Complex Wave Vector Research Articles

RF dispersion relations in FRC geometries and HHFW regime

Field reversed configurations (FRC) are characterized by a magnetic field topology, which exhibits the inversion of the external magnetic field through plasma sustained current and the subsequent presence of a null field surface. A monotonical radial decrease in the longitudinal magnetic field leads to the potential presence of harmonics of the ion cyclotron frequency of any order in the region included between the outer wall and the null field surface. What is the effective hot-plasma dispersion relation obtained through the convolution of a large ensemble of high harmonics fast waves (HHFW) confined in a finite radial region represents an open question that we attempt to address here. In particular, we discuss a combination of analytical modeling and numerical treatment, which allows us to retrieve the resulting high harmonic fast wave complex wavevector for any radial location of any FRC radial profile. Moreover, we show how the obtained hot-plasma HHFW wavevector relates to the cold-plasma solution, and how it depends on the plasma parameters.

On symmetric Tetranacci polynomials in mathematics and physics

In this manuscript, we introduce (symmetric) Tetranacci polynomials ξ j as a twofold generalization of ordinary Tetranacci numbers, considering both non unity coefficients and generic initial values. We derive a complete closed form expression for any ξ j with the key feature of a decomposition in terms of generalized Fibonacci polynomials. For suitable conditions, ξ j can be understood as the superposition of standing waves. The issue of Tetranacci polynomials originated from their application in condensed matter physics. We explicitly demonstrate the approach for the spectrum, eigenvectors, Green’s functions and transmission probability for an atomic tight binding chain exhibiting both nearest and next nearest neighbor processes. We demonstrate that in topological trivial models, complex wavevectors can form bulk states as a result of the open boundary conditions. We describe how effective next nearest neighbor bonding is engineered in state of the art theory/experiment exploiting onsite degrees of freedom and close range hopping. We argue about experimental tune ability and on-demand complex wavevectors.

Propagating analysis of Airy beams via complex phase space and ray method.

A complex phase space and ray method is proposed to present intuitive interpretation of unique intensity profiles, parabolic accelerating trajectories, and distinctive phase shifts of Airy beam and its exponentially decaying version (i.e., finite-energy Airy beam). In the complex phase space, finite-energy Airy beam manifests itself as a complex parabolic phase space curve (PSC) which represents a cluster of light rays with complex wave vectors. The complex ray cluster converges to a complex parabolic caustic curve in complex coordinate space. For infinite-energy Airy beam, phase space, PSC, light ray and caustic curve change to real values. In the paraxial condition, Airy beams can maintain parabolic form of PSC and keep constant ray density, which guarantees the non-diffraction property of Airy beam and approximate non-diffraction property of finite-energy Airy beam. From the evolution of vertexes of parabolic PSC along a parabolic trajectory in phase space, one can also give the parabolic caustic curve for Airy beam and the complex parabolic caustic curve for exponentially decaying Airy beam. Further, the special phase and decay factors in the propagating solution of complex amplitude of Airy beams can be directly derived from the phase shifts of light ray cluster along transverse and longitudinal directions. The proposed phase space and ray method can present intuitive comprehension to distinctive propagating characteristics of Airy beams including intensity and phase, without resort to solving wave equation or diffracting integral formula.

Non-Bloch bands in two-dimensional non-Hermitian systems

The non-Bloch band theory can describe energy bands in a one-dimensional (1D) non-Hermitian system. On the other hand, whether the non-Bloch band theory can be extended to higher-dimensional non-Hermitian systems is nontrivial. In this work, we construct the non-Bloch band theory in two classes of two-dimensional non-Hermitian systems by reducing the problem to that for a 1D non-Hermitian model. In these classes of systems, we get the generalized Brillouin zone for a complex wave vector and investigate topological properties. In the model of the non-Hermitian Chern insulator, as an example, we show the bulk-edge correspondence between the Chern number defined from the generalized Brillouin zone and the appearance of the edge states.

Critical Angle Formulation of Nonuniform Plane Waves for Determining Correct Refraction Angles at Planar Interface

The expressions and properties of the critical angles of nonuniform plane waves for determining the correct refraction angle at the infinite planar interface of two linear, isotropic, and homogeneous media are presented. The two media could be lossless or lossy. For the case where the complex wave vector of Adler–Chu–Fano formulation and the normal vector to the interface are coplanar, a critical angle equation, which is simpler than the extant one, under the condition that one or two critical angles exist, is formulated. In addition, for the case where the complex wave vector and normal vector to the interface are noncoplanar, a critical angle equation is formulated, rendering the 3-D nonuniform plane wave refraction feasible for utilization in many areas of optics. The presented critical angle equations were validated for various nonuniform plane waves and media.

Evidencing non-Bloch dynamics in temporal topolectrical circuits

One of the core concepts from the non-Hermitian skin effect is the extended complex wavevectors (CW) in the generalized Brillouin zone (GBZ), while the origin of CW remains elusive, and further experimental demonstration of GBZ is still lacking. We show that the bulk states of an open quantum system dynamically governed by the Lindblad master equation exhibit non-Bloch evolution which results in CW. Experimentally, we present temporal topolectrical circuits to serve as simulators for the dynamics of an open system. By reconstructing the correspondence between the bulk states of an open system and circuit voltage modes through gauge scale potentials in the circuit, the non-Bloch evolution is demonstrated. Facilitated by the simulators and proper approach to characterize the non-Bloch band proposed here, the GBZ is confirmed. Our work may advance the investigation of the dissipative topological modes and provide a versatile platform for exploring the unique evolution and topology for both closed and open systems.

Calculation of lateral optical energy flow in bound modes in organic light-emitting diodes

We investigate the lateral energy flow of light bound to an OLED (i.e., light that cannot be coupled to the substrate or air mode), with a focus on the attenuation characteristics. A methodology is developed for approximating the total bound field as a superposition of the individual guided modes constructed from multilayer eigenmodes with complex wave vectors. The field distribution and surface energy flow density obtained from our approximate expressions are found to be well matched to those calculated using the Green’s function formalism. The method presented in this study provides mode-based interpretation of the surface energy flow density versus distance characteristics, and can be used in designing OLEDs for applications where the optical energy transported by the bound field is of interest. • The complex photonic band structure of an OLED is calculated. • The bound-mode field of an OLED is constructed from eigenmodes with complex wave vectors. • The attenuation characteristics of the surface energy flow density are quantitatively described.

Estimating dispersion relations of ultrasonic guided waves in bone using a modified matrix pencil algorithm

Guided wave ultrasound technology is well recognized for non-destructive testing. The technology is increasingly applied in bone characterization and imaging to evaluate bone strength and fracture risk. Cortical bone with porous microstructure induces substantial dispersion and attenuation effects on ultrasonic guided waves (UGW). Estimating frequency-dependent propagation characteristics of co-excited wave modes is significant to studying UGW propagation and developing wave-based approaches. This work implements a modified matrix pencil method to simultaneously compute modal wavenumber and attenuation coefficient from dispersive bone UGW signals with improved convergence rate and noise reduction ability. The dispersion estimation is formulated as a matrix pencil or generalized eigenvalue problem with Loewner matrices. The extracted eigenvalues are estimated complex wavevectors, in which the wavenumber and attenuation can be deduced from the real and imaginary components respectively. The performance of the proposed algorithm is demonstrated with low signal-to-noise-ratio synthetic and experimental datasets acquired using axial-transmission measurement settings. The computed dispersive features are validated via comparison with the theoretically calculated dispersion curves by semi-analytical finite-element simulation. The dispersive wave properties are accurately reconstructed in a computationally efficient manner and can be further utilized for bone parametric analyses and subsequently clinical bone health assessment.

Spin accumulation and longitudinal spin diffusion of magnets

We extend to the longitudinal component of the magnetization the spintronics idea that a magnet near equilibrium can be described by two magnetic variables. One is the usual magnetization $\stackrel{P\vec}{M}$. The other is the nonequilibrium quantity $\stackrel{P\vec}{m}$, called the spin accumulation, by which the nonequilibrium spin current can be transported. $\stackrel{P\vec}{M}$ represents a correlated distribution of a very large number of degrees of freedom, as expressed in some equilibrium distribution function for the excitations; we therefore forbid $\stackrel{P\vec}{M}$ to diffuse, but we permit $\stackrel{P\vec}{M}$ to decay. On the other hand, we permit $\stackrel{P\vec}{m}$, due to spin excitations, to both diffuse and decay. For this physical picture, diffusion from a given region occurs by decay of $\stackrel{P\vec}{M}$ to $\stackrel{P\vec}{m}$, then by diffusion of $\stackrel{P\vec}{m}$, and finally by decay of $\stackrel{P\vec}{m}$ to $\stackrel{P\vec}{M}$ in another region. This somewhat slows down the diffusion process. Restricting ourselves to the longitudinal variables $M$ and $m$ with equilibrium properties ${M}_{eq}={M}_{0}+{\ensuremath{\chi}}_{M\ensuremath{\parallel}}H$ and ${m}_{eq}=0$, we argue that the effective energy density must include a new, thermodynamically required exchange constant ${\ensuremath{\lambda}}_{M}=\ensuremath{-}1/{\ensuremath{\chi}}_{M\ensuremath{\parallel}}$. We then develop the appropriate macroscopic equations by applying Onsager's irreversible thermodynamics and use the resulting equations to study the space and time response. At fixed real frequency $\ensuremath{\omega}$ there is, as usual, a single pair of complex wave vectors $\ifmmode\pm\else\textpm\fi{}k$ but with an unusual dependence on $\ensuremath{\omega}$. At fixed real wave vector, there are two decay constants, as opposed to one in the usual case. Extending the idea that nonequilibrium diffusion in other ordered systems involves a nonequilibrium quantity, this work suggests that, in a superconductor, the order parameter $\mathrm{\ensuremath{\Delta}}$ can decay but not diffuse, but a nonequilibrium gap-like $\ensuremath{\delta}$, due to pair excitations, can both decay and diffuse.

Interpreting non-semielliptical complex bands

Complex band structure (CBS) emerges when translational symmetry is broken and material states with complex wavevectors become admissible. The resulting complex bands continuously connect conventional bands and their shapes are directly related to measurable physical quantities. To date, interpretations of complex bands usually assume they are semielliptical because this is the shape produced by the Su–Schrieffer–Heeger (SSH) model. However, numerous studies have reported CBSs with distinctly non-semielliptical shapes, including loops (essentially deformed, asymmetric semiellipses), spikes, and vertical lines. The primary goal of this work is to explore the phenomenology of these shapes such that deeper physical insight can be obtained from a qualitative inspection of a material’s CBS. By using several variations on the SSH model, we find that (i) vertical lines are unphysical numerical artifacts, (ii) spikes indicate perfectly evanescent states in the material that couple adjacent layers but do not transfer amplitude, and (iii) asymmetric loops result from hybridization. Secondarily, we also develop a strategy for eliminating any unphysical vertical lines from calculations, thereby improving computational techniques for CBS.

Extended frequency range of transverse-electric surface plasmon polaritons in graphene

The dispersion relation of surface plasmon polaritons in graphene that includes optical losses is often obtained for complex wave vectors while the frequencies are assumed to be real. This approach, however, is not suitable for describing the temporal dynamics of optical excitations and the spectral properties of graphene. Here we propose an alternative approach that calculates the dispersion relation in the complex frequency and real wave vector space. This approach provides a clearer insight into the optical properties of a graphene layer and allows us to find the surface plasmon modes of a graphene sheet in the full frequency range, thus removing the earlier reported limitation $(1.667<\ensuremath{\hbar}\ensuremath{\omega}/\ensuremath{\mu}<2)$ for the transverse-electric mode. We further develop a simple analytic approximation which accurately describes the dispersion of the surface plasmon polariton modes in graphene. Using this approximation, we show that transverse-electric surface plasmon polaritons propagate along the graphene sheet without losses even at finite temperature.

Efficient Bloch mode calculation of periodic systems with arbitrary geometry and open boundary conditions in the complex wavevector domain.

We show how existing iterative methods can be used to efficiently and accurately calculate Bloch periodic solutions of Maxwell's equations in arbitrary geometries. This is carried out in the complex-wavevector domain using a commercial frequency-domain finite-element solver that is available to the general user. The method is capable of dealing with leaky Bloch mode solutions, and is extremely efficient even for 3D geometries with non-trivial material distributions. We perform independent finite-difference time-domain simulations of Maxwell's equations to confirm our results. This comparison demonstrates that the iterative mode finder is more accurate, since it provides the true solutions in the complex-wavevector domain and removes the need for additional signal processing and fitting. Due to its efficiency, generality and reliability, this technique is well suited for complex and novel design tasks in integrated photonics, and also for a wider range of photonics problems.

ЗАДАЧА О РАСПРОСТРАНЕНИИ АКУСТИЧЕСКИХ ВОЛН В ПОРИСТОЙ СРЕДЕ, НАСЫЩЕННОЙ ПУЗЫРЬКОВОЙ ЖИДКОСТЬЮ

In many cases, formation fluids contain gas. For example, a gas-liquid mixture in a porous medium is formed during acid treatment of low-permeability zones, during water-gas stimulation of reservoirs, etc. Therefore, it seems relevant to take into account the presence of gas bubbles when studying wave processes in porous media saturated with liquid. In this work, the propagation of acoustic waves in a porous medium saturated with a gas-liquid mixture is theoretically investigated taking into account the interfacial forces of interaction between the liquid and the matrix, and heat transfer between the gas and the liquid. A general system of equations and physical relations is written, which describes the propagation of waves in a porous medium filled with bubble liquid. A dispersion relation is obtained that describes the dependence of the complex wave vector on the frequency, on the basis of which the dependence of the phase velocity and attenuation coefficient on the frequency for “fast” and “slow” waves is investigated. The calculation results make it possible to evaluate the effect of gas bubbles on the propagation of sound waves in a porous medium saturated with bubble liquid. In addition, the results of the work can be used to interpret the data of acoustic sounding of porous media.

ACOUSTIC WAVES IN POROUS MEDIA SATURATED WITH WATER WITH GAS BUBBLES ON THE PORE WALLS

The study of the propagation of disturbances in a bubble liquid is of considerable scientific and practical interest due to the wide spread of such systems in nature and in industry. In this study, wave processes are theoretically investigated in a two-speed porous medium saturated with water with gas bubbles on the pore walls. A dispersion relation describing the dependence of the complex wave vector on frequency is obtained, which is used to study the dependence of the phase velocity and the attenuation coefficient on frequency for «fast» and «slow» waves. It is established that for both types of waves there are three characteristic frequency ranges, where not only the quantitative but also the qualitative picture of the dispersion curves and the dependence of the phase velocity differ significantly. The influence of the characteristics of gas bubbles on the propagation of sound waves in a porous medium saturated with water with gas bubbles on the pore walls is studied.

The problem spreading acoustic waves in a porous environment with air bubbles on por walls

The propagation of acoustic waves in a porous medium with air bubbles on the pore walls is numerically studied. Based on the equations of mechanics of multiphase media, a dispersion relation is obtained that describes the dependence of the complex wave vector on frequency, on the basis of which the dependence of the phase velocity and attenuation coefficient on frequency for “fast” and “slow” waves is studied. The calculation results make it possible to evaluate the effect of gas bubbles on the propagation of sound waves in a porous medium with bubbles on the pore walls. The study was carried out with the financial support of the Russian Federal Property Fund in the framework of the scientific project No. 19-31-60015.

Enhancement of Resolution and Propagation Length by Sources with Temporal Decay in Plasmonic Devices

Highly lossy nature of metals has severely limited the scope of practical applications of plasmonics. The conventional approach to circumvent this limitation has been to search for new materials with more favorable dielectric properties (e.g., reduced loss), or to incorporate gain media to overcome the inherent loss. In this study, however, we turn our attention to the source and show that the wealth of new SPP modes with simultaneous complex frequencies and complex wave vectors that are otherwise unreachable can be excited by imposing temporal decay on the excitation. Therefore, to understand the possible implications of these new modes and how to be able to tune them for specific applications, we propose a framework of pseudo-monochromatic modes that are generated by introducing exponential decays into otherwise monochromatic sources. Within this framework, the dispersion relation of complex SPPs is re-evaluated and cast to be a surface rather than a curve, depicting all possible ω − k pairs (both complex in general) that are supported by the given geometry. To demonstrate the potentials of the complex modes and the use of the framework to study them selectively, we have chosen two important, and somewhat limiting, features of SPPs to investigate; resolution in plasmonic lenses and propagation length in SPP waveguides. While the former is mainly used to validate the proposed method and the framework on the recent improvement of resolution in plasmonic superlenses, the latter provides a novel approach to extend the propagation length of the SPP modes in planar waveguides significantly. Since the improvement in propagation length due to the introduction of temporal decay to the excitation is rather counter-intuitive, the dispersion-based theoretical predictions (the proposed approach) have been validated via the FDTD simulations of Maxwell’s equations in the same geometry without any a priori assumptions on the frequency or the wave vector.

Bound states of spin–orbit coupled cold atoms in a Dirac delta-function potential

Dirac delta-function potential is widely studied in quantum mechanics because it usually can be exactly solved and at the same time is useful in modeling various physical systems. Here we study a system of delta-potential trapped spin–orbit coupled cold atoms. The spin–orbit coupled atomic matter wave has two kinds of evanescent modes, one of which has a pure imaginary wavevector and is an ordinary evanescent wave; while the other with a complex number wave vector is recognized as an oscillating evanescent wave. We identified the eigenenergy spectra and the existence of bound states in this system. The bound states can be constructed analytically using the two kinds of evanescent modes and we found that they exhibit typical features of stripe phase, separated phase or zero-momentum phase. In addition to that, the properties of semi-bound states are also discussed, which is a localized wave packet on a plane wave background.

Reducible Fermi surfaces for non-symmetric bilayer quantum-graph operators

This work constructs a class of non-symmetric periodic Schrödinger operators on metric graphs (quantum graphs) whose Fermi, or Floquet, surface is reducible. The Floquet surface at an energy level is an algebraic set that describes all complex wave vectors admissible by the periodic operator at the given energy. The graphs in this study are obtained by coupling two identical copies of a periodic quantum graph by edges to form a bilayer graph. Reducibility of the Floquet surface for all energies ensues when the coupling edges have potentials belonging to the same asymmetry class. The notion of asymmetry class is defined in this article through the introduction of an entire spectral A-function a(\lambda) associated with a potential – two potentials belong to the same asymmetry class if their A-functions are identical. Symmetric potentials correspond to a(\lambda) \equiv 0 . If the potentials of the connecting edges belong to different asymmetry classes, then typically the Floquet surface is not reducible. An exception occurs when two copies of certain bipartite graphs are coupled; the Floquet surface in this case is always reducible. This includes AA-stacked bilayer graphene.

Torus solutions to the Weierstrass-Enneper representation of surfaces

In this paper, we present a torus solution to the generalized Weierstrass-Enneper representation of surfaces in R4. The key analytical technique will be Bloch wave functions with complex wave vectors. We will also discuss some possible uses of these solutions which derive from their explicit nature, such as Dehn surgery and the physics of exotic smooth structure.

Negative Refraction at a Lossy Interface and a Bold Hypothesis via Complex Frequency

We investigate the negative refraction effect at a planar interface of a highly absorptive material, where the direct experimental verification is difficult because of the loss-induced skin depth effect. An apparent contradiction occurs when we try to determine the group velocity direction by the method of equifrequency contours (EFCs) in detail. This contradiction forbids any physical solution to be found for negative refraction. We conclude that this paradox is mainly caused by the definition of complex wavevector ~k which is conventionally adopted in the case of complex permittivity. The complex wavevector may result in ambiguously defined optical path, which limits the application of the classical Snell’s law. We propose a bold suggestion that the complex wavevector ~k should be replaced by a complex frequency ~ω. Therefore, the optical path can always be defined as real. The proposed hypothesis is capable of resolving the contradiction about the loss-induced negative refraction, and the obtained theoretical prediction fits well with the reported experimental results.