Light scattering by a spheroidal bubble with geometrical optics approximation

Abstract

This paper proposes the spheroidal model for analyzing the light scattering characteristics of an air bubble. The angular distributions of light scattered by a large spheroidal bubble with end-on incidence are calculated using geometrical optics approximation. The divergence factor, diffraction, and phase shift are considered in the computation. The MATLAB code was developed and verified using the Mie result for a spherical bubble, and the scattering patterns of the two methods agreed well. The effects on the scattering properties are analyzed in terms of the size and shape parameter of the bubble and the incident beam width. The relations between the deviation angle and incident angle, emergent light intensity, and scattering angle are analyzed and used to explain the scattering patterns of a spheroidal bubble.

Introduction

The light intensity scattered by bubbles underwater has been widely researched over the past decades. Applications of their scattering properties in non-intrusive optical instruments have been invented and developed in various fields of science and engineering as diagnostic tools. The Mie theory gives the exact solution for light scattering by a homogeneous spherical particle. However, geometrical optics approximation (GOA) is often preferred because it offers a clear interpretation of the scattering mechanisms, as well as much better computational efficiency for large particles (i.e., the size of the particle is much larger than the wavelength of the light).

Many authors have contributed to the development of the GOA of light scattered by bubbles [1], [2], [3], [4], [5], [6], [7], [8]. In the previous studies, the spherical model was used to approximate the bubble shape to analyze light scattering. However, the bubble shape is more like a spheroid rather than a sphere in some special circumstances, for example, turbulent bubbly flow [9], [10], [11], [12]. The spherical model is not suitable for describing the scattering characteristics of these bubbles. Therefore, a further study on the light scattering of a spheroidal bubble needs to be developed.

The rigorous theory of plane-wave scattering by a spheroidal particle was derived by solving Maxwell's equation, for example, the separation of variables [13], [14], [15], [16] and T-matrix [17], [18], [19], [20]. However, they failed to describe the scattering light intensity with emergent rays experiencing an arbitrary number of internal reflections. Debye series [21] succeeds in this problem. All of the above methods lack computational efficiency for large particles. The ray-tracing method [22], [23], [24], which targets larger particles, consumes less time to evaluate scattering properties with random orientations. However, most of the reports focus on the particle of which relative refractive index is larger than 1.

In this work, the angular distributions of the light scattered by a large spheroid bubble in water, which has a relative refractive index smaller than 1, were calculated using GOA. The scattering light is obtained as a superposition of reflected, refracted, and diffracted components with end-on incidence. In this instance, the incident field and bubble share the same symmetric axis. Thus, the calculation can be simplified. The two polarizations remain separate, and no cross-polarization effects exist [25], [26]. The divergence factor, phase shift, and amplitude of the emergent rays experiencing an arbitrary number of internal reflections can be evaluated in the two-dimensional domain. The decomposition of scattering pattern in a series of surface interactions can be clearly illustrated in the scattering plane.

This paper is organized as follows: the description of the GOA method is presented in Section 2, the numerical results are given in Section 3, including the verification of the spherical bubble using the Mie theory and the comparison between the scattering patterns of the spherical and spheroidal models and the conclusions are given in Section 4.