Arbitrary Inhomogeneous Media Research Articles

Path integral formulation of light propagation in a static collisionless plasma, and its application to dynamic plasma.

In many studies on the laser impinging on a plasma surface, an assumption is made that the reflection of a laser pulse propagating to a plasma surface takes place only at the turning point, at which the plasma density exceeds the critical one. A general reflection amplitude of light R from an arbitrary inhomogeneous medium can be obtained by solving a Riccati-type integral equation, which can be solved analytically in low-reflection conditions, i.e., |R|2 ≪ 1. In this work, we derive an intuitive analytic solution for the reflection amplitude of light R from a plasma surface by integrating all possible reflection paths given by the Fresnel equation. In the low-reflection condition, reflection paths having only one reflection event can be used. By considering the higher-order reflection paths, our analytic expression can describe reflection in the high-reflection condition. We show the results of a one-dimensional particle-in-cell simulation to support our discussions. Since our model derived for static plasmas is well corroborated by the simulation results, it can be a useful tool for analyzing light reflection from dynamically varying plasmas.

Modelling the electromagnetic field distribution in arbitrary inhomogeneous media using the Green’s function approach

The equations of electric vector and its spectrum for the electromagnetic field in a stationary isotropic inhomogeneous medium with conductivity, magnetic and permittivity with its arbitrary dependence from coordinates are presented. The method of solving this equations is based Green’s function together with the method of iterations. This approach is also applicable to optical frequencies.

Propagation Characteristics of Dielectric Waveguides with Arbitrary Inhomogeneous Media along the Middle Layer

In this paper, we apply an improved method for the guiding problem of dielectric waveguide with arbitrary inhomogeneous media along the middle layer introduced the defect layers, and analyzed the propagation characteristics of dielectric waveguide composed of dielectric circular cylinders and the arbitrary inhomogeneous media in the middle layer by using a combination of the improved Fourier series expansion method and multilayer method. Numerical results are given for the propagation constants in the first stop band regions, and the distribution of energy flow for both TE0 and TM0 modes. The influence of permittivity and the shape variation of arbitrary inhomogeneous media are discussed in the first stop band regions.

PROPAGATING COUPLED ALFVÉN AND KINK OSCILLATIONS IN AN ARBITRARY INHOMOGENEOUS CORONA

Observations have revealed ubiquitous transverse velocity perturbation waves propagating in the solar corona. We perform three-dimensional numerical simulations of footpoint-driven transverse waves propagating in a low β plasma. We consider the cases of distorted cylindrical flux tubes and a randomly generated inhomogeneous medium. When density structuring is present, mode coupling in inhomogeneous regions leads to the coupling of the kink mode to the Alfven mode. The decay of the propagating kink wave is observed as energy is transferred to the local Alfven mode. In all cases considered, modest changes in density were capable of efficiently converting energy from the driving footpoint motion to localized Alfven modes. We have demonstrated that mode coupling efficiently couples propagating kink perturbations to Alfven modes in an arbitrary inhomogeneous medium. This has the consequence that transverse footpoint motions at the base of the corona will deposit energy to Alfven modes in the corona.

Tutorial on seismic interferometry: Part 1 — Basic principles and applications

Seismic interferometry involves the crosscorrelation of responses at different receivers to obtain the Green’s function between these receivers. For the simple situation of an impulsive plane wave propagating along the [Formula: see text]-axis, the crosscorrelation of the responses at two receivers along the [Formula: see text]-axis gives the Green’s function of the direct wave between these receivers. When the source function of the plane wave is a transient (as in exploration seismology) or a noise signal (as in passive seismology), then the crosscorrelation gives the Green’s function, convolved with the autocorrelation of the source function. Direct-wave interferometry also holds for 2D and 3D situations, assuming the receivers are surrounded by a uniform distribution of sources. In this case, the main contributions to the retrieved direct wave between the receivers come from sources in Fresnel zones around stationary points. The main application of direct-wave interferometry is theretrieval of seismic surface-wave responses from ambient noise and the subsequent tomographic determination of the surface-wave velocity distribution of the subsurface. Seismic interferometry is not restricted to retrieving direct waves between receivers. In a classic paper, Claerbout shows that the autocorrelation of the transmission response of a layered medium gives the plane-wave reflection response of that medium. This is essentially 1D reflected-wave interferometry. Similarly, the crosscorrelation of the transmission responses, observed at two receivers, of an arbitrary inhomogeneous medium gives the 3D reflection response of that medium. One of the main applications of reflected-wave interferometry is retrieving the seismic reflection response from ambient noise and imaging of the reflectors in the subsurface. A common aspect of direct- and reflected-wave interferometry is that virtual sources are created at positions where there are only receivers without requiring knowledge of the subsurface medium parameters or of the positions of the actual sources.

A finite-difference contrast source inversion method

We present a contrast source inversion (CSI) algorithm using a finite-difference (FD) approach as its backbone for reconstructing the unknown material properties of inhomogeneous objects embedded in a known inhomogeneous background medium. Unlike the CSI method using the integral equation (IE) approach, the FD-CSI method can readily employ an arbitrary inhomogeneous medium as its background. The ability to use an inhomogeneous background medium has made this algorithm very suitable to be used in through-wall imaging and time-lapse inversion applications. Similar to the IE-CSI algorithm the unknown contrast sources and contrast function are updated alternately to reconstruct the unknown objects without requiring the solution of the full forward problem at each iteration step in the optimization process. The FD solver is formulated in the frequency domain and it is equipped with a perfectly matched layer (PML) absorbing boundary condition. The FD operator used in the FD-CSI method is only dependent on the background medium and the frequency of operation, thus it does not change throughout the inversion process. Therefore, at least for the two-dimensional (2D) configurations, where the size of the stiffness matrix is manageable, the FD stiffness matrix can be inverted using a non-iterative inversion matrix approach such as a Gauss elimination method for the sparse matrix. In this case, an LU decomposition needs to be done only once and can then be reused for multiple source positions and in successive iterations of the inversion. Numerical experiments show that this FD-CSI algorithm has an excellent performance for inverting inhomogeneous objects embedded in an inhomogeneous background medium.

Theoretical background of retrieving Green's function by cross-correlation: one-dimensional case

SUMMARY Recently, an assertion has been verified experimentally and theoretically that Green’s function between two receivers can be reproduced by cross-correlating the records at the receivers. In this paper, we have theoretically proved the assertion for 1-D media with the free surface by using the Thomson‐Haskell matrix method. Strictly speaking, one side of the cross-correlation between records at two receivers is the convolution between Green’s function and the autocorrelation function of the source wavelet. This study extends the geometry considered by Claerbout to two receivers vertically apart, and is a special case of the proof by Wapenaar et al. which dealt with 3-D arbitrary inhomogeneous media. However, a simple geometry in 1-D problems enables us to make the proof without any approximations and to better understand the physical background with more ease. That is the main advantage of this study. Though a 1-D geometry seems far from reality, it may be sufficient if an appropriate combination of receivers and earthquakes is selected. In fact, such a geometry is often seen in seismological observations by a vertical array of seismographs in the shallow subsurface. Therefore, we refer to a possibility that the proof in this paper is applied to the estimation of site amplification factors by using records of a vertical seismographic array.

The ADI Multi-Domain Pseudospectral Time-Domain Algorithm For 2-D Arbitrary Inhomogeneous Media

A new time domain algorithm which combines alternating-direction implicit (ADI) technique and the multi-domain pseudospectral time domain (PSTD) algorithm is presented in the paper. This algorithm not only maintains accuracy and flexibility of PSTD for an accurate treatment of arbitrarily curved objects but also effectively improves the stability problem. Some numerical examples of two dimensional scattering are given to illustrate the accuracy and stability of the proposed approach in this paper.

On application of the reciprocity theorem to calculation of a microwave radiation signal in inhomogeneous hot magnetized plasmas

A general expression for a radiation signal received by a wave-guide antenna from an arbitrary inhomogeneous medium with the spatial dispersion is obtained using the electrodynamics reciprocity theorem. The radiation may be either a spontaneous noise emission or it may result from the scattering of the incident wave off fluctuations. The signal calculation is reduced to finding the electromagnetic field radiated by the receiving antenna into a physically realizable substance whose wave properties differ, to some extent from these of the real medium. This problem is usually much simpler than direct signal calculation with the use of the Green function. The reciprocity approach is especially useful in the presence of cut-offs and resonances or/and when the radiation source is located in the antenna near-field zone.

Split-field and anisotropic-medium PML-FDTD implementations for inhomogeneous media

In this paper, we present three-dimensional finite-difference time-domain (FDTD) algorithms for transient simulation of electromagnetic-wave propagation in arbitrary inhomogeneous media, which incorporate the perfectly matched layer (PML) absorbing boundary condition. We discuss the choice of constitutive parameters inside the PML layers to match the interior inhomogeneous media for planar interfaces and corner regions. We illustrate the method using both a split-fleld PML-FDTD formulation in Cartesian coordinates and an anisotropic medium (unsplit) PML-FDTD formulation in cylindrical coordinates.

An accurate model of wave refraction over shallow water

A computer model of wave refraction is desirable, in the context of landscape modeling, to generate the familiar wave patterns seen near coastlines. In this article, we present a new method for the calculation of shallow water wave refraction. The method resorts to Fermat's principle of the shortest path and computes the propagation of wavefronts over an arbitrary inhomogeneous medium. The propagation of wavefronts produces a phase map for each terrain. This phase map is then coupled with a geometric model of waves to generate a heightfield representation of the sea surface. The method is accurate and provides realistic wave refraction effects. The degree of accuracy provided by this method has enabled us to identify two other wave transport mechanisms, namely diffraction and caustics, for which no solution yet exists in the computer graphics literature. Our method does not attempt, at present, to solve these transport mechanisms.

Modelling of electromagnetic fields in thin heterogeneous layers with application to field generation by volcanoes-theory and example

When interpreting electromagnetic fields observed at the Earth’s surface in a realistic geophysical environment it is often necessary to pay special attention to the effects caused by inhomogeneities of the subsurface sedimentary and/or water layer and by inhomogeneities of the Earth’s crust. The inhomogeneities of the Earth’s crust are expected to be especially important when the electromagnetic field is generated by a source located in a magma chamber of a volcano. The simulation of such effects can be carried out using generalized thin-sheet models, which were independently introduced by Dmitriev (1969) and Ranganayaki & Madden (1980). In the first part of the paper, a system of integral equations is derived for the horizontal current that flows in the subsurface inhomogeneous conductive layer and for the vertical current crossing the inhomogeneous resistive layer representing the Earth’s mantle. The terms relating to the finite thickness of the laterally inhomogeneous part of the model are retained in the equations. This only marginally complicates the equations, whilst allowing for a significant expansion of the approximation limits. The system of integral equations is solved using the iterative dissipative method developed by the authors in the period from 1978 to 1988. The method can be applied to the simulation of the electromagnetic field in an arbitrary inhomogeneous medium that dissipates the electromagnetic energy. When considered on a finite numerical grid, the integral equations are reduced to a system of linear equations that possess the same contraction properties as the original equations. As a result, the rate at which the iterative-perturbation sequence converges to the solution remains independent of the numerical grid used for the calculations. In contrast to previous publications on the method, aspects of the algorithm implementation that guarantee its effectiveness and robustness are discussed here.

Interference blots and fringe dislocations in optics of twisted birefringent media

The appearance of interference blots and of the dislocation of interference fringes is studied by means of modeling an arbitrary inhomogeneous birefringent medium with rotating principal optical axes by a system of two birefringent plates. A theoretical explanation of these irregularities is given. Conditions for the appearance of interference blots and fringe dislocations are derived. A method of classification of singular points in the fringe patterns is presented.

A new construction for spinor wave equations

The construction of discrete scalar wave propagation equations in arbitrary inhomogeneous media was recently achieved by using elementary dynamical processes realizing a discrete counterpart of the Huygens principle. In this paper, we generalize this approach to spinor wave propagation. Although the construction can be formulated on a discrete lattice of any dimension, for simplicity we focus on spinors living in 1+1 space-time dimensions. The Dirac equation in the Majorana-Weyl representation is directly recovered by incorporating appropriate symmetries of the elementary processes. The Dirac equation in the standard representation is also obtained by using its relationship with the Majorana-Weyl representation.

An efficient numerical modeling of the DC resistivity logging: Theory and applications

The numerical solution of Green’s function for the potential in 2-D arbitrary inhomogeneous media with axial symmetry has been given by use of efficient half-analytical, halfnumerical hybrid method. Then the logging responses of various kinds of the DC resistivity log with axisymmetric excitation have been obtained by using surface integral equation method to match the boundary conditions on the electrodes of the logging sonde. Comparing the results with that obtained by other methods, one can see good precision and efficiency of the given method. Some applications of the numerical modeling have been also discussed.

Perturbation methods in geometric optics

Perturbation theories for the rays and for the eikonal are discussed. It is shown that these theories give completely equivalent results in the calculation of any arbitrarily chosen value. The problem of refraction in an arbitrary inhomogeneous medium is considered as an example.

Shell function method for computing the acoustic pressure field in arbitrary inhomogeneous media

The shell function method is presented as a new technique to evaluate the acoustic pressure field produced by a practical source of finite area, in a medium having an arbitrary spatially varying sound speed in three dimensions. Instead of initially subdividing the source surface area into many area elements, the shell function method determines the surface element of source area associated with the ray received at the observation point for a specified travel time. Special attention is given to the deficiency of conventional ray tracing for the case of a caustic, where an infinite pressure is predicted. On the other hand, the shell function method takes into account both the convergent state of a sound beam and the corresponding surface element of the source area. As a consequence, the sound pressure at and in the vicinity of caustics can be given an appropriate value directly. Numerical examples are included to show the computational procedure of the shell function method and the ensuing results.

Scattering of elastic waves near an explosion

Abstract We use the method of small perturbation to study the scattered waves generated by an arbitrary 3D inhomogeneous medium around a spherically symmetric compressional source. We consider two models of the medium inside the source: a homogeneous solid and a fluid. The results from these two models differ only when scattering occurs within a few source's radii from the explosion. We find that there is a simple relation between the structure of the first order scattered waves and the structure of the medium, namely that a given harmonic of the medium parameters excites only the same harmonic of the two spheroidal potentials. When scattering occurs within a wavelength from the source, we find that the quadrantal terms in the spherical harmonic decomposition of the field have the lowest frequency dependence. They depend on frequency only through the spectrum of the source. Thus, in the far field, the dominant scattered waves generated near an explosion are similar to the primary waves generated by an earthquake. However, when the displacement field is observed in the near field of the explosion, the static solution reveals that a complete set of harmonics may be required to properly account for the displacement field. We compare the perturbation solution with the exact solution of the scattering by a sphere located within a wavelength from the source. This suggests that the perturbation solution has a fairly wide domain of practical applicability. We attempt to apply these results to the Love wave generated near the Boxcar nuclear explosion.

Development of generalized nonlinear behavior of elastic waveforms in an arbitrary inhomogeneous, lossy medium

A general method of developing the nonlinear wavefield for an arbitrary inhomogeneous medium incorporating energy loss mechanisms is presented. The equations of motion for an inhomogeneous medium with physical constants that are arbitrary functions of position, are modified by the incorporation of a generalized loss function, L. This acts as a sink function by introducing a dissipative force field in the force balance equations that accounts for losses in the medium. A general power series solution is developed and the nonlinear wavefield solution is presented that incorporates the effects of losses and higher order interactions in its functional form. The effect of loss on the dispersion relationship is seen as a frequency shift of the wave and loss effects introduce harmonics which lead to waveform distortion. The classical theory of attenuation is extended by this treatment. [Work supported by Navy Long-term Training Committee.

Perturbational solution of the Helmholtz equation in arbitrary inhomogeneous media

The one-dimensional Helmholtz equation for a medium with arbitrary inhomogeneities is solved by a uniformly valid iteration technique. Unlike the Born-Newman series approach, this technique does not depend on the assumption of small inhomogeneities. Here, the Helmholtz equation is first manipulated to yield, in terms of a transformation function, a Volterra integral equation whose series solution still converges when the Born-Newman series breaks down. For the case of small inhomogeneities, it is shown that the series solution, using the present technique, reduces to that obtained by a Born-Newman series-solution. Formulas for the reflection and transmission coefficients are set in terms of the transformation function.