To each
pair of symmetrical spots corresponds a pair of two particles. Then the number of spots
quantifies the number of particles in the control volume. More accurately the number of spots
Mc is the number of way to choose 2 particles in a cloud of N particles which is equal to Mc =
N (N-1). From Fig. 6 (left or right) 30 spots can be identified, corresponding to 6 particles, as
it must be.
Figure 5 displays the images recorded at

Table 1. The Particle Computation Parameters
Particle number
1
2
3
4
5
6

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Diameter (µm)
30.08
30.88
30.77
30.17
30.80
30.61

xi (µm)
12
6
71
116
137
128

yi (µm)
176
134
76
117
153
3.0

zi (µm)
179
170
75
98
75
189

Received 25 Feb 2011; revised 31 May 2011; accepted 1 Jun 2011; published 16 Jun 2011

20 June 2011 / Vol. 19, No. 13 / OPTICS EXPRESS 12709

Fig. 5. Images simulated for six particles located in the control volume. The left image
corresponds to a recording at  0  40° while the right image corresponds to a recording at

 0  90°.

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Fig. 6. The 2D-FFT associated to images of Fig. 5. First line displays a 2D-FFT with Harris
windowing while the second line displays the 2D-FFT with Gaussian windowing.

The first step is to identify each spot, to give it a number and to record its coordinates η
and ξ. Remember that for the coordinate η only the mean location is used in that paper (see
Fig. 3)
To number the spots, we use the properties that ξ coordinate is directly proportional to the
distance between the particles along the y coordinate for both 40 and 90° images (in this paper
only the case where all the particles have different ξ coordinates is presented without loss of
generality). For example, the left most spot in image 6 left and right corresponds to the same
pair of particles. Then, the spots are ordered with respect to their ξ coordinate and numbered
accordingly. The images of Fig. 6 give the map of spots given in Fig. 7.

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Received 25 Feb 2011; revised 31 May 2011; accepted 1 Jun 2011; published 16 Jun 2011

20 June 2011 / Vol. 19, No. 13 / OPTICS EXPRESS 12711

Fig. 7. Identification and spot numbering.

The second step is devoted to identifying all the spots corresponding to a group of three
particles interacting two by two. To reach this aim, remark that for three particles noted P1 ,

P2 and P3 there are six interactions: P1P2 , P2 P1 , P1P3 , P3 P1 , P3 P2 and P2 P3 . The coordinates of
the particles verify the relations:

 X1  X 2    X1  X 3    X 3  X 2 

(29)

Y1  Y2   Y1  Y3   Y3  Y2 

(30)

 Z1  Z2    Z1  Z3    Z3  Z 2 

(31)

As the distances and the frequencies are linearly connected, then as in the 2D-FFT plane the
spot locations are also linearly connected to the frequencies the following relations for the η
and ξ coordinates exist:

1,2 
1,3 2,3 
       
 1,2 
 1,3 2,3 
Where

(32)

i , j is the frequency of the fringes created by the interference between the light

scattered by particle i and j.
By using this relation, all the groups of three particles interacting two by two are
identified. The identification is carried out as follow:
a) Two spots are selected
b) From the coordinates η and ξ of these two spots, by using Eq. (32) and central
symmetry, the coordinates of 4 possible spots are computes
c) If these spots exist: it means that all the interactions between three particles have been
identified.
d) Two other spots are select and steps b and c are repeated up to all the spots are
selected.
The Fig. 8 displays such a group of three particles interacting two by two for the two cameras.

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20 June 2011 / Vol. 19, No. 13 / OPTICS EXPRESS 12712

Fig. 8. A group of three particles interacting two by two. Left at  0  40°, right at  0  90°.

In the third step, we start from a system of three interacting particles called A, then
another system of three interacting particles with a common interaction with system A is
researched, this system is called B. Finally a last system of three particles with one common
interaction with group A and another with group B is researched. Such groups are displayed in
Fig. 9.

Fig. 9. Three groups of three particles view at  0  40°. Group B has a common interaction
with group A. Group C has a common interaction with group B and one with group A.

The merging of these three groups of three particles gives a group of four particles
interacting two by two as exemplified in Fig. 10.

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20 June 2011 / Vol. 19, No. 13 / OPTICS EXPRESS 12713

150

5

100

 coordinate

50

6

7
1

0

2

23

12
10 14
11
8

28
29

15 19
16
13 17

-50

3

21
18 22
20

4

-100

9

-150
-100

-80

-60

-40

-20

0

31
30

24
25

20

27
26
40

60

80

100

 coordinate

Fig. 10. The group of four particles interacting two by two determined from three groups of
three particles interacting two by two.

The process is repeated on the other groups of three particles until all the groups of three
particles are merged in groups of four particles. When all the groups of four particles
interacting two by two are identified, these groups are merged to obtain the groups of five
particles interacting two by two and so on, up to groups of (N-1) particles interacting two by
two (where N is the number of particles in the probe volume, N is six in our example). The
complement to a group of (N-1) particles interacting two by two corresponds to one particle
interacting with the (N-1) others as displayed in Fig. 11.

Fig. 11. a) Interaction for a group of five particles interacting two by two. b) Interaction for one
particle interacting with all the others five particles in the control volume.

Six such groups of one particle interacting with the five other exist.
When a set of interactions between one particle and all the other particles is determined,
the 2D locations η and ξ are inversed by using Eq. (28) to obtained the 3D relative positions
x , y and z between of all the other particles relatively to this one. Because of the non-

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20 June 2011 / Vol. 19, No. 13 / OPTICS EXPRESS 12714

sensitivity of the fringes frequencies to the sign of the particles relative positions (only the
relative distances are pertinent) eight sets of possible coordinates are obtained.
The next step is to reduce the number of solutions. For each of the previous sets of relative
3D-coordinates a fringe pattern is computed, then its 2D-FFT is compared to the original one
(displayed in Fig. 6). Figure 12 displays some 2D-FFT images of reconstructed particle fields.
In Fig. 12, the notations x  , y  and z  correspond to a symmetry on the relative
coordinates.

Fig. 12. The 40° 2D-FFT computed from the extracted 3D Location. These 2D-FFT are
compared to the one displayed in Fig. 6.

This comparison permits to select only 2 solutions, which are ( x  , y  , z  ) and ( x  ,

y  , z  ) and are given in Table 2.

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20 June 2011 / Vol. 19, No. 13 / OPTICS EXPRESS 12715

Table 2. Original and Extracted 3D Particle Locations
Particle number
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6

Particle locations relatively to particle 4: original data
Relative x coordinates
Relative y coordinates
Relative y coordinates
59
277
104
17
122
72
173
187
193
0
0
0
36
173
253
12
120
91
Particle locations relatively to particle 4: FII first solution
58
276
105
15
120
73
173
187
191
0
0
0
36
173
251
11
118
91
Particle locations relatively to particle 4: FII second solution
105
58
276
120
73
15
187
191
173
0
0
0
251
36
173
91
11
118

These two solutions can’t be distinguished from the associated 2D-FFT as well as from the
fringe fields (displayed in Fig. 13). Note that the fringe fields corresponding to the two
selected solutions are identical but differ from the original one (displayed in Fig. 5). The cause
of this behavior is the moiré effect; a small change on the particle locations has a large effect
on the fringe field sampling but not on the 2D FFT. It is the reason why we did the
comparison in the associated 2D-FFT space.

Fig. 13. The fringes field reconstructed from the selected extracted particle fields.

Figure 14 is a 3D representation of the original particle field (in red) and of the two
selected solutions (in yellow and blue). The fact that the red and yellow spheres are nearly
perfectly merged is a signature of the quality in the particle 3D location determination. The
blue and yellow spheres are perfectly symmetrical relatively to the particle from which the
distances are determined (the particle from which the relative distance are computed is
assumed to be at 0; 0; 0 and is represented by a small red sphere).

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Received 25 Feb 2011; revised 31 May 2011; accepted 1 Jun 2011; published 16 Jun 2011

20 June 2011 / Vol. 19, No. 13 / OPTICS EXPRESS 12716

Fig. 14. The original locations (in red) and the extracted locations (in yellow and blue).

The next test involved creating and processing sixty systems of six particles randomly
located. Each system of six particles has been created by a random process, assuming that the
particle coordinates are uniformly distributed in a cube of 1000 x 1000 x 1000 µm3. The
relative coordinates of the particles have been extracted by using the scheme previously
described and the errors between the original and extracted relative distances have been
computed. The results (corresponding to 900 pair of particles) are summarized in Table 3.
Table 3. Statistical Errors on the Relative Distance Between Particles
Coordinate
x
y
z

Mean error
0.0689 µm
0.0472 µm
0.0450 µm

Error rms
2.37
2.015
2.43

Largest error
8 µm
5 µm
6 µm

From Table 3, the average errors on the relative distances are smallest than 0.1 µm. and
comparatively with digital holography it is possible to say that the relative locations are
determined with the same accuracy in the three directions of the space.
Conclusion
This article is a proof of concept of a new method to characterize 3D fields of spherical
droplets. This interferometric method is based on the recording of two images of the fringe
pattern at two different collection angles. The numerical effort consists in:
• The computation of two 2D-FFT,
• The determination of spot locations
• The analytical manipulation of the spot coordinates (2D),
• The computation of the possible 3D coordinates by the resolution of two equations
linking 2D-FFT coordinates and 3D coordinates.
• The selection of the best solution by computing the associated 2D-FFT (8 2D-FFT).

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Received 25 Feb 2011; revised 31 May 2011; accepted 1 Jun 2011; published 16 Jun 2011

20 June 2011 / Vol. 19, No. 13 / OPTICS EXPRESS 12717

Comparatively to classical holography technique, the method is characterized by:
• the same accuracy for the three space directions
• a reduced sensitivity to the Moiré effect (the interference are between spherical waves
coming from nearly the same point and with close curvatures, contrary to classical
holography where interference between a spherical wave and a plane wave are
recorded)
• only the particles relative locations are extracted.
The next step will be the experimental implementation of this approach.
Acknowledgments
The authors gratefully acknowledge the financial support from the European program
INTERREG IVa-C5: Cross-Channel Center for Low Carbon Combustion, the Chinese
Program of Introducing Talents of Discipline to University (B08026), the National Natural
Science Foundation of China (NSFC) projects (grants 5080606) and the National Basic
Research Program of China (grants 2009CB219802).

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Received 25 Feb 2011; revised 31 May 2011; accepted 1 Jun 2011; published 16 Jun 2011

20 June 2011 / Vol. 19, No. 13 / OPTICS EXPRESS 12718


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