Stability and Numerical Dispersion Analysis of a Fourth-Order Accurate FDTD Method

In order to obtain high-order accuracy, a fourth-order accurate finite difference time-domain (FDTD) method is presented by Kyu-Pyung Hwang. Unlike conventional FDTD methods, a staggered backward differentiation scheme instead of the leapfrog scheme is used to approximate the temporal partial differential operator. However, the high order of its characteristic equation derived by the Von Neumann method makes the analysis of its numerical dispersion and stability very difficult. In automatic control theory, there are two effective methods for the stability analysis, i.e., the Routh–Hurwitz test and the Jury test. The combination of the Von Neumann method with each of the two can strictly derive the stability condition, which only makes use of the coefficients of its characteristic equation without numerically solving it. The method of analysis in this paper is also applicable in the stability and numerical analysis of other high-order accurate FDTD methods.     

Reducing the phase error for finite-difference methods withoutincreasing the order

The phase error in finite-difference (FD) methods is related to the spatial resolution and thus limits the maximum grid size for a desired accuracy. Greater accuracy is typically achieved by defining finer resolutions or implementing higher order methods. Both these techniques require more memory and longer computation times. In this paper, new modified methods are presented which are optimized to problems of electromagnetics. Simple methods are presented that reduce numerical phase error without additional processing time or memory requirements. Furthermore, these methods are applied to both the Helmholtz equation in the frequency domain and the finite-difference time-domain (FDTD) method. Both analytical and numerical results are presented to demonstrate the accuracy of these new methods     

A single scatter improvement for beam propagation methods

The beam propagation methods are widely used in numerical simulations of optical waveguides that change gradually in the propagation direction. However, their accuracy is still limited if the one-way models are used over a large propagation distance. The single scatter approximation can be used to improve the accuracy of the beam propagation method, but its usual discrete form is not convenient for numerical implementation. We obtain an improved one-way equation as the continuous limit of the discrete single scatter approximation. Alternative discretization of this equation leads to efficient numerical implementations with the same improved accuracy.     

High accuracy numerical solutions for band structures in strained quantum well semiconductor optical amplifiers

In this paper,we have calculated the band structure of strained quantum well (QW) semiconductor optical amplifiers (SOAs) by using plane wave expansion method (PWEM) and finite difference method (FDM),respectively.The difference between these two numerical methods is presented.First,the solution of Schr(o)dinger's equation in a conduction band for parabolic potential well is used to check the validity and accuracy of these two numerical methods.For the PWEM,its stability and computational speed are investigated as a function of the number of plane waves and the period of QW.For FDM,effects of mesh size and QW width on its accuracy and calculation time are discussed.Finally,we find that the computational speed of FDM generally is faster than that of PWEM.However,the PWEM is more efficient than the FDM when wider SOAs are needed to be calculated.Therefore,to obtain high accuracy and efficient numerical solutions for band structures,numerical methods should be selected depending on required accuracy,device structure and further applications.     

Reconstruction of cylindrically layered media using an iterative algorithm with a stable solution

The reconstruction of cylindrically layered media is investigated in this article. The inverse problem is modeled using a source-type integral equation with a series of cylindrical waves as incidences, and a conventional Born iterative procedure is modified for solving the integral equation. In the modified iterative procedure, a conventional single-point approximation for the calculation of the field inside media is replaced by a multi-points approximation to improve the numerical stability of its solution. Numerical simulations for different permittivity distributions are demon- strated in terms of artificial scattering data with the procedure. The result shows that the procedure enjoys both accuracy and stability in the numerical computation.     

Modified Fabry–Perot and Rate Equation Methods for the Nonlinear Dynamics of an Optically Injected Semiconductor Laser

Modifications have been introduced to the Fabry– Perot (FP) and the rate equation methods to improve the accuracy of the analysis of the nonlinear dynamics of a laser with external optical injection. Comparison between the modified methods and the more accurate transmission-line laser model (TLLM) shows good agreement, while the computational time of the latter is larger by two or three orders of magnitude. In the FP method, the stimulated recombination term in the carrier density evolution equation is modified to include the backward propagating wave and the exponential longitudinal dependence of the electromagnetic field. In the rate equation method, the optical injection term is modified to account for the contribution of the amplification and losses of the injected light inside the cavity to the average photon density. The derivation explaining the validity of these changes and the mathematical relationship between the two methods is presented. Improved stability maps for different values of the injected optical power and frequency detuning are demonstrated and compared with those obtained by the TLLM. The gain compression effect is included in the FP model, and its effect on the stability properties is discussed.      

基于抛物方程的无线电环境地图快速构建方法

研究一种基于三维抛物方程的无线电环境地图快速构建方法，并探讨几种空间插值方法的精确度和适用性，为网络优化和管理提供一种有效工具。利用抛物方程快速计算大区域复杂环境电场分布，参考地理学中最佳统计单元选取四种典型研究区域，分别采用Kriging、反距离加权法(IDW)、双调和样条插值和基于三角剖分的几种插值法对研究区域进行分析。结果表明：a) 几种三角剖分插值和双调和样条插值在各区域RMSE最小且时间最短，但空间拟合程度最低；b) Kriging在各区域中空间拟合程度最高，RMSE和时间随半方差模型有所差异；c) IDW在各区域中RMSE和空间拟合程度较好，但需要时间最长。几种空间插值方法中，指数模型Kriging能以最高精确度和适用性构建基于三维抛物方程的无线电环境地图。     

Numerical solution of the continuous waveguide transition problem

A simple transition between two sizes of rectangular waveguide is analyzed using the generalized telegraphist's equation. Solutions are obtained using a new moment method technique, a Runge-Kutta algorithm, and an iterative numerical integration technique. The results are compared to previously published experimental and numerical data. It is found that the numerical stability, accuracy, and consistency of the results are critically dependent on the choice of weighting and expansion functions. The best results for a simple rectangular-to-rectangular transition were obtained when Galerkin's method and triangle expansion functions were applied to several short sections which were then cascaded. Unlike the Runge-Kutta technique or the integration technique, the Galerkin's method procedure did not result in instabilities with the inclusion of evanescent modes. The programs can, in fact, be extended to any number of modes, the only apparent limitations being the obvious ones of computer time and memory     

球谐波展开法求解玻尔兹曼方程综述

介绍3种常用的直接求解玻尔兹曼传输方程(BTE)的球谐波展开法(SHE),分别是项匹配法、Galerkin法和投影法,以及求解过程中常用的几种能带模型和H-转换、最大熵消散两种数值稳定技术,并对比了以上3种方法的异同点。综合已发表文献中的模拟结果,得出结论:高阶球谐波展开和能带模型的选取对模拟器件的传输特性具有至关重要的作用。     

On the Entropy Rate of Hidden Markov Processes Observed Through Arbitrary Memoryless Channels

This paper studies the entropy rate of hidden Markov processes (HMPs) which are generated by observing a discrete-time binary homogeneous Markov chain through an arbitrary memoryless channel. A fixed-point functional equation is derived for the stationary distribution of an input symbol conditioned on all past observations. While the existence of a solution to the fixed-point functional equation is guaranteed by martingale theory, its uniqueness follows from the fact that the solution is the fixed point of a contraction mapping. The entropy or differential entropy rate of the HMP can then be obtained through computing the average entropy of each input symbol conditioned on past observations. In absence of an analytical solution to the fixed-point functional equation, a numerical method is proposed in which the fixed-point functional equation is first converted to a discrete linear system using uniform quantization and then solved efficiently. The accuracy of the computed entropy rate is shown to be proportional to the quantization interval. Unlike many other numerical methods, this numerical solution is not based on averaging over a sample path of the HMP.      

一种快速有限周期阵列特征模分析法北大核心CSCD

周期特征模分析(PCMA)方法是一种有限周期阵列特征模分析数值算法。该方法使用特征向量在每个阵元上定义若干组全域基函数,然后基于全域基函数得到一个压缩广义特征值方程。利用该方程可实现该阵列的特征模分析。与传统方法相比,该方法未知数量显著减少,节省了存储成本和求解时间。然而,广义特征值方程中的矩阵元素计算涉及二重面积分,非常耗时。文章采用等效偶极矩法计算矩阵元素,避免了二重面积分运算,从而节省了计算时间。数值结果表明新方法的结果与PCMA法以及传统分析法结果吻合很好,验证了所提方法的精度。同时,该方法显著降低了求解时间。     

深度学习点衍射干涉三维坐标定位技术

为了提高现有的三维坐标定位技术的测量精度、稳定性和测量效率,提出了基于深度学习的点衍射干涉三维坐标定位方法。该方法设计了一个深度神经网络用于点衍射干涉场的坐标重构,将相位差矩阵作为输入,构建训练数据集,将点衍射源坐标作为输出,训练神经网络模型。利用训练有素的神经网络对测量到的相位分布进行初步处理,将相位信息转换为点衍射源坐标,根据得到的点衍射源坐标进一步修改粒子群算法的初始粒子,进而重构出高精度的三维坐标值。该神经网络为建立干涉场相位分布与点衍射源坐标之间的非线性关系提供了一种可行的方法,显著提高了三维坐标定位的精度、稳定性和测量效率。为验证所提方法的可行性,进行了数值仿真和实验验证,采用不同的方法进行反复对比与分析。结果表明：所提方法的单次测量时间均在0.05 s左右,其实验精度能够达到亚微米量级,重复性实验的均值和RMS值分别为0.05μm和0.05μm,充分证明了该方法的可行性,并证明了其良好的测量精度和可重复性,为三维坐标定位提供了一种有效可行的方法。     

Numerical Stability and Numerical Dispersion of Compact Conformal Mapping 2D-FDTD Method Used for the Dispersion Analysis of Waveguides

In dispersion analysis of waveguides with particular cross-section by compact 2D-FDTD method, using conformal-boundary coordinates can obtain high computational accuracy. The transformation from conformal-boundary coordinates to rectangular coordinates can be done by conformal mapping technique in order to match Yee algorithm. In this paper, numerical stability and numerical dispersion equation of compact conformal mapping 2D-FDTD (CCM-2D-FDTD) method are derived. It is shown that the upper limit of Courant number for CCM-2D-FDTD is always smaller than 1/√2. As an example, the dispersion equation is used to examine the impact of number of cell for circular waveguide.     

Accuracy Controllable Method for Oblique Incidence on Bodies of Revolution

In the methods, for solving problems of bodies of revolution (BOR-problems), Fourier series are used to convert an original three-dimensional problem illuminated by plane wave into a series of 2-D problems, which we call Fourier components or harmonic series, illuminated by cylindrical plane waves. When the plane wave illuminates obliquely, the number of Fourier components should be larger than one. The quantitative relationship between this number and the accuracy of the results has not been well established yet. In this paper, a simple and accuracy controllable method based on a partly iterative procedure is proposed, which can be used to determine the number of Fourier components accurately for a desired accuracy of the results with the most economic computational cost. Although this method is introduced through the method of moments, it can work equally well to other numerical methods for solving BOR-problems. The validity of this method is confirmed by several numerical examples.     

An analysis of new and existing FDTD methods for isotropic coldplasma and a method for improving their accuracy

Over the past few years, a number of different finite-difference time-domain (FDTD) methods for modeling electromagnetic propagation in an isotropic cold plasma have been published. We have analyzed the accuracy and stability of these methods to determine which method provides the greatest accuracy for a given computation time. For completeness, two new FDTD methods for cold plasma, one of which is based on the concept of exponential fitting, are introduced and evaluated along with the existing methods. We also introduce the concept of cutoff modification which can be easily applied to most of the FDTD methods, and which we show can improve the accuracy of these methods with no additional computational cost. Von Neumann's stability analysis is used to evaluate the stability of the various methods, and their accuracy is determined from a straightforward time-and-space harmonic analysis of the dispersion and dissipation errors. Results of numerical experiments to verify the accuracy analysis are presented. It is found that for low-loss plasma, the piecewise linear recursive convolution method (PLRC) method is the most accurate, but the method of Young (see Radio Sci., vol.29, p.1513-22, 1994) can use less memory and is nearly as accurate. In this low-loss plasma regime, cutoff modification can significantly reduce the error near cutoff at the expense of slightly greater error at lower frequencies. For strongly collisional plasmas, the PLRC method also provides the most accurate solution     

Illumination robust face recognition using random projection and sparse representation

Under uneven illumination, the performances degrade significantly for some existing face recognition methods. It is a challenge for face recognition methods to work effectively under different illumination conditions. In this paper, an illumination robust face recognition method, based on random projection and sparse representation, is proposed. In the proposed method, face images are preliminary illumination normalized by gamma correction and difference of Gaussian filtering, and then several projection spaces are obtained by iterative random projection, followed by constructing an initial sample space using Fisher discrimination analysis. This scheme enriches the discrimination abilities of sample features and achieves the security and completeness for biometric template. Test samples are sparsely decomposed into each subspace, and based on statistical average residual, a modified sparse representation method is proposed to realize face recognition with higher stability and illumination robustness. Experimental results indicate that the proposed method provides competitive performance with acceptable computational efficiency. Specifically, for the five subsets of Yale B database, our approach achieves 99.74% average recognition rate, which performs higher accuracy than that of comparative methods.     

Treating the instabilities in marching-on-in-time method from adifferent perspective [electromagnetic scattering]

A general method is presented to treat the instabilities which are frequently observed in the electromagnetic transient solutions using the marching-on-in-time method. The basic idea is to apply an finite impulse response (FIR) filter with a constant group delay during the course of marching-in-time. An electric field integral equation (EFIE) formulation for perfectly conducting bodies is used as a vessel to demonstrate the method. Sample numerical results are presented and discussed. The computed results, while showing good agreement with the data obtained from other methods, present great stability improvement     

基于化学物质释放的五点傅里叶变换求解泊松方程的方法

在电离层化学物质释放数值模拟中,傅里叶变换是一种有效的求解泊松方程的方法,但其仿真精度有待进一步提高.文章提出一种改进的傅里叶变换求解泊松方程的方法,对泊松方程的五点差分形式做傅里叶变换,并引入误差修正项,弥补二阶差分带来的误差,模型中给出了详细推导过程,并与常用方法进行了对比分析.研究结果表明,改进方法求得电势的均方误差（Mean Square Error,MSE）相对Birdsall方法和电势求导方法小6个量级,电势空间分布更加接近解析结果.文中所提方法满足化学物质释放数值仿真的要求,为等离子体粒子模拟中泊松方程的求解提供了一种新的方法.     

Large signal design of broadband monolithic microwave frequencydividers and phase-locked oscillators

This paper presents a design method for phaselocked devices such as frequency dividers or injection-locked oscillators. The method requires a full nonlinear analysis of the circuit. This analysis relies upon harmonic balance techniques and is suitable for monolithic circuits simulation. First, a modified formulation of the general harmonic balance equation is proposed which includes the presence of probes. These probes allow us to suppress the degenerated solution of the HB equation in autonomous cases. Moreover, a global stability analysis of phaselocked regimes is carried out. It provides invaluable information on the nonlinear behavior of the device. In particular, synchronization bandwidths as well as power ranges for which the circuit can be synchronized are obtained from the stability loci drawn in the parameter space. All these features have been used to design a broadband monolithic frequency divider, and the simulated and experimental results have been compared with very good accuracy. Therefore, the method proposed is a very useful tool for the design of potentially unstable circuits     

An iterative measured equation technique for electromagneticproblems

An iterative measured equation technique (IMET) is presented for a numerical solution of electromagnetic problems. This technique is an extension and improvement of the method of measured equation of invariance (MEI). In this technique, an iterative scheme is designed in such away that a new set of metrons used to generate the measured equations is formed in each iteration based on the solution of the previous iteration. The new metrons are more meaningful in that they converge to the physical quantity of interest such as the surface current density for electrodynamic problems and the surface charge density for electrostatic problems. The IMET offers several advantages over the MEI method because it requires only two mesh layers, resulting in a significant reduction in the memory requirement and computing time. More importantly, it provides a means for a systematic improvement of the accuracy of solution. The IMET is applied successfully to two-dimensional (2-D) electrodynamic and three-dimensional (3-D) electrostatic problems. Numerical results show that the technique is highly accurate and the iterative process converges very quickly, usually within two iterations