Azimuthal Coordinate Research Articles

High-dimensional entanglement with orbital-angular-momentum states of light

We engineer high-dimensional orbital-angular-momentum entanglement of photon pairsthat emerge from a parametric down-conversion source. By means of two angular stateanalysers, in essence composed of a rotatable multi-sector phase plate and a single-mode fibre,we perform selective projective measurements that maximize the Shannon dimensionalityD of the measured entanglement. The multi-sector phase plates have a binary phase profilealong the azimuthal coordinate, and the arc sector sizes are optimized so as to maximizeD. We find that the maximum dimensionality increases linearly with the number of sectorsN. The potential of our method is illustrated with an experiment forN = 4, yieldingD = 16.5.

Restoring the azimuthal symmetry of lateral distributions of charged particles in the range of the KASCADE-Grande experiment

The reconstruction of Extensive Air Showers (EAS) observed by particle detectors at the ground is based on the characteristics of observables like the lateral particle density and the arrival times. The lateral densities, inferred for different EAS components from detector data, are usually parameterised by applying various lateral distribution functions (LDFs). The LDFs are used in turn for evaluating quantities like the total number of particles or the density at particular radial distances. Typical expressions for LDFs anticipate azimuthal symmetry of the density around the shower axis. The deviations of the lateral particle density from this assumption arising from various reasons are smoothed out in the case of compact arrays like KASCADE, but not in the case of arrays like Grande, which only sample a smaller part of the azimuthal variation. KASCADE-Grande, an extension of the former KASCADE experiment, is a multi-component Extensive Air Shower (EAS) experiment located at the Karlsruhe Institute of Technology (Campus North), Germany. The lateral distributions of charged particles are deduced from the basic information provided by the Grande scintillators – the energy deposits – first in the observation plane, then in the intrinsic shower plane. In all steps azimuthal dependences should be taken into account. As the energy deposit in the scintillators is dependent on the angles of incidence of the particles, azimuthal dependences are already involved in the first step: the conversion from the energy deposits to the charged particle density. This is done by using the Lateral Energy Correction Function (LECF) that evaluates the mean energy deposited by a charged particle taking into account the contribution of other particles (e.g. photons) to the energy deposit. By using a very fast procedure for the evaluation of the energy deposited by various particles we prepared realistic LECFs depending on the angle of incidence of the shower and on the radial and azimuthal coordinates of the location of the detector. Mapping the lateral density from the observation plane onto the intrinsic shower plane does not remove the azimuthal dependences arising from geometric and attenuation effects, in particular for inclined showers. Realistic procedures for applying correction factors are developed. Specific examples of the bias due to neglecting the azimuthal asymmetries in the conversion from the energy deposit in the Grande detectors to the lateral density of charged particles in the intrinsic shower plane are given.

Three-Dimensional SAR Focusing From Multipass Signals Using Compressive Sampling

Three-dimensional synthetic aperture radar (SAR) image formation provides the scene reflectivity estimation along azimuth, range, and elevation coordinates. It is based on multipass SAR data obtained usually by nonuniformly spaced acquisition orbits. A common 3-D SAR focusing approach is Fourier-based SAR tomography, but this technique brings about image quality problems because of the low number of acquisitions and their not regular spacing. Moreover, attained resolution in elevation is limited by the overall acquisitions baseline extent. In this paper, a novel 3-D SAR data imaging based on Compressive Sampling theory is presented. It is shown that since the image to be focused has usually a sparse representation along the elevation direction (i.e., only few scatterers with different elevation are present in the same range-azimuth resolution cell), it suffices to have a small number of measurements to construct the 3-D image. Furthermore, the method allows super-resolution imaging, overcoming the limitation imposed by the overall baseline span. Tomographic imaging is performed by solving an optimization problem which enforces sparsity through l1-norm minimization. Numerical results on simulated and real data validate the method and have been compared with the truncated singular value decomposition technique.

Dynamic behavior of the circular membrane of an electrostatic microphone: Effect of holes in the backing electrode

Today, several applications require using electrostatic microphones in environments and/or in frequency ranges, which are significantly different from those they were designed for. When low uncertainties on the behavior of acoustic fields, generated or measured by these transducers, are required, the displacement field of the diaphragm of the transducers (which can be highly nonuniform in the highest frequency range) must be characterized with an appropriate accuracy. An analytical approach, which leads to results depending on the location of the holes in the backing electrode (i.e., depending on the azimuthal coordinate) not available until now (regarding the displacement field of the membrane in the highest frequency range, up to 100 kHz), is presented here. The holes and the slit surrounding the electrode are considered as localized sources described by their volume velocity in the propagation equation governing the pressure field in the air gap (not by nonuniform boundary conditions on the surface of the backing electrode as usual). Experimental results, obtained from measurements of the displacement field of the membrane using a laser scanning vibrometer, are presented and compared to the theoretical results.

Unifying Experiment Design and Convex Regularization Techniques for Enhanced Imaging With Uncertain Remote Sensing Data—Part II: Adaptive Implementation and Performance Issues

The unified descriptive experiment design regularization (DEDR) method from a companion paper provides a rigorous theoretical formalism for robust estimation of the power spatial spectrum pattern of the wavefield scattered from an extended scene observed in the uncertain remote sensing (RS) environment. For the considered here imaging synthetic aperture radar (SAR) application, the proposed DEDR approach is aimed at performing, in a single optimized processing, SAR focusing, speckle reduction, and RS scene image enhancement and accounts for the possible presence of uncertain trajectory deviations. Being nonlinear and solution dependent, the optimal DEDR estimator requires rather complex signal processing operations ruled by the fixed-point iterative implementation process. To simplify further the processing, in this paper, we propose to incorporate the descriptive regularization via constructing the projections onto convex sets that enable us to factorize and parallelize the reconstructive image processing over the range and azimuth coordinates, design a family of such regularized easy-to-implement iterative algorithms, and provide the relevant computational recipes for their application to fractional imaging SAR. We also comment on the adaptive adjustment of the DEDR operational parameters directly from the actual speckle-corrupted scene images and demonstrate the effectiveness of the proposed adaptive DEDR techniques.

Tensor Green’s functions for axisymmetrical models of magnetotelluric marine sounding of locally nonhomogeneous layered media

A method for computing the electromagnetic fields of point (dipole) sources is proposed for magnetotelluric sounding (MTS) problems in axisymmetrical conducting layered media. The method expands the tensor Green’s functions of the layered medium in Fourier series in the azimuthal coordinate. For an arbitrary system of point sources we construct algorithms to compute the electromagnetic fields propagating across the plane interface of two conducting half-spaces with different constant conductivities.

Charge dissolution and heat and mass transfer during hydrothermal crystal growth with heat field rotation

Analytical modeling of laminar thermal gravitational convection in a vertical porous cylinder of radius R and height H (R is much smaller than H) with a low-compressed fluid (a model of the charge-filled bottom part of autoclave) is performed by the Poincare method. Boundary thermal conditions of the first kind are set at the cylinder boundary. Along with the main thermal field (stationary, homogeneous along the azimuthal coordinate, and linear along the vertical; the temperature gradient is set at the top), an additional cyclically rotating azimuthal inhomogeneous thermal field is applied to the cylinder wall. Asymptotic representations of the temperature, velocity, and concentration fields in the porous layer are obtained in the criteria form. It is shown that applying an additional temperature field increases the charge dissolution rate without increasing the average temperature gradient.

THE STELLAR VELOCITY DISTRIBUTION IN THE SOLAR NEIGHBORHOOD: DEVIATIONS FROM THE SCHWARZSCHILD DISTRIBUTION

The idea of the collisionless relaxation of the stellar disk of the Milky Way is elaborated. Jeans' gravitationally unstable stellar disk is considered by applying the procedure of the quasi-linear kinetic approach. It is shown that unstable gravity perturbations (e.g., those produced by a spontaneous disturbance) grow almost aperiodically in the main part of the disk between the inner and outer Lindblad resonances and affect the in-plane averaged velocity distribution of stars. The diffusion equation in velocity space is derived describing the secular increase in the dispersion of the peculiar velocities of stars with time after star formation in the rotationally supported Galactic disk. Our previous investigation is extended by including the minor second-order terms of the theory to describe the small distortion in the local distribution function of stars away from the standard Schwarzschild distribution. It is shown that the residual velocity distributions along the radial and azimuthal coordinates in velocity space are non-Gaussian in the sense that the central peaks are more populated and the higher energy portions of the distributions are somewhat underpopulated. For a subsystem of relatively old stars with ages (2-3)×109 years, the quantitative change is of the order of 10%-15%. The observational test of the theory is also suggested.

Extremal static AdS black hole/CFT correspondence in gauged supergravities

A recently proposed holographic duality allows the Bekenstein–Hawking entropy of extremal rotating black holes to be calculated microscopically, by applying the Cardy formula to the two-dimensional chiral CFTs associated with certain reparameterisations of azimuthal angular coordinates in the solutions. The central charges are proportional to the angular momenta of the black hole, and so the method degenerates in the case of static (non-rotating) black holes. We show that the method can be extended to encompass such charged static extremal AdS black holes by using consistent Kaluza–Klein sphere reduction ansatze to lift them to exact solutions in the low-energy limits of string theory or M-theory, where the electric charges become reinterpreted as angular momenta associated with internal rotations in the reduction sphere. We illustrate the procedure for the examples of extremal charged static AdS black holes in four, five, six and seven dimensions.

An alternative method for calculation of Rayleigh and Love wave phase velocities by using three-component records on a single circular array without a central station

SUMMARY A method for the computation of phase velocities of surface waves from microtremor waveforms is shown. The technique starts from simultaneous three-component records obtained in a circular array without a central station. Then, Fourier spectra of vertical, radial and tangential components of motion are calculated for each station and considered as complex-valuated functions of the azimuthal coordinate. A couple of intermediate real physical quantities, B and C, can be defined from the 0- and ±1-order coefficients of the Fourier series expansion of such functions. Finally, phase velocities of Rayleigh and Love waves can be retrieved from B and C by solving respective one-unknown equations. The basic assumption is the possibility of expanding the wavefield as a sum of plane surface waves with Rayleigh and Love wavenumbers being univocal functions of the circular frequency. The method is tested in synthetic ambient noise wavefields confirming its reliability and robustness for passive seismic surveying.

Instability of optimal non-axisymmetric base-flow deviations in pipe Poiseuille flow

The stability of pipe flow when mildly deviating from the developed Poiseuille profile by a non-axisymmetric azimuthally periodic distortion is examined. The motivation for this is to consider deviations, the origin of which may be attributed to small-amplitude disturbances having sinusoidal periodicity along the azimuthal coordinate, which are known to be the ones most amplified by the transient growth linear mechanism. A mathematical technique for finding the minimum energy density of azimuthally periodic deviations triggering exponential instability is presented. The results show that owing to bifurcations multiple solutions of optimal deviations exist. As the Reynolds number is increased additional bifurcations appear and create more distinct solutions. The different solutions correspond to different radial distributions of the deviations, and at Reynolds numbers of about 2000 they are distributed over less than a half of the pipe radius. It is found that the dependence of the optimal deviation velocity leading to instability on the Reynolds number Re is approximately 20/Re. A comparison to axisymmetric base-flow deviations shows that the minimum energy required for an azimuthally periodic deviation to trigger instability is almost twice that for the axisymmetric flow. However, azimuthally periodic deviations, which are shown to have a streaky pattern, may have a role in the self-sustaining process. They may be formed as a result of a transient growth amplification of initial streamwise rolls and can produce, via self-interactions between the resulting growing waves, patterns of streamwise rolls as well.

Generation of Alfvén waves by a plasma inhomogeneity moving in the Earth’s magnetosphere

The generation of an Alfven wave by an azimuthally drifting cloud of high-energy particles injected in the Earth's magnetosphere is studied analytically. In contrast to the previous studies where the generation mechanisms associated with the resonant wave-particle interaction were considered, a nonresonant mechanism is investigated in which the wave is excited by the alternating current produced by drifting particles. It is shown that, at a point with a given azimuthal coordinate, a poloidally polarized wave, in which the magnetic field lines oscillate predominantly in the radial direction, is excited immediately after the passage of the particle cloud through this point. As the cloud moves away from that point, the wave polarization becomes toroidal (the mag- netic field lines oscillate predominantly in the azimuthal direction). The azimuthal wavenumber m is defined as the ratio of the wave eigenfrequency to the angular velocity of the cloud (the drift velocity of the particles). It is shown that the amplitudes of the waves so generated are close to those obtained under realistic assumptions about the density and energy of the particles.

Feature-Independent Aperture Evaluator for the Curvilinear SAR

Curvilinear synthetic aperture radar (SAR), as a more practicable 3-D SAR imaging system, utilizes parametric target feature estimates extracted from the received data to reconstruct the target image. The reconstructed image quality is then impacted by the estimation accuracy of the features. In this letter, through discussing the correlation between the system parameters and the estimation performance of the curvilinear SAR, a conclusion can be drawn on how the overall location accuracy of a target is determined by the correlation between the azimuth and elevation coordinates of the flight path, compactly characterizing the curvilinear aperture. Consequently, a new index, determined only with the aperture parameters, is proposed as an aperture evaluator, which is referred to as the feature-independent aperture evaluator (FAE). FAE can be used for guiding the operational aperture design

Self-trapped electron states in nanotubes

We study numerically self-trapped (polaron) states of quasiparticles (electrons or holes) in a deformable nanotube formed by a hexagonal lattice, wrapped into a cylinder (carbon- and boron nitride-type nanotube structures). We present a Hamiltonian for such a system taking into account an electron–phonon interaction, and determine conditions under which the lowest energy states are polarons. We compute a large class of numerical solutions of this model for a wide range of the parameters. We show that at not-too-strong electron–phonon coupling, the system admits ring-like localized solutions wrapped around the nanotube (the charge carrier is localized along the nanotube axis and is uniformly distributed with respect to the azimuthal coordinate). At stronger coupling, solutions are localized on very few lattice sites in both directions of the nanotube. The transition from one type of solution to the other one depends on the diameter of the nanotube. We show that for the values of the carbon nanotube parameters, the polarons have a ring-like structure wrapped around the nanotube, with a profile resembling that of the nonlinear Schrödinger soliton.

Matter-wave solitons in radially periodic potentials

We investigate two-dimensional (2D) states in Bose-Einstein condensates with self-attraction or self-repulsion, trapped in an axially symmetric optical-lattice potential periodic along the radius. The states trapped both in the central potential well and in remote circular troughs are studied. In the repulsive mode, a new soliton species is found, in the form of radial gap solitons. The latter solitons are completely stable if they carry zero vorticity (l=0) , while with l not equal 0 they develop a weak azimuthal modulation, which makes them rotating patterns, that persist indefinitely long. In addition, annular gap solitons may support stable azimuthal dark-soliton pairs on their crests. In remote troughs of the attractive model, stable localized states may assume a ringlike shape with weak azimuthal modulation, or shrink into solitons strongly localized in the azimuthal direction, which is explained in the framework of an averaged 1D equation with the cyclic azimuthal coordinate. Numerical simulations of the attractive model also reveal stable necklacelike patterns, built of several strongly localized peaks. Dynamics of strongly localized solitons circulating in the troughs is studied too. While the solitons with sufficiently small velocities are completely stable, fast solitons gradually decay, due to the leakage of matter into the adjacent trough, under the action of the centrifugal force. Investigation of head-on collisions between strongly localized solitons traveling in circular troughs shows that collisions between in-phase solitons in a common trough lead to collapse, while pi-out-of-phase solitons bounce many times, but eventually merge into a single one, without collapsing. In-phase solitons colliding in adjacent circular troughs also tend to merge into a single soliton.

Fourier reconstruction of marine-streamer data in four spatial coordinates

Many methods exist for interpolation of seismic data in one and two spatial dimensions, but few can interpolate properly in three or four spatial dimensions. Marine multi-streamer data typically are sampled relatively well in the midpoint and absolute offset coordinates but not in the azimuth because the crossline shot coordinate is significantly under sampled. We approach the problem of interpolation of marine-streamer data in four spatial dimensions by splitting the problem into a 1D interpolation along the densely sampled streamers and a 3D Fourier reconstruction for the remaining spatial coordinates. In Fourier reconstruction, the Fourier coefficients that synthesize the nonuniformly sampled seismic data are estimated in a least-squares inversion. The method is computationally efficient, requires no subsurface information, and can handle uniform grids with missing data as well as nonuniform grids or random sampling. The output grid of the 1D interpolation in the first step is arbitrary. When the output grid has uniform inline midpoints spacing, the 3D Fourier reconstruction in the second step is performed in the crossline midpoint, absolute offset, and azimuth coordinates. When the first step outputs to uniform absolute offset, the 3D Fourier reconstruction handles the crossline/inline midpoint and the azimuth coordinates. In both cases, the main innovation is the inclusion of the azimuthal coordinate in the Fourier reconstruction. The azimuth multiplicity must be increased for the method to be successful, which means that overlap shooting is required. We have tested the algorithm on synthetic streamer data for which the proposed method outperforms an approach where the azimuthal coordinate is ignored. Potential applications are interpolation of marine streamer data to decrease the crossline source sampling for the benefit of 3D multiple prediction and regularization to reduce sampling-related differences in processing of time-lapse data.

Acoustic backscattering from fluid‐filled submerged prolate spheroidal shells

The equations of motion for nonaxisymmetric vibration of prolate spheroidal shells of constant thickness were derived using Hamilton’s principle [S. I. Hayek and J. E. Boisvert, J. Acoust. Soc. Am. 114, 2799–2811 (2003)]. The shell theory used in this derivation includes shear deformations and rotatory inertias. The shell displacements and rotations were expanded in infinite series of comparison functions. These include associated Legendre functions in terms of the prolate spheroidal angular coordinate and circular functions in the azimuthal angle coordinate. The fluid‐filled shell is insonified by a plane wave with an arbitrary angle of incidence. The external and internal fluid loading impedances were computed using an eigenfunction expansion of prolate spheroidal wavefunctions. Far‐field backscattered acoustic pressure spectra are presented as a function of the angle of incidence for several shell thickness‐to‐half‐length ratios ranging from 0.005 to 0.1, and for various shape parameters, a, ranging from an elongated spheroidal shell (a=1.01) to a spherical shell (a∼100). A comparison of the backscattering from fluid‐filled and empty shells is presented at selected plane wave incident angles. [Work supported by the ONR/ASEE Summer Faculty Research Program and the NAVSEA Newport ILIR Program.]

Stability of gravity currents generated by finite-volume releases

We generalize the linear stability analysis of the axisymmetric self-similar solution of gravity currents from finite-volume releases to include perturbations that depend on both radial and azimuthal coordinates. We show that the similarity solution is stable to sufficiently small perturbations by proving that all perturbation eigenfunctions decay in time. Moreover, asymmetric perturbations are shown to decay more rapidly than axisymmetric perturbations in general. An asymptotic formula for the eigenvalues is derived, which indicates that asymptotic rates of decay of perturbations are given by

Nonaxisymmetric acoustic scattering from submerged prolate spheroidal shells

The equations of motion for nonaxisymmetric vibration of prolate spheroidal shells of constant thickness were derived using Hamiltons principle [S. I. Hayek and J. E. Boisvert, J. Acoust. Soc. Am. 114, 2799–2811 (2003)]. The shell theory used in this derivation includes shear deformations and rotatory inertias. The shell displacements and rotations were expanded in infinite series of comparison functions. These include associated Legendre functions in terms of the prolate spheroidal angular coordinate and circular functions in the azimuthal angle coordinate. The shell is insonified by a plane wave with arbitrary incident angle. The external (heavy) fluid-loading impedance was computed using an eigenfunction expansion of prolate spheroidal wavefunctions. Far-field backscattered acoustic pressure spectra are presented as a function of incident angle for several shell thickness-to-half-length ratios ranging from .005 to 0.1, and for various shape parameters, a, ranging from an elongated spheroidal shell (a=1.01) to a spherical shell (a∼100). The far-field directivity of acoustic scattering is presented at selected frequencies and plane wave incident angles. [Work supported by the ONR/ASEE Summer Faculty Research Program and the NAVSEA Newport ILIR Program.]

Semiclassical strings in electric and magnetic field deformedAdS5×S5spacetimes

We first apply the transformation of mixing azimuthal and internal coordinate or mixing time and internal coordinate to the 11D M-theory with a stack N M2-branes to find the spacetime of a stack of N D2-branes with magnetic or electric flux in 10 D IIA string theory, after the Kaluza-Klein reduction. We then perform the T-duality to the spacetime to find the background of a stack of N D3-branes with magnetic or electric flux. In the near-horizon limit the background becomes the magnetic or electric field deformed $\mathrm{Ad}{\mathrm{S}}_{5}\ifmmode\times\else\texttimes\fi{}{S}^{5}$. We adopt an ansatz to find the classical string solution which is rotating in the deformed ${S}^{5}$ with three angular momenta in the three rotation planes. The relations between the classical string energy and its angular momenta are found and results show that the external magnetic and electric fluxes will increase the string energy. Therefore, from the AdS/CFT point of view, the corrections of the anomalous dimensions of operators in the dual SYM theory will be positive. We also investigate the small fluctuations in these solutions and discuss the effects of magnetic and electric fields on the stability of these classical rotating string solutions. Finally, we find the possible solutions of string pulsating on the deformed spacetimes and show that the corrections to the anomalous dimensions of operators in the dual SYM theory are non-negative.