Finite Expansion Research Articles

Compact 4-bit all optical digital to analog converter based on photonic crystal ring resonators

A compact all optical 4-bit digital-to-analog converter (DAC) based on 2D photonic crystals is proposed to operate in the III transmission window 1550 nm of optical communication systems. The DAC consists of a photonic crystal ring resonator whose operation is based on the nonlinear Kerr effect. The overall operation of the proposed DAC is analyzed using the finite difference time domain method and plane wave expansion method. The total footprint of the proposed structure is 1150 µm2 with silicon rods of refractive index equal to 3.4641 placed in an air substrate with a 2D square lattice, exhibiting an insertion loss of −11 dB and absolute error of 2.79% which is suitable for photonic integrated circuits.

Elastic Wave Propagation in a Bio-Inspired Phononic Crystal

The wave propagation in a two-dimensional bio-inspired phononic crystal (PC) is analysed. When composite materials and structures consist of two or more different materials periodically, there will be stop band characteristic, in which there are no mechanical propagating waves. These periodic structures are known as PCs. PCs have shown an excellent potential in many disciplines of science and technology in the last decade. They have generated lots of interests due to their ability to manipulate mechanical waves like sound waves and thermal properties which are not available in nature. The physical properties of PCs are not essentially determined by chemical elements and bonds in the materials, but rather on the internal specific structures. Structures of this type have the ability to inhibit the propagation of vibrational energy over certain ranges of frequencies forming band gaps. The main purpose of this study is to investigate the band structure and especially the location and width of band gaps. For this analysis, it is used the finite element method (FEM) and plane wave expansion (PWE). The results are shown in the form of band structure and wave modes. Band structures calculated by FEM and PWE present good agreement. We suggest that the bio-inspired PC considered should be feasible for elastic vibration control.

Kinematical and dynamical aspects of ghost-matter cosmologies

We consider the kinematical and dynamical evolution of Friedmann Universes with a mixture of non-interacting matter and a ghost-like field, in a scenario analogous to that advocated by the Quintom model. Assuming that the conventional matter dominates today, we find that the ghost component can bring the future expansion and the past contraction of the model to a finite halt. Moreover, at the moment the expansion or contraction stops, we find that the tendency of the Universe is to bounce back and re-collapse or re-expand. Therefore, the presence of a (never dominant) ghost-field with negative density could, in principle, drive the Universe into an eternal cycle of finite expansion, collapse, and re-expansion. Our study outlines the key features of such a scenario and provides a simple condition for it to occur. We also derive an autonomous set of differential equations and employ dynamical-system techniques to identify two families of fixed points, with and without spatial curvature respectively. The members of the first family correspond to coasting Universes and are stable in the Lyapunov sense. Those of the latter family are unstable repellers when their matter satisfies the strong energy condition and Lyapunov stable in the opposite case.

Predicting 3D Motions from Single-Camera Optical Test Data

In a typical optical test, a stereo camera pair is required to measure the three-dimensional motion of a test article; one camera typically only measures motions in the image plane of the camera, and measurements in the out-of-plane direction are missing. Finite element expansion techniques provide a path to estimate responses from a test at unmeasured degrees of freedom. Treating the case of a single camera as a measurement with unmeasured degrees of freedom, a finite element model is used to expand to the missing third dimension of the image data, allowing a full-field, three-dimensional measurement to be obtained from a set of images from a single camera. The key to this technique relies on the mapping of finite element deformations to image deformations, creating a set of mode shape images that are used to filter the response in the image into modal responses. These modal responses are then applied to the finite element model to estimate physical responses at all finite element model degrees of freedom. The mapping from finite element model to image is achieved using synthetic images produced by a rendering software. The technique is applied first to a synthetic deformation image, and then is validated using an experimental set of images.

A comparative study of the delta-Eddington and Galerkin quadrature methods for highly forward scattering of photons in random media

A versatile and accurate treatment for the highly forward-peaked phase function in the three-dimensional (3D) radiative transfer equation (RTE) based on the discrete ordinates method (DOM) is crucial for biomedical optics. Our first objective was to compare the delta-Eddington (dE) and Galerkin quadrature (GQ) methods. The dE method decomposes the phase function into a purely forward-peaked component and the other component, and expands the other component by Legendre polynomials as well as the finite order Legendre expansion (FL) method does. The GQ method conducts the weighting procedure in addition to the Legendre expansion. Although it was reported that both methods can provide the accurate results for calculations of the RTE, the versatility of both methods is still unclear. The second objective was to examine a possibility of a conjunction of the GQ method with the dE method, called as the GQ-dE method, which has the advantages of both methods. We examined numerical errors in the moment conditions of the phase function using the FL, dE, GQ, and GQ-dE methods at various types and orders of the quadrature sets, mainly in the region of the errors induced by the angular discretization using the DOM. The errors were reduced by the dE method from those by the FL method, however the error reduction depended on the types and orders of the quadrature sets. Meanwhile, the errors were significantly reduced by the GQ and GQ-dE methods, regardless of the quadrature sets. We also verified the numerical calculations of the time-dependent 3D RTE by the analytical solution of the RTE for homogeneous media in the region of the scattering length scale, where the highly forward-peaked phase function strongly influences the RTE-results. The errors in the RTE-results were similar to those in the moment conditions. Our results suggest the higher versatility and accuracy of the GQ and GQ-dE methods than those of the FL and dE methods.

The Effect of Barodiffusion on the Distribution of Biological Fluid Velocity and Concentration During Filtration Through a Cylindrical Layer

A model of filtration of a biological fluid for a cylindrical layer in a stationary regime is presented. The model takes into consideration finite compressibility and concentration expansion associated with barodiffusion. Dimensionless complexes relating characteristic physical scales of different phenomena are singled out. The influence of the revealed effects on the flow characteristics are studied numerically. The flow can be either convective or diffusive, depending on the relation between the dimensionless complexes. New qualitative relationships are distinguished in the component concentration and flow velocity distributions. A significant influence of the thickness of the porous cylinder wall is shown. It is found out that barodiffusion has a significant effect on the flow in the convective mode and a small cylinder wall thickness.

Locally polynomially integrable surfaces and finite stationary phase expansions

Let $M$ be a strictly convex smooth connected hypersurface in $\mathbb R^n$ and $\widehat{M}$ its convex hull. We say that $M$ is locally polynomially integrable if the $(n-1)-$ dimensional volumes of the sections of $\widehat M$ by hyperplanes, sufficiently close to the tangent hyperplanes to $M,$ depend polynomially on the distance of the hyperplanes to the origin. It is conjectured that only quadrics in odd dimensional spaces possess such a property. The main result of this article partially confirms the conjecture. The study of integrable domains and surfaces is motivated by a conjecture of V.I. Arnold about algebraically integrable domains. The result and the proof are related to study oscillating integrals for which the asymptotic stationary phase expansions consist of finite number of terms.

Singularity modal analysis of frictional interfaces in orthotropic composites

Shear crack or sliding contact within or between orthotropic composites contains innovative local solutions in terms of stress singularity, which are concentrated in this study. For this purpose, a finite element asymptotic expansion is proposed, which extracts the singular point solution for different frictional crack or contact problems in orthotropic composite design.The investigations illustrate the fact that many situations in orthotropic fracture and contact mechanics deal with stress singularities of order higher than the conventional cracks in isotropic materials and even the free crack conditions in the same orthotropic cases.In this paper, the significance of different situations for shear crack and various configuration of frictional contact between orthotropic composites are discussed in terms of the order of singularity. The consistent singular enrichment function derived may be utilized in numerical simulations of fracture and contact phenomena. Direct eXtended Finite Element Method (Direct-XFEM) proposed facilitates an efficient algorithm for implementation of the singularity modal analysis results.

Quadratic axial expansion function with sub-plane acceleration scheme for the high-fidelity transport code PROTEUS-MOC

PROTEUS-MOC is a pin-resolved high-fidelity transport code that employs the method of characteristics to discretize the radial variables and the discontinuous finite element method to discretize the axial variable. In this study, we extended the hybrid transport solution method to axial quadratic trial functions from the current linear functions. Compared with the linear approximation, the quadratic approximation reduces the memory requirement by 50% by allowing coarser axial meshes, but the coarser axial meshes deteriorate the performance of the CMFD and pCMFD acceleration schemes. To remedy the deteriorated performance of the acceleration schemes, a sub-plane acceleration method was implemented. The sub-plane acceleration scheme can improve the computational performance of the quadratic approximation in conjunction with the GMRES method. It was also observed that for a fixed radial mesh configuration, the flat leakage source approximation error increases with axial mesh refinement while the truncation error of the finite functional expansion decreases.

Optical soliton solutions to a (2+1) dimensional Schrödinger equation using a couple of integration architectures

Abstract In this work, we consider a (2+1) dimensional nonlinear Schrödinger system which appears in the theory of nonlinear optics and describe transmission of the optical pulses in optical fibers. We attain certain special type traveling wave solutions of the under investigated model by help of finite series expansion and auxiliary differential equations. In this manner, we exploit exp(−ϕ(ε)) and modified Kudryashov approaches as solution procedures. Moreover, we make tanh ansatz because of the being even order of the reduced ordinary differential equation. The obtained solutions are in the form of dark soliton, combined soliton, symmetrical Lucas sine, Lucas cosine functions, and periodic wave solutions. We present also some graphical simulations of the solutions corresponding to values of parameters which leads to a better understanding the phenomena.

Perturbative Algebraic Quantum Field Theory on Quantum Spacetime: Adiabatic and Ultraviolet Convergence

The quantum structure of Spacetime at the Planck scale suggests the use, in defining interactions between fields, of the Quantum Wick product. The resulting theory is ultraviolet finite, but subject to an adiabatic cutoff in time which seems difficult to remove. We solve this problem here by another strategy: the fields at a point in the interaction Lagrangian are replaced by the fields at a quantum point, described by an optimally localized state on QST; the resulting Lagrangian density agrees with the previous one after spacetime integration, but gives rise to a different interaction hamiltonian. But now the methods of perturbative Algebraic Quantum Field Theory can be applied, and produce an ultraviolet finite perturbation expansion of the interacting observables. If the obtained theory is tested in an equilibrium state at finite temperature the adiabatic cutoff in time becomes immaterial, namely it has no effect on the correlation function at any order in perturbation theory. Moreover, the interacting vacuum state can be obtained in the vanishing temperature limit. It is nevertheless important to stress that the use of states which are optimally localized for a given observer brakes Lorentz invariance at the very beginning.

Optical sensor based on two-dimensional photonic crystals for measuring glucose in urine

The design and simulation of two bio-optical sensors for the noninvasive measurement of glucose levels in urine of diabetic patients are presented. While the sensors are of the same shape and structure, their quality factors (QFs), sensitivities, and sizes are different. The structure developed is based on two-dimensional (2D)-hexagonal-latticed photonic crystals containing silicon rods immersed in urine. By creating line and point defects in the structure, two resonant modes are localized in the photonic bandgap (the photonic bandgap is a frequency range corresponding to mirror action for incident light of frequency within that range). Sensor 1 design was based on localized resonant mode 1, while sensor 2 uses localized resonant mode 2. The mechanism of these sensors relies on the change of the refractive index of urine as a result of the change in glucose concentration, which in turn causes shifts in resonant modes of the sensors. For tuning in the wavelength of 1.55 μm, the scaling method is utilized. The major difference between the proposed sensors and previously reported bio-optical sensors is the determination of glucose level based on the shift in resonant wavelengths. Previous works relied on the change of the amplitude of emerging waves for the same purpose. Consequently, the sensors presented here exhibit a higher QF and operate within the wavelength window of 1.55 μm. QF calculations and sensitivity simulations for sensor 1 yielded approximate values of 3800 and 500 nm/refractive index unit (RIU), respectively. For sensor 2, QF and sensitivity were 5540 and 500 nm / RIU, respectively. The sizes of sensor 1 and sensor 2 are 14.93 × 8.8 μm2 and 15.7 × 9.47 μm2, respectively. The 2D finite difference time domain and plane wave expansion methods were relied upon in the simulations.

Perceptible exciton-to-trion conversion and signature of defect mediated vibronic modes and spin relaxation in nanoscale WS2 exposed to γ-rays

In this work, we report manifested optical, optoelectronic and spin–spin relaxation features of a few layered tungsten disulphide (WS2) nanosheets subjected to energetic γ-photons (∼1.3 MeV) emitted from a Co60 source. Upon intense irradiation (dose = 96 kGy), a slight departure from the pure hexagonal phase was realized with the introduction of the trigonal phase at large. Moreover, in the Raman spectra, as a consequence of the radiation-induced effect, an apparent improvement of the E-to-A mode intensity and a reduction in phonon lifetimes have been realized, with the latter being dependent on the linewidths. The emergence of the new peak (D) maxima observable at ∼406 cm−1 in the Raman spectra and ∼680 nm in the photoluminescence (PL) spectra can be attributed to the introduction of defect centres owing to realization of sulphur vacancies (V S) in the irradiated nanoscale WS2. Additionally, neutral exciton to charged exciton (trion) conversion is anticipated in the overall PL characteristics. The PL decay dynamics, while following bi-exponential trends, have revealed ample improvement in both the fast parameter (0.39 ± 0.01 ns to 1.88 ± 0.03 ns) and the slow parameter (2.36 ± 0.03 ns to 12.1 ± 0.4 ns) after γ-impact. We attribute this to the finite band gap expansion and the incorporation of new localized states within the gap, respectively. A declining exciton annihilation rate is also witnessed. The isotropic nature of the electron paramagnetic resonance spectra as a consequence of γ-exposure would essentially characterize a uniform distribution of the paramagnetic species in the system, while predicting a three-fold improvement of relative spin density at 96 kGy. Exploring defect dynamics and spin dynamics in 2D nanoscale systems does not only strengthen fundamental insight but can also offer ample scope for designing suitable components in the areas of miniaturized optoelectronic and spintronic devices.

Absence of a Vainshtein radius in torsion bigravity

It was pointed out long ago by Vainshtein [Phys. Lett. 39B, 393 (1972)] that the weak-field perturbation expansion of generic theories (of the nonlinear Fierz-Pauli type) involving massive spin-2 excitations breaks down below a certain distance around a material source ("Vainshtein radius"), scaling as some inverse power of the spin-2 mass $m_2$, i.e., some positive power of the range $m_2^{-1}$. Here we prove that this conclusion does not apply in a generalized Einstein-Cartan theory (called "torsion bigravity") whose spectrum is made (like that of bimetric gravity) of a massless spin-2 excitation and a massive spin-2 one. Working within a static spherically symmetric ansatz, we prove, by reformulating the field equations in terms of new variables, that one can construct an all-order weak-field perturbative expansion where no denominators involving $m_2$ ever appear in the region $r \ll m_2^{-1}$. In particular, we show how the formal large-range limit, $m_2 \to 0$, leads to a well-defined, finite perturbation expansion, whose all-order structure is discussed in some detail.

Highly accurate technique for solving distributed-order time-fractional-sub-diffusion equations of fourth order

This paper presents a new method for calculating the numerical solution of distributed-order time-fractional-sub-diffusion equations (DO-TFSDE) of fourth order. The method extends the shifted fractional Jacobi (SFJ) collocation scheme for discretizing both the time and space variables. The approximate solution is expressed as a finite expansion of SFJ polynomials whose derivatives are evaluated at the SFJ quadrature points. The process yields a system of algebraic equations that are solved analytically. The new method is compared with alternative numerical algorithms when solving different types of DO-TFSDE. The results show that the proposed method exhibits superior accuracy with an exponential convergence rate.

Quasi-Herglotz functions and convex optimization.

We introduce the set of quasi-Herglotz functions and demonstrate that it has properties useful in the modelling of non-passive systems. The linear space of quasi-Herglotz functions constitutes a natural extension of the convex cone of Herglotz functions. It consists of differences of Herglotz functions and we show that several of the important properties and modelling perspectives are inherited by the new set of quasi-Herglotz functions. In particular, this applies to their integral representations, the associated integral identities or sum rules (with adequate additional assumptions), their boundary values on the real axis and the associated approximation theory. Numerical examples are included to demonstrate the modelling of a non-passive gain medium formulated as a convex optimization problem, where the generating measure is modelled by using a finite expansion of B-splines and point masses.

Design of a highly sensitive photonic crystal refractive index sensor incorporating ring-shaped GaAs cavity

Two highly sensitive optical sensor topologies are proposed and simulated in this paper. The proposed structures are optimized to provide better performance characteristics such as sensitivity, detection limit, and quality factor. They are based on two-dimensional photonic crystals consisting of rectangular arrays of GaAs rods in SiO 2 substrates. Such lattices have bandgaps for transverse magnetic modes. Two-dimensional finite difference time domain and plane wave expansion methods are used for the simulation and analysis of the refractive index sensors and particle swarm optimization method is used to optimize the structural parameters. The designed structures show a high sensitivity to refractive index variations. They are able to detect refractive indices from 1.33 to 1.5. An excellent figure of merit equal to 737 RIU −1 is observed for the proposed structure and a significant improvement is observed compared to the structures reported in the literature.

Finite TYCZ expansions and cscK metrics

Let (M,g) be a Kähler manifold whose associated Kähler form ω is integral and let (L,h)→(M,ω) be a quantization hermitian line bundle. In this paper we study those Kähler manifolds (M,g) admitting a finite TYCZ expansion, namely those for which the associated Kempf distortion function Tmg is of the form:Tmg(p)=fs(p)ms+fs−1(p)ms−1+⋯+fr(p)mr,fj∈C∞(M),s,r∈Z. We show that if the TYCZ expansion is finite then Tmg is indeed a polynomial in m of degree n, n=dimC⁡M, and the log-term of the Szegö kernel of the disc bundle D⊂L⁎ vanishes (where L⁎ is the dual bundle of L). Moreover, we provide a complete classification of the Kähler manifolds admitting finite TYCZ expansion either when M is a complex curve or when M is a complex surface with a cscK metric which admits a radial Kähler potential.

Slow light enhanced bio sensing properties of silicon sensors

In this paper, we investigated slow light to increase sensitivity and quality factor of our recently designed bio-sensors. A finite difference time domain and Plane Wave Expansion methods are employed to figure out sensors’ performances. We proposed slow light based photonic crystal structures with elliptical cells and we obtained higher sensitivity and quality factor as previously reported. For the first proposed sensor, the results showed that quality factor increased from 570 to 723 and a detection limit below 10−3. For the second sensor, the sensitivity increased from 550 to 650 nm/RIU and the detection limit is around 10−3. These two proposed structures would offer great potential for novel compact biosensors in medical application.

The hysteretic behavior of recycled aggregate concrete-filled square steel tube columns

The research purpose was to explore the hysteretic behavior of recycled aggregate concrete-filled square steel tube (S-RACFST) columns. With the replacement ratio of recycled coarse aggregate and axial compression ratio as the variation parameters, 6 S-RACFST columns were carried out for the hysteretic behavior test, and the hysteretic behavior indexes such as failure mode, hysteretic curve and backbone curve were investigated through the test. Based on the experimental research, a feasible finite element analysis model was established. Meanwhile, the finite element expansion analysis of 32 full-scale specimens was carried out to reveal the influence of the design parameters on the hysteretic behavior indexes. It is shown that the damage of the specimen is mainly concentrated at the column roots, the square steel tubes bulge outwards in a “lantern” shape, and the interior recycled concrete has a large area of crushing and spalling. The hysteretic curves of all the test specimens are full and stable, the backbone curves are relatively complete with a smooth descending segment. The hysteretic behavior indexes of all full-scale specimens are less affected by the replacement ratio of recycled coarse aggregate, while the parameters such as steel ratio, aspect ratio and axial compression ratio have an enormous impact on the hysteretic behavior indexes such as peak load-bearing capacity, ductility and energy dissipation.