Full-vectorial finite element beam propagation method withperfectly matched layers for anisotropic optical waveguides

Perfectly matched layer (PML) boundary conditions are incorporated into the full-vectorial beam propagation method (BPM) based on a finite element scheme for the three-dimensional (3-D) anisotropic optical waveguide analysis. In the present approach, edge elements based on linear-tangential and quadratic-normal vector basis functions are used for the transverse field components. To show the validity and usefulness of this approach, numerical examples are shown for Gaussian beam propagation in proton-exchanged LiNbO₃ optical waveguides. Numerical accuracy of the present PML boundary condition is investigated in detail by comparing the results with those of the conventional absorbing boundary condition (ABC)     

A finite element beam propagation method for strongly guiding andlongitudinally varying optical waveguides

A unified finite element beam propagation method is described for both TE and TM waves propagating in strongly guiding and longitudinally varying optical waveguides with magnetooptic materials. In order to avoid nonphysical reflections from the computational window edges, the transparent boundary condition is introduced for both polarizations. In order to show the validity and usefulness of this approach, numerical examples are presented for a directional coupler composed of two parallel identical waveguides, an S-bend, a Y-branching optical isolator, and a 4-ports optical circulator. The present algorithm is, to our knowledge, the first beam propagation method for modeling nonreciprocal magnetooptic components     

An iterative finite difference beam propagation method for modelingsecond-order nonlinear effects in optical waveguides

An iterative finite difference beam propagation method based on the Crank-Nicholson scheme is presented to simulate continuous wave (CW) second-order nonlinear effects in optical waveguides with the depletion of the pump wave taken into account. This method is an extension of the linear finite difference beam propagation method and preserves the same order of accuracy. Comparisons with the previously published explicit finite difference beam propagation method and the rectangular approximation method are presented. Quasi-phase matched difference frequency generation in AlGaAs and quasi-phase-matched second harmonic generation in LiNbO₃ are considered in the evaluation, showing that one iteration for the IFD-BPM is sufficient for the simulation with good accuracy and without increasing much computation time     

Full-vector finite-element beam propagation method for three-dimensional nonlinear optical waveguides

A full-vector finite-element beam propagation method (VFE-BPM) in terms of all the components of slowly varying electric fields is described for the analysis of three-dimensional (3-D) nonlinear optical waveguides. Electric fields obtained with this approach can be directly utilized for evaluating nonlinear refractive index distributions. To eliminate nonphysical, spurious solutions, hybrid edge/nodal elements are introduced. Furthermore, to avoid spurious reflections from the computational window edges, anisotropic perfectly matched layer boundary conditions are implemented, and to reduce computational effort for the nonlinear optical waveguide analysis, an iterative algorithm is also introduced. The effectiveness of the present approach is verified by way of numerical examples: nonlinear directional couplers, spatial soliton emission phenomena, and soliton couplers.     

Modeling second-order nonlinear effects in optical waveguides usinga parallel-processing beam propagation method

In this work, we present a simple efficient numerical solution for the three-dimensional coupled wave equations containing a second-order nonlinearity, using an explicit finite difference beam propagation method (EFD BPM). The linear EFD-BPM is known to be very efficient and to gain large speed up when implemented on parallel computers. The new nonlinear version of the EFD-BPM has the same features of the linear counterpart in using two separate computational windows, one for the fundamental field and the other for the second-harmonic field. We demonstrate the implementation and discuss the application of this method to a nonlinear rib waveguide using the quasi-phase-matching technique     

A wide-angle vector beam propagation method

A wide-angle vector beam propagation method is developed and presented. Considerable improvement in accuracy over the paraxial vector propagating algorithms is achieved with virtually no additional computation     

A new finite-difference-based method for wide-angle beam propagation

A novel split-step finite-difference method for wide-angle beam propagation is presented. The formulation allows 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 requiring only simple matrix multiplication for propagation.     

改进的宽角光束传播法

依据Ｄｏｕｇａｌｓ算子，阐述了一种改进的宽角光束传播法（ＷＢＰＭ），此方法和传统的宽角光束传播法相比，在几乎不增加计算时间和计算机资源的情况下，把原数值计算时横向截断误差从ｏ（Δｘ）＾２减小到ｏ（Δｘ）４。该方法在宽角光束传播计算中，更显示出其优越性。     

有限元光束传输法分析锥形线性波导

文章介绍了有限元光束传输法(FE BPM)和完美匹配层法的结合,并用该法分析了不同折射率比和仰角的锥形线性波导,说明了这种方法的有效性和实用性.     

Improved full-vectorial beam propagation method with high accuracy for arbitrary optical waveguides

An improved finite-difference (FD) full-vectorial beam propagation method is introduced for arbitrary optical waveguides with a dramatic improvement in accuracy. This method is developed based on the generalized Douglas scheme and a novel FD formula for the cross-coupling terms which, expressed in explicit form, is independent of specific types of waveguides. The present method is demonstrated for a strongly guiding rib waveguide.     

Vector finite element solution of saturable nonlinear strip-loaded optical waveguides

A numerical solution is given for nonlinear optical waveguides with power confinement in both transverse dimensions. Self-consistent solutions are obtained by using an accurate vector-finite-element formulation along with the penalty technique. Numerical results for the first quasi-transverse-magnetic power-dependent mode are presented for strip-loaded waveguides with saturable self-focusing media. The variations with total power are illustrated for the modal index and for the fraction of the total power carried by different regions, showing interesting abrupt power switching for realistic geometries. It is shown that the switching effect is maintained in the presence of saturation and over a range of two-transverse-dimensions geometries. This switching effect can be controlled with variation of some of the parameters of the guide.<>     

A bidirectional beam propagation method for periodic waveguides

For waveguiding structures with multiple longitudinal reflecting interfaces, a bidirectional beam propagation method has been developed using a scattering operator formulation and rational approximations to the square root and the propagation operators. For periodic waveguides, it is possible to significantly speed up the method when scattering operators for a waveguide with s₁+s₂ periods are obtained directly from those operators associated with waveguides having s₁ and s₂ periods. The required computing time is then proportional to the logarithm of the total number of periods     

Three-dimensional semi-vectorial wide-angle beam propagation method

A wide-angle Beam Propagation Method (BPM) based on the series expansion technique is proposed. The method employs a finite difference approach and allows extremely dense discretizations. A theoretical analysis supporting this numerical technique is demonstrated. Semi-vectorial formulations are included in this BPM algorithm. It is easily applicable to 3-D simulations. Following an accuracy test with a 2-D problem, several 3-D examples for quasi-TE and -TM modes are demonstrated     

A full-vectorial beam propagation method for anisotropic waveguides

An extension of the full-vectorial beam propagation method to anisotropic media is presented. Optical waveguides made of anisotropic materials can be modeled and simulated. The polarization dependence and coupling due to both the material and the geometric effects are considered     

Accurate finite difference beam propagation method for complexintegrated optical structures

A simple and effective finite-difference beam propagation method in a z-varying nonuniform mesh is developed. The accuracy and computation time for this method are compared with a standard finite-difference method for both the 3-D and 2-D versions     

Bidirectional finite-element method-of-line beam propagation method(FE-MoL-BPM) for analyzing optical waveguides with discontinuities

A bidirectional finite-element method-of-line beam propagation method (FE-MoL-BPM) is newly proposed for analyzing optical waveguides with discontinuities including transmissions, reflections and radiations. In this approach, the finite-element method (FEM) is introduced to discretize the derivatives of the variable perpendicular to the propagation direction. Since the proposed method is accurate and stable, only a small number of nodal points are required     

Semi-vectorial finite element method for arbitrary step-indexoptical waveguides

A simple semi-vectorial finite element method using a new formula is presented which treats only a linearly polarised transversal electric field E_t. The electric field should satisfy the conditions that either E_t or n²E_t is continuous, whether E_t is parallel or perpendicular to the refractive index discontinuous interface in an optical waveguide. This allows accurate and efficient analyses of the polarisation dependence of arbitrary step-index optical waveguides     

Matrix method for determining propagation characteristics of optical waveguides

We have transformed the scalar wave equation into a matrix eigenvalue equation, the diagonalization of which gives the guided modes and a discrete representation of the radiation modes. The method is simple and can be used for arbitrary refractive index profiles.     

Finite element beam propagation method for anisotropic opticalwaveguides

A finite element beam propagation method (BPM) for anisotropic optical waveguides is newly formulated. In order to treat a wide-angle beam propagation, a Pade approximant operator is employed and to avoid nonphysical reflection from computational window edges, a transparent boundary condition is extended to anisotropic materials. To show the validity and usefulness of this approach, the numerical results for an anisotropic planar waveguide and a magnetooptic channel waveguide are presented