Experimental validation of the vectorial complex ray model on the inter-caustics scattering of oblate droplets

Three-dimensional rainbow refractometry

We propose a new, to the best of our knowledge, rainbow technique called three-dimensional rainbow refractometry (TDRR), with a cylindrical lens in the signal collecting system. With a TDRR model based on the ray transfer matrix developed, it is proved that the tilt angle of the rainbow signal is related to the axial position of the droplet, which helps to obtain the 3D position. By converting rainbow scattering angle calibration into the system parameter calibration, a new rainbow data processing program is written in combination with the model to obtain the refractive index and the particle size. With TDRR, we measured a monodisperse droplet stream of deionized water at room temperature for experimental validation and obtained the refractive index with an absolute error of less than 0.0015, the droplet size with an error within ±5%, and the axial position with an error within ±3%, which demonstrated a high accuracy of TDRR.

Numerical implementation of three-dimensional vectorial complex ray model and application to rainbow scattering of spheroidal drops

The rainbow patterns of oblate spheroidal drops have been observed in experiments nearly forty years ago [Nature312, 529 (1984)10.1038/312529a0]. However, the prediction for those complex patterns has been a challenge for conventional light scattering models. The vectorial complex ray model (VCRM) allows to account for the direction, the polarization, the phase, the amplitude and the wavefront curvature of waves and provides a powerful tool for the study of the light/electromagnetic wave interaction with a homogeneous object of any shape with smooth surface. In [Opt. Lett.46, 4585 (2021)10.1364/OL.434149], the authors have reported an important breakthrough of VCRM for the three-dimensional scattering (VCRM3D) and the simulated rainbow patterns of oblate drops. The present paper is devoted to the detailed description of the numerical implementation allowing the simulation of the 3D scattering field by a nonspherical particle. Its ability to predict both the fine and coarse intensity structures of the rainbows and the near-backward scattering patterns of spheroids is demonstrated. This work opens perspectives for exploring the 3D scattering characteristics of large objects with any smooth shape and developing relevant optical techniques for particle characterization.

A method for scattering angle calibration in the rainbow region using a droplet stream

Accurate quantification of scattering angle versus detector pixel strongly determines the measurement accuracy of rainbow refractometry. This is an emerging measurement technique operating at backscatter angles and characterizing droplets or complex droplets in terms of size and refractive index. A novel method for calibration of the rainbow scattering angle using a monodisperse droplet stream is introduced and the achievable accuracy is estimated. The assumption of a linear pixel-to-angle relation is derived, and a calibration procedure is proposed based on global fit of calibration data to the theoretically known rainbow signal. The accuracy of this method was examined by simulations and experiments, where the uncertainties of a priori parameters of droplets were also considered and validated using shadowgraphy as a ground truth. The results confirm the feasibility of this method with a maximum absolute error of 0.032°and 3.9E-5°/pixel respectively for the intercept and slope of the linear relationship. These values translate into maximum uncertainties in diameter and refractive index of approx. 0.67% and 2.8 × 10^-4.

Synthetic aperture rainbow refractometry

This work proposed a synthetic aperture rainbow refractometry (SARR) by synthesizing rainbow signals of the same droplet with dual-wavelength laser beams, in order to increase the aperture of rainbow refractometry. In this way, the SARR can apply to long distance and small droplets measurement. An achromatic imaging system, which simultaneously records while separating the two rainbow signals in two channels of a color image, is elaborately designed. A data processing algorithm is developed to retrieve the optimal droplet refractive index and size. Numerical simulations of different droplet sizes from 10 μm to 200 μm certify the viability of the SARR. Proof-of-concept experiments of micron-sized ethanol droplets are performed with 1650 mm measurement distance. Results show that the SARR can accurately measure droplet refractive index and size with uncertainties of 2.3 × 10^-4 and 2μm, respectively. The feasibility and accuracy of the proposed SARR are successfully demonstrated, paving the way for rainbow refractometry applied to large-scale industrial applications.

Polarization-dependent LIF/Mie ratio for sizing of micrometric ethanol droplets doped with Nile red

The present study deals with droplet sizing based on laser-induced fluorescence (LIF) and Mie scattering for varied polarization of the utilized laser (parallel or perpendicular). The polarization-dependent LIF/Mie ratio is studied for micrometric droplets (25-60 µm) produced with a droplet generator. The investigations were carried out with the dye Nile red dissolved in ethanol and ethanol/iso-octane mixtures. A spectral absorption and fluorescence characterization at various dye and ethanol concentrations is carried out in a cuvette in order to identify reabsorption effects. The LIF_|| droplet images (index ||: parallel polarization) show a more homogeneous intensity distribution in the droplets and slightly stronger morphology-dependent resonances (MDRs) in comparison to LIF_⊥ (index ⊥: perpendicular polarization). The spectral LIF emissions reveal a dependence of the MDR on the ethanol admixture. The larger the ethanol content, the lower the MDR peak, which is also shifted further to the red part of the spectrum. The Mie droplet signal images are mainly characterized by two distinct glare points, one at the entrance of the laser light (reflection) and one at the exit (first-order refraction). The Mie_⊥ images show a more pronounced entrance glare point, in comparison to Mie_||, where the exit glare point is more pronounced. These observations are in accordance with the theory. The calibration curve of the micro droplet signals revealed a volumetric trend of the LIF signals and a slightly higher LIF_⊥ signal and sensitivity in comparison to LIF_||. The signal Mie⊥ follows roughly a quadratic trend on average, while Mie|| follows a linear trend. Consequently, the calculated LIF⊥/Mie⊥ ratio shows a linear trend, whereas the LIF||/Mie|| ratio shows a quadratic trend, which confirms theoretical calculations. A numerical simulation of the Mie signal at various detection angles shows a good agreement with the experimental data at large apertures.

Simultaneous determination of the shape and refractive index of a deformed microjet cavity from its resonances

Measuring the boundary shape of a deformed liquid microjet is of great importance for using it as an optical resonator for various applications. However, there have been technical challenges due to transparency and uncertainty in the refractive index of the liquid. In this study, we have developed a spectroscopic technique that enables simultaneous determination of the boundary shape and the refractive index of a liquid deformed microjet. A detailed procedure of the technique based on imposition of one-to-one correspondence between experimentally observed resonances and numerically calculated ones are presented along with the measurement results including the refractive index of ethanol between a wavelength of 550 nm and 670 nm.

Analyzing light-structuring features of droplet lenses on liquid-repelling surfaces

The complete understanding of the formation of seemingly levitating droplets on liquid-repelling surfaces provides the basis for further development of applications requiring friction-free liquid transport. For the investigation of these droplets and, thereby, the underlying surface properties, standard techniques typically only reveal a fraction of droplet or surface information. Here, we propose to exploit the light-shaping features of liquid droplets when interpreted as thick biconvex elliptical lenses. This approach has the potential to decode a plethora of droplet information from a passing laser beam, by transforming the information into a structured light field. Here, we explore this potential by analyzing the three-dimensional intensity structures sculpted by the droplet lenses, revealing the transfer of the characteristics of the underlying liquid-repelling effect onto the light field. As illustrative complementary examples, we study droplet lenses formed on a non-wetting Taro (Colocasia esculenta) leaf surface and by the Leidenfrost effect on a heated plate. Our approach may reveal even typically "invisible" droplet properties as the refractive index or internal flow dynamics and, hence, will be of interest to augment conventional tools for droplet and surface investigation.

Planar rainbow refractometry

Rainbow refractometry has been used in the past to measure size and refractive index of spherical particles, typically droplets in a spray. In the present study, conventional optical configurations for point measurements or line measurements have been extended to allow also the particle position in a plane to be determined, and hence, the designation planar rainbow refractometry. However, this extension introduces challenges in accurately calibrating the 2D scattering angles with the image coordinates. This challenge has been met using a novel calibration method, employing a monodispersed droplet stream traversed through the measurement plane. Experiments confirm achievable horizontal and vertical position accuracies of 0.42 mm and 0.36 mm, respectively, and a refractive index uncertainty of 2×10^-4.

Generalized rainbow patterns of oblate drops simulated by a ray model in three dimensions

The scattering patterns near the primary rainbow of oblate drops are simulated by extending the vectorial complex ray model (VCRM) [Opt. Lett.36, 370 (2011)OPLEDP0146-959210.1364/OL.36.000370] to three-dimensional (3D) calculations. With the curvature of a wavefront as an intrinsic property of a ray, this advanced ray model permits, in principle, to predict the amplitudes and phases of all emergent rays with a rigorous algebraic formalism. This Letter reports a breakthrough of VCRM for 3D scattering with a line-by-line triangulation interpolation algorithm allowing to calculate the total complex amplitude of a scattered field. This makes possible to simulate not only the skeleton (geometrical rainbow angles, hyperbolic-umbilic caustics), but also the coarse (Airy bows, lattice) and fine (ripple fringes) structures of the generalized rainbow patterns (GRPs) of oblate drops. The simulated results are found qualitatively and quantitatively in good agreement with experimental scattering patterns for drops of different aspect ratios. The physical interpretation of the GRPs is also given. This work opens up prominent perspectives for simulating and understanding the 3D scattering of large particles of any shape with a smooth surface by VCRM.

Remote sensing of raindrop size distribution using the coherent Doppler lidar

The coherent Doppler wind lidar (CDL) shows capability in precipitation detection. Retrieval of the raindrop size distribution (DSD) using CDL is still challenging work, as both accurate backscattering cross section at the working wavelength and reflectivity spectrum of raindrop are required. Firstly, the Mie theory and the vectorial complex ray model (VCRM) are applied to calculate backscattering cross section for small spheric raindrops and large oblate raindrops, respectively. Secondly, an iterative deconvolution method is proposed to separate the reflectivity spectrum of raindrop from the lidar power spectrum, which is a superposition of rain and aerosol components. An accompanying aerosol signal model considering the effect of temporal window, from the same height and time, is used to improve the accuracy and robustness of the iteration. In experiment, a co-located micro rain radar (MRR) is used for comparison. Good agreements are obtained despite tremendous differences in wavelength and scattering characteristics. As an example, at 600 m height, the R² of linear fitting to the mean rain velocity and mean raindrop diameter between CDL and MRR are 0.96 and 0.93, respectively.

Analysis of the LIF/Mie Ratio from Individual Droplets for Planar Droplet Sizing: Application to Gasoline Fuels and Their Mixtures with Ethanol

In this work, the possibility of using planar droplet sizing (PDS) based on laser-induced fluorescence (LIF) and Mie scattering was investigated within the framework of measuring the droplet Sauter mean diameter (SMD) of direct-injection spark-ignition (DISI) spray systems. For this purpose, LIF and Mie signals of monodisperse fuel droplets produced by a droplet generator were studied at engine relevant diameters (20–50 µm). The surrogate gasoline fuel Toliso (consisting of 65 vol. % isooctane, 35 vol. % toluene) and the biofuel blend E20 (consisting of 80 vol. % Toliso, 20 vol. % ethanol) were used and which were doped with the fluorescence dye “nile red”. The effects of ethanol admixture, dye concentration, laser power, and temperature variation on the LIF/Mie ratio were studied simultaneously at both macroscopic and microscopic scale. The deduced calibration curves of the LIF and Mie signals of both fuels showed volumetric and surface dependent behaviors, respectively, in accordance with the assumptions in the literature. The existence of glare points and morphology-dependent resonances (MDRs) lead to slightly higher LIF and Mie exponents of E20 in comparison to Toliso. In principle, these calibration curves enable the determination of the SMD from LIF/Mie ratio images of transient fuel sprays.

Droplet sizing and mixture fraction measurement in liquid-liquid flows with rainbow-angle diffractometry

The capabilities and resolution of the rainbow technique were extended to estimate the size distribution and composition of droplets in liquid-liquid systems. For these droplets, essentially characterized by a low relative refractive index (m≈1.001-1.20), the first-order rainbow is localized in the near-forward to sideways region. It exhibits an unusually higher contrast in the parallel polarization due to the vicinity of the rainbow and the Brewster angles. A numerical study revealed that a few thousand to ten thousand droplets were necessary to obtain reliable estimations of the first moments of typical droplet size distributions when the diffractometer is operated as an ensemble averaging technique. The importance of the accuracy of the light scattering model and the inverse methods used are also documented. Experimental results performed on free-rising submillimeter to millimeter droplets of various compositions showed that a global resolution of 1% to 5% of their mean diameter and about 1.6×10^-4 of the dispersion on their refractive index (i.e., 3% in the mixture fraction of oily droplets in water) could be achieved, which enhances the perspectives on mixing and extraction studies in liquid-liquid systems.

Simulation of optical caustics associated with the tertiary rainbow of oblate droplets

In this paper, a vector ray tracing (VRT) model is used to simulate optical caustic structures, including rainbow and hyperbolic umbilic (HU) fringes, in the tertiary rainbow region of light scattering from oblate spheroidal droplets. In order to apply the optical caustic structures to particle diagnostics, the evolution of rainbow and HU fringes with an increase in the aspect ratio of oblate spheroidal droplets is investigated in detail, and the curvature of rainbow fringes are calculated. Next, on the basis of the VRT model, the location of cusp caustics is calculated and compared with theoretical prediction.

Catastrophe optics of sharp-edge diffraction

A classical problem of diffraction theory, namely plane wave diffraction by sharp-edge apertures, is here reformulated from the viewpoint of the fairly new subject of catastrophe optics. On using purely geometrical arguments, properly embedded into a wave optics context, uniform analytical estimates of the diffracted wavefield at points close to fold caustics are obtained, within paraxial approximation, in terms of the Airy function and its first derivative. Diffraction from parabolic apertures is proposed to test reliability and accuracy of our theoretical predictions.