Equations In Fourier Space Research Articles

Kinetic Theory of Drift‐Mirror Mode

AbstractWe present a nonlocal gyrokinetic theory for the drift‐mirror mode in high‐ anisotropic plasmas. Here, represents the ratio of plasma pressure to magnetic pressure. The equilibrium distribution is established self‐consistently via guiding‐center Hamiltonian theory. To keep the nonuniformity and finite Larmor radius (FLR) effects on an equal footing, we derive the three‐field nonlocal eigenmode equations in Fourier space under the assumption of weak nonuniformity. It is found that the drift‐mirror mode is essentially a dissipative instability confined within the potential well due to the FLR effect. Numerical analyses demonstrate that the drift‐mirror mode originates from the coupling between shear Alfvén and mirror branch, where the ion‐sound wave component is negligible. Furthermore, the threshold of this nonlocal drift‐mirror mode is significantly lower than that of the conventional drift‐mirror mode.

Fourier transform approach to homogenization of frequency-dependent heat transfer in porous media

PurposeThe purpose of this paper is to solve the local problem involving strong contrast heterogeneous conductive material, with application to gas-filled porous media with both perfect and imperfect Kapitza boundary conditions at the bi-material interface. The effective parameters like the dynamic conductivity and the thermal permeability in the acoustics of porous media are also derived from the cell solution.Design/methodology/approachThe Fourier transform method is used to solve frequency-dependent heat transfer problems. The periodic Lippmann–Schwinger integral equation in Fourier space with source term is first formulated using discrete Green operators and modified wavevectors, which can then be solved by iteration schemes.FindingsNumerical examples show that the schemes converge fast and yield accurate results when compared with analytical solution for benchmark problems.Originality/valueThe formulation of the method is constructed using static and dynamic Green operators and can be applied to pixelized microstructure issued from tomography images.

Equações de Maxwell no espaço de Fourier e acoplamento entre excitações elementares em sólidos

Maxwell’s equations are fundamental to the understanding of electromagnetic events, as they describe the totality of experimentally verified electromagnetic phenomena from its boundary conditions and constitutive relations. Although, they are normally presented in the time and space domain, many physical effects and their associated mathematical manipulation are easier to understand when we use Maxwell’s equations in Fourier space. In this work we give a general introduction about Maxwell’s equations in Fourier domain in material media. The simplicity of the obtained equations in comparison with the ones in space-time domain is emphasized, and the link between them and the elementary excitations in solids is presented and explored. In particular, the coupling between various types of quasiparticles in solids is explained.

Intrinsic Optical Spatial Differentiation Enabled Quantum Dark-Field Microscopy.

By solving the Maxwell's equations in Fourier space, we find that the cross-polarized component of the dipole scattering field can be written as the second-order spatial differentiation of the copolarized component. This differential operation can be regarded as intrinsic which naturally arises as consequence of the transversality of electromagnetic fields. By introducing the intrinsic spatial differentiation into heralded single-photon microscopy imaging technique, it makes the structure of pure-phase object clearly visible at low photon level, avoiding any biophysical damages to living cells. Based on the polarization entanglement, the switch between dark-field imaging and bright-field imaging is remotely controlled in the heralding arm. This research enriches both fields of optical analog computing and quantum microscopy, opening a promising route toward a nondestructive imaging of living biological systems.

A model order reduction technique for FFT‐based microstructure simulation using a geometrically adapted reduced set of frequencies

AbstractThe FFT‐based method introduced by Moulinec and Suquet [9] serves as an alternative for the classical finite element based simulation of periodic microstructures. This simulation approach makes use of fast Fourier transforms (FFT) as well as fixed‐point iterations to solve the microscopic boundary value problem which is captured by the Lippmann‐Schwinger equation. Kochmann et al. [5] introduced a model order reduction technique using a reduced set of frequencies to decrease the computational effort of solving the Lippmann‐Schwinger equation in Fourier space. This earlier proposed method is based on a fixed sampling pattern, which determines the reduced set of frequencies. Instead of the fixed sampling pattern, we propose to use a geometrically adapted choice of frequencies, which corresponds to the representation of phases within the considered microstructure.

A geometrically adapted reduced set of frequencies for a FFT-based microstructure simulation

We present a modified model order reduction (MOR) technique for the FFT-based simulation of composite microstructures. It utilizes the earlier introduced MOR technique (Kochmann et al., 2019), which is based on solving the Lippmann–Schwinger equation in Fourier space by a reduced set of frequencies. Crucial for the accuracy of this MOR technique is on the one hand the amount of used frequencies and on the other hand the choice of frequencies used within the simulation. Kochmann et al. (2019) defined the reduced set of frequencies by using a fixed sampling pattern, which is most general but leads to poor microstructural results when considering only a few frequencies. Consequently, a reconstruction algorithm based on the TV1-algorithm (Candes et al., 2006) was used in a post-processing step to generate highly resolved micromechanical fields. The present work deals with a modified sampling pattern generation for this MOR technique. Based on the idea, that the micromechanical material response strongly depends on the phase-wise material behavior, we propose the usage of sampling patterns adapted to the spatial arrangement of the individual phases. This leads to significantly improved microscopic and overall results. Hence, the time-consuming reconstruction in the post-processing step that was necessary in the earlier work is no longer required. To show the adaptability and robustness of this new choice of sampling patterns, several two dimensional examples are investigated. In addition, also the 3D extension of the algorithm is presented.

High Light Outcoupling Efficiency from Periodically Corrugated OLEDs.

Organic light-emitting diodes (OLEDs) suffer from poor light outcoupling efficiency (ηout < 20%) due to large internal waveguiding in the high-index layers/substrate, and plasmonic losses at the metal cathode interface. A promising approach to enhance light outcoupling is to utilize internal periodic corrugations that can diffract waveguided and plasmonic modes back to the air cone. Although corrugations can strongly diffract trapped modes, the optimal geometry of corrugations and limits to ηout are not well-established. We develop a general rigorous scattering matrix theory for light emission from corrugated OLEDs, by solving Maxwell’s equations in Fourier space, incorporating the environment-induced modification of the optical emission rate (Purcell effect). We computationally obtain the spectrally emissive power inside and outside the OLED. We find conformally corrugated OLEDs, where all OLED interfaces are conformal with a photonic crystal substrate, having triangular lattice symmetry, exhibit high light outcoupling ηout ∼60–65%, and an enhancement factor exceeding 3 for optimal pitch values between 1 and 2.5 μm. Waveguided and surface plasmon modes are strongly diffracted to the air cone through first-order diffraction. ηout is insensitive to corrugation heights larger than 100 nm. There is a gradual roll-off in ηout for a larger pitch and sharper decreases for small pitch values. Plasmonic losses remain below 10% for all corrugation pitch values. Our predicted OLED designs provide a pathway for achieving very high light outcoupling over the full optical spectrum that can advance organic optoelectronic science and solid-state lighting.

Asymptotic behavior of critical dissipative quasi-geostrophic equation in Fourier space

In this paper we show the global existence for critical dissipative quasi-geostrophic equations if ‖θ0ˆ‖L1 is small enough; among others we prove the analyticity of such a solution. If in addition the initial condition verifies |D|−δθ0∈L1(R2) with 0<δ<1, then the solution remains regular and limt→∞⁡tδ‖θˆ(t)‖L1=0. Fourier analysis and standard techniques are used.

Musical pitch quantization as an eigenvalue problem

How can discrete pitches and chords emerge from the continuum of sound? Using a quantum cognition model of tonal music, we prove that the associated Schrödinger equation in Fourier space is invariant under continuous pitch transpositions. However, this symmetry is broken in the case of transpositions of chords, entailing a discrete cyclic group as transposition symmetry. Our research relates quantum mechanics with music and is consistent with music theory and seminal insights by Hermann von Helmholtz.

Selection of Quasi-stationary States in the Stochastically Forced Navier–Stokes Equation on the Torus

The stochastically forced vorticity equation associated with the two dimensional incompressible Navier-Stokes equation on $D_\delta:=[0,2\pi\delta]\times [0,2\pi]$ is considered for $\delta\approx 1$, periodic boundary conditions, and viscocity $0<\nu\ll 1$. An explicit family of quasi-stationary states of the deterministic vorticity equation is known to play an important role in the long-time evolution of solutions both in the presence of and without noise. Recent results show the parameter $\delta$ plays a central role in selecting which of the quasi-stationary states is most important. In this paper, we aim to develop a finite dimensional model that captures this selection mechanism for the stochastic vorticity equation. This is done by projecting the vorticity equation in Fourier space onto a center manifold corresponding to the lowest eight Fourier modes. Through Monte Carlo simulation, the vorticity equation and the model are shown to be in agreement regarding key aspects of the long-time dynamics. Following this comparison, perturbation analysis is performed on the model via averaging and homogenization techniques to determine the leading order dynamics for statistics of interest for $\delta\approx1$.

Modeling Imbalanced Collisionless Alfvén Wave Turbulence with Nonlinear Diffusion Equations

Abstract A pair of nonlinear diffusion equations in Fourier space is used to study the dynamics of strong Alfvén wave turbulence, from MHD to electron scales. Special attention is paid to the regime of imbalance between the energies of counter-propagating waves commonly observed in the solar wind (SW), especially in regions relatively close to the Sun. In the collisionless regime where dispersive effects arise at scales comparable to or larger than those where dissipation becomes effective, the imbalance produced by a given injection rate of generalized cross-helicity (GCH), which is an invariant, is much larger than in the corresponding collisional regime described by the usual (or reduced) magnetohydrodynamics. The combined effect of high imbalance and ion Landau damping induces a steep energy spectrum for the transverse magnetic field at sub-ion scales. This spectrum is consistent with observations in highly Alfvénic regions of the SW, such as trailing edges, but does not take the form of a transition range continued at smaller scales by a shallower spectrum. This suggests that the observed spectra displaying such a transition result from the superposition of contributions originating from various streams with different degrees of imbalance. Furthermore, when imbalanced energy injection is supplemented at small scales in an already fully developed turbulence, for example under the effect of magnetic reconnection, a significant enhancement of the imbalance at all scales is observed.

Poisson structure of the three-dimensional Euler equations in Fourier space

We derive a simple Poisson structure in the space of Fourier modes for the vorticity formulation of the Euler equations on a three-dimensional periodic domain. This allows us to analyse the structure of the Euler equations using a Hamiltonian framework. The Poisson structure is valid on the divergence free subspace only, and we show that using a projection operator it can be extended to be valid in the full space. We then restrict the simple Poisson structure to the divergence-free subspace on which the dynamics of the Euler equations take place, reducing the size of the system of ordinary differential equations by a third. The projected and the restricted Poisson structures are shown to have the helicity as a Casimir invariant. We conclude by showing that periodic shear flows in three dimensions are equilibria that correspond to singular points of the projected Poisson structure, and hence that the usual approach to study their nonlinear stability through the energy-Casimir method fails.

Turbulent kinetic energy budgets in wall bounded flows with pressure gradients and separation

Numerical simulations are employed to investigate the turbulent kinetic energy (TKE) budgets in turbulent channel flows with pressure gradients and separation. Incompressible, highly resolved large eddy simulations are performed for Reτ = 170 and 615 to investigate the flow developing along a convergent-divergent channel. The aim of this work is to analyze the TKE budgets both in physical and Fourier spaces to characterize the important scales in the individual processes in such turbulent flows. The study is performed for different positions along the channel where favorable and adverse pressure gradients are present. Proper orthogonal decomposition is employed to understand the role of the most energetic structures in the TKE budgets. Results indicate that such structures account for most of the turbulent effects present in the flow, except for the transport term. A spectral TKE equation in Fourier space is developed for flows with one homogeneous direction to characterize the turbulent processes as a function of the wavelength in the channel spanwise direction. The results show that viscous effects occur at the same range of wavelengths for which production is found and that TKE is transported to the near-wall region, being dissipated by large spanwise scale motion. They also show that favorable pressure gradients change the distribution of processes along the spanwise wavelengths. In the adverse pressure gradient region, TKE is transported both toward the wall and toward the center of the channel, where it is balanced by the advection term.

Numerical Computation of Doubly-Periodic Stokes Flow Bounded by a Plane with Applications to Nodal Cilia

AbstractA numerical method is presented for the computation of externally forced Stokes flows bounded by the planez=0 and satisfying periodic boundary conditions in thexandydirections. The motivation for this work is the simulation of flows generated by cilia, which are hair-like structures attached to the surface of cells that generate flows through coordinated beating. Large collections of cilia on a surface can be modeled using a doubly-periodic domain. The approach presented here is to derive a regularized version of the fundamental solution of the incompressible Stokes equations in Fourier space for the periodic directions and physical space for thezdirection. This analytical expression forû(k,m;z) can then be used to compute the fluid velocityu(x,y,z) via a two-dimensional inverse fast Fourier transform for any fixed value ofz. Repeating the computation for multiple values ofzleads to the fluid velocity on a uniform grid in physical space. The zero-flow condition at the planez=0 is enforced through the use of images. The performance of the method is illustrated by numerical examples of particle transport by nodal cilia, which verify optimal particle transport for parameters consistent with previous studies. The results also show that for two cilia in the periodic box, out-of-phase beating produces considerablemore particle transport than in-phase beating.

Nested polyhedra model of turbulence.

A discretization of the wave-number space is proposed, using nested polyhedra, in the form of alternating dodecahedra and icosahedra that are self-similarly scaled. This particular choice allows the possibility of forming triangles using only discretized wave vectors when the scaling between two consecutive dodecahedra is equal to the golden ratio and the icosahedron between the two dodecahedra is the dual of the inner dodecahedron. Alternatively, the same discretization can be described as a logarithmically spaced (with a scaling equal to the golden ratio), nested dodecahedron-icosahedron compounds. A wave vector which points from the origin to a vertex of such a mesh, can always find two other discretized wave vectors that are also on the vertices of the mesh (which is not true for an arbitrary mesh). Thus, the nested polyhedra grid can be thought of as a reduction (or decimation) of the Fourier space using a particular set of self-similar triads arranged approximately in a spherical form. For each vertex (i.e., discretized wave vector) in this space, there are either 9 or 15 pairs of vertices (i.e., wave vectors) with which the initial vertex can interact to form a triangle. This allows the reduction of the convolution integral in the Navier-Stokes equation to a sum over 9 or 15 interaction pairs, transforming the equation in Fourier space to a network of "interacting" nodes that can be constructed as a numerical model, which evolves each component of the velocity vector on each node of the network. This model gives the usual Kolmogorov spectrum of k^{-5/3}. Since the scaling is logarithmic, and the number of nodes for each scale is constant, a very large inertial range (i.e., a very high Reynolds number) with a much lower number of degrees of freedom can be considered. Incidentally, by assuming isotropy and a certain relation between the phases, the model can be used to systematically derive shell models.

Inverse energy cascade in nonlocal helical shell models of turbulence

Following the exact decomposition in eigenstates of helicity for the Navier-Stokes equations in Fourier space [F. Waleffe, Phys. Fluids A 4, 350 (1992)], we introduce a modified version of helical shell models for turbulence with nonlocal triadic interactions. By using both an analytical argument and numerical simulation, we show that there exists a class of models, with a specific helical structure, that exhibits a statistically stable inverse energy cascade, in close analogy with that predicted for the Navier-Stokes equations restricted to the same helical interactions. We further support the idea that turbulent energy transfer is the result of a strong entanglement among triads possessing different transfer properties.

Integral method for the calculation of Hawking radiation in dispersive media. I. Symmetric asymptotics.

Hawking radiation has become experimentally testable thanks to the many analog systems which mimic the effects of the event horizon on wave propagation. These systems are typically dominated by dispersion and give rise to a numerically soluble and stable ordinary differential equation only if the rest-frame dispersion relation Ω^{2}(k) is a polynomial of relatively low degree. Here we present a new method for the calculation of wave scattering in a one-dimensional medium of arbitrary dispersion. It views the wave equation as an integral equation in Fourier space, which can be solved using standard and efficient numerical techniques.

Fourier Spectral Methods for Numerical Solving of the Kuramoto-Sivashinsky Equation

In this paper we present a numerical technique for solving Kuramoto-Sivashinsky equation, based on spectral Fourier methods. This equation describes reaction diffusion problems, and the dynamics of viscous-fuid films flowing along walls. After we wrote the equation in Fourier space, we get a system. In this case, the exponential time differencing methods integrate the system very much more accurately than other methods since the exponential time differencing methods assume in their derivation that the solution varies slowly in time. When evaluating the coefficients of the exponential time differencing and the exponential time differencing Runge Kutta methods via the”Cauchy integral”. All computational work is done with Matlab package.

Nano-photonic and nano-plasmonic enhancements in thin film silicon solar cells

We design, simulate and fabricate enhanced thin film silicon solar cells using periodically textured photonic crystal substrates. We utilize a thin film absorber layer consisting of a superlattice of alternating hydrogenated amorphous silicon (a-Si:H) and nano-crystalline silicon (nc-Si). Rigorous vectorial simulations optimized the periodically patterned solar cells by solving Maxwell׳s equations in Fourier space. Simulations found optimized architectures for a triangular lattice of metallic nano-cones as a back-reflector, and a conformal solar cell geometry. The periodically patterned photonic crystal based substrates achieve (1) high diffraction, enhancing the path length of light in thin absorber layers and (2) plasmonic concentration of light intensity. Simulations predict an absorption enhancement of 43% for a 12-period superlattice of 800nm thickness. The optimized pitch of the photonic lattice is near 700nm. Experimentally the periodically patterned substrates were fabricated with nano-imprint lithography, and utilized as a substrate for the superlattice cells. We measured a large photo-current enhancement between the textured photonic crystal based superlattice cell and the flat cell of 21%, together with long-wavelength quantum efficiency enhancements beyond 600nm. This is an approach to achieving thin film solar cells with high currents through advanced light-trapping techniques on novel materials.

Modeling of laser wakefield acceleration in Lorentz boosted frame using EM-PIC code with spectral solver

Simulating laser wakefield acceleration (LWFA) in a Lorentz boosted frame in which the plasma drifts towards the laser with vb can speed up the simulation by factors of γb2=(1−vb2/c2)−1. In these simulations the relativistic drifting plasma inevitably induces a high frequency numerical instability that contaminates the interesting physics. Various approaches have been proposed to mitigate this instability. One approach is to solve Maxwell equations in Fourier space (a spectral solver) as this has been shown to suppress the fastest growing modes of this instability in simple test problems using a simple low pass or “ring” or “shell” like filters in Fourier space. We describe the development of a fully parallelized, multi-dimensional, particle-in-cell code that uses a spectral solver to solve Maxwell's equations and that includes the ability to launch a laser using a moving antenna. This new EM-PIC code is called UPIC-EMMA and it is based on the components of the UCLA PIC framework (UPIC). We show that by using UPIC-EMMA, LWFA simulations in the boosted frames with arbitrary γb can be conducted without the presence of the numerical instability. We also compare the results of a few LWFA cases for several values of γb, including lab frame simulations using OSIRIS, an EM-PIC code with a finite-difference time domain (FDTD) Maxwell solver. These comparisons include cases in both linear and nonlinear regimes. We also investigate some issues associated with numerical dispersion in lab and boosted frame simulations and between FDTD and spectral solvers.