Infinite Linear Array Research Articles

Exact solution of the two-dimensional scattering problem for a class of δ-function potentials supported on subsets of a line

We use the transfer matrix formulation of scattering theory in two-dimensions (2D) to treat the scattering problem for a potential of the form where , and b are constants, is the Dirac δ function, and g is a real- or complex-valued function. We map this problem to that of and give its exact (nonapproximate) and analytic (closed-form) solution for the following choices of : (i) a linear combination of δ functions, in which case is a finite linear array of 2D δ functions; (ii) a linear combination of with real; (iii) a general periodic function that has the form of a complex Fourier series. In particular we solve the scattering problem for a potential consisting of an infinite linear periodic array of 2D δ functions. We also prove a general theorem that gives a sufficient condition for different choices of to produce the same scattering amplitude within specific ranges of values of the wavelength λ. For example, we show that for arbitrary real and complex parameters, a and , the potentials and have the same scattering amplitude for .

Nearly bound states in the radiation continuum in a circular array of dielectric rods

We consider E polarized bound states in the radiation continuum (BICs) in circular periodical arrays of $N$ infinitely long dielectric rods. We find that each true BIC which occurs in an infinite linear array has its counterpart in the circular array as a near-BIC with extremely large quality factor. We argue analytically as well as numerically that the quality factor of the symmetry protected near-BICs diverges as $e^{\lambda N}$ where $\lambda$ is a material parameter dependent on the radius and the refraction index of the rods. By tuning of the radius of rods we also find numerically non-symmetry protected near-BICs. These near-BICs are localized with exponential accuracy outside the circular array but fill the whole inner space of the array carrying orbital angular momentum.

Propagating Bloch bound states with orbital angular momentum above the light line in the array of dielectric spheres.

We present propagating Bloch bound states in the radiation continuum with orbital angular momentum in an infinite linear periodical array of dielectric spheres. The bound states in the continuum demonstrate a giant Poynting vector spiraling around the array. They can be excited by a plane wave with incident linear polarization with a small tilt relative to the axis of the array.

Radiation from Structured-Ring Resonators

We investigate the scalar-wave resonances of systems composed of identical Neumann-type inclusions arranged periodically around a circular ring. Drawing on natural similarities with the undamped Rayleigh--Bloch waves supported by infinite linear arrays, we deduce asymptotically the exponentially small radiative damping in the limit where the ring radius is large relative to the periodicity. In our asymptotic approach, locally linear Rayleigh--Bloch waves that attenuate exponentially away from the ring are matched to a ring-scale WKB-type wave field. The latter provides a descriptive physical picture of how the mode energy is transferred via tunnelling to a circular evanescent-to-propagating transition region a finite distance away from the ring, from where radiative grazing rays emanate to the far field. Excluding the zeroth-order standing-wave modes, the position of the transition circle bifurcates with respect to clockwise and counterclockwise contributions, resulting in striking spiral wavefronts.

Ignition and combustion of a gaseous fuel pocket array in an oxidizing environment

ABSTRACTIgnition and combustion of an infinite linear array of gaseous fuel pockets in a stagnant oxidizing environment under the microgravity condition is studied by a numerical approach. The combustion process is considered isobaric and the fluid motion is induced by density gradients due to the heat and mass transfer processes. A simple finite chemical reaction mechanism and the ideal gas equation of state are considered. The thermophysical properties, except density, are assumed constant. The Finite Volume Method is used with a hybrid non-staggered grid in a generalized system of coordinates. The SIMPLEC algorithm solves the modified pressure–velocity coupling. The Damköhler number effects on flame dynamics and on the fuel consumption are analyzed. Three stages in the burning processes: the induction time, the flame propagation and the diffusive burning are identified. The merging processes of the fuel pockets and of the flames are depicted.

Circularly Polarized Stacked Patch Antenna With Perpendicular Feed Substrate

A circularly polarized stacked patch antenna with a perpendicular feed substrate is presented. The antenna uses a new type of transition to couple from the patch to the microstrip feed line, and is demonstrated to operate in a scanning linear array at 20 GHz. Coupling from the stacked patch to the perpendicular microstrip feed is through a slot in the ground plane of both the stacked patch and the perpendicular substrate. Performance of the antenna element has been optimized in an infinite linear array configuration. A finite array consisting of eight elements has also been analyzed for various scan angles, built and tested. Computed reflection coefficient, realized gain, axial ratio and radiation patterns are provided for the linear array element, and measured performance presented for an eight-element linear array prototype.

Syntheses, structures, and properties of zinc(II), cadmium(II), cobalt(II), and manganese(II) coordination polymers with tetraiodoterephthalate

The coordination chemistry of a rigid periodinated ligand, 2,3,5,6-tetraiodo-1,4-benzenedicarboxylic acid (H2BDC-I4), with a series of transition metal ions has been explored to afford five new coordination polymers {[M(BDC-I4)(MeOH)4](H2BDC-I4)(MeOH)2}n (M = ZnII for 1, CdII for 2, CoII for 3 and MnII for 4) and {[Mn(BDC-I4)(MeOH)4](DMF)}n (5). All these complexes have been characterized by elemental analysis, IR spectroscopy, thermogravimetric (TG) analysis, and X-ray crystallography. Single-crystal X-ray diffraction reveals that complexes 1–4 are isostructural and have a one-dimensional chain structure. Upon the addition of the solvent DMF, the infinite linear chain array in 4 is converted to a 1-D wave-like chain motif in 5 with a different space group (\( P\overline{1} \) for 4 and P21/c for 5). The difference between structures 1–4 and 5 can be attributed to the coordination mode of carboxylate changing from trans to cis fashion. The ZnII and CdII complexes 1 and 2 display similar emissions in the solid state, which essentially are intraligand transitions.

Traveling waves in two parallel infinite linear point-scatterer arrays

Traveling waves in two coupled parallel infinite linear point-scatterer arrays are studied analytically for the first time to our knowledge. The two arrays are considered to be generally offset in the axial direction. It is found that slow quasi-even/odd supermodes are supported, as a result of the coupling-induced splitting of the modes of the single array, in direct analogy to standard optical waveguide couplers. Exactly even/odd supermodes are supported when the axial offset is zero. Mode splitting, dispersion curves, and coupling length are numerically investigated versus the inter-element spacing, the inter-array distance, and the axial offset. Potential applications of the concept are in directional optical couplers made of metallic or dielectric nanoparticle chains.

Nonunitary Lewis Number Effects on the Combustion of a Linear Array of Gaseous Fuel Pockets

The numerical solution for the combustion of an infinite linear array of gaseous fuel pockets in a stagnant oxidizing environment under microgravity conditions is discussed. The gas pocket combustion is described using the generalized Shvab-Zel'dovich formulation with nonunitary Lewis number. The combustion process is considered isobaric and the flow is induced by density gradients due to the heat and mass transfer processes (Stefan flow). The model is based on mass, momentum, excess enthalpy, and mixture fraction conservation equations and considers the Burke-Schumann reaction mechanism and ideal gas behavior. The thermophysical properties, except the density, are assumed constant. The finite-volume method is employed in the numerical solution, using a generalized system of coordinates. A nonstaggered grid is used and the SIMPLEC algorithm is employed to solve the modified pressure-velocity coupling. Nonunitary Lewis number and interaction effects on flame behavior and on the fuel consumption are analyzed. Results show that the nonunitary Lewis number can modify the interaction effects on gas pocket linear array combustion. During the combustion process, the flame can evolve from individual flames around each gas fuel pocket to a merged flame, surrounding the merged fuel region. However, under certain conditions, the merged flame and the merged fuel region can be broken, returning to individual flames around each gas fuel pocket.

Pseudopotentials: A Simplified Model for Certain Types of Array Elements

The results obtained by modeling an infinite linear array of electrically short dipoles by an array of pseudopotentials (the pseudopotential is a quantum-mechanical analog of the dipole) can be used for the analysis of finite linear dipole arrays. This idea is utilized to investigate properties of finite arrays such as resonances, grating lobes, and end effects. For a wide range of parameter values, it is found that the pseudopotentials can quantitatively describe all aforementioned properties of the actual finite array. Certain extentions to waveguide arrays are discussed.

Interaction Effects During Combustion of Linear Arrays of Gaseous Fuel Pockets

The numerical solution for the combustion of an infinite linear array of dense fuel gas pockets in a quiescent ambient is discussed in the present work. Fuel mass, flame, and gas pocket shape behaviors are analyzed in order to quantify the interference effects. The combustion process is considered isobaric. The model is based on mass, momentum, energy, and species conservation equations and considers the simple chemical reaction mechanism and ideal gas behavior. Coupling potentials are used in the solution of the energy and species conservations equations. The thermophysical properties, except the density, are assumed constant. The finite-volume method is used for the numerical solution, using a generalized system of coordinates and a nonstaggered grid. The SIMPLEC algorithm is applied to solve the modified pressure-velocity coupling. The flame and gas pocket morphological evolutions as well as the transient fuel consumption and the flame position behaviors are presented. The effects of the initial fuel and oxidant temperatures, of the interpocket distance, of the stoichiometric condition, of the heat of reaction, and of the mixture thermophysical properties on the combustion process are considered. Results show that the interactive effects on gas pocket linear array combustion are relevant only for small interpocket distances. Results also show flame shape temporal evolutions that indicate an oscillatory behavior dependent on the gas pocket burning conditions.

Transient gas phase interaction effects during droplet-stream combustion

Gas-phase transient effects during the combustion of an infinite linear array of droplets are studied in the limit of infinite Damkoler number and in the absence of surrounding convective effects. The present study stems from the quantification of truncation-distance effects on the droplet mass vaporization rate, flame position and flame shape in quasi-steady numerical simulations. The solution domain is initially split into near and far-field subdomains. Within each subdomain, analytical grid generation techniques are applied allowing the development of appropriate finite-difference expressions, the control of grid point distribution and the treatment of outflow boundary conditions. The interdroplet distance effect is studied and results for the droplet mass vaporization rate and flame behavior are presented. The results show the existence of different flame regimes ranging from isolated to merged flame conditions and similar vaporization rate evolution for different interdroplet distances.

Subharmonic resonance of a trapped wave near a vertical cylinder by narrow-banded random incident waves

It is well-known that near an infinite linear array of periodically spaced cylinders trapped waves of certain eigenfrequencies can exist. If there are only a finite number of cylinders in an infinite sea, trapping is imperfect. Simple harmonic incident waves can excite a nearly trapped wave at one of the eigen frequencies through a linear mechanism. However, the maximum amplification ratio increases monotonically with the number of the cylinders; hence the solution is singular in the limit of infinitely many cylinders. Recently, a nonlinear theory of subharmonic resonance of perfectly trapped waves has been completed. In this article the theory is further extended to random incident waves with a narrow spectrum centered near twice the natural frequency of the trapped wave. The effects of detuning and bandwidth of the spectrum are examined.

Efficient computation of the 3D Green’s function for the Helmholtz operator for a linear array of point sources using the Ewald method

The Ewald method is applied to accelerate the evaluation of the Green’s function (GF) of an infinite equispaced linear array of point sources with linear phasing. Only a few terms are needed to evaluate Ewald sums, which are cast in terms of error functions and exponential integrals, to high accuracy. It is shown analytically that the choice of the standard “optimal” Ewald splitting parameter E 0 causes overflow errors at high frequencies (period large compared to the wavelength), and convergence rates are analyzed. A recipe for selecting the Ewald splitting parameter is provided.

Two-Term Theory for Infinite Linear Array and Application to Study of Resonances

The "two-term theory" is an approximate solution to the integral equations for the currents on arrays of parallel, electrically short cylindrical dipoles. For finite arrays, this theory is concisely summarized in a recent book. The present paper draws upon that summary to extend the two-term theory to uniform linear arrays with an infinite number of elements. Certain desirable features of the formulas thus developed are pointed out. The formulas are studied in detail with respect to the phenomenon of resonances, and representative numerical results illustrating resonances and associated phenomena are presented; these studies parallel previous ones for circular arrays. Similarities and differences between circular and infinite arrays are discussed.

Double-Stranded Metal−Organic Networks for One-Dimensional Mixed Valence Coordination Polymers

The design of new types of metal-organic networks and the search for unusual crystal architecture represents an important task for modern inorganic and materials chemistry research. A group of new monosubstituted phenylcyanoximes, containing F, Cl, and Br atoms at the 2, 3, or 4 positions, were synthesized using the high yield nitrosation reaction with CH3-ONO and were spectroscopically (1H NMR, 13C NMR, UV-visible, IR, mass spectrometry) and structurally characterized. Results of X-ray analysis revealed nonplanar trans-anti geometry for 2-chlorophenyl(oximino)acetonitrile, H(2Cl-PhCO); a nonplanar anti configuration for 4-chlorophenyl(oximino)acetonitrile, H(4Cl-PhCO); and planar cis-syn geometry for 3-fluorophenyl(oximino)acetonitrile, H(3F-PhCO). All arylcyanoximes undergo deprotonation in solutions with the formation of colored anions exhibiting pronounced negative solvatochromism in a series of polar protic and aprotic solvents. Nine thallium(I) cyanoximates were obtained using the reaction between hot (approximately 95 degrees C) aqueous solutions of Tl2CO3 and solid powdery monohalogenated arylcyanoximes HL. Crystal structures of two Tl(I) cyanoximates [Tl(2Cl-PhCO) and Tl(4Br-PhCO)] contained centrosymmetric dimeric units (TlL)2 that are connected to a coordination polymer by means of an oxygen atom of the oxime group of the neighboring molecule. Cyanoxime anions act as bridging ligands in both structures where the polymeric motif consists of double-stranded Tl-O chains interconnected with the formation of zigzagging Tl2O2 planar rhombes. Thallium atoms form infinite linear arrays with close intermetallic separations. The nearest Tl(I)...Tl(I) distances are 3.838 and 4.058 angstroms in the Tl(2Cl-PhCO) and Tl(4Br-PhCO) structures, respectively, close to that in metallic thallium (3.456 angstroms). Monosubstituted phenyl groups are well aligned in pi-stacking columns that are perpendicular to the array of Tl(I) atoms and stabilize formed structures. Coordination polyhedrons of thallium(I) in these complexes represent distorted trigonal pyramids with stereoactive lone pair.

Traveling Electromagnetic Waves on Linear Periodic Arrays of Lossless Penetrable Spheres

Traveling electromagnetic waves on infinite linear periodic arrays of lossless penetrable spheres can be conveniently analyzed using the source scattering-matrix framework and vector spherical wave functions. It is assumed that either the spheres are sufficiently small, or the frequency such, that the sphere scattering can be treated using only electric and magnetic dipole vector spherical waves, the electric and magnetic dipoles being orthogonal to each other and to the array axis. The analysis simplifies because there is no cross-coupling of the modes in the scattering matrix equations. However, the electric and magnetic dipoles in the array are coupled through the fields scattered by the spheres. The assumption that a dipolar traveling wave along the array axis can be supported by the array of spheres yields a pair of equations for determining the traveling wave propagation constant as a function of the sphere size, inter-sphere separation distance, the sphere permittivity and permeability, and the free-space wave number. These equations are obtained by equating the electric (magnetic) field incident on any sphere of the array with the sum of the electric (magnetic) fields scattered from all the other spheres in the array. Both equations include a parameter equal to the ratio of the unknown normalized coefficients of the electric and magnetic dipole fields. By eliminating this parameter between the two equations, a single transcendental equation is obtained that can be easily solved numerically for the traveling wave propagation constant. Plots of the k - β diagram for different types and sizes of spheres are shown. Interestingly, for certain spheres and separations it is possible to have multiple traveling waves supported by the array. Backward traveling waves are also shown to exist in narrow frequency bands for arrays of spheres with suitable permittivity and permeability.

Travelling electromagnetic waves on linear periodic arrays of lossless spheres

Source scattering-matrix analyses of travelling waves on infinite linear periodic arrays of lossless spheres and of small electric wires are applied to determine the propagation constants of dipolar travelling waves on an infinite linear periodic chain of metallic nanospheres.

On the receiving cross section of an antenna in infinite linear and planar arrays

A rigorous derivation is given for the receiving cross section of an antenna located in infinite linear and planar arrays. For linear arrays the result involves the active input impedance and the azimuthal directivity of the array. For planar arrays it depends on the active input impedance and on geometrical parameters of the array. The expression of the receiving cross section of an element can be used to derive the generally accepted expression for the active element pattern without, however, making use of intuitive arguments. Results are first presented for conditions without grating lobes; they are then generalized to conditions where grating lobes enter the visible space.

(E)-1-(2-Hydroxy-4-methoxyphenyl)-2-(3,4,5-trimethoxyphenyl)ethene.

In the crystal structure of the title compound, C(18)H(20)O(5), all geometric parameters fall within experimental error of the expected values. Analysis of the molecular-packing plots reveals an infinite one-dimensional linear array running parallel to the c axis, formed by an O-H...O intermolecular hydrogen-bonding interaction. The stilbene framework and most of the substituents are approximately coplanar.