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.     

Interface between transparent isotropic and uniaxial absorbing dielectric media: equations for ray tracing and for the direction of propagation

In this work we analyze the propagation of a plane wave that passes from an isotropic transparent medium to a uniaxial absorbing medium. Detailed expressions that give the real directions of propagation of the wave and the energy of the reflected and refracted ordinary and extraordinary waves are obtained. These expressions are valid for every orientation of the optic axis of the uniaxial medium and for every direction of propagation of the incident wave. Expressions are tested in the case of an interface between a transparent and an absorbing isotropic media and for the air-rutile (TiO2) interface. The effect of absorption has been evaluated by comparing the results obtained in rutile with the results obtained in a transparent uniaxial medium with the same real refractive indices. Results are presented for different values of the angle of incidence and the orientation of the plane of incidence.     

Scattering by a dense layer of infinite cylinders at normal incidence

The solution for scattering by a layer of densely distributed infinite cylinders is presented. The layer is irradiated by an arbitrarily polarized plane wave that propagates in the plane perpendicular to the axes of the cylinders. The theoretical formulation utilized the effective field and quasi-crystalline approximation to treat the multiple scattering interactions in the dense finite medium. Governing equations for the propagation constants and amplitudes of the effective fields are derived for TM and TE mode incident waves, from which the scattered intensity distribution and scattering cross section for arbitrary polarization are obtained. The dense medium gives rise to coherent and incoherent scattered radiation that propagates in the plane normal to the axes of the cylinders. The coherent scattered radiation includes the forward component in the direction of the incident wave and the backward component in the direction of specular reflection. The incoherent scattered intensity distribution shows a pronounced forward peak that coincides with the angle of refraction of the effective waves inside the medium. Numerical results are presented to illustrate the scattering characteristics of a dense layer of cylinders as a function of layer thickness for a given solid volume fraction.     

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.     

Effects of fiber ellipticity and orientation on dynamic stress concentrations in porous fiber-reinforced composites

Interaction of time harmonic fast longitudinal and shear incident plane waves with an elliptical fiber embedded in a porous elastic matrix is studied. The novel features of Biot dynamic theory of poroelasticity along with the classical method of eigen-function expansion and the pertinent boundary conditions are employed to develop a closed form series solution involving Mathieu and modified Mathieu functions of complex arguments. The complications arising due to the non-orthogonality of angular Mathieu functions corresponding to distinct wave numbers in addition to the problems associated with appearance of additional angular dependent terms in the boundary conditions are all avoided by expansion of the angular Mathieu functions in terms of transcendental functions and subsequent integration, leading to a linear set of independent equations in terms of the unknown scattering coefficients. A MATHEMATICA code is developed for computing the Mathieu functions in terms of complex Fourier coefficients which are themselves calculated by numerically solving appropriate sets of eigen-systems. The analytical results are illustrated with numerical examples in which an elastic fiber of elliptic cross section is insonified by a plane fast compressional or shear wave at normal incidence. The effects of fiber cross sectional ellipticity, angle of incidence (fiber two-dimensional orientation), and incident wave polarization (P, SV, SH) on dynamic stress concentrations are studied in a relatively wide frequency range. Limiting cases are considered and fair agreements with well-known solutions are established.     

纤维复合材料的热膨胀系数

提出了一种利用压电光声技术测量材料热膨胀系数的实验方法,并测试了单向复合材料C/C、C/Al的横向、纵向的热膨胀系数。根据已有的理论计算方法与实验结果对该方法的测试结果进行验证,证明了该检测方法的可靠性,进而又测量了C/C、C/Al材料在任一方向上的热膨胀系数。这种方法克服了理论计算过程复杂以及常规手段无法测量任一方向上热膨胀系数的缺陷。     

Propagation of inhomogeneous plane waves in anisotropic viscoelastic media

A new technique is explained to study the propagation of inhomogeneous waves in a general anisotropic medium. The harmonic plane waves are considered in a viscoelastic anisotropic medium. The complex slowness vector is decomposed into propagation vector and attenuation vector for the given directions of propagation and attenuation of waves in an unbounded medium. The attenuation is further separated into the contributions from homogeneous and inhomogeneous waves. A non-dimensional inhomogeneity parameter is defined to represent the deviation of an inhomogeneous wave from its homogeneous version. Such a partition of slowness vector of a plane wave is obtained with the help of an algebraic method for solving a cubic equation and a numerical method for solving a real transcendental equation. Derived specifications enable to study the 3D propagation of inhomogeneous plane waves in a viscoelastic medium of arbitrary anisotropy. The whole procedure is wave-specific and obtains the propagation characteristics for each of the three inhomogeneous waves in the anisotropic medium. Numerical examples analyze the variations in propagation characteristics of each of the three waves with propagation direction and inhomogeneity strength.     

Representation of birefringent filters by directly composed 4 × 4 matrices

The interaction of an obliquely incident plane wave with an arbitrary birefringent plane structure can be described exactly by a so-called 4 × 4 layer matrix. However, its formation is rather complex, and therefore a simplified method to set up this matrix is proposed. It is deduced from the boundary conditions for the tangential components of the E and the H fields at the interfaces of the birefringent layer and the phase shifts of the four waves within the layer between both interfaces. The refractive indices and the directions of polarization of the four waves are calculated from the well-known dispersion relation in the principal coordinate system and a coordinate transformation. Transmission curves of Lyot filters made of uniaxial and biaxial materials are calculated in the cases of plane waves of finite or infinite lateral extension that are incident under Brewster's angle upon the filter positioned outside or within a laser resonator. The special examples given for a quartz filter and a YAlO(3) filter are characterized by a high selectivity within a broad bandwidth if Fabry-Perot effects can be neglected. Transmission curves of a quartz filter were verified by measurements.     

Forbidden gaps in periodic anisotropic layered media

The general case of obliquely incident plane-wave propagation in periodic anisotropic layered media is presented. Arbitrary permittivity tensors of the two alternating anisotropic layers are considered. An immersion model is used with the assumption that each layer is embedded between two isotropic regions that have the same index of refraction as the isotropic medium of incidence and a thickness that is set equal to zero. Then explicit relations are presented for normally incident plane waves in periodic structures that consist of alternating biaxial layers of arbitrary principal-axis orientation. Specific cases of alternating isotropic and biaxial layers are also considered. Unit cell translation matrices are presented for both traveling directions, from the left to the right and vice versa. Dispersion relations that contain information regarding the propagation bands and the forbidden gaps in periodic anisotropic structures are presented.     

The reflection and transmission of elastic waves through a plane of spheres in periodic arrangement

The work presented in this paper focuses on the reflection and transmission coefficients of an incident plane wave which impinges obliquely a plane of identical spheres arranged periodically in a homogeneous host with infinite extension. The Bloch theorem of periodic structure and the addition theorem of spherical wave functions are used to obtain the total scattering wave from all spherical scatterers periodically arranged in a plane. The total scattering wave in series form of spherical wave functions is then transformed into plane wave form in order to derive the reflection and transmission coefficients. Some numerical examples are given for different size, material constants and array patterns of spherical scatterers, and their influences on the reflection and transmission coefficients of a plane of spheres are discussed based on the numerical results. This study implies that a plane of spheres can be elaborately designed to serve as a sound barrier at a certain frequency range.     

Complex unitary vectors for the direction of propagation and for the polarization of electromagnetic waves in uniaxial and absorbing dielectric media

The propagation of inhomogeneous and elliptically polarized plane waves in absorbing uniaxial anisotropic media is described using complex unitary vectors to represent the direction of propagation and the direction of polarization. Detailed expressions for electric displacement, electric field, and magnetic field vectors are obtained for the ordinary and extraordinary waves, and their geometry is discussed. According to the complex direction of propagation, three particular cases are studied: the real case (homogeneous wave), the case perpendicular to the optical axis, and the case coplanar with the optic axis. The case of isotropic media is also analyzed.     

Transverse-electric/transverse-magnetic polarization converter using 1D finite biaxial photonic crystal

We show that by using a one-dimensional anisotropic photonic structure, it is possible to realize optical wave polarization conversion by reflection and transmission processes. Thus a single incident S(P) polarized plane wave can produce a single reflected P(S) polarized wave and a single transmitted P(S) polarized wave. This polarization conversion property can be fulfilled with a simple finite superlattice (SL) constituted of anisotropic dielectric materials. We discuss the appropriate choices of the material and geometrical properties to realize such structures. The transmission and reflection coefficients are calculated in the framework of the Green's function method. The amplitude and the polarization characteristics of reflected and transmitted waves are determined as functions of frequency, wave vector k(parallel) (parallel to the interface), and the orientations of the principal axes of the layers constituting the SL. Specific applications of these results are given for a SL consisting of alternating biaxial anisotropic layers NaNO(2)/SbSI sandwiched between two identical semi-infinite isotropic media.     

Existence of longitudinal waves in anisotropic poroelastic solids

In anisotropic fluid-saturated porous solids, four waves can propagate along a general phase direction. However, solid particles in different waves may not vibrate in mutually orthogonal directions. In the propagation of each of these waves, the displacement of pore–fluid particles may not be parallel to that of solid particles. The polarization for a wave is the direction of aggregate displacement of the particles of the two constituents of a porous aggregate. These polarizations, for different waves, are not mutually orthogonal. Out of the four waves in anisotropic poroelastic medium, two are termed as quasi-longitudinal waves. The prefix ‘quasi’ refers to their polarization being nearly, but not exactly, parallel to the direction of propagation. The existence of purely longitudinal waves in an anisotropic poroelastic medium is ensured by the stationary characters of two expressions. These expressions involve the elastic (stiffness and coupling) coefficients of a porous aggregate and the components of phase direction. Necessary and sufficient conditions for the existence of longitudinal waves are discussed for different anisotropic symmetries. Conditions are also discussed for the existence of the apparent longitudinal waves, i.e., the propagation of wave motion with the particle displacement parallel to the ray direction instead of the phase direction. A graphical solution of a numerical example is shown to check the existence of these apparent longitudinal waves for general directions of phase propagation.     

Double refraction of a Gaussian beam into a uniaxial crystal

For a linearly polarized three-dimensional Gaussian beam in air that is normally incident upon a plane interface with a uniaxial crystal with optic axis in an arbitrary direction, we present integral representations for the transmitted field suitable for asymptotic analysis and efficient numerical evaluation and derive analytical expressions for transmitted nontruncated Gaussian beams for the cases in which the incident beam is polarized parallel to the plane containing the optic axis and the interface normal and transverse to it. The general solution for an arbitrary polarization state of an incident Gaussian beam follows by superposition of these two solutions.     

Nonpolarizing guided-mode resonant grating filter for oblique incidence

A new type of guided-mode resonant grating filter is described. The filter is independent of polarization state for oblique incidence. The filter has a crossed grating structure, and the plane of incidence on the filter contains the symmetric axis of the grating structure. Theoretical considerations and numerical calculations using two-dimensional rigorous coupled-wave analysis show that a rhombic lattice structure is suitable to such filters. In this configuration an incident light wave is diffracted into the waveguide and is divided into two propagation modes whose directions are symmetric with respect to the plane of incidence. In particular, when the propagation directions of the two modes are perpendicular to each other, the fill factor of grating structure can be approximately 50%. The filter was designed for an incident angle of 45 degrees. Tolerances of setting errors and fabrication errors for this filter were estimated by numerical calculations.     

Diffraction theory in TM polarization: application of the fast Fourier factorization method to cylindrical devices with arbitrary cross section

The diffraction of an electromagnetic wave by a cylindrical object with arbitrary cross section is studied by taking advantage of recent progress in grating theories. The fast Fourier factorization method previously developed in Cartesian coordinates is extended to cylindrical coordinates thanks to the periodicity of both the diffracting object and the incident wave with respect to the polar angle theta. Thus Maxwell equations in a truncated Fourier space are derived and separated in TE and TM polarization cases. The new set of equations for TM polarization is resolved numerically with the S-matrix propagation algorithm. Examples of elliptic cross sections and cross sections including couples of nonconcentric circles show fast convergence of the results, for both dielectric and metallic materials, as well as good agreement with previous published results. Thus the method is suitable for an extension to conical (out-of-plane) diffraction, which will allow studying mode propagation along microstructured fibers.     

Dispersion of linearly polarized light propagating in a thin birefringent plate

We analyze theoretically the dispersion of linearly polarized light propagating in a uniaxial anisotropic medium where multibeam interference is present. Explicit expressions of the group-delay dispersion for transmitting waves are derived for the simplest situation, and the effect of dispersion on pulse broadening is analyzed for a few selected cases. Our results reveal that at normal incidence and in the situation where the optic axis is parallel to the surface of birefringent plate (in the x-y plane), the dispersion of the refracted wave decreases with the extent of birefringence. In particular, the dispersion for the electric field parallel to the polarization direction of the incident light changes with the rotation angle between the optic axis and the polarization direction of the incident field, whereas the dispersion for the refracted field whose direction is vertical to the polarization of incident light is independent of this angle. For oblique incidence, dispersion varies substantially for different incident angles. In the situation where the optic axis is in the x-z plane at either normal or oblique incidence, the dispersion increases in a periodically oscillating manner as a function of the relative thickness of the birefringent plate.     

Propagation properties of an elliptical anisotropic metamaterial cylinder

The propagation characteristics were investigated of an open elliptical cylinder comprising a uniaxial anisotropic metamaterial. The dispersion relation for hybrid modes in this cylinder was examined theoretically. It is shown that forward and backward waves can be supported by such a metamaterial cylinder. Even and odd modes have different losses in a realistic anisotropic elliptical cylinder. The forward wave can propagate longer distances than the backward wave because of its lower loss. The light can propagate backward on the cylinder without the limit of structure size. The odd modes have higher operating wavelengths than the even modes. The extraordinary properties in this structure are identified as multilayer plasmons. The validity of the effective medium approximation of this structure is examined.     

Non-absorbing isotropic–uniaxial interfaces: refraction in ordinary and extraordinary total reflection

We study characteristics of refraction of plane waves in a non-absorbing isotropic–uniaxial interface in conditions of total and partial reflection when the incident wave has arbitrary direction with regard to the optical axis of the crystal and polarization. We analyse the ordinary and extraordinary fields and a phase shift that appears in the propagating wave which subsists in ordinary or extraordinary total reflection. We obtain that, unlike what happens in isotropic interfaces, under conditions of ordinary total reflection, the ordinary time-averaged Poynting vector changes its direction with the angle of incidence which does not depend on the existence of extraordinary total reflection. Extraordinary total reflection gives place to an analogous behaviour of the extraordinary time-averaged Poynting vector. In addition, we define and calculate in an explicit way complex coefficients of transmission for geometries with and without symmetry in partial and total reflection.     

Ultrasonic Backscattering in Cubic Polycrystals with Ellipsoidal Grains and Texture

An ultrasonic model for backscattering from polycrystalline microstructure is developed for polycrystals with uniaxial texture and elongated cubic crystallites. The uniaxial texture or crystallographic orientation of the grains is described by a modified Gaussian orientation distribution function (ODF) with a texture parameter. Macroscopically such a textured polycrystalline medium exhibits hexagonal symmetry. The preferred texture direction and elongation are independently defined in a global system. The dependence of backscattering coefficients and their directional ratios on both texture and grain anisotropy are discussed. Attenuation coefficients in the high frequency range for arbitrary wave propagation direction are obtained and then the ratios in the three axis directions are studied. The model is compared with experimental data available in the literature for Al rolled alloys and shows good agreement when accounting for both texture and grain anisotropy effects.