Complex Dispersion Relation Research Articles

Theoretical and numerical analysis of propagation and scattering of eigen- and non-eigenmodes of an irregular integrated-optical waveguide

We consider theoretical and numerical methods for studying propagation and scattering of laser radiation of eigenmodes and non-eigenmodes in an irregular integrated-optical waveguide. Scattering of non-eigenmodes in an irregular integratedoptical waveguide is investigated for the first time. We present thecalculated dispersion curves for TE and TM eigenmodes and TE non-eigenmodes. For the leaky TE0 modes we plot the dependence of the complex dispersion relation and show the vertical complex profile of the field. The dependence of the scattered laser radiation field on the effective refractive index is obtained for the given parameters of the waveguide. We compare for the first time thecalculated complex scattering diagram of laser radiation outside the waveguide layer in the plane, perpendicular to the plane of incidence, for the leaky and guided TE0 modes.

A new analytical solution for wave scattering by a submerged horizontal porous plate with finite thickness

This study gives a new analytical solution for water wave scattering by a submerged horizontal porous plate breakwater with finite thickness in the context of linear potential theory. The original velocity potential is split into a symmetric part and an antisymmetric part. Each part is further written as the sum of two artificial potentials satisfying appropriate boundary conditions. All the artificial potentials are determined by the matched eigenfunction expansions. The symmetric solution is also the solution for wave reflection by a porous plate wave absorber in laboratory flume. A multi-domain boundary element method solution is developed to confirm the analytical solution. The agreement between the results calculated by the analytical solution and the boundary element method solution is excellent. Also the analytical solution is confirmed by experimental data. The new analytical solution needs no complex dispersion relations for water wave motion over porous mediums. Thus the particular difficulties due to complex dispersion relations are avoided.

Analytical solutions to zeroth-order dispersion relations of a cylindrical metallic nanowire near the backbending point

Zeroth-order complex dispersion relations of a cylindrical metallic nanowire have been solved out analytically with approximate methods. The analytical solutions are valid for the sections of the dispersion relations whose frequencies are close to the Surface Plasmon frequency. The back bending of the Surface Plasmon-Polaritons(SPPs) can be well described by the analytical solutions, confirming that the back bending is originated from the metal Ohmic loss. The utility of the back bending point in the dispersion relation for the measurement of the metallic Ohimc loss has also been suggested.

A new approximate analytic solution for water wave scattering by a submerged horizontal porous disk

This study gives a new approximate analytic solution for water wave scattering by a submerged horizontal porous disk in the context of the linear potential theory. The solution is based on the domain decomposition method. The velocity potentials are determined by two different approaches. One approach is to adopt decompositions for velocity potentials, and the other is to expand the vertical derivative of the velocity potential on the porous disk along the radial direction. Hence the velocity potentials are determined by the matched eigenfunction expansions. Differing from previous solutions with respect to the porous disk, the present solution needs no complex dispersion relations. Thus the new solution is easier for numerical implementation. According to numerical examples, the convergence of the present solution is satisfactory. In addition, the present predictions of the wave surface elevation and the vertical wave force on the disk agree very well with previous results by different approaches. The present solution can also be extended to other structures involving disks, such as a fish cage, a porous disk with finite thickness, and a submerged elastic disk.

Complex dispersion relation of a double chain of lossy metal nanoparticles

We study the propagation characteristics of optical signals in waveguides composed of a double chain of metallic nanoparticles embedded in a dielectric host. We find that the complex Bloch band diagram for the guided modes, derived by the Mie scattering theory including material losses, exhibits strong differences with respect to the previously studied single chain. The results of the model are validated through the finite element solution of the Maxwell equations.

Complex dispersion relation calculations with the symmetric interior penalty method

AbstractA high‐order discontinuous Galerkin method for calculations of complex dispersion relations of two‐dimensional photonic crystals is presented. The medium is characterized by a complex‐valued permittivity and we relate for this absorptive system the spectral parameter to the time frequency. We transform the non‐linear eigenvalue problem for a Lorentz material in air into a non‐Hermitian linear eigenvalue problem and uses a Krylov space method to compute approximate eigenvalues. Moreover, we study the impact of the penalty term numerically and illustrate the high convergence rate of the method. Copyright © 2010 John Wiley & Sons, Ltd.

Analysis of the reflection properties of sonic crystal devices.

The phenomenon of band-gaps typical of wave propagation in periodic materials has been a subject of extensive investigations. In acoustics, the interest is mainly related with the ability of such structures, the so-called sonic crystals, of blocking the propagation of sound at certain frequencies, acting as filters. Recently, there is an increasing interest in the use of sonic crystals to control the beam propagation, and a number of interesting features as focusing and collimation have been predicted and observed. At the root of these phenomena is the angle-dependent dispersion introduced by the crystal. In propagating regimes, for frequencies outside the gap, the particular form of the dispersion relations in the wavenumber (reciprocal) space allows us to determine the spatial characteristics of the transmitted field. In this work, we merge the two ideas above, and investigate the spatial characteristics of the reflected acoustic field when its frequency belongs to a band-gap. In a band-gap, the propagating modes are substituted by evanescent ones, defined by complex wavenumbers. We evaluate the complex dispersion relation, and relate the particular form of the isofrequency contours in k-space with the numerically evaluated spatial distribution of the reflected field. Different reflection regimes are presented and discussed.

Evanescent modes in sonic crystals: Complex dispersion relation and supercell approximation

Evanescent modes in complete sonic crystals (SCs) and SC with point defects are reported both theoretically and experimentally in this paper. Plane wave expansion (PWE) and in general, ω(k) methods have been used to calculate band structures showing gaps that have been interpreted as ranges of frequencies where no real k exists. In this work, we extend PWE to solve the complex k(ω) problem applied to SC, introducing the supercell approximation for studying one vacancy. Explicit matrix formulation of the equations is given. This k(ω) method enables the calculation of complex band structures, as well as enabling an analysis of the propagating modes related with real values of the function k(ω), and the evanescent modes related with imaginary values of k(ω). This paper shows theoretical results and experimental evidences of the evanescent behavior of modes inside the SC band gap. Experimental data and numerical results using the finite elements method are in very good agreement with the predictions obtained using the k(ω) method.

Modeling propagation in negative index media using causal complex dispersion relations

Starting from the causality of the permittivity and permeability of a medium, we investigate the causality of the propagation constant. We show that a reduced dispersion relation, obtained from the frequency dependence of the propagation constant by neglecting a linear frequency dependent term, obeys causality. The propagation constant is identical to the reduced propagation constant under appropriate limiting values of the physical parameters. We illustrate the causality of the reduced propagation constant through examples of (a) a nonmagnetic material where the permittivity is given by the Lorentz model, (b) a material where the permittivity and permeability are both Lorentz-type, and (c) an effective medium comprising a nonmagnetic material with Lorentz-type permittivity in a dispersionless host medium, where the effective permittivity is given by the Maxwell–Garnett rule. Causality of the propagation constant enables the use of simple operator formalisms to derive the underlying partial differential equations for baseband and envelope wave propagation, as demonstrated through an illustrative example of a negative index medium with gain.

Measurements of Inertial Limit Alfvén Wave Dispersion for Finite Perpendicular Wave Number

Measurements of the dispersion relation for shear Alfvén waves as a function of perpendicular wave number are reported for the inertial regime for which V{A}>V{Te}. The parallel phase velocity and damping are determined as k{ perpendicular} varies and the measurements are compared to theoretical predictions. The comparison shows that the best agreement between theory and experiment is achieved for a fully complex plasma dispersion relation which includes the effects of electron collisions.

DISPERSION RELATIONS FOR COLD PLASMA AROUND THE HORIZON OF SCHWARZSCHILD–DE SITTER BLACK HOLE

In this paper, we investigate the wave properties of cold plasma in the vicinity of Schawarzchild–de Sitter black hole horizon using 3 + 1 formalism. The general relativistic magnetohydrodynamical equations are formulated for this space–time with the use of Rindler coordinates. We consider both the rotating and nonrotating surroundings with magnetized and nonmagnetized plasmas. Linear perturbation and Fourier analysis techniques are applied by introducing simple harmonic waves. We derive complex dispersion relation from the determinant of Fourier analyzed equations for each case which provides real and complex values of the wave number. From the wave number we determine the phase and group velocities, the refractive index etc., which are used to discuss the characteristics of the waves around the event horizon.

Correction to “Three‐dimensional nonlinear evolution of equatorial ionospheric bubbles with gravity wave seeding and tidal wind effects”

[1] In the paper ‘‘Three-dimensional nonlinear evolution of equatorial ionospheric bubbles with gravity wave seeding and tidal wind effects’’ by M. J. Keskinen and Sharon L. Vadas (Geophysical Research Letters, 36, L12102, doi:10.1029/2009GL037892), Keskinen and Vadas [2009, hereafter KV09] presented the three-dimensional nonlinear evolution of equatorial ionospheric bubbles using a realistic lower atmospheric gravity wave (GW) source. There are a number of misrepresentations and errors by KV09 in regards to the presentation and discussion of the GW source function that was used in the plasma model. The figures and discussion of KV09 refer to a different GW specification, despite the fact that the correct GW specification was used in the plasma model, results of which are presented in Figure 4 of KV09. As a result, the model data shown in Figures 1, 2, and 3 of KV09 were not used to seed the 3D plasma model described by KV09. Correct Figures 1–3 are supplied in this erratum. Additionally, part of paragraph 8 and all of paragraphs 9, 10, and 11 of KV09 should be disregarded and replaced with the text below. [2] The dispersion relation discussed in the latter portion of paragraph 8 in KV09 was not the GW dispersion relation, but was rather the full complex dispersion relation derived from the compressible fluid equations by Vadas and Fritts [2005, hereafter VF2005]. Due to formula errors in KV09, readers interested in the compressible dispersion relation should refer directly to VF2005. The last half of paragraph 8 in VF09 should be replaced by the following: For anelastic GWs with horizontal phase speeds much less than the speed of sound, the GW dispersion relation is given by equation (26) from VF2005 or by equation (47) from Vadas and Fritts [2009, hereafter VF2009]. [3] Vadas and Liu [2009, Figure 1] shows a GOES-12 satellite image on 01 October, 2005, at 21:22 UT, in Brazil. For the KV09 study, we simulated the GWs excited by the energetic convective cluster at 59.0 W and 13.5 S. This cluster contained several tightly-clumped convective plumes, each having an approximate full-width horizontal diameter of 20 km and updraft velocities of 40 m/s. We defined a convective cluster to be composed of 3 convective plumes in an equilateral triangle configuration, with a separation of 50 km between the plume centers (VF2009). Figure 1 shows the GW spectrum excited by this cluster. Figure 1 replaces KV09’s Figure 1. This cluster excites GWs with horizontal wavelengths of lH > 100 km and vertical wavelengths of lz > 50 km, which are capable of penetrating well into the thermosphere prior to dissipating [Vadas, 2007; Fritts and Vadas, 2008]. [4] The ray trace model we used was described by VF2009. The hyperbolic tanh zonal wind and temperature models shown in Figure 2 of KV09 were not used in either the ray trace or plasma models. Instead, the wind and temperature models we used were determined mainly from the Thermosphere-Ionosphere-Mesosphere-Electrodynamics General Circulation Model. Vadas et al. [2009] describe the background wind and temperature models used. Because they include semidiurnal and diurnal tides, both zonal and meridional background wind components were present. [5] The GWs whose spectrum is plotted in Figure 1 were located at 59.0 Wand 13.5 S at an altitude of z = 13.6 km at 21:22 UT. Those GWs were then ray traced into the middle and upper atmosphere through the spatially and temporallyvarying background winds and temperatures. We then reconstructed the GW wind, temperature, and density perturbations as a function of (x, y, z, t) using the method described by VF2009. We extracted the vertical profiles (cuts) of this solution at 10 different locations and times. These profiles were inputted into the 3D plasma model

Optimized implicit finite-difference and Fourier finite-difference migration for VTI media

Propagation velocity of seismic waves in heterogeneous VTI media depends not only on spatial location but also on their propagation direction, which leads to a much more complex dispersion relation than in isotropic media. As a result, designing implicit finite-difference (FD) schemes for wavefield extrapolation in anisotropic media through analytic Taylor-series expansion is more difficult. Implicit FD and Fourier finite-difference (FFD) schemes are developed for vertical transversely isotropic (VTI) media based on function fitting. The dispersion relation of VTI media is approximated with a rational function and its coefficients are estimated by weighted least-squares optimization. Because these coefficients are functions of Thomsen anisotropy parameters ([Formula: see text] and [Formula: see text]) and vary laterally in heterogeneous VTI media, they are calculated before wavefield extrapolation and stored in a table. Implicit FD and FFD schemes for VTI media are almost the same as for isotropic media, except that coefficients are looked up in a precalculated table. Impulse responses and relative dispersion-relation error show that accuracy of the FD scheme for VTI media is similar to its counterpart in isotropic media. Application to a synthetic data set showed that implicit FD and FFD can handle laterally varying VTI media.

INVESTIGATION OF WAVES PROPAGATING IN ISOTHERMAL PLASMA AROUND DE SITTER BLACK HOLE

We investigate the wave properties for isothermal plasma state around to the de Sitter black hole’s horizon using 3+1 split of spacetime. The corresponding Fourier analyzed perturbed perfect GRMHD equations are used to obtain the complex dispersion relations. We obtain the real values of the wave number k, from these relations, which are used to evaluate the quantities like phase and group velocities etc. These have been analyzed graphically in the neighborhood of the horizon.

Wave Attenuation over a Rigid Porous Medium on a Sandy Seabed

In this study, an analytic solution of wave interaction with a rigid porous medium above a poro-elastic sandy bottom is derived to investigate the attenuation of the surface wave and the wave-induced soil response. In the model, both inertial and damping effects of the flow are considered in the rigid porous region using the potential theory, while the consolidation theory is adopted in the sand region. A new complex dispersion relation involving parameters of the rigid porous and the poro-elastic medium is obtained. The analytic solutions are verified by some special cases, such as wave interaction with a porous structure over an impermeable bottom or wave interaction with a poro-elastic medium only. Numerical results indicate that the wave attenuation is highly dependent upon the thickness of the rigid porous layer, the soil stiffness, and their respective coefficients of permeability. Increasing the thickness of the rigid porous layer will shorten the wavelength of the surface wave regardless of the sand coarseness. The pore pressure in fine-sand is larger than in coarse sand, with both decaying with wave progression. It is also found that increasing the thickness of the rigid porous medium will effectively reduce the pore pressure in the sand. For the applications, an extended hyperbolic mild-slope equation is finally obtained, based on the basic analytic solutions. Examples of the wave height transformation over submerged permeable breakwaters on a slope sandy seabed are given. The simulated results show that the wave decay of the coarse sand seabed is larger than those of fine-sand and impermeable seabeds when waves pass after the submerged porous breakwater. The wave damping versus the friction factor for various height of the submerged breakwater is discussed.

Wave properties of plasma surrounding the event horizon of a nonrotating black hole

We study the wave properties of a cold isothermal plasma in the vicinity of a Schwarzschild black-hole event horizon. The Fourier-analyzed perturbed 3+1 general relativistic magnetohydrodynamics equations are examined such that the complex dispersion relations are obtained for nonrotating, rotating nonmagnetized, and rotating magnetized backgrounds. The propagation and attenuation vectors along with the refractive index are obtained (shown graphically) to study the dispersive properties of the medium near the event horizon. The results show that no information can be obtained from the Schwarzschild magnetosphere. Further, the pressure stops the existence of normal dispersion of waves.PACS Nos.: 95.30.Sf, 95.30.Qd, 04.30.Nk

Complex dispersion relation of 1D dielectric photonic crystal with thin metallic layers

We have carried out the analytical and numerical analysis of metallic insets embedded in a dielectric photonic crystal (DPC). The corresponding one-dimensional metallo-dielectric photonic crystal (MDPC) is studied in relation to the substratum DPC. We describe the complex MDPC dispersion relation curves and the transmission coefficient at the different propagation regimens trough the finite stack. While in the DPC, the band gap argument corresponds to undefined complex solutions of a real dispersion relation, in a MDPC corresponds to a definite complex solution, function of the metal thickness; and analytic continuation of the dispersion curve on the imaginary K plane. The metal absorption is described by the Drude model, and as a result of the absorption in the inset metallic films, complex Bloch vectors are produced for all frequencies as evanescent waves that differ from the straightforward metal absorption.

Damping and drive of low-frequency modes in tokamak plasmas

The ability to predict the stability of fast-particle-driven Alfvén eigenmodes in burning fusion plasmas requires a detailed understanding of the dissipative mechanisms that damp these modes. In order to address this question, the linear gyro-kinetic, electromagnetic code LIGKA (Lauber 2003 Linear gyrokinetic description of fast particle effects on the MHD stability in tokamaks PhD Thesis Technical University München, Lauber et al 2007 J. Comp. Phys. 226 447) is employed to investigate their behaviour in realistic tokamak geometry.Recently, the model and the implementation of LIGKA were extended in order to capture rigorously the coupling of the shear Alfvén wave to the drift and sound waves. This coupling becomes important for the investigation of low-frequency modes such as the Alfvén cascade modes (AC), the beta-induced Alfvén eigenmodes (BAEs), the geodesic acoustic modes (GAMs), the energetic particle modes (EPMs) and—at even lower frequencies—the resistive wall modes (RWMs).The authors report on an effort to close the gap between high-n micro-scale turbulence codes (in their linear phase) and low-n global MHD codes. More precisely, the aim is to describe both regimes within the same framework of equations and with the same numerical implementation by improving the range of applicability and validity of LIGKA: an eigenvalue code allows one to explore both the local complex dispersion relations, e.g. kinetic Alfvén waves (electromagnetic), ion acoustic waves (electrostatic) or drift waves in realistic geometry and also their global eigenfunctions.As an application of this extension, an investigation of the kinetic RWM damping mechanisms is carried out.

The theory of surface waves in waveguides filled with a smoothly inhomogeneous plasma

Electrostatic surface waves in a smoothly inhomogeneous plasma contained in a waveguide are considered. For the plasma with a linear-constant density profile, four types of surface waves are demonstrated for various wave-number regions. The existence conditions, the complex dispersion relations, and the structures of such waves are determined. Complex frequencies of weakly damped surface waves are obtained for the plasma with an arbitrary density profile in the long-wavelength approximation.

Baseband and envelope propagation in media modeled by a class of complex dispersion relations

We demonstrate that a class of simplified complex dispersion relations, which obey causality, can model baseband and envelope propagation in conventional and negative index materials. One such dispersion relation is a special case of the Drude model, another yields the Kuramoto-Shivashinsky equation, while a third, in the limit, yields the simplest dispersion for negative index materials.