Nonlinear Propagation Research Articles

Numerical analysis of the limits of effective medium theory for nonlinearly propagating dispersive waves

Acoustic metameterials are an increasingly popular approach to control both linear and nonlinear sound propagation. In this approach, heterogeneous structures are engineered to behave as predetermined continuous media. A key requirement for a system to be accurately approximated as a homogeneous material for wave motion is that the smallest length scale of the waveform be much larger than any length scale associated with the microscale structure of the system. However, the fact that the characteristic length scales of a finite-amplitude waveform can change due to nonlinear effects means that care must be taken when analyzing such signals in an acoustic metamaterial. This talk presents recent work toward understanding the breakdown of effective medium theory as finite-amplitude waves propagate in a heterogeneous system and weak shocks or solitons begin to form. A finite-difference time-domain code was used to model the propagation of a pulse of sound in a one-dimensional propagation domain using both a full description of the system and an effective-medium description accounting for effective mass density, compressibility, coefficient of nonlinearity, and dispersion. The difference between the solutions of the two models provides a method by which one can estimate the error associated with using effective properties.

High-order singular value decomposition of contrast pulsing sequences for improved clutter suppression

Contrast-enhanced ultrasound is a diagnostic tool used to visualize blood flow in the cardiovascular system. The use of ultrasound contrast agent (microbubbles) in combination with contrast pulsing scheme (CPS) improves the sensitivity and specificity of ultrasound flow imaging by enhancing the signal in the blood compartment. The commonly used CPS are pulse inversion (PI), amplitude modulation (AM), and amplitude-modulated pulse inversion (AMPI). Using differences in phase or amplitude of multiple pulses, the linear tissue clutter signal can be suppressed. However, this process can be degraded by motion or non-linear propagation of the ultrasound wave. These effects cause the cancellation of linear clutter signal to be ineffective We propose using higher-order singular value decomposition (HOSVD) with spatial, temporal, and pulsing dimensions as the input to improve clutter suppression under the conditions of motion and non-linear propagation. We performed systematic in-vitro- experiment emulating these conditions, as well as in-vivo cardiac measurements. The results showed that HOSVD increases the clutter suppression of all the 3 CPS compared to the conventional linear processing. The improvement of clutter reduction could be beneficial to various cardiac evaluation like myocardial perfusion or intra ventricular flow assessment.

From Paul Langevin’s quartz sensor to ultrasound metrology of today

This talk focuses on the remarkable impact of Paul Langevin’s research since he (in 1917) proposed to use the piezoelectric properties of quartz for underwater detection of ultrasound signals. Not only has his pioneering idea opened the field of quantitative measurements of acoustic waves, but it also had far-reaching implications for the development of modern ultrasound metrology, driven by the need to ensure the safety of ultrasound used in visualizing of internal human organs for diagnostic applications and maximizing the efficacy and precision of treatment in therapeutic application of ultrasound waves. In summarizing the current advances in ultrasound metrology, a succinct background explaining why, initially, both the scientific community and industry were skeptical about the existence of the nonlinear (NL) wave propagation in tissue will be given and the design of an adequately wideband piezoelectric polymer hydrophone probe that was eventually used to verify that the 1–5 MHz probing wave then used in diagnostic ultrasound imaging was undergoing nonlinear distortion and generated harmonics in tissue will be discussed. [ Work supported by the NIHNINR through Grant No. 5R01NR015995. The contents of this presentation are solely the responsibility of the authors and do not necessarily represent the official views of the NIH.]

Multi‐Ion Oscillitons—Origin of Coherent Magnetospheric EMIC Waves

AbstractThe recent spacecraft observations by MMS and Van Allen Probes associated with electromagnetic ion cyclotron (EMIC) waves in the Earth magnetosphere emphasize the important role of multi‐ion plasma composition for generation and characteristics of these emissions. We show that main properties of the coherent EMIC waves can be explained with the concept of “multi‐ion oscillitons” (Sauer et al., 2001, https://doi.org/10.1029/2001GL013047). In a plasma with two types of ions of different masses (e.g., protons and oxygen ions), oscillitons arise from the exchange of momentum and energy between the two ion components, with the electromagnetic field acting as a mediator. At frequencies near cross‐over frequencies of different wave modes in the multi‐ion plasma the nonlinear resonance which strongly amplifies the seed unstable mode can be excited. A small phase difference in oscillations of different ion species leads to a nonlinear wave beating and generation of wave packets. The “resonance” frequency is characterized by a local maximum of the phase velocity and the coincidence of phase and group velocity. It is suggested that the oscillitons are triggered by the instability due to the proton temperature anisotropy and may survive outside the source region for long distances. The generation of coherent waves by oscillitons is of a general nature and may contribute to understand the manifold of phenomena in other space plasma environments in which the dynamics of minor ion admixtures cannot be neglected. The concept of oscillitons can also be applied to the momentum exchange between particle groups of the same mass, but different temperature.

Effective monitoring of power system with phasor measurement unit and effective data storage system

In the recent years the monitoring and operation of the power system became complex, due to the more demand from the different linear and non-linear loads and generation from the different sources. For the effective monitoring and operation of the power system, existing power system monitoring methods need to improve or new technologies are required. For the effective monitoring and operation of the power system phasor measurement unit (PMU) based monitoring is suitable, because it provide the dynamic state monitoring system. In this paper PMU based monitoring is proposed with effective data storage system and protection. With this method phasor values of voltage and current signals are calculated at the location of PMU and with the help of software based program effective data storage also possible. With this proposed model the phasor values in the power system at different locations monitoring also possible and required phasor data only stored and total data is only monitored. The phasor values of signals are calculated with direct phasor measurement technique in LabVIEW and by adding time stamping to the each phasor value accurate measurement of power flow is possible.

Contribution of initial bubble radius distribution to weakly nonlinear waves with a long wavelength in bubbly liquids

In this study, the weakly nonlinear propagation of plane progressive pressure waves in an initially quiescent liquid was theoretically investigated. This liquid contains several small uniformly distributed spherical polydisperse gas bubbles. The polydispersity considered here represents various types of initial bubble radii, and the liquid contains multiple bubbles, each with an initial radius. Using the method of multiple scales, we first derived the Korteweg–de Vries–Burgers (KdVB) equation with a correction term as a nonlinear wave equation. This equation describes the long-range wave propagation with weak nonlinearity, low frequency, and long wavelength in the polydisperse bubbly liquid using the basic equations in a two-fluid model. The utilization of the two-fluid model incorporates the dependence of an initial void fraction on each coefficient in the nonlinear, dissipation, and dispersion terms in the KdVB equation. Furthermore, unlike previous studies on waves in polydisperse bubbly liquids, we achieved the formulation without assuming an explicit form of the polydispersity function. Consequently, we discovered the contribution of polydispersity to the various effects of wave propagation, that is, the nonlinear, dissipation, and dispersion effects. In particular, the dispersion effect of the waves was found to be strongly influenced by polydispersity.

Simulation of Internal Undular Bores in Rapidly Varying Topography

This paper intends to look at the influence of rapidly varying regions on the propagation of internal undular bores in a two-layer fluid flow. Internal undular bores have been observed in the ocean around the world. The appropriate mathematical model to describe the evolution of internal undular bores in a stratified fluid is the variable-coefficient extended Korteweg-de Vries equation. The governing equation is solved numerically using the method of lines to simulate the propagation of internal undular bores. Our numerical results show that the effects of rapidly varying topography lead to adiabatic and non-adiabatic deformation of the internal undular bores, including the generation of solitary wavetrain, generation of nonlinear wavetrain and generation of rarefaction wave. Besides, multi-phase behaviour is also observed during the evolution.

Identification approach of acoustic cavitation via frequency spectrum of sound pressure wave signals in numerical simulation

In a sono-reactor, complex ultrasound pressure wave signal can be detected, containing multiple information related to acoustic cavitation. In this present study, acoustic cavitation in a cylinder is investigated numerically. Via Fast Fourier Transfer (FFT), the sound pressure signals from sonotrode emitting surface are separated into harmonics, sub/ultra-harmonics and cavitation white noise: (1) the appearance of harmonics proved the non-linear propagation of ultrasound, (2) at the vibratory amplitude from 5∼20μm, only harmonics exists in the frequency spectra, corresponding to expansion and compression of non-condensable gas (NCG), (3) at the vibratory amplitude range of 30∼50μm, the occurrence of sub/ultra-harmonics demonstrated gaseous cavitation occurred, and (4) at the vibratory amplitude higher than 55μm, cavitation white noise arose, pointing out the initiation of vaporous cavitation. Based on the combination of frequency spectra and cavitation zones distribution, the acoustic cavitation state in water liquid is determined.

Tunable optical nonlinearity and self-collimation of light in food dye solutions

Food dyes are widely used in scientific research apart from the food colorant industry. In this work, we investigate nonlinear propagation of optical beams in food dye solutions of different colors, experimentally observing tunable nonlinearity and self-collimating effects of optical beams during propagation. We discuss possible mechanisms leading to optical nonlinearity by numerical simulation and theoretical analysis of the optical forces, and show that the nonlinearity in such solutions is mainly attributed to the thermo-optic effects. In addition, with appropriately mixed food dye solutions, we observe that nonlinear response arises at otherwise inactive wavelengths, leading to coupling of two self-collimated beams of different wavelengths. These results illustrate the possibility of employing low-cost food dye solutions for optical limiting and switching applications.

Nonlinear dispersive wave propagation pattern in optical fiber system

Nonlinear fractional-order partial differential equations are an important tool in science and engineering for explaining a wide range of nonlinear processes. We consider the nonlinear space-time fractional modified Korteweg de Vries and sine-Gordon equations in this article and extract diverse sorts of traveling wave as well as soliton solutions using a typical approach, namely the generalized G′/G-expansion approach. These equations are used to model a wide range of nonlinear phenomena, including plasma physics, high-intensity laser-generated plasma, ultra-small electronic devices, optical fibers, control theory, turbulence, acoustics, and others. The fractional derivative is defined in the sense of the beta-derivative established by Atangana and Baleanu. Some standard shapes of waveforms, including kink type, singular-kink type, bell-shaped, periodic-type, single soliton, multiple soliton type, and several other types of solitons, have been established. To validate the physical compatibility of the results, the 3D, contour, and vector plots have been delineated using consistent values of the parameters. The approach used in this study to extract inclusive and standard solutions is approachable, efficient, and speedier in computing.

Variational estimates for the speed propagation of fronts in a nonlinear diffusive Fisher equation

We examine non-linear diffusive front propagation in the frame of the Fisher-type equation: ∂tu=∂xD(u)∂xu+u(1−u). We study the problem of a sudden jump in diffusivity motivated by models of glassy polymers. It is shown that this problem differs substantially from the problem of front propagation in the usual Fisher equation which was solved by Kolmogorov, Petrovsky, and Piskunov (KPP) in 1937. As in the Fisher, Kolmogorov, Petrovsky, Piskunov (FKPP) problem, the asymptotic dynamics of the non linear diffusive front propagation is reduced to the study of a nonlinear ordinary differential equation with adequate boundary conditions. Since this problem does not allow an exact result for the propagation speed, we use a variational approach to estimate the front speed and compare it with direct time-dependent numerical simulations showing an excellent agreement.

The nonlinear second harmonic generation and third harmonic generation in GaAs/GaAlAs exponentially confined quantum well: Electric and magnetic field effects

The second harmonic generation(SHG) and third harmonic generation(THG) of a GaAs/GaAlAs exponential quantum well(QW) are numerically investigated in the presence of applied electric and magnetic field, which are parallel to the QW growth direction. The eigenvalues and corresponding eigenfunctions of the system are obtained by using the finite difference technique . The expression for the second- and third-order susceptibility is given by the density matrix and iterative approach. It is showed that, the SHG and THG not only depend strongly on quantum confinement, but also the intensity of external fields. We have systematically studied the SHG and THG of different confinement QWs by adjusting the structural parameters, as well as the effects of external electric and magnetic fields. We give the optimal system parameters in which the external field effects play important role.

Nonlinear Employment Effects of Tax Policy

AbstractWe study the nonlinear propagation mechanism of tax policy in a heterogeneous‐agent equilibrium business cycle model with search frictions in the labor market and an extensive margin of employment adjustment. The model exhibits endogenous job destruction and endogenous hiring standards in the form of occasionally‐binding zero‐surplus constraints. After parameterizing the model using U.S. data, we find that the dynamic response of employment to a temporary change in the labor income tax is highly nonlinear, displaying sizable asymmetries and state dependence. Notably, the response to a tax rate cut is at least twice as large in a recession as in an expansion.

Optimizing supercontinuum spectro-temporal properties by leveraging machine learning towards multi-photon microscopy

Multi-photon microscopy has played a significant role in biological imaging since it allows to observe living tissues with improved penetration depth and excellent sectioning effect. Multi-photon microscopy relies on multi-photon absorption, enabling the use of different imaging modalities that strongly depends on the properties of the sample structure, the selected fluorophore and the excitation laser. However, versatile and tunable laser excitation for multi-photon absorption is still a challenge, limited by e.g. the narrow bandwidth of typical laser gain medium or by the tunability of wavelength conversion offered by optical parametric oscillators or amplifiers. As an alternative, supercontinuum generation can provide broadband excitations spanning from the ultra-violet to far infrared domains and integrating numerous fluorophore absorption peaks, in turn enabling different imaging modalities or potential multiplexed spectroscopy. Here, we report on the use of machine learning to optimize the spectro-temporal properties of supercontinuum generation in order to selectively enhance multi-photon excitation signals compatible with a variety of fluorophores (or modalities) for multi-photon microscopy. Specifically, we numerically explore how the use of reconfigurable (femtosecond) pulse patterns can be readily exploited to control the nonlinear propagation dynamics and associated spectral broadening occurring in a highly-nonlinear fiber. In this framework, we show that the use of multiple pulses to seed optical fiber propagation can trigger a variety of nonlinear interactions and complex propagation scenarios. This approach, exploiting the temporal dimension as an extended degree of freedom, is used to maximize typical multi-photon excitations at selected wavelengths, here obtained in a versatile and reconfigurable manner suitable for imaging applications. We expect these results to pave the way towards on-demand and real time supercontinuum shaping, with further multi-photon microscopy improvements in terms of spatial 3D resolution, optical toxicity, and wavelength selectivity.

Substantial frequency conversion at long-wavelength limit in metamaterial with weakly nonlinear local electromechanical resonators: Analytical, computational, and experimental study

Recent studies of nonlinear metamaterials have shown interesting wave propagation phenomena including the birth of soliton, tunable bandgap, acoustic nonreciprocity and broadband energy harvesting. However, most studies are limited to nonlinear mechanical metamaterials and their applications in the short-wavelength limit. There is no study of nonlinear electromechanical metamaterials at the long-wavelength limit. The present work fills this gap through investigating a metamaterial with nonlinear local electromechanical resonators. An approximate analytical solution for the dispersion relations is obtained by the method of multiple scales (MMS). This analytical solution is used to study the role of different system’s mechanical and electromechanical parameters on the band structure. Other important nonlinear wave propagation characteristics are deduced by exploring the spectro-spatial features through different signal processing methods of the numerical results. The output voltage results demonstrate a significant wave distortion and frequency shift, particularly at the long-wavelength limit and other wavelength limits. The numerical results also show the birth of solitary waves and significant increase in the output voltage. Computational study via COMSOL Multiphysics and physical experiments are carried out to qualitatively demonstrate the theoretical findings. The overall observations suggest that the proposed nonlinear electromechanical metamaterial can enhance sensing and increase the operation range of electromechanical diode.

Numerical and experimental study of a FORM-based design wave applying the HOS-NWT nonlinear wave solver

This paper presents a comparative study of long-time irregular waves and equivalent design waves (EDW) in terms of geometric similarity and probability of exceedance (POE) distribution of the wave crest. For a proper comparison between the two wave types, the same nonlinear model was applied in the wave generation by application of the Higher-Order Spectral-Numerical Wave Tank (HOS-NWT), a fully nonlinear propagation solver.Numerical and experimental Monte-Carlo results were obtained through a number of realizations of a given sea state, and results were used as a reference for the EDW results. The experimental measurements and numerical simulations for a given sea state were analyzed and compared in terms of the wave spectrum estimation and the POE distribution of wave crests. The agreement between experimental and numerical results in wave quality seemed sufficient for it to be used as a reference for EDW cases.The First-Order Reliability Method (FORM) approach was applied in the calculation of EDW. The geometrical similarity between the measured EDW wave signal and the corresponding irregular wave signals measured in a given sea state was reviewed. It confirmed that the FORM-based EDW generates a comparable wave profile. In the statistical analysis, however, the results showed that for some EDW cases in relatively severe sea states, the POE estimates by the FORM method appeared to have conservative values compared to the Monte-Carlo reference POE distribution, whereas for the EDW cases in moderate sea states, the estimated POEs were in very good agreement with the empirical wave crest POE distribution of a given sea state.

Passive symmetry breaking of the space–time propagation in cavity dissipative solitons

Dissipative solitons are fundamental wave-pulses that preserve their form in the presence of periodic loss and gain. The canonical realization of dissipative solitons is Kerr-lens mode locking in lasers, which delicately balance nonlinear and linear propagation in both time and space to generate ultrashort optical pulses. This linear-nonlinear balance dictates a unique pulse energy, which cannot be increased (say by elevated pumping), indicating that excess energy is expected to be radiated in the form of dispersive or diffractive waves. Here we show that Kerr-lens mode-locked lasers can overcome this expectation. Specifically, by breaking the spatial symmetry between the forward and backward halves of the round-trip in a linear cavity, the laser can modify the soliton in space to incorporate the excess energy. Increasing the pump power leads therefore to a different soliton solution, rather than to dispersive/diffractive loss. We predict this symmetry breaking by a complete numerical simulation of the spatio-temporal dynamics in the cavity, and confirm it experimentally in a Kerr-lens mode-locked Ti:Sapphire laser with quantitative agreement to the simulation. The simulation opens a window to directly observe the nonlinear space-time dynamics that molds the soliton pulse, and possibly to optimize it.

Phase-Matching in Nonlinear Crystal-Based Monochromatic Terahertz-Wave Generation

Optically pumped nonlinear frequency down conversion is a proven approach for monochromatic terahertz (THz)-wave generation that provides superior properties such as continuous and wide tunability as well as laser-like linewidth and beam quality. Phase-matching (PM) is an important connection between the pump sources and nonlinear crystals and determines the direction of energy flow (as well as the output power). In past decades, a variety of peculiar PM configurations in the THz region have been invented and are different from the traditional ones in the optical region. We summarize the configurations that have been applied in nonlinear THz-wave generation, which mainly fall in two categories: scalar (collinear) PM and vector PM (including macroscopic noncollinear PM and microscopic vector PM). The development of this technique could relax the matching conditions in a wide range of nonlinear crystals and pump wavelengths and could finally promote the improvement of coherent THz sources.

Nonlinear Generation of Fluting Perturbations by Kink Mode in a Twisted Magnetic Tube

We study the excitation of fluting perturbations in a magnetic tube by an initially imposed kink mode. We use the ideal magnetohydrodynamic (MHD) equations in the cold-plasma approximation. We also use the thin-tube approximation and scale the dependent and independent variables accordingly. Then we assume that the dimensionless amplitude of the kink mode is small and use it as an expansion parameter in the regular perturbation method. We obtain the expression for the tube boundary perturbation in the second-order approximation. This perturbation is a superposition of sausage and fluting perturbations.

Chirped self-similar localized pulses on a continuous wave background in presence of cubic–quintic nonlinearity and self-frequency shift

We investigate the nonlinear pulse propagation through an inhomogeneous single-mode optical fiber where the evolution dynamics is governed by the generalized higher-order nonlinear Schrödinger equation with varying group velocity dispersion, cubic and quintic nonlinearities, self-frequency shift arising from stimulated Raman scattering, and linear gain or loss. A new class of soliton pulse solutions for a wave field is constructed based on the similarity transformation connecting with the constant-coefficient Kundu–Eckhaus equation. This class describes soliton structures that propagate self-similarity on a continuous-wave (cw) background, and acquires a nonlinear chirp as it propagates in the inhomogeneous fiber medium. It is found that the nonlinear chirp associated with these self-similar pulses originates essentially from the self-frequency shift effect. The conditions on fiber parameters for the existence of those nonlinearly chirped self-similar localized pulses are also given. As a practical example, we studied their evolution dynamics in a specified soliton control system. The solution stability against perturbation is checked by direct numerical simulation.