Paraxial Direction Research Articles

Light propagation in a three-dimensional Rydberg gas with a nonlocal optical response.

We theoretically investigate the linear susceptibility and propagation of light in a three-dimensional (3-D) Rydberg gas under conditions of electromagnetically induced transparency. Rydberg atoms with two relevant S states are coupled via exchange interactions. When the gas is initially prepared in an entangled spin-wave state, this coupling induces a strong, nonlocal susceptibility whereby the photon field at one point of the medium acts as a source at a distant position. The nonlocal propagation occurs not only in the propagation direction but also in the paraxial direction. We discuss the absorption features and numerically simulate the 3-D propagation of probe laser light. Combined with the long-range exchange interaction, we show that the 3-D Rydberg gas is an ideal medium for studying nonlocal wave phenomena, in which the strength, range, and sign of the nonlocal interaction kernel can be widely tuned.

A Body-of-Revolution Implementation of the Parabolic Wave Equation With Application to Rocket Plume Attenuation Modeling

The parabolic wave equation has been used extensively to model long-range paraxial wave propagation due to its high computational efficiency. Typical radar, sonar, and communication geometries are most naturally treated using cylindrical coordinates with the radial direction chosen as the paraxial direction. Here, a method is described for modeling waves with 3-D variation propagating through an inhomogeneous body of revolution with the paraxial direction chosen to be the axis of symmetry. The technique decomposes the initial field into independent modes that are propagated in parallel using 2-D solvers. The method is benchmarked by comparison to finite-element calculations. An example application—modeling the interaction of a radio communications system with a rocket exhaust plume—is then considered.

Two-Way Propagation Modeling of Expressway With Vehicles by Using the 3-D ADI-PE Method

The two-way (2W), 3-D parabolic equation (PE) is first derived and solved by the finite differential (FD) method of alternating direction implicit (ADI) scheme in this communication. By using the 2W PE, both the forward and backward propagating effects can be taken into consideration. Moreover, the calculation can be taken in a marching manner by applying the FD method along the paraxial direction, and the unknowns are computed line by line in each transverse plane. In this way, the computational resources can be saved significantly. The proposed 2W 3-D ADI-PE method is used to analyze the electromagnetic propagating in the expressway with vehicles. The vehicles can be modeled with high accuracy by using the finite-difference scheme. Moreover, the proper impedance boundary conditions are added to the road of expressway and vehicles are regarded as the perfectly electrical conducting targets. Numerical results are given to demonstrate the electromagnetic propagation of expressway with vehicles.

Frequency-domain and time-domain solvers of parabolic equation for rotationally symmetric geometries

Both the frequency-domain and the time-domain body of revolution parabolic equations (FD-BoR-PE, TD-BoR-PE) are derived in this paper. By taking advantage of the rotationally symmetrical property, better performance of the PE method can be achieved for the analysis of bodies of revolution (BoRs). In each transverse plane, only the unknowns of a line, which starts from the center and passes through the node on the generatrix, need to be calculated. Then the unknowns for each transverse plane can be obtained by Fourier series summation for each Fourier mode and the calculation can be taken with a marching manner along the paraxial direction of the PE. As a result, the computational resources can be reduced greatly when compared with the traditional CN, alternating direction implicit (ADI) and alternating group explicit (AGE) finite difference schemes. Both the propagating and scattering problems are given to demonstrate the validity and efficiency of the proposed methods.

Pade Second-Order Parabolic Equation Modeling for Propagation over Irregular Terrain

The Pade second-order parabolic equation (PE) method is developed to predict radio propagation over irregular terrain. This solution is solved by the finite-difference method, which produces a stable pentadiagonal difference scheme. It allows propagation angles up to about 55° from the paraxial direction, which is able to give better results than the Fourier split-step method. This solution can be extended to an arbitrary general terrain profile by the boundary shift approach. We investigate the important problems about propagation angles of the different approximations of the PE models. It is found that Pade second-order PE model has the greater propagation angles, and thus can provide more accurate prediction of radio propagation. The calculated results are compared with the measurements in a fir forest environment, and a good agreement is obtained.

A Novel Marching-on-in-Degree Solver of Time Domain Parabolic Equation for Transient EM Scattering Analysis

A novel marching-on-in-degree (MOD) solver of 3-D time domain parabolic equation is proposed to solve the transient electromagnetic scattering from electrically large perfect electric conductor (PEC) targets. The finite difference (FD) scheme is applied to the spatial discretization, while the weighted Laguerre polynomials are used as the temporal basis functions. In this way, a large number of computational resources can be saved by using the FD scheme along the paraxial direction and the late-time stability can be guaranteed by the MOD method. Both the FD schemes of Crank–Nicolson and the alternating direction implicit types are discussed in this communication. Numerical results are given to demonstrate the accuracy and efficiency of the proposed method.

A Novel Parallel Parabolic Equation Method for Electromagnetic Scatterings

The alternating group explicit (AGE) scheme, together with the alternating direction implicit (ADI) scheme, is used for the parabolic equation (PE) method to analyze the electromagnetic scattering from electrically large conducting targets. The calculation can be taken plane by plane by using the finite difference scheme along the paraxial direction. Furthermore, each transverse plane is divided into two regions, which are a small region around the target and an outer region combined with the air layer and a perfect matching layer. The ADI scheme is applied to the small region, whereas the AGE scheme is introduced to the outer region. In this manner, only a few matrix equations are needed to be solved for the small region and the unknowns of the outer region can be directly obtained without solving any matrix equations. Therefore, the CPU time can be reduced significantly by the proposed method and it can be easily adapted to the parallel technology. The numerical results verify the accuracy and efficiency of the proposed method compared with the Crank–Nicolson-based PE method and the ADI-based PE method. Moreover, the parallel efficiency is given as the last numerical example.

A Wide-Angle Alternate Direction Implicit–Parabolic Equation Method for Electromagnetic Scattering from Electrically Large Targets

ABSTRACTThe alternate direction implicit scheme is introduced to the wide-angle parabolic equation method for the analysis of electromagnetic scattering from electrically large targets. Modified non-homogeneous boundary conditions are added on the surface of the scattering target. In this way, a marching solution is used along the paraxial direction and the fields in each marching plane can be calculated direction by direction in a parallel manner. The accurate results can be obtained at wider angles than the traditional narrow-angle alternate direction implicit–parabolic equation method within 30° along the paraxial direction by the proposed method. Moreover, encouraging accuracy can be obtained with fewer computing resources than the traditional wide-angle Crank–Nicolson–parabolic equation method. Numerical results are given to demonstrate the validity, stability, and efficiency of the proposed method, and the parallel efficiency is given at the same time.

Spectral element method-based parabolic equation for EM-scattering problems

The traditional parabolic equation (PE) method is based on the finite difference (FD) scheme. However, the scattering object cannot be well approximated for complex geometries. As a result, a large number of meshes are needed to discretize the complex scattering objects. In this paper, the spectral element method is introduced to better approximate the complex geometry in each transverse plane, while the FD scheme is used along the paraxial direction. This proposed algorithm begins with expanding the reduced scattered fields with the Gauss–Lobatto–Legendre polynomials and testing them by the Galerkin’s method in each transverse plane. Then, the calculation can be taken plane by plane along the paraxial direction. Numerical results demonstrate that the accuracy can be improved by the proposed method with larger meshes when compared with the traditional PE method.

Angular dependence of screen speckle and fiber speckle of coupled output of nine high-power blue laser diodes through a multi-mode fiber

Angular dependence of speckle contrast of speckle pattern projected out of a multi-mode fiber connected to a high-power blue laser module is investigated. The laser module has nine high-power InGaN/GaN blue laser diodes arranged in a three-by-three array. Each of the arrayed laser diodes have slightly different incident angle to the fiber. We have successfully extracted the fine screen speckle pattern from the projected pattern mixed up with the coarse fiber speckle pattern by processing the measured data. It is found that the speckle contrast of the both screen and fiber speckles are larger around the center area where the guided-light component closest to the paraxial direction is projected. This is because the output of the center laser in the array is likely to couple into the paraxial mode more than the rest. Speckle contrast behaviors when applying the speckle reduction methods, fiber vibration, diffuser, and spinning diffuser are also investigated.

Dynamics of the Abrikosov Vortices on Cylindrical Microtubes

We consider the special features at nano- and microscales of the vortex dynamics on superconducting cylindrical Nb tubes produced by the roll-up (self-rolling) technique. A transport current enters the tube through electrodes placed on both sides of a cut (in the paraxial direction) of the tube. The system is in the magnetic field perpendicular to the tube axis. The vortex dynamics is described by means of characteristic times: time (Δt1) needed for a vortex to move from one edge of the tube to another and time (Δt2) between two consecutive vortex nucleation events at one edge of the tube. A range of magnetic field values is analyzed where Δt1 as a function of the magnetic field has a highly nonlinear and non-monotonic behavior. For certain values of the magnetic field, two different trajectories are possible for a moving vortex, i.e., a bifurcation phenomenon occurs. We explain the reason of this bifurcation.

Paraxial propagation in disclinated amorphous media

We study paraxial beam propagation along the wedge axis of a disclinated amorphous medium. The defect-induced inhomogeneity results in Berry phase and curvature that are affected by the induced uniaxial anisotropy. The Berry phase manifests itself as a precession of the polarization vector. The Berry curvature is responsible for the optical spin Hall effect in the disclinated medium, where beam deflection varies sinusoidally along the paraxial direction. Its application in determining the birefringence and the magnitude of the Frank vector is explained.

Experiments with a plasma-filled rod-pinch diode on the MIG generator

Experimental data for creating an X-ray source several millimeters in size on the basis of a plasmafilled rod-pinch diode are reported. Experiments are carried out on the MIG high-current generator. A voltage pulse applied to the diode is sharpened and shortened by injecting plasma into the interelectrode gap up to the plasma-filled diode. Radiation extraction from the vacuum chamber in the paraxial direction is provided by inverting the positions of the cathode and anode using a post-hole vacuum convolute. It is demonstrated that the energy of a low-impedance high-current generator can be deposited with a high efficiency into an electron beam focused on the 1-mm2 tip of the rod-shaped anode. With a tapered tungsten rod 1.5 mm in diameter used as an anode, a pulsed X-ray source about 1 mm in size generating 30-ns-long pulses is obtained. The absorbed dose per pulse measured with a LiF thermoluminescent dosimeter behind a 5-mm-thick aluminum screen 1 m away from the source reaches 0.042 Gy.

Recursive Two-Way Parabolic Equation Approach for Modeling Terrain Effects in Tropospheric Propagation

The Fourier split-step method is a one-way marching-type algorithm to efficiently solve the parabolic equation for modeling electromagnetic propagation in troposphere. The main drawback of this method is that it characterizes only forward-propagating waves, and neglects backward-propagating waves, which become important especially in the presence of irregular surfaces. Although ground reflecting boundaries are inherently incorporated into the split-step algorithm, irregular surfaces (such as sharp edges) introduce a formidable challenge. In this paper, a recursive two-way split-step algorithm is presented to model both forward and backward propagation in the presence of multiple knife-edges. The algorithm starts marching in the forward direction until the wave reaches a knife-edge. The wave arriving at the knife-edge is partially-reflected by imposing the boundary conditions at the edge, and is propagated in the backward direction by reversing the paraxial direction in the parabolic equation. In other words, the wave is split into two components, and the components travel in their corresponding directions. The reflected wave is added to the forward-wave in each range step to obtain the total wave. The wave-splitting is performed each time a wave is incident on one of the knife-edges. This procedure is repeated until convergence is achieved inside the entire domain.

PLANE WAVE DIFFRACTION BY A STRONGLY ELONGATED OBJECT ILLUMINATED IN THE PARAXIAL DIRETION

After a short presentation of the boundary layer method extended to strongly elongated objects by Andronov and Bouche (1), the author develops some techniques for deriving explicit formulas for the asymptotic currents on a strongly elongated object of revolution excited by an electromagnetic plane wave propagating in the paraxial direction. The performance of the different techniques are demonstrated by comparing numerical results obtained for the asymptotic currents on an elongated prolate ellipsoid with those obtained by solving the EFIE.

Wide-Angle Shift-Map PE for a Piecewise Linear Terrain—A Finite-Difference Approach

A wide-angle parabolic wave equation solution is presented using shift-map and finite-difference techniques. The corresponding split-step Fourier solution is well known. The solution using finite-difference technique, where the standard parabolic wave equation is modified into the so-called Claerbout equation allowing propagation angles up to 45deg from the paraxial direction, is also well known. Here, we present an extension to that solution in which the shift-map technique is incorporated into the finite-difference scheme allowing a varying terrain to be considered. The result is a solution that corresponds to the well known split-step solution, which is believed to perform well for terrain slopes up to 10deg-15deg and discontinuous slope changes on the order of 15deg-20deg. This solution is a first-order one with respect to the terrain slope. However, when using the finite-difference technique, it is also possible to find a second-order solution with respect to the terrain slope. This new solution performs well for slopes up to about 15deg and discontinuous slope changes up to about 30deg, which is an improvement.

Solving multi-object radar cross section based on wide-angle parabolic equation method

Based on a Padé approximation, a wide-angle parabolic equation method is introduced for computing the multiobject radar cross section (RCS) for the first time. The method is a paraxial version of the scalar wave equation, which solves the field by marching them along the paraxial direction. Numerical results show that a single wide-angle parabolic equation run can compute multi-object RCS efficiently for angles up to 45°. The method provides a new and efficient numerical method for computation electromagnetics.

Influence of attachment wear on retention of mandibular overdenture

The aims of this study were: (i) to evaluate and compare retention of two-teeth (implant) supported mandibular overdenture with either stud or magnetic attachments during linear (axial) and rotational (paraxial) dislodgements; (ii) to compare retentive properties before and after wear simulation. The test group consisted of five magnetic and four stud overdenture attachments (n = 12 specimens for each attachment type). Retention in axial direction was evaluated on one-tooth (implant) model by measuring maximum retentive force (N) and range of retention (mm) during the linear dislodgement. Retention in the paraxial direction was evaluated on mandibular-overdenture model by measuring the maximum retentive force (N) during three types of rotational dislodgements - anterior, lateral and posterior. The minimum number of cycles required to simulate wear was determined by special wear test. Afterwards, the wear was simulated in the test group, and retention in axial and paraxial directions was measured again. one-way anova, Scheffe post hoc and paired-samples t-tests (P < 0.05). Initially, studs had higher retention (4-11 N) than magnets (4.5-6 N) in axial direction. After the wear simulation, it had decreased from 76% to 48% for some of the studs and had become similar to the retention of magnetic attachments. Magnets had lower retention range (0.2-0.3 mm) than studs (0.5-1.1 mm). Studs provided similar or higher retention in paraxial directions than magnetic attachments both before and after wear simulation. Retentive properties of magnets decreased mostly with posterior rotational dislodgement. Retentive properties of stud overdenture attachments were less constant.

Parabolic equation techniques for scattering

Parabolic equation techniques for scattering are outlined for both acoustic and electromagnetic waves. Angular limitations are overcome by rotating the paraxial direction and solving for scattered fields that are accurate in sectors centered about the paraxial direction. These fields are then combined to obtain a scattered field that is accurate in all directions. This approach is efficient for bistatic scattering by objects that can be large compared to the wavelength. Parabolic equation techniques are applied to electromagnetic scattering by a finitely conducting sphere and a large idealised aircraft shape and to acoustic scattering in the presence of an interface.

Bistatic RCS calculations with the vector parabolic equation method

The vector parabolic equation (PE) method provides accurate solutions for electromagnetic scattering from three-dimensional (3-D) objects ranging from a size comparable to the wavelength of the incident wave to several tens of wavelengths. A paraxial version of Maxwell's equations is solved with a marching solution that only requires limited computing resources, even for large scatterers. By decoupling the PE paraxial direction from the direction of incidence, the bistatic radar cross section (RCS) can be computed at all scattering angles. A sparse-matrix formulation is used to implement electromagnetic boundary conditions, ensuring that polarization effects are treated fully. Computing costs are kept to a minimum through the use of a double-pass method so that calculations can be carried out on a desktop computer for realistic targets and radar frequencies. The method has been validated on simple canonical shapes and tested on complex targets.