Weak Spatial Dispersion Research Articles

Spatial Dispersion in Hypercrystal Distributed Feedback Lasing.

This work is a first approach to investigate the role of spatial dispersion in photonic hypercrystals (PHCs). The scope of the presented analysis is focused on exploiting nonlocality, which can be controlled by appropriate design of the structure, to obtain new light generation effects in a distributed feedback (DFB) laser based on PHC, which are not observable under weak spatial dispersion. Here, we use effective medium approximation and our original model of threshold laser generation based on anisotropic transfer matrix method. To unequivocally identify nonlocal generation phenomena, the scope of our analysis includes comparison between local and nonlocal threshold generation spectra, which may be obtained for different geometries of PHC structure. In particular, we have presented that, in the presence of strong spatial dispersion, it is possible to obtain spectrally shifted Bragg wavelengths of TE- and TM-polarization spectra, lowered generation threshold levels for both light polarizations, generation of light of selected light polarization (TE or TM), or simultaneous generation of TE- and TM-polarized waves at different frequencies with controllable spectral separation, instead of single mode operation anticipated with local approach.

Extension of Lorentz reciprocity and Poynting theorems for spatially dispersive media with quadrupolar responses

We provide a self-consistent extension of the Lorentz reciprocity theorem and the Poynting theorem for media possessing electric and magnetic dipolar and quadrupolar responses related to electric and magnetic fields and field gradients thus corresponding to weak spatial dispersion. Using these two theorems, we respectively deduce the conditions of reciprocity and gainlessness and losslessness that apply to the various tensors mediating the interactions of these multipole moments and the associated fields and field gradients. We expect that these conditions will play an essential role in developing advanced metamaterial modeling techniques that include quadrupolar and spatially dispersive responses.

Gradient catastrophe of nonlinear photonic valley-Hall edge pulses

We derive nonlinear wave equations describing the propagation of slowly-varying wavepackets formed by topological valley-Hall edge states. We show that edge pulses break up even in the absence of spatial dispersion due to nonlinear self-steepening. Self-steepening leads to the previously-unattended effect of a gradient catastrophe, which develops in a finite time determined by the ratio between the pulse's nonlinear frequency shift and the size of the topological band gap. Taking the weak spatial dispersion into account results then in the formation of stable edge quasi-solitons. Our findings are generic to systems governed by Dirac-like Hamiltonians and validated by numerical modeling of pulse propagation along a valley-Hall domain wall in staggered honeycomb waveguide lattices with Kerr nonlinearity.

Ducting of upper-hybrid waves by density depletions in a magnetoplasma with weak spatial dispersion

The effects of wave trapping and ducting inside a plasma density depletion are studied in a large laboratory magnetoplasma in the upper-hybrid (UH) range of frequencies. A field-aligned density depletion is generated via localized rf plasma heating and subsequent plasma thermal diffusion. Test UH waves are emitted and detected by small-size electric monopole antennas. For a given set of experimental parameters, propagation of UH waves can be considered as collisionless. At the same time, wave surface topology in the UH range is determined by weak spatial dispersion, which is conditioned by the thermal motion of electrons and the ambient magnetic field. For various density depletion depths and diameters, ducting of UH waves belonging to different characteristic parts of the “dumbbell-shaped” wave surface can be observed. Particularly, quasiparallel Langmuir (L-mode) waves and oblique resonance cone (X-mode) waves can be ducted in different regimes. A qualitative explanation of ducting regimes is given based on ray tracing analysis.

Beyond local effective material properties for metamaterials

To discuss the properties of metamaterials on physical grounds and to consider them in applications, effective material parameters are usually introduced and assigned to a given metamaterial. In most cases, only weak spatial dispersion is considered. It allows to assign local material properties, i.e. a permittivity and a permeability. However, this turned out to be insufficient. To solve this problem, we study here the effective properties of metamaterials with constitutive relations beyond a local response and take strong spatial dispersion into account. The isofrequency surfaces of the dispersion relation are investigated and compared to those of an actual metamaterial. The significant improvement provides evidence for the necessity to use nonlocal material laws in the effective description of metamaterials. The general formulation we choose here renders our approach applicable to a wide class of metamaterials.

Nonlocal effect on the transmission of a transverse electric electromagnetic wave through periodic dielectric-metal multilayers

The problem of electromagnetic response of a periodic dielectric-metal structure to transverse electric electromagnetic excitation is solved analytically on the basis of the Boltzmann kinetic equation formalism. We analyze the dispersion relation, transmission coefficient, and absorption for different angles of incidence. We show that, due to nonlocal dependence of the electron current density on the electric field in metal layers, the THz and near-infrared photonic properties of dielectric-metal structures are strikingly different from those obtained with the use of the simplified description based on the Drude–Lorentz model. In the limit of weak spatial dispersion, the developed theory reproduces the known results for transverse electric electromagnetic waves scattered by layered structures consisting of spatially local layers.

Perfect control of reflection and refraction using spatially dispersive metasurfaces

Non-uniform metasurfaces (electrically thin composite layers) can be used for shaping refracted and reflected electromagnetic waves. However, known design approaches based on the generalized refraction and reflection laws do not allow realization of perfectly performing devices: there are always some parasitic reflections into undesired directions. In this paper we introduce and discuss a general approach to the synthesis of metasurfaces for full control of transmitted and reflected plane waves and show that perfect performance can be realized. The method is based on the use of an equivalent impedance matrix model which connects the tangential field components at the two sides on the metasurface. With this approach we are able to understand what physical properties of the metasurface are needed in order to perfectly realize the desired response. Furthermore, we determine the required polarizabilities of the metasurface unit cells and discuss suitable cell structures. It appears that only spatially dispersive metasurfaces allow realization of perfect refraction and reflection of incident plane waves into arbitrary directions. In particular, ideal refraction is possible only if the metasurface is bianisotropic (weak spatial dispersion), and ideal reflection without polarization transformation requires spatial dispersion with a specific, strongly non-local response to the fields.

Nonlocal Effects on Surface Plasmon Polariton Propagation in Graphene Nanoribbons

Fundamental properties of surface plasmon polariton modes propagating along graphene nanoribbons are investigated by means of a full-wave method-of-moments approach adopting a spatially dispersive dyadic model for graphene conductivity. Modal propagation, current profiles, field distributions, and characteristic impedances are studied in detail showing the dramatic effects of spatial dispersion and the need to go beyond the usual weak (low- $q$ ) spatial dispersion formulation.

Phenomenological theory of magnetoelectric coupling in granular multiferroics

We study coupling between the ferroelectric polarization and magnetization of granular ferromagnetic film using a phenomenological model of combined multiferroic system consisting of granular ferromagnetic film placed above the ferroelectric (FE) layer. The coupling is due to screening of Coulomb interaction in the granular film by the FE layer. Below the FE Curie temperature the magnetization has hysteresis as a function of electric field. Below the magnetic ordering temperature the polarization has hysteresis as a function of magnetic field. We study the magneto-electric coupling for weak and strong spatial dispersion of the FE layer. The effect of mutual influence decreases with increasing the spatial dispersion of the FE layer. For weak dispersion the strongest coupling occurs in the vicinity of the ferroelectric-paraelectric phase transition. For strong dispersion the situation is the opposite. We study the magneto-electric coupling as a function of distance between the FE layer and the granular film. For large distances the coupling decays exponentially due to the exponential decrease of electric field produced by the oscillating charges in the granular ferromagnetic film.

Nonlocal effective parameters of a coated sphere medium

An analytical theory has been developed to find the general effective parameters of a nonlocal medium. The medium is nonlocal due to presence of spatial dispersion. The proposed theory is based upon the dipolar scattering model of the inclusions comprising the medium. The bianisotropy stemming from the magnetoelectric coupling at the inclusion and lattice level has been discussed. The developed theory is then applied to a medium which consists of coated spheres with realistic materials. Effects of different values of wavevector upon the effective permittivity, effective permeability, and magnetoelectric coefficient stemming from lattice effects have been studied for a coated sphere medium. It is shown that a coated sphere medium with a weak spatial dispersion gives rise to a broader range of frequencies, where real parts of the effective permittivity and permeability are negative. On the other hand, this range of frequencies becomes smaller when the spatial dispersion of the medium is not weak.

Fractional Gradient Elasticity from Spatial Dispersion Law

Nonlocal elasticity models in continuum mechanics can be treated with two different approaches: the gradient elasticity models (weak nonlocality) and the integral nonlocal models (strong nonlocality). This paper focuses on the fractional generalization of gradient elasticity that allows us to describe a weak nonlocality of power-law type. We suggest a lattice model with spatial dispersion of power-law type as a microscopic model of fractional gradient elastic continuum. We demonstrate how the continuum limit transforms the equations for lattice with this spatial dispersion into the continuum equations with fractional Laplacians in Riesz's form. A weak nonlocality of power-law type in the nonlocal elasticity theory is derived from the fractional weak spatial dispersion in the lattice model. The continuum equations with derivatives of noninteger orders, which are obtained from the lattice model, can be considered as a fractional generalization of the gradient elasticity. These equations of fractional elasticity are solved for some special cases: subgradient elasticity and supergradient elasticity.

Importance of Mueller matrix characterization of bianisotropic metamaterials

We present a study on the optical properties of metamaterial layers containing U-shaped resonators. Using Finite Element Methods, we show that the description of the optical properties on one isolated resonator requires taking into account permittivity, permeability and magneto-electric tensors. The effect of weak spatial dispersion in arrays of resonators can be reproduced by adding two elements in the permittivity and magneto-electric coupling tensors. We show how the symmetries of the Mueller matrix of the sample can be used to discriminate the contributions of the different tensor elements. Generalized ellipsometry measurements made on a sample realized by e-beam lithography confirm the qualitative findings of the theoretical study. A quantitative determination of the different elements of permittivity, permeability and magneto-electric coupling is obtained by using oscillators with Lorentzian lineshape, to fit to the measurements.

Parametric excitation of surface electron cyclotron O-modes by an external alternating electric field

Using the model of uniform semi-bounded plasma, it is shown that the influence of an external alternating electric field in the range of electron cyclotron frequencies leads to excitation of eigen ordinary polarized waves at the first harmonic of electron cyclotron frequency. The external electric field was assumed to be uniform and monochromatic; electro-dynamical properties of the plasma were described by the kinetic Vlasov–Boltzmann equation in the approach of a weak spatial dispersion of plasma. Two components of these modes’ wave vector oriented across an external magnetic field are taken into consideration. Fields of these electron cyclotron O-modes are described by the Maxwell equations; their phase velocity is assumed to be less than the velocity of light. Nonlinear boundary condition-described discontinuity of the modes’ tangential magnetic field is formulated using the concept of a nonlinear surface electric current. The infinite set of equations for harmonics of these O-modes’ tangential electric field has been derived with the aid of this nonlinear boundary condition. It is solved using a wave packet consisting of the basic harmonic and two nearest satellite harmonics. A simple analytical expression for growth rates of these surface waves’ parametric instability is obtained; the influence of plasma parameters on their excitation is numerically analyzed.

Parametric excitation of surface ion cyclotron extraordinary polarized waves by an external alternating electric field

An external alternating electric field in the range of ion–cyclotron frequencies is shown to excite extraordinary polarized surface electromagnetic waves at the second harmonic of ion–cyclotron frequencies. The kinetic model of a uniform semi-bounded plasma is applied. Plasma electrodynamical properties are described by using the Vlasov–Boltzmann kinetic equation in the approach of weak spatial plasma dispersion. Two components of these modes’ wave vector, which are oriented across an external steady magnetic field, are taken into account. Fields of these ion–cyclotron modes are described by Maxwell equations. Their phase velocity is assumed to be much lower than light velocity. The external alternating electric field is assumed to be uniform and monochromatic. The nonlinear boundary condition is formulated for the tangential magnetic field of the studied surface waves (SW). An infinite set of equations is derived for the values of tangential electric field harmonics on the plasma interface. This set is solved using the approach to the wave packet consisting of the main harmonic and two nearest satellite harmonics. Simple analytical expressions for the growth rates of parametric instability of these SW are derived. The influence of plasma parameters on their parametric excitation is numerically analyzed.

On effective electromagnetic parameters of artificial nanostructured magnetic materials

In this paper we discuss effective material parameter description of new nanostructures designed to perform as artificial magnetic materials for visible light. Among these structures there are various split-ring resonators, dual-bar structures, fishnet layers and other geometries. Artificial magnetic response in these structures appears due to weak spatial dispersion effects, and it is important to study the conditions under which the magnetic response can be adequately measured with effective permeability tensor. On the examples of dual bars and split rings we show that this is possible only under some quite restrictive conditions. In the general case, more complicated constitutive relations with more effective material parameters need to be developed.

A possible interpretation of the zebra pattern in solar radiation

The nature of the zebra pattern in continual type-IV solar radio bursts is discussed. It is shown that, when a weakly relativistic monoenergetic proton beam propagates in a highly nonisothermal plasma, the energy of the slow beam mode can be negative and explosive instability can develop due to the interaction of the slow and fast beam modes with ion sound. Due to weak spatial dispersion, ion sound generation is accompanied by cascade merging, which leads to stabilization of explosive instability. The zebra pattern forms due to the scattering of fast protons by ion sound harmonics. The efficiency of the new mechanism is compared with that of previously discussed mechanisms.

Theory of propagation of ordinary surface cyclotron waves

The dispersion properties of ordinary surface cyclotron waves in a semiinfinite nonuniform plasma are investigated. The waves propagate across the external magnetic field directed along the plasma surface in a metal waveguide the internal surface of which is covered with a dielectric. The problem is solved analytically in the framework of a kinetic model for plasma particles under the assumption of weak spatial dispersion. The influence of the parameters of the dielectric layer separating the plasma from the metal wall, the shape of the plasma density profile, and the value of the external magnetic field on the dispersion properties of surface cyclotron waves is studied both numerically and analytically.

Wave arrival singularities at cuspidal points in the acoustic wave surfaces of anisotropic solids and their unfolding under weak spatial dispersion

This paper is concerned with wave arrival singularities in the elastodynamic Green's functions of infinite anisotropic elastic solids, and their unfolding into smooth wave trains, known as quasi-arrivals, through spatial dispersion. The wave arrivals treated here are those occurring in (i) the displacement response to a suddenly applied point force or three-dimensional Green's function,, and (ii) the displacement response to an impulsive line force or two-dimensional Green's function,. These arrivals take on various analytical forms, including step function and logarithmic and power-law divergences. They travel outwards from the source at the group velocities in each direction, and their locus defines the three- and two-dimensional acoustic wave surfaces, respectively. The main focus of this paper is on the form of the wave arrivals in the neighbourhood of cuspidal points in the wave surfaces, and how these arrivals unfold into quasi-arrivals under the first onset of spatial dispersion. This regime of weak spatial dispersion, where the acoustic wavelength,λ, begins to approach the natural length scale,l, of the medium, is characterized by a correction to the phase velocity, which is quadratic in the wavevector,k, and the presence of fourth-order spatial derivatives of the displacement field in the wave equation. Integral expressions are established for the quasi-arrivals near to cuspidal points, involving the Airy function in the case ofand the Scorer function in the case of. Numerical results are presented, illustrating the oscillatory nature of the quasi-arrivals and the interference effects that occur near to cuspidal points in the wave surface.

Spatial dispersion and energy in strong chiral medium

Since the discovery of backward-wave materials, people have tried to realize a strong chiral medium, which is traditionally thought impossible mainly for the reason of energy and spatial dispersion.We compare the two most popular descriptions of a chiral medium. After analyzing several possible reasons for the traditional restriction, we show that a strong chirality parameter leads to positive energy without any frequency-band limitation in the weak spatial dispersion. Moreover, strong chirality does not result in a strong spatial dispersion, which occurs only around the traditional limit point. For strong spatial dispersion where higher-order terms of spatial dispersion need to be considered, the energy conservation is also valid. Finally, we show that realization of strong chirality requires the conjugated type of spatial dispersion.

Unfolding of wave-arrival singularities in the elastodynamic Green’s functions of anisotropic solids under weak spatial dispersion

In this paper we examine the effects on transient wave propagation of weak spatial dispersion, i.e., the first onset of dispersion as the acoustic wavelength approaches the natural scale of length of a medium. In centrosymmetric solids, and also away from acoustic axes in noncentrosymmetric crystals, this takes the form of a correction to the phase velocity which is quadratic in the wave vector $\mathbf{k}$. Our analysis is developed for centrosymmetric solids for which weak spatial dispersion expresses itself through the presence of fourth order spatial derivatives of the displacement field in the wave equation. We examine the effect that spatial dispersion has on wave-arrival singularities in the elastodynamic Green's functions of anisotropic solids. These are singular features that propagate at the group velocities in each direction, and are interpreted on the basis of the stationary phase approximation. The focus of this paper is on wave arrivals associated with elliptic and hyperbolic points on the acoustic slowness surface. In the response to a suddenly applied point force, the wave arrival pertaining to an elliptical point takes the form of a step function, or where symmetry intervenes, a ramp function. The wave arrival for a hyperbolic point takes the form of a logarithmic singularity, or where symmetry intervenes, a softer logarithmic singularity. We report how, under the influence of weak spatial dispersion, these arrivals unfold into wave trains known as quasiarrivals, which conform to integral expressions involving the Airy function. For positive dispersion, i.e., the more common situation of downward curving dispersion relation, $\ensuremath{\omega}(\mathbf{k})$, the ripples of the wave train follow the arrival, whereas for negative dispersion they precede the arrival. The results presented here are of relevance in the broad context of transient acoustic waves in any situation where the characteristic wavelength approaches the natural scale of length of the medium, whether that be the interatomic spacing in a crystal, the repeat distance in a layered solid, or whatever. They are expected to be of particular interest in the topical field of picosecond laser ultrasonics.