Component Of Wave Vector Research Articles

Lattice fractional calculus

Integration and differentiation of non-integer orders for N-dimensional physical lattices with long-range particle interactions are suggested. The proposed lattice fractional derivatives and integrals are represented by kernels of lattice long-range interactions, such that their Fourier series transformations have a power-law form with respect to components of wave vector. Continuous limits for these lattice fractional derivatives and integrals give the continuum derivatives and integrals of non-integer orders with respect to coordinates. Lattice analogs of fractional differential equations that include suggested lattice differential and integral operators can serve as an important element of microscopic approach to nonlocal continuum models in mechanics and physics.

The inherently three-dimensional nature of magnetized plasma turbulence

It is often asserted or implicitly assumed, without justification, that the results of two-dimensional investigations of plasma turbulence are applicable to the three-dimensional plasma environments of interest. A projection method is applied to derive two scalar equations that govern the nonlinear evolution of the Alfvénic and pseudo-Alfvénic components of ideal incompressible magnetohydrodynamic (MHD) plasma turbulence. The mathematical form of these equations makes clear the inherently three-dimensional nature of plasma turbulence, enabling an analysis of the nonlinear properties of two-dimensional limits often used to study plasma turbulence. In the anisotropic limit, k⊥ ≫ k∥, that naturally arises in magnetized plasma systems, the perpendicular 2D limit retains the dominant nonlinearities that are mediated only by the Alfvénic fluctuations but lacks the wave physics associated with the linear term that is necessary to capture the anisotropic cascade of turbulent energy. In the in-plane 2D limit, the nonlinear energy transfer is controlled instead by the pseudo-Alfvén waves, with the Alfvén waves relegated to a passive role. In the oblique 2D limit, an unavoidable azimuthal dependence connecting the wavevector components will likely cause artificial azimuthal asymmetries in the resulting turbulent dynamics. Therefore, none of these 2D limits is sufficient to capture fully the rich three-dimensional nonlinear dynamics critical to the evolution of plasma turbulence.

2-d finite barrier rectangular quantum dots II: Dirac description

The Schrödinger description of 2-d finite barrier rectangular quantum dots [1] is expanded to Dirac description through transfer matrices and reflection and rotation symmetries of the dot system. Inexactness of wave vector components of spinors is then reduced to two relations which lead to two different bispinors and four quantized transcendental energy relations corresponding to even–even, odd–odd and even–odd, odd–even factorizing functions of each bispinor. In order to show the spin effect on the dot energy levels, the solutions of the transcendental relations of the Schrödinger and Dirac descriptions are plotted.

The Radiation Problem from a Vertical Short Dipole Antenna above Flat and Lossy Ground: Novel Formulation in the Spectral Domain with Closed – Form Analytical Solution in the High Frequency Regime

In this paper we consider the problem of radiation from a vertical short Hertzian dipole above flat lossy ground, which represents the well known in the literature Sommerfeld radiation problem. The problem is formulated in a novel spectral domain approach, and by inverse three dimensional Fourier transformation the expressions for the received electric and magnetic field in the physical space are derived as one dimensional integrals over the radial component of wavevector, in cylindrical coordinates. Subsequent use of the Stationary Phase Method in the high frequency regime yields closed form analytical solutions for the received EM field vectors, which coincide with the corresponding reflected EM field originating from the image point. In this way, we conclude that the so called in the literature space wave, i.e. line of sight plus reflected EM field, represents the total solution of the Sommerfeld problem in the high frequency regime, in which case the surface wave can be ignored. Finally, numerical results in the high frequency regime are presented in this paper, in comparison with corresponding numerical results based on Norton solution of the problem, i.e. space and surface waves.

Surface polaritons in a dielectric – anisotropic nanocomposite system

We have studied the peculiarities of propagation of surface polaritons at a planar interface between an isotropic dielectric and anisotropic nanocomposite with metal inclusions of ellipsoidal shape. For the case when the axes of all nanoellipsoids of revolution are perpendicular to the propagation direction and parallel to the medium interface, we have obtained frequency dependences of the propagation constant and transverse wave vector components, penetration depth and path length, longitudinal and transverse energy fluxes for surface polaritons. The shape of nanoellipsoids is shown to have an effect on the wave characteristics of surface polaritons.

Mobility as controlled by the transverse component of phonon wave vector in quantum layers at low temperatures

Without resorting to either the Kawaji’s simplified model of interaction with only two-dimensional phonons or to the equipartition approximation for the phonon distribution, the characteristics of the momentum relaxation time of the conduction electrons in a quantized surface layer for interaction with intravalley acoustic phonons have been analysed under the condition of low temperature. The scattering and the mobility characteristics thus obtained for an n-channel (100)-oriented Si inversion layer are apparently quite different from what follows in the traditional framework.

Toward lattice fractional vector calculus

An analog of fractional vector calculus for physical lattice models is suggested. We use an approach based on the models of three-dimensional lattices with long-range inter-particle interactions. The lattice analogs of fractional partial derivatives are represented by kernels of lattice long-range interactions, where the Fourier series transformations of these kernels have a power-law form with respect to wave vector components. In the continuum limit, these lattice partial derivatives give derivatives of non-integer order with respect to coordinates. In the three-dimensional description of the non-local continuum, the fractional differential operators have the form of fractional partial derivatives of the Riesz type. As examples of the applications of the suggested lattice fractional vector calculus, we give lattice models with long-range interactions for the fractional Maxwell equations of non-local continuous media and for the fractional generalization of the Mindlin and Aifantis continuum models of gradient elasticity.

Freeze Frames vs. Movies

Sometimes, model building in crystallography is like resolving a puzzle: All obvious symmetrical or methodological errors are excluded, you apparently understand the measured patterns in 3D, but the structure solution and/or refinement is just not working. One such nerve-stretching problem arises from metrically commensurate structures (MCS). This expression means that the observed values of the components of the modulation wave vectors are rational by chance and not because of a lock-in. Hence, it is not a superstructure - although the boundaries between the two descriptions are blurry. Using a superstructure model for a MCS decreases the degrees of freedom, and forces the atomic arrangement to an artificial state of ordering. Just imagine it as looking at a freeze frame from a movie instead of watching the whole film. The consequences in structure solution and refinement of MCS are not always as dramatically as stated in the beginning. On the contrary, treating a superstructure like a MCS might be a worthwhile idea. Converting from a superstructure model to a superspace model may lead to a substantial decrease in the number of parameters needed to model the structure. Further, it can permit for the refinement of parameters that the paucity of data does not allow in a conventional description. However, it is well known that families of superstructures can be described elegantly by the use of superspace models that collectively treat a whole range of structures, commensurate and incommensurate. Nevertheless, practical complications in the refinement are not uncommon. Instances are overlapping satellites from different orders and parameter correlations. Notably, MCS occur in intermetallic compounds that are important for the performance of next-generation electronic devices. Based on examples of their (pseudo)hexagonal 3+1D and 3+2D structures, we will discuss the detection and occurrence of MCS as well as the benefits and limitations of implementing them artificially.

A continuous excitation approach to determine time-dependent dispersion diagrams in 2D magnonic crystals

We propose an alternative micromagnetic approach to determine the spin wave dispersion relations in magnonic structures. Characteristic of the method is that a limited area of the system is continuously excited with a spatially uniform oscillating field, tuned at a given frequency. After a transitory time, the regime magnetization dynamics is collected and a spatial Fourier analysis on it determines the frequency vs wave vector relation. Combining several simulations in any predetermined range of frequencies, at any resolution, we investigate the dispersion relations for different kinds of magnonic crystals: a dot array, an antidot array and a bicomponent film. Especially compared to traditional pulse-excitation methods, this technique has many advantages. First, the excitation power is concentrated at a single frequency, allowing the corresponding spin waves to propagate with very low attenuation, resulting in a higher k-space resolution. Second, the model allows to include very large wave vector components, necessary to describe the high-frequency response of non-quantized spin waves in quasi-continuous systems. Finally, we address some possible experimental opportunities with respect to excitation/detection techniques over large distances and the observation of the odd/even symmetry of spin waves using Brillouin light scattering.

Energetic-particle-induced electromagnetic geodesic acoustic mode in tokamak plasmas

Energetic-particle-induced kinetic electromagnetic geodesic acoustic modes (EKEGAMs) are numerically studied in low β (=plasma pressure/magnetic pressure) tokamak plasmas. The parallel component of the perturbed vector potential is considered along with the electrostatic potential perturbation. The effects of finite Larmor radius and finite orbit width of the bulk and energetic ions as well as electron parallel dynamics are all taken into account in the dispersion relation. Systematic harmonic and ordering analysis are performed for frequency and growth rate spectra of the EKEGAMs, assuming (kρi)∼q−3∼β≪1, where q, k, and ρi are the safety factor, radial component of the EKEGAMs wave vector, and the Larmor radius of the ions, respectively. It is found that there exist critical βh/βi values, which depend, in particular, on pitch angle of energetic ions and safety factor, for the mode to be driven unstable. The EKEGAMs may also be unstable for pitch angle λ0B<0.4 in certain parameter regions. Finite β effect of the bulk ions is shown to have damping effect on the EKEGAMs. Modes with higher radial wave vectors have higher growth rates. The damping from electron dynamics is found decreasing with decrease of the temperature ratio Te/Ti. The modes are easily to be driven unstable in low safety factor q region and high temperature ratio Th/Ti region. The harmonic features of the EKEGAMs are discussed as well.

Selective excitation of higher order modes in hollow-core PCF via prism-coupling.

Prism-coupling through the microstructured cladding is used to selectively excite individual higher order modes in hollow-core photonic crystal fibers (PCFs). Mode selection is achieved by varying the angle between the incoming beam and the fiber axis, in order to match the axial wavevector component to that of the desired mode. The technique allows accurate measurement of the effective indices and transmission losses of modes of arbitrary order, even those with highly complex transverse field distributions that would be extremely difficult to excite by conventional endfire coupling.

Spatial control of surface plasmon polariton excitation at planar metal surface.

We illustrate that the surface plasmon polariton (SPP) excitation through the prism coupling method is fundamentally limited by destructive interference of spatial light components. We propose that the destructive interference can be canceled out by tailoring the relative phase for the different wave-vector components. As a numerical demonstration, we show that through the phase modulation the excited SPP field is concentrated to a hot energy spot, and the SPP field intensity is dramatically enhanced about three-fold in comparison with a conventional Gaussian beam illumination. The proposed phase-shaped beam approach provides a new degree of freedom to fundamentally control the SPP excitation and benefits the development of surface-enhanced applications.

THREE-DIMENSIONAL HYBRID SIMULATION STUDY OF ANISOTROPIC TURBULENCE IN THE PROTON KINETIC REGIME

Three-dimensional numerical hybrid simulations with particle protons and quasi-neutralizing fluid electrons are conducted for a freely decaying turbulence that is anisotropic with respect to the background magnetic field. The turbulence evolution is determined by both the combined root-mean-square (rms) amplitude for fluctuating proton bulk velocity and magnetic field and by the ratio of perpendicular to parallel wavenumbers. This kind of relationship had been considered in the past with regard to interplanetary turbulence. The fluctuations nonlinearly evolve to a turbulent phase whose net wave vector anisotropy is usually more perpendicular than the initial one, irrespective of the initial ratio of perpendicular to parallel wavenumbers. Self-similar anisotropy evolution is found as a function of the rms amplitude and parallel wavenumber. Proton heating rates in the turbulent phase vary strongly with the rms amplitude but only weakly with the initial wave vector anisotropy. Even in the limit where wave vectors are confined to the plane perpendicular to the background magnetic field, the heating rate remains close to the corresponding case with finite parallel wave vector components. Simulation results obtained as a function of proton plasma to background magnetic pressure ratio β p in the range 0.1-0.5 show that the wave vector anisotropy also weakly depends on β p .

Transitional Intermittency Exponents Through Deterministic Boundary-Layer Structures and Empirical Entropic Indices

A computational procedure is developed to determine initial instabilities within a three-dimensional laminar boundary layer and to follow these instabilities in the streamwise direction through to the resulting intermittency exponents within a fully developed turbulent flow. The fluctuating velocity wave vector component equations are arranged into a Lorenz-type system of equations. The nonlinear time series solution of these equations at the fifth station downstream of the initial instabilities indicates a sequential outward burst process, while the results for the eleventh station predict a strong sequential inward sweep process. The results for the thirteenth station indicate a return to the original instability autogeneration process. The nonlinear time series solutions indicate regions of order and disorder within the solutions. Empirical entropies are defined from decomposition modes obtained from singular value decomposition techniques applied to the nonlinear time series solutions. Empirical entropic indices are obtained from the empirical entropies for two streamwise stations. The intermittency exponents are then obtained from the entropic indices for these streamwise stations that indicate the burst and autogeneration processes.

The Alfvén resonance in pair plasmas

Waves propagating obliquely in a magnetized cold pair plasma experience an approximate resonance in the wavevector component perpendicular to the magnetic field, which is the analogue of the Alfvén resonance in normal electron-ion plasmas. Wave absorption at the resonance can take place via mode conversion to the analogue of the short wavelength inertial Alfvén wave. The Alfvén resonance could play a role in wave propagation in the pulsar magnetosphere leading to pulsar radio emission. Ducting of waves in strong plasma gradients may occur in the pulsar magnetosphere, which leads to the consideration of Alfvén surface waves, whose energy is concentrated in the region of strong gradients.

Photonic-magnonic crystals: Multifunctional periodic structures for magnonic and photonic applications

We investigate the properties of a photonic-magnonic crystal, a complex multifunctional one-dimensional structure with magnonic and photonic band gaps in the GHz and PHz frequency ranges for spin waves and light, respectively. The system consists of periodically distributed dielectric magnetic slabs of yttrium iron garnet and nonmagnetic spacers with an internal structure of alternating TiO2 and SiO2 layers which form finite-size dielectric photonic crystals. We show that the spin-wave coupling between the magnetic layers, and thus the formation of the magnonic band structure, necessitates a nonzero in-plane component of the spin-wave wave vector. A more complex structure perceived by light is evidenced by the photonic miniband structure and the transmission spectra in which we have observed transmission peaks related to the repetition of the magnetic slabs in the frequency ranges corresponding to the photonic band gaps of the TiO2/SiO2 stack. Moreover, we show that these modes split to very high sharp (a few THz wide) subpeaks in the transmittance spectra. The proposed novel multifunctional artificial crystals can have interesting applications and be used for creating common resonant cavities for spin waves and light to enhance the mutual influence between them.

Resonant waves in elastic structured media: Dynamic homogenisation versus Green’s functions

We address an important issue of dynamic homogenisation in vector elasticity for a doubly periodic mass-spring elastic lattice. The notion of logarithmically growing resonant waves is used in the analysis of star-shaped wave forms induced by an oscillating point force. We note that the dispersion surfaces for Floquet–Bloch waves in the elastic lattice may contain critical points of the saddle type. Based on the local quadratic approximations of a dispersion surface, where the radian frequency is considered as a function of wave vector components, we deduce properties of a transient asymptotic solution associated with the contribution of the point source to the wave form. The notion of local Green’s functions is used to describe localised wave forms corresponding to the resonant frequency. The special feature of the problem is that, at the same resonant frequency, the Taylor quadratic approximations for different groups of the critical points on the dispersion surfaces (and hence different Floquet–Bloch vectors) are different. Thus, it is shown that for the vector case of micro-structured elastic medium there is no uniformly defined dynamic homogenisation procedure for a given resonant frequency. Instead, the continuous approximation of the wave field can be obtained through the asymptotic analysis of the lattice Green’s functions, presented in this paper.

The Radiation Problem from a Vertical Hertzian Dipole Antenna above Flat and Lossy Ground: Novel Formulation in the Spectral Domain with Closed-Form Analytical Solution in the High Frequency Regime

We consider the problem of radiation from a vertical short (Hertzian) dipole above flat lossy ground, which represents the well-known “Sommerfeld radiation problem” in the literature. The problem is formulated in a novel spectral domain approach, and by inverse three-dimensional Fourier transformation the expressions for the received electric and magnetic (EM) field in the physical space are derived as one-dimensional integrals over the radial component of wavevector, in cylindrical coordinates. This formulation appears to have inherent advantages over the classical formulation by Sommerfeld, performed in the spatial domain, since it avoids the use of the so-called Hertz potential and its subsequent differentiation for the calculation of the received EM field. Subsequent use of the stationary phase method in the high frequency regime yields closed-form analytical solutions for the received EM field vectors, which coincide with the corresponding reflected EM field originating from the image point. In this way, we conclude that the so-called “space wave” in the literature represents the total solution of the Sommerfeld problem in the high frequency regime, in which case the surface wave can be ignored. Finally, numerical results are presented, in comparison with corresponding numerical results based on Norton’s solution of the problem.

Models for electrostatic drift waves with density variations along magnetic field lines

Drift waves with vertical magnetic fields in gravitational ionospheres are considered where the unperturbed plasma density is enhanced in a magnetic flux tube. The gravitational field gives rise to an overall decrease of plasma density for increasing altitude. Simple models predict that drift waves with finite vertical wave vector components can increase in amplitude merely due to a conservation of energy density flux of the waves. Field‐aligned currents are some of the mechanisms that can give rise to fluctuations that are truly unstable. We suggest a self‐consistent generator or “battery” mechanism that in the polar ionospheres can give rise to magnetic field‐aligned currents even in the absence of electron precipitation. The free energy here is supplied by steady state electric fields imposed in the direction perpendicular to the magnetic field in the collisional lower parts of the ionosphere or by neutral winds that have similar effects.

Radiative decay engineering 7: Tamm state-coupled emission using a hybrid plasmonic–photonic structure

There is a continuing need to increase the brightness and photostability of fluorophores for use in biotechnology, medical diagnostics, and cell imaging. One approach developed during the past decade is to use metallic surfaces and nanostructures. It is now known that excited state fluorophores display interactions with surface plasmons, which can increase the radiative decay rates, modify the spatial distribution of emission, and result in directional emission. One important example is surface plasmon-coupled emission (SPCE). In this phenomenon, the fluorophores at close distances from a thin metal film, typically silver, display emission over a small range of angles into the substrate. A disadvantage of SPCE is that the emission occurs at large angles relative to the surface normal and at angles that are larger than the critical angle for the glass substrate. The large angles make it difficult to collect all of the coupled emission and have prevented the use of SPCE with high-throughput and/or array applications. In the current article, we describe a simple multilayer metal–dielectric structure that allows excitation with light that is perpendicular (normal) to the plane and provides emission within a narrow angular distribution that is normal to the plane. This structure consists of a thin silver film on top of a multilayer dielectric Bragg grating, with no nanoscale features except for the metal or dielectric layer thicknesses. Our structure is designed to support optical Tamm states, which are trapped electromagnetic modes between the metal film and the underlying Bragg grating. We used simulations with the transfer matrix method to understand the optical properties of Tamm states and localization of the modes or electric fields in the structure. Tamm states can exist with zero in-plane wavevector components and can be created without the use of a coupling prism. We show that fluorophores on top of the metal film can interact with the Tamm state under the metal film and display Tamm state-coupled emission (TSCE). In contrast to SPCE, the Tamm states can display either S or P polarization. The TSCE angle is highly sensitive to wavelength, which suggests the use of Tamm structures to provide both directional emission and wavelength dispersion. Metallic structures can modify fluorophore decay rates but also have high losses. Photonic crystals have low losses but may lack the enhanced light-induced fields near metals. The combination of plasmonic and photonic structures offers the opportunity for radiative decay engineering to design new formats for clinical testing and other fluorescence-based applications.