Method In Cylindrical Coordinates Research Articles

Dispersion characteristics of arc-axis waveguides

Ultrasonic single mode excitation methods, mode separation algorithms and damage detection applications all require the calculation of the dispersion characteristics of the waveguide. However, existing dispersion calculation methods are mainly applicable to straight-axis waveguides. Therefore, in this paper, a novel semi-analytical finite element method in cylindrical coordinates (SAFEM-CC) is established and we focus on the dispersion characteristics of arc-axis waveguides. The cross-section of the waveguide can have arbitrary complex shapes, and the correctness of this method is verified by the finite element eigenfrequency method. At the same time, this paper takes the example of an arc-axis structure with a square cross-section and investigates the convergence of SAFEM-CC under different numbers of elements, element types, and element orders. This paper also establishes a finite element simulation model to study the waveform transmission law over a wide frequency range and the wave amplitude distribution law within the cross-section of arc-axis waveguides. The dispersion characteristics of arc-axis waveguides at different radii are also investigated. Finally, the correctness of the proposed method and the dispersion characteristics of arc-axis waveguides are verified by a large number of experiments. The main conclusions are as follows: the waveform transmission laws of the guided wave in arc-axis waveguides at each frequency agree with the dispersion curve; the theoretical wavestructure gives the distribution law of the amplitude within the cross-section; when the axis is arc, the displacement of the wavestructure on the inner and outer sides does not have symmetry, and the difference between the left and right sides gradually increases as the radius decreases.

Optical radiation force and torque of light-sheets on a cylindrical particle near an infinite boundary

This work aims to extend the previous studies on the optical radiation force and torque for a cylindrical dielectric particle to the case near an infinite boundary. Without loss of generality, the two-dimensional cylindrical object is assumed to have an arbitrary shape and be immersed in any light-sheet beam of arbitrary wave front. The partial-wave series expansion method in cylindrical coordinates and the method of images are utilized to derive the exact expressions for the axial and transverse radiation force and torque functions, which are dependent on the beam shape coefficients of the incident light-sheet and the scattering coefficients characterized by the boundary conditions at the surface. Numerical computations are performed for a circular cylindrical particle illuminated by a two-dimensional plane wave and a two-dimensional Gauss beam, respectively, with particular emphases on the size parameter, the refractive index, the particle-boundary distance, the beam waist and the offset shifts. The radiation force will increase in magnitude and reverse its direction under selected conditions. When the absorptive cylindrical particle is shifted off-axially with respect to the beam axis, it will be rotated counterclockwise or clockwise by the radiation torque. The results obtained in this work have potential applications in non-contact particle manipulation and transportation using optical tweezers.

Nanoparticles optimal shape designing in the cylindrical coordinate to use for optical devices

Nanoparticles have unique optical properties that can enhance the performance of optical devices. In this regard size and shape of nanoparticles are very important factors that influence these properties. Although there are many different sample shapes of nanoparticles but researchers have not yet identified the best shape to use, especially in general. This article presents a mathematical model to determine the optimal shape of nanoparticles for use in various optical devices. We use the shape-measure method in cylindrical coordinates to solve this problem. This method has many advantages like it is independent of the initial shape and transforming the optimal shape design problem into a simple optimization one, even in strongly nonlinear cases. The use of cylindrical coordinates also makes calculations faster and more straightforward. First the optical properties of nanoparticles is described by using the discrete dipole method; then, simultaneously the optimal surface and image for the optimal shape and size of the nanoparticles are determined. Numerical simulations are also provided to demonstrate the effectiveness and benefits of the new approach.

Analysis of Wave Force and Run-Up Acting on an Impermeable Vertical Circular Cylinder Surrounded by Multiple Thick Porous Layers

Abstract In this work, we analyze the effect of multiple thick porous layers, fitted around a rigid vertical circular cylinder, on the wave forces acting on the rigid structure. Using the eigenfunction expansion method in cylindrical coordinates, we derive the expressions for velocity potentials in the respective domains and finally calculate the wave force acting on the rigid structure by integrating the pressure term. Consideration of the multiple porous layers, each with different porosities, gives rise to a very basic question to answer: what will be the appropriate arrangement of the porous layers to reduce the wave impact? Hence, for numerical study, we consider three different arrangements of the porous layers. For such arrangements of the porous layers, we also analyze the effects of the other crucial parameters, such as the number and the thickness of the porous layers, on the wave force. The key finding of the analysis is that the wave force acting on the rigid structure can be minimized by increasing the number of porous layers with decreasing porosity from the innermost layer to the outermost layer. The wave breaking and forced oscillations for certain values of the porous impedance parameter are some of the interesting observations. The present model is also verified against an existing work in the literature which shows an excellent agreement.

Elastic full-wave field simulation in 3D tunnel space with the variable staggered-grid finite-difference method in cylindrical coordinates

Tunnel advance detection technology is an important method for determining the structure of complex geological bodies in front of the tunnel face. Among the tunnel advance detection technologies, the seismic method is one of the most accurate methods with long detection distances. In seismic tunnel advance detection, the cylindrical configuration aggravates the complexity of the wave field in the 3D tunnel space and significantly influences the accuracy of the detection results. Thus, it is crucial to simulate an accurate seismic full-wave field of the tunnel space and to understand the propagation and wave-field characteristics of individual seismic waves for seismic tunnel advance detection. Usually, in 3D Cartesian coordinates, the tunnel wall is approximated with a staircase boundary, but it is not sufficiently accurate in shape and generates numerical dispersion in the simulation, especially in the presence of surface waves. Therefore, we developed a variable staggered-grid finite-difference method in cylindrical coordinates to simulate the elastic full-wave field in a 3D tunnel space. Properly setting free-surface boundary conditions solves the propagation of surface waves on the tunnel wall and face. The interference of the instability and discontinuity of the pole axis in the seismic wave field simulation was eliminated using our method. According to practical engineering situations, models containing three types of geological bodies in front of the tunnel face were designed. Their seismic wave propagation and wave field characteristics were analyzed. Compared with the simulation results in Cartesian coordinates, the results in cylindrical coordinates show that numerical dispersion is negligible and conclude that a higher signal-to-noise ratio and more accurate seismic wave field can be simulated with cylindrical coordinates in the 3D tunnel space. Our simulation method provides theoretical and practical guidance for analyzing and interpreting seismic wave fields in tunnel advance detection.

A Novel Structure of an Eddy Current Sensor for Speed Measurement of Rotating Shafts

This paper presents a novel structure of an eddy current sensor for the measurement of rotational speed. The sensor has a perpendicular configuration of the excitation coil and the pick-up coil. Its maximum nonlinearity error is 0.15%. The excitation coils and the pick-up coils are divided into two identical parts with a coil span of 180 degrees. Various solid iron cylinders with different electrical and magnetic characteristics, one solid aluminum cylinder and one solid copper cylinder, were used as the rotating shafts. A novel 2D analytical method in cylindrical coordinates has been developed for fast parametric analysis of this sensor. 2D and 3D time-stepping finite-element methods that take into consideration the shaft motion are used for a performance analysis of the eddy current sensor. The measurements of the speed sensor are performed at various frequencies and speeds and are compared with the calculations for detailed analysis. The effects of the rotating shaft materials are evaluated both in the measurements and in the calculations.

Curved photonic nanojet generated by a rotating cylinder.

The curved photonic nanojet (CPNJ) produced due to the interaction between a dielectric circular cylinder rotating at a stable angular velocity and a plane wave is investigated. Based on this model, the optical Magnus effect of a dielectric circular cylinder is verified. And the analytical expression of both internal and external electric field are given based on the instantaneous rest-frame theory and the partial-wave series expansion method in cylindrical coordinates. The influence of the size parameter, the relative refractive index, and the rotating dimensionless parameter on the CPNJ are analyzed and discussed in numerical results. The "photonic nanojet curved" effect is highlighted, which can be used to generate the off-axis photonic nanojet (PNJ) controlling particles by adjusting the angular velocity of the dielectric cylinder. The results of this manuscript have promising application prospects in optical tweezers, particle manipulation, and optical trapping. Moreover, it also provides theoretical support for the particle spinning and generation of the off-axis CPNJ.

Use of the Alternating Directions Method in Cylindrical Coordinates, Applied to the Two-Dimension Unsteady Conduction Heat Transfer

Abstract Finding analytical solutions for the unsteady conduction heat transfer equation is generally very difficult due to the intrinsic complexity of the problem. In most cases, the equation has no analytical solutions. A numerical approach that surmounts these difficulties consists of the finite difference technique. The alternate directions finite-difference technique especially suits to the two-dimension heat conduction. The paper considers a hollow cylinder consisting of a conduction heat transfer medium, and the two-direction unsteady heat propagation within this medium is studied by means of the alternate directions method. The approach simulates the heat conduction process during a charge-discharge cycle of a single borehole heat storage.

Lagrangian PAFs in multiple optical scattering by two absorptive dielectric parallel cylinders

The objective of this work is to derive semi-analytical integral expressions for the Lagrangian longitudinal (L) and transverse (T) photophoretic asymmetry factors (PAFs) for an aggregate pair of parallel absorptive dielectric cylinders of arbitrary radii in plane waves with arbitrary incidence angles and polarizations. Based on the multiple scattering theory of waves and its rigorous mathematical formalism, the components of the internal electric field vectors in cylindrical coordinates are determined and used subsequently to compute the PAFs. The L- and T-PAFs are directly proportional to the L and T components of the photophoretic (known also as radiometric) force vector, respectively, induced by light absorption inside each dielectric cylinder. The modal expansion method in cylindrical coordinates and adequate boundary matching at the surface of each particle are used to determine the internal coefficients to compute the PAFs. Subsequently, the integral expressions are derived and evaluated assuming TE- and TM-polarized plane waves with arbitrary angles in the polar plane. Additional computations for the dimensionless intensity function are performed, and the corresponding results provide quantitative assessment of the internal heated portions of the absorptive dielectric cylinders at different interparticle distances while illuminated by plane waves with variable incidence angles and polarizations. The results are of some importance in electromagnetic/optical multiple scattering theory and related applications in optical binding, optical tweezers, particle manipulation, and photophoresis.

Optical Magnus radiation force and torque on a dielectric layered cylinder with a spinning absorptive dielectric core.

A dielectric cylindrical shell material with an inner stationary absorptive core (or, equivalently, a concentric layered cylinder) illuminated by an axisymmetric wave field (in 2D) with arbitrary polarization will experience a time-averaged radiation force along the direction of wave propagation, whereas the transverse component vanishes as required by symmetry. Counterintuitively, the present analysis shows that when the inner absorptive core material rotates with an initial angular velocity Ω0 around its main geometrical axis, a transverse radiation force component arises (in addition to a longitudinal force) as well as an axial radiation torque component in plane waves. This phenomenon is the analog of the classical hydrodynamic Magnus effect. In this analysis, the instantaneous rest-frame theory and the modal series expansion method in cylindrical coordinates are utilized to formulate the EM/optical scattering from a cylindrical shell with arbitrary thickness, and to compute the optical radiation force and torque vector components. Particular emphases are given on the size parameter of the cylindrical shell, the layer thickness, the angular rotation of the inner absorptive core, and the polarization (TM or TE) of the incident plane waves. Numerical computations illustrate the analysis and explicate the behaviors of the longitudinal, transverse, and axial radiation force and torque components. Related applications for the optical Magnus effect in radiation force and torque investigations can benefit from the results of the present analysis in spin-optics, rotational Doppler shift for optical waves, optical tweezers, optical manipulation of elongated spinning objects, and particle rotation.

Optical Magnus effect in the photophoresis of a spinning absorptive dielectric circular cylinder.

When a stationary absorptive dielectric cylinder suspended in a gas (such as air) is illuminated by an axisymmetric wave field (such as plane waves), the transverse (T) photophoretic asymmetry factor (PAF) vanishes as required by geometrical symmetry [Appl. Opt.60, 7937 (2021) APOPAI0003-693510.1364/AO.435306]. Counter-intuitively, when the cylinder possesses an initial angular velocity Ω0 with a sufficiently small acceleration and spins around its main axis in the illuminating field of axisymmetric plane waves, it is shown here that the T-PAF (which is directly proportional to the T-photophoretic force vector component) is quantifiable, in analogy with the Magnus effect in hydrodynamics where a force perpendicular to the axis of the cylinder and to the propagation direction arises. Based upon the instantaneous rest-frame theory and the partial-wave series expansion method in cylindrical coordinates, the internal electric field of the spinning absorptive dielectric cylinder is determined and utilized to compute both the longitudinal (L) and T-PAFs. Particular emphases are given on the size parameter of the cylinder, its angular rotation, the light absorption inside its core material and the polarization (TM or TE) of the incident plane waves. The dimensionless intensity function (DIF) is also computed, which reveals quantitative information on the heated portions within the internal absorptive core material of the cylinder. Numerical computations illustrate the analysis and explicate the behaviors of the DIF and the L- and T-PAFs, which predict the emergence of the forward, neutral, and reverse optical/electromagnetic Magnus effect in the photophoresis of an absorptive dielectric cylinder and related applications in spin optics, optical tweezers, optical manipulation of elongated objects, and radiative transfer research.

Effect of a perfectly conducting corner space on the PAFs for an absorptive dielectric circular cylinder

The photophoretic asymmetry factors (PAFs) for an absorptive dielectric circular cylinder, located near a perfectly conducting and totally reflecting corner space are derived and computed. The method used in this analysis relies on the modal expansion method in cylindrical coordinates, the classical method of images, and the translational addition theorem of cylindrical wave functions. Initially, the components of the internal electric field vector are obtained stemming from an analysis of the scattering. Subsequently, the solution is used to integrate to the normalized intensity function over the cylinder’s volume to obtain the longitudinal (L) and transverse (T) PAFs. Both TM- and TE-polarized plane progressive waves with arbitrary incidence (in the polar plane) are considered. Attention is given to varying the dimensionless size parameter of the cylinder, the angle of incidence of the incoming waves, and the dimensionless distance parameters from the corner space. Numerical examples illustrate the analysis and demonstrate the net effect of the totally reflecting corner space on the L- and T-PAFs, where negative, positive, and neutral values have been predicted. The results are relevant in applications related to the emergence of the photophoretic force and torque on an absorptive particle located near surfaces and topics in electromagnetic/optical scattering, particle manipulation and assembly, optically bound matter, light–matter interactions, and photopheresis.

Jacobian-free Newton Krylov nodal expansion method in three-dimensional cylindrical coordinates

A cylindrical version of Jaocbian-free Newton Krylov nodal expansion methods in cylindrical coordinates (CyliNEM_JFNK) are successfully developed to solve the three-dimensional and multigroup k-eigenvalue reactor problems in cylindrical geometry. Unlike nodal methods in Cartesian geometry, CyliNEM_JFNK adopts the special polynomial-expansion approximation and some separate treatments to solve the transverse-integrated equations in each direction and it is not a simple and trivial extension from Cartesian geometry to cylindrical geometry. To reduce the computational cost and the scale of residual systems constructed in the CyliNEM_JFNK as much as possible, the outgoing interface partial currents, node-averaged fluxes, third and fourth expansion coefficients are only chosen as the final-solving variables to obtain the residual functions by eliminating all the other variables including incoming interface partial currents, first and second expansion coefficients, source and leakage moments. Then the efficient physics-based preconditioners are successfully developed by making full use of the efficient Row-Column iteration and original power iteration framework. The cylindrical version of 3D LWR IAEA benchmark problem and a series of three-dimensional high temperature gas-cooled pebble bed reactor models are chosen to verify the numerical accuracy and computational efficiency of the CyliNEM_JFNK method. Numerical solutions show that the proposed CyliNEM_JFNK method can track the reference solutions very well and obtain higher computational efficiency than the CyliNEM method with efficient Row-Column iteration in three-dimensional cylindrical coordinates, which indicates the advantages of the developed CyliNEM_JFNK method that can serve as a potential nodal diffusion analysis tool for three-dimensional cylindrical advanced reactor problems.

Longitudinal and transverse PAFs for an absorptive magneto-dielectric circular cylinder in light-sheets of arbitrary wavefronts and polarization.

Based upon the expression of the heat source function in photophoresis, generalized mathematical expressions for the longitudinal (L) and transverse (T) photophoretic asymmetry factors (PAFs) for a light-absorptive magneto-dielectric circular cylinder of arbitrary relative permittivity and permeability, illuminated by an arbitrarily shaped polarized light-sheet, are derived and computed. The L- and T-PAFs are directly proportional to the L and T components of the photophoretic force vector, respectively, induced by light absorption inside the particle, and their sign predicts the behavior of the force (pulling/attractive or pushing/repulsive). The partial-wave series expansion method in cylindrical coordinates is used, and the obtained mathematical expressions for the L- and T-PAFs depend on the beam-shape coefficients and the internal coefficients of the cylinder. Numerical examples illustrate the theory for TE and TM polarized plane waves, and nonparaxial Airy light-sheets with particular emphasis on absorption inside the cylinder and varying the Airy light-sheet parameters. The generalized expressions presented here are applicable to any light-sheet of an arbitrary wavefront, and offer additional quantitative observables for the analysis of the photophoretic force in applications in electromagnetic scattering, optical light-sheet tweezers, particle manipulation, radiative transfer, and other research fields.

A Perfectly-Matched Layer Formulation Capable of Absorbing Multi-Modal Waves in Cylindrical FDTD

The perfectly-matched-layer (PML) absorbing boundary condition as applied to the finite-difference time-domain method in cylindrical coordinates is investigated. A critical error in implementing coordinate stretching which is present in various published works is identified and shown to diminish wave absorption efficiency for all wave modes except rotationally invariant ones. A correction to the underlying coordinate stretching formulation is proposed and confirmed to improve wave absorption by approximately two orders of magnitude for higher cylindrical wave modes even when the PML absorber is placed within the near-field region to the axis of rotation.

Longitudinal and lateral interparticle optical binding and extrinsic radiation force and torque on a pair of lossless dielectric cylinders of arbitrary sizes and the acoustical analogue

The purpose of this work is to develop an exact formalism and analytical equations for the interparticle electromagnetic/optical (longitudinal and lateral) binding forces arising between two lossless dielectric cylinders with different permittivities in a TE-polarized plane wave-field with an arbitrary incidence angle (in the polar plane). Additionally, the extrinsic/external optical radiation force and torque acting on the aggregate (i.e. the cluster of particle pair) are determined, and their exact mathematical expressions are derived. Based on the modal series expansion method in cylindrical coordinates in conjunction with the translational addition theorem, numerical simulations and computations for the longitudinal and lateral components of the interparticle and extrinsic forces as well as the axial component of the extrinsic torque show that attraction/repulsion, rotation clockwise or counter-clockwise, or neutrality, arise depending on the interparticle distance, the incidence angle, particle sizes and their dielectric permittivities. This investigation offers a complete analytical formalism without any approximations related to particle sizes. The numerical results computed based on exact mathematical expressions for interparticle and extrinsic physical observables can be used in the design and optimization of optical tweezers and related applications in optically-bound matter, particle manipulation, cloaking, and optically-assembled metamaterials with adjustable periodicities to name some examples. Also, the relevant analogy with the acoustic counterpart is noted.

Longitudinal and transverse optical scattering asymmetry parameters for a dielectric cylinder in light-sheets of arbitrary wavefronts and polarization.

The asymmetry parameter is an important quantity used in radiative transfer modeling and scattering. This parameter specifies the amount of energy scattered by the particle along the direction of the incident illuminating field. However, a rigorous and complete analysis of the energetic scattering requires determining the energy scattered in the lateral direction as well. As such, the present work introduces generalized expressions for the scattering asymmetry parameters for a dielectric cylinder in arbitrary-shaped light-sheets, both along and perpendicular to the direction of the incident radiation. Both longitudinal and transverse scattering asymmetry parameters are defined, and their generalized expressions are obtained based on the (spatial) average cosine and sine of the scattering angle θ and the expression of the scattering cross section (or energy efficiency). The partial-wave series expansion method in cylindrical coordinates is used, and the resulting mathematical expressions depend on the beam-shaped coefficients and the scattering coefficients of the dielectric cylinder. Numerical results for arbitrary-shaped light-sheets illuminating a dielectric cylinder cross section located arbitrarily in space are presented and discussed. The longitudinal and transverse scattering asymmetry parameters defined here offer additional quantitative (quadratic) observables for the analysis of the energetic scattering in applications in electromagnetic scattering, optical light-sheet tweezers, radiative transfer computations, and remote sensing, to name a few examples.

1D porous silicon photonic crystals comprising Tamm/Fano resonance as high performing optical sensors

We theoretically discuss a new design for the sensing and detection of chemical and biomedical fluids based on the change in their indices of refraction in the refractive index range 1−1.5. The transfer matrix method in cylindrical coordinates and the Bruggeman's effective medium approximation represent the theoretical overview of our study. The metallic cap in front of a ternary porous silicon photonic crystal induces Tamm / Fano states inside the photonic band gap that take part of the resonant peaks in conventional defected PCs. Analysis shows that the spectral positions of these states are very sensitive to changes in refractive index of the void fluids. In particular, the proposed sensor could provide higher performance than many previous 1D photonic crystals sensors. At a void refractive index of 1.1, our design exhibits a sensitivity of 6770 nm/RIU and a quality factor of 662. Therefore, the detection and sensing procedures of chemical and biomedical fluids could be investigated in a rather much efficient, stable and novel technique that has not been applied before.

Novel Design for the Temperature Sensing Using Annular Photonic Crystals

In this research, a high-performance temperature sensor of a one-dimensional defective annular photonic crystal is proposed. The structure design of our proposed defective annular photonic crystal is: [(Si/SiO2) N/2 TiO2 (Si/SiO2) N/2] with TiO2 as a defect layer. The transmission spectrum of the defective annular photonic crystal is calculated by using the well-modified transfer matrix method in cylindrical coordinates. Thermal characteristics of the defect mode that emerged in the transmission spectrum are studied in the visible region. Dependence of the defect mode frequency on the defect layer thickness is also discussed. As temperature increases, the defect mode is shifted to new positions due to thermal expansion and thermo–optical effects. Numerical results show that the core radius has an important effect on the transmission intensity of the defect mode. Moreover, our proposed temperature sensor showed a sensitivity of about 11 nm / 1000 °C, figure of merit of 0.218 / oC and a very small full width at half maximum of about 0.049 nm. The proposed annular photonic crystals temperature sensor could present a novel method to overcome the limited performance of the planar photonic crystals temperature sensors besides the possibility of using in many potential filtering applications.

Radiation force and torque of light-sheets illuminating a cylindrical particle of arbitrary geometrical cross-section exhibiting circular dichroism

Unlike standard dielectric materials, the interaction of light-sheets of arbitrary wavefronts with a particle exhibiting circular dichroism, such as chiral, topological insulator, liquid crystal, or metamaterials to name a few examples, requires additional fundamental developments from the standpoint of scattering, radiation force and torque theories.The purpose of this work is therefore directed toward developing an exact analytical formalism applicable to a 2D object exhibiting TM ⇄ TE mode conversion and possessing an arbitrary geometrical cross-section. Exact mathematical expressions for the longitudinal and transverse optical radiation force and axial torque components are derived stemming from series expansions for the incident and scattered electromagnetic fields using the mode matching method in cylindrical coordinates. The generalized radiation force and torque expressions depend on the beam-shape coefficients (BSCs) of the incident light-sheet, and the scattering coefficients of the object. Extra new terms describing the generation of the scattered cross-polarized waves induced due to TM ⇄ TE mode conversion contribute to the radiation force and torque series. Numerical illustrative examples for a circular lossy electromagnetic conductor (LEMC) cylinder are provided assuming illumination by various wavefronts, ranging from plane progressive, standing and quasi-standing waves to other non-paraxial Airy light-sheets with linear and circular polarization. In essence, the present theoretical formalism provides a complete analysis in the framework of the generalized Lorenz–Mie theory in 2D for any particle exhibiting circular dichroism in any structured light-sheet. Potential applications are in the design, optimization and the numerical predictions and computation of the optical radiation force and torque for particle transport and rotation in the realm of optical tweezers, particle manipulation and stabilization. Some perspectives are discussed and replies to misleading comments made in the literature on the publication [F.G. Mitri, J Quant Spectro Rad Transf 2020;250:106994] given by Ref. [33] are provided.