Multiple scattering of electromagnetic waves by an aggregate of uniaxial anisotropic spheres

An exact analytical solution is obtained for the scattering of electromagnetic waves from a plane wave with arbitrary directions of propagation and polarization by an aggregate of interacting homogeneous uniaxial anisotropic spheres with parallel primary optical axes. The expansion coefficients of a plane wave with arbitrary directions of propagation and polarization, for both TM and TE modes, are derived in terms of spherical vector wave functions. The effects of the incident angle α and the polarization angle β on the radar cross sections (RCSs) of several types of collective uniaxial anisotropic spheres are numerically analyzed in detail. The characteristics of the forward and backward RCSs in relation to the incident wavelength are also numerically studied. Selected results on the forward and backward RCSs of several types of square arrays of SiO? spheres illuminated by a plane wave with different incident angles are described. The accuracy of the expansion coefficients of the incident fields is verified by comparing them with the results obtained from references when the plane wave is degenerated to a z-propagating and x- or y-polarized plane wave. The validity of the theory is also confirmed by comparing the numerical results with those provided by a CST simulation.     

Peculiarities in light scattering by spherical particles with radial anisotropy

Light scattering by a spherical particle with radial anisotropy is discussed by extending Mie theory to diffraction by an anisotropic sphere, including both the electric and the magnetic anisotropy ratio. It is shown that radial anisotropy may lead to great modifications in scattering efficiencies and field enhancement, elucidating the importance of anisotropies in the control of scattering. The capacity for nondissipating damping is demonstrated for anisotropic spheres with different signs in radial and transversal permittivities.     

Scattering of a plane wave by an anisotropic ferrite-coated conducting sphere

On the basis of the three-dimensional spherical vector wave functions in ferrite anisotropic media, and the fact that the first and second spherical vector wave functions in ferrite anisotropic media satisfy the same differential equations, the electromagnetic fields in homogeneous ferrite anisotropic media can be expressed as the addition of the first and second spherical vector wave functions in ferrite anisotropic media. Applying the continue boundary condition of the tangential component of electromagnetic fields in the interface between the ferrite anisotropic medium and free space, and the tangential electric field vanishing in the interface of the conducting sphere, the expansion coefficients of electromagnetic fields in terms of spherical vector wave function in ferrite medium and the scattering fields in free space can be derived. The theoretical analysis and numerical result show that when the radius of a conducting sphere approaches zero, the present method can be reduced to that of the homogeneous ferrite anisotropic sphere. The present method can be applied to the analyses of related microwave devices, antennas and the character of radar targets.     

Effects of multi-scattering on the performance of a single-beam acoustic manipulation device

The effects of multiple scattering on acoustic manipulation of spherical particles using helicoidal Bessel-beams are discussed. A closed-form analytical solution is developed to calculate the acoustic radiation force resulting from a Besselbeam on an acoustically reflective sphere, in the presence of an adjacent spherical particle, immersed in an unbounded fluid medium. The solution is based on the standard Fourier decomposition method and the effect of multi-scattering is taken into account using the addition theorem for spherical coordinates. Of particular interest here is the investigation of the effects of multiple scattering on the emergence of negative axial forces. To investigate the effects, the radiation force applied on the target particle resulting from a helicoidal Bessel-beam of different azimuthal indexes (m = 1 to 4), at different conical angles, is computed. Results are presented for soft and rigid spheres of various sizes, separated by a finite distance. Results have shown that the emergence of negative force regions is very sensitive to the level of cross-scattering between the particles. It has also been shown that in multiple scattering media, the negative axial force may occur at much smaller conical angles than previously reported for single particles, and that acoustic manipulation of soft spheres in such media may also become possible.     

Coupled Dielectric Nanoparticles Manipulating Metamaterials Optical Characteristics

In this paper, we investigate the concept and theory of all-dielectric metapatterned structures that manipulate electric and magnetic optical characteristics. A 3-D array of dielectric particles is designed, where the spheres operate in their magnetic modes and their couplings offer electric modes. An analytical solution for the problem of plane wave scattering by 3-D array of dielectric nanospheres is presented. FW multipole expansion method is applied to express the optical fields in terms of the electric and magnetic dipole modes and the higher order moments. By enforcing the boundary conditions at the surface of each sphere, with the use of the translational addition theorem for vector spherical wave functions, required equations to determine the scattering coefficients are obtained. Novel materials features in optics are demonstrated. Electric and magnetic scattering coefficient resonances around the same frequency band are obtained. It is highlighted how a metapatterned structure constructed from dielectric nanosphere unit cells can provide electric and magnetic modes resulting in backward wave phenomenon. A comprehensive circuit model based on the RLC (resistor, inductor, and capacitor) realization is presented to successfully analyze the scattering performance of a dielectric nanosphere. To better understand the physics of an array of spheres, circuit models for the interactions, and couplings between spheres are also accomplished. The engineered dispersion diagram for a 3-D array of identical highly coupled nanospheres is scrutinized, verifying that the high couplings between spheres can offer the backward wave characteristics.     

Hydrodynamics of deformable contiguous spherical shapes in an incompressible inviscid fluid

Summary An exact solution to the two-body interaction problem is presented for the case of spherical shapes moving in an incompressible and inviscid fluid. The spheres are assumed to translate in an arbitrary manner and to undergo radial deformation (or pulsation). The problem is formulated in terms of spherical harmonics and the force experienced by the spheres is obtained by employing the Lagally theorem. The expressions for the force are given as an infinite sum of coefficients which are found by solving an infinite set of linear equations. Three main geometries are considered, namely, two spheres exterior to each other, one sphere in the interior of the other and sphere in a rectangular channel. Numerical values for the added-mass coefficients as well as for the hydrodynamic forces are found for the case of rigid sphere moving toward or parallel to a rigid wall or a free surface, and a pulsating sphere in the proximity of these boundaries. Also given are numerical values for the transverse and the longitudinal addedmass coefficients for a sphere moving in a rectangular channel for different channel-blockage ratios.     

Scattering of laser pulses (plane wave and focused gaussian beam) by spheres

The scattering of laser pulses (in the femtosecond-picosecond range) by large spheres is investigated. We call a sphere large when its diameter is larger than the length associated with the pulse duration, allowing one to observe the temporal separation of scattering modes including surface waves.     

Electromagnetic scattering from a multilayered sphere located in an arbitrary beam

A solution is given for the problem of scattering of an arbitrary shaped beam by a multilayered sphere. Starting from Bromwich potentials and using the appropriate boundary conditions, we give expressions for the external and the internal fields. It is shown that the scattering coefficients can be generated from those established for a plane-wave illumination. Some numerical results that describe the scattering patterns and the radiation-pressure behavior when an incident Gaussian beam or a plane wave impinges on a multilayered sphere are presented.     

Hydrodynamic interaction of two neutrally-buoyant smooth spheres suspended in plane Poiseuille flow: the BEM simulations versus the MoR approximations

The hydrodynamic interaction between two particles suspended in shear flows is fundamental to the macroscopic characterization of suspension flows. Although such interaction in quiescent or linear shear flow is well understood, studies on that in a nonlinear shear field are rare. In this study, the hydrodynamic interaction between two neutrally-buoyant smooth spheres moving at negligible Reynolds numbers in an unbounded plane Poiseuille flow has been calculated by three-dimensional boundary element method (BEM) simulations. The BEM results have been compared with the analytical results obtained with the method-of-reflection (MoR) approximations. The BEM simulations have been found to provide satisfactory predictions if the number of elements on the spheres are more than 200, whereas the MoR approximations provide satisfactory predictions only when the minimum separation between the spheres is relatively large although this MoR method has the advantage to easily calculate the hydrodynamic interaction between two spheres freely moving at negligible Reynolds numbers in unbounded quadratic flow by solving ordinary differential equations. Furthermore, it is found that there is a preferential cross-streamline migration of the center-of-gravity of the sphere-pair in the plane of shear in plane Poiseuille flow which does not arise in simple shear flow. This migration is always directed towards low shear regions when the sphere having larger translational velocity approaches the other sphere, and reverses towards high shear regions when the faster sphere leads the other sphere in the plane of shear. There is also a cross-streamline migration of the center-of-gravity of the sphere-pair in the plane of vorticity, but this migration does not have a preferential direction. These migrations are symmetric about the point where the spheres are at the minimum separation, and are only significant when the hydrodynamic interaction of the spheres is strong. These results show that the migration of the center-of-gravity of the sphere-pair can be attributed to the nonlinearity of the shear field, which agrees with the MoR approximations. The hydrodynamic interaction between the two spheres has been quantified under various conditions by the BEM simulations for both identical and disparate spheres.     

Water-wave scattering by two submerged plane vertical barriers—Abel integral-equation approach

The classical problem of surface water-wave scattering by two identical thin vertical barriers submerged in deep water and extending infinitely downwards from the same depth below the mean free surface, is reinvestigated here by an approach leading to the problem of solving a system of Abel integral equations. The reflection and transmission coefficients are obtained in terms of computable integrals. Known results for a single barrier are recovered as a limiting case as the separation distance between the two barriers tends to zero. The coefficients are depicted graphically in a number of figures which are identical with the corresponding figures given by Jarvis (J Inst Math Appl 7:207–215, 1971) who employed a completely different approach involving a Schwarz–Christoffel transformation of complex-variable theory to solve the problem.     

Mean-field approximation of Mie scattering by fractal aggregates of identical spheres

We apply the recent exact theory of multiple electromagnetic scattering by sphere aggregates to statistically isotropic finite fractal clusters of identical spheres. In the mean-field approximation the usual Mie expansion of the scattered wave is shown to be still valid, with renormalized Mie coefficients as the multipolar terms. We give an efficient method of computing these coefficients, and we compare this mean-field approach with exact results for silica aggregates of fractal dimension 2.     

Angle-resolved reflectance of obliquely aligned silver nanorods

Arrays of silver nanorods (AgNRs) formed by oblique-angle deposition (OAD) are strongly anisotropic, with either metallic or dielectric characteristics depending on the polarization of incident light, and may be used to enhance Raman scattering and surface plasmon polaritons. This work investigates the polarization-dependent reflectance of inclined AgNR arrays at the wavelengths of 635 and 977 nm. The specular reflectance at various incidence angles and the bidirectional reflectance distribution function were measured with a laser scatterometer, while the directional-hemispherical reflectance was measured with an integrating sphere. The AgNR layer is modeled as an effectively homogenous, optically uniaxial material using the effective medium theory to elucidate the dielectric or metallic response for differently polarized incidence. The thin-film optics formulation is modified considering optical anisotropy and surface scattering. This study helps gain a better understanding of optical properties of nanostructured materials.     

Electromagnetic scattering by an aggregate of spheres

We present a comprehensive solution to the classical problem of electromagnetic scattering by aggregates of an arbitrary number of arbitrarily configured spheres that are isotropic and homogeneous but may be of different size and composition. The profile of incident electromagnetic waves is arbitrary. The analysis is based on the framework of the Mie theory for a single sphere and the existing addition theorems for spherical vector wave functions. The classic Mie theory is generalized. Applying the extended Mie theory to all the spherical constituents in an aggregate simultaneously leads to a set of coupled linear equations in the unknown interactive coefficients. We propose an asymptotic iteration technique to solve for these coefficients. The total scattered field of the entire ensemble is constructed with the interactive scattering coefficients by the use of the translational addition theorem a second time. Rigorous analytical expressions are derived for the cross sections in a general case and for all the elements of the amplitude-scattering matrix in a special case of a plane-incident wave propagating along the z axis. As an illustration, we present some of our preliminary numerical results and compare them with previously published laboratory scattering measurements.     

Nonstationary heat transfer in anisotropic half-space under the conditions of heat exchange with the environment having a specified temperature

An analytical solution to the problem of heating anisotropic half-space by the environment with a spatially and temporally variable temperature (boundary conditions of the third kind at an anisotropic body) has been obtained for the first time based on the construction of the boundary influence function (the Green’s function), which is determined using the Fourier and Laplace integral transforms. Nonstationary temperature fields in anisotropic blunt bodies have been found under the conditions of aero-gas-dynamic heating of hypersonic aircrafts with different heat-transfer coefficients and incoming-flow temperatures. The solution obtained is recommended for determining the state of thermal protection fabricated from composites which are generally anisotropic.     

Study of the axially symmetric motion of an incompressible viscous fluid between two concentric rotating spheres

The research reported herein involves the study of the steady state and transient motion of a system consisting of an incompressible, Newtonian fluid in an annulus between two concentric, rotating, rigid spheres. The primary purpose of the research is to study the use of an approximate analytical method for analyzing the transient motion of the fluid in the annulus and the spheres which are started suddenly due to the action of prescribed torques. The problems include cases where: (a) one (or both) spheres rotate with prescribed constant angular velocities and (b) one sphere rotates due to the action of an applied constant or impulsive torque.In this research, the coupled solid and fluid equations of motion are linearized by employing the perturbation technique. The meridional dependence in these equations is removed by expanding the dependent variables in a series of Gegenbauer functions with variable coefficients and employing the orthogonality property of these functions. The equations for the variable coefficients are solved by separation of variables and Laplace transform methods. Results for the stream function, circumferential function, angular velocity of the spheres and torque coefficient are presented as a function of time for various values of the dimensionless system parameters.     

An analysis of transport phenomena for multi-solid particle systems at higher Reynolds numbers by a 5th order polynomial Karman-Pohlhausen method

Using the concentric spheres free surface model and a 5th order polynomial Karman-Pohlhausen method of the laminar boundary layer theory, the dimensionless tangential stress distributions, the dimensionless pressure distributions around a solid sphere in a swarm and the viscous, form and total drag coefficients for multi-solid sphere systems were numerically computed at higher Reynolds numbers, based on the first assumption that the pressure distribution equals that of potential flow between concentric spheres up to the separation point, and behind the separation point in the wake region the pressure does not recover and keeps constant, and on the second assumption that the pressure distribution varies according to the measurement of Flachsbart.The theoretical drag coefficient of single solid spheres in an infinite medium based on the second assumption agreed with the experimental data in the range of the Reynolds numbers from $\text{3 × 10}{}^2\text{to 10}{}^5$.The friction factor for multi-solid particle systems based on the first assumption is almost the same as that on the 4th order polynomial and agreed with the experimental data of packed and distended beds.The void functions obtained from the drag coefficients for multi-solid particle systems based on both first and second assumptions were almost the same as the one on the 4th order polynomial.Using the velocity profiles based on concentric spheres free surface model and a 5th order polynomial Karman-Pohlhausen method of the laminar boundary layer obtained previously, the diffusion equation was solved numerically at higher Reynolds numbers on the first assumption of the pressure distribution around a solid sphere in a swarm equals that of potential flow between concentric spheres from the frontal stagnation point to the separation one, and the pressure does not recover, but keeps constant behind the separation point in the wake region. The mass transfer rate for multi-solid particle systems so computed was almost the same as that on the 4th order polynominal and agreed with the experimental data of single Solid spheres, and packed and particulate fluidized beds at higher Reynolds numbers.     

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.     

Effect of Constitutive Parameters on Cavity Formation and Growth in a Class of Incompressible Transversely Isotropic Nonlinearly Elastic Solid Spheres

Cavity formation and growth in a class of incompressible transversely isotropic nonlinearly elastic solid spheres are described as a bifurcation problem, for which the strain energy density is expressed as a nonlinear function of the invariants of the right Cauchy-Green deformation tensor. A bifurcation equation that describes cavity formation and growth is obtained. Some interesting qualitative properties of the bifurcation equation are presented. In particular, cavitated bifurcation is examined for a solid sphere composed of an incompressible anisotropic Gent-Thomas material model with a transversely isotropy about the radial direction. The effect of constitutive parameters on cavity formation and growth is then carried out. It is proved that a cavity forms in the interior of the sphere earlier or later than that in the isotropic Gent-Thomas sphere as the anisotropic parameter takes certain values. The condition for the bifurcation to the left or to the right of the cavity solution is proposed. The stability and the catastrophe of the equilibrium solutions are discussed by using the minimal potential energy principle. Whereas, in contrast to other isotropic nonlinear elastic spheres, cavitated bifurcation in the isotropic Gent-Thomas sphere is quite different, it is proved that the cavity solution can bifurcate locally to the left. The growth of a pre-existing micro-void in the sphere is examined, which interprets the physical implications of the preceding bifurcation problem.     

T matrix of the homogeneous anisotropic sphere: applications to orientation-averaged resonant scattering

We illustrate some numerical applications of a recently derived semianalytic method for calculating the T matrix of a sphere composed of an arbitrary anisotropic medium with or without losses. This theory is essentially an extension of Mie theory of the diffraction by an isotropic sphere. We use this theory to verify a long-standing conjecture by Bohren and Huffman that the extinction cross section of an orientation-averaged anisotropic sphere is not simply the average of the extinction cross sections of three isotropic spheres, each having a refractive index equal to that of one of the principal axes.     

Transformation of Reference Spheres by an Aberration Free and Infinitely Large Optical System in the Fresnel Approximation

Imaging of a complex distribution by a thin lens is studied and generalized by analysing the imaging process of a complex distribution defined on an off-axis object sphere in an optical system. First, imaging of an object in an elementary system is considered, and then the considerations for the system being a combination of the elementary systems are presented. The general rule of reference spheres transformation is introduced. It is shown that complex distributions given on an off-axis object sphere are fully reconstructed on an image sphere of the system with a change of scale only. The relations between the parameters of the object and image spheres are determined. Some examples of the applications of the transforming rule are given.