2D Inhomogeneous Medium Research Articles

Transient characteristics of induced polarization in inhomogeneous media (from results of 2D numerical simulation)

This paper presents the results of 2D mathematical modeling of induced polarization (IP) in the cases of profiling and electrical tomography for piecewise homogeneous media containing ore bodies with different structures, sizes, depths, relaxation times, chargeability, and electrical resistivity. The standard and spectral approaches to data analysis are compared. The cases where analysis of transient IP characteristics yields new information of practical importance are considered. The main features of transient IP characteristics in 2D inhomogeneous media are determined from the modeling results. Analysis of transient IP characteristics is found to yield the best results when the depth of occurrence of polarizable bodies is minimal and their electrical conductivity is an order of magnitude lower than that of the host tock.

Application of the homogeneous function and modeling methods for reconstruction of a Tibetan geological section from a travel-time inversion

New structures were revealed in Central Tibet after reinterpretation of the first arrivals of refracted seismic waves along the INDEPTH III profile. The method that was used for travel-time inversion is the numerical generalization of the Herglotz-Wiechert formula for the case of a 2D inhomogeneous medium (the homogeneous-function method). The new section shows that the Bangong-Nujiang Suture (BNS) is a contact in the upper crust, while its continuation in the middle crust is a listric thrust; at the top of the lower crust, which has risen to a 32-km depth in the northern part of the section, this thrust become gentler. Below the middle crust top, a layer with a lower velocity of seismic waves is found. The middle crust is deformed and forms oblique southward listric thrusts. In the upper crust south of the BNS, higher-velocity zones (probable zones of magmatism) are distinguished. North of the BNS, undisturbed outlines of the structures are observed.

Gaussian beam evolution in logarithmically saturable nonlinear media

The method of paraxial complex geometrical optics (PCGO) is presented, which describes Gaussian beam (GB) diffraction and self-focusing in smoothly inhomogeneous and nonlinear media of cylindrical symmetry. PCGO reduces the problem of Gaussian beam diffraction in nonlinear and inhomogeneous media to the system of the first order ordinary differential equations for the complex curvature of the wave front and for GB amplitude, which can be readily solved both analytically and numerically. The power of PCGO method is presented on the example of GB evolution in logarithmically saturable medium with defocusing refractive profile. The solutions obtained by PCGO method are compared with numerical results of Nonlinear Schrodinger Equation by beam propagation method (BPM). Full Text: PDF References Yu.A. Kravtsov, "Complex rays and complex caustics", Radiophys. Quantum Electronics 10, 719 (1967). CrossRef .B. Keller, W. Streifer, "Complex Rays with an Application to Gaussian Beams", J. Opt. Soc. Am. 61, 40 (1971). CrossRef G. A. Deschamps, "Gaussian beam as a bundle of complex rays", Electron. Lett. 7(23), 684 (1971). CrossRef Yu.A. Kravtsov, G.W. Forbes, A.A. Asatryan, Theory and applications of complex rays, in Progress in Optics, edited by E. Wolf, 39, 3 (Elsevier, Amsterdam, 1999). S.J. Chapman, J.M. Lawry, J.R. Ockendon, R.H. Tew, "On the Theory of Complex Rays", SIAM Review 41, 417 (1999). CrossRef Yu.A. Kravtsov, P. Berczynski, "Gaussian beams in inhomogeneous media: A review", Stud. Geophys. Geod. 51(1), 1 (2007). CrossRef Yu.A. Kravtsov N.Y. Zhu. Theory of diffraction: Heuristic Approaches (Alpha Science International, ISBN 1842653725, 2009). P. Berczynski, Yu.A. Kravtsov, "Theory for Gaussian beam diffraction in 2D inhomogeneous medium, based on the eikonal form of complex geometrical optics", Phys. Lett. A 331(3-4), 265 (2004). CrossRef P. Berczynski, K.Yu. Bliokh, Yu.A. Kravtsov, A. Stateczny, "Diffraction of a Gaussian beam in a three-dimensional smoothly inhomogeneous medium: an eikonal-based complex geometrical-optics approach", J. Opt. Soc. Am. A, 23(6), 1442 (2006). CrossRef P. Berczyński, Yu. Kravtsov, "Gaussian beam evolution in nonlinear inhomogeneous fibres", Phot. Lett. Poland 4(1), 26 (2012). CrossRef P. Berczynski, Yu. A. Kravtsov, G. Zeglinski, "Gaussian beam diffraction in inhomogeneous media of cylindrical symmetry", Optica Applicata 40(3), 705 (2010). DirectLink A.W. Snyder, J.D. Mitchell, "Mighty morphing spatial solitons and bullets", Optics Letters 22(1), 16 (1997). CrossRef Y.L. Tang, J.G Chen, D.Y Li, J. Kang, Y. Li, "Specific gaussian beams propagating inside logarithmically saturable nonlinear media", J. Modern Optics 46(8), 1177 (1999). CrossRef T. Hansson, D. Anderson, M. Lisak, "Soliton interaction in logarithmically saturable media", Opt. Comm. 283(2), 318 (2010). CrossRef

A refraction method for the diagnostics of 2D inhomogeneous media

A refraction method is proposed for the determination of the refractive index in a 2D inhomogeneous medium. The method involves one-direction sounding of the medium under study by means of a set of narrow beams. A solution to the inverse geometric-optics problem is used to develop a recurrent algorithm for determining the values of the refractive index at a discrete set of points on the plane boundary of the medium. The algorithm is tested for a plane-layer medium and a 2D inhomogeneous medium, and the test results are reported.

The field of a point source in a horizontally inhomogeneous short-wave radio channel with inaccurate parameters. Part I. General relationships

The reference problem method is used to construct an expression for the field of a point source in a 2D inhomogeneous medium as a function of the distance from the source. The short-wave fields in an ionispheric reflective channel with allowance for horizontal electron density gradients of the ionosphere are described. The results make it possible to solve the problem of a short-wave radio channel with inaccurate parameters.

Tilted Gaussian beam propagation in inhomogeneous media

The present work is concerned with applying a ray-centered non-orthogonal coordinate system which is a priori matched to linearly-phased localized aperture field distributions. The resulting beam-waveobjects serve as the building blocks for beam-type spectral expansions of aperture fields in 2D inhomogeneous media that are characterized by a generic wave-velocity profile. By applying a rigorous paraxial-asymptotic analysis, a novel parabolic wave equation is obtained and termed "Non-orthogonal domain parabolic equation"--NoDope. Tilted Gaussian beams, which are exact solutions to this equation, match Gaussian aperture distributions over a plane that is tilted with respect to the beam-axes initial directions. A numerical example, which demonstrates the enhanced accuracy of the tilted Gaussian beams over the conventional ones, is presented as well.

Theory for Gaussian beam diffraction in 2D inhomogeneous medium, based on the eikonal form of complex geometrical optics

A simple and effective method to describe Gaussian beams propagation and diffraction in arbitrary smoothly inhomogeneous 2D medium has been developed based on the eikonal form of complex geometrical optics. The method assumes the eikonal equation can be solved in paraxial approximation in curvilinear frame of references, connected with the central ray. The Riccati-type ordinary differential equation is derived for complex parameter characterizing the Gaussian beam width and phase front curvature. The same parameter was proved to define both the modulus and the argument of the complex amplitude. As a result, the problem of the Gaussian beam diffraction in inhomogeneous media has been reduced to the solution of the ordinary differential equation of the first order, which can be readily calculated numerically for arbitrary profile of dielectric permittivity.

Darboux transformation for classical acoustic spectral problem

We study discrete isospectral symmetries for the classical acoustic spectral problem in spatial dimensions one and two by developing a Darboux (Moutard) transformation formalism for this problem. The procedure follows steps similar to those for the Schrödinger operator. However, there is no one‐to‐one correspondence between the two problems. The technique developed enables one to construct new families of integrable potentials for the acoustic problem, in addition to those already known. The acoustic problem produces a nonlinear Harry Dym PDE. Using the technique, we reproduce a pair of simple soliton solutions of this equation. These solutions are further used to construct a new positon solution for this PDE. Furthermore, using the dressing‐chain approach, we build a modified Harry Dym equation together with its LA pair. As an application, we construct some singular and nonsingular integrable potentials (dielectric permitivity) for the Maxwell equations in a 2D inhomogeneous medium.