A fast method for the solution of the Helmholtz equation

In this paper, we consider the numerical solution of the Helmholtz equation, arising from the study of the wave equation in the frequency domain. The approach proposed here differs from those recently considered in the literature, in that it is based on a decomposition that is exact when considered analytically, so the only degradation in computational performance is due to discretization and roundoff errors. In particular, we make use of a multiplicative decomposition of the solution of the Helmholtz equation into an analytical plane wave and a multiplier, which is the solution of a complex-valued advection–diffusion–reaction equation. The use of fast multigrid methods for the solution of this equation is investigated. Numerical results show that this is an efficient solution algorithm for a reasonable range of frequencies.     

Finite element method resolution of non-linear Helmholtz equation

A non-paraxial beam propagation method for non-linear media is presented. It directly implements the non-linear Helmholtz equation without introducing the slowing varying envelope approximation. The finite element method has been used to describe the field and the medium characteristics on the transverse cross-section as well as along the longitudinal direction. The finite element capabilities as, for example, the non-uniform mesh distribution, the use of adaptive mesh techniques and the high sparsity of the system matrices, allow one to obtain a fast, versatile and accurate tool for beam propagation analysis. Examples of spatial soliton evolution describe phenomena not predicted in the frame of the slowing varying envelope approximation.     

On a new one-dimensional equation in the method of phase functions. I

A one-dimensional phase equation is formulated which is useful in the development of quasiclassical scattering. The relation between quasi-classical scattering and perturbation theory is established.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 87–91, 1978.I wish to use this opportunity to express my gratitude to N. I. Zhirnov, Lecturer in the Department of Theoretical Physics of the V. I. Lenin Moscow State Pedagogical Institute (Moscow), for consultations regarding questions of the principles of quasi-classical scattering.     

Inverse scattering for the one-dimensional Helmholtz equation: fast numerical method

The inverse scattering problem for the one-dimensional Helmholtz wave equation is studied. The equation is reduced to a Fresnel set that describes multiple bulk reflection and is similar to the coupled-wave equations. The inverse scattering problem is equivalent to coupled Gel'fand-Levitan-Marchenko integral equations. In the discrete representation its matrix has T?plitz symmetry, and the fast inner bordering method can be applied for its inversion. Previously the method was developed for the design of fiber Bragg gratings. The testing example of a short Bragg reflector with deep modulation demonstrates the high efficiency of refractive-index reconstruction.     

一种新的基于最小平方逼近的广角光束传播方法

用最小平方逼近展开传播算子，实现了一种新的半矢量显式高阶有限差分光束传播方法-这种方法中不需要选择参考折射率，并在整个传播常数（包括辐射模的传播常数）分布区域进行逼近，解决了在泰勒展开和庞德逼近中存在的参考折射率选择和远离展开点误差增大等问题-用这种方法对几种典型波导结构进行了数值模拟，模拟结果验证了算法的正确性和可靠性- 关键词： 光束传播方法 有限差分 集成光学 数值方法     

A high-order numerical method for the nonlinear Helmholtz equation in multidimensional layered media

We present a novel computational methodology for solving the scalar nonlinear Helmholtz equation (NLH) that governs the propagation of laser light in Kerr dielectrics.     

Fast integral equation methods for the modified Helmholtz equation

We present integral equation methods for the solution to the two-dimensional, modified Helmholtz equation, u(x) ? α²Δu(x) = 0, in bounded or unbounded multiply-connected domains. We consider both Dirichlet and Neumann problems. We derive well-conditioned Fredholm integral equations of the second kind, which are discretized using high-order, hybrid Gauss-trapezoid rules. Our fast multipole-based iterative solution procedure requires only O(N) operations, where N is the number of nodes in the discretization of the boundary. We demonstrate the performance of our methods on several numerical examples, and we show that they have both the ability to handle highly complex geometry and the potential to solve large-scale problems.     

On exterior boundary-value problems of the Helmholtz equation not deduced from waves

A boundary integral equation is applied to describe a special kind of exterior Helmholtz boundary-value problem that is not deduced from waves. Then the asymptotic property of O(r ^–2) decay at infinity and the uniqueness of the solution as well as its finite energy property are discussed.     

Fast multipole accelerated boundary element method for the Helmholtz equation in acoustic scattering problems

We apply the fast multipole method (FMM) accelerated boundary element method (BEM) for the three-dimensional (3D) Helmholtz equation, and as a result, large-scale acoustic scattering problems involving 400000 elements are solved efficiently. This is an extension of the fast multipole BEM for two-dimensional (2D) acoustic problems developed by authors recently. Some new improvements are obtained. In this new technique, the improved Burton-Miller formulation is employed to over-come non-uniqueness difficultie...     

Calculating the Green's function for the Helmholtz equation by the method of the fifth parameter

Moscow State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 31, No. 12, pp. 1507–1512, December, 1988.     

Exact nonparaxial beams of the scalar Helmholtz equation

It is shown that three-dimensional nonparaxial beams are described by the oblate spheroidal exact solutions of the Helmholtz equation. For what is believed to be the first time, their beam behavior is investigated and their corresponding parameters are defined. Using the fact that the beam width of the family of paraxial Gaussian beams is described by a hyperbola, we formally establish the connection between the physical parameters of nonparaxial spheroidal beam solutions and those of paraxial beams. These results are also helpful for investigating exact vector nonparaxial beams.     

High-order numerical method for the nonlinear Helmholtz equation with material discontinuities in one space dimension

The nonlinear Helmholtz equation (NLH) models the propagation of electromagnetic waves in Kerr media, and describes a range of important phenomena in nonlinear optics and in other areas. In our previous work, we developed a fourth order method for its numerical solution that involved an iterative solver based on freezing the nonlinearity. The method enabled a direct simulation of nonlinear self-focusing in the nonparaxial regime, and a quantitative prediction of backscattering. However, our simulations showed that there is a threshold value for the magnitude of the nonlinearity, above which the iterations diverge.In this study, we numerically solve the one-dimensional NLH using a Newton-type nonlinear solver. Because the Kerr nonlinearity contains absolute values of the field, the NLH has to be recast as a system of two real equations in order to apply Newton’s method. Our numerical simulations show that Newton’s method converges rapidly and, in contradistinction with the iterations based on freezing the nonlinearity, enables computations for very high levels of nonlinearity.In addition, we introduce a novel compact finite-volume fourth order discretization for the NLH with material discontinuities. Our computations corroborate the design fourth order convergence of the method.The one-dimensional results of the current paper create a foundation for the analysis of multidimensional problems in the future.     

A polynomial expansion method for solving the laser Fokker-Planck equation

Distribution functions of the laser amplitude and intensity can be determined by solving the laser Fokker-Planck equation. By a suitable expansion of the distribution functions in Laguerre polynomials, a system of ordinary differential equations for the coefficients of the expansion is derived and is shown to have the form of a recurrence relation with length four. Applying it to the transient solution, the averaged amplitude and the first four cumulants of the intensity distribution are obtained even for those pump parameters where the hitherto known numerical solution is not applicable.     

Some remarkableR-separable coordinate systems for the Helmholtz equation

We present examples to show that the phenomenon ofR-separation occurs nontrivially for the Helmholtz equation on a pseudo-Riemannian manifold.R-separable coordinate systems can be both orthogonal and non-orthogonal and a given coordinate system mayR-separate in more than one way. A satisfactory theory of variable separation for the Helmholtz equation must incorporateR-separation.Supported in part by NSF Grant MCS 78-26216.     

On the Helmholtz resonator

This paper develops the theory of the excitation of a Helmholtz resonator by external disturbances located arbitrarily close to the mouth of the resonator. The classical approach of Rayleigh is thereby extended to situations in which the disturbance at the mouth is not necessarily equivalent to a uniform, time dependent pressure perturbation. The analysis involves the derivation of the Green function of the resonator in a manner similar to that described in an earlier paper. The use of the Green function is illustrated by two examples in which the resonator is excited by a low Mach number stream of air. In the first case the air stream has a periodic large scale structure such as may be caused by a Kelvin-Helmholtz type of instability. The second example models the case of excitation by a shear layer possessing a continuous spectrum of turbulent eddies. In both of these applications the orders of magnitude of the sound pressure levels involved are illustrated for a typical resonator.     

Light scattering by a “soft” layered sphere: Analysis by the method of spectral expansion in Kotel’nikov-Shannon choice functions

Light scattering by a two-layer “soft” sphere is studied. The analysis is made by the method of expansion in the spatial spectrum in Kotel’nikov-Shannon choice functions. In the case where changes of relative refractive indices of the first (Δn ₁ and the second ((Δn ₂) layers have the same sign, the scattering indicatrices are found to be close to the indicatrices for a homogeneous sphere with averaged effective parameters, and the integrated cross sections should be close as well. However, in the case where Δn ₁ and Δn ₂ have opposite signs, changes in the indicatrix are stronger. The major lobe becomes weaker and, in certain cases, even vanishes, whereas the side lobes and backward scattering increase in intensity. The physical interpretation of the results that can be used for the analysis of experimental data is given.     

A simple approximate method for the solution of the Helmholtz equation in the case of rectangular,non-homogeneous domains

The dynamic behavior of non-homogeneous continuous media is of interest in many fields, from geophysics to everyday mechanical engineering systems and especially in bioengineering.Discontinuous types of non-homogeneities are considered in this paper and it is shown that the Ritz method is convenient for this complicated type of structural element.     

Generalized Riccati equation expansion method and its application to the Bogoyavlenskii's generalized breaking soliton equation

Based on the computerized symbolic system Maple and a Riccati equation, a Riccati equation expansion method is presented by a general ansatz. Compared with most of the existing tanh methods, the extended tanh-function method, the modified extended tanh-function method and generalized hyperbolic-function method, the proposed method is more powerful. By use of the method, we not only can successfully recover the previously known formal solutions but also construct new and more general formal solutions for some nonlinear differential equations. Making use of the method, we study the Bogoyavlenskii's generalized breaking soliton equation and obtain rich new families of the exact solutions, including the non-travelling wave and coefficient functions' soliton-like solutions, singular soliton-like solutions, periodic form solutions.