Effective Permeability Tensor Research Articles

Effective Constitutive Parameters Extraction of Bianisotropic Metamaterials Using Bloch Modes

In this article, a method is proposed to extract the effective constitutive parameters of bianisotropic metamaterials using Bloch modes. We consider 3-D structures that are realized by stacking identical unit layers, where each unit layer is a 2-D periodic array of metallic or dielectric elements of arbitrary shape embedded in a background dielectric medium. Bianisotropic metamaterials do not satisfy the condition of reflection symmetry of unit layers considered in previous studies. The eigenvectors and eigenvalues of the generalized transfer matrix of a unit layer are related to the Bloch modes of the structure. When only two dominant Bloch modes exist, closed-form expressions are found for effective permittivity, permeability, and magnetoelectric tensors of the medium. The effects of nondominant Bloch modes are taken into account as interface surface impedance matrices. The calculated scattering parameters of a slab of bianisotropic metamaterial are shown to be in good agreement with the results obtained from full-wave electromagnetic simulations using commercial solvers.

EQUIVARIANT GEOMETRIC LEARNING FOR DIGITAL ROCK PHYSICS: ESTIMATING FORMATION FACTOR AND EFFECTIVE PERMEABILITY TENSORS FROM MORSE GRAPH

We present a SE(3)-equivariant graph neural network (GNN) approach that directly predicts the formation factor and effective permeability from micro-CT images. Fast Fourier Transform (FFT) solvers are established to compute both the formation factor and effective permeability, while the topology and geometry of the pore space are represented by a persistence-based Morse graph. Together, they constitute the database for training, validating, and testing the neural networks. While the graph and Euclidean convolutional approaches both employ neural networks to generate low-dimensional latent space to represent the features of the microstructures for forward predictions, the SE(3) equivariant neural network is found to generate more accurate predictions, especially when the training data are limited. Numerical experiments have also shown that the new SE(3) approach leads to predictions that fulfill the material frame indifference whereas the predictions from classical convolutional neural networks (CNNs) may suffer from spurious dependence on the coordinate system of the training data. Comparisons among predictions inferred from training the CNN and those from graph convolutional neural networks with and without the equivariant constraint indicate that the equivariant graph neural network seems to perform better than the CNN and GNN without enforcing equivariant constraints.

Asymptotic homogenization in the determination of effective intrinsic magnetic properties of composites

We present a computational framework for two-scale asymptotic homogenization to determine the intrinsic magnetic permeability of composites. To this end, considering linear magnetostatics, both vector and scalar potential formulations are used. Our homogenization algorithm for solving the cell problem is based on the displacement method presented in Lukkassen et al. 1995, Composites Engineering, 5(5), 519-531. We propose the use of the meridional eccentricity of the permeability tensor ellipsoid as an anisotropy index quantifying the degree of directionality in the linear magnetic response. As application problems, 2D regular and random microstructures with overlapping and nonoverlapping monodisperse disks, all of which are periodic, are considered. We show that, for the vanishing corrector function, the derived effective magnetic permeability tensor gives the (lower) Reuss and (upper) Voigt bounds with the vector and scalar potential formulations, respectively. Our results with periodic boundary conditions show an excellent agreement with analytical solutions for regular composites, whereas, for random heterogeneous materials, their convergence with volume element size is fast. Predictions for material systems with monodisperse overlapping disks for a given inclusion volume fraction provide the highest magnetic permeability with the most increased inclusion interaction. In contrast, the disk arrangements in regular square lattices result in the lowest magnetic permeability and inadequate inclusion interaction. Such differences are beyond the reach of the isotropic effective medium theories, which use only the phase volume fraction and shape as mere statistical microstructural descriptors.

Prediction of numerical homogenization using deep learning for the Richards equation

For the nonlinear Richards equation as an unsaturated flow through heterogeneous media, we build a new coarse-scale approximation algorithm utilizing numerical homogenization. This approach follows deep neural networks (DNNs) to quickly and frequently calculate macroscopic parameters. More specifically, we train neural networks with a training set consisting of stochastic permeability realizations and corresponding computed macroscopic targets (effective permeability tensor, homogenized stiffness matrix, and right-hand side vector). Our proposed deep learning scheme develops nonlinear maps between such permeability fields and macroscopic characteristics, and the treatment for Richards equation’s nonlinearity is included in the predicted coarse-scale homogenized stiffness matrix, which is a novelty. This strategy’s good performance is demonstrated by several numerical tests in two-dimensional model problems, for predictions of the macroscopic properties and consequently solutions.

Integral effective medium approach for a metamaterial with radially-inhomogeneous spherical inclusions

In this study, a novel integral effective medium approach (IEMA) to obtain the complex effective permittivity and permeability tensors of 3-D metal-dielectric composite is developed. An infinite isotropic homogenous dielectric medium with periodically imbedded spherical inclusions with spherical inhomogeneity is considered here as an artificial dielectric in subwavelength. Within full-scattering theory, it is shown that such composite belongs to a general class of metamaterials with spherical inclusions.The obtained expressions for the elements of the effective tensors are valid in the entire subwavelength range. In order to test the proposed IEMA, the problem for metallic spherical particles coated by a layer of dielectric is solved analytically in greater detail by using the approach proposed earlier by the first author of the study.We were able to show that a metal-dielectric composite/metamaterial with spherical inhomogeneous inclusions can be considered as a quasi-periodic artificial crystal with a tunable band gap in the subwavelength range. In certain sub-ranges in subwavelength, such crystals can exhibit the properties of homogeneous ferrite material.

Field enhancement in hydrogen storage by periodic layered structures

This paper investigates, through the field enhancement factor, the increase of hydrogen adsorption around the interface between a layer of hydrogen and a periodic layered structure under different incidence angles of an applied transverse magnetic polarized electromagnetic field. This periodic layered structure is composed of n-binary unit cells based on alternating a thin layer of gold with a thin layer of a metamaterial with equal negative relative permittivity while the relative permeability of the first and the second material's unit cell is considered equal to 1 and -1, respectively. We apply the effective medium theory to replace this layered structure with a single slab of a homogeneous material with an effective permittivity tensor and an effective permeability tensor. We use the Transfer Matrix Method to analyze the reflectivity spectra at the hydrogen/slab interface for adjustable layers’ thicknesses and then we derive the field enhancement factor. We obtain a significant increase of the field enhancement factor of the structure in comparison with the field enhancement factor around the interface between a layer of hydrogen and a structure composed of one gold layer.

Observation of optical gyromagnetic properties in a magneto-plasmonic metamaterial

Metamaterials with artificial optical properties have attracted significant research interest. In particular, artificial magnetic resonances with non-unity permeability tensor at optical frequencies in metamaterials have been reported. However, only non-unity diagonal elements of the permeability tensor have been demonstrated to date. A gyromagnetic permeability tensor with non-zero off-diagonal elements has not been observed at the optical frequencies. Here we report the observation of gyromagnetic properties in the near-infrared wavelength range in a magneto-plasmonic metamaterial. The non-zero off-diagonal permeability tensor element causes the transverse magneto-optical Kerr effect under s-polarized incidence that otherwise vanishes if the permeability tensor is not gyromagnetic. By retrieving the permeability tensor elements from reflection, transmission, and transverse magneto-optical Kerr effect spectra, we show that the effective off-diagonal permeability tensor elements reach 10−3 level at the resonance wavelength (~900 nm) of the split-ring resonators, which is at least two orders of magnitude higher than magneto-optical materials at the same wavelength. The artificial gyromagnetic permeability is attributed to the change in the local electric field direction modulated by the split-ring resonators. Our study demonstrates the possibility of engineering the permeability and permittivity tensors in metamaterials at arbitrary frequencies, thereby promising a variety of applications of next-generation nonreciprocal photonic devices, magneto-plasmonic sensors, and active metamaterials.

Broadband Applications of a Tunable Nano-Rod Metaferrite

The rigorous analytical model of broadband effective electromagnetic response of the two-component metaferrite is presented in this study in the range up to THz frequencies. The metaferrite is a square array of ferric or ferromagnetic infinitely long cylinders of circular cross section symmetrically imbedded into homogenous isotropic dielectric host medium. It is assumed that the cylinders are fully or partially magnetized by a dc biasing magnetic field directed along the cylinder axes. Expressions of the complex effective permeability and permittivity tensors are obtained in the study. Expressions of the complex effective relative permeability and permittivity transverse to the direction of magnetization are derived and analyzed as functions of the frequency of incident electromagnetic wave and constitutive parameters of the metaferrite.It is shown that tuning a dc biasing magnetic field enables to effectively change the transmittivity and reflectivity of the metaferrites in the considered frequency range. We were also able to show that a layer of the metaferrite can be used for exciting surface plasmons at GHz frequencies. Finally, we have discussed the principal possibility of usage of the considered metaferrite for fabricating the substrates of miniaturized patch antennas with improved performance.

Efficient radiational outcoupling of electromagnetic energy from hyperbolic metamaterial resonators

Hyperbolic metamaterials were initially proposed in optics to boost radiation efficiencies of quantum emitters. Adopting this concept for antenna design allows approaching long-standing contests in radio physics. For example, broadband impedance matching, accompanied with moderately high antenna gain, is among the existent challenges. Here we propose employing hyperbolic metamaterials for a broadband impedance matching, while a structured layer on top of a metamaterials slab ensures an efficient and directive energy outcoupling to a free space. In particular, a subwavelength loop antenna, placed underneath the matching layer, efficiently excites bulk metamaterial modes, which have well-resolved spatial–temporal separation owing to the hypebolicity of effective permeability tensor. Interplaying chromatic and modal dispersions enable to map different frequencies into non overlapping spatial locations within a compact subwavelength hyperbolic slab. The outcoupling of energy to the free space is obtained by patterning the slab with additional resonant elements, e.g. high index dielectric spheres. As the result, two-order of magnitude improvement in linear gain of the device is predicted. The proposed new architecture can find a use in applications, where multiband or broadband compact devices are required.

Homogenization method for one-dimensional photonic crystals with magnetic and chiral inclusions

We have theoretically investigated the electromagnetic properties for one-dimensional (1D) photonic crystals with magnetic and artificial chiral inclusions in the quasi-static limit, that is when the size of the unit cell of the crystal is small with respect to the wavelength of the operating wave. We suggest a homogenization theory to determine the effective tensors of the optical response, to achieve this objective, we apply the Bloch’s plane waves method to describe the electromagnetic modes that can propagate in the periodic structure under consideration. Subsequently, the Maxwell’s “microscopic” equations are homogenized replacing the Bloch waves by plane waves that attenuate the fast oscillations of the electromagnetic fields within the unit cell (macroscopic level), i.e., the average-macroscopic electromagnetic fields are determined by the component corresponding to the null reciprocal lattice vector in the expansion in plane waves. The numerical implementation of our theory of homogenization allow us to study the effective bianisotropic electromagnetic response (effective dielectric permittivity, magnetic permeability and electric-magnetic coupling tensors, this last is described by an effective chiral parameter) for 1D photonic crystals whose constituents in its unit cell are a dielectric layer and another magnetic (both isotropic and anisotropic) or chiral inclusion layer. The results are illustrated and discussed by the effective parameters as a function of the filling fraction of the inclusion and show for each case of homogeneous effective medium different components of anisotropy in the electromagnetic response of permittivity, permeability and chirality with the increase of the filling fraction. Besides, the behaviors of the obtained graphs agree well with Rytov’s formulas of effective medium. The relevant results of this theory will be very useful for the study and better understanding of the nature and design of metamaterials with predetermined anisotropic optical properties.

Flow of shear-thinning fluids through porous media

Pseudo-plastic fluids exhibit a non-linear stress-strain relationship which can provoke large, localized viscosity gradients. For the flow of such fluids in porous media the consequence is a strong variability of the effective permeability with porosity, angle of the macroscopic pressure gradient, and rheological parameters of the fluid. Such a variability is investigated on the basis of adjoint homogenization theory for a Carreau fluid in an idealized porous medium geometry, highlighting differences with respect to the Newtonian case. It is shown in particular that the more we depart from Newtonian conditions, the more the (often used) hypothesis of an effective viscosity in Darcy’s law is a poor approximation, for the effective permeability tensor becomes strongly anisotropic.

Effective medium properties of a ferromagnetic microwire grid

The work presented in this paper involves the analysis of effective permittivity and permeability of a ferromagnetic microwire grid consisting of an infinite number of ferromagnetic microwires. Analytical solution is obtained through satisfying the tangential boundary conditions at the surface of a reference microwire. Subsequently, the local and average fields and hence, effective permittivity and permeability are obtained within a unit cell. The analysis reported in the literature allows to calculate only and components of the effective permittivity and permeability for the case of polarization with normal incidence. Proposed analysis provides a most generalized solution for the three diagonal elements of the effective permittivity and permeability tensors for the case of arbitrary polarization with arbitrarily incident uniform plane wave. Numerical results are obtained through MATLAB to show the impact of frequency, radius of the microwires, periodicity and applied external magnetization on the effective permittivity and permeability for and polarization. A comparative analysis is also done with the numerical results available in the literature to validate the proposed method of analysis.

Effective Parameters for 1D Photonic Crystals with Isotropic and Anisotropic Magnetic Inclusions: Coherent Wave Homogenization Theory.

A homogenization theory that can go beyond the regime of long wavelengths is proposed, namely, a theory that is still valid for vectors of waves near the edge of the first zone of Brillouin. In this paper, we consider that the displacement vector and the magnetic induction fields have averages in the volume of the cell associated with the values of the electric and magnetic fields in the edges of the cell, so they satisfy Maxwell’s equations. Applying Fourier formalism, explicit expressions were obtained for the case of a photonic crystal with arbitrary periodicity. In the case of one-dimensional (1D) photonic crystals, the expressions for the tensor of the effective bianisotropic response (effective permittivity, permeability and crossed magneto-electric tensors) are remarkably simplified. Specifically, the effective permittivity and permeability tensors are calculated for the case of 1D photonic crystals with isotropic and anisotropic magnetic inclusions. Through a numerical calculation, the dependence of these effective tensors upon the filling fraction of the magnetic inclusion is shown and analyzed. Our results show good correspondence with the approach solution of Rytov’s effective medium. The derived formulas can be very useful for the design of anisotropic systems with specific optical properties that exhibit metamaterial behavior.

Effective permeability tensor of confined flows with wall grooves of arbitrary shape

Pressure and shear-driven flows of a confined film of fluid overlying a periodic one-dimensional topography of arbitrary shape are considered for prediction of the effective hydraulic permeability in the Stokes flow regime. The other surface confining the fluid may be a planar no-slip wall, an identically patterned wall, a free surface or a surface with prescribed shear. Analytical predictions are obtained using spectral analysis and the domain perturbation method under the assumption of small pattern size to pitch ratio. Using a novel decomposition of the channel height effects into exponentially and algebraically decaying components, a simple surface-metrology-dependent relationship which connects the eigenvalues of the effective permeability tensor is obtained. Two representative topographies are assessed numerically: the infinitely differentiable topography of a phase-modulated sinusoid which has multiple local extrema and zero crossings and the non-differentiable triangular-wave topography. Non-differentiability in the form of corners of triangular patterns and the cusps of scalloped patterns are not found to be an impediment to meaningful and numerically accurate asymptotic predictions of effective permeability and effective slip, contradicting an earlier suggestion from the literature. Several distinct applications of the theory to the friction-reduction and shear-stability performance of the recently developed lubricant impregnated patterned surfaces as well as to scalloped and trapezoidal drag-reduction riblets are discussed, with comparison to experimental data from the literature for the last application. Analytical approximations which have an extended domain of numerical accuracy are also proposed.

MAGNETIC PROPERTIES OF MULTICOMPONENT HETEROGENEOUS MEDIA WITH A DOUBLY PERIODIC STRUCTURE

Heterogeneous media have a wide range of practical applications. Media with a doubly periodic structure (matrices of high-gradient magnetic separators, etc.) occupy an important place. Their study is usually based on experimental and approximate methods and is limited to simple two-phase systems. The development of universal and accurate methods of mathematical modelling of electrophysical processes in such environments is an urgent task. The aim of the paper is to develop a method for calculating local and effective parameters of a magnetostatic field with minimal restrictions on the number of phases, their geometry, concentration, and magnetic properties. Based on the theory of elliptic functions and secondary sources, an integral equation is formulated with respect to the magnetization vector of the elements of the main parallelogram of the periods. The calculated expressions for the complex potential, field strength, and components of the effective magnetic permeability tensor are obtained. The results of a series of computational experiments confirming the universality and effectiveness of the method are presented. As an example of a practical application, a detailed study of the field of the magnetic forces of the matrix is carried out: the lines of magnetic isodine and potential extraction areas for a complex version of the matrix are constructed. Within the framework of the developed method, the calculation of local and effective field characteristics is carried out by solving the field problem in the field of an arbitrary parallelogram of periods without specifying boundary conditions on its sides with a comprehensive consideration of significant interdependent factors. The practical value of the method is to create new opportunities for improving the technical characteristics of electrophysical devices for which the universality and accuracy of calculating local and effective field characteristics is decisive. An algorithm for optimizing the characteristics of the separator is proposed.

Terahertz routing with graphene magnetic metamaterials

Terahertz wireless communication has emerged as a fast-growing research area. Routing, which is designed to control the propagation of terahertz wave, is of great importance for building the terahertz all-optical communication network. Metamaterials provide a good method for manipulating the terahertz wave, which can be used to realize the wave router. To steer the input wave in different directions, for example negative refraction angle, some structures have been proposed, such as stacked structures and nanowire arrays. However, in these metamaterials, the incident wave is generally limited to transverse magnetic polarization. Here, a new terahertz routing method for transverse electric wave is put forward with using graphene magnetic metamaterials. Theoretical analysis is performed to show that this metamaterial has negative components not only in effective permittivity tensor but also in effective permeability tensor. Routing is realized by topological transitions between different dispersion regimes in this metamaterial. Our finding provides a new candidate for signal routing of terahertz wave.

Dynamic regulation of nonlocal effect and nonlocal degree in gyromagnetic metamaterials with an applied magnetic field.

We report dynamic regulation of nonlocal degree, nonlocal effects and spatial dispersion characteristics for transverse electric (TE) waves in periodic layered gyromagnetic metamaterials (PLGMs) by an applied magnetic field. A nonlocal effective permeability tensor, relying on both frequency and wave vector, is derived by expanding the accurate dispersion relation obtained by the transfer-matrix method (TMM) to high-order terms. The numerical results indicate that the degree of nonlocality of electromagnetic response in such PLGMs is closely dependent on the ratio between the period of PLGMs and the working wavelength. There are giant spatial nonlocality and strong spatial dispersion near the center or boundary of the first Brillouin zone, which leads two or three propagating modes to appear in these regions for a fixed frequency. Interestingly, the degree of nonlocality, nonlocal effects and spatial dispersion properties in such PLGMs can be manipulated dynamically by an applied static magnetic field. In addition, it is possible that a quasi-straight isofrequency contour occurs in the case of linear response. These properties make the PLGMs become excellent candidates for designing photonic devices in information communication, storage, nondiffraction transmission, and so on.

Extraction of effective constitutive parameters of artificial media using Bloch modes

The effective constitutive parameters of a three-dimensional periodic structure are calculated using its Bloch modes. These modes and their propagation constants are obtained from eigenvectors and eigenvalues of the generalized transfer matrix of a unit layer of the structure. Effective bulk permittivity and permeability tensors of the medium are obtained when two of the Bloch modes are dominant, i.e., propagate without significant decay inside the medium. The effect of the strongly decaying Bloch modes, which are excited at the interface with a conventional medium, are included by means of surface impedance matrices. The results are in excellent agreement with full-wave electromagnetic simulations.

Engineering Electromagnetic Wave Propagation in Periodically Layered Gyromagnetic Metamaterials with an External Magnetic Field

Tunable transmission characteristics of electromagnetic waves in periodically layered metamaterials (PLMMs), constructed by alternating dielectric layer and gyromagnetic layer such as yttrium-iron-garnet (YIG), are investigated through an external magnetic field. Based on transfer matrix method (TMM) and effective medium theory (EMT), we compare the dispersion curves of TE waves propagating in the PLMMs, and obtain the effective permeability and permittivity tensors of the homogenized gyromagnetic PLMMs by expanding the exact dispersion relation in long-wavelength limit. Then, we calculate some propagating parameters when TE waves incident on the PLMMs by EMT in detail. Numerical results show that the refraction state of TE waves in the PLMMs can be dynamically changed between positive refraction state and cutoff state by controlling the external magnetic field. These exotic properties of gyromagnetic PLMMs may have wide potential applications in many fields such as sub-wavelength all-optical switches and wave cutoff devices.

Effective Elastic and Hydraulic Properties of Fractured Rock Masses with High Contrast of Permeability: Numerical Calculation by an Embedded Fracture Continuum Approach

In this work, the hydromechanical modeling of the fractured rock masses was conducted based on a new numerical simulation method named as embedded fracture continuum (EFC) approach. As the principal advantage, this approach allows to simplify the meshing procedure by using the simple Cartesian meshes to model the fractures that can be explicitly introduced in the porous medium based on the notion of fracture cells. These last elements represent the grid cells intersected by at least one fracture in the medium. Each fracture cell in the EFC approach present a continuum porous medium whose hydromechanical properties are calculated from ones of the matrix and ones of the intersected fractures, thanks for using the well‐known solution of the joint model. The determination of the hydromechanical properties of the fracture cells as presented in this work allows to provide the theoretical base and to complete some simple approximations introduced in the literature. Through different verification tests, the capability of the developed EFC approach to model the hydromechanical behavior of fractured rock was highlighted. An analysis of different parameters notably the influence of the fracture cell size on the precision of the proposed approach was also conducted. This novel approach was then applied to investigate the effective permeability and elastic compliance tensor of a fractured rock masses taken from a real field, the Sellafield site. The comparison of the results calculated from this approach with ones conducted in the literature based on the distinct element code (UDEC) presents a good agreement. However, unlike the previous studies using UDEC, which limits only in the case of fractured rock masses without dead‐end fractures, our approach allows accounting for this kind of fractures in the medium. The numerical simulations show that the dead‐end fractures could have a considerable contribution on the effective compliance moduli, while their effect can be neglected to calculate the overall permeability of the of fractured rock masses.