General Dispersive Media Research Articles

Dispersion Relations Alone Cannot Guarantee Causality.

We show that linear superpositions of plane waves involving a single-valued, covariantly stable dispersion relation ω(k) always propagate outside the light cone unless ω(k)=a+bk. This implies that there is no notion of causality for individual dispersion relations since no mathematical condition on the function ω(k) (such as the front velocity or the asymptotic group velocity conditions) can serve as a sufficient condition for subluminal propagation in dispersive media. Instead, causality can only emerge from a careful cancellation that occurs when one superimposes all the excitation branches of a physical model. This happens automatically in local theories of matter that are covariantly stable. Hence, we find that the need for nonhydrodynamic modes in relativistic fluid mechanics is analogous to the need for antiparticles in relativistic quantum mechanics.

Propagation Properties of Generalized Schell-Model Pulse Sources in Dispersive Media

A model of a generalized pulse source, whose complex degree of temporal coherence is described by a function of the nth power difference of two instants, was constructed. As examples, we consider the generalized Gaussian and multi-Gaussian Schell-model pulse sources and study their propagation in dispersive media. It is indicated that such pulse beams present unique self-focusing, off-axis self-shifting and asymmetric self-splitting characteristics by adjusting the power exponent and phase parameters. Further, we explicitly discuss how the coherence time, summation factor as well as the dispersive coefficient significantly affect the self-focusing and self-shifting behaviors of the pulse beam. The results will benefit some applications involving pulse shaping, optical trapping and remote sensing.

Fractional derivative-based approximation of acoustic waveform dispersion measured in bubbly water beyond resonance frequency.

Acoustic pulses transmitted across air bubbles in water are usually analyzed in terms of attenuation coefficient and phase velocity in the frequency domain. The present work expresses an analytical approximation of the acoustic waveform in the time domain. It is introduced by experiments performed with a Gaussian derivative source wavelet, S0(t), with a derivative order, β0 = 4, and a peak frequency, νp0, much larger than the bubble resonance frequency. The measurements highlight a significant shape variability of the waveform Bx(t), measured at x≤ 0.74 m and characterized by a peak frequency νpx≃νp0. The results are in good agreement with the approximation Bx(t)∝(dγx/dtγx)S0(δxt - T), where γx is an additional fractional-derivative order determined by an optimization procedure and T is related to the travel time. The time-scale parameter, δx=β0/(β0+γx), becomes a free parameter for more general source signals. The correlation coefficient between Bx(t) and the approximated waveform is used to identify the applicability of the method for a wide range of bubbly waters. The results may be of potential interest in characterizing gas bubbles in the ocean water column and, more generally, in modeling wave propagation in dispersive media with fractional-derivative orders in the time domain.

A novel frequency-dependent lattice Boltzmann model with a single force term for electromagnetic wave propagation in dispersive media

Electromagnetic wave simulation is of pivotal importance in the design and implementation of photonic nano-structures. In this study, we developed a lattice Boltzmann model with a single extended force term (LBM-SEF) to simulate the propagation of electromagnetic waves in dispersive media. By reconstructing the solution of the macroscopic Maxwell equations using the lattice Boltzmann equation, the final form only involves an equilibrium term and a non-equilibrium force term. The two terms are evaluated using the macroscopic electromagnetic variables and the dispersive effect, respectively. The LBM-SEF scheme is capable of directly tracking the evolution of macroscopic electromagnetic variables, leading to lower virtual memory requirement and facilitating the implementation of physical boundary conditions. The mathematical consistency of the LBM-SEF with the Maxwell equations was validated by using the Champman-Enskog expansion; while three practical models were used to benchmark the numerical accuracy, stability, and flexibility of the proposed method.

Approximation of High-Frequency Wave Propagation in Dispersive Media

Approximation of High-Frequency Wave Propagation in Dispersive Media

Random pulse source with the temporal multi-complex degree of coherence

Based on the coherence theory, the non-stationary optical fields with the complex degree of coherence (CDOC) are expanded from stationary fields, and random pulse sources with temporal multi-complex degree of coherence (TMCDOC) is introduced. Theoretical model for the random pulse sources with TMCDOC is established. The analytical transmission formulas for the temporally spectral density and TMCDOC of the pulse beam spreading through dispersive media are derived and its propagation properties in dispersive media are explored. One can find from the numerical results that the TMCDOC pulse beam will evolve from a single pulse at the source plane into N subpulses in the far field or produce far field spectral density with flat-topped profiles. The variation of the non-trivial phase of the CDOC can regulate the propagation angle of the TMCDOC pulse beam. Moreover, in case of the propagation in dispersive media with chirp coefficient s < 0, the TMCDOC pulse beam will exhibit self-focusing process, and its coherence curves will be reversed from the clockwise rotation in the near field to anticlockwise rotation in the far field.

Effective Analytical Computational Technique for Conformable Time-Fractional Nonlinear Gardner Equation and Cahn-Hilliard Equations of Fourth and Sixth Order Emerging in Dispersive Media

The aim of this paper is to investigate the approximate solutions of nonlinear temporal fractional models of Gardner and Cahn-Hilliard equations. The fractional models of the Gardner and Cahn-Hilliard equations play an important role in pulse propagation in dispersive media. The time-fractional derivative is observed in the conformable framework. In this orientation, a reliable computationally algorithm is designed and developed by following a residual error and multivariable power series expansion. Basically, the approximate solutions of pulse wave function of the fractional higher-order Gardner and Cahn-Hilliard equations are obtained in the form of a conformable convergent fractional series. Relevant consequences are theoretically and numerically investigated under the conformable sense. Besides, the analysis of the error and convergence of the developed technique are discussed. Some of the unidirectional homogeneous physical applications of the posed models in a finite compact regime are tested to confirm the theoretical aspects, demonstrate different evolutionary dynamics, and highlight the superiority of the novel developed algorithm compared to other existing analytical methods. For this purpose, associated graphs are displayed in two and three dimensions. Growing and decaying modes of the fractional parameters are analyzed for several α values. From a numerical viewpoint, the simulations and results declare that the proposed iterative algorithm is indeed straightforward and appropriate with efficiency for long-wavelength solutions of nonlinear partial differential equations.

Outage Probability and BER Estimation for FSO Links with Truncated Normal Time Jitter and Longitudinal Gaussian Pulse Propagation in Dispersive Media

FSO is one of the most widespread, low-cost, wireless, optical communicational technologies with sufficiently high throughput, transmission reliability, and high-level security. Nevertheless, many fading effects act on the optical pulses used, during their propagation, causing performance degradation. In this work, group velocity dispersion and time jitter, modeled by the truncated normal distribution, are jointly investigated analytically and numerically. The availability of the studied model is expressed in terms of outage probability, while its reliability is given in terms of its average bit error rate, through the derived novel mathematical expressions. To the best of authors’ knowledge, this is the first time that the outage and the BER performance are estimated analytically, through specific approximations, taking into account the abovementioned physical effects. Furthermore, using the obtained mathematical forms, the corresponding numerical results are presented by assuming typical parameter values for realistic FSO links.

Second-order symplectic PRK scheme for ground-penetrating radar simulations in dispersive materials

An accurate and effective forward model is helpful in interpreting measured ground-penetrating radar (GPR) data. In this paper, an explicit second-order symplectic partitioned Runge–Kutta scheme (SPRK) with Higdon's absorbing boundary condition is developed to simulate GPR wave propagation in dispersive media. For this purpose, the first-order Debye dispersion model is successfully applied to the second-order SPRK method. A single-layer dispersive medium model and a three-layer pavement structure model are simulated, in order to verify the accuracy and efficiency of the second-order SPRK method, as well as to clarify the effect of dispersion on the GPR echo signal. The numerical simulation results demonstrate that the proposed algorithm not only can maintain the simulation accuracy, but can also save about 19% of the calculation time, when compared with the FDTD algorithm, in a dispersive medium. We also obtain the GPR profile of a pavement structure with a circular cavity disease and an underground pipe structure in a dispersive medium, and the simulation results show that the dispersion characteristics of the materials lead to high absorption of the GPR electromagnetic waves. Finally, field experiment data are used to further verify the applicability of the algorithm. The numerical simulations and experiments indicate that the dispersion of the medium must be considered in GPR numerical simulation, and the authenticity of the simulated image must be ensured.

Gaussian dispersion analysis in the time domain: Efficient conversion with Padé approximants

We present an approach for adapting the Gaussian dispersion analysis (GDA) of optical materials to time-domain simulations. Within a GDA model, the imaginary part of a measured dielectric function is presented as a sum of Gaussian absorption terms. Such a simple model is valid for materials where inhomogeneous broadening is substantially larger than the homogeneous linewidth. The GDA model is the essential broadband approximation for the dielectric function of many glasses, polymers, and other natural and artificial materials with disorder. However, efficient implementation of this model in time-domain full-wave electromagnetic solvers has never been fully achieved. We start with a causal form of an isolated oscillator with Gaussian-type absorption — Causal Dawson-Gauss oscillator. Then, we derive explicit analytical formulas to implement the Gaussian oscillator in a finite-difference time-domain (FDTD) solver with minimal use of memory and floating point operations. The derivation and FDTD implementation employ our generalized dispersive material (GDM) model — a universal, modular approach to describing optical dispersion with Padé approximants. We share the FDTD prototype codes that include automated generation of the approximants and a universal FDTD dispersion implementation that employs various second-order accurate numerical schemes. The codes can be used with non-commercial solvers and commercial software for time-domain simulations of light propagation in dispersive media, which are experimentally characterized with GDA models. Program summaryProgram Title:MADISCPC Library link to program files:https://doi.org/10.17632/x69b4krsy8.1Licensing provisions: GPLv3Programming language: MATLABNature of problem: The problem of efficient time-domain simulation of the Gaussian absorption is essential for wideband modeling of the optical response from materials with inhomogeneous spectral broadening, such as glasses, polymers, and other natural and artificial composite materials with structural or phase disorder.Solution method: A Coupled Oscillator (CO) approximation to the Gaussian absorption in both the frequency and time domains is derived to solve this problem. The time-domain CO approximation is coupled to the finite-difference time-domain (FDTD) solver for the Maxwell equations in the MADIS (MAterial DIspersion Simulator) package. Verification of the Maxwell solvers' accuracy and stability is performed with the FDTD solver coupled to a compact universal implementation of the CO model, employing second-order schemes (either ADE or RC). Code prototypes of these efficient CO schemes can be ported to other methods and platforms for implementing the Gaussian absorption in open-source codes or commercial software.Additional comments including restrictions and unusual features: Patent pending. Restrict any commercial use, including for profit reproduction.

Steeping and dispersive effects analysis of a couple of long-wave equations in dispersive media

Dispersive wave propagations are described by nonlinear models, such as regularized long waves and modified regularized long waves, where nonlinearity and dispersion are important aspects of wave evolution to model long-wave propagation in dispersive media with small amplitudes. In this article, solitary wave solutions to the formerly indicated nonlinear equations are computed as exponential, hyperbolic, and trigonometric structures and their integration by balancing the steeping and dispersion terms of the highest order, and different classes of definitive localized coherent structures with their interaction properties, such as soliton solutions, are extracted by assigning appropriate functions. We have also investigated the impact of steeping and wave spread on waveforms. The wave solutions are assessed using the enhanced modified simple equation approach, which organizes the soliton solutions based on the features of the dispersive interconnections with the nonlinear cubic and quadratic characters that exist in the equations. The consistent 3D and 2D soliton configurations show that the wave profile is modulated by the wave velocity function and the parameters associated with it, and most of the modulation is due to linear effects.

Conservation laws and series solutions of variable coefficient time fractional Kawahara equation

This paper investigates explicit power series solutions and conservation laws of variable coefficient time fractional Kawahara equation with Riemann–Liouville derivative which describs oscillatory solitary wave propagation in dispersive media. The similarity reductions and explicit solutions are derived by using Lie symmetry method. Further, using Ibragimov's theorem conservation laws for this equation are discussed.

A Parallel Conformal Symplectic Euler Algorithm for GPR Numerical Simulation on Dispersive Media

This letter presents a ground-penetrating radar (GPR) forward model on dispersive media based on parallel conformal symplectic Euler algorithm. We developed a symplectic Euler algorithm combined with conformal meshes and graphics processing unit (GPU) acceleration technology, which proved to be an accurate and effective method to simulating GPR electromagnetic wave propagation in dispersive media.

Efficient Modeling of Dispersive Media in Leapfrog WCS-FDTD Method

A leapfrog weakly conditionally stable finite-difference time-domain method is extended to simulate general dispersive media modeled through a partial fraction model via a vector fitting tool and auxiliary difference equation. The computational accuracy and efficiency of this algorithm are demonstrated by numerical examples of metal and graphene.

Newmark-FDTD Formulation for Modified Lorentz Dispersive Medium and Its Equivalence to Auxiliary Differential Equation-FDTD with Bilinear Transformation

The finite-difference time-domain (FDTD) method has been popularly utilized to analyze the electromagnetic (EM) wave propagation in dispersive media. Various dispersion models were introduced to consider the frequency-dependent permittivity, including Debye, Drude, Lorentz, quadratic complex rational function, complex-conjugate pole-residue, and critical point models. The Newmark-FDTD method was recently proposed for the EM analysis of dispersive media and it was shown that the proposed Newmark-FDTD method can give higher stability and better accuracy compared to the conventional auxiliary differential equation- (ADE-) FDTD method. In this work, we extend the Newmark-FDTD method to modified Lorentz medium, which can simply unify aforementioned dispersion models. Moreover, it is found that the ADE-FDTD formulation based on the bilinear transformation is exactly the same as the Newmark-FDTD formulation which can have higher stability and better accuracy compared to the conventional ADE-FDTD. Numerical stability, numerical permittivity, and numerical examples are employed to validate our work.

Edge-based finite-element modeling of 3D frequency-domain electromagnetic data in general dispersive medium

The geophysical electromagnetic (EM) theories are commonly based on the assumption that the conductivity of underground media is frequency independent. However, due to the existence of induced polarization (IP) effect, many earth materials are dispersive, and their electrical conductivity varies significantly with frequency. Therefore, the conventional numerical techniques are not proper for EM forward modeling in general dispersive medium. We present a new algorithm for modeling three-dimensional (3D) EM data containing IP phenomena in frequency domain by using an edge-based finite element algorithm. In this research, we describe the dispersion behavior of earth media by using a Cole–Cole complex conductivity model. Our algorithm not only models land and airborne EM surveys but also provides more flexibility in describing the surface topography with irregular hexahedral grids. We have validated the developed algorithm using an analytic solution over a half-space model with and without IP effect. The capabilities of our code were demonstrated by modeling coupled EM induction and IP responses in controlled-source audio magnetotelluric (CSAMT) and airborne electromagnetic (AEM) examples. This algorithm will have important guiding significance for survey planning in the dispersive areas, and it could be taken as a forward solver for practical 3D inversion incorporated IP parameters.

Wide-Band Modeling On-Chip Spiral Inductors Using Frequency-Dependent Conformal ADI-FDTD Method

To analyze on-chip spiral inductors efficiently, an alternating direction implicit (ADI) finite difference time-domain (FDTD) method is proposed for general dispersive media. The time-domain recursive-convolution iteration equation can be used to calculate the Debye, cold Plasma, and Lorentz media in the same formula. A locally conformal technique, named medium parameters weighting method is proposed to modeling complex structure accurately. The spiral inductor integrated on the silicon substrate has been analyzed by using conformal ADI-FDTD method. And the simulation results are in agreement with the measured results. These data have been obtained over a wide frequency range from 0.1–20 GHz. The proposed method can easily be used as an accurate computer-aided design tool for radio-frequency integrated circuits.

Modal analysis of wave propagation in dispersive media

Surveys on wave propagation in dispersive media have been limited since the pioneering work of Sommerfeld [Ann. Phys. 349, 177 (1914)] by the presence of branches in the integral expression of the wave function. In this article, a method is proposed to eliminate these critical branches and hence to establish a modal expansion of the time-dependent wave function. The different components of the transient waves are physically interpreted as the contributions of distinct sets of modes and characterized accordingly. Then, the modal expansion is used to derive a modified analytical expression of the Sommerfeld precursor improving significantly the description of the amplitude and the oscillating period up to the arrival of the Brillouin precursor. The proposed method and results apply to all waves governed by the Helmholtz equations.

What is the Wigner Function Closest to a Given Square Integrable Function?

We consider an arbitrary square integrable function $F$ on the phase space and look for the Wigner function closest to it with respect to the $L^2$ norm. It is well known that the minimizing solution is the Wigner function of any eigenvector associated with the largest eigenvalue of the Hilbert-Schmidt operator with Weyl symbol $F$. We solve the particular case of radial functions on the two-dimensional phase space exactly. For more general cases, one has to solve an infinite dimensional eigenvalue problem. To avoid this difficulty, we consider a finite dimensional approximation and estimate the errors for the eigenvalues and eigenvectors. As an application, we address the so-called Wigner approximation suggested by some of us for the propagation of a pulse in a general dispersive medium. We prove that this approximation never leads to a {\it bona fide} Wigner function. This is our prime motivation for our optimization problem. As a by-product of our results we are able to estimate the eigenvalues and Schatten norms of certain Schatten-class operators. The techniques presented here may be potentially interesting for estimating eigenvalues of localization operators in time-frequency analysis and quantum mechanics.

Finite-element time-domain modeling of electromagnetic data in general dispersive medium using adaptive Padé series

The induced polarization (IP) method has been widely used in geophysical exploration to identify the chargeable targets such as mineral deposits. The inversion of the IP data requires modeling the IP response of 3D dispersive conductive structures. We have developed an edge-based finite-element time-domain (FETD) modeling method to simulate the electromagnetic (EM) fields in 3D dispersive medium. We solve the vector Helmholtz equation for total electric field using the edge-based finite-element method with an unstructured tetrahedral mesh. We adopt the backward propagation Euler method, which is unconditionally stable, with semi-adaptive time stepping for the time domain discretization. We use the direct solver based on a sparse LU decomposition to solve the system of equations. We consider the Cole-Cole model in order to take into account the frequency-dependent conductivity dispersion. The Cole-Cole conductivity model in frequency domain is expanded using a truncated Padé series with adaptive selection of the center frequency of the series for early and late time. This approach can significantly increase the accuracy of FETD modeling.