Hyperbolic Modes Research Articles

Application Conditions of Effective Medium Theory in Near-Field Radiative Heat Transfer Between Multilayered Metamaterials

This work addresses the validity of the local effective medium theory (EMT) in predicting the near-field radiative heat transfer between multilayered metamaterials, separated by a vacuum gap. Doped silicon and germanium are used to form the metallodielectric superlattice. Different configurations are considered by setting the layers adjacent to the vacuum spacer as metal–metal (MM), metal–dielectric (MD), or dielectric–dielectric (DD) (where M refers to metallic doped silicon and D refers to dielectric germanium). The calculation is based on fluctuational electrodynamics using the Green's function formulation. The cutoff wave vectors for surface plasmon polaritons (SPPs) and hyperbolic modes are evaluated. Combining the Bloch theory with the cutoff wave vector, the application condition of EMT in predicting near-field radiative heat transfer is presented quantitatively and is verified by exact calculations based on the multilayer formulation.

Energy streamlines in near-field radiative heat transfer between hyperbolic metamaterials.

Metallodielectric photonic crystals having hyperbolic dispersions are called indefinite materials because of their ability to guide modes with extremely large lateral wavevectors. While this is useful for enhancing near-field radiative heat transfer, it could also give rise to large lateral displacements of the energy pathways. The energy streamlines can be used to depict the flow of electromagnetic energy through a structure when wave propagation does not follow ray optics. We obtain the energy streamlines through two semi-infinite uniaxial anisotropic effective medium structures, separated by a small vacuum gap, using the Green functions and fluctuation-dissipation theorem. The lateral shifts are determined from the streamlines within two penetration depths. For hyperbolic modes, the predicted lateral shift can be several thousand times of the vacuum gap width.

Large penetration depth of near-field heat flux in hyperbolic media

We compare super-Planckian thermal radiation between phonon-polaritonic media and hyperbolic metamaterials. In particular, we determine the penetration depth of thermal photons inside the absorbing medium for three different structures: two semi-infinite phonon-polaritonic media supporting surface modes, two multilayer hyperbolic metamaterials and two nanowire hyperbolic metamaterials. We show that for hyperbolic modes the penetration depth can be orders of magnitude larger than for surface modes suggesting that hyperbolic materials are much more preferable for near-field thermophotovoltaic applications than pure phonon-polaritonic materials.

Near-field radiative heat transfer with doped-silicon nanostructured metamaterials

The objective of this work is to evaluate different practically achievable doped-silicon (D-Si) nanostructured metamaterials (including nanowires and nanoholes, multilayers, and one-dimensional gratings) in terms of their potential for enhancing near-field radiative heat transfer at ambient temperature. It is found that both doped silicon nanowires and nanoholes may achieve an enhancement over bulk doped silicon by more than one order of magnitude in the deep submicron gap region. The enhancement is attributed to either the broadband hyperbolic modes or low-loss surface modes or a combination of both. On the other hand, polarization coupling, which can occur in the grating configuration, contributes little to the radiative transfer at the nanometer scale. This work will facilitate the application of nanostructures in more efficient non-contact thermal management, thermal imaging, and near-field thermophotovoltaics.

The linear hyperbolic initial and boundary value problems in a domain with corners

In this article, we consider linear hyperbolic Initial and Boundary Value Problems (IBVP) in a rectangle (or possibly curvilinear polygonal domains) in both the constant and variable coefficients cases. We use semigroup method instead of Fourier analysis to achieve the well-posedness of the linear hyperbolic system, and we find by diagonalization that there are only two elementary modes in the system which we call hyperbolic and elliptic modes. The hyperbolic system in consideration is either symmetric or Friedrichs-symmetrizable.

Viscous boundary layers in hyperbolic-parabolic systems with Neumann boundary conditions

We initiate the study of noncharacteristic boundary layers in hyperbolic-parabolic problems with Neumann boundary conditions. More generally, we study boundary layers with mixed Dirichlet--Neumann boundary conditions where the number of Dirichlet conditions is fewer than the number of hyperbolic characteristic modes entering the domain, that is, the number of boundary conditions needed to specify an outer hyperbolic solution. We have shown previously that this situation prevents the usual WKB approximation involving an outer solution with pure Dirichlet conditions. It also rules out the usual maximal estimates for the linearization of the hyperbolic-parabolic problem about the boundary layer. Here we show that for linear, constant-coefficient, hyperbolic-parabolic problems one obtains a reduced hyperbolic problem satisfying Neumann or mixed Dirichlet--Neumann rather than Dirichlet boundary conditions. When this hyperbolic problem can be solved, a unique formal boundary-layer expansion can be constructed. In the extreme case of pure Neumann conditions and totally incoming characteristics, we carry out a full analysis of the quasilinear case, obtaining a boundary-layer approximation to all orders with a rigorous error analysis. As a corollary we characterize the small viscosity limit for this problem. The analysis shows that although the associated linearized hyperbolic and hyperbolic--parabolic problems do not satisfy the usual maximal estimates for Dirichlet conditions, they do satisfy analogous versions with losses.

Near-field thermal radiation between hyperbolic metamaterials: Graphite and carbon nanotubes

The near-field radiative heat transfer for two hyperbolic metamaterials, namely, graphite and vertically aligned carbon nanotubes (CNTs), is investigated. Graphite is a naturally existing uniaxial medium, while CNT arrays can be modeled as an effective anisotropic medium. Different hyperbolic modes can be separately supported by these materials in certain infrared regions, resulting in a strong enhancement in near-field heat transfer. It is predicted that the heat flux between two CNT arrays can exceed that between SiC plates at any vacuum gap distance and is about 10 times higher with a 10 nm gap.

On the limits of the effective description of hyperbolic materials in the presence of surface waves

Here, we address the question of the validity of an effective description for hyperbolic metamaterials in the near-field region. We show that the presence of localized modes such as surface waves drastically limits the validity of the effective description, and requires revisiting the concept of homogenization in the near-field. We demonstrate, from exact scattering matrix calculations for multilayer hyperbolic structures, that one can find surface modes in spectral regions where the effective approach predicts hyperbolic modes only. Hence, the presence of surface modes which are not accounted for in the effective description can lead to physical misinterpretations in the description of hyperbolic materials and their related properties. In particular, we discuss in detail how the choice of the topmost layer affects the validity of the effective medium approach for calculating the local density of states and the super-Planckian thermal radiation.

Chaotic dynamics of the Bianchi IX universe in Gauss-Bonnet gravity

We investigate the dynamics of closed FRW universe and anisotropic Bianchi type-IX universe characterized by two scale factors in a gravity theory including a higher curvature (Gauss-Bonnet) term. The presence of the cosmological constant creates a critical point of saddle type in the phase space of the system. An orbit starting from a neighborhood of the separatrix will evolve toward the critical point, and it eventually either expands to the de Sitter space or collapses to the big crunch. In the closed FRW model, the dynamics is reduced to hyperbolic motions in the two-dimensional center manifold, and the system is not chaotic. In the anisotropic model, anisotropy introduces the rotational mode, which interacts with the hyperbolic mode to present a cylindrical structure of unstable periodic orbits in the neighborhood of the critical point. Due to the non-integrability of the system, the interaction of rotational and hyperbolic modes makes the system chaotic, making it impossible for us to predict the final fate of the universe. We find that the chaotic dynamics arises from the fact that orbits with even small perturbations around the separatrix oscillate in the neighborhood of the critical point before finally expanding or collapsing. The chaotic character is also evidenced by the fractal structures in the basins of attraction.

Super-Planckian near-field thermal emission with phonon-polaritonic hyperbolic metamaterials

We study super-Planckian near-field heat exchanges for multilayer hyperbolic metamaterials using exact scattering-matrix (S-matrix) calculations. We investigate heat exchanges between two multilayer hyperbolic metamaterial structures. We show that the super-Planckian emission of such metamaterials can either come from the presence of surface phonon-polariton modes or from a continuum of hyperbolic modes depending on the choice of composite materials as well as the structural configuration.

Equivalent and nonequivalent barotropic modes for rotating stratified flows

All the possible equivalent barotropic (EB) laminar solutions are firstly explored, and all the possible non-EB elliptic circulations and hyperbolic laminar modes of rotating stratified fluids are discovered in this paper. The EB circulations (including the vortex streets and hurricane like vortices) possess rich structures, because either the arbitrary solutions of arbitrary nonlinear Poisson equations can be used or an arbitrary two-dimensional stream function is revealed which may be broadly applied in atmospheric and oceanic dynamics, plasma physics, astrophysics and so on. The discovery of the non-EB modes disproves a known conjecture.

Study of resonance-cone propagation in truncated hyperbolic metamaterial grids using transmission-line matrix simulations

In this paper, we utilize transmission-line matrix simulations to study the electromagnetic properties of hyperbolic metamaterial grids, which are synthesized by arranging unloaded transmission lines over electrical ground planes, such that the unit cells are rectangular in shape. The hyperbolic grids support Bragg and Hyperbolic resonant modes that have rectangular and hyperbolic dispersion surfaces, respectively. We demonstrate the formation of these resonant modes in the hyperbolic grids and their refraction, and spatial scanning properties via circuit simulations. The simulation results are compared with experimental results of a two-layer anisotropic hyperbolic metamaterial.

The stability of the variable-density Kelvin–Helmholtz billow

We perform a three-dimensional stability analysis of the Kelvin–Helmholtz (KH) billow, developing in a shear layer between two fluids with different density. We begin with two-dimensional simulations of the temporally evolving mixing layer, yielding the unsteady base flow fields. The Reynolds number is 1500 while the Schmidt and Froude numbers are infinite. Then exponentially unstable modes are extracted from a linear stability analysis performed at the saturation of the primary mode kinetic energy. The spectrum of the least stable modes exhibits two main classes. The first class comprises three-dimensional core-centred and braid-centred modes already present in the homogeneous case. The baroclinic vorticity concentration in the braid lying on the light side of the KH billow turns the flow into a sharp vorticity ridge holding high shear levels. The hyperbolic modes benefit from the enhanced level of shear in the braid whereas elliptic modes remain quite insensitive to the modifications of the base flow. In the second class, we found typical two-dimensional modes resulting from a shear instability of the curved vorticity-enhanced braid. For a density contrast of 0.5, the wavelength of the two-dimensional instability is about ten times shorter than that of the primary wave. Its amplification rate competes well against those of the hyperbolic three-dimensional modes. The vorticity-enhanced braid thus becomes the preferred location for the development of secondary instabilities. This stands as the key feature of the transition of the variable-density mixing layer. We carry out a fully resolved numerical continuation of the nonlinear development of the two-dimensional braid-mode. Secondary roll-ups due to a small-scale Kelvin–Helmholtz mechanism are promoted by the underlying strain field and develop rapidly in the compression part of the braid. Originally analysed by Reinoud et al. (Phys. Fluids, vol. 12, 2000, p. 2489) from two-dimensional non-viscous numerical simulations, this instability is shown to substantially increase the mixing.

Viscous boundary value problems for symmetric systems with variable multiplicities

Extending investigations of Métivier and Zumbrun in the hyperbolic case, we treat stability of viscous shock and boundary layers for viscous perturbations of multidimensional hyperbolic systems with characteristics of variable multiplicity, specifically the construction of symmetrizers in the low-frequency regime where variable multiplicity plays a role. At the same time, we extend the boundary-layer theory to “real” or partially parabolic viscosities, Neumann or mixed-type parabolic boundary conditions, and systems with nonconservative form, in addition proving a more fundamental version of the Zumbrun–Serre–Rousset theorem, valid for variable multiplicities, characterizing the limiting hyperbolic system and boundary conditions as a nonsingular limit of a reduced viscous system. The new effects of viscosity are seen to be surprisingly subtle; in particular, viscous coupling of crossing hyperbolic modes may induce a destabilizing effect. We illustrate the theory with applications to magnetohydrodynamics.

Metamaterial slabs coupling

AbstractMetamaterial slab coupling to another metamaterial slab or to a dielectric slab is studied, taking into consideration regular trigonometric modes as well as hyperbolic modes. The first TM even mode has been given special attention, since it exhibits interesting filtering properties. The coupling coefficient frequency variation of the structures has been investigated. It is found that, for a given frequency, the thinner the width of the slabs or the lower their permittivity, the higher the coupling between them. © 2004 Wiley Periodicals, Inc. Microwave Opt Technol Lett 40: 340–344, 2004; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.11373

Stability of Large-Amplitude Viscous Shock Profiles of Hyperbolic-Parabolic Systems

We establish nonlinear L1∩H3→Lp orbital stability, 2≦p≤∞, with sharp rates of decay, of large-amplitude Lax-type shock profiles for a class of symmetric hyperbolic-parabolic systems including compressible gas dynamics and magnetohydrodynamics (MHD) under the necessary conditions of strong spectral stability, i.e., a stable point spectrum of the linearized operator about the wave, transversality of the profile, and hyperbolic stability of the associated ideal shock. This yields in particular, together with the spectral stability results of [50], the nonlinear stability of arbitrarily large-amplitude shock profiles of isentropic Navier–Stokes equations for a gamma-law gas as γ→1: the first complete large-amplitude stability result for a shock profile of a system with real (i.e., partial) viscosity. A corresponding small-amplitude result was established in [53, 54] for general systems of Kawashima class by a combination of ‘‘Kawashima-type’’ energy estimates and pointwise Green function bounds, where the small-amplitude assumption was used only to close the energy estimates. Here, under the mild additional assumption that hyperbolic characteristic speeds (relative to the shock) are not only nonzero but of a common sign, we close the estimates instead by use of a Goodman-type weighted norm [25, 26] designed to control estimates in the crucial hyperbolic modes.

Observation of transverse patterns in an isotropic microchip laser

An isotropic microchip laser is used to study the characteristics of high-order wave functions in a two-dimensional (2D) quantum harmonic oscillator based on the identical functional forms. With a doughnut pump profile, the spontaneous transverse modes are found to, generally, be elliptic and hyperbolic transverse modes. Theoretical analyses reveal that the elliptic transverse modes are analogous to the coherent states of a 2D harmonic oscillator; the formation of hyperbolic transverse modes is a spontaneous mode locking between two identical Hermite-Gaussian modes.

An analytical solution for the zero frequency hyperbolic RUM modes of distortion of SiO 2-tridymite

The constraint equations controlling the existence of soft, Rigid Unit Mode (RUM) modes of distortion of the ideal tridymite tetrahedral framework structure are derived and used to obtain an analytical solution for the potential soft phonon mode wave-vectors and their associated displacement patterns. One of two types of RUM known to exist are characterized by modulation wave-vectors perpendicular to 〈110〉 and atomic displacement patterns involving tetrahedral edge rotation while a second type of RUM is found to be soft on a hyperbolic reciprocal space distribution and to involve tetrahedral rotation about in-plane rotation axes. The analytical form of this hyperbolic diffuse distribution is derived as is an analytical expression for the associated RUM eigenvector of each point on the diffuse distribution.

Particle hydrodynamic moment models in biology and microelectronics: Singular relaxation limits

The hydrodynamic model has become important in the simulation of semiconductor devices during the last decade, as an alternative to the more specialized drift-diffusion model. The more comprehensive hydrodynamic model includes features of general carrier heating, velocity overshoot, and various small device features, and is based upon moments of the Boltzmann equation. It employs a macroscopic relaxation time approximation, which incorporates averaged collision mechanisms. Unlike the drift-diffusion model, the hydrodynamic model contains hyperbolic modes related to the momentum subsystem. It is therefore much more complex to analyze and simulate than the corresponding driftdiffusion model. A detailed introduction to both models may be found in the book [9]. In particular, it is demonstrated how an averaging process over group velocity space, coupled with identification of particle density as one such average, and velocity as a first moment, leads to a system whose macroscopic characterization is a conservation law system, much in the tradition of classical gas dynamics, with forcing terms due to electrical, mechanical, and heating effects, as well as frictional dissipation due to collisions. When the microscopic and macroscopic derivations are correlated, the moment closure assumptions emerge more clearly. For example, isotropy leads to the ideal gas law for the carriers. Readers may wish to consult the reference [2] for an in depth survey of some of these ideas. The reader interested in a closer explanation of the engineering aspects of semiconductor devices can consult the book [15]. In recent years, there has been an attempt to understand the relationship between the two classes of models. In the biological context of the open ionic channel, it was noticed phenomenologically that small variations of the saturation velocity in the energy relaxation term led to significant variation in the computed temperature (cf. [3]). Indeed, this interpretation suggests that the temperature tends to that of the isothermal drift-diffusion regime, which is seen as a high friction regime. Some mathematical studies suggest this also (cf. [14]). Another way of viewing high friction is in the framework of the inertial approximation, which here would assert that the drift-diffusion regime (nonisothermal) is the limit of increasingly small momentum rate of change, with respect to frequency of collisions. This approach is discussed in [10]. In this paper, we shall study the reduced hydrodynamic model, sometimes referred to as the perturbed isentropic model, with adiabatic pressure density relationship. In fact, for the most general initial conditions, rigorous mathematical results exist only in this case (see [18] for the initialboundary value problem), though recent studies have extended the models to include multi-species and geometric structure, thus permitting multi-dimensional transport with symmetry to be analyzed (cf. [5]). The paper [5] includes comprehensive simulations based on the shock capturing algorithm, ENO (cf. [16] for description of the algorithm). In particular, [5] analyzes in depth the two-valley

Primordial gravitational waves from open inflation

We calculate the spectrum of gravitational waves generated during inflation in open $({\ensuremath{\Omega}}_{0}<1)$ inflationary models. In such models an initial epoch of old inflation solves the horizon and flatness problems, and during this first epoch of inflation the quantum state of the graviton field rapidly approaches the Bunch-Davies vacuum. Then the old inflation ends by the nucleation of a single bubble, inside of which there is a shortened epoch of slow-roll inflation giving ${\ensuremath{\Omega}}_{0}<1$ today. In this paper we reexpress the Bunch-Davies vacuum for the graviton field in terms of the hyperbolic modes inside the bubble and propagate these modes forward in time into the present era. We derive the expression for the contribution from these gravity waves to the cosmic microwave background anisotropy including the effect of a finite-energy difference across the bubble wall.