Nonlinear Dispersion Properties Research Articles

Nonlinearity enhanced wave bandgaps in metamaterial honeycombs embedding spider web-like resonators

Wave propagation in metamaterial honeycombs endowed with periodically distributed nonlinear resonators is addressed. The linear and nonlinear dispersion properties of the metamaterial are investigated. The nonlinear wave propagation equations obtained via a projection method and the Floquet–Bloch theorem are attacked by the method of multiple scales to obtain in closed form the nonlinear manifolds parametrized by the amplitudes, the frequency, and the wave numbers. The effects of the nonlinearity on the frequency bandgaps are thoroughly investigated and the optimization problem of the resonators nonlinearity towards increased bandgap size is tackled to provide a significant practical framework for the design of nonlinear metamaterials.

Nonlinear dispersion properties of metamaterial beams hosting nonlinear resonators and stop band optimization

The nonlinear dispersion properties of a metamaterial beam embedding nonlinear resonators are investigated. The metamaterial beam incorporates a distributed array of local resonators exhibiting softening or hardening cubic nonlinearity. The nonlinear dynamic behavior of the metamaterial beam is first investigated asymptotically via the method of multiple scales, which yields the closed-form nonlinear dispersion functions with the associated stop bands and the nonlinear dispersive waves as a combination of acoustic and optical bending waves. The effects of the resonators nonlinearity and the hosting beam nonlinear bending curvature on the stop bands are investigated in terms of sensitivity with respect to the strength and type of resonators nonlinearity. A full multi-variable optimization is carried out to study the nonlinear stop band size variations with respect to the resonators nonlinearity and the acoustic/optical wave amplitudes. The results show how the exploitation of the nonlinearities offers great possibilities to enlarge the size of the stop bands compared to the corresponding linear case and, hence, greatly enhance the energy absorption bandwidth.

Dispersive nonlinear acoustic waves

The main phenomena and mathematical models of the theory of nonlinear dispersive acoustic waves are described. For the physical dispersion of the relaxation type, the form of the kernels of the integro-differential equation for several relaxation times and the continuous spectrum of such times is indicated. The possibility of obtaining simple equations for media with a finite “memory time” is indicated. As a medium with resonant dispersion, where the kernels have an oscillatory character, a liquid with gas bubbles is considered. The dispersion curves are analyzed. It is argued that the introduction of resonant elements into a homogeneous matrix is a special case of creating metamaterials with unusual nonlinear-dispersion properties. As an example of the geometric dispersion, important problems of nonlinear focusing are considered based on the Ostrovsky–Vakhnenko equation, which is interpreted here as the projection of the Khokhlov–Zabolotskaya equation onto the acoustic beam axis. When analyzing the nonlinear dispersion, it was indicated that in acoustics the dependence of the wave propagation velocity on the magnitude of the perturbation is a common phenomenon. It leads to a distortion of the wave profile in time, as well as to a deformation of the spatial shape of the beam — to effects like self-refraction. In acoustics, nonlinearities of odd degrees are encountered much less frequently than in optics; cubic nonlinearity is possessed, for example, by shear waves in solids. Nevertheless, it leads to original effects — the formation of trapezoidal sawtooth waves and unusual self-focusing. The article systematically presents both known and new results.

Nonlinear wave propagation in locally dissipative metamaterials via Hamiltonian perturbation approach

The cellular microstructure of periodic architected materials can be enriched by local intracellular mechanisms providing innovative distributed functionalities. Specifically, high-performing mechanical metamaterials can be realized by coupling the low-dissipative cellular microstructure with a periodic distribution of tunable damped oscillators, or resonators, vibrating at relatively high amplitudes. The benefit is the actual possibility of combining the design of wave-stopping bands with enhanced energy dissipation properties. This paper investigates the nonlinear dispersion properties of an archetypal mechanical metamaterial, represented by a one-dimensional lattice model characterized by a diatomic periodic cell. The intracellular interatomic interactions feature geometric and constitutive nonlinearities, which determine cubic coupling between the lattice and the resonators. The non-dissipative part of the coupling can be designed to exhibit a softening or a hardening behavior, by independently tuning the geometric and elastic stiffnesses. The nonlinear wavefrequencies and waveforms away from internal resonances are analytically determined by adopting a perturbation technique. The employed approach makes use of tools borrowed from Hamiltonian perturbation theory, together with techniques often used in the context of nearly-integrable Hamiltonian systems.The dispersion spectra are determined in closed, asymptotically approximate, form as a nonlinear function of the time-dependent decreasing amplitude decrement. The invariant manifolds defined by the harmonic periodic motions are also analytically determined. The asymptotic results are further validated numerically.

Nonlinear dispersion properties of one-dimensional mechanical metamaterials with inertia amplification

Architected metamaterials offering superior dynamic performances can be conceived by inducing local mechanisms of inertia amplification in the periodic microstructure. A one-dimensional cellular lattice characterized by a pantograph mechanism in the tetra-atomic cell is proposed as minimal physical realization of inertially amplified metamaterial. A discrete model is formulated to describe the undamped free dynamics of the cell microstructure. The ordinary differential equations of motion feature quadratic and cubic inertial nonlinearities, induced by the indeformability of the pantograph arms connecting the principal atoms with the secondary atoms, serving as inertial amplifiers. An asymptotic approach is employed to analytically determine the dispersion properties governing the free propagation of harmonic waves in the pantographic metamaterial. First, the linear wavefrequencies and waveforms are obtained by solving the eigenproblem governing the lowest asymptotic order. An invariant parametric form is achieved for the pass and stop band structure, corresponding to propagation and attenuation branches of the dispersion spectrum in the plane of complex wavenumbers. The major effects due to the mass ratio of the inertial amplifiers are discussed. Particularly, the existence conditions, amplitude and centerfrequency of the band gap separating the acoustic and optical pass bands are determined analytically. Second, the nonlinear wavefrequencies and waveforms are obtained by solving the hierarchical linear problems governing the higher asymptotic orders. Analytical, although asymptotically approximate, functions are achieved for the nonlinear wavefrequencies and waveforms, which show quadratic dependence on the oscillation amplitudes. The mechanical conditions for the softening/hardening behaviour of the nonlinear wavefrequencies and the different topological properties of the invariant manifolds associated to the nonlinear waveforms are discussed. Finally, numerical simulations are provided to validate the analytical results.

Third-order Nonlinear Dispersion Properties of Surface Plasmon Resonance in Gold Nanoparticles

Third-order Nonlinear Dispersion Properties of Surface Plasmon Resonance in Gold Nanoparticles

Wave propagation properties of one-dimensional acoustic metamaterials with nonlinear diatomic microstructure

Acoustic metamaterials are artificial microstructured media, typically characterized by a periodic locally resonant cell. The cellular microstructure can be functionally customized to govern the propagation of elastic waves. A one-dimensional diatomic lattice with cubic inter-atomic coupling—described by a Lagrangian model—is assumed as minimal mechanical system simulating the essential undamped dynamics of nonlinear acoustic metamaterials. The linear dispersion properties are analytically determined by solving the linearized eigenproblem governing the free wave propagation in the small-amplitude oscillation range. The dispersion spectrum is composed by a low-frequency acoustic branch and a high-frequency optical branch. The two frequency branches are systematically separated by a stop band, whose amplitude is analytically derived. Superharmonic 3:1 internal resonances can occur within a wavenumber-dependent locus defined in the mechanical parameter space. A general asymptotic approach, based on the multiple scale method, is employed to determine the nonlinear dispersion properties. Accordingly, the nonlinear frequencies and waveforms are obtained for the two fundamental cases of non-resonant and superharmonically 3:1 resonant or nearly resonant lattices. Moreover, the invariant manifolds associated with the nonlinear waveforms are parametrically determined in the space of the two principal coordinates. Finally, some examples of non-resonant and resonant lattices are selected to discuss their nonlinear dispersion properties from a qualitative and quantitative viewpoint.

Tunable self-focusing and self-defocusing effects in a triple quantum dot via the tunnel-enhanced cross-Kerr nonlinearity.

Kerr-nonlinearity induced self-focusing or self-defocusing effect provides the opportunity for exploring fundamental phenomena related to the light-matter interactions. Here we show that the linear and nonlinear dispersion responses are significantly sensitive to both the detunings and the tunneling strengths of the indirect-excitonic (IX) states in an asymmetric triple quantum dot system. In particular, the nonlinear dispersion properties are dominated by the tunnel-enhanced cross-Kerr nonlinearity from one of the IX states. Meanwhile, by varying the detunings of other IX states, we reveal that the tunnel-enhanced cross-Kerr nonlinearity gives rise to the realization of the self-focusing and self-defocusing effects. Moreover, by taking into account the effect of the longitudinal-acoustic-phonon induced dephasing of the IX states, it is possible to modulate the height and position of the peak of the self-focusing or self-defocusing effect. Our results may have potential applications in nonlinear-optics and quantum-optics devices based on the tunnel-enhanced nonlinearities in this solid-state system.

Phase-sensitive Kerr nonlinearity of V-type atoms-doped dielectric slab

We investigate the linear and nonlinear dispersion and absorption properties of a weak probe field in a slab doped with V-type atoms. The effect of relative phase between applied fields on linear and nonlinear dispersion and absorption is discussed. It is demonstrated that the system is phase-sensitive when spontaneously generated coherence (SGC) exists. The effects of coupling field intensity, incoherent pump, and phase difference between probe and coupling fields on Kerr nonlinearity are also presented.

Phase-dependent Kerr nonlinearity of a dielectric slab doped by Λ-type atoms

We investigate the linear and nonlinear dispersion and absorption properties of a weak probe field in a dielectric slab doped with -type atoms. The effect of a relative phase between applied fields on linear and nonlinear dispersion, absorption and Kerr nonlinearity is discussed. It is demonstrated that the system is phase sensitive when spontaneously generated coherence and an incoherent pump exist simultaneously. The effect of the coupling field intensity on the Kerr nonlinearity is also presented.

Nonlinear plasmonic dispersion and coupling analysis in the symmetric graphene sheets waveguide.

We study the nonlinear dispersion and coupling properties of the graphene-bounded dielectric slab waveguide at near-THz/THz frequency range, and then reveal the mechanism of symmetry breaking in nonlinear graphene waveguide. We analyze the influence of field intensity and chemical potential on dispersion relation, and find that the nonlinearity of graphene affects strongly the dispersion relation. As the chemical potential decreases, the dispersion properties change significantly. Antisymmetric and asymmetric branches disappear and only symmetric one remains. A nonlinear coupled mode theory is established to describe the dispersion relations and its variation, which agrees with the numerical results well. Using the nonlinear couple model we reveal the reason of occurrence of asymmetric mode in the nonlinear waveguide.

New Compact antenna based on simplified CRLH-TL for UWB wireless communication systems

This letter presents the experimental results of a novel planar antenna design which is synthesized using simplified composite left/right-handed transmission-line (SCRLH-TL), which is a version of a conventional composite left/right handed-transmission-lines (CRLH-TL), however, with the omission of shunt-inductance in the unit-cell. SCRLH-TL exhibits a right-handed response with nonlinear dispersion properties and a smooth Bloch-impedance distribution. Arranged within the inner slot of the antenna are three smaller rectangular patch radiators. Each patch radiator is embedded with an E-shaped notch, and located in the antenna conductor is a larger E-shaped notch next to the 50-Ω termination. The E-shaped notches constitute SCRLH-TL property. The gap in the slot between the smaller patches and the conductor next to the larger E-shaped notch determines the impedance bandwidth of the antenna. The dimensions of the smaller patches determine the radiation characteristics of the antenna. The antenna is excited using a conductor-backed coplanar waveguide transmission-line. The antenna covers a bandwidth of 7.3 GHz between 0.7 GHz and 8GHz, which corresponds to 167.81%. In this band, the antenna resonates at 4.75 GHz and 7 GHz; the gain and radiation efficiency at these frequencies are 4 dBi—80% and 3.6 dBi—73%, respectively. The antenna's performance was validated through measurement. The antenna has dimensions of 0.0504λ0 × 0.0462λ0 × 0.0018λ0, where λ0 is free-space wavelength at 700 MHz.

Dispersion and absorption properties of the Y-type-four-level system with electromagnetically induced transparency

The linear and nonlinear dispersion and absorption properties of a Y-type four-level system, which driven by a weak probing field and two coupling fields, are investigated. Using the perturbation method, third-order perturbation iterative solutions of the steady density matrix equations are obtained. Numerical results and the analysis with the dressed state show that when the two coupling fields are in two-photon resonance, the dispersion and absorption spectra are symmetrical, there is only one transparency window in the system. When the two coupling fields are off two-photon resonance, the dispersion and absorption spectra are no longer symmetrical, and there are two transparency windows in the system. The third-order nonlinear properties of the system are mainly determined by the population difference between the levels which associate with the probing field, and the contributions of the transitions associated with the coupling fields are negligible.

Nonlinear dispersion properties of high-frequency waves in the gradient theory of elasticity

The dispersion law ceases to be linear already at ultrasonic frequencies of elastic vibrations of particles as mechanical perturbation waves propagate through the medium. A variant of the continuum model of an elastic medium is proposed which is based on the assumption of pair and triplet potential interaction between infinitely small particles; this allows one to represent the dispersion law with any required accuracy. The corresponding wave equation, which is still linear, can have an arbitrarily large order of partial derivatives with respect to the coordinates. It is suggested that the results of comparing the representations of the dispersion law from the elasticity and solid-state physics viewpoints should be used to determine nonclassical characteristics of the elastic state of the medium. The theoretical conclusions are illustrated with calculations performed for plane waves propagating through aluminum.

Electron–Phonon Coupling Effects on the Karpman–Washimi Ponderomotive Magnetization in a Semiconductor Plasma

In recent years, there has been a considerable interest in the physical properties of semiconductor quantum plasmas since the ion implantation technique has been widely used as an effective nano-fabrication tool for manufacture of various semiconductor components. In quantum plasmas, the linear and nonlinear dispersion properties of plasma waves including the Bohm potential term due to the collective behavior have been considerably investigated by using the quantum hydrodynamic model. Very recently, the lattice electron– phonon effects are explored in a homogeneous piezoelectric semiconductor quantum plasma and is also found that the electron–phonon coupling generates the far-field dynamically oscillatory wake field. In various plasmas, the wave propagation and trapping have been extensively explored by the ponderomotive interactions between the electromagnetic field and plasma since the ponderomotive force provides the spectral characteristics of the plasma. It has been also shown that the ponderomotive interaction in plasmas would be obtained as the space–time derivatives of the stress tensor. However, the physical characteristics of the Karpman– Washimi ponderomotive magnetization caused by the timevariation of the intensity of the electromagnetic field in a homogeneous piezoelectric semiconductor quantum plasma including the influence of the electron–phonon interactions have not been investigated as yet. Hence, in this paper we have investigated the electron–phonon interaction effects on the non-stationary Karpman–Washimi ponderomotive magnetization in a semiconductor quantum plasma. Hence, the induced electron cyclotron motion due to the Karpman– Washimi ponderomotive force of the electromagnetic wave has been investigated as a function of the electron plasma frequency, quantum angular frequency, speed of the acoustic phonon, and electron–phonon coupling coefficient. From the Karpman–Washimi procedure, the total Karpman–Washimi ponderomotive force FKW of the electromagnetic field ~ Eðr; tÞ 1⁄21⁄4 ðEðr; tÞeiðk r !tÞ þ E ðr; tÞe iðk r !tÞÞ=2 in unmagnetized plasmas would be expressed as follows:

Electromagnetically induced spatial nonlinear dispersion of four-wave mixing

Spatial displacements of the probe and generated four-wave mixing beams are observed in a three-level V-type, as well as a two-level atomic system near resonance. The observed spatial shift curves reflect the typical enhanced cross-Kerr nonlinear dispersion properties in the electromagnetically induced transparency (EIT) systems. The spatial beam displacements are controlled by the strong control laser beam and the atomic density. Studying such controlled spatial beam shifts can be important in image storage and in generating spatially correlated (entangled) laser beams in multilevel EIT systems.

Modeling of micro-diameter-scale liquid core optical fiber filled with various liquids

This paper gives the simulation results on micro-diameter-scale liquid core optical fiber (LCOF) filled with different kinds of liquids. The nonlinear and group velocity dispersion (GVD) properties of the microdiameter- scale LCOF are achieved. The simulation of supercontinuum generation of LCOF is also obtained. The calculations show that LOCF can provide huge nonlinear parameter and large span of slow varying GVD characteristics in the infrared region, which have potential applications in optical communications and nonlinear optics. Besides, LOCF has advantage of easy fabricating and robustness compared with silica nano-wire.

Impacts of imperfect geometry structure on the nonlinear and chromatic dispersion properties of a microstructure fiber

We numerically investigated the impacts of the imperfect geometry structure on the nonlinear and chromatic dispersion properties of a microstructure fiber (MF). The statistical results show that the imperfect geometry structure degrades the high nonlinearity and fluctuates the chromatic dispersion in a MF. Moreover, the smaller air holes and the larger pitch are more likely to maintain the properties of nonlinearity and chromatic dispersion. Finally, the nonlinearity and chromatic dispersion are more insensitive to air-hole nonuniformity than to air-hole disorder. All of these will provide references for designing and fabricating MF.

NONLINEAR TE SURFACE WAVES IN A LEFT-HANDED MATERIAL AND SUPER LATTICES WAVE-GUIDE STRUCTURE

The paper is concerned with the propagation characteristics of TE surface waves in a planar wave-guide structure of a lateral antiferromagnetic-non magnetic super lattices (LANS) film bounded by a nonlinear dielectric cover and a left-handed substrate (LHM). In LHM substrate both permittivity and magnetic permeability are negative in definite frequency range. We study nonlinear dispersion properties of the TE surface waves and illustrate the power flow variation with the wave index in three different cases, i.e., when both permittivity and magnetic permeability are negative, when permittivity is negative and permeability is positive, and when both permittivity and magnetic permeability are positive. In the first case, we found that surface waves are backward traveling and the wave power variation with the wave index shows bistability behavior. We also found that by increasing the film thickness the waves turn from forward to backward traveling in the second case. In the third case, surface waves are always forward traveling.

Nonlinear guided waves and symmetry breaking in left-handed waveguides

We analyze nonlinear guided waves in a planar waveguide made of a left-handed material surrounded by a Kerr-like nonlinear dielectric, and predict that such a waveguide can support fast and slow symmetric and antisymmetric nonlinear modes. We study the symmetry breaking bifurcation and asymmetric modes in such a symmetric structure. The analysis of nonlinear dispersion properties of the guided waves shows that the modes can be either forward or backward.