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.     

Scattering of a spheroidal particle illuminated by a gaussian beam

An approach to expanding a Gaussian beam in terms of the spheroidal wave functions in spheroidal coordinates is presented. The beam-shape coefficients of the Gaussian beam in spheroidal coordinates can be computed conveniently by use of the known expression for beam-shape coefficients, g(n), in spherical coordinates. The unknown expansion coefficients of scattered and internal electromagnetic fields are determined by a system of equations derived from the boundary conditions for continuity of the tangential components of the electric and magnetic vectors across the surface of the spheroid. A solution to the problem of scattering of a Gaussian beam by a homogeneous prolate (or oblate) spheroidal particle is obtained. The numerical values of the expansion coefficients and the scattered intensity distribution for incidence of an on-axis Gaussian beam are given.     

Can simple particle shapes be used to model scalar optical properties of an ensemble of wavelength-sized particles with complex shapes?

We compute the scalar optical properties of size-shape distributions of wavelength-sized randomly oriented homogeneous particles with different nonaxially symmetric geometries and investigate how well they can be modeled with a simple spherical, spheroidal, or cylindrical particle model. We find that a spherical particle model can be used to determine the extinction and scattering cross sections, the single-scattering albedo, and the asymmetry parameter with an error of less than 2%, whereas the extinction-to-backscatter ratio Reb is reproduced only with an error of 9%. The cylindrical and spheroidal particle models yield slightly improved results for Reb that deviate from those obtained for the complex particle ensemble by 7% and 5%, respectively. Large discrepancies between results of the different models are observed for the linear depolarization ratio, thus indicating limitations of models based on simple particle shapes.     

Expansion of an arbitrarily oriented, located, and shaped beam in spheroidal coordinates

Within the framework of the generalized Lorenz-Mie theory (GLMT), the incident shaped beam of an arbitrary orientation and location is expanded in terms of the spheroidal vector wave functions in given spheroidal coordinates. The beam shape coefficients (BSCs) in spheroidal coordinates are computed by the quadrature method. The classical localization approximation method for BSC evaluation is found to be inapplicable when the Cartesian coordinates of the beam and the particle are not parallel to each other. Once they are parallel, all the symmetry relationships existing for the BSCs in spherical coordinates (spherical BSCs) [J. Opt. Soc. Am. A11, 1812 (1994)] still pertain to the BSCs in spheroidal coordinates (spheroidal BSCs). In addition, the spheroidal BSCs computed by our method are verified by comparing them with those evaluated by Asano and Yamamoto for plane wave incidence [Appl. Opt.14, 29 (1975)]. Furthermore, formulas are given for field reconstruction from the spheroidal BSCs, and consistency is found between the original incident fields and the reconstructed ones.     

亚微米级扁椭球颗粒群的散射特性

采用T矩阵方法计算亚微米级扁椭球随机取向分布颗粒群的散射特性,研究消光截面、散射截面、吸光截面、单散射反照率、非对称因子以及散射矩阵元素与颗粒的大小、折射率、长短轴比之间的关系。结果表明,随颗粒粒径增大,消光截面、散射截面、吸光截面、非对称因子都单调增加,散射相函数F11的角分布曲线特征可以区分颗粒的大小;颗粒越偏离球形,颗粒对入射光的衰减效率越低,后向散射光强越强,在轴比不大时,前向50°内的F22/F11值可以区分颗粒的形状;折射率变化主要是对后向散射光的分布产生影响,实部、虚部的变化可分别通过F34/F11的角分布曲线、F12/F11的第一个峰值来体现。     

Application of Mie theory to assess structure of spheroidal scattering in backscattering geometries

Inverse light scattering analysis seeks to associate measured scattering properties with the most probable theoretical scattering distribution. Although Mie theory is a spherical scattering model, it has been used successfully for discerning the geometry of spheroidal scatterers. The goal of this study was an in-depth evaluation of the consequences of analyzing the structure of spheroidal geometries, which are relevant to cell and tissue studies in biology, by employing Mie-theory-based inverse light scattering analysis. As a basis for this study, the scattering from spheroidal geometries was modeled using T-matrix theory and used as test data. In a previous study, we used this technique to investigate the case of spheroidal scatterers aligned with the optical axis. In the present study, we look at a broader scope which includes the effects of aspect ratio, orientation, refractive index, and incident light polarization. Over this wide range of parameters, our results indicate that this method provides a good estimate of spheroidal structure.     

Electromagnetic field calculations for a sphere illuminated by a higher-order Gaussian beam. II. Far-field scattering

A previously developed theoretical procedure for determination of electromagnetic fields associated with the interaction of a higher-order Gaussian beam with a homogeneous spherical particle is used to investigate the effects of incident beam type on far-field scattering. Far-field scattering patterns are calculated for (0,0), (0,1), and (1,1) mode Hermite-Gaussian beams and for the helix doughnut mode beam. The effects of incident beam type on the angular distribution of far-field scattering, for both on-sphere-center and off-sphere-center focusing, are examined.     

Scattering of electromagnetic waves by spheroidal particles: a novel approach exploiting the T matrix computed in spheroidal coordinates

A method other than the extended-boundary-condition method (EBCM) to compute the T matrix for electromagnetic scattering is presented. The separation-of-variables method (SVM) is used to solve the electromagnetic scattering problem for a spheroidal particle and to derive its T matrix in spheroidal coordinates. A transformation is developed for transforming the T matrix in spheroidal coordinates into the corresponding T matrix in spherical coordinates. The T matrix so obtained can be used for analytical calculation of the optical properties of ensembles of randomly oriented spheroids of arbitrary shape by use of an existing method to average over orientational angles. The optical properties obtained with the SVM and the EBCM are compared for different test cases. For mildly aspherical particles the two methods yield indistinguishable results. Small differences appear for highly aspherical particles. The new approach can be used to compute optical properties for arbitrary values of the aspect ratio. To test the accuracy of the expansion coefficients of the spheroidal functions for arbitrary arguments, a new testing method based on the completeness relation of the spheroidal functions is developed.     

Geometrical-optics approximation of forward scattering by gradient-index spheres

By means of geometrical optics we present an approximation method for acceleration of the computation of the scattering intensity distribution within a forward angular range (0-60 degrees ) for gradient-index spheres illuminated by a plane wave. The incident angle of reflected light is determined by the scattering angle, thus improving the approximation accuracy. The scattering angle and the optical path length are numerically integrated by a general-purpose integrator. With some special index models, the scattering angle and the optical path length can be expressed by a unique function and the calculation is faster. This method is proved effective for transparent particles with size parameters greater than 50. It fails to give good approximation results at scattering angles whose refractive rays are in the backward direction. For different index models, the geometrical-optics approximation is effective only for forward angles, typically those less than 60 degrees or when the refractive-index difference of a particle is less than a certain value.     

Forward scattering of a Gaussian beam by a nonabsorbing sphere

The forward scattering of a Gaussian laser beam by a spherical particle located along the beam axis is analyzed with the generalized Lorenz-Mie theory (GLMT) and with diffraction theory. Forwardscattering and near-forward-scattering profiles from electrodynamically levitated droplets, 51.6 μm in diameter, are also presented and compared with GLMT-based predictions. The total intensity in the forward direction, formed by the superposition of the incident and the scattered fields, is found to correlate with the particle-extinction cross section, the particle diameter, and the beam width. Based on comparison with the GLMT, the diffraction solution is accurate when beam widths that are approximately greater than or equal to the particle diameter are considered and when large particles that have an extinction efficiency near the asymptotic value of 2 are considered. However, diffraction fails to describe the forward intensity for more tightly focused beams. The experimental observations, which are in good agreement with GLMT-based predictions, reveal that the total intensity profile about the forward direction is quite sensitive to particle axial position within a Gaussian beam. These finite beam effects are significant when the ratio of the beam to the particle diameter is less than approximately 5:1. For larger beam-to-particle-diameter ratios, the total field in the forward direction is dominated by the incident beam.     

焦区附近球形粒子对高斯光束的弹性散射

本文应用复源球面波理论,将高斯光束场按矢量球波函数展开,对高斯光束入射到单个球形粒子上时的弹性散射问题进行了理论分折。并对球形粒子在波束传播轴上时的远场散射光强角分布进行了数值计算,同时还与平面波散射的结果进行了比较。     

Scattering of a transversely confined Neumann beam by a spherical particle

Various properties of an electromagnetic wave whose spherical multipole expansion contains only Riccati-Neumann functions are examined. In particular, the novel behavior of the beam phase during diffractive spreading is discussed. When a Neumann beam is scattered by a spherical particle, the diffraction and external reflection portions of the scattering amplitude constructively interfere for large partial waves. As a result, a set of rapidly decreasing beam shape coefficients is required to cut off the partial wave sum in the scattering amplitudes. Because of its strong singularity at the origin, a Neumann beam can be produced by a point source of radiation at the center of a spherical cavity in a high conductivity metal, and Neumann beam scattering by a spherical particle can occur for certain partial waves if the sphere is placed at the center of the cavity as well.     

Far-field scattering of a non-Gaussian off-axis axisymmetric laser beam by a spherical particle

Experimental laser beam profiles often deviate somewhat from the ideal Gaussian shape of the axisymmetric TEM(00) laser mode. To take these deviations into account when calculating light scattering of an off-axis beam by a spherical particle, we use our phase-modeling method to approximate the beam-shape coefficients in the partial wave expansion of an experimental laser beam. We then use these beam-shape coefficients to compute the near-forward direction scattering of the off-axis beam by the particle. Our results are compared with laboratory data, and we give a physical interpretation of the various features observed in the angular scattering patterns.     

Geometrical-optics approximation of forward scattering by coated particles

By means of geometrical optics we present an approximation algorithm with which to accelerate the computation of scattering intensity distribution within a forward angular range (0 degrees-60 degrees) for coated particles illuminated by a collimated incident beam. Phases of emerging rays are exactly calculated to improve the approximation precision. This method proves effective for transparent and tiny absorbent particles with size parameters larger than 75 but fails to give good approximation results at scattering angles at which refractive rays are absent. When the absorption coefficient of a particle is greater than 0.01, the geometrical optics approximation is effective only for forward small angles, typically less than 10 degrees or so.     

Far-field scattering of an axisymmetric laser beam of arbitrary profile by an on-axis spherical particle

Experimental laser beam profiles often deviate somewhat from the ideal Gaussian shape of the TEM(00) laser mode. In order to take these deviations into account when calculating light scattering, we propose a method for approximating the beam shape coefficients in the partial wave expansion of an experimental laser beam. We then compute scattering by a single dielectric spherical particle placed on the beam's axis using this method and compare our results to laboratory data. Our model calculations fit the laboratory data well.     

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.     

The analytic solution for hydrodynamic diffraction by submerged prolate spheroids in infinite water depth

This study investigates the linear hydrodynamic scattering problem by stationary prolate spheroidal bodies and aims at providing an analytic solution for the associated boundary value problem. It extends the work of the present author on the hydrodynamics of oblate spheroidal bodies following the same procedure. The structural model under consideration is a spheroid with its polar axis greater than its equatorial diameter, subjected to the action of monochromatic incident waves. The polar axis is assumed to be perpendicular to the free surface that leads to the axisymmetric case concept. The analytic solution is sought using the method of multipole expansions constructed by employing Thorne’s formulas (Multipole expansions in the theory of surface waves. Proc Cam Philos Soc 49:707–716, 1953) that describe the velocity potential at singular points within a fluid domain with free upper surface and infinite water depth. The final stage of the solution process is the application of the zero velocity condition on the wetted surface of the spheroid. Inevitably this task requires the transformation of the involved velocity potentials, originally expressed with respect to spherical and polar coordinates, into prolate spheroidal coordinates. To this end, the appropriate addition theorems are derived, which recast Thorne’s expressions into infinite series of associated Legendre functions.     

Wall effects on a spherical particle

The motion of a solid spherical particle at the instant it passes the centre of a spheroidal container has been investigated using the no-slip condition at the surfaces. The shape of this spheroidal container is assumed to deviate slightly from that of a sphere at the surface. An expression for the stream function describing the flow field has been obtained to the first order in the parameter characterizing the deviation and this in turn has been utilized to evaluate the drag on the particle. Wall effects are then examined and special cases deduced.     

Scattering of a spherical acoustic field from an eccentric spheroidal structure simulating the kidney-stone system

Summary.  We consider the acoustic scattering of time-harmonic spherical waves from an eccentric non coaxial spheroidal structure simulating the kidney-stone system. The proposed analysis is based on the application of the translational addition theorem for spheroidal wave functions. The resulting theoretical model is frequency-independent. Numerical results concerning the applicability of our approach are also presented. Received September 20, 2002 Published online: March 20, 2003     

Debye series for Gaussian beam scattering by a multilayered sphere

The Debye series has been a key tool for the understanding of light scattering features, and it is also a convenient method for understanding and improving the design of optical instruments aimed at optical particle sizing. Gouesbet has derived the Debye series formulation for generalized Lorenz-Mie theory (GLMT). However, the scattering object is a homogeneous sphere, and no numerical result is provided. The Debye series formula for plane-wave scattering by multilayered spheres has been derived before. We have devoted our work to the Debye series of Gaussian beam scattering by multilayered spheres. The integral localized approximation is employed in the calculation of beam-shape coefficients (BSCs) and allows the study of the scattering characteristics of particles illuminated by the strongly focused beams. The formula and code are verified by the comparison with the results produced by GLMT and also by the comparison with the result for the case of plane-wave incidence. The formula is also employed in the simulation of the first rainbow by illuminating the particle with one or several narrow beams.