Stratified Anisotropic Media Research Articles

Second harmonic generation in anisotropic stratified media: a generalization of the Berreman method and its application to photonic materials.

We have developed a numerical method for calculating the second-harmonic generation (SHG) generated by an anisotropic material whose optical properties present an arbitrary modulation in one dimension. The method is based on the Berreman 4 × 4 matrix formalism, which is generalized to include nonlinear optical phenomena. It can be used under oblique incidences of the input beam, and is valid even when the SHG frequency is close to photonic bands, where the usual slowly-varying-amplitude approximation breaks down. As an example of application, we have studied the SHG performance of ferroelectric and helielectric fluids. The obtained results indicate that the present procedure may contribute to improving the structural design and enlarging the variety of nonlinear optical materials for application in optical devices.

Spectral properties of exterior transmission problem for spherically stratified anisotropic media

ABSTRACT The exterior transmission eigenvalue problem for anisotropic media with spherical symmetry assumptions is considered. First, we discuss the existence and the asymptotic distribution of exterior transmission eigenvalues. Then, we investigate the inverse problem of determining the index of refraction from a knowledge of two sets of spectral data under appropriate conditions.

Horizontally polarized shear waves in stratified anisotropic (monoclinic) media

The dispersion relations are derived for the horizontally polarized shear (SH) waves in stratified anisotropic plates with arbitrary elastic anisotropy. Analytical expressions for the vectorial group and ray velocities of SH waves propagating in anisotropic layers with monoclinic symmetry are obtained. Closed form relations between velocities and specific kinetic and strain energy for SH waves are derived and analyzed.

Azimuthal seismic responses from shale formation based on anisotropic rock physics and reflectivity method: a case study from south-west China

Due to intrinsic anisotropy related to preferred alignment of clay particles and the existence of vertical or high angle fractures, shales usually present orthorhombic anisotropy. The objective of this study was to build anisotropic rock physics models for shales at the seismic scale. Based on the well-log and Formation Micro Imager (FMI) log data from a shale formation in the Sichuan Basin in south-west China, we derive an orthorhombic model at the seismic scale by using Schoenberg and Helbig’s method and generalised Backus averaging method in the rock physics workflow. In order to understand the relationship between physical properties of the rock physics model and seismic wave propagation, we apply the simplified reflectivity method to calculate seismic responses for amplitude variation with azimuth (AVAz) analysis. The method is based on the scheme of anisotropic reflectivity method which is commonly used to simulate the full-wave field in stratified anisotropic media, in analogy with the formula of horizontal slowness components in Schoenberg and Protázio’s method. The AVAz analysis is conducted on the seismograms of PP-wave, radial and transverse components of PS-wave. The results show that overburden effects caused by wave propagation in anisotropic media can’t be ignored. Azimuthal variations in amplitudes of both PP-wave and radial component of PS-wave can be used to indicate strikes of fractures, while PS-wave appears to be more sensitive.

Optical Characterization of a Thin-Film Material Based on Light Intensity Measurements

Light interacts with materials in a variety of ways; this article focuses on determination of refraction and absorption characterized by a material’s refractive index. We discuss some of the useful models for the frequency dependence of the refractive index, and practical approaches to calculating refractive indices of thin films and thick substrates. The efficiency of manufacturing of existing and successful creation of new devices of solid-state micro- and nanoelectronics largely depends on the level of development of the technology for manufacturing layers of various materials with a thickness of several nanometers to tens of micrometers. A high degree of perfection of layered structures and particularly structures based on dielectric and/or metallic films with nanometer thickness is needed for their successful application in micro-, nano-, acousto-, microwave and optoelectronics. It is impossible to achieve high degree of perfection without the use of high-precision methods of measuring electrophysical parameters of dielectric and semiconductor materials and structures, metallic films. We have developed the program “Multilayer”, which serves both to simulate the propagation of light through multilayer thin-film layered media, and to determine the dielectric (permittivity tensor of anisotropic films) and geometric (physical and optical thicknesses of the film) parameters of various thin-film coatings. The base mathematical models applied for the description of the light wave propagation through a homogeneous optical medium and for the determination of the optical characteristics of thin layers of optical materials based on the results of light intensity measurements are described. The main mathematical formalism employed in the program is based on solving the Maxwell’s equations for propagation of light through anisotropic stratified media. The algorithm uses the Berreman matrices of order

Resonant absorption of electromagnetic waves in transition anisotropic media.

We study the mode conversion and resonant absorption phenomena occurring in a slab of a stratified anisotropic medium, optical axes of which are tilted with respect to the direction of inhomogeneity, using the invariant imbedding theory of wave propagation. When the tilt angle is zero, mode conversion occurs if the longitudinal component of the permittivity tensor, which is the one in the direction of inhomogeneity in the non-tilted case, varies from positive to negative values within the medium, while the transverse component plays no role. When the tilt angle is nonzero, the wave transmission and absorption show an asymmetry under the sign change of the incident angle in a range of the tilt angle, while the reflection is always symmetric. We calculate the reflectance, the transmittance and the absorptance for several configurations of the permittivity tensor and find that resonant absorption is greatly enhanced when the medium from the incident surface to the resonance region is hyperbolic than when it is elliptic. For certain configurations, the transmittance and absorptance curves display sharp peaks at some incident angles determined by the tilt angle.

Apparent attenuation of shear waves propagating through a randomly stratified anisotropic medium

Waves propagating through heterogeneous media experience scattering that can convert a coherent pulse into small incoherent fluctuations. This may appear as attenuation for the transmitted front pulse. The classic O’Doherty–Anstey theory describes such a transformation for scalar waves in finely layered media. Recent observations for seismic waves in the earth suggest that this theory can explain a significant component of seismic attenuation. An important question to answer is then how the O’Doherty–Anstey theory generalizes to seismic waves when several wave modes, possibly with the same velocity, interact. An important aspect of the O’Doherty–Anstey theory is the statistical stability property, which means that the transmitted front pulse is actually deterministic and depends only on the statistics of the medium but not on the particular medium realization when the medium is modeled as a random process. It is shown in this paper that this property generalizes in the case of elastic waves in a nontrivial way: the energy of the transmitted front pulse, but not the pulse shape itself, is statistically stable. This result is based on a separation of scales technique and a diffusion-approximation theorem that characterize the transmitted front pulse as the solution of a stochastic partial differential equation driven by two Brownian motions.

Frequency-domain inversion of marine CSEM data in one-dimensional vertically anisotropic structures

The marine controlled-source electromagnetic (CSEM) method has proven to be an important addition to seismic imaging techniques in exploration of offshore hydrocarbon reservoirs and near-surface investigations. In inversion and interpretation of industrial CSEM data sets, this method can be used in a number of situations by assuming isotropy. However, the presence of electric anisotropy in the Earth's crust, due to thin layer interbedding or grain alignments in the sediments, can significantly alter the response measured by the EM receivers. Ignoring anisotropy in interpreting marine CSEM data may lead to distorted images of seabed conductivity structures, even misinterpretation.#br#In this paper, we present an inversion method for frequency domain marine controlled-source electromagnetic data generated by a titled dipole source in vertically anisotropic stratified media. This approach is based on the Gauss-Newton scheme. We extend and generalize the formulation of marine CSEM fields to calculation of the electromagnetic fields excited by arbitrarily oriented dipole sources. The partial derivatives of the electromagnetic fields with respect to both the horizontal and vertical resistivity are analytically calculated. Based on the relationship between the horizontal resistivity (ρh) and the vertical resistivity (ρv) of the inversion model, an adaptive selection method for regularization factors is proposed to balance the effects of the data misfit and the structural constraint.#br#The synthetic and real data inversion tests indicate that our inversion method can reconstruct the anisotropic resistivity of the overburden layer and the basement, and the burial depth, the thickness and the vertical resistivity of the reservoir layer can be well recovered. A better reconstruction can be obtained with multiple frequency and multiple component data sets. Combing inline and broadside geometry data sets can provide high resolution in reconstructing the burial depth of the reservoir layer and resistivity of the anisotropic basement.

Light propagation in stratified anisotropic media

Light propagation in stratified anisotropic media with arbitrary orientation of optic axes smoothly varying from layer to layer is considered. In the WKB approximation, a general expression for the field is obtained. For the case of a uniaxial medium, the normal waves are found and specific features of the light propagation are analyzed. General conditions are obtained that determine the turning points and forbidden zones. It is shown that the developed approach allows one to find trajectories of rays in anisotropic media with arbitrary layered structure.

Numerical Modeling of Multicomponent Induction Well-Logging Tools in the Cylindrically Stratified Anisotropic Media

In this paper, we present an efficient algorithm to simulate the response of a multicomponent induction well-logging (MCIL) tool in a cylindrically layered anisotropic formation (transversely isotropic media). The tool consists of three pairs of orthogonally arranged transmitter-receiver coils for measuring the magnetic field around a borehole for oil and gas exploration. First, we will derive the EM field components to present the coupled transverse magnetic and transverse electric modes in the spectral domain for an arbitrary observation layer from the Maxwell equation and obtain the spectral-domain Green's function. The new local transmission and both the local and generalized reflections for the media are given. After that, we use a cubic spline interpolation to realize the inverse Fourier transformation and obtain the spatial-domain Green's function for the investigation of the response characteristics of the MCIL tool. Finally, we give some numerical results to explore the response characteristics of the tool in several different environments.

A 2.5D finite-element-modeling difference method for marine CSEM modeling in stratified anisotropic media

We have developed a 2.5D finite-element modeling (FEM) method for marine controlled-source electromagnetic (CSEM) applications in stratified anisotropic media. The main feature of the method is that delta sources are used to solve the governing partial differential equations for cases with and without a resistive target and to obtain the difference of these two solutions as the scattered field from the target. The total field is then the sum of the analytical background field calculated with a 1D modeling method and the difference or scattered field mentioned above. Compared with a conventional direct solution (using delta sources directly in a 2.5D formulation), the new method has smaller near-field error as a result of the source singularity and smaller boundary reflections. The new method does not require a dense mesh in the source region, which thereby reduces the total number of variables to be solved. In this way, the modeling time can be kept within a few minutes for some cases. We show that the maximum relative error of the calculation can be kept within 2% for targets at depths of approximately [Formula: see text]. The method is valid for stratified anisotropic media. The anisotropic modeling examples show that (1) marine CSEM is predominantly sensitive to target vertical resistivity and not to target horizontal resistivity, provided that the targets are thin, horizontal, high-resistivity layers and (2) marine CSEM is sensitive to the horizontal resistivity of the conductive sediments surrounding the target (e.g., the overburden).

Reflection of light from nanoscopically stratified anisotropic media and optical probing of dielectric nanofilms

The reflection of linearly polarized light from a layered nanoscopic system of anisotropic dielectric films on homogeneous isotropic substrate is investigated in the long-wavelength limit. The expressions for reflection characteristics of an N-layer system of arbitrarily anisotropic ultrathin films are derived. The analytical results are supported by computer-aided analysis made on the basis of general wave propagation theory for anisotropic layered media. We show that the accuracy of the long-wavelength approximation for nanoscopic anisotropic multilayers is quite satisfactory: if the thickness of a multilayer divided by the wavelength comprises a few hundredths, then the errors of approximate formulas do not exceed a few percent. The most useful feature of obtained approximate expressions is that they are simply invertible, allowing a direct calculation of the parameters of ultrathin layers.

Anisotropy from full waveform inversion of multicomponent seismic data using a hybrid optimization method

ABSTRACTSeismic waveforms contain valuable information about the media, but the waveform inversion is a non‐linear problem. We present a waveform inversion method that combines a local optimization method with a genetic algorithm to determine the anisotropic parameters of a horizontally stratified medium. Synthetic seismograms for a horizontally stratified anisotropic medium are calculated using the reflectivity technique. In the initial stage of the inversion, the global space‐sampling properties of the genetic algorithm are used to direct the search to the region close to the global solution. This solution is then further improved using a conjugate‐gradient method. The numerical experiments performed with noisy synthetic data show that our hybrid optimization method satisfactorily reconstructs the anisotropic parameters at a reasonable computing cost while the range of slowness is adequate. We found that (i) for small‐angle data neither single‐ nor multiple‐component data are sufficient to determine the anisotropic parameters uniquely; (ii) for medium‐angle data the multiple‐component data are sufficient to determine the anisotropic parameters exactly whereas the single‐component data are not sufficient; and (iii) for wide‐angle data, either single‐ or multiple‐component data are sufficient to determine the anisotropic parameters accurately.

An algorithm for solving the optical problem for stratified anisotropic media

An algorithm for solving the Maxwell equations for propagation of light through anisotropic stratified media is considered. The algorithm uses the Berreman matrices of order 4 × 4. In contrast to the numerical methods suggested by Berreman, the new method is exact. The Sylvester theorem for calculating functions of a matrix and the Laguerre method for determining eigenvalues provide the basis for an algorithm with an efficiency comparable to that of the algorithms based on analytic solutions, which exist only in the case of uniaxial media. The method suggested in this paper allows for the analysis of complex optical systems where the effects of biaxiality, magnetic anisotropy, and optical activity play an important role.

Beam Propagation Method Modelling of Zenithal Bistable Nematic Devices: Analysis and Assessment

Modelling of liquid crystal optics is commonly based on variants of matrix-type solvers, which are ideally suited for addressing light wave propagation in stratified anisotropic media. Liquid crystal devices with complex director structures on the micron scale require more rigorous methods for predicting their optical response. Two potential candidate methods exist: the Beam Propagation Method and the Finite-Difference Time-Domain Method. The scope of this paper is to apply the former and assess its accuracy against the latter, which is considered to be more general. The analysis and assessment is focused on Zenithal Bistable Nematic devices.

Eikonal groups of photon and phonon propagators in one-dimensional structures

A general tensor eikonal method is proposed for solving wave equations of the optics and acoustics of stratified anisotropic media in the case of non-commutation of the tensor eikonals. Such an approach allows one to operate with electromagnetic and acoustic fields in inhomogeneous anisotropic media without any division of them into partial waves and not referring to a particular coordinate systems. The tensor eikonals which are determined by the polarization of waves are involved in the evolution operators (propagators) and are expressed in terms of the normal refraction tensors. These tensors are non-commutative in the general case and disentangling the evolution operators is necessary, including ones for waves in isotropic media. Such disentangling is performed with the use of known standard operator procedures. An example of the calculation of the photon propagator is given for a stratified medium with the inhomogeneity profile e(z) = a + b/z2.

New method for wave radiation and propagation in anisotropic stratified media

A matrix-exponent method is proposed to treat wave radiation and propagation in anisotropic stratified media. By using matrix-exponent formulations, the present method is elegant for expressions and reliable for computations. As an example, the method is used to calculate the ELF excitation of the Earth-ionosphere waveguide by an equivalent dipole resulting from heating the lower ionosphere with modulated HF waves.

An alternative approach for magneto-optic calculations involving layered media

A generalized 4 × 4 matrix method for solving for the propagation of electromagnetic plane waves in optically isotropic and anisotropic stratified media has been extended to calculate the relative Kerr or Faraday intensities for most optical arrangements. The utility of this method lies in its versatility. The layered structures may consist of magnetic metals, non-magnetic metals, insulators, or semiconductors. For a given light scattering arrangement, the reflectivity, Kerr rotation, Kerr ellipticity, Faraday rotation, and Faraday ellipticity may be calculated as functions of the incidence angle, laser frequency, overlayer coverage, or underlayer thickness. Experimental intensity measurements can be checked for different configurations of the light-polarizing and beam-modulating optics.

Efficient numerical computation of the spectral transverse dyadic Green's function in stratified anisotropic media

Several new ideas are provided for computing the spectral transverse dyadic Green's function (STDGF) of a stratified anisotropic medium in an accurate and efficient way. A modified version of the equivalent boundary method is developed which makes it possible to compute the asymptotic values of the aforementioned STDGF without employing exponential terms. This new technique for the computation of the STDGF of a stratified anisotropic medium has more numerical stability than the other techniques proposed so far, in which the use of exponential terms for the computation of the STDGF in all the range of variation of the spectral variables unavoidably leads to numerical problems for high values of the spectral variables. Also in this paper, it is shown how a two-step interpolation approach can be used for obtaining simple closed-form expressions of the STDGF of a stratified anisotropic medium in wide ranges of the spectral variables. It is expected that the use of this interpolation approach will introduce im...

Travel time computation in stratified anisotropic media: Generalized ray method

The generalized ray, combined with the system matrix, can be utilized to calculate the travel time of the elastic waves in stratified anisotropic media. The method proposed by Van der Hijden is modified to calculate the horizontal slowness (corresponding to travel time) of the Cagniard path. The accuracy of this method is verified by comparing the results with those of perturbation and ray tracing methods.