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

Lorenz-Mie theory-type solution for light scattering by spheroids with small-to-large size parameters and aspect ratios

There has been a long-term endeavor in the light-scattering research community to develop a Lorenz-Mie theory-type method for simulating light scattering by spheroidal particles with small-to-large sizes. A spheroid is a very important nonspherical shape in modeling the optical properties of many natural particles. For the first time, we develop a computationally feasible separation of variables method (SVM) in spheroidal coordinates to compute optical properties of spheroids with small-to-large sizes compared to the wavelength of the incident light (λ). The method is applicable to spheroids with size parameters (2π/λ times the major semiaxis) up to at least 600, and is not restricted by particle aspect ratios. Therefore, the work reported here represents a breakthrough in solving the optical properties of a nonspherical particle in an analytical form.

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.

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.

Identify the limits of geometric optics ray tracing by numerically solving the vector Kirchhoff integral

The properties of a pencil of light as defined approximately in the geometric optics ray tracing method are investigated. The vector Kirchhoff integral is utilized to accurately compute the electromagnetic near field in and around the pencil of light with various beam base sizes, shapes, propagation directions and medium refractive indices. If a pencil of light has geometric mean cross section size of the order p times the wavelength, it can propagate independently to a distance p² times the wavelength, where most of the beam energy diffuses out of the beam region. This is consistent with a statement that van de Hulst made in a classical text on light scattering. The electromagnetic near fields in the pencil of light are not uniform, have complicated patterns within short distances from the beam base, and the fields tend to converge to Fraunhofer diffraction fields far away from the base.

Rainbows by elliptically deformed drops. I. Möbius shift for high-order rainbows

Using ray theory, the Möbius shift of the (p-1)-order rainbow angle for a particle having an elliptical cross section is obtained to first order in the ellipticity as a function of the tilt of the ellipse with respect to the propagation direction of the incoming rays. The result is then adapted to the geometry of scattering of light rays from the sun by a falling water drop as a function of sun height angle. The variation in the angular spacing between the supernumeraries is determined as a function of location along the rainbow arc, the conditions under which the rainbow angle is insensitive to drop flattening were determined, and the dependence of the Möbius shift on the drop refractive index is shown for rainbows up to fourth order (p=5).

Transmission bows of radially inhomogeneous spheres

We consider transmission scattering of a plane wave by a radially inhomogeneous sphere containing a localized region of refractive index decrease. In ray theory, the boundary conditions on the deflection angle at axial and grazing incidence determine that transmission scattering gives rise to an even number of bows, half of them being relative maximum bows and half being relative minimum bows. For a model refractive index profile, we determine the conditions under which different numbers of bows occur, and we suggest physical mechanisms responsible for producing them. We also verify that these bows occur in wave scattering in the short wavelength limit, both in the frequency domain and time domain.

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.

Near-critical-angle scattering for the characterization of clouds of bubbles: particular effects

We report experimental investigations on the influence of various optical effects on the far-field scattering pattern produced by a cloud of optical bubbles near the critical scattering angle. Among the effects considered, there is the change of the relative refractive index of the bubbles (gas bubbles or some liquid-liquid droplets), the influence of intensity gradients induced by the laser beam intensity profile and by the spatial filtering of the collection optics, the coherent and multiple scattering effects occurring for densely packed bubbles, and the tilt angle of spheroidal optical bubbles. The results obtained herein are thought to be fundamental for the development of future works to model these effects and for the extension of the range of applicability of an inverse technique (referenced herein as the critical angle refractometry and sizing technique), which is used to determine the size distribution and composition of bubbly flows.

Realistic 3D coherent transfer function inverse filtering of complex fields

We present a novel technique for three-dimensional (3D) image processing of complex fields. It consists in inverting the coherent image formation by filtering the complex spectrum with a realistic 3D coherent transfer function (CTF) of a high-NA digital holographic microscope. By combining scattering theory and signal processing, the method is demonstrated to yield the reconstruction of a scattering object field. Experimental reconstructions in phase and amplitude are presented under non-design imaging conditions. The suggested technique is best suited for an implementation in high-resolution diffraction tomography based on sample or illumination rotation.

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.

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

On the basis of our previous work on the extension of the geometrical-optics approximation to Gaussian beam scattering by a spherical particle, we present a further extension of the method to the scattering of a transparent or absorbing spheroidal particle with the same symmetric axis as the incident beam. As was done for the spherical particle, the phase shifts of the emerging rays due to focal lines, optical path, and total reflection are carefully considered. The angular position of the geometric rainbow of primary order is theoretically predicted. Compared with our results, the Möbius prediction of the rainbow angle has a discrepancy of less than 0.5 degrees for a spheroidal droplet of aspect radio kappa within 0.95 and 1.05 and less than 2 degrees for kappa within 0.89 and 1.11. The flux ratio index F, which qualitatively indicates the effect of a surface wave, is also studied and found to be dependent on the size, refractive index, and surface curvature of the particle.

Model for optical forward scattering by nonspherical raindrops

We describe a numerical model for the interaction of light with large raindrops using realistic nonspherical drop shapes. We apply geometrical optics and a Monte Carlo technique to perform ray traces through the drops. We solve the problem of diffraction independently by approximating the drops with area-equivalent ellipsoids. Scattering patterns are obtained for different polarizations of the incident light. They exhibit varying degrees of asymmetry and depolarization that can be linked to the distortion and thus the size of the drops. The model is extended to give a simplified long-path integration.

Semiclassical theory to optical resonant modes of a transparent dielectric spheroidal cavity

We study the resonant scattering of light by a transparent dielectric spheroid. We try to understand the features of the resonant modes of a spheroidal optical cavity. In this way, we use an analogy between optics and quantum mechanics. Through this analogy it is possible to interpret resonances as quasibound states of light. Using semiclassical methods such as the WKB method and a uniform asymptotic expansion for spheroidal radial functions, we developed algorithms that permit us to calculate the resonance position as well as the resonance width.

Geometrical optics calculation of inelastic scattering on large particles

A geometrical optics approximation was used for calculations of inelastic (Raman and fluorescent) scattering on particles with large size parameters. The inelastic part of the radiation was obtained by use of the principle of ray reversibility. The technique presented simplifies the computations and provides a geometric interpretation of how far-field patterns can be calculated by use of the internal field distributions. The numerical results for homogeneous spherical particles are compared with the classic dipole solution.

Ray scattering by an arbitrarily oriented spheroid. I. Diffraction and specular reflection

Diffraction and reflection of an arbitrarily polarized plane wave by an arbitrarily oriented spheroid in the short-wavelength limit are considered in the context of ray theory. A closed-form solution for both diffraction and reflection is obtained, and the polarization character of the diffracted plus reflected electric field is obtained. It is found that the magnitude of the reflected electric field is multivalued for forward scattering. This is interpreted in terms of the variation of the spheroid's Gaussian curvature at the points where grazing ray incidence occurs.