Weighted-Residual Methods for the Solution of Two-Particle Lippmann–Schwinger Equation Without Partial-Wave Decomposition

Recently there has been a growing interest in computational methods for quantum scattering equations that avoid the traditional decomposition of wave functions and scattering amplitudes into partial waves. The aim of the present work is to show that the weighted-residual approach in combination with local basis functions give rise to convenient computational schemes for the solution of the multi-variable integral equations without the partial wave expansion. The weighted-residual approach provides a unifying framework for various variational and degenerate-kernel methods for integral equations of scattering theory. Using a direct-product basis of localized quadratic interpolation polynomials, Galerkin, collocation and Schwinger variational realizations of the weighted-residual approach have been implemented for a model potential. It is demonstrated that, for a given expansion basis, Schwinger variational method exhibits better convergence with basis size than Galerkin and collocation methods. A novel hybrid-collocation method is implemented with promising results as well.     

Modelling of photonic crystal waveguides: a simple and accurate approach

We present an improved variational effective index method for reduction of 2-D Bragg grating problems to 1-D and show significant improvements particularly at smaller wavelengths. The method is based on the optimal variational (Vopt) method, which we have earlier used successfully for conventional waveguides. A 1-D transverse profile along with a longitudinal index variation are reduced to a 1-D longitudinal distribution, reflection and transmission spectra of which have been studied for both the TE and TM modes by transfer matrix methods. An accurate modeling for the out of plane scattering losses has been presented which occur when a guided wave propagating in a conventional waveguide impinges on a photonic crystal waveguide. Taking these losses into account brings the results pretty close to those of a rigorous 2-D Helmholtz solver QUEP, improving them remarkably over the variational EIM (vEIM) results.     

类铍离子激发态1s22P2 3Pe的能量、精细结构和跃迁波长的计算

采用多组态相互作用方法及Rayleigh-Ritz变分法,计算了类铍离子等电子系列(Z=4-10)激发态1s22p2 3Pe的非相对论能量,利用截断变分方法得到能量改进量,进一步考虑相对论效应和质量极化效应,从而获得了高精度的能量计算值.给出了类铍离子等电子系列激发态1s22p2 3Pe的相对论能量修正和质量极化效应随核电荷数Z变化的情况.同时还计算了激发态1s22p23Pe的精细结构能级和劈裂,以及1s22s2p3Po到1s22p2 3Pe态的辐射跃迁波长.计算结果与其他理论和实验符合得很好.     

三维海洋波导中多声散射的数值解

研究三维海洋波导中多障碍物的声散射问题.基于Galerkin变分原理和DtN映射方法,发展一种自然边界元与有限元耦合的求解方法,提出有效的数值实现方法,避免常规有限元中需要采用吸收边界的缺点.对多种形状多障碍物的数值计算结果表明,所建立的理论方法是有效的.     

A High-Efficiency Spectral Method for Two-Dimensional Ocean Acoustic Propagation Calculations

The accuracy and efficiency of sound field calculations highly concern issues of hydroacoustics. Recently, one-dimensional spectral methods have shown high-precision characteristics when solving the sound field but can solve only simplified models of underwater acoustic propagation, thus their application range is small. Therefore, it is necessary to directly calculate the two-dimensional Helmholtz equation of ocean acoustic propagation. Here, we use the Chebyshev–Galerkin and Chebyshev collocation methods to solve the two-dimensional Helmholtz model equation. Then, the Chebyshev collocation method is used to model ocean acoustic propagation because, unlike the Galerkin method, the collocation method does not need stringent boundary conditions. Compared with the mature Kraken program, the Chebyshev collocation method exhibits a higher numerical accuracy. However, the shortcoming of the collocation method is that the computational efficiency cannot satisfy the requirements of real-time applications due to the large number of calculations. Then, we implemented the parallel code of the collocation method, which could effectively improve calculation effectiveness.     

Analysis of non-linear multilayer waveguides with Kerr-like permittivities

This paper presents a semi-analytical method for calculating the dispersion relationships for stationary non-linear TE waves guided by general multilayer waveguides with Kerr-like permittivities. The non-linear wave equations are solved rigorously for each layer, and the mode index and field distribution are systematically determined so as to satisfy the boundary conditions at all interfaces. To verify the validity of the present method, a non-linear graded-index waveguide is approximated by a number of step layers and the resultant multilayer waveguide is analysed numerically. The convergence properties of mode index and total power flow are presented, and these are compared with direct solutions of the non-linear wave equation by the numerical integration method. The dispersion relationships for a non-linear multiple-quantum-well waveguide have also been investigated under two simple permittivity models. The present method is useful for the analysis of non-linear graded-index waveguides in addition to non-linear multilayer waveguides such as the multiple-quantum-well structure.     

铍原子的能级和超精细结构的研究

采用多组态相互作用方法及Rayleigh Ritz变分法 ,并考虑相对论修正、质量极化效应等 ,获得了铍原子低激发态 1s2 2s2p3 Po和 1s2 2p2 3 P高精度的相对论能量 .同时还计算了铍原子超精细结构常数 ,与其他理论和实验结果符合得很好. The Rayleigh-Ritz variational method is used with a multiconfiguration-interaction function and restricted variation method to obtain the relativistic energies of 1s 22s2p 3P o and 1s 22p 2 3Pin beryllium, including the mass polarization and relativistic corrections. Hyperfine structure is also studied to compared with theoretical and experiment data in the literature.     

Exact and Analytical Methods for Finding Complex Roots of Moving Waveguides with Absorbing Layers

We present here two methods of analysis of the complex moving waveguide. One approach is a numerical technique based on Davidenko's method, the other is a perturbation method. Numerical results by Davidenko's method are compared with these obtained from the perturbation method. It is found that the results of the perturbation method are in a good agreement with the numerical one. A comparative study of the propagation and attenuation characteristics of the stationary and moving waveguides has also been investigated and discussed. These results could be used in designing many optical moving sensors.     

Scattering from an abruptly terminated planar metamaterial waveguide

We use a variational method to study the problem of reflection of a guided mode from an abruptly terminated planar dielectric waveguide made of a metamaterial. The theory is illustrated by problems of abruptly terminated three-layer waveguides with piecewise-constant and variable permittivity profiles. Differences in the scattering characteristics for systems with metamaterials and conventional media are discussed.     

Modelling of guided-wave Bragg gratings and grating sensors using the collocation method

We have used the collocation method to model the characteristics of guided-wave Bragg gratings. The collocation method being a total field method, takes into account all modes, guided as well as radiation. We have first studied the effect of the grating structural parameters such as the grating profile and the duty cycle of periodic variation and have shown that these can have significant effect on the Bragg wavelength and the reflection spectrum. We have then obtained the response characteristics of gratings for their use in strain, temperature and pressure sensing. Our results compare very well with available experimental results. Comparisons with the coupled mode theory have also been included.     

掩埋型离子交换玻璃光波导的变分分析

用变分法对离子交换法制备的掺铒光波导的传播特性进行了分析,推导出了适用于掩埋型离子交换玻璃沟道光波导中场分布传播常量的变分表达式,构建了场分布的厄米-高斯型试探解,在两种不同实验条件下,采用变分法确定了试探解中的待定参量,获得光波导中的场分布,利用传播常量的变分公式和已确定的场分布计算得到了传播常量和有效折射率。计算数据表明:导波区域的有效折射率稍稍地大于限制层的折射率,说明离子交换法制备的波导器件是弱波导;高的辅助退火电场强度和适当的退火时间下,所制备的光波导可以支持更多光模式的传输。该方法计算过程简洁、快速,计算结果与实验结果吻合。     

类锂离子内壳激发态的能量、俄歇宽度和俄歇分支率（英文）

采用鞍点变分方法和鞍点复数转动方法并考虑相对论修正和质量极化效应,计算了类锂离子内壳激发态的能量、俄歇宽度、俄歇分支率和俄歇电子能量.进一步采用截断变分方法饱和空间波函数.计算结果与其他理论结果以及实验数据符合得很好.     

Basis sets for the study of point defects in solids using green's-function methods

A number of Green's-function methods have recently been used to study the electronic structure of point defects and impurities in semiconductors. These methods are based on the same underlying operator formalism. An important factor controlling the efficiency of any of these methods is the basis set used to represent the operators in matrix form. Here we point out that it is the physical character of the perturbation potential that should suggest the nature of suitable basis sets. For point defects, the perturbation potential has recently been demonstrated by self-consistent calculations to have a predominantly single-center character. A single-center basis set, therefore, appears to be more suitable than multicenter basis sets. We present formal arguments and calculations that indicate that a set of harmonic-oscillator eigenfunctions centered at the point-defect site is particularly suitable, largely because such functions concentrate their variational flexibility in a small volume.     

A simple variational analysis of diffused channel waveguides

The variational analysis of Ti:LiNbO₃ strip waveguides is resolved into two related one-dimension problems, which provides a new iterative approach for the calculation of the mode parameters and the propagation constant. The analytical expressions of the local effective index profiles can be obtained from the variational analysis. The parameters of four trial solutions for the fundamental mode in a diffused-channel waveguide have been determined by using the present method. Comparison with results using the effective method shows that these four approximate expressions are accurate enough. It is also shown that this method converges very quickly and is suitable for both diffused channel waveguides and optical waveguides with rectangular cross-section.     

Comment on: On shallow-deep instability of impurity level in semiconductors

It is pointed out that a correct form and also a more general form of the model introduced by Mukhopadhyay in the problem of shallow-deep instability of impurity ground level in semiconductors have, in fact, been introduced long ago by various authors to study the same problem. The models have also been solved for the ground state by various methods such as variational (with better variational trial function than that of Mukhopadhyay) and numerical. Recently we have solved the problem analytically for all the states.     

Cutoff frequencies in planar optical waveguides with arbitrary index profiles: An efficient numerical method

A numerical method for obtaining mode cutoffs for planar waveguides with arbitrary index profiles is developed. The method is based on the Galerkin method in which the wave equation for modes at cutoff is converted to a matrix eigenvalue equation using a set of orthogonal basis functions. Due to different boundary conditions, we have identified two separate cases; one, in which the cover and the substrate indices are identical leading to same behavior of the field at cutoff in these two regions and, the other, in which the two indices are different and hence, the field behaves differently. We have accordingly chosen different basis functions for the two cases. The method results in a generalized matrix eigenvalue problem which has been converted to a standard symmetric matrix eigenvalue analytically. The method has been used to obtain mode cutoffs for waveguides with a variety of index profiles. Comparisons with available exact results show that very good accuracies can be obtained with moderate matrix sizes.     

类铍CⅢ 和 OⅤ离子激发态的相对论能量和跃迁几率

利用多组态相互作用方法及Rayleigh-Ritz变分法,计算了类铍CⅢ和OⅤ离子激发态1s22s2p1P0,1s22s2p3P0,and 1s22p2 3P的能量,包括了相对论能量和质量极化效应。计算了相关态的振子强度、跃迁几率和波长。计算结果与其它理论和实验结果符合得很好。     

Accurate calculation of the eigenvalues of a new simple class of superpotentials in SUSY quantum mechanics

We obtain accurate eigenvalues for two recently derived SUSY partner Hamiltonians. We improve the Rayleigh-Ritz variational method proposed by the authors and show how to apply the Riccati-Padé method to those particular partner potentials.     

高离化类锂原子体系(Z=21～30)1s23p态的能级结构和振子强度

用全实加关联方法计算了类锂原子体系(Z=21～30)偶极跃迁1s22s -1s23p的跃迁能、振子强度以及1s23p态的精细结构劈裂.非相对论能量用Rayleigh -Ritz变分法确定;相对论修正和质量极化效应用微扰论计算;同时考虑了来自量子电动力学(QED)效应的修正.得到的理论结果与实验数据及物理规律符合的很好.     

A new variational method to calculate escape rates in bistable systems

We apply the Rayleigh-Ritz variational principle to the inverse of the Smoluchowsky operator in order to calculate the lowest non-vanishing eigenvalue of the Smoluchowsky equation. In contrast to former variational calculations the new method is very insensitive to the choice of the test function. We obtain the asymptotic expansion of the eigenvalue in the limit of vanishing diffusion.