Measurements of 3D relative locations of particles by Fourier Interferometry Imaging (FII)

Paul Briard; Sawitree Saengkaew; Xuecheng Wu; Siegfried Meunier-Guttin-Cluzel; Linghong Chen; Kefa Cen; Gérard Grehan

doi:10.1364/OE.19.012700

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 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.

Principle, tools and numerical analysis

Geometry and measurement principle

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 ( $θ_{0}$ ), 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 ${x_{i}, y_{i}, z_{i}}$ . 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 x_i, y_i and z_i 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 ( $θ_{0}$ ) 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.

Fig. 1 Configuration under study.

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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 components (noted $V_{k}^{s, j}$ 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 $θ_{j}, φ_{j}$ and at a distance $r_{j}$ from the particle center are given by:

E_{r}^{s, j} = E_{0} \cos φ_{j} \sum_{n = 1}^{\infty} i^{n + 1} {(- 1)}^{n} \frac{2 n + 1}{n (n + 1)} a_{n} [ξ_{n}^{''} (k r_{j}) + ξ_{n} (k r_{j})] P_{n}^{1} (\cos θ_{j})

E_{θ}^{s, j} = \frac{E_{0}}{k r_{j}} \cos φ_{j} \sum_{n = 1}^{\infty} i^{n + 1} {(- 1)}^{n} \frac{2 n + 1}{n (n + 1)} [\begin{array}{l} a_{n} ξ_{n}^{'} (k r_{j}) τ_{n} (\cos θ_{j}) \\ - i b_{n} ξ_{n} (k r_{j}) π_{n} (\cos θ_{j}) \end{array}]

E_{φ}^{s, j} = \frac{- E_{0}}{k r_{j}} \sin φ_{j} \sum_{n = 1}^{\infty} i^{n + 1} {(- 1)}^{n} \frac{2 n + 1}{n (n + 1)} [\begin{array}{l} a_{n} ξ_{n}^{'} (k r_{j}) π_{n} (\cos θ_{j}) \\ - i b_{n} ξ_{n} (k r_{j}) τ_{n} (\cos θ_{j}) \end{array}]

H_{r}^{s, j} = E_{0} {(\frac{ε}{μ})}^{1 / 2} \sin φ_{j} \sum_{n = 1}^{\infty} i^{n + 1} {(- 1)}^{n} \frac{2 n + 1}{n (n + 1)} b_{n} [ξ_{n}^{''} (k r_{j}) + ξ_{n} (k r_{j})] P_{n}^{1} (\cos θ_{j})

H_{θ}^{s, j} = \frac{E_{0}}{k r_{j}} {(\frac{ε}{μ})}^{1 / 2} \sin φ_{j} \sum_{n = 1}^{\infty} i^{n + 1} {(- 1)}^{n} \frac{2 n + 1}{n (n + 1)} [\begin{array}{l} - a_{n} ξ_{n}^{'} (k r_{j}) π_{n} (\cos θ_{j}) \\ + i b_{n} ξ_{n} (k r_{j}) τ_{n} (\cos θ_{j}) \end{array}]

H_{φ}^{s, j} = \frac{E_{0}}{k r_{j}} {(\frac{ε}{μ})}^{1 / 2} \cos φ_{j} \sum_{n = 1}^{\infty} i^{n + 1} {(- 1)}^{n} \frac{2 n + 1}{n (n + 1)} [\begin{array}{l} - i a_{n} ξ_{n}^{'} (k r_{j}) τ_{n} (\cos θ_{j}) \\ + b_{n} ξ_{n} (k r_{j}) π_{n} (\cos θ_{j}) \end{array}]

Where

E_{0}

is the amplitude of the incident plane wave and the Legendre functions

π_{n}

and

τ_{n}

are defined by:

π_{n} (\cos θ) = P_{n}^{1} (\cos θ) / \sin θ

τ_{n} (\cos θ) = d P_{n}^{1} (\cos θ) / d θ

P_{n}^{1} (\cos θ) = - \sin θ \frac{d P_{n} (\cos θ)}{d \cos θ}

where the

P_{n} (\cos θ)

are the classical Legendre polynomials.

The functions $ξ_{n} (k r)$ are given by:

ξ_{n} (k r) = ψ_{n} (k r) + i χ_{n} (k r)

Where

ψ_{n} (k r) = {(\frac{π k r}{2})}^{1 / 2} J_{n + 1 / 2} (k r)

χ_{n} (k r) = {(- 1)}^{n} {(\frac{π k r}{2})}^{1 / 2} J_{- n - 1 / 2} (k r)

Where the

J_{n + 1 / 2} (k r)

are the classical half-order Bessel functions, and k is the wave-number

2 π / λ

.

The scattering coefficients read

a_{n} = \frac{ψ_{n} (α) ψ'_{n} (β) - m ψ'_{n} (α) ψ_{n} (β)}{ξ_{n} (α) ψ'_{n} (β) - m ξ'_{n} (α) ψ_{n} (β)}

b_{n} = \frac{m ψ_{n} (α) ψ'_{n}^{} (β) - ψ'_{n}^{} (α) ψ_{n} (β)}{m ξ_{n} (α) ψ'_{n}^{} (β) - ξ'_{n}^{} (α) ψ_{n} (β)}

Where

α = π d / λ

is the size parameter of the particle under study and

β = m α

. 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:

V_{w}^{t} = \sum_{j = 1}^{N} Y_{w}^{s, j}

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:

S = \frac{1}{2} R e [E^{t} . H^{t *}]

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:

• 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.
• 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 ${x_{1}, y_{1}, z_{1}}$ and ${x_{2}, y_{2}, z_{2}}$ , respectively. A first assumption is that each particle can be viewed as the center of a scattered spherical wave, described by the following relations:

E_{1} = \frac{E_{0}}{r_{1}} e x p^{- j (k r_{1} - ω t + φ_{1})}

E_{2} = \frac{E_{0}}{r_{2}} e x p^{- j (k r_{2} - ω t + φ_{2})}

Where k is the wave number (

k = \frac{2 π}{λ}

) where λ is the wavelength,

φ_{1}

and

φ_{2}

are the initial phase, that is to say that

φ_{2} - φ_{1} = k (z_{2} - z_{1})

and r₁ and r₂ are the distance from particles G₁ and G₂ to point M, given by:

r_{1} = \sqrt{{(x_{M} - x_{1})}^{2} + {(y_{M} - y_{1})}^{2} + {(z_{M} - z_{1})}^{2}}

r_{2} = \sqrt{{(x_{M} - x_{2})}^{2} + {(y_{M} - y_{2})}^{2} + {(z_{M} - z_{2})}^{2}}

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

I_{M} = | \frac{E_{0}^{2}}{r_{1} r_{2}} (1 + 2 c o s {k [r_{2} - r_{1}] + φ_{2} - φ_{1}}) |

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

ρ_{M G_{i}}

by

ρ_{M G_{i}} = \sqrt{{(x_{M} - x_{i})}^{2} + {(z_{M} - z_{i})}^{2}}

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

r_{1} = {[ρ_{M G_{1}}^{2} [1 + \frac{{(y_{M} - y_{1})}^{2}}{ρ_{M G_{1}}^{2}}]]}^{1 / 2}

r_{2} = {[ρ_{M G_{2}}^{2} [1 + \frac{{(y_{M} - y_{2})}^{2}}{ρ_{M G_{2}}^{2}}]]}^{1 / 2}

As

ρ_{M G_{i}}^{2} ≫ {(y_{M} - y_{i})}^{2}

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

r_{1} \approx ρ_{M G_{1}} + \frac{{(y_{M} - y_{1})}^{2}}{2 ρ_{M G_{1}}}

r_{2} \approx ρ_{M G_{2}} + \frac{{(y_{M} - y_{2})}^{2}}{2 ρ_{M G_{2}}}

Starting from expressions (24) and (25), the difference

r_{2} - r_{1}

can be written as depending on two terms A and B, with

r_{2} - r_{1} = A + B

and:

A = ρ_{M G_{2}}^{} - ρ_{M G_{1}}^{}

B = \frac{ρ_{M G_{1}}^{} {(y_{M} - y_{2})}^{2} - ρ_{M G_{2}}^{} {(y_{M} - y_{1})}^{2}}{2 ρ_{M G_{1}_{}}^{} ρ_{M G_{2}}^{}}

These expressions can be developed, using again the Fresnel approximation and the fact that

ρ_{M G_{1}} \approx ρ_{M G_{2}} \approx R_{M}

where

R_{M}

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

x_{M}, y_{M}, z_{M}

to its 2D-surface coordinates

η_{M}, ξ_{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

R_{M}

from point O and iii) that the detector center position is defined by its angular position

θ_{0}

.

Then, Eq. (21) is written in this new coordinate system, using all the previously introduced approximations, and leading to:

I_{η, ξ} = | \frac{E_{0}^{2}}{r_{1} r_{2}} (1 + 2 c o s (k (\frac{η_{M}}{R_{M}}) [- (x_{2} - x_{1}) \cos θ_{0} + (z_{2} - z_{1}) \sin θ_{0}] + k \frac{ξ_{M}}{R_{M}} (y_{2} - y_{1}))) |

This equation can be analyzed as follows:

• $k \frac{ξ_{M}}{R_{M}} (y_{2} - y_{1})$ 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 $k (\frac{η_{M}}{R_{M}}) [- (x_{2} - x_{1}) \cos θ_{0} + (z_{2} - z_{1}) \sin θ_{0}]$ shows that the fringe spacing 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 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:

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 $θ_{0}$ 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 2D-FFT with Gaussian windowing associated to Fig. 2(b).

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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

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 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.

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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 θ₀ = 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/R_M sinθ₀ (k/R_M 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.

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

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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 θ₀, in order to extract the three particle relative coordinates $(\bar{x}, \bar{y}, \bar{z})$ .

Figure 5 displays the images recorded at $θ_{0} =$ 40° and $θ_{0} =$ 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 M_c is the number of way to choose 2 particles in a cloud of N particles which is equal to M_c = N (N-1). From Fig. 6 (left or right) 30 spots can be identified, corresponding to 6 particles, as it must be.

Fig. 5 Images simulated for six particles located in the control volume. The left image corresponds to a recording at $θ_{0} =$ 40° while the right image corresponds to a recording at $θ_{0} =$ 90°.

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Table 1. The Particle Computation Parameters

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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.

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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 .

Fig. 7 Identification and spot numbering.

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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 $P_{1}$ , $P_{2}$ and $P_{3}$ there are six interactions: $P_{1} P_{2}, P_{2} P_{1}, P_{1} P_{3}, P_{3} P_{1}, P_{3} P_{2}$ and $P_{2} P_{3}$ . The coordinates of the particles verify the relations:

(X_{1} - X_{2}) = (X_{1} - X_{3}) + (X_{3} - X_{2})

(Y_{1} - Y_{2}) = (Y_{1} - Y_{3}) + (Y_{3} - Y_{2})

(Z_{1} - Z_{2}) = (Z_{1} - Z_{3}) + (Z_{3} - Z_{2})

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:

[\begin{matrix} η_{1, 2} \\ ξ_{1, 2} \end{matrix}] = \pm [\begin{matrix} η_{1, 3} - η_{2, 3} \\ ξ_{1, 3} - ξ_{2, 3} \end{matrix}]

Where

η_{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.

Fig. 8 A group of three particles interacting two by two. Left at $θ_{0} =$ 40°, right at $θ_{0} =$ 90°.

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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 $θ_{0} =$ 40°. Group B has a common interaction with group A. Group C has a common interaction with group B and one with group A.

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

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

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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 .Six such groups of one particle interacting with the five other exist.

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.

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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 $\bar{x}$ , $\bar{y}$ and $\bar{z}$ between of all the other particles relatively to this one. Because of the non-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 $\bar{x \pm}$ , $\bar{y \pm}$ and $\bar{z \pm}$ correspond to a symmetry on the relative coordinates.This comparison permits to select only 2 solutions, which are ( $\bar{x +}$ , $\bar{y +}$ , $\bar{z -}$ ) and ( $\bar{x -}$ , $\bar{y -}$ , $\bar{z +}$ ) and are given in Table 2 .

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

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Table 2. Original and Extracted 3D Particle Locations

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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.

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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).

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

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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 .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.

Table 3. Statistical Errors on the Relative Distance Between Particles

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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).

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).

References and links

1. 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). [CrossRef]

2. 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). [CrossRef]

3. G. E. Elsinga, F. Scarano, B. Wieneke, and B. W. Oudheusden, “Tomographic particle image velocimetry,” Exp. Fluids 41(6), 933–947 (2006). [CrossRef]

4. 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). [CrossRef]

5. 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). [CrossRef]

6. G. Pan and H. Meng, “Digital holography of particle fields: reconstruction by use of complex amplitude,” Appl. Opt. 42(5), 827–833 (2003). [CrossRef] [PubMed]

Particle number	Diameter (µm)	x_i (µm)	y_i (µm)	z_i (µm)
1	30.08	12	176	179
2	30.88	−6	134	−170
3	30.77	−71	−76	75
4	30.17	116	117	−98
5	30.80	−137	153	75
6	30.61	128	−3.0	−189

Particle locations relatively to particle 4: original data
Particle number	Relative x coordinates	Relative y coordinates	Relative y coordinates
1	−104	59	277
2	−122	17	−72
3	−187	−193	173
4	0	0	0
5	−253	36	173
6	12	120	−91

Particle locations relatively to particle 4: FII first solution
1	−105	58	276
2	−120	15	−73
3	−187	−191	173
4	0	0	0
5	−251	36	173
6	11	118	−91

Particle locations relatively to particle 4: FII second solution
1	105	−58	−276
2	120	−15	73
3	187	191	−173
4	0	0	0
5	251	−36	−173
6	−11	−118	91

Coordinate	Mean error	Error rms	Largest error
x	0.0689 µm	2.37	8 µm
y	−0.0472 µm	2.015	5 µm
z	0.0450 µm	2.43	6 µm

Measurements of 3D relative locations of particles by Fourier Interferometry Imaging (FII)

Abstract

Introduction

Principle, tools and numerical analysis

Geometry and measurement principle

Numerical computation of fringe patterns

The field scattered by one particle

The total field from several particles

Simplified analytical relationship

Associated 2D-FFT

Image Processing

Conclusion

Acknowledgments

References and links

Cited By

Figures (14)

Tables (3)

Equations (32)

Optics Express