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# Liquid volume fraction and droplet sizing in atomizing sprays using polarization ratio with dual structured laser illumination planar imaging

Sebastien Garcia, Vassily Kornienko, Mehdi Stiti, Elias Kristensson, Edouard Berrocal

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Sebastien Garcia, Vassily Kornienko, Mehdi Stiti, Elias Kristensson, Edouard Berrocal. Liquid volume fraction and droplet sizing in atomizing sprays using polarization ratio with dual structured laser illumination planar imaging. Optics Letters, 2025, 50 (19), pp.6233. 10.1364/ol.563173. hal-05337672

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# Optics Letters

Liquid volume fraction and droplet sizing in
atomizing sprays using polarization ratio with dual
structured laser illumination planar imaging

SEBASTIEN GARCIA,<fnref n="1" />,<fnref n="3" />,† VASSILY KORNIENKO,<fnref n="1" />,†,\* MEHDI STITI,<fnref n="2" /> ELIAS KRISTENSSON,<fnref n="1" /> AND EDOUARD BERROCAL<fnref n="1" />

Received 25 March 2025; revised 30 August 2025; accepted 16 September 2025; posted 16 September 2025; published 1 October 2025

It is essential that the design of atomizing sprays be tailored to the specific requirements of various industrial processes within, e.g., the agricultural or energy sector. However, the measurement of key atomizer parameters—central to the optimization of such systems—suffers from time-intensive techniques, complex calibration routines, and/or the need to include additives in the injected liquid. Here, we present an additive-less, two-dimensional optical technique for determining droplet sizes, number densities, and liquid volume fractions of optically dense atomizing sprays. This method is applied to characterize two distinct hollow-cone sprays injected at high pressure and flow rate. Our results reveal local inhomogeneities in various key atomizer parameters, demonstrating the potential for efficient, in situ characterization of atomizing spray systems. Published by Optica Publishing Group under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

https://doi.org/10.1364/OL.563173

Atomizing sprays are integral to various industrial processes, including agricultural crop treatment, painting, internal combustion engines, or powder production in the food and pharmaceutical industries. To effectively characterize a spray, many parameters related to performance and external characteristics are detailed in Lefebvre's book [1]. Among these, key atomizer parameters such as droplet size (D) and concentration (N) must be optimized to meet the specific demands of these applications. The combination of these two parameters defines the liquid volume fraction (LVF), which should be considered during the design phase, as it directly influences an atomizer's effectiveness for its intended purpose [2].

This non-exhaustive list of characteristics can be determined via the use of various optical measurement techniques, where Phase Doppler Analyzer (PDA) is often utilized as a standard [3]. This off-the-shelf method has the advantage of not requiring a calibration; however, being a point measurement (its

measurement volume is limited to a few hundred cubic micrometers), applying it to the characterization of an entire spray becomes severely time-consuming. Moreover, due to the phenomenon of multiple light scattering, the application of PDA on optically dense spray systems becomes challenging.

In contrast, laser-based imaging techniques, which inherently sample events in two dimensions, offer an approach for significantly reducing measurement times [4]. In conjunction with Structured Laser Illumination Planar Imaging (SLIPI), diverse approaches exist in the literature for determining the characteristics of optically dense sprays. For example, by combining SLIPI, Planar Laser-Induced Fluorescence (PLIF), and image analysis algorithms, Deshmukh and Ravikrishna were able to measure mean droplet diameters (Dᵢⱼ), N, and LVF [5]. Alternatively, these characteristics can be measured via the application of SLIPI-LIF/Mie in conjunction with a scanning-type strategy, dubbed a SLIPI-scan [6]. However, these approaches require complex post-processing algorithms in conjunction with intricate calibration steps rendering them challenging to apply. Furthermore, the addition of dyes in these methods is necessary to induce fluorescence and, as such, is restricted to the study of non-evaporating sprays. This addition of a dye in significant proportions can also alter the physical properties of the fluid, which, in turn, affects droplet sizes and distributions. Hence, the use of fluorescence-based methods can result in skewed representations of, e.g., the LVF of a given spray system, thereby undermining the choice of design for efficient atomizer operation in industrial settings.

Recently, SLIPI has been combined with the Polarization Ratio technique (SLIPI-PR) [7,8]. This technique, devoid of any additives and solely based on Lorenz–Mie theory, aims to measure the ratio between the perpendicularly and parallel-polarized scattered light from the droplets. By analyzing this signal at an observation angle of 90°, the authors previously demonstrated that over a wide range of droplet sizes (10–200 µm), the S–pol signal (β = 90° from the incident plane, i.e., perpendicularly polarized) is proportional to the projected surface area (D<fnref n="2" />), while the P–pol signal (β = 0° from the incident plane, i.e., parallel polarized) is proportional to the diameter (D).

<footnote n="1">¹Department of Physics, Division of Combustion Physics, Lund University, Sweden</footnote>

<footnote n="2">²Institut de Mécanique des Fluides de Toulouse (IMFT), CNRS, Université de Toulouse, France</footnote>

<footnote n="3">³sebastien.garcia@fysik.lu.se</footnote>

<footnote>†These authors contributed equally to this work.</footnote>

<footnote>*vassily.kornienko@fysik.lu.se</footnote>
<figure><img src="image_5.png" /><figcaption>Fig. 1. Experimental setup for Dual-SLIPI Polarization Ratio (Dual-SLIPI-PR). The modulated and polarized light sheet is generated using a combination of spherical (SL) and cylindrical (CL) lenses along with a diffractive optical element (DOE) and half-wave plate (HWP), set for a polarization β = 15°. The scattered signal is collected by two large telecentric lenses (0.066x, FOV: 133.2 × 193.8 mm²) positioned face-to-face at θ = 90° in relation to the spatially modulated light sheet. They are each fixed with a rotating polarization filter for the capture of polarization sensitive images.</figcaption></figure>

This ratio thus provides a statistical surface mean diameter ($D_{21}$). In that previous work, the spray was observed from a single viewing direction. In the present study, an additional camera was introduced on the opposite side, and a pixel-to-pixel calibration between the two views was performed. The inclusion of a second face-to-face camera enables the acquisition of depth-resolved extinction coefficient maps ($\mu_e$) [9]. On this basis, an additive-free, two-dimensional technique is presented for the simultaneous estimation of the $D_{21}$, $N$, and $LVF$ within dense inhomogeneous sprays. We apply this to a range of pressure swirl atomizers (Delavan SDX V spray drying series) resulting in consistent two-dimensional information of these quantities, opening up for reliable diagnostics in harsh in situ environments.

Figure 1 presents the setup of the Polarization Ratio with Dual Structured Laser Illumination Planar Imaging (Dual-SLIPI-PR) technique. A CW laser (Spectra-Physics Millenia Prime, λ = 532 nm, 5 W) illuminates the atomizing spray under consideration with a polarization angle of β = 15° from the horizontal plane (meticulously adjusted with a half-wave plate, WPHSM05-532). The laser beam is shaped into a sheet, approximately 12 cm in height and 0.5 mm thick, using a combination of spherical and cylindrical lenses (SL and CL, respectively). As the sphericity of droplets is essential to the application of Lorenz–Mie theory, the top of the laser sheet is positioned 2 cm downstream from the injector exit, thereby minimizing the presence of areas where the spray is not fully formed from the analysis. A two-spot diffractive optical element (DOE) induces a spatial modulation upon the light sheet, and the 3p-SLIPI method is implemented. In short, three images, modulated and phase-shifted by $2\pi/3$, are acquired to reconstruct the final image where multiple light scattering is suppressed. Each phase is displaced vertically by shifting the DOE with a piezoelectric motor (PiMotor).

The light that is scattered by the droplets is recorded perpendicular to the sheet with two sCMOS cameras (Andor Zyla 5.5, 2180 × 2560). These are positioned face-to-face and are equidistant from the laser sheet. They are fitted with large telecentric objectives (TitanTL, 0.066x, FOV = 133.2 × 193.8 mm², 1" C-Mount), with aperture angles of $\Delta\theta = 5°$. An in-house pixel-to-pixel calibration procedure is performed with checkerboards and USAF targets. S-pol and P-pol images are acquired sequentially with a rotating polarizing filter (LPVISC100b, OD6) placed in front of each camera. To accurately characterize highly dense sprays exhibiting significant optical extinction (see OD measurements in Fig. 1), measurements were acquired by averaging a total of 40 images per SLIPI phase, each with an exposure time of 10 ms. This process was then repeated three times.

Within the detection volume of this imaging system, the signals from individual droplets (k) recorded by each pixel (i, j) are ensemble-averaged over the entire acquisition time. Considering the signal emitted by the droplets as a scattered irradiance, the ratio of the perpendicular and parallel polarized scattered intensities (γ) can be expressed as follows [10]:

$$ (\gamma)_{i,j} = \left( \frac{I_{1(S-pol, x)}}{I_{1(P-pol, x)}} \right)_{i,j} = \frac{C_{(S-pol)}}{C_{(P-pol)}} \left( \frac{\sum N_k D_k^2}{\sum N_k D_k} \right)_{i,j} \\ = C \cdot (D_{21})_{i,j}, \quad (1) $$

where $I_{1(S-pol, x)}$ and $I_{1(P-pol, x)}$ are the S-pol and P-pol images from camera 1 acquired at a distance x from the plane subtended by the laser sheet (YZ). Figure S1 in Supplement 1 shows the post-processing details of the S-pol and P-pol images, and thus the determination of γ.
<figure><img src="image_6.png" /><figcaption>**Fig. 2.** (a) Schematic of the spray morphology with the PDA measurement path (in red). Two droplet size distributions obtained from PDA are presented. (b) Calibration of the SLIPI polarization ratio, ($\gamma$)$_{i,j}$, as a function of $D_{21}$. (c) Calculation of the extinction cross-section for an aperture angle of $\Delta\theta = 5^{\circ}$ (defined by the telecentrical lens used in the experiment).</figcaption></figure>

In practice, intensity-based techniques depend on parameters such as the incident laser wavelength and intensity, refractive index, scattering angle, and particle size—each of which affects the proportionality between the light intensity ratio and the mean diameter. Thus, $C_{(S\_pol)}$ and $C_{(P\_pol)}$ are two constants, experimentally determined via calibration, whose purpose is to maintain the proportionality of Eq. (1). As previously demonstrated [7,8], the calibration procedure is applied by comparing the acquired intensity ratio with results obtained by a PDA instrument at equivalent positions of a steady-state spray. Since the PDA algorithms account for the principal detection-volume biases, particularly the droplet-size dependence of collection versus excitation volume, the PDA-based calibration provides a direct, unbiased relationship between $D_{21}$ and the measured polarization ratio. Figures 2(a) and 2(b) illustrate this calibration performed with water injected at $P_{inj} = 50$ bar through the SD30 atomizer, corresponding to a low optical density injection and thereby minimizing the effects of multiple light scattering. Unlike the relationship between the LIF/Mie ratio and $D_{32}$ [6], the main advantage of this relationship, between polarization ratio and $D_{21}$, is its direct correspondence to Lorenz-Mie theory along with its linearity which homogenizes the sensitivity across a large range of droplet sizes. The planar data are processed using a ratio-based metric, incorporating flat-field correction for each channel and telecentric collection, which together minimize illumination and volume-sampling biases; furthermore, the PDA-anchored calibration is conducted at a single reference point and subsequently applied across the entire planar field.

In order to advance beyond the measurement of droplet diameters toward the LVF, it becomes necessary to obtain the local extinction coefficient within the spray. This is achieved using Dual-SLIPI, a calibration-free method which is solely based on side scattering detection. To proceed, the intensity

reduction caused by signal attenuation between two laser sheets separated by $\Delta x$ [mm] is recorded using two face-to-face cameras, as described by Kristensson et al. [9]. By acquiring 4 images and following Eq. (2), it is possible to determine $\mu_e$ for each pixel (i, j) across the entire two-dimensional spray:

$$ (\mu_e)_{i,j} = \frac{1}{2\Delta x} \ln \left( \frac{I_{1(P\_pol, x)} I_{2(P\_pol, x+\Delta x)}}{I_{2(P\_pol, x)} I_{1(P\_pol, x+\Delta x)}} \right)_{i,j} \quad (2) $$

Here, $I_{1(P\_pol, x)}$ and $I_{1(P\_pol, x+\Delta x)}$ are the $P\_pol$ images recorded by camera 1 at the focal positions x and $x + \Delta x$, whereas $I_{2(P\_pol, x)}$ and $I_{2(P\_pol, x+\Delta x)}$ are recorded by camera 2 in an equivalent manner. Within the context of the Beer-Lambert law in spray applications, $\mu_e$ can be described as the probability of interaction between photons and droplets per unit distance. When the droplet size is uniform in a spray (e.g., a monodisperse system or an approximation using mean diameters), $\mu_e$ can be defined as the product of the droplet concentration ($N$) and the extinction cross-section ($\sigma_e$):

$$ (\mu_e)_{i,j} = (N \sigma_e)_{i,j} \quad (3) $$

Lorenz-Mie theory dictates that $\sigma_e$ is a function of wavelength ($\lambda$), scattering angle ($\theta$), droplet diameter ($D$), and complex refractive index ($n_r + i\kappa$). In practice, using a single diameter, $\sigma_e$ can be interpreted as a geometrical cross-section ($\sigma_g = \pi D^2/4$) that scatters incident energy with an efficiency factor $Q$. However, when using a surface mean diameter, such as $D_{21}$, the relationship becomes more complex, as it depends strongly on the shape of the scattering particle size distribution [8]. Thus, the main challenge is to express $\sigma_e \sim f(D_{21})$. To address this, accurate values of $\sigma_e$ for different $D_{21}$ values were calculated using Lorenz-Mie theory in combination with PDA data (real droplet size distributions) within the MiePlot software. Based on these results, a look-up table was constructed, as illustrated in Fig. 2(c). Building on this, a value of the number density $N$, can be directly extracted from this calculation resulting in:

$$ (N)_{i,j} = \left( \frac{\mu_e}{1.09 D_{21}^{1.65}} \right)_{i,j}, \quad (4) $$

where $\sigma_e$ is equal to $D_{21}$ raised to the power of 1.65 and multiplied by 1.09. Finally, in order to calculate the LVF of the spray, the droplet volume ($V$) and $N$ are required. The LVF is then determined through pixel-wise multiplication of $N$ and an approximate volume derived from the $D_{21}$ measurement:

$$ (LVF)_{i,j} = \frac{\pi}{6} (N D_{21}^3)_{i,j} \quad (5) $$

The application of this methodology onto the characterization of two steady-state water hollow-cone sprays results in a series of images containing information on the atomizer performance in terms of droplet sizes and distributions (Fig. 3). Both these sprays are injected at high pressures (100 bars for the SD30 and 80 bars for the SE50) and volume flow rates (1.9 l/min and 4.2 l/min, respectively) resulting in optically dense measurement situations where multiple light scattering dominates, limiting light-based techniques to the use of SLIPI. Nonetheless, at such high OD, a rotation of the spray was necessary to image both sides of the spray (see Supplement 1, Section 1). For the SD30 (upper row), a high concentration of small droplets is observed in the center, whereas a low concentration of large droplets is observed on the edge, a characteristic of hollow-cone sprays.
<figure><img src="image_7.png" /><figcaption>**Fig. 3.** Two-dimensional maps of $D_{21}$, $\mu_e$, $N$, and $LVF$, for two steady-state water hollow-cone sprays generated by the pressure-swirl atomizers SD30 (upper row), and the SE50 (lower row), from the Delavan SDX V Spray Drying series. All mappings correspond to a laser sheet position of $x = 0$ mm and have dimensions of $14 \times 16$ cm$^2$ (each square on the grid represents $2 \times 2$ cm$^2$). The resulting spatial structure of the various parameters show local inhomogeneities, phenomena that DUAL-SLIPI-PR is sensitive to.</figcaption></figure>

Similar structures, albeit with values reflecting the larger flow rate and cone angle, can be observed for the SE50 spray on the second row. Merging all the images together according to the presented routine, an $LVF$ can be extracted, a quantity that gives information about the spatial distribution of the liquid during operation. Along with the observation of a hollow-cone structure, these images reveal local inhomogeneities in the liquid distributions. This is especially evident at the center of the SD30 spray, where streaks are visible along the central axis.

To conclude, the Dual-SLIPI Polarization Ratio technique combines Lorenz–Mie theory with the Beer–Lambert law enabling the direct measurement of local mean surface diameter $D_{21}$ and the extinction coefficient $\mu_e$. This, in turn, allows for the determination of atomizer performance and external spray parameters such as the number density $N$ and liquid volume fraction $LVF$ across an entire, optically dense inhomogenous spray. There are multiple advantages associated with the Dual-SLIPI-PR technique. First, with only a single calibration measurement for $D_{21}$, a full two-dimensional spray characterization can be performed within a single Dual-SLIPI-PR measurement. Indeed, with the presented spatial resolution of 99 µm (see Supplement 1, Section 2), millions of single PDA measurements would be necessary in order to see the large-scale inhomogeneities of these atomizers, an extremely time-consuming task. Second, the act of acquiring SLIPI images on both sides of the laser sheet cancels out the unknown distribution of scattering particles between the light sheet and camera. This reveals local inhomogeneous structures deep within the dense spray, previously observable only via synchrotron X-rays [11] or complex reconstruction algorithms based on intricate experimental procedures. Finally, since the imaging can occur at a distance (in this case ~40 cm away from the spray), no additives are necessary and

a single PDA calibration suffices to provide accurate measurements across multiple cases, in situ measurements of atomizers during regular operation, e.g., in gas turbine combustors, are made possible.

**Funding.** Vetenskapsrådet (2019-05183, 2021-04542); European Research Council (803634); HORIZON EUROPE European Innovation Council (grant agreement: "E-COOL" 101130315).

**Acknowledgment.** This work was supported by Tetra Pak Processing Systems, Swedish Research Council.

**Disclosures.** The authors declare no conflicts of interest.

**Data availability.** Data may be obtained from the authors upon reasonable request.

**Supplemental document.** See Supplement 1 for supporting content.

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