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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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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 12700

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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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 12701

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)

ann' krj n cos j

E0

n 2n 1

cos j i n 1 1

krj

n(n 1) ibnn (krj ) n cos j

n 1

(2)

ann' krj n cos j

E0

n 2n 1

n 1

sin j i 1

krj

n(n 1) ibnn (krj ) n cos j

n 1

(3)

Es , 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 n1 i

krj

E

0

krj

n 1

1

n

'

2n 1 ann krj n cos j

n ( n 1) ib ( kr ) cos

j

n n j n

1/2

iann' krj n cos j

n 2n 1

n 1

cos j i 1

n(n 1) bnn (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 n1/2 (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

mn '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 cos0 /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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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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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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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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(C) 2011 OSA

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 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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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 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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(C) 2011 OSA

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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(C) 2011 OSA

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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