Parabolic Wave Equation Research Articles

A comparison of time-domain solutions for the full-wave equation and the parabolic wave equation for a diagnostic ultrasound transducer

A study of numerical solutions to the linear wave equation and the parabolic wave equation is presented. Finite-difference, time-domain methods are used to calculate the acoustic field emitted from a phased diagnostic ultrasound transducer in a non-attenuating medium. Results are compared to Field II, a simulation package that has been used extensively to linearly model transducers in ultrasound. The simulation of the parabolic equation can accurately predict the lateral beamplot for large f/#s, but, it exhibits 2-3 dB errors for small f/#s. It also overestimates the depth at which the focus occurs. For the considered array, it is shown that the finite-difference solution of the wave equation is accurate for a small and large f/#. The lateral beamplots and axial intensities are in excellent agreement with the Field II simulations.

X-ray propagation in tapered waveguides: Simulation and optimization

We use the parabolic wave equation to study the propagation of X-rays in tapered waveguides by numerical simulation and optimization. The goal of the study is to elucidate how beam concentration can be best achieved in X-ray optical nanostructures. Such optimized waveguides can e.g. be used to investigate single biomolecules. Here, we compare tapering geometries, which can be parametrized by linear and third-order (Bézier-type) functions and can be fabricated using standard e-beam lithography units. These geometries can be described in two and four-dimensional parameter spaces, respectively. In both geometries, we observe a rugged structure of the optimization problem’s “gain landscape”. Thus, the optimization of X-ray nanostructures in general will be a highly nontrivial optimization problem.

Modeling of Shock Wave Propagation in Large Amplitude Ultrasound

The Rankine-Hugoniot relation for shock wave propagation describes the shock speed of a nonlinear wave. This paper investigates time-domain numerical methods that solve the nonlinear parabolic wave equation, or the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation, and the conditions they require to satisfy the Rankine-Hugoniot relation. Two numerical methods commonly used in hyperbolic conservation laws are adapted to solve the KZK equation: Godunov's method and the monotonic upwind scheme for conservation laws (MUSCL). It is shown that they satisfy the Rankine-Hugoniot relation regardless of attenuation. These two methods are compared with the current implicit solution based method. When the attenuation is small, such as in water, the current method requires a degree of grid refinement that is computationally impractical. All three numerical methods are compared in simulations for lithotripters and high intensity focused ultrasound (HIFU) where the attenuation is small compared to the nonlinearity because much of the propagation occurs in water. The simulations are performed on grid sizes that are consistent with present-day computational resources but are not sufficiently refined for the current method to satisfy the Rankine-Hugoniot condition. It is shown that satisfying the Rankine-Hugoniot conditions has a significant impact on metrics relevant to lithotripsy (such as peak pressures) and HIFU (intensity). Because the Godunov and MUSCL schemes satisfy the Rankine-Hugoniot conditions on coarse grids, they are particularly advantageous for three-dimensional simulations.

Wide-Angle Shift-Map PE for a Piecewise Linear Terrain—A Finite-Difference Approach

A wide-angle parabolic wave equation solution is presented using shift-map and finite-difference techniques. The corresponding split-step Fourier solution is well known. The solution using finite-difference technique, where the standard parabolic wave equation is modified into the so-called Claerbout equation allowing propagation angles up to 45deg from the paraxial direction, is also well known. Here, we present an extension to that solution in which the shift-map technique is incorporated into the finite-difference scheme allowing a varying terrain to be considered. The result is a solution that corresponds to the well known split-step solution, which is believed to perform well for terrain slopes up to 10deg-15deg and discontinuous slope changes on the order of 15deg-20deg. This solution is a first-order one with respect to the terrain slope. However, when using the finite-difference technique, it is also possible to find a second-order solution with respect to the terrain slope. This new solution performs well for slopes up to about 15deg and discontinuous slope changes up to about 30deg, which is an improvement.

Analytical Study of HChG-Laser Beam Propagation in Collisional and Collisionless Plasmas

We have studied the propagation of m=0, 1 and 2 mode Hermite-cosh-Gaussian (HChG) laser beams in collisional and collisionless plasmas. The field distribution in the medium is expressed in terms of beam-width parameter f and decentred parameter b. The differential equations for f-parameter are established by parabolic wave equation approach under paraxial approximation. Analytical solutions are obtained under the condition Rn < Rd where, Rn is the self-focusing length and Rd is the diffraction length. The behaviour of f-parameter with the dimensionless distance of propagation η for various b-values is examined by numerical estimates. The results are presented graphically.

APPLICATION OF THE PARABOLIC EQUATION METHOD TO SEVERAL CANONICAL INDOOR PROPAGATION PROBLEMS

The parabolic wave equation is derived for general lossy dielectric media and applied to the analysis of radio wave propagation problems inside buildings. The paraxial version of the wave equation is solved with a marching algorithm that requires limited computing resources as compared with its full version. Using this site-specific approach, several canonical indoor propagation problems such as propagation through windows, reinforced concrete walls and inside corridors are investigated. Good agreement between the parabolic equation numerical simulation results and those reported in the literature validates the proposed method.

Study of the slow fading in indoor environment using parabolic equations

AbstractThis article compares slow fading component of the propagation loss calculated through parabolic wave equation and some models of the literature with results obtained during measurement campaign, in indoor environment. The frequency used was 850 MHz; a complex refractive index was considered. The implicit finite difference scheme of the Crank‐Nicolson type was applied in order to get the solution of the parabolic equation. The propagation was considered in 15° with direction paraxial. © 2007 Wiley Periodicals, Inc. Microwave Opt Technol Lett 49: 1676–1679, 2007; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.22530

Computer simulations and experimental results on air-gap X-ray waveguides

X-ray Waveguides (WG) are optical elements able to provide nanometer size coherent X-ray beams. Both Resonant beam Coupling (RBC) and Front Coupling (FC) can be implemented to couple the incident radiation with the WG. In this paper we implemented a computer code based on finite difference scheme solutions of parabolic wave equations to simulate the coupling and the propagation of electromagnetic waves into air-gap waveguides in Front Coupling mode. Both a synchrotron source and a table-top laboratory sources have been considered. In this last case the degree of coherence of the incoming beam has been explicitly taken into consideration. Air-gap WGs have been fabricated following simple procedures, and tested at the Elettra synchrotron source. The results are in good agreement with theoretical expectations.

An Efficient Hybrid Parabolic Equation – Integral Equation Method for the Analysis of Wave Propagation in Highly Complex Indoor Communication Environments

An efficient, full-wave computational technique to investigate the electromagnetic wave propagation within a complex building environment, resulting from contemporary indoor communication systems, is proposed. Unlike a standard ray-tracing technique, this new methodology is based on the parabolic wave equation (PE), appropriately modified to deal with the extremely wide-angle propagation cases, encountered in a typical wireless system of this kind. It is also successfully applied to model the field in the presence of walls, doors or other penetrable structures, taking into account the exact geometric configuration of the environment under consideration. Next, the PE technique is significantly enhanced by an integral equation formulation, in which the computed field in the interior of the walls and other obstacles is used as a secondary equivalent current source and a corrected version of the electromagnetic field is recalculated in the whole indoor environment. This combined approach has all the advantages of a full wave method, does not call for a highly dense mesh, and it also has moderate requirements of computational resources.

A single-scattering correction for large contrasts in elastic layers

The single-scattering solution is implemented in a formulation that makes it possible to accurately handle solid-solid interfaces with the parabolic equation method. Problems involving large contrasts across sloping stratigraphy can be handled by subdividing a vertical interface into a series of two or more scattering problems. The approach can handle complex layering and is applicable to a large class of seismic problems. The solution of the scattering problem is based on an iteration formula, which has improved convergence in the new formulation, and the transverse operator of the parabolic wave equation, which is implemented efficiently in terms of banded matrices. Accurate solutions can often be obtained by using only one iteration.

Computation of paraxial wave fields using transparent boundary conditions

The parabolic wave equation is solved numerically by applying transparent conditions used to confine the computational domain. A numerical implementation of the boundary conditions is proposed based on representing the incident wave as a superposition of Gaussian beams. A modification of the transparent conditions in the case of dielectric objects extending beyond the computational domain is described. Numerical examples are presented.

A transformation of the environment eliminates parabolic equation phase errors

It is shown that phase errors associated with the standard parabolic wave equation can be eliminated in a range-independent environment by appropriately transforming the environment. Solutions to the standard parabolic equation (PE) in the transformed environment are close to solutions to the Helmholtz equation in the original environment in the sense that ray and asymptotic mode contributions to the PE wave fields have no phase errors relative to their Helmholtz equation counterparts. Although phase errors are eliminated in the PE wave fields, complete asymptotic equivalence to the Helmholtz equation wave fields is not attained. Ray- and mode-based wave-field expansions are used to illustrate similarities and differences between PE and Helmholtz equation wave fields. Numerical simulations show excellent agreement between PE wave fields in the transformed environment and mode-based solutions to the Helmholtz equation in the true environment.

Finite-difference field calculations for two-dimensionally confined x-ray waveguides

A numerical method for calculation of the electromagnetic field in two-dimensionally confined x-ray waveguides is presented. It is based on the parabolic wave equation, which is solved by means of a finite-difference scheme. The results are verified by a comparison to analytical theory, namely, Fresnel reflectivity and the weakly guiding optical fiber.

Numerical simulation of wave processes in inhomogeneous media with impedance boundaries

The paper is concerned with numerical simulation and properties of wave processes in infinite inhomogeneous media with impedance boundaries. A numerical method for solving a boundary-value problem and optimal control problem for a parabolic Schrodinger-type wave equation with a complex nonself-adjoint operator is proposed and analyzed.

Synthesis of vector wave envelopes in three‐dimensional random elastic media characterized by a Gaussian autocorrelation function based on the Markov approximation: Plane wave case

High‐frequency seismograms of microearthquakes observed at long travel distances have longer apparent durations than the source durations caused by scattering due to lithospheric inhomogeneity. Frequency dependence of the peak delay from the onset and the broadening of seismogram envelope well reflect the spectra of random velocity inhomogeneity. Here we propose a formulation for the envelope synthesis of vector waves in three‐dimensional random elastic media in the case that wavelengths are smaller than the characteristic scale of random inhomogeneity. A stochastic master equation for the two‐frequency mutual coherence function (TFMCF) of potential field is derived on the basis of the Markov approximation, which is a stochastic extension of the phase screen method for solving the parabolic wave equation. From the Fourier transform of TFMCF at a given travel distance, we are able to synthesize the mean square traces of three vector wave components. For the incidence of an impulsive plane P wavelet to random elastic media characterized by a Gaussian autocorrelation function, the mean square envelope of each component at a long travel distance is analytically written by using an elliptic theta function. Each synthesized envelope shows peak delay from the onset and broadening with travel distance increasing; however, the peak delay of the transverse component is larger than that of the longitudinal component. For the same fractional velocity fluctuation, the envelope broadening for S waves is larger than that for P waves.

Radiative transfer of elastic waves versus finite difference simulations in two‐dimensional random media

High‐frequency seismograms mainly consist of incoherently scattered waves. Although their phases are more or less random, their envelopes show smooth and stable variations depending on frequency and distance. Envelope modeling can thus be used to infer stochastic parameters of the heterogeneous Earth medium. Radiative transfer theory (RTT) describes energy transport through a random heterogeneous medium neglecting phase information and has been frequently used to simulate observed mean square (MS) envelopes of high‐frequency waves. The radiative transfer equations can be numerically solved by Monte Carlo simulations. So far, mostly isotropic scattering and acoustic approximations have been used. Here we present an extension of the Monte Carlo method to the full elastic case including P, S, and conversion scattering where the single scattering events are described by angular‐dependent scattering coefficients in random media which follow from the Born approximation. In order to validate the method, the simulated envelopes are compared to average envelopes obtained by full waveform modeling with a finite difference method in two‐dimensional random media with Gaussian and exponential correlation functions. Envelope shapes agree remarkably well for both short and long lapse times and for a broad range of scattering parameters. We conclude that the use of Born scattering coefficients in RTT does not pose severe limits on its validity range. Even in the strong forward scattering regime, envelope broadening and peak amplitude delays can be successfully modeled if one includes the wandering effect as obtained from the parabolic wave equation and Markov approximation into RTT.

Forward Radar Propagation Over a Rough Sea Surface: A Numerical Assessment of the Miller-Brown Approximation Using a Horizontally Polarized 3-GHz Line Source

In order to reliably predict long-range radar coverage within an ocean environment, one must accurately account for the effects of ocean roughness on radar propagation. The method known as the Miller-Brown approximation is a widely used technique for computing the coherent field reflected by a rough sea surface and it is often incorporated into sophisticated numerical propagation models such as the Fourier split step algorithm of the parabolic wave equation. Nevertheless, the accuracy of the Miller-Brown approximation has not been systematically or rigorously assessed for realistic scenarios of interest to shipboard radars. We present here a first step toward this assessment by using a method of moments solution to compute the propagation factor, /spl eta/, for an ensemble of one-dimensional (1-D) rough sea surface realizations having dielectric properties and roughness profiles that are consistent with a wind roughened ocean surface. The particular method of moments (MoM) technique used combines an accelerated spectral method and a multigrid iterative approach (MGIA). The MGIA results for the propagation factor /spl eta//sub MGIA/ are compared to the corresponding predictions /spl eta//sub MB/ obtained via the Miller-Brown approximation. In addition, the well-known Ament approximation is used to compute results for the propagation factor, /spl eta//sub A/, and the results predicted by /spl eta//sub A/ are compared with those of /spl eta//sub MB/. A heuristic numerical fitting scheme is also presented that improves the agreement between /spl eta//sub MGIA/ and /spl eta//sub MB/ and is used to facilitate the comparison between /spl eta//sub MB/ and /spl eta//sub A/.

Parabolic Wave Equation Method Applied to the Tropospheric Ducting Propagation Problem: A Survey

A survey is made of one of the most widely used approximation methods in the wave propagation studies—the parabolic wave equation method—applied for the specific case of microwave propagation assessment under tropospheric ducting conditions. A brief review of the methods for tropospheric refractivity profiling, the average refractivity modeling, and the applicability of the often-assumed lateral homogeneity of the refractivity is given. The parabolic wave equation derivation and the numerical methods for solving it are summarized with special emphasis on transparent boundary conditions for the upper computational window boundary. Results are reported which illustrate the application of the finite element method and split-step Fourier-based solutions of parabolic wave equations for different cases of tropospheric ducts, concerning radar and communications links performance.

Least Square Solution of the 3-D Vector Parabolic Equation

In this paper a least square method is proposed for the solution of the 3-dimensional (3-D) vector parabolic wave equation. This method can correctly incorporate the impedance boundary conditions of the propagation environment. A proper basis function is introduced to expand electromagnetic field components in the successive computation steps. The proposed method needs much less memory size against its finite-difference counterpart.

Synthesis of plane vector wave envelopes in two-dimensional random elastic media based on the Markov approximation and comparison with finite-difference simulations

SUMMARY High-frequency seismograms mainly consist of incoherently scattered waves. Their envelopes are a stable measure exhibiting characteristic features like peak amplitude decay and envelope broadening with increasing travel distance which can be used to infer stochastic parameters of the inhomogeneous Earth medium. As a simple model we study the propagation of plane P and S waves through a 2-D random elastic medium. If the wavelength is less than the correlation distance and the medium inhomogeneity is weak conversion scattering can be neglected, and a stochastic parabolic wave equation is derived. Solving the master equation for the two-frequency mutual coherence function and taking the Fourier transform, we obtain the temporal change of the mean squared amplitude at a fixed distance. The distribution of energy between longitudinal and transverse components can then be calculated by using the angular spectrum representation. For the case of a Gaussian autocorrelation function the solution is completely analytical. The theoretical envelopes are compared with the results of 2-D elastic finite-difference (FD) simulations. For a stable estimate of mean-squared envelopes the squared FD traces from different receiver positions and several realizations of the random medium have been averaged. The theoretical curves well explain the delay of the peak arrival from the onset and the broadening of the envelope with increasing propagation distance. Also the transverse component amplitude for P-wave incidence and the longitudinal component amplitude for S-wave incidence are precisely explained by the theory. The time integral of the mean-squared transverse component for P-wave incidence and of the mean-squared longitudinal component for S-wave incidence linearly increases with travel distance. The linear coefficient is a measure of the ratio of mean-squared fractional fluctuation to correlation distance.