A numerical study of solitary waves in a non-linear Bragg reflector

A two-dimensional time domain finite difference beam propagation method, based on the slowly varying envelope approximation is presented. Expressions to correct this approximation and to include third-order non-linear effects are given. The method is applied on solitary waves in a non-linear Bragg reflector, assuming realistic materials parameters.     

等束腰宽度超短脉冲光束传输的脉冲修正方法

利用角谱分析和傅里叶变换的方法,得到一种描述几个周期的等束腰宽度脉冲光束传输的脉冲修正方法。以准单色光束传输的结果为出发点,通过对准单色光束的解进行泰勒级数展开,得到了一种相对简单的修正方法,可以精确的描述具有任意时间波形和横向光束分布的不短于一个周期的超短脉冲光束的传输行为。给出等束腰宽度超短脉冲的近似解,具体研究高斯脉冲光束的传输特性,分析几种不同的频谱对脉冲光束传输行为的影响。     

Non-paraxial Split-step Finite-difference Method for Beam Propagation

A method based on symmetrized splitting of the propagation operator in the finite difference scheme for non-paraxial beam propagation is presented. The formulation allows the solution of the second order scalar wave equation without having to make the slowly varying envelope and one-way propagation approximations. The method is highly accurate and numerically efficient. Unlike most Padé approximant based methods, it is non-iterative in nature and requires less computation. The method can be used for bi-directional propagation as well.     

Semi-analytical finite element analysis of elastic waveguides subjected to axial loads

Predicting the influence of axial loads on the wave propagation in structures such as rails requires numerical analysis. Conventional three-dimensional finite element analysis has previously been applied to this problem. The process is tedious as it requires that a number of different length models be solved and that the user identify the computed modes of propagation. In this paper, the more specialised semi-analytical finite element method is extended to account for the effect of axial load. The semi-analytical finite element method includes the wave propagation as a complex exponential in the element formulation and therefore only a two-dimensional mesh of the cross-section of the waveguide is required. It was found that the stiffness matrix required to describe the effect of axial load is proportional to the mass matrix, which makes the extension to existing software trivial. The method was verified by application to an aluminium rod, where after phase and group velocities of propagating waves in a rail were computed to demonstrate the method.     

Steady state self-focusing, self-modulation of laser beam in an inhomogeneous plasma

Laboratory as well as PIC simulation experiments reveal strong flow of energetic electrons co-moving with laser beam in laser plasma interaction. Equation governing the evolution of complex envelope in slowly varying envelope approximation is nonlinear parabolic equation. Variational approach is used to solve this problem and a Lagrangian for the problem is set up. Assuming a trial Gaussian profile, authors solve the reduced Lagrangian problem for beam width and curvature. Two scale lengths for inhomogenity along the direction of propagation, one for nonlinearity and other for diffraction management are introduced. Self-focusing, self-modulation as well as self-trapping of the laser-electron-beam plasma system is studied under variety of parameters.     

On using moving windows in finite element time domain simulation for long accelerator structures

A finite element moving window technique is developed to simulate the propagation of electromagnetic waves induced by the transit of a charged particle beam inside large and long structures. The window moving along with the beam in the computational domain adopts high-order finite element basis functions through p refinement and/or a high-resolution mesh through h refinement so that a sufficient accuracy is attained with substantially reduced computational costs. Algorithms to transfer discretized fields from one mesh to another, which are the keys to implementing a moving window in a finite element unstructured mesh, are presented. Numerical experiments are carried out using the moving window technique to compute short-range wakefields in long accelerator structures. The results are compared with those obtained from the normal finite element time domain (FETD) method and the advantages of using the moving window technique are discussed.     

Modelling of self-aligned total internal reflection waveguide mirrors: an interlaboratory comparison

Results of modelling of light propagation in 45° self-aligned total internal reflection rib waveguide mirrors on InP substrate are compared. Six laboratories participated in the comparison with the following six modelling methods: the standard fast-Fouriertransform beam propagation method (BPM), the standard finite-difference (FD) BPM using the Crank-Nicholson scheme (two laboratories), the FD-BPM with the correction for the slowly varying envelope approximation, the method of lines, the eigenmode expansion and propagation method, and a simple method based on the field overlap. All the laboratories used the effective-index method to reduce the three-dimensional problem to two dimensions. The differences among the results obtained by different methods are briefly discussed and qualitatively compared to measured values.     

An FD-BPM with adaptive mesh for longitudinally varying optical structures

A finite difference beam propagation method (FD-BPM) algorithm using an adaptive mesh is proposed for optimal use of computer memory and minimization of CPU times in three-dimensional longitudinally varying guiding structures. Preliminary results confirming its feasibility and the advantages of the new algorithm both in CPU and computer memory requirements are described in a 2D case using a Y-junction.     

Propagation of soliton waves in nonlinear dielectric slab waveguides of parabolic index of refraction

The pulse propagation in a non-linear slab waveguide of parabolic index of refraction is treated by using differential equation techniques. A graded index dielectric slab waveguide free of material dispersion with a cubic order non-linearity is considered. The electromagnetic wave inside the waveguide is described in terms of a non-linear equation. Slowly varying envelope function representation is employed to develop a non-linear partial differential equation for the unknown envelope function of the electric field. An averaging method over the transverse direction is applied to reduce the unknown envelope function non-linear differential equation into a form resembling the well known non-linear Schrödinger differential equation. This equation is solved by applying the Inverse Scattering Method. The N-soliton solution is developed and presented explicitly for the practical case of the single mode dielectric slab waveguide. Numerical results presenting single and double soliton propagation are also given.     

A finite element method with mesh adaptivity for computing vortex states in fast-rotating Bose–Einstein condensates

Numerical computations of stationary states of fast-rotating Bose–Einstein condensates require high spatial resolution due to the presence of a large number of quantized vortices. In this paper we propose a low-order finite element method with mesh adaptivity by metric control, as an alternative approach to the commonly used high-order (finite difference or spectral) approximation methods. The mesh adaptivity is used with two different numerical algorithms to compute stationary vortex states: an imaginary time propagation method and a Sobolev gradient descent method. We first address the basic issue of the choice of the variable used to compute new metrics for the mesh adaptivity and show that refinement using simultaneously the real and imaginary part of the solution is successful. Mesh refinement using only the modulus of the solution as adaptivity variable fails for complicated test cases. Then we suggest an optimized algorithm for adapting the mesh during the evolution of the solution towards the equilibrium state. Considerable computational time saving is obtained compared to uniform mesh computations. The new method is applied to compute difficult cases relevant for physical experiments (large nonlinear interaction constant and high rotation rates).     

A semi-analytical finite element formulation for modeling stress wave propagation in axisymmetric damped waveguides

A semi-analytical finite element (SAFE) method is presented for analyzing the wave propagation in viscoelastic axisymmetric waveguides. The approach extends a recent study presented by the authors, in which the general SAFE method was extended to account for material damping. The formulation presented in this paper uses the cylindrical coordinates to reduce the finite element discretization over the waveguide cross-section to a mono-dimensional mesh. The algorithm is validated by comparing the dispersion results with viscoelastic cases for which a Superposition of Partial Bulk Waves solution is known. The formulation accurately predicts dispersion properties and does not show any missing root. Applications to viscoelastic axisymmetric waveguides with varying mechanical and geometrical properties are presented.     

超短脉冲复宗量厄米高斯光束

从超短高斯脉冲光束出发,根据复宗量厄米函数的性质,用理论解析推导的方法,给出了一组新的超短脉冲光束的解析解,称为超短复宗量厄米高斯脉冲光束。此脉冲光束解的每个频率分量都是复宗量厄米高斯光束,时间脉冲的形状是任意的,具有相同的衍射距离参量,并且可以描述短于一个光学周期的超短脉冲。对这种超短脉冲光束及其在自由空间中的传输过程进行了较为细致的研究,分析了超短复宗量厄米高斯脉冲光束的轴上光强、光强的横向分布、等衍射性质、脉冲极性反转、脉冲延迟等基本性质。讨论了引入缓变包络近似后出现的时空奇异性。     

A Novel Three-Dimensional Wide-Angle Beam Propagation Method Based on Split-Step Fast Fourier Transform

A novel three-dimensional wide-angle beam propagation method based on the split-step fast Fourier transform is developed. The formulation is based on the three-dimensional Helmholtz wave equation. Each propagation step is performed by utilizing both the FFT and split-step scheme. The solution of Helmholtz wave equation does not make the slowly varying envelope and one-way propagation approximations. To validate the efficiency and accuracy, numerical results for a propagation beam in a tilted step-index optical waveguide are compared with other beam propagation algorithms.     

Derivation of a wave equation for pulse propagation beyond a slowly varying envelope approximation

An iterative method is used to derive an accurate weve equation that governs ultrashort pulse propagation in a single mode fibre. In the derivation, the slowly varying evelope approximation (SVEA) is not required. Therefore, the wave equation can be used to describe the propagation of an ultrashort pulse of a few optical cycles. It is found that, compared with the wave equation obtained by making an SVEA, an accurate wave equation has the same coefficients of the linear dispersion terms and different coefficients of the high-order time derivatives of the non-linear terms.     

Propagation of cosh Gaussian laser beam in plasma with density ripple in relativistic – ponderomotive regime

Self-focusing of cosh Gaussian laser beam in plasma with periodic density ripple has been investigated. The pondermotive force on electron and the relativistic oscillation of the electron mass causes periodic self-focusing/defocusing of the cosh Gaussian laser beam. The beam converges in the region of high plasma density due to dominance of self-focusing effect over diffraction effect and diverges in the low density region. Non-linear partial differential equation governing the evolution of complex envelope in slowly varying approximation is solved using paraxial ray approximation. The variation of beam-width parameter is studied with distance of propagation for different values of ripple wave number d and decentred parameter b. In order to get strong self-focusing, wavelength and intensity parameters of cosh Gaussian laser beam are optimized.     

超短脉冲复宗量拉盖尔-高斯光束

从超短高斯脉冲光束出发。根据拉盖尔多项式的性质,用理论解析推导的方法。给出了一组新的超短脉冲光束的解析解,称为超短复宗量拉盖尔-高斯脉冲光束。此脉冲光束解的每个频率分量都是复宗量拉盖尔-高斯光束。时间脉冲的形状是任意的,具有相同的衍射距离参量,并且可以描述短于一个光学周期的超短脉冲。对这种超短脉冲光束及其在自由空间中的传输过程进行了较为细致的研究,分析了超短复宗量拉盖尔一高斯脉冲光束的轴上光强、光强的横向分布、衍射性质、脉冲极性反转、脉冲延迟等。讨论了引入缓变包络近似后出现的时空奇异性。     

Computational techniques for the analysis and design of dielectric-loaded plasmonic circuitry

The finite element and the beam propagation method, two widely used methods in photonics, are utilized for the analysis of plasmonic components based on the dielectric-loaded plasmonic waveguide. Two components are chosen as examples and are subsequently numerically investigated by employing the aforementioned methods, in order to demonstrate their applicability in plasmonics. Specifically, a microring resonator add-drop filter and a Mach–Zehnder interferometric switch are analyzed by means of the finite element and the beam propagation method, respectively. The formulation adopted is clearly presented in both cases and the case-dependent implementation details are thoroughly discussed.     

Finite element approximation of the Fokker-Planck equation for diffuse optical tomography

In diffuse optical tomography, light transport theory is used to describe photon propagation inside turbid medium. A commonly used simplification for the radiative transport equation is the diffusion approximation due to computational feasibility. However, it is known that the diffusion approximation is not valid close to the sources and boundary and in low-scattering regions. Fokker-Planck equation describes light propagation when scattering is forward-peaked. In this article a numerical solution of the Fokker-Planck equation using finite element method is developed. Approach is validated against Monte Carlo simulation and compared with the diffusion approximation. The results show that the Fokker-Planck equation gives equal or better results than the diffusion approximation on the boundary of a homogeneous medium and in turbid medium containing a low-scattering region when scattering is forward-peaked.     

Meshless formulation using NURBS basis functions for eigenfrequency changes of beam having multiple open-cracks

This paper presents a meshless formulation using non-uniform rational B-spline (NURBS) basis functions, and its applications to evaluate natural frequencies of a beam having multiple open-cracks. Node-based NURBS basis functions are used to construct the approximation function. The characteristic differentiability of the NURBS basis functions allows it to represent a function having specific degrees of smoothness and/or discontinuity. The discontinuity can be incorporated simply by assigning multiple knots at those locations. Hence, it can yield exact solutions having interior discontinuous derivatives. These advantages of NURBS are well known, and have been used extensively in graphical approximation of geometrical surfaces. However, it is seldom used in other engineering applications. To model the multiple open-cracks in a beam, quartic NURBS basis functions are employed and quadruplicate knots are assigned at the crack locations. Hence, it is capable to model the abrupt changes of slope (the first derivative of displacement) across a crack. In the present applications, additional equivalent massless rotational springs are inserted at the crack locations to represent the local flexibility caused by the cracks. As such, the cracked beam can be treated in the usual manner as a continuous beam. By adopting the meshless Petrov–Galerkin formulation, a generalized stiffness matrix for the cracked beam can be derived. Compared to the conventional finite element method, the present method does not require a finite element mesh for the purposes of interpolation and numerical integration. The advantages and effectiveness of the present method is illustrated in solving the eigenfrequencies of a beam having multiple open-cracks of different depths.     

Spatio-temporal pulse propagation in nonlinear dispersive optical media

We discuss state-of-art approaches to modeling of propagation of ultrashort optical pulses in one and three spatial dimensions. We operate with the analytic signal formulation for the electric field rather than using the slowly varying envelope approximation, because the latter becomes questionable for few-cycle pulses. Suitable propagation models are naturally derived in terms of unidirectional approximation.