E(6) diffraction catastrophe of the primary rainbow of oblate water drops: observations with white-light and laser illumination

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.

Simulation of the optical caustics associated with the primary rainbow for oblate spheroidal drops illuminated by a Gaussian beam

A vector ray-tracing model (VRT) has been developed to compute the optical caustics associated with the primary rainbow for an oblate spheroidal water drop illuminated by a Gaussian beam. By comparing the optical caustic structures (in terms of limiting rainbow and hyperbolic umbilic fringes) for a water drop with a Gaussian beam (GB) illumination with that for the same drop, but with parallel beam (PB) illumination, the influence of the Gaussian beam on the optical caustics is investigated. For a water drop with GB illumination and different drop/beam ratios (i.e., the ratio between the drop equatorial radius and the Gaussian beam waist), the location of cusp points and the curvature of the limiting rainbow fringe are also studied. We anticipate that these results not only confirm the approach to compute optical caustics for oblate spheroidal drops illuminated by a shaped beam, but may also lead to a new method for measuring the aspect ratio of spheroidal drops.

Model for computing optical caustic partitions for the primary rainbow from tilted spheriodal drops

A model is proposed to compute the salient optical caustic partitions occurring in the primary rainbow for oblate spheroidal drops. By computing the boundary limits of outgoing rays, the optical caustic structures (termed rainbow and hyperbolic umbilic fringes) for tilted drops are calculated and compared with those for aligned (untilted) drops. The curvature of the rainbow fringe and the shifts of cusp caustics are discussed as well. The observed properties of the caustics can potentially be used for drop measurements. The model could also be applied to compute the optical caustics for drops with arbitrary shape, arbitrary orientation, and shaped beam illumination.

Light scattering from sessile water drops and raindrop-shaped glass beads as a validation tool for rainbow simulations

The shape deviation of falling raindrops from exact spheres is known to affect the appearance of natural rainbows, e.g., by enhancing the visibility of supernumerary arcs around the top or by creating branching effects known as "twinned rainbows." To check the accuracy of numerical optical models for rainbow scattering from such nonspherical drops, two simple and low-cost experiments are presented in this paper: (1) sessile, i.e., sitting, drops on ultrahydrophobic surfaces, and (2) glass beads in the shape of falling raindrops. The experimental results are compared to polarization-resolved Monte Carlo ray-tracing simulations, with special emphasis on circular polarization, which results from total internal reflections in these nonspherical scatterers.

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.

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

We report the first experimental validation of the Vectorial Complex Ray Model (VCRM) using the scattering patterns of large oblate droplets trapped in an acoustic field. The two principal radii and refractive index of the droplets are retrieved with a minimization method that involves VCRM predictions and experimental light scattering patterns. The latter are recorded in the droplet equatorial plane between the primary rainbow region and the associated hyperbolic-umbilic diffraction catastrophe. The results demonstrate that the VCRM can predict the fine and coarse stuctures of scattering patterns with good precision, opening up perspectives for the characterization of large non-spherical particles.

Optical caustics associated with the primary rainbow of oblate droplets: simulation and application in non-sphericity measurement

A vector ray tracing (VRT) model is developed to simulate optical caustic structures in the primary rainbow region of light scattering from oblate droplets. The changes of the optical caustic structures in response to shape deformation of oblate droplets are investigated. Then the curvature calculated from the simulated rainbow fringes is compared with that from the measured rainbow fringes and good agreement is found. Furthermore, according to the generalized rainbow patterns and the relation between aspect ratio and curvature of the rainbow fringe, non-sphericities in terms of aspect ratio of an oblate water droplet is measured with high measurement accuracy.

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

A vector-ray tracing (VRT) model is employed to simulate optical caustic structures [in terms of rainbow and hyperbolic umbilic (HU) fringes] in the secondary rainbow region of light scattering from oblate droplets. The changes of the rainbow fringe and HU fringe in response to shape deformation of oblate droplets are investigated and the curvature of the rainbow fringe is calculated. Then the location of cusp caustics is calculated from VRT simulations and compared with analytic solutions.

Analysis of rainbow scattering by a chiral sphere

Based on the scattering theory of a chiral sphere, rainbow phenomenon of a chiral sphere is numerically analyzed in this paper. For chiral spheres illuminated by a linearly polarized wave, there are three first-order rainbows, with whose rainbow angles varying with the chirality parameter. The spectrum of each rainbow structure is presented and the ripple frequencies are found associated with the size and refractive indices of the chiral sphere. Only two rainbow structures remain when the chiral sphere is illuminated by a circularly polarized plane wave. Finally, the rainbows of chiral spheres with slight chirality parameters are found appearing alternately in E-plane and H-plane with the variation of the chirality.

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.

Optical caustics observed in light scattered by an oblate spheroid

The electromagnetic fields scattered when a plane wave is incident on an oblate spheroid in the side-on orientation may be calculated using a generalization of Mie theory, and the results may be decomposed in a Debye series expansion. A number of optical caustics are observed in the computed scattered intensity for the one internal reflection portion of the Debye series for scattering angles in the vicinity of the first-order rainbow, and are analyzed in terms of the rainbow, transverse cusp, and hyperbolic umbilic caustics of catastrophe optics. The specific features of these three caustics are described, as is their assembly into the global structure of the observed caustics for spheroid scattering. It is found that, for a spheroid whose radius is an order of magnitude larger than the wavelength of the incident light, the interference structure accompanying the transverse cusp and hyperbolic umbilic caustics is only partially formed.

Experimental observation of rainbow scattering by a coated cylinder: twin primary rainbows and thin-film interference

We experimentally examine the primary rainbow created by the illumination of a coated cylinder. We present a simple technique for varying the coating thickness over a wide range of values, and we see evidence for two different scattering regimes. In one, where the coating thickness is large, twin rainbows are produced. In the second, where the coating is thin enough to act as a thin film, a single rainbow is produced whose intensity varies periodically as the coating thickness varies. We find good agreement with previous theoretical predictions.

Generalized tertiary rainbow of slightly oblate drops: observations with laser illumination

The tertiary rainbow of acoustically levitated water drops was observed in the laboratory. Nontrivial caustics were observed for relatively small values of eccentricity. The angular locations of caustics were modeled with matrix methods of generalized ray tracing. Photographs of the scattering were in general agreement with models. Possible effects on the appearance of natural tertiary bow features are discussed.

Ray scattering by an arbitrarily oriented spheroid. II. Transmission and cross-polarization effects

Transmission of an arbitrarily polarized plane wave by an arbitrarily oriented spheroid in the short-wavelength limit is considered in the context of ray theory. The transmitted electric field is added to the diffracted plus reflected ray-theory electric field that was previously derived to obtain an approximation to the far-zone scattered intensity in the forward hemisphere. Two different types of cross-polarization effects are found. These are (a) a rotation of the polarization state of the transmitted rays from when they are referenced with respect to their entrance into the spheroid to when they are referenced with respect to their exit from it and (b) a rotation of the polarization state of the transmitted rays when they are referenced with respect to the polarization state of the diffracted plus reflected rays.

Hyperbolic umbilic and E(6) diffraction catastrophes associated with the secondary rainbow of oblate water drops: observations with laser illumination

Previous observations of oblate drops of water illuminated perpendicular to their axis of symmetry exhibit catastrophe patterns near the primary-rainbow scattering angle [see, e.g., Appl. Opt. 30, 3468 (1991). The present research concerns observation of diffraction catastrophes near the secondary-rainbow scattering angle under similar experimental conditions. Illumination with laser light exhibits similar caustic structures in the secondary rainbow including the hyperbolic umbilic focal section and the E(6) or symbolic umbilic focal section. The range of drop aspect ratios explored includes aspect ratios as small as those found for freely falling drops in air as well as highly flattened drops. The new features of the secondary rainbow occur for highly flattened drops.