Extension of geometrical-optics approximation to on-axis Gaussian beam scattering. II. By a spheroidal particle with end-on incidence

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.

Möbius shifts associated with the third-order and the fourth-order rainbows of a spheroidal droplet computation

Vector-ray tracing (VRT) is employed to calculate Möbius shifts of the third-order and the fourth-order rainbows for a spheroidal droplet. When the aspect ratio of a spheroidal droplet is small, approximation expressions for calculating the Möbius shift (i.e., deviation of the geometrical rainbow angle for a spheroidal droplet and that for a spherical droplet) were given by Lock and Können [Appl. Opt.56, G88 (2017)APOPAI0003-693510.1364/AO.56.000G88]. The assessment of applicability ranges of the Lock approximation is obtained by comparing with a VRT simulation for a water droplet with the refractive index m=1.333. For this, a parameter ΔD is defined that measures the disagreement between the two methods. A threshold value of 5% for ΔD is chosen below which the agreement is considered to be good. For the third-order rainbow, it is shown that this is the case for the Lock approximation for water droplets (m=1.333) with aspect ratios in the range of 0.97≤a/c≤1.03. For the fourth-order rainbow, the application range of the Lock approximation is 0.99≤a/c≤1.01 for water droplets. For the first-order and second-order rainbows, the application ranges are briefly revisited with the current method. The influence of the droplet refractive index on the Möbius shift is also investigated.

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.

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.

Application of vector ray tracing to the computation of Möbius shifts for the primary and secondary rainbows

The Möbius approximation for the primary rainbow and the Können approximation for the secondary rainbow have been modified to yield consistent predictions of the Möbius shift of the top and bottom rainbows, respectively. The applicability ranges of the Möbius and Können approximations are investigated by comparison to vector ray tracing (VRT) simulations. For the primary rainbow, these results indicate that the Möbius approximation is valid for spheroidal water droplets (m=1.333) in the range of aspect ratios 0.98≤a/c≤1.02. For the secondary rainbow, the Können approximation predicts the Möbius shift well for spheroidal water droplets within the range 0.99≤a/c≤1.01. For a spheroidal droplet with side-on incidence, the difference between the approximations and VRT simulations are discussed. Furthermore, the dependence of Möbius shifts on the relative refractive index of droplet is discussed.

Measuring the light scattering and orientation of a spheroidal particle using in-line holography

The light scattering properties of a horizontally and vertically oriented spheroidal particle under laser illumination are experimentally investigated using digital in-line holography. The reconstructed wave field shows the bright singular points as a result of the condensed beam formed by a transparent spheroidal particle acting as a lens. The in-plane (θ) and out-of-plane (ϕ) rotating angles of an arbitrarily oriented spheroidal particle are measured by using these scattering properties. As a feasibility test, the 3D orientation of a transparent spheroidal particle suspended in a microscale pipe flow is successfully reconstructed by adapting the proposed method.

High-frequency extinction efficiencies of spheroids: rigorous T-matrix solutions and semi-empirical approximations

A semi-empirical high-frequency formula is developed to efficiently and accurately compute the extinction efficiencies of spheroids in the cases of moderate and large size parameters under either fixed or random orientation condition. The formula incorporates the semi-classical scattering concepts formulated by extending the complex angular momentum approximation of the Lorenz-Mie theory to the spheroid case on the basis of the physical rationales associated with changing the particle morphology from a sphere to a spheroid. The asymptotic edge-effect expansion is truncated with an optimal number of terms based on a priori knowledge obtained from comparing the semi-classical Mie extinction efficiencies with the Lorenz-Mie solutions. The present formula is fully tested in comparison with the T-matrix results for spheroids with the aspect ratios from 0.5 to 2.0, and for various refractive indices m(r) + im(i), with m(r) from 1.0 to 2.0 and m(i) from 0 to 0.5.

Scattering of a Gaussian beam by an elliptical cylinder using the vectorial complex ray model

The scattered waves of a shaped beam by an infinite cylinder in the far field are, stricto sensu, neither cylindrical nor spherical, so the asymptotic form of special functions involved in the theories based on the rigorous solution of Maxwell equations cannot be used to evaluate scattered intensities, even in the most simple case of Gaussian beam scattering by an infinite circular cylinder. Thus, although theories exist for the scattering of a shaped beam by infinite cylinders with circular and elliptical sections, the numerical calculations are limited to the near field. The vectorial complex ray model (VCRM) developed by Ren et al. describes waves by rays with a new property: the curvature of the wavefront. It is suitable to deal with the scattering of an arbitrarily shaped beam by a particle with a smooth surface of any form. In this paper, we apply this method to the scattering of an infinite elliptical cylinder illuminated by a Gaussian beam at normal incidence with an arbitrary position and orientation relative to the symmetric axis of the elliptical section of the cylinder. The method for calculating the curvature of an arbitrary surface is given and applied in the determination of the two curvature radii of the Gaussian beam wavefront at any point. Scattered intensities for different parameters of the beam and the particle as well as observation distance are presented to reveal the scattering properties and new phenomena observed in the beam scattering by an infinite elliptical cylinder.

Vectorial complex ray model and application to two-dimensional scattering of plane wave by a spheroidal particle

A vectorial complex ray model is introduced to describe the scattering of a smooth surface object of arbitrary shape. In this model, all waves are considered as vectorial complex rays of four parameters: amplitude, phase, direction of propagation, and polarization. The ray direction and the wave divergence/convergence after each interaction of the wave with a dioptric surface as well as the phase shifts of each ray are determined by the vector Snell law and the wavefront equation according to the curvatures of the surfaces. The total scattered field is the superposition of the complex amplitude of all orders of the rays emergent from the object. Thanks to the simple representation of the wave, this model is very suitable for the description of the interaction of an arbitrary wave with an object of smooth surface and complex shape. The application of the model to two-dimensional scattering of a plane wave by a spheroid particle is presented as a demonstration.

Debye series for light scattering by a spheroid

The Debye series is developed for electromagnetic scattering by a spheroid in order to decompose the far-zone fields into various physical processes. The geometrical rainbow angle and supernumerary spacing parameter are determined from the Debye intensity by fitting the results to an Airy function and comparing them to their assumed values in ray optics and Airy theory, respectively. Eccentricity-related scattering phenomena including the rainbow's angular shift, the disappearance of the rainbow, and the rainbow-enhanced glory are quantitatively demonstrated and analyzed.

Critical angle refractometry and sizing of bubble clouds

The principle of the critical angle refractometry and sizing technique is extended to characterize the size distribution and the mean refractive index of clouds of bubbles. For a log-normal bubble-size distribution, simulations show that the mean size, the relative width of the size distribution, and the mean refractive index of the bubbles have a particular and easily identified influence on the critical scattering patterns. Preliminary experimental results on air bubble/water flows clearly demonstrate the potential and robustness of this new technique for bubbly flow characterization.

Extension of geometrical-optics approximation to on-axis Gaussian beam scattering. I. By a spherical particle

The geometrical-optics approximation of light scattering by a transparent or absorbing spherical particle is extended from plane wave to Gaussian beam incidence. The formulas for the calculation of the phase of each ray and the divergence factor are revised, and the interference of all the emerging rays is taken into account. The extended geometrical-optics approximation (EGOA) permits one to calculate the scattering diagram in all directions from 0 degrees to 180 degrees. The intensities of the scattered field calculated by the EGOA are compared with those calculated by the generalized Lorenz-Mie theory, and good agreement is found. The surface wave effect in Gaussian beam scattering is also qualitatively analyzed by introducing a flux ratio factor. The approach proposed is particularly important to the further extension of the geometrical-optics approximation to the scattering of large spheroidal particles.