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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 23 — Nov. 18, 2013
  • pp: 27964–27980
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Accurate 3D tracking and size measurement of evaporating droplets using in-line digital holography and “inverse problems” reconstruction approach

Mozhdeh Seifi, Corinne Fournier, Nathalie Grosjean, Loic Méès, Jean-Louis Marié, and Loic Denis  »View Author Affiliations


Optics Express, Vol. 21, Issue 23, pp. 27964-27980 (2013)
http://dx.doi.org/10.1364/OE.21.027964


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Abstract

Digital in-line holography was used to study a fast dynamic 3D phenomenon: the evaporation of free-falling diethyl ether droplets. We describe an unsupervised reconstruction algorithm based on an “inverse problems” approach previously developed by our team to accurately reconstruct 3D trajectories and to estimate the droplets’ size in a field of view of 7 × 11 × 20 mm3. A first experiment with non-evaporating droplets established that the radius estimates were accurate to better than 0.1 μm. With evaporating droplets, the vapor around the droplet distorts the diffraction patterns in the holograms. We showed that areas with the strongest distortions can be discarded using an exclusion mask. We achieved radius estimates better than 0.5 μm accuracy for evaporating droplets. Our estimates of the evaporation rate fell within the range predicted by theoretical models.

© 2013 Optical Society of America

1. Introduction

In fluid mechanics, the study of bubbles and droplets carried by flows essentially relies on velocity and size measurements. The most widely used measurement technique in this context is Phase Doppler Anemometry (PDA) [1

1. M. Sommerfeld and H.-H. Qiu, “Experimental studies of spray evaporation in turbulent flow,” International Journal of Heat and Fluid Flow 19, 10–22 (1998). [CrossRef]

]. PDA provides only single-point measurements and therefore is incompatible with the tracking of individual particles (i.e., Lagrangian tracking). Interferometric Laser Imaging for Droplet Sizing (ILIDS) [2

2. S. M. Skippon and Y. Tagaki, “ILIDS measurements of the evaporation of fuel droplets during the intake and compression strokes in a firing lean burn engine,” Technical Report 960830, SAE International, Warrendale, PA (1996).

] and Global Phase Doppler Anemometry [3

3. N. Damaschke, H. Nobach, and C. Tropea, “Optical limits of particle concentration for multi-dimensional particle sizing techniques in fluid mechanics,” Experiments in fluids 32, 143–152 (2002). [CrossRef]

] provide alternative solutions to measure droplet (or bubble) size and location in a whole flow section. In principle, these techniques can be extended to estimate 3D location of particles, but in practice their depth of field is limited by the small thickness of a laser sheet. Small displacements in depth (i.e., parallel to the optical axis) can be considered in order to measure a third component of velocity [4

4. D. Sugimoto, K. Zarogoulidis, T. Kawaguchi, K. Matsuura, Y. Hardalupas, A. Taylor, and K. Hishida, “Extension of the compressed interferometric particle sizing technique for three component velocity measurements,” in “13th international symposium on applications of laser techniques to fluid mechanicsLisbon, Portugal,”, 26–29 (2006).

]. Such measurements still remain essentially 2D as the measurement volume is flat, with a field of view much wider along the transversal directions than in depth. Lagrangian tracking of evaporating droplets requires time-resolved 3D imaging system with a full 3D measurement volume, large enough to contain a significant part of a droplet trajectory. To this purpose, digital holography is a very promising technique, allowing both 3D location and droplet size tracking with a good temporal resolution and a wide/deep field of view.

Digital holography is being increasingly used in applications that require micro-objects tracking (e.g., [5

5. F. Verpillat, F. Joud, P. Desbiolles, and M. Gross, “Dark-field digital holographic microscopy for 3D-tracking of gold nanoparticles,” Optics Express 19, 26044–26055 (2011). [CrossRef]

, 6

6. L. Dixon, F. C. Cheong, and D. G. Grier, “Holographic deconvolution microscopy for high-resolution particle tracking,” Optics Express 19, 16410–16417 (2011). [CrossRef] [PubMed]

, 7

7. X. Zhang, I. Khimji, U. A. Gurkan, H. Safaee, P. N. Catalano, H. O. Keles, E. Kayaalp, and U. Demirci, “Lensless imaging for simultaneous microfluidic sperm monitoring and sorting,” Lab on a Chip 11, 2535–2540 (2011). [CrossRef] [PubMed]

] ). In contrast to off-axis setups, the in-line setup (i.e., the Gabor setup) is less sensitive to vibrations because it does not involve beam splitters, mirrors or lenses. It also exploits the whole frequency bandwidth of the sensor to encode accurately the depth and the size of the objects on the holograms. This imaging technique is also called “lensless imaging” [8

8. L. Repetto, E. Piano, and C. Pontiggia, “Lensless digital holographic microscope with light-emitting diode illumination,” Optics letters 29, 1132–1134 (2004). [CrossRef] [PubMed]

, 9

9. C. P. Allier, G. Hiernard, V. Poher, and J. M. Dinten, “Bacteria detection with thin wetting film lensless imaging,” Biomedical optics express 1, 762–770 (2010). [CrossRef]

, 10

10. J. R. Fienup, “Coherent lensless imaging,” in “Imaging Systems,” (2010). [CrossRef]

] , as it involves no lens between the object and the sensor. A key issue is to design hologram processing algorithms that achieve the best possible accuracy in 3D location and sizing. Over the past decade, numerous algorithms for the analysis of digital holograms have been proposed and several journal special issues were published on the subject, e.g., [11

11. T. C. Poon, T. Yatagai, and W. Juptner, “Digital holography-coherent optics of the 21st century: introduction,” Applied Optics 45, 821 (2006). [CrossRef]

, 12

12. J. Coupland and J. Lobera, “Special issue : Optical tomography and digital holography,” Measurement Science and Technology 19, 070101 (2008). [CrossRef]

, 13

13. M. K. Kim, Y. Hayasaki, P. Picart, and J. Rosen, “Digital holography and 3D imaging: introduction to feature issue,” Appied Optics 52, DH1 (2013). [CrossRef]

]. The most widely used approach is based on the simulation of the optical reconstruction of the hologram (using, e.g., Fresnel functions to back-propagate the hologram), following by segmentation of this 3D image and estimation of the object parameters. These techniques have a limited accuracy due to the following characteristics of the imaging technique [14

14. J. Gire, L. Denis, C. Fournier, E. Thiébaut, F. Soulez, and C. Ducottet, “Digital holography of particles: benefits of the ‘inverse problem’ approach,” Measurement Science and Technology 19, 074005 (2008). [CrossRef]

, 15

15. C. Fournier, L. Denis, E. Thiebaut, T. Fournel, and M. Seifi, “Inverse problem approaches for digital hologram reconstruction,” in “Proceedings of SPIE ,”, 80430S (2011). [CrossRef]

] : (i) the signal is truncated resulting in biased estimations, especially for objects located at the borders of the field, (ii) the spatial resolution of the sensor is low which either results in the presence of ghost images in the reconstruction or forces the experimenter to increase the working distance which reduces the hologram signal to noise ratio and thus degrades the accuracy of estimation. Furthermore, most of these reconstruction algorithms require the user to tune several parameters. These methods already achieve very interesting results [16

16. J. Sheng, E. Malkiel, and J. Katz, “Digital holographic microscope for measuring three-dimensional particle distributions and motions,” Applied optics 45, 3893–3901 (2006). [CrossRef] [PubMed]

, 17

17. J. Lu, J. P. Fugal, H. Nordsiek, E. W. Saw, R. A Shaw, and W. Yang, “Lagrangian particle tracking in three dimensions via single-camera in-line digital holography,” New Journal of Physics 10, 125013 (2008). [CrossRef]

, 9

9. C. P. Allier, G. Hiernard, V. Poher, and J. M. Dinten, “Bacteria detection with thin wetting film lensless imaging,” Biomedical optics express 1, 762–770 (2010). [CrossRef]

, 18

18. D. Nguyen, D. Honnery, and J. Soria, “Measuring evaporation of micro-fuel droplets using magnified DIH and DPIV,” Experiments in Fluids 1–11 (2010).

].

Another family of hologram processing methods follow a signal processing approach to perform the detection and sizing of the objects directly from the hologram rather than from the back-propagated optical field [19

19. S. H. Lee, Y. Roichman, G. R. Yi, S. H. Kim, S. M. Yang, A. van Blaaderen, P. van Oostrum, and D. G. Grier, “Characterizing and tracking single colloidal particles with video holographic microscopy,” Optics Express 15, 18275–18282 (2007). [CrossRef] [PubMed]

, 20

20. F. C. Cheong, K. Xiao, D. J. Pine, and D. G. Grier, “Holographic characterization of individual colloidal spheres’ porosities,” Soft Matter 7, 6816–6819 (2011). [CrossRef]

, 21

21. D. Chareyron, J. L. Marie, C. Fournier, J. Gire, N. Grosjean, L. Denis, M. Lance, and L. Mees, “Testing an in-line digital holography “inverse method” for the lagrangian tracking of evaporating droplets in homogeneous nearly isotropic turbulence,” New Journal of Physics 14, 043039 (2012). [CrossRef]

, 22

22. J. Fung, K. E. Martin, R. W. Perry, D. M. Kaz, R. McGorty, and V. N. Manoharan, “Measuring translational, rotational, and vibrational dynamics in colloids with digital holographic microscopy,” Optics express 19, 8051–8065 (2011). [CrossRef] [PubMed]

]. In this article, an image processing algorithm based on the “inverse problems” approach [23

23. F. Soulez, L. Denis, C. Fournier, É. Thiébaut, and C. Goepfert, “Inverse-problem approach for particle digital holography: accurate location based on local optimization,” Journal of the Optical Society of America . A 24, 1164–1171 (2007).

, 24

24. F. Soulez, L. Denis, E. Thiébaut, C. Fournier, and C. Goepfert, “Inverse problem approach in particle digital holography: out-of-field particle detection made possible,” Journal of the Optical Society of America. A, Optics, Image Science, and Vision 24, 3708–3716 (2007). [CrossRef] [PubMed]

] is used to perform an accurate estimation of both size and 3D position of evaporating diethyl ether droplets. As the Gabor holographic setup is restricted to small objects at low concentration (satisfying Royer criterion [25

25. H. Royer, “An application of high-speed microholography: the metrology of fogs,” Nouvelle Revue d’ Optique 5, 87–93 (1974). [CrossRef]

]), the hologram can be approximated as the sum of the diffraction patterns produced by the objects. The “inverse problems” methods aim to invert the hologram formation finding a set of diffraction patterns that closely fit the experimental data. In the case of spherical objects, finding the best fit corresponds to estimating the size and 3D position parameters. In contrast to the light back-propagation approaches, these calculated patterns intrinsically take into account signal truncation and the low spatial resolution of the sensor. In addition to an improved accuracy of estimation, such unsupervised approaches can greatly expand the field of view outside of the sensor area [24

24. F. Soulez, L. Denis, E. Thiébaut, C. Fournier, and C. Goepfert, “Inverse problem approach in particle digital holography: out-of-field particle detection made possible,” Journal of the Optical Society of America. A, Optics, Image Science, and Vision 24, 3708–3716 (2007). [CrossRef] [PubMed]

, 26

26. L. Denis, D. Lorenz, E. Thiébaut, C. Fournier, and D. Trede, “Inline hologram reconstruction with sparsity constraints,” Optics Letters 34, 3475–3477 (2009). [CrossRef] [PubMed]

].

In this paper we employ a previously introduced parameter estimation method [23

23. F. Soulez, L. Denis, C. Fournier, É. Thiébaut, and C. Goepfert, “Inverse-problem approach for particle digital holography: accurate location based on local optimization,” Journal of the Optical Society of America . A 24, 1164–1171 (2007).

, 24

24. F. Soulez, L. Denis, E. Thiébaut, C. Fournier, and C. Goepfert, “Inverse problem approach in particle digital holography: out-of-field particle detection made possible,” Journal of the Optical Society of America. A, Optics, Image Science, and Vision 24, 3708–3716 (2007). [CrossRef] [PubMed]

, 21

21. D. Chareyron, J. L. Marie, C. Fournier, J. Gire, N. Grosjean, L. Denis, M. Lance, and L. Mees, “Testing an in-line digital holography “inverse method” for the lagrangian tracking of evaporating droplets in homogeneous nearly isotropic turbulence,” New Journal of Physics 14, 043039 (2012). [CrossRef]

] , to study the evaporation of diethyl ether droplets from hologram videos. The droplets are assumed to be spherical. The parametric description of each object then relies on four parameters: three spatial coordinates and a single shape parameter (the radius). We chose ether to evaluate the performance of the method, because it evaporates very fast.

The structure of this paper is as follows: the recording setup is detailed in Sec. 2. In Sec. 3 the image formation model and the iterative model fitting algorithm are presented as well as two improvements of the algorithm regarding accuracy and time costs. The reconstruction of experimental holograms, presented in Sec. 4, shows that in-line digital holography and the reconstruction algorithm are suitable for the study of evaporation processes. The paper is concluded in Sec. 5.

2. Description of the experimental setup

An in-line digital holography setup, also called the Gabor setup, is used in this paper to record the holograms. The diethyl ether droplets are generated by a piezoelectric jetting device manufactured by MicroFab Technologies (see Fig. 1). This injector generates mono-dispersed droplets with radii of 31 μm ± 0.25 μm [21

21. D. Chareyron, J. L. Marie, C. Fournier, J. Gire, N. Grosjean, L. Denis, M. Lance, and L. Mees, “Testing an in-line digital holography “inverse method” for the lagrangian tracking of evaporating droplets in homogeneous nearly isotropic turbulence,” New Journal of Physics 14, 043039 (2012). [CrossRef]

]. The droplets are injected at distances of 45 cm to 52 cm from a 800 × 1280 pixel Phantom V611 camera with the pixel size of 20 μm and the fill-factor (i.e., active area over total area of pixel) of 0.56. The frame rate is set to 620 frames per second. The illuminating laser beam is produced using a Nd:YVO4 laser (Spectra-Physics, Millenia). The laser beam divergence introduces a magnification factor of approximately 2.4 in the system (see Sec. 4.1). The experimental holograms have a signal to noise ratio (i.e., the ratio of the magnitude of the signal to the standard deviation of noise) ranging from 5 to 9. Given the magnification factor and in order to investigate the whole evaporation process, it proved necessary to capture three sets of holograms to investigate the whole evaporation process. The set “0” captures the holograms of the droplets being injected into the air. The injector’s diffraction pattern is visible on this set. The sets “1” and “2” capture the holograms with relative camera translations of 7.5 mm and 15 mm compared to the first setup (see Fig. 2). For each set, a video of 100 holograms is recorded. One hologram of each set after removing its background is shown in Fig. 2.

Fig. 1 A picture of the droplet jet
Fig. 2 The setup for capturing three sets of holograms containing evaporating diethyl ether droplets. The droplets radii are scaled up for the sake of visualization.

3. Automated estimation of particles parameters

Signal processing methods can be employed in the case of parametric objects like droplets to estimate the size and 3D position of every droplet directly from the hologram. In the case of low density object fields (i.e., satisfying the Royer criterion [25

25. H. Royer, “An application of high-speed microholography: the metrology of fogs,” Nouvelle Revue d’ Optique 5, 87–93 (1974). [CrossRef]

] ), the image formation model can be considered as linear and the hologram can be decomposed into the sum of the diffraction patterns of the objects [23

23. F. Soulez, L. Denis, C. Fournier, É. Thiébaut, and C. Goepfert, “Inverse-problem approach for particle digital holography: accurate location based on local optimization,” Journal of the Optical Society of America . A 24, 1164–1171 (2007).

]. When objects are modeled with a parametric shape, the model of their diffraction pattern is parametric, too. It depends on the shape and 3D position of the object as well as on the experimental setup parameters. A previously introduced iterative algorithm [23

23. F. Soulez, L. Denis, C. Fournier, É. Thiébaut, and C. Goepfert, “Inverse-problem approach for particle digital holography: accurate location based on local optimization,” Journal of the Optical Society of America . A 24, 1164–1171 (2007).

, 24

24. F. Soulez, L. Denis, E. Thiébaut, C. Fournier, and C. Goepfert, “Inverse problem approach in particle digital holography: out-of-field particle detection made possible,” Journal of the Optical Society of America. A, Optics, Image Science, and Vision 24, 3708–3716 (2007). [CrossRef] [PubMed]

] finds the size and the coordinates of the droplets with high accuracy.

3.1. The parametric diffraction pattern model

We begin by describing the forward model of hologram formation in order to build the parametric model of diffraction patterns. The diffraction pattern of a spherical droplet of radius rn located at distance z n from the sensor is here assumed to be identical to the diffraction pattern of a circular opaque disk of the same radius at the same distance from the sensor. Let us define the aperture of the opaque disk ϑn as equal to 1 inside a disk (with radius rn) centered on 2D position (xe, ye) and 0 outside this disk. Under the far field condition (4 πrn2/(λzn) ≪ 1), the complex amplitude of the signal Aholo (x, y) on every pixel (x, y) of the camera can be written as [27

27. G. B. Parrent and B. J. Thompson, “On the fraunhofer (far field) diffraction patterns of opaque and transparent objects with coherent background,” Journal of Modern Optics 11, 183–193 (1964).

]:
Aholo(x,y)Aref0(x,y)Arefzn(xn,yn)(ϑn*hzn)(x,y),
(1)
where Aref0(x,y) stands for the complex amplitude of the reference wave on the hologram plane and Arefzn(xn,yn) refers to the complex amplitude of the reference wave on the object aperture plane and is assumed to be uniform on the object’s aperture. In this formulation, *corresponds to the two dimensional convolution operator. Besides, h zn (x, y) represents the Fresnel function:
hzn(x,y)=[1/(iλzn)]exp[iπ(x2+y2)/(λzn)],
(2)
where λ stands for the laser wavelength and i=1.

The captured intensity on the sensor I holo (x, y) is calculated as
Iholo(x,y)=|Aholo(x,y)|2,
(3)
leading to:
Iholo(x,y)=Iref0(x,y)2{Aref0*(x,y)Arefzn(xn,yn)[ϑn*hzn](x,y)}+β.
(4)

In this expression β refers to the second order term which can be considered negligible for small particles with rnλzn.

In the case of a spherical diverging reference beam with a point source located on the optical axis at a distance z s from the sensor, this expression can be simplified (see Appendix. A ):
Iholo(x,y)Iref0(x,y)Iref0(x,y)αe.[ϑe*(hze)](x,y).
(5)

In this equation αe is a proportionality factor accounting for the non-uniformity of the illuminating beam and ze = mzn, where m=zszszn is called the magnification of the system and can be estimated through a calibration procedure (see Sec. 4.1).

A spherical droplet with an aperture ϑn is therefore reconstructed as an object of aperture ϑe with a radius of re = mrn located at xe = mx n , y e = my n and z e. Defining the set of parameters θ e = ( x e , y e , z e , r e ) and the oscillating term g θe = −[ ϑ e ℜ (h z e )]( x , y ), the intensity expression is simplified to:
Iholo(x,y)=Iref0(x,y)+αeIref0(x,y)gθe(x,y).
(6)

Under the assumption of having reλze , the diffraction model can be mathematically formulated as [28

28. M. Seifi, C. Fournier, L. Denis, D. Chareyron, and J.-L. Marié, “Three-dimensional reconstruction of particle holograms: a fast and accurate multiscale approach,” Journal of the Optical Society of America A 29, 1808–1817 (2012). [CrossRef]

] :
gθe(x,y)=πre2λzesin(πρe2λze)J1c(2πreρeλze)Pixθe(x,y),
(7)
with ρe=(xxe)2+(yye)2. In this formulation, Pix θe ( x , y ) accounts for the pixel integration performed by the sensor on the active area of the pixel:
Pixθe(x,y)=sinc(κπ(xxe)λze)sinc(κπ(yye)λze),
(8)
where κ denotes the width of the square active area.

3.2. Non-uniform background removal

In practice, having holograms with non-uniform backgrounds is common. Such a background image can degrade the accuracy of parameter estimation by introducing correlated noise. Nonuniform background can be produced by non-uniform laser beams, scratches or dust on the cover glass of the camera or environmental noise. A non-uniform background is classically removed either by subtraction or division. This background image is either captured in the absence of object or is estimated as the mean image of a video of holograms.

An effective background removal method can be derived from the image formation model of Eq. (5) : the background image I ref is first subtracted from the hologram and the resulting intensity is divided by the amplitude of the background, leading to the pre-processed hologram d ( x , y ):
d(x,y)=Iholo(x,y)Iref0(x,y)Iref0(x,y)=αe.gθ(x,y).
(9)

For each measurement set in this paper, a distinct background image is calculated corresponding to the mean of 100 holograms. An example of a droplet hologram after background removal is provided in Fig. 3(a). Such an operation leads to holograms with uniform backgrounds if objects are distributed randomly in the volume. In our case, a dark mark can be noticed along the trajectory of the droplets, because droplets locations do no cover uniformly the sensor during the time sequence, but rather are aligned along the same trajectory (see Fig. 3(a-1) ).

Fig. 3 Illustration of a pixel region exclusion mask application on a hologram of set “1”. (a-1) an experimental hologram, (a-2) the experimental hologram being masked, (b-1) cleaned hologram, (b-2) masked cleaned hologram. The colored pixels show the masked pixels which were not included in the parameter estimation.

3.3. Iterative parameter estimation algorithm

In the case of volumes with a low density of objects, an iterative algorithm [23

23. F. Soulez, L. Denis, C. Fournier, É. Thiébaut, and C. Goepfert, “Inverse-problem approach for particle digital holography: accurate location based on local optimization,” Journal of the Optical Society of America . A 24, 1164–1171 (2007).

, 24

24. F. Soulez, L. Denis, E. Thiébaut, C. Fournier, and C. Goepfert, “Inverse problem approach in particle digital holography: out-of-field particle detection made possible,” Journal of the Optical Society of America. A, Optics, Image Science, and Vision 24, 3708–3716 (2007). [CrossRef] [PubMed]

] can be used to estimate the parameters of the diffraction pattern of each object. This algorithm performs the parameter estimation task, object by object, by minimizing the weighted least square difference between the experimental data and a parametric model. The method is optimal from the signal processing point of view if noise can be considered as white and Gaussian. Having such assumptions on noise, we introduce a binary mask w to exclude, from the analysis, some regions of the sensor that do not contain meaningful data (see Sec. 3.4 for a detailed discussion on the mask). The binary mask can be generalized to a weighting mask to account for non-uniformity of the noise variance in the image, as described in [24

24. F. Soulez, L. Denis, E. Thiébaut, C. Fournier, and C. Goepfert, “Inverse problem approach in particle digital holography: out-of-field particle detection made possible,” Journal of the Optical Society of America. A, Optics, Image Science, and Vision 24, 3708–3716 (2007). [CrossRef] [PubMed]

].

Let us note that a last refinement step can be performed by cleaning all the detected particles except one, and then carrying out step (ii) in order to refine the parameters of this particle with an improved signal to noise ratio.

To speed up the weighted normalized correlation computations (of step (i)), fast Fourier transforms are used [23

23. F. Soulez, L. Denis, C. Fournier, É. Thiébaut, and C. Goepfert, “Inverse-problem approach for particle digital holography: accurate location based on local optimization,” Journal of the Optical Society of America . A 24, 1164–1171 (2007).

, 24

24. F. Soulez, L. Denis, E. Thiébaut, C. Fournier, and C. Goepfert, “Inverse problem approach in particle digital holography: out-of-field particle detection made possible,” Journal of the Optical Society of America. A, Optics, Image Science, and Vision 24, 3708–3716 (2007). [CrossRef] [PubMed]

]. A free Matlab ® toolbox called “HoloRec3D” implements the iterative algorithm for digital hologram reconstruction of spherical micro-particles [29

29. M. Seifi, C. Fournier, and L. Denis, “HoloRec3D”. http://labh-curien.univ-st-etienne.fr/wiki-reconstruction/index.php/Main_Page, [Online; accessed 12-April-2013].

, 30

30. M. Seifi, C. Fournier, and L. Denis, “HoloRec3D : A free Matlab toolbox for digital holography,” Hal:ujm (2012).

].

3.4. Pixel region exclusion mask

The first advantage of the binary mask w used in step (i) (see Eq. (12) and Eq. (14) ) is to take into account the truncation of the data using FFTs for faster calculations. The second advantage is to exclude pixel regions [in steps (i) and (ii)] where the parametric model can’t explain the data. A closer look to Eq. (7) reveals that the intensity of the model is equal to zero at the center of the pattern (i.e., x = x e and y = y e ). However the experimental holograms of evaporating droplets show high intensities in the pattern centers (see Fig. 3(a-1) ) which may be caused by a film of diethyl ether vapor surrounding the droplet during evaporation. The diffraction pattern produced by the vapor film is not included in the diffraction model used here. Because it is spatially well localized, this effect can be removed from consideration by using a binary weighting mask. The binary mask contains the values ”one” for the useful pixels, and ”zero” for the pixels in a circular neighborhood of the center of evaporating patterns, in order to discard regions where our model is too coarse (see Fig. 3(a-2) for a masked hologram). The fit between the experimental data and the model is thus performed only on the high frequencies of the pattern.

To create the mask, a first rough estimation of the parameters is performed to obtain the centers of the masking areas (i.e., ( x n , y n )). The size of the circular neighborhood is the same for all diffraction patterns. The best size is the one that gives the highest weighted normalized correlation value between a test hologram and the fitted model. One residual hologram from which the estimated models have been subtracted (during the cleaning step), is shown in Fig. 3(b-1) . The high frequency part of the diffraction patterns is well cleaned. It illustrates that the model accurately fits the data (see also Fig. 3(b-2) ) in high frequency regions. Radial mean profiles of experimental data and the diffraction pattern model are presented in Fig. 4 together with the residuals.

Fig. 4 Comparison of the radial mean profile of one evaporating droplet’s signature (calculated from a hologram of set “2”) with the fitted model, the residuals and the masked part of the signal. (b) shows a zoomed-in version of (a).

The exclusion mask allows parameter estimation for evaporating droplets, using a simple hologram model. As a future work, another approach would consist in using a more appropriate model, taking into account the effect of the surrounding vapor film. Such a model has been very recently proposed by our group [31

31. L. Mees, N. Grosjean, D. Chareyron, J-L. Marie, M. Seifi, and C. Fournier, “Evaporating Droplet Hologram Simulation for Digital In-line Holography set-up with Divergent Beam,” Journal of Optical Society of America A 30, 2021–2028 (2013). [CrossRef]

]. Using this new model would require the estimation of more parameters, which would be computationally more demanding, yet it could lead to a richer description of the evaporation process.

3.5. Rough parameter prediction

4. Experimental results

In this section, the results of volume reconstruction using the iterative algorithm are presented. First, in Sec. 4.1, the calibration process is described. In Sec. 4.2 water droplet trajectories are reconstructed and size measurements are compared to PDA measurements. In Sec. 4.3 the iterative algorithm is applied to evaporating ether droplets, showing that the technique provides results that are accurate enough to study the evaporation phenomenon (e.g., the evaporation rate).

4.1. Calibration

To perform the calibration step, holograms of a glass reticle with a linear scale (Edmunds Optics, #62–252) are captured with varying distance between the reticle and the sensor. The reticle is first placed at a distance of z 0 = 473.5 mm from the sensor and moved back and forth in a depth range of 10 mm by steps of 1 mm.

Fig. 5 Calibration process, (a) a sample of a reconstructed calibration hologram, (b) the fitted regression curve used to estimate the magnification.

4.2. Non-evaporating droplets

PDA can be considered as a reference method for spherical particle size measurement. We use it to validate the accuracy of size estimations obtained by our digital holography method. PDA measurements are performed at the outlet of the injector just before recording the holograms. The mean droplet diameter, estimated by PDA from 3500 measurements, is 31.07 μm ± 0.078 μm. These values are close to those found in the same conditions in another work [ 21

21. D. Chareyron, J. L. Marie, C. Fournier, J. Gire, N. Grosjean, L. Denis, M. Lance, and L. Mees, “Testing an in-line digital holography “inverse method” for the lagrangian tracking of evaporating droplets in homogeneous nearly isotropic turbulence,” New Journal of Physics 14, 043039 (2012). [CrossRef]

] , showing that the injection is well reproducible. Using the setup described in Sec. 2, 310 droplets are reconstructed. The mean radius and the standard deviation of the radius are estimated as 30.57 μm and 0.11 μm, respectively. This proves that the accuracy of digital holography is comparable to that of PDA, with a relative standard deviation of 0.3%. One of the holograms of such droplets is shown in Fig. 6(a). Figure 6(b) shows the residuals after cleaning the diffraction pattern of the droplet.

Fig. 6 A hologram of non-evaporating droplets (a) and the residuals after cleaning the diffraction pattern of the droplet (b).

4.3. Evaporating droplets

The trajectories containing more than 10 droplets are reconstructed from set “1” (17 trajectories with an average of 34 droplet positions) and set “2” (13 trajectories with an average of 16 droplet positions). 3D visualization of the trajectories for set “1” and set “2” are shown using videos in Figs. 7(a)–7(b) respectively. Some examples of the squared radius evolution over time are presented in Figs. 8(a)–8(b) for set “1” and set “2” respectively. We note that after some time, the radius stabilizes around a constant value, on the order of 10 micrometers for all the runs (see Figs. 8(a)–8(b) ). We suppose that the evaporation stops at this stage, when all the ether has evaporated. The remaining droplet would then be composed of only water and would not evaporate anymore. The percentage of water given by the manufacturer, and experimentally verified, is 0.2%, while based on the final radius estimation, an initial concentration of about 4% is measured. Because these experiments are performed in humid air (with relative humidity of 31.8%), a plausible explanation for the higher percentage of water is that the fast evaporation of the diethyl ether cools the humid air around the droplet, thus causing some condensation at the surface. The existence of such condensation phenomenon was reported by Law, [33

33. C. K. Law, T. Y. Xiong, and C. Wang, “Alcohol droplet vaporization in humid air,” International Journal of heat and mass transfer 30, 1435–1443 (1987). [CrossRef]

] for alcohol droplets vaporizing in humid air. The repeatability of the droplet size evolution over time for various 3D trajectories can be noted from Figs. 8(a)–8(b).

Fig. 7 3D visualization of reconstructed trajectories (a) movie for set “1” ( Media 1 ), (b) movie for set “2” ( Media 2 ). Scale of the sphere radius has been increased to visualize the evaporation phenomenon.
Fig. 8 Evolution of the radius squared of the droplets over time (a) set “1”, (b) set “2”

The droplets that contain only water (i.e., the ones with constant radius) can be seen in Fig. 3(a-1). They correspond to the four patterns located at the bottom, which do not exhibit the high contrasted central disturbances created by the vapor film, confirming our belief that evaporation has stopped. The use of the mask is not necessary for such water droplets and we note that they are well cleaned from the residual image (see Fig. 3(b-1) ). Further investigation could be done to confirm this assumption, by measuring the refractive index of the droplets along their trajectories or by varying the humidity and temperature of ambient room conditions.

The droplet trajectories are then analyzed and results are compared to a simple evaporation model. When evaporation has reached stationary conditions (i.e., reaching constant droplet temperature), the squared diameter of a droplet generally decreases linearly with time. In terms of squared radius, this so-called “ d 2 ” law [34

34. C. K. Law, “Recent advances in droplet vaporization and combustion,” Progress in energy and combustion science 8, 171–201 (1982). [CrossRef]

] is expressed by
r(t)2=r(0)2+Kt,
(19)
where K < 0 is the evaporation rate. This linear decrease can be observed for big evaporating droplets in Fig. 8 . A theoretical value of K can be calculated as a function of the physical properties of diethyl ether, temperature and pressure. As the temperature of the droplets at the output of the injector is difficult to estimate, we can only give a range for K values: [−7.5, −6.2] μm 2 / ms (see Appendix B for the detailed calculation of this range).

To estimate K from the results obtained from digital holography, a least squares linear fit was performed on the first part of the squared-radius-over-time curves (i.e., the part of the curves that presents evaporation). Figure 8 shows the regression lines which closely fit the data. The average K is found equal to −7.7 μm 2 / ms and −6.2 μm 2 / ms for set “1” and set “2” respectively.

To calculate the accuracy of the measurements, each trajectory is used to compute the distances between the estimated radii and the fitted curve [35

35. W. H. Press, T. A. Saul, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, The Art of Scientific Computing (Cambridge University Press, 1992), cambridge university press ed.

]. The standard deviation of these distances gives a rough estimation of the accuracy on the radius measurement. For set “1”, the standard deviation is 0.5 μm for evaporating droplets with the radii in the range [9

9. C. P. Allier, G. Hiernard, V. Poher, and J. M. Dinten, “Bacteria detection with thin wetting film lensless imaging,” Biomedical optics express 1, 762–770 (2010). [CrossRef]

, 26

26. L. Denis, D. Lorenz, E. Thiébaut, C. Fournier, and D. Trede, “Inline hologram reconstruction with sparsity constraints,” Optics Letters 34, 3475–3477 (2009). [CrossRef] [PubMed]

] μm. The same study on set “2” gives a standard deviation of 0.4 μm in a radius range [13

13. M. K. Kim, Y. Hayasaki, P. Picart, and J. Rosen, “Digital holography and 3D imaging: introduction to feature issue,” Appied Optics 52, DH1 (2013). [CrossRef]

, 18

18. D. Nguyen, D. Honnery, and J. Soria, “Measuring evaporation of micro-fuel droplets using magnified DIH and DPIV,” Experiments in Fluids 1–11 (2010).

] μm.

This evaluation of the precision overestimates the actual precision, since all deviations from the evaporation model are attributed to measurement errors [35

35. W. H. Press, T. A. Saul, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, The Art of Scientific Computing (Cambridge University Press, 1992), cambridge university press ed.

]. This precision is still much larger than the precision estimated with non-evaporating droplets. The most reasonable explanation is that our model of the diffraction pattern of an evaporating droplet is coarse. The exclusion mask w accounts for the large misfit of the model at the center of the diffraction pattern. The higher frequency part of the model (i.e., corresponding to the larger diffraction rings) is also distorted, probably because of the presence of vapor, leading to a poor fit in this region [see Fig. 3(b-2)], and to coarser estimates of the depth and radius parameters. By excluding some pixels from the estimation procedure (about 7% of the pixels), the mask also degrades the accuracy.

5. Conclusion

As mentioned above, a part of the droplet signature is caused by evaporation and is not explained by the diffraction model of non-evaporating droplets. A future work can be directed towards using a more rigorous light scattering model (e.g., generalized Lorenz-Mie theory for multi-layered sphere [31

31. L. Mees, N. Grosjean, D. Chareyron, J-L. Marie, M. Seifi, and C. Fournier, “Evaporating Droplet Hologram Simulation for Digital In-line Holography set-up with Divergent Beam,” Journal of Optical Society of America A 30, 2021–2028 (2013). [CrossRef]

] ), to take the vapor film effect into account in the iterative algorithm. Such a model could provide additional information about the evaporation process.

Finally the prediction step could be further improved in the case of more complex trajectories using Kalman filters.

Appendix A

This appendix shows that, in the case of a spherical reference beam (produced by a point source located on the optical axis at distance z s ), the mathematical model of hologram formation:
Iholo(x,y)=Iref0(x,y)2{Aref0*(x,y)1Arefzn(xn,yn)2(ϑn*hzn)(x,y)3}
(20)
can be approximated as
Iholo(x,y)Iref0(x,y)+αnIref0(x,y).πre2λzeJ1c(2πreρeλze)sin(πρe2λze)
(21)
introducing a magnification factor m=zszszn by means of a new parameter set θ e = m θ n with ρe=(xxe)2+(yye)2.

For simplicity, we consider the particle to be located on the optical axis i.e., x e = y e = 0 (see [36

36. C. S. Vikram and M. L. Billet, “Some salient features of in-line fraunhofer holography with divergent beams,” Optik 78, 80–83 (1988).

] for the general case). Let us consider, the terms 1, 2 and 3 of equation Eq. (20).

The first term corresponds to the reference wave amplitude Aref0*(x,y) on the hologram plane. The modulus of this term can be non-uniform. However it is equal to the square root of the background intensity: Iref0(x,y). The phase of this term is non-uniform too and depends on laser point source location. Under Fresnel approximation and considering only the phase modulation in x and y directions, the first term of Eq. (20) can be expressed as:
Aref0*(x,y)=Iref0(x,y)eiπρe2λzs.
(22)

The second term is the amplitude of the reference wave across the particle’s aperture Arefzn(xn,yn). As the particle is located on the optical axis, the phase of this term is zero (we omit the phase modulation in z direction). We therefore obtain:
Arefzn(xn,yn)=Arefzn(0,0)=A0zn.
(23)

The third term is the object’s diffraction pattern. Assuming rnλzn , this term can be written as (see [28

28. M. Seifi, C. Fournier, L. Denis, D. Chareyron, and J.-L. Marié, “Three-dimensional reconstruction of particle holograms: a fast and accurate multiscale approach,” Journal of the Optical Society of America A 29, 1808–1817 (2012). [CrossRef]

] ):
[ϑn*hzn](ρ)πrn2iλznJ1c(2πrnρeλzn)eiπρe2λzn
(24)

By using these simplifications in Eq. (20), we obtain:
Iholo(x,y)=Iref0(x,y)2Iref0(x,y)A0zn{eiπρe2λzsπrn2iλznJ1c(2πrnρeλzn)eiπρe2λzn}.
(25)

The product of the two chirp functions can be simplified to eiπρe2λzseiπρe2λzn=eiπρe2λze , with z e = mz n. Defining αe=2A0znm , we obtain
Iholo(x,y)=Iref0(x,y)+αeIref0(x,y){πre2iλzeJ1c(2πreρeλze)eiπρe2λze}.
(26)

Appendix B

In this appendix, the estimated range for the theoretical value of the evaporating rate K is detailed. When the evaporation law is formulated in terms of squared radius, the evaporation rate is given by
K=(ρgρdDdShln(BM+1))
(27)
where ρ g is the density of the gas around the droplet, ρ d = 731.8 [ kg/m 3 ] is the density of of the diethyl ether, D d = 0.918 × 10 −5 [ m 2 / s ] is the diffusion coefficient of the diethyl ether (vapor) in the air, S h ≈ 2 is the Sherwood number and B M is the Spalding mass number.

Since the real composition of the gas around the droplet is unknown, this gas is assimilated for simplicity to air, and its density ρ g is assimilated to the density of dry air, that is not very different from that of humid air for our experimental conditions. A simple approximation of ρ g is given by the law of ideal gas:
ρg=ρair=PatmRdaTr
(28)
where P atm = 101325[ Pa ], R da = 287[ J/Kg/K ] is the constant for dry air, and T r is the temperature of reference which can be approximated as the ambient temperature ( T r = 25° C ).

The correlation relating P satd as a function of T has been taken from the Cheric Data Bank. It is written as:
Psatd=1000exp(12.43790log(T)6340.514T+95.14704+1.412198×105T2)
(31)
where T ≈ 253° K to 258° K. The theoretical value of K is therefore ∈ [−7.5, −6.2] × 10 −9 [ m 2 /s ].

Acknowledgments

This work was funded by the MORIN project (3D Optical Measurements for Research and INdustry), which is supported by the french government through the “Agence Nationale de la Recherche” (ANR) and the “Programme Avenir Lyon-Saint-Etienne” (PAL-SE). The authors would like to thank Rolf Clackdoyle for his fruitful comments on the manuscript.

References and links

1.

M. Sommerfeld and H.-H. Qiu, “Experimental studies of spray evaporation in turbulent flow,” International Journal of Heat and Fluid Flow 19, 10–22 (1998). [CrossRef]

2.

S. M. Skippon and Y. Tagaki, “ILIDS measurements of the evaporation of fuel droplets during the intake and compression strokes in a firing lean burn engine,” Technical Report 960830, SAE International, Warrendale, PA (1996).

3.

N. Damaschke, H. Nobach, and C. Tropea, “Optical limits of particle concentration for multi-dimensional particle sizing techniques in fluid mechanics,” Experiments in fluids 32, 143–152 (2002). [CrossRef]

4.

D. Sugimoto, K. Zarogoulidis, T. Kawaguchi, K. Matsuura, Y. Hardalupas, A. Taylor, and K. Hishida, “Extension of the compressed interferometric particle sizing technique for three component velocity measurements,” in “13th international symposium on applications of laser techniques to fluid mechanicsLisbon, Portugal,”, 26–29 (2006).

5.

F. Verpillat, F. Joud, P. Desbiolles, and M. Gross, “Dark-field digital holographic microscopy for 3D-tracking of gold nanoparticles,” Optics Express 19, 26044–26055 (2011). [CrossRef]

6.

L. Dixon, F. C. Cheong, and D. G. Grier, “Holographic deconvolution microscopy for high-resolution particle tracking,” Optics Express 19, 16410–16417 (2011). [CrossRef] [PubMed]

7.

X. Zhang, I. Khimji, U. A. Gurkan, H. Safaee, P. N. Catalano, H. O. Keles, E. Kayaalp, and U. Demirci, “Lensless imaging for simultaneous microfluidic sperm monitoring and sorting,” Lab on a Chip 11, 2535–2540 (2011). [CrossRef] [PubMed]

8.

L. Repetto, E. Piano, and C. Pontiggia, “Lensless digital holographic microscope with light-emitting diode illumination,” Optics letters 29, 1132–1134 (2004). [CrossRef] [PubMed]

9.

C. P. Allier, G. Hiernard, V. Poher, and J. M. Dinten, “Bacteria detection with thin wetting film lensless imaging,” Biomedical optics express 1, 762–770 (2010). [CrossRef]

10.

J. R. Fienup, “Coherent lensless imaging,” in “Imaging Systems,” (2010). [CrossRef]

11.

T. C. Poon, T. Yatagai, and W. Juptner, “Digital holography-coherent optics of the 21st century: introduction,” Applied Optics 45, 821 (2006). [CrossRef]

12.

J. Coupland and J. Lobera, “Special issue : Optical tomography and digital holography,” Measurement Science and Technology 19, 070101 (2008). [CrossRef]

13.

M. K. Kim, Y. Hayasaki, P. Picart, and J. Rosen, “Digital holography and 3D imaging: introduction to feature issue,” Appied Optics 52, DH1 (2013). [CrossRef]

14.

J. Gire, L. Denis, C. Fournier, E. Thiébaut, F. Soulez, and C. Ducottet, “Digital holography of particles: benefits of the ‘inverse problem’ approach,” Measurement Science and Technology 19, 074005 (2008). [CrossRef]

15.

C. Fournier, L. Denis, E. Thiebaut, T. Fournel, and M. Seifi, “Inverse problem approaches for digital hologram reconstruction,” in “Proceedings of SPIE ,”, 80430S (2011). [CrossRef]

16.

J. Sheng, E. Malkiel, and J. Katz, “Digital holographic microscope for measuring three-dimensional particle distributions and motions,” Applied optics 45, 3893–3901 (2006). [CrossRef] [PubMed]

17.

J. Lu, J. P. Fugal, H. Nordsiek, E. W. Saw, R. A Shaw, and W. Yang, “Lagrangian particle tracking in three dimensions via single-camera in-line digital holography,” New Journal of Physics 10, 125013 (2008). [CrossRef]

18.

D. Nguyen, D. Honnery, and J. Soria, “Measuring evaporation of micro-fuel droplets using magnified DIH and DPIV,” Experiments in Fluids 1–11 (2010).

19.

S. H. Lee, Y. Roichman, G. R. Yi, S. H. Kim, S. M. Yang, A. van Blaaderen, P. van Oostrum, and D. G. Grier, “Characterizing and tracking single colloidal particles with video holographic microscopy,” Optics Express 15, 18275–18282 (2007). [CrossRef] [PubMed]

20.

F. C. Cheong, K. Xiao, D. J. Pine, and D. G. Grier, “Holographic characterization of individual colloidal spheres’ porosities,” Soft Matter 7, 6816–6819 (2011). [CrossRef]

21.

D. Chareyron, J. L. Marie, C. Fournier, J. Gire, N. Grosjean, L. Denis, M. Lance, and L. Mees, “Testing an in-line digital holography “inverse method” for the lagrangian tracking of evaporating droplets in homogeneous nearly isotropic turbulence,” New Journal of Physics 14, 043039 (2012). [CrossRef]

22.

J. Fung, K. E. Martin, R. W. Perry, D. M. Kaz, R. McGorty, and V. N. Manoharan, “Measuring translational, rotational, and vibrational dynamics in colloids with digital holographic microscopy,” Optics express 19, 8051–8065 (2011). [CrossRef] [PubMed]

23.

F. Soulez, L. Denis, C. Fournier, É. Thiébaut, and C. Goepfert, “Inverse-problem approach for particle digital holography: accurate location based on local optimization,” Journal of the Optical Society of America . A 24, 1164–1171 (2007).

24.

F. Soulez, L. Denis, E. Thiébaut, C. Fournier, and C. Goepfert, “Inverse problem approach in particle digital holography: out-of-field particle detection made possible,” Journal of the Optical Society of America. A, Optics, Image Science, and Vision 24, 3708–3716 (2007). [CrossRef] [PubMed]

25.

H. Royer, “An application of high-speed microholography: the metrology of fogs,” Nouvelle Revue d’ Optique 5, 87–93 (1974). [CrossRef]

26.

L. Denis, D. Lorenz, E. Thiébaut, C. Fournier, and D. Trede, “Inline hologram reconstruction with sparsity constraints,” Optics Letters 34, 3475–3477 (2009). [CrossRef] [PubMed]

27.

G. B. Parrent and B. J. Thompson, “On the fraunhofer (far field) diffraction patterns of opaque and transparent objects with coherent background,” Journal of Modern Optics 11, 183–193 (1964).

28.

M. Seifi, C. Fournier, L. Denis, D. Chareyron, and J.-L. Marié, “Three-dimensional reconstruction of particle holograms: a fast and accurate multiscale approach,” Journal of the Optical Society of America A 29, 1808–1817 (2012). [CrossRef]

29.

M. Seifi, C. Fournier, and L. Denis, “HoloRec3D”. http://labh-curien.univ-st-etienne.fr/wiki-reconstruction/index.php/Main_Page, [Online; accessed 12-April-2013].

30.

M. Seifi, C. Fournier, and L. Denis, “HoloRec3D : A free Matlab toolbox for digital holography,” Hal:ujm (2012).

31.

L. Mees, N. Grosjean, D. Chareyron, J-L. Marie, M. Seifi, and C. Fournier, “Evaporating Droplet Hologram Simulation for Digital In-line Holography set-up with Divergent Beam,” Journal of Optical Society of America A 30, 2021–2028 (2013). [CrossRef]

32.

M. Seifi, L. Denis, and C. Fournier, “Fast and accurate 3D object recognition directly from digital holograms,” Journal of the Optical Society of America A 30, 2216–2224 (2013). [CrossRef]

33.

C. K. Law, T. Y. Xiong, and C. Wang, “Alcohol droplet vaporization in humid air,” International Journal of heat and mass transfer 30, 1435–1443 (1987). [CrossRef]

34.

C. K. Law, “Recent advances in droplet vaporization and combustion,” Progress in energy and combustion science 8, 171–201 (1982). [CrossRef]

35.

W. H. Press, T. A. Saul, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, The Art of Scientific Computing (Cambridge University Press, 1992), cambridge university press ed.

36.

C. S. Vikram and M. L. Billet, “Some salient features of in-line fraunhofer holography with divergent beams,” Optik 78, 80–83 (1988).

OCIS Codes
(100.3010) Image processing : Image reconstruction techniques
(100.3190) Image processing : Inverse problems
(100.6640) Image processing : Superresolution
(120.3940) Instrumentation, measurement, and metrology : Metrology
(280.2490) Remote sensing and sensors : Flow diagnostics
(090.1995) Holography : Digital holography

ToC Category:
Image Processing

History
Original Manuscript: June 17, 2013
Revised Manuscript: September 10, 2013
Manuscript Accepted: September 11, 2013
Published: November 7, 2013

Citation
Mozhdeh Seifi, Corinne Fournier, Nathalie Grosjean, Loic Méès, Jean-Louis Marié, and Loic Denis, "Accurate 3D tracking and size measurement of evaporating droplets using in-line digital holography and “inverse problems” reconstruction approach," Opt. Express 21, 27964-27980 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-23-27964


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References

  1. M. Sommerfeld, H.-H. Qiu, “Experimental studies of spray evaporation in turbulent flow,” International Journal of Heat and Fluid Flow 19, 10–22 (1998). [CrossRef]
  2. S. M. Skippon, Y. Tagaki, “ILIDS measurements of the evaporation of fuel droplets during the intake and compression strokes in a firing lean burn engine,” Technical Report 960830, SAE International, Warrendale, PA (1996).
  3. N. Damaschke, H. Nobach, C. Tropea, “Optical limits of particle concentration for multi-dimensional particle sizing techniques in fluid mechanics,” Experiments in fluids 32, 143–152 (2002). [CrossRef]
  4. D. Sugimoto, K. Zarogoulidis, T. Kawaguchi, K. Matsuura, Y. Hardalupas, A. Taylor, K. Hishida, “Extension of the compressed interferometric particle sizing technique for three component velocity measurements,” in “13th international symposium on applications of laser techniques to fluid mechanicsLisbon, Portugal,”, 26–29 (2006).
  5. F. Verpillat, F. Joud, P. Desbiolles, M. Gross, “Dark-field digital holographic microscopy for 3D-tracking of gold nanoparticles,” Optics Express 19, 26044–26055 (2011). [CrossRef]
  6. L. Dixon, F. C. Cheong, D. G. Grier, “Holographic deconvolution microscopy for high-resolution particle tracking,” Optics Express 19, 16410–16417 (2011). [CrossRef] [PubMed]
  7. X. Zhang, I. Khimji, U. A. Gurkan, H. Safaee, P. N. Catalano, H. O. Keles, E. Kayaalp, U. Demirci, “Lensless imaging for simultaneous microfluidic sperm monitoring and sorting,” Lab on a Chip 11, 2535–2540 (2011). [CrossRef] [PubMed]
  8. L. Repetto, E. Piano, C. Pontiggia, “Lensless digital holographic microscope with light-emitting diode illumination,” Optics letters 29, 1132–1134 (2004). [CrossRef] [PubMed]
  9. C. P. Allier, G. Hiernard, V. Poher, J. M. Dinten, “Bacteria detection with thin wetting film lensless imaging,” Biomedical optics express 1, 762–770 (2010). [CrossRef]
  10. J. R. Fienup, “Coherent lensless imaging,” in “Imaging Systems,” (2010). [CrossRef]
  11. T. C. Poon, T. Yatagai, W. Juptner, “Digital holography-coherent optics of the 21st century: introduction,” Applied Optics 45, 821 (2006). [CrossRef]
  12. J. Coupland, J. Lobera, “Special issue : Optical tomography and digital holography,” Measurement Science and Technology 19, 070101 (2008). [CrossRef]
  13. M. K. Kim, Y. Hayasaki, P. Picart, J. Rosen, “Digital holography and 3D imaging: introduction to feature issue,” Appied Optics 52, DH1 (2013). [CrossRef]
  14. J. Gire, L. Denis, C. Fournier, E. Thiébaut, F. Soulez, C. Ducottet, “Digital holography of particles: benefits of the ‘inverse problem’ approach,” Measurement Science and Technology 19, 074005 (2008). [CrossRef]
  15. C. Fournier, L. Denis, E. Thiebaut, T. Fournel, M. Seifi, “Inverse problem approaches for digital hologram reconstruction,” in “Proceedings of SPIE,”, 80430S (2011). [CrossRef]
  16. J. Sheng, E. Malkiel, J. Katz, “Digital holographic microscope for measuring three-dimensional particle distributions and motions,” Applied optics 45, 3893–3901 (2006). [CrossRef] [PubMed]
  17. J. Lu, J. P. Fugal, H. Nordsiek, E. W. Saw, R. A Shaw, W. Yang, “Lagrangian particle tracking in three dimensions via single-camera in-line digital holography,” New Journal of Physics 10, 125013 (2008). [CrossRef]
  18. D. Nguyen, D. Honnery, J. Soria, “Measuring evaporation of micro-fuel droplets using magnified DIH and DPIV,” Experiments in Fluids1–11 (2010).
  19. S. H. Lee, Y. Roichman, G. R. Yi, S. H. Kim, S. M. Yang, A. van Blaaderen, P. van Oostrum, D. G. Grier, “Characterizing and tracking single colloidal particles with video holographic microscopy,” Optics Express 15, 18275–18282 (2007). [CrossRef] [PubMed]
  20. F. C. Cheong, K. Xiao, D. J. Pine, D. G. Grier, “Holographic characterization of individual colloidal spheres’ porosities,” Soft Matter 7, 6816–6819 (2011). [CrossRef]
  21. D. Chareyron, J. L. Marie, C. Fournier, J. Gire, N. Grosjean, L. Denis, M. Lance, L. Mees, “Testing an in-line digital holography “inverse method” for the lagrangian tracking of evaporating droplets in homogeneous nearly isotropic turbulence,” New Journal of Physics 14, 043039 (2012). [CrossRef]
  22. J. Fung, K. E. Martin, R. W. Perry, D. M. Kaz, R. McGorty, V. N. Manoharan, “Measuring translational, rotational, and vibrational dynamics in colloids with digital holographic microscopy,” Optics express 19, 8051–8065 (2011). [CrossRef] [PubMed]
  23. F. Soulez, L. Denis, C. Fournier, É. Thiébaut, C. Goepfert, “Inverse-problem approach for particle digital holography: accurate location based on local optimization,” Journal of the Optical Society of America. A 24, 1164–1171 (2007).
  24. F. Soulez, L. Denis, E. Thiébaut, C. Fournier, C. Goepfert, “Inverse problem approach in particle digital holography: out-of-field particle detection made possible,” Journal of the Optical Society of America. A, Optics, Image Science, and Vision 24, 3708–3716 (2007). [CrossRef] [PubMed]
  25. H. Royer, “An application of high-speed microholography: the metrology of fogs,” Nouvelle Revue d’ Optique 5, 87–93 (1974). [CrossRef]
  26. L. Denis, D. Lorenz, E. Thiébaut, C. Fournier, D. Trede, “Inline hologram reconstruction with sparsity constraints,” Optics Letters 34, 3475–3477 (2009). [CrossRef] [PubMed]
  27. G. B. Parrent, B. J. Thompson, “On the fraunhofer (far field) diffraction patterns of opaque and transparent objects with coherent background,” Journal of Modern Optics 11, 183–193 (1964).
  28. M. Seifi, C. Fournier, L. Denis, D. Chareyron, J.-L. Marié, “Three-dimensional reconstruction of particle holograms: a fast and accurate multiscale approach,” Journal of the Optical Society of America A 29, 1808–1817 (2012). [CrossRef]
  29. M. Seifi, C. Fournier, L. Denis, “HoloRec3D”. http://labh-curien.univ-st-etienne.fr/wiki-reconstruction/index.php/Main_Page , [Online; accessed 12-April-2013].
  30. M. Seifi, C. Fournier, L. Denis, “HoloRec3D : A free Matlab toolbox for digital holography,” Hal:ujm (2012).
  31. L. Mees, N. Grosjean, D. Chareyron, J-L. Marie, M. Seifi, C. Fournier, “Evaporating Droplet Hologram Simulation for Digital In-line Holography set-up with Divergent Beam,” Journal of Optical Society of America A 30, 2021–2028 (2013). [CrossRef]
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