Analysis of electromagnetic scattering by uniaxial anisotropic bispheres

Based on the generalized multiparticle Mie theory and the Fourier transformation approach, electromagnetic (EM) scattering of two interacting homogeneous uniaxial anisotropic spheres with parallel primary optical axes is investigated. By introducing the Fourier transformation, the EM fields in the uniaxial anisotropic spheres are expanded in terms of the spherical vector wave functions. The interactive scattering coefficients and the expansion coefficients of the internal fields are derived through the continuous boundary conditions on which the interaction of the bispheres is considered. Some selected calculations on the effects of the size parameter, the uniaxial anisotropic absorbing dielectric, and the sphere separation distance are described. The backward radar cross section of two uniaxial anisotropic spheres with a complex permittivity tensor changing with the sphere separation distance is numerically studied. The authors are hopeful that the work in this paper will help provide an effective calibration for further research on the scattering characteristic of an aggregate of anisotropic spheres or other shaped anisotropic particles.     

Generalized Lorenz–Mie scattering theory for focused radiation and finite solids of revolution: case I: symmetrically polarized beams

Interaction of focused radiation with spherical and finite cylindrical homogeneous particles is considered. The aim of this investigation is to calculate the structure of the electromagnetic (EM) fields scattered by and propagated within the scattering objects. The incident EM fields are assumed to be focused fields in the image space of an aplanatic system with or without aberrations of category one. The radiation in the object space is assumed to be symmetrically polarized. The incident fields in the neighbourhood of the focus are calculated using the well-known theory of Richards and Wolf and a methodology developed by the author. At the interface between the homogeneous and the image space of the aplanatic system, the continuity conditions of the tangential components of the electric displacement and magnetic moment vectors are satisfied. The procedure results in dual discretized-Fredholm integral equations that are solved using orthogonal expansions. It is assumed that the scattered field, at large distances from the focus, is a spherical wave propagating away from the focus. Scattering by objects of various materials ranging from dielectric to perfect conductor is studied. The theory and its solution developed here allow for the scattering objects to be located anywhere along the optical axis in the image space. One of the main objectives is to calculate the energy distribution at the tip of the cylindrical homogeneous particle. Numerical calculations suggest that energy density at the tip is further enhanced if the cylindrical homogeneous particle is placed away from focus.     

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.     

Mie scattering by an anisotropic object. Part I. Homogeneous sphere

Establishing a vector spherical harmonic expansion of the electromagnetic field propagating inside an arbitrary anisotropic medium, we extend Mie theory to the diffraction by an anisotropic sphere, with or without losses. The particular case of a uniaxial material leads to a simpler analysis. This work opens the way to the construction of a differential theory of diffraction by a three-dimensional object with arbitrary shape, filled by an arbitrary anisotropic material.     

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.     

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.     

Binary inclusion model for the overall elasticity of imperfectly bonded composites

Summary. A micromechanical approach is developed to estimate the overall elastic moduli of composite materials with imperfectly bonded spherical fillers. The randomly dispersed particles are assumed to satisfy linear interfacial conditions where both tangential and normal interface displacement discontinuities are linearly related to the respective surface tractions. Using the generalized version of Eshelbys equivalent inclusion method proposed by Furuhashi et al. [6] the analysis of the heterogeneous medium reduces to the study of a corresponding homogeneous medium containing spherical inclusions with a proper distribution of eigenstrain and Somigliana dislocation fields. Based on the estimated pair-wise average of strain fields in two interacting imperfect fillers embedded in the homogeneous infinite matrix, the ensemble phase volume average of field quantities has been evaluated within a representative volume element containing a finite number of imperfect particles. For the case of a constant radial distribution function, results are in reasonable agreement with those based on the generalized self-consistent method and composite sphere assemblage proposed by Hashin [11].     

Scattering of electromagnetic plane wave from a perfect electric conducting strip placed at free space–chiral interface

The scattering of electromagnetic plane wave from a perfect electric conducting strip of finite width is investigated in this study. The strip is placed at the planar interface of free space and chiral medium. The Kobayashi potential method is used to determine the scattering from the strip. The dual integral equations are acquired through the boundary conditions. These equations as well as the edge conditions are satisfied through the properties of Weber–Schafheitlin integrals and the Jacobi polynomials. The projection of the Jacobi polynomials is used to get the matrix equations which are solved numerically to determine the unknown coefficients. Monostatic and bistatic scattering widths are examined in the free space region of the geometry. The far-zone scattered fields are analyzed by changing the values of different parameters, i.e. chirality and incident angle.     

Nonparaxial description of reflection and transmission at the interface between an isotropic medium and a uniaxial crystal

Angular spectra of reflected and transmitted fields, induced by an arbitrary electromagnetic beam passing through the planar interface between a homogeneous medium and a uniaxially anisotropic medium, are derived and related to the incident medium. By using these formulas, we obtain the expressions for paraxial and slightly nonparaxial fields. The reflected paraxial field is related to the incident one by means of Fresnel relations; the transmitted paraxial field is the superposition of an ordinary and an extraordinary beam, multiplied by the Fresnel coefficient. We find that the nonparaxial corrections, owing to the medium discontinuity, are larger than their free-propagation counterparts and that they are very simply related to the paraxial solutions of the incident beam. The case of two homogeneous media with different refractive indices is also discussed. The general expressions obtained are applied to the case of a nonparaxial Gaussian beam.     

Time‐domain vector potential technique for the meshless radial point interpolation method

A time‐domain meshless algorithm based on vector potentials is introduced for the analysis of transient electromagnetic fields. The proposed numerical algorithm is a modification of the radial point interpolation method, where radial basis functions are used for local interpolation of the vector potentials and their derivatives. In the proposed implementation, solving the second‐order vector potential wave equation intrinsically enforces the divergence‐free property of the electric and magnetic fields. Furthermore, the computational effort associated with the generation of a dual node distribution (as required for solving the first‐order Maxwell's equations) is avoided. The proposed method is validated with several examples of 2D waveguides and filters, and the convergence is empirically demonstrated in terms of node density or size of local support domains. It is further shown that inhomogeneous node distributions can provide increased convergence rates, that is, the same accuracy with smaller number of nodes compared with a solution for homogeneous node distribution. A comparison of the magnetic vector potential technique with conventional radial point interpolation method is performed, highlighting the superiority of the divergence‐free formulation. Copyright © 2015 John Wiley & Sons, Ltd.     

Generalized Investigations into the Brewster Angle

The standard elementary theory of the Brewster angle generated by a plane interface separating two homogeneous isotropic media, through which incident and transmitted electromagnetic waves are propagating, is well known. We introduce here a more general inhomogeneous medium for which a Brewster wave and a Brewster angle can be defined. The properties of this wave are investigated, together with a model whose field can be expressed in terms of a special degenerate hypergeometric function. Other angles of incidence also yield zero reflectivity, but these are carefully distinguished from the Brewster angle.     

The scattering of electro-elastic waves by a spherical piezoelectric particle in a polymer matrix

This paper is devoted to the study of scattering of plane harmonic waves by a piezoelectric sphere with spherical isotropy embedded in an unbounded isotropic polymer matrix. The scattered displacement field and the electric potential in the matrix are expressed in terms of spherical vector wave functions and spherical harmonic functions, respectively. For the field points inside the inhomogeneity, new displacement functions are introduced. Expansion of the new displacement functions and the electric potential in terms of spherical harmonic functions, the equations of motion and electrostatic lead to four second order ordinary differential equations (odes), where three of them are coupled. The coupled system of odes is solved by the generalized Frobenius series. This approach is readily used to handle low and high frequencies. Three different types of piezoelectric inhomogeneities, PZT-4, PZT-5H, and BaTiO₃ are considered and the associated piezoelectric effects on the electro-mechanical fields, differential and total scattering cross-sections are addressed.     

Scattering of SH-waves by an interface cavity

Summary. The scattering of the SH-wave and dynamic stress concentrations near an arbitrary cavity situated at the planar interface separating two different elastic media are investigated. The total wave field can be obtained by superposition of the free field and the scattered field. The free field is composed of the incident, reflected and refracted waves. The scattered wave fields in adjacent media are expressed respectively, and the method of wave functions expansion is applied to obtain the solutions for these fields. The scattered wave functions can be expanded into Hankel-Fourier series with unknown coefficients. In solving for the unknown coefficients according to the boundary conditions for the total wave field at the interface and at the cavity wall, the non-orthogonality makes the system of equations for the unknown coefficients infinite and coupling each other. Another key point is to extend each scattered wave field from its own half-plane domain into the full plane domain by a certain way keeping the total wave field unchanged for the non-orthogonal Fourier integrals around the cavity. Finally, the scattering of the SH wave by an interface ellipse with different ratios between long and short axis is considered, and the distributions of dynamic stress concentration factors at the cavity wall are presented.     

On the quantization of the electromagnetic field in conducting media

In this work we investigate the quantization of electromagnetic waves propagating through homogeneous conducting linear media with no charge density. We use Coulomb's gauge to reduce the problem to that of a time-dependent harmonic oscillator, which is described by the Caldirola–Kanai Hamiltonian. Furthermore, we obtain the corresponding exact wave functions with the help of quadratic invariants and of the dynamic invariant method. These wave functions are written in terms of a particular solution of the Milne–Pinney equation. We also construct coherent and squeezed states for the quantized electromagnetic waves and evaluate the quantum fluctuations in coordinates and momentum as well as the uncertainty product for each mode of the electromagnetic field.     

Reflection and refraction of an arbitrary electromagnetic wave at a plane interface separating an isotropic and a biaxial medium.

Exact solutions are obtained for the reflected and transmitted fields resulting when an arbitrary electromagnetic field is incident on a plane interface separating an isotropic medium and a biaxially anisotropic medium in which one of the principal axes is along the interface normal. From our exact solutions for the reflected fields resulting when a plane TE or TM wave is incident on the plane interface, it can be inferred that the reflected field contains both a TE and a TM component. This gives a change in polarization that can be utilized to determine the properties of the biaxial medium. The time-harmonic solution for the reflected field is in the form of two quadruple integrals, one of which is a superposition of plane waves polarized perpendicular to the plane of incidence and the other a superposition of plane waves polarized parallel to the plane of incidence. The time-harmonic solution for the transmitted field is also in the form of two quadruple integrals. Each of these is a superposition of extraordinary plane waves with displacement vectors that are perpendicular to the direction of phase propagation.     

Scattering of a Hermite-Gaussian beam field by a chiral sphere.

Scattering of a Hermite-Gaussian beam field by a chiral sphere is analyzed. A Hermite-Gaussian beam field is expressed as a superposition of multipole fields at complex-source points. Electromagnetic fields are expanded in terms of the spherical vector wave functions. The unknown expansion coefficients for the scattered field and the internal field are determined by the boundary conditions. As numerical examples, the scattered near fields of the beam incidence are calculated, and the effects of the chirality and the radius of the chiral sphere on the fields are examined. The results for a Gaussian beam incidence are also compared with those of a plane-wave incidence.     

Electromagnetic applications of a new finite-difference calculus

The accuracy of finite-difference analysis in electromagnetics can be qualitatively improved by employing arbitrary local approximating functions, not limited to Taylor expansion polynomials. In the proposed new class of flexible local approximation methods (FLAME), desirable local analytical approximations (such as harmonic polynomials, plane waves, and cylindrical or spherical harmonics) are directly incorporated into the finite-difference scheme. Although the method usually (but not necessarily) operates on regular Cartesian grids, it is in some cases much more accurate than the finite-element method with its complex meshes. This paper reviews the theory of FLAME and gives a tutorial-style explanation of its usage. While one motivation for the new approach is to minimize the notorious "staircase" effect at curved and slanted interface boundaries, it has much broader applications and implications. As illustrative examples, the paper examines the simulation of: 1) electrostatic fields of finite-size dielectric particles in free space or in a solvent with or without salt; 2) scattering of electromagnetic waves; 3) plasmon resonances; and 4) wave propagation in a photonic crystal.     

Coupled problem of thermomechanics for a conducting layer subjected to a uniform pulsed electromagnetic action

A coupled dynamic problem of thermomechanics is formulated for a conducting layer with plane-parallel boundaries subjected to a nonstationary electromagnetic action. As key functions of the problem, we use the tangential component of the vector of magnetic-field intensity, temperature, and the normal components of the tensor of dynamic stresses. We propose a procedure for the determination of these functions based on their cubic approximation in the thickness coordinate and the Laplace transformation with respect to time. On the basis of the obtained general solutions, we present the solutions of the problem for a sinusoidal electromagnetic action with pulsed modulating signal. The influence of the parameter of coupling of the strain and temperature fields on the stresses and temperature is analyzed.     

Experimental verification of nondiffracting X waves

The propagation of acoustic waves in isotropic/homogeneous media and electromagnetic waves in free space is governed by the isotropic/homogeneous (or free space) scalar wave equation. A zeroth-order acoustic X wave (axially symmetric) was experimentally produced with an acoustic annular array transducer. The generalized expression includes a term for the frequency response of the system and parameters for varying depth of field versus beam width of the resulting family of beams. Excellent agreement between theoretical predictions and experiment was obtained. An X wave of finite aperture driven with realizable (causal, finite energy) pulses is found to travel with a large depth of field (nondiffracting length).     

Thermoelastic State of a Conducting Plate under the Action of an Electromagnetic Field in the Form of Damped Sinusoid

We determine the temperature fields and stresses formed in an infinite nonferromagnetic conducting plate of constant thickness under the action of a pulsed electromagnetic field specified by the values of the tangential component of magnetic vector on the surfaces. In the case of electromagnetic action obeying the law of damped sinusoid, we perform the comparative numerical analysis of ponderomotive forces, temperature, and components of the dynamic stress tensor for plates made of stainless steel and copper. It is shown that the influence of electric and thermal conductivity, thermal diffusivity, the coefficient of linear thermal expansion, and Young's modulus on the quantitative and qualitative behavior of the analyzed parameters as functions of time is significant.