On the morphology of irregular rough particles from the analysis of speckle-like interferometric out-of-focus images Marc Brunel a,n , Sara González Ruiz b , Justin Jacquot a , Jeroen van Beeck b a UMR CNRS 6614 CORIA, 675 Av. de l'Université, 76801 Saint-Etienne du Rouvray cedex, France b Von Karman Institute for Fluid Dynamics, Chaussée de Waterloo, 72, 1640 Rhode-Saint-Genèse, Belgium article info Article history: Received 1 July 2014 Received in revised form 21 October 2014 Accepted 24 October 2014 Available online 30 October 2014 Keywords: Interferometric out-of focus imaging Optical transfer matrices Rough objects Morphology abstract Using a simplified description, we show that the Fourier transform of the speckle-like out-of-focus image of an irregular rough particle is given by the 2 dimensional-autocorrelation of the shape of the particle. Experiments confirm well this result. Using a matrix transfer based-formulation, we further determine the exact scaling factors between both functions, whatever the imaging system is. & Elsevier B.V. All rights reserved. 1. Introduction Interferometric out-of-focus imaging is a robust technique currently used to characterize droplets or bubbles. With this technique, droplets or bubbles are observed in a defocused plane, and their size is determined from the analysis of the interfero- metric fringes present in their out-of-focus spot [1–5]. Fast image processing can be developed to analyze globally a complete image [6]. Specific set-ups can be further designed for the characteriza- tion of droplets or bubbles in a 3D volume [7,8]. Enhanced algo- rithms are used to improve the accuracy of the fringe's frequency determination [9]. Combining interferometric out-of-focus ima- ging and Particle Image Velocimetry, simultaneous planar mea- surements of droplet velocity and size with gas phase velocities in a spray can be realized [10]. Simultaneous size and velocity mea- surements of cavitating microbubbles can be further realized [11]. Recently, it was demonstrated that the analysis of irregular rough particles could be done with this technique [12]. In this case, the-out-of-focus images are speckle-like patterns. It was shown that the size or the global ellipticity of a particle can be linked analytically to the size of the speck of light in well-controlled configurations. One can however wonder whether there are no general relations which link theoretically the speckle-like pattern to the exact morphology of the irregular particle. The analysis of speckle-like patterns to characterize irregular particles as ice crystals or ashes has important applications in airborne research [13]. The aim of this paper is to establish and verify such a relation with experimental data. As in reference [12], particles will be modeled as a collection of a large number of coherent point emitters (the so-called glare points in the case of droplets, which are then only two [14–17]). Under this assumption we will show theoretically that the 2-dimensional Fourier transform of the speckle-like out-of-focus pattern is directly linked to the 2-di- mensional autocorrelation of the particle's shape. This theoretical formulation will then be verified experimentally in the case of different irregular rough sand particles. 2. Theoretical formulation In our simplified approach, the field emitted by the irregular rough particle is approximated by an ensemble of N gp punctual light sources located over the global form of the particle. This assumption will be confirmed in the experimental part. The elec- tric field emitted by the particle can thus be written: ∑ α δ = − − φ = ( ) G x y e x a y b ( , ) , (1) j N j i j j 0 0 0 1 0 0 gp j where the δ functions are Dirac functions. a j and b j are the transverse coordinates of glare point denoted j. N gp is the number of point emitters to be considered. z will represent the optical longitudinal axis of the imaging set-up. We assume that all glare Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/optcom Optics Communications http://dx.doi.org/10.1016/j.optcom.2014.10.053 0030-4018/& Elsevier B.V. All rights reserved. n Corresponding author. E-mail address: marc.brunel@coria.fr (M. Brunel). Optics Communications 338 (2015) 193–198