Anisotropic Media Research Articles

Polynomial Solutions of Electric Field Equations in Anisotropic Media

The time-dependent system of partial differential equations of the second order describing the electric wave propagation in electrically and magnetically anisotropic homogeneous media is considered in the paper. A method for the computation of the polynomial solutions of the initial value problem for the considered system is proposed. Symbolic computations are used and these symbolic computations are implemented in Maple. It is proved also that these polynomial solutions are approximate solutions of the considered initial value problem with smooth initial data and the inhomogeneous term. The computational experiments confirm the robustness of the suggested method for the computation of electric fields in general electrically and magnetically anisotropic media.

Singularity points and their degeneracies in anisotropic media

Summary We define double (S1S2) and triple (PS1S2) singularity points and their degeneracy classes in triclinic anisotropic media. We derive equations for the group velocity image for all these cases. The degeneracy classes are defined by factorization of quadratic (double singularity point) and cubic (triple singularity point) forms with three variables.

Diffraction of light by a slit in anisotropic media with closed and open wave vector surface

Fraunhofer diffraction of an electromagnetic wave by a slit on an opaque screen located between vacuum and an anisotropic medium with closed and open wave vectors surface is considered. A formula for diffraction in a material medium is obtained, including the case of diffraction into a uniaxial anisotropic medium in the absence of absorption, and the formation of diffraction minima and maxima is studied.

Higher Order Spectral Element Method for Rapid Analysis of Eccentric Coaxial Waveguides Filled With Lossy Anisotropic Media

Higher Order Spectral Element Method for Rapid Analysis of Eccentric Coaxial Waveguides Filled With Lossy Anisotropic Media

Response of Semicircular Canyons and Movable Cylindrical Cavities to SH Waves in Anisotropic Half-space Geology

The development of tunnels or the laying of underground pipelines are essential engineering projects in modern society, and in canyon tunnels and underground pipeline projects, the surface motion and cavity edge motion have been topics of concern in ground vibration problems. In this paper, we investigate the wave scattering problem in an elastic half-space anisotropic medium containing a semicircular canyon and a subsurface movable cylindrical cavity by using the wave function expansion method, the complex function method and the mirror method. By deriving the governing equation and transforming it into the standard form of the Helmholtz equation satisfying the zero-stress boundary condition, we solve the corresponding displacement functions. Introducing a position correction coefficient, the scattered wave field in a half-space anisotropic medium is constructed by the mirror method, which improves the problem of scattered wave source singularity in anisotropic half-space medium. Then, combining the free boundary conditions with a Fourier series expansion method we solve for the unknown coefficients in the equations. The correctness of the method is verified by degenerating it to a classical analytic solution. Finally, using frequency and time domain analysis, we investigate the effects of the relevant parameters on the surface motion| w1|( w), the dynamic stress concentration factor (DSCF) and the displacement amplitude| w2|. The results show that rock anisotropy and the presence of semicircular canyons have a significant effect on the dynamic response of subsurface structures. This not only provides a theoretical basis for practical unlined tunnels or pipeline projects, but can also provide a basis for seismic design of underground structures.

Anisotropy in Stratified and Equivalent Media: Kinematic and Amplitude Analysis

We implemented an elastic-anisotropic seismic modeling algorithm via synthetic models to evaluate different geological scenarios, where intercalations of sandstones and shales, carbonates and shales and halite and high-speed salt were simulated. These lithologies were selected based on their abundance in the sedimentary record and the importance they have in the composition of petroleum systems. We analyzed the results relative to kinematic and dynamic aspects to broaden the understanding of how these elements act in the presence of anisotropy. Additionally, the elastic and anisotropic parameters of intercalations composed by the referred lithologies were analytically estimated, thus obtaining the characteristics of several equivalent media. The results confirm previous postulations that the intercalation of layers with different properties can generate extrinsic anisotropy, contributing to the perspective of different wavelength(λ)/thickness ratios favorable to such effects. The results indicate that λ/thickness ratios greater than 6 are propitious for extrinsic anisotropy generation and the use of equivalent media as substitutes for stratified layers are ratified for thicknesses lower than 10 meters. Also, it was observed that in most cases, intrinsic anisotropy has a main role over extrinsic anisotropy.

Negative reflection and total internal reflection at the internal surface of lithium niobate crystal

In this paper we analytically find out the expression for the wave vector and Poynting vector in the reflection domain for negative uniaxial material. We investigate the impact of the optical axis rotation and incident angle on the wave vector and Poynting vector reflected from the internal surface of lithium niobate crystal when the light is propagating from the denser anisotropic medium to the rarer isotropic medium. We derive the Brewster angle condition and discuss the walk-off between TE and TM polarized light. We explore where the wave vector and Poynting vector are parallel after reflection. In further analysis, we look into the phase differences between ordinary and extraordinary reflected light after total internal reflection from the internal surface of lithium niobate crystal. The necessary conditions for linear polarization and circular polarization have been established. The impact of optical axis orientation is also taken into consideration. The refractive index for the isotropic material for the desired polarization response has been found as 2.18, 2.1, and 2.03 at 632 nm, 1550 nm, and 3500 nm, respectively.

Quantifying anisotropy parameters in weakly anisotropic media through diffraction traveltime parameter clusters

The seismic response datasets obtained from anisotropic media present several challenges for established seismic processing methods. To determine whether wavefront interference stems from anisotropic effects, velocity model heterogeneity, or both remains a key challenge. While reflection signatures may be insufficient for distinguishing these attributes, diffraction information in seismic datasets often provides richer insights into subsurface structures. In response to these challenges, we propose a framework that explores the feasibility of using diffraction traveltime parameters as indicators of anisotropy. By introducing an average measurement velocity derived from a cluster of diffraction traveltime responses, defined as a function of the traveltime slopes estimated from the dataset, we aim to discern the prevalence of anisotropy, heterogeneity, or both within a target region. Experiments conducted with synthetic data designed to simulate realistic scenarios have yielded promising results.

DC3DPAFEM: An efficient and accurate 3-D direct current resistivity anisotropic forward modeling software for complex geological settings

Nowadays, there is a growing trend that direct current (DC) field surveys are shifting towards challenging areas characterized by mountainous topography and electrical anisotropy. Given these complex geological settings, there is an urgent need for 3-D DC forward modeling software capable of effectively addressing large-scale problems and delivering accurate modeling results to interpret field data. However, most open-source software packages face certain limitations, such as the high numerical cost to handle complex surface topography, the lack of consideration for anisotropic conductivity, the absence of mesh refinement techniques to guarantee accuracy in forward modeling, and the lack of parallel computing techniques to solve large-scale problems. In this study, we develop an efficient and highly accurate 3-D DC anisotropic forward modeling software, namely DC3DPAFEM, using the adaptive finite element algorithm based on the unstructured tetrahedral mesh. Firstly, we construct a strong compatible boundary value problem (BVP) for 3-D anisotropic DC problems by adopting a specialized secondary potential approach to handle the surface topography efficiently. Then, we develop a goal-oriented adaptive mesh refinement (AMR) technique to ensure accurate forward modeling results, even with a coarse initial mesh. To ensure time and memory efficiency, we employ a robust conjugate gradient (CG) algorithm preconditioned by the algebraic multigrid (AMG) solver to solve the large-scale linear system of equations resulting from complex geological structures. We aim to investigate the performance of the AMG scheme in anisotropic DC cases. Furthermore, we incorporate the domain decomposition technique into the iterative solution scheme for further efficiency gains. This technique significantly improves computing efficiency for large-scale problems in parallel clusters. Finally, we conduct comprehensive performance tests for DC3DPAFEM using a two-layer anisotropic model and a 3-D complex model with undulating terrain. The results of both examples validate the accuracy of DC3DPAFEM, as they closely align with the analytical solutions and the solutions obtained from the existing 3-D DC forward modeling code. Compared to traditional direct solver MUMPS and ILU-preconditioned iterative solvers, DC3DPAFEM exhibits highly scalable performance for large-scale problems, offering significant advantages in terms of memory and time consumption. Overall, DC3DPAFEM demonstrates substantial advances in efficiency, accuracy, and practicality through a series of numerical examples. This open-source code provides an efficient and available tool for developing a 3-D DC inversion method that can deal with large-scale problems involving intricate topography and anisotropic media.

Singular solutions for complex second order elliptic equations and their application to time-harmonic diffuse optical tomography

We construct singular solutions of a complex elliptic equation of second order, having an isolated singularity of any order. In particular, we extend results obtained for the real partial differential equation in divergence form by Alessandrini in 1990. Our solutions can be applied to the determination of the optical properties of an anisotropic medium in time-harmonic Diffuse Optical Tomography (DOT).

Green's functions for anisotropic elastic plates containing polygonal holes

Two sets of Green's functions for anisotropic elastic media containing a polygonal hole with rounded corners are presented. Analyzing non-elliptical holes poses a formidable mathematical challenge for anisotropic materials because existing mapping functions generally fail to meet certain criteria. The nonconformal mapping scheme and the perturbation method with conformal mapping are applied to derive two sets of solutions, respectively. They are solved using the method of analytical continuation. For the nonconformal approach, the hole is mapped onto a unit circle with nonconformal mapping functions, and the solution form is identical to elliptical holes only with a different mapping function. The perturbation solution is expressed in a series form based on the conformal mapping functions and solutions for elliptical holes. Proper adjustments of branch cuts of complex logarithm functions are introduced for both solutions. These adjustments are essential in numerical computation, as they ensure the continuity of an elastic body across branch cuts. Also, the two solutions presented in this paper are compared in-depth. Based upon their pros and cons, a combined use of these two solutions is suggested for application in boundary element method.

Efficient modeling of heat conduction in multiply bonded composites covered with thermal barrier coating

In engineering practice, 3D (three-dimensional) composites consisting of multiply thin layers of anisotropic materials have been extensively applied. For enhancing their thermal endurance, the composites are often coated with a very thin layer of medium with low conductivity, usually referred to as the thermal barrier coating (TBC). This paper presents an efficient algorithm for the boundary element method (BEM) to accurately compute the nearly singular integrals in modeling 3D anisotropic heat conduction in TBC-coated composites, consisting of several thin anisotropic media. This algorithm is especially efficient because no analytical transformation of the integrands is involved as proposed in the past. When the TBC is prescribed with Dirichlet condition, a new equivalent convective condition is proposed to replace the TBC modeling. Moreover, thin films of adhesives on interfaces are considered by introducing interfacial conductance. In the end, several examples are presented.

P-wave propagation in attenuative elliptical VTI media

Elliptical anisotropy is convenient to use as the reference medium in perturbation methods designed to study P-wave propagation for transverse isotropy (TI). We make the elliptically anisotropic TI model attenuative and discuss the corresponding P-wave dispersion relation and the wave equation. Our analysis leads to two conditions in terms of the Thomsen-type parameters, which guarantee that the P-wave slowness surface and the dispersion relation satisfy elliptical equations. We also obtain the viscoacoustic wave equation for such elliptically anisotropic media and solve it for point-source radiation using the correspondence principle. For the constant- Q TI model, we use the weighting function method to derive the viscoacoustic wave equation in differential form. Numerical examples validate the proposed elliptical conditions and illustrate the behavior of the P wavefield in attenuative elliptical TI models.

Scintillation analysis of gaussian beams in anisotropic and uniaxial media

Scintillation of Gaussian beams both in anisotropic and uniaxial media is examined numerically utilizing the split step approach. Both the point-like and the power scintillation indices of Gaussian beams that propagate for a 5 Km in anisotropic and uniaxial media are investigated. The effect of turbulence strength on the scintillation level is also studied. In this regard, the scintillation level of the three components of Gaussian beam both the anisotropic and the uniaxial are analysed and compared with the scintillation of the fundamental Gaussian beam. The provided results show that the x-component of the Gaussian beam as propagate in both the anisotropic and the uniaxial media has the lowest scintillation. In addition, in uniaxial medium, the three components of the Gaussian beam have a power scintillation level less than the fundamental Gaussian.

An efficient multi-field dynamic model for 3D wave propagation in saturated anisotropic porous media

An efficient multi–field coupled dynamic model for saturated anisotropic porous materials is proposed here. The mathematical formulation of the dynamic deformation–diffusion problem is developed starting from the mixture theory and the definition of the effective stress for anisotropic poro–elasticity, taking into account also the fluid phase compressibility and anisotropic permeability. The effective stress principle is properly extended through the Biot tensor to predict the coupling between the shear stress of the solid skeleton and the pore fluid pressure. Numerical solution of the coupled problem is obtained by inf-sup stable Finite Element spaces. A fundamental issue for the computational efficiency of the coupled model is the numerical solution of the resulting large–size and non–symmetric discrete problem. In this work, we develop a fully implicit monolithic solver based on the Bi–Conjugate Gradient Stabilized (Bi-CGStab) algorithm accelerated via an ad–hoc Multi–Physics Reduction (MPR) preconditioning technique. The proposed advances are implemented in the 3D GeoMatFem research code and several numerical analyses are performed to test the potential and computational efficiency of the proposed tool. In particular, we focus on 3D wave propagation applications in fully saturated single and multi-layered anisotropic media. Numerical results show that our implementation is able to identify the shear wave splitting phenomenon due to the different degree of material symmetry between the solid phase and the whole porous material.

Advancing electromagnetic field modeling in axially anisotropic media with conformal method in 3D cylindrical coordinate systems

A conformal methodology for solving electromagnetic fields in axially anisotropic media in 3D cylindrical coordinate systems is proposed for the investigation of high-power microwave devices filled with such media. This method leverages the material properties of axially anisotropic media and employs the finite integral technique for solution. A simplified conformal algorithm is utilized for the boundaries of the computational region, extending the traditional conformal method for axially anisotropic media in 3D cylindrical coordinates. The proposed methodology is validated through simulations of TM01 and TE01 mode propagation in a circular waveguide filled with the medium. Subsequently, the method is integrated into particle-in-cell simulations, and an X-band relativistic backward wave oscillator filled with an axially anisotropic medium exhibiting azimuthal conductivity is numerically modeled. The results demonstrate that the medium with azimuthal conductivity effectively mitigates asymmetric modes in the relativistic backward wave oscillator, aligning with published literature. This study offers technical insights for research on high-power microwave devices incorporating axially anisotropic media, potentially facilitating the development of innovative microwave devices with enhanced performance characteristics.

Exact closed forms for the transmittance of electromagnetic waves in one-dimensional anisotropic periodic media

In this work, we obtain closed expressions for the transfer matrix and the transmittance of electromagnetic waves propagating in finite one-dimensional anisotropic periodic stratified media with an arbitrary number of cells. By invoking the Cayley–Hamilton theorem on the transfer matrix for the electromagnetic field in a periodic stratified media formed by N cells, we obtain a fourth-order recursive relation for the matrix coefficients that defines the so-called Tetranacci polynomials (TPs). In the symmetric case, corresponding to a unit-cell transfer matrix with a characteristic polynomial where the coefficients of the linear and cubic terms are equal, closed expressions for the solutions to the recursive relation, known as symmetric TPs, have recently been derived, allowing us to write the transfer matrix and transmittance in a closed form. We show as sufficient conditions that the 4×4 differential propagation matrix of each layer in the binary unit cell, Δ , a) has eigenvalues of the form ±p1 , ±p2 , with p1≠p2 , and b) its off-diagonal 2×2 block matrices possess the same symmetric structure in both layers. Otherwise, the recursive relations are still solvable for any 4×4 matrix and provide an algorithm to compute the Nth power of the transfer matrix without carrying out explicitly the matrix multiplication of N matrices. We obtain analytical expressions for the dispersion relation and transmittance, in closed form, for two finite periodic systems: the first one consists of two birefringent uniaxial media with their optical axis perpendicular to the z-axis, and the second consists of two isotropic media subject to an external magnetic field oriented along the z-axis and exhibiting the Faraday effect. Our formalism applies also to lossy media, magnetic anisotropy or optical activity.

Spin-wave self-imaging: Experimental and numerical demonstration of caustic and Talbot-like diffraction patterns

Extending the scope of the self-imaging phenomenon, traditionally associated with linear optics, to the domain of magnonics, this study presents the experimental demonstration and numerical analysis of spin-wave (SW) self-imaging in an in-plane magnetized yttrium iron garnet film. We explore this phenomenon using a setup in which a plane SW passes through a diffraction grating, and the resulting interference pattern is detected using Brillouin light scattering. We have varied the frequencies of the source dynamic magnetic field to discern the influence of the anisotropic dispersion relation and the caustic effect on the analyzed phenomenon. We found that at low frequencies and diffraction fields, the caustics determine the interference pattern. However, at large distances from the grating, when the waves of high diffraction order and number of slits contribute to the interference pattern, the self-imaging phenomenon and Talbot-like patterns are formed. This methodological approach not only sheds light on the behavior of SW interference under different conditions but also enhances our understanding of the SW self-imaging process in both isotropic and anisotropic media.

Cascaded chiral birefringent media enabled planar lens with programable chromatic aberration

Wavefront control is the fundamental requirement in optical informatics. Planar optics have drawn intensive attention due to the merits of compactness and light weight. However, it remains a challenge to freely manipulate the dispersion, hindering practical applications, especially in imaging. Here, we propose the concept of frequency-synthesized phase engineering to solve this problem. A phasefront-frequency matrix is properly designed to encode different spatial phases to separate frequencies, thus makes arbitrary dispersion tailoring and even frequency-separated functionalization possible. The periodically rotated director endows cholesteric liquid crystal with a spin and frequency selective reflection. Moreover, via presetting the local initial orientation of liquid crystal, geometric phase is encoded to the reflected light. We verify the proposed strategy by cascading the chiral anisotropic optical media of specifically designed helical pitches and initial director orientations. By this means, planar lenses with RGB achromatic, enhanced chromatic aberration and color routing properties are demonstrated. Inch-sized and high-efficient lenses are fabricated with low crosstalk among colors. It releases the freedom of dispersion control of planar optics, and even enables frequency decoupled phase modulations. This work brings new insights to functional planar optics and may upgrade the performance of existing optical apparatuses.

A new highly accurate and efficient pure visco‐acoustic wave equation for tilted transversely isotropic attenuating media

AbstractThe propagation of seismic waves in attenuating anisotropic media exhibits amplitude dissipation and phase dispersion. To describe its effects, the fractional Laplacian pure visco‐acoustic wave equations capable of producing stable and noise‐free wavefields have been derived. However, except for acoustic approximation, previous wave equations utilize the approximations with lower accuracy in simplifying the denominator of the approximate complex‐valued dispersion relation, resulting in reduced accuracy. To address this concern, we use a combination of complex stiffness coefficients to replace the denominator term of the approximate complex‐valued dispersion relation. This approximation effectively reduces the loss of accuracy caused by ignoring the influence of the velocity anisotropy parameter ε and the attenuation anisotropy parameter εQ in the denominator term, leading to a wave equation with high accuracy in media with large anisotropic parameters ε and δ. In addition, the new wave equation only contains two high‐order spatial partial derivatives and has high computational efficiency. Theoretical analysis and numerical examples demonstrate that the proposed pure visco‐acoustic tilted transversely isotropic wave equation outperforms the previous pure visco‐acoustic wave equation in terms of simulation accuracy. The newly developed wave equation is well suited for the application of Q‐compensated reverse time migration and full waveform inversion in attenuating anisotropic media.