Approximate scalar finite-element beam-propagation method withperfectly matched layers for anisotropic optical waveguides

The perfectly matched layer boundary condition for arbitrary anisotropic media is incorporated into the approximate scalar beam propagation method. The procedure is based on a finite-element method for three-dimensional anisotropic optical waveguides with off-diagonal elements in a permittivity tensor. In order to treat a wide-angle beam propagation, the Pade approximant operator is employed. To show the validity and usefulness of this approach, numerical results are presented for Gaussian beam propagation in free space and Gaussian beam excitation on a three-dimensional anisotropic optical waveguide     

Finite element analysis of diffused anisotropic optical waveguides

Results are presented for anisotropic waveguides, with an arbitrary permittivity tensor, being diffused in both the transverse directions and by using the finite element method with the vector H-field formulation for the analysis. The importance of considering the waveguide core dimensions to be greater than the diffusion depth in both the transverse directions, the use of extrapolation techniques and of a symmetry plane for anisotropic waveguides are also discussed     

Eigenmode analysis of anisotropic optical fibres or integrated optical waveguides

Using the finite-element technique, the eigenmodes propagating in an inhomogeneous, anisotropic optical waveguide with arbitrary cross-section are computed. The method is based on a variational principle for the modes of an anisotropic dielectric waveguide. It is shown that very accurate results are obtained, even for the guides formed by very small changes in the refractive index. As an example, the propagation characteristics of a channel guide embedded in LiNbO3 are computed.     

Nonlinear finite-element semivectorial propagation method forthree-dimensional optical waveguides

A nonlinear semivectorial beam propagation method based on the finite element method is presented. It applies to three-dimensional optical waveguides and accounts for polarization effects. To show the usefulness of this approach numerical results regarding both E^x (quasi-TE) and E^y (quasi-TM) waves are presented for directional couplers with nonlinear cores. The limits of scalar formulations are highlighted especially when working near and above the coupler threshold power     

Full vectorial finite-element solution of nonlinear bistable optical waveguides

The accurate computation of the propagation constants and field distributions of different modes in nonlinear optical dielectric waveguides is addressed in this paper. Using the vector finite-element formulation of the beam propagation method, combined with the imaginary distance propagation technique, both linear and nonlinear modes can be accurately calculated. The proposed technique is applied to obtain the fundamental TE nonlinear mode of a strip-loaded waveguide, and the excellent agreement seen with published results shows its high numerical precision.     

Improved coupled-mode theory of anisotropic optical waveguides

A coupled-mode equation for anisotropic waveguide systems of arbitrary cross section and general dielectric distribution is derived. Numerical results comparing the exact calculations to those of the method of Hardy et al. (Opt. Lett., vol.11, 742-4, 1986) show that the same accuracy can be obtained not only for TE, but also for TM mode coupling in the case of anisotropic waveguides, and the improved coupled-mode theory is applicable to the situation when moderately strong coupling occurs under the condition where the edge-to-edge separation of two coupled guides D₂ is about 0.1 μm     

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.     

Modes in anisotropic optical waveguides formed by diffusion

Solutions of the wave equation are obtained for planar guides whose dielectric constant ε is anisotropic and decreases exponentially in the direction perpendicular to that of wave propagation. Although for TE modes the solutions are quite similar to those obtained earlier for exponentially varying but isotropic ε, there are some singificant differences for TM modes. The condition for single-mode operation and the number of modes of such guides are given in terms of the exponential decay rate, the difference between the surface and bulk values of ε, and the wavelength. The results are illustrated for the cases of Se-diffused CdS and out-diffused LiNbO3.     

Finite-element solution of planar arbitrarily anisotropic diffused optical waveguides

The finite-element method for propagation in planar anisotropic diffused optical waveguides with arbitrary permittivity tensor is presented. A Galerkin procedure has been introduced to the finite-element formulation, to study both the nonleaky and leaky surface waves. The complex propagation constants are determined as a function of frequency for possible modes of propagation. The accuracy of the method has been checked by calculating the nonleaky and leaky surface waves of Ti-diffused LiNbO3waveguides with Gaussian index profiles. The numerical results of Ti-diffused LiNbO3waveguides with dielectric overlays are also presented and the effects of dielectric overlays on the propagation characteristics for the nonleaky and leaky surface waves are examined.     

Simple and efficient finite-element analysis of microwave andoptical waveguides

A simple and efficient finite-element method for the analysis of microwave and optical waveguiding problems is formulated using three components of the electric or magnetic field. In order to eliminate spurious solutions, edge elements are introduced. In the edge element approach the nodal parameters are not limited to the magnetic field as in the conventional three-component formulation for the dielectric waveguiding problem. An eigenvalue equation that involves only the edge variables in the transversal plane and can provide a direct solution for the propagation constant is derived. To show the validity and usefulness of this approach, computed results are illustrated for microstrip transmission lines and dielectric waveguides     

Comparison of two interactive finite-element programs for analysisof optical and microwave waveguides

This paper presents a comparison of two- finite-element programs developed for accurate cross-sectional analysis of arbitrary optical and microwave waveguiding structures. Both techniques were developed in response to the growing demand from scientists and engineers for application-specific expert systems, and employ advanced interactive pre- and post-processing facilities. They show good agreement, strongly complement one another, and could form the core of a whole library of similar “intelligent” software for computer-assisted analysis and design, with a wide range of possible applications within both supercomputer and workstation environments     

Sensitivity analysis of optical waveguides

The change in the characteristics of dielectric slab and fiber waveguides due to dimensional tolerances of their parameters has been analyzed. The analysis is helpful in fabrication and use of such guides in optical communication systems.     

Failure of scalar analyses in dielectric optical waveguides with perturbed refractive-index profile

It is theoretically shown that scalar analyses cannot properly predict the propagation constant of a particular mode in circular dielectric optical waveguides when the angular order of the mode satisfies some condition with respect to the order of the Fourier spectrum of the perturbed refractive-index profile.     

Coupled-mode analysis of anisotropic dielectric planar branching waveguides

We present a theoretical study of mode propagation in anisotropic planar branching waveguides consisting of isotropic and anisotropic layers. The characteristic equation of anisotropic five-layered structure is introduced to calculate the propagation constants and field patterns of the branching-waveguide modes. Coupled-mode analysis shows that the normalized parameter is effective for the evaluation of mode-coupling effects in the anisotropie branching waveguides as well as in the isotropic branching waveguides.     

Novel vectorial analysis of optical waveguides

A nonlinear iterative (NLI) method, originally developed by Hewson-Browne in geomagnetism, is applied to the vectorial analysis of optical waveguides. The method explicitly shows the interrelations between the scalar, polarized and vectorial operators and can be conveniently implemented using finite difference methods. Excellent accuracy in the normalized propagation constant is claimed along with agreement with earlier work and field distributions. The method presented enables vector results to be obtained simply by using computer programs, often available, for either polarized or scalar modes. This procedure proves more efficient than a standard vectorial finite difference technique     

Hermitian finite-element method for inhomogeneous waveguides

A finite-element method (FEM) based on Hermitian fifth-degree polynomials is established in order to determine the field within a closed waveguide filled with inhomogeneous material. As with the method based on the Lagrangian approximation, spurious solutions are eliminated when the divergence-free constraint is satisfied and the boundary conditions are explicitly enforced. However, the smooth (C¹) Hermitian approximation allows the direct elimination of the axial field component in each triangle element. This procedure results in a reduction of the computer memory needed and in programming efficiency. As the Hermitian FEM uses smooth basis functions, the method also increases the quality of the field solutions. The method has been applied to mode characterization in waveguides. Several comparisons with Lagrangian FEM demonstrate the advantages of the Hermitian FEM. Some difficulties arising in cases of waveguides with sharp edges are discussed. A solution based on mesh refinement near the sharp edges is proposed     

Full vectorial finite-element analysis of sharp optical waveguidecorners

The transmission efficiency of optical waves through a guided-wave structure, incorporating a sharp corner, is investigated by using rigorous numerical approaches based on the finite-element method. To show the versatility of the proposed numerical approaches, the modal power transmission coefficient of two rib waveguides is calculated, and the results are compared with those obtained by using other approaches     

Numerical analysis of three-parallel embedded optical waveguides

Three parallel embedded dielectric waveguides are investigated numerically. The mode-matching method that matches the boundary conditions in the sense of least squares is applied to this problem, using the hybrid-modal representation. Precise numerical results for the dispersion relations and field distributions are presented for the lowest three modes near the cutoff. The frequency range in which only the three modes can propagate and the differences in their propagation constants are discussed at length, in order to highlight the available frequency range and the coupling length of the configuration with respect to its use as a directional coupler     

Numerical analysis of nonlinear bistable optical waveguides

Numerical solutions illustrating the onset of bistability and hysteresis are presented for the symmetrical step index, asymmetrical step index, and asymmetrical diffused slab nonlinear optical waveguides. Two different numerical techniques have been used independently-a finite-element method and a variational method. Both methods produce numerically stable solutions, and agreement between them is good for both increasing and decreasing total power from below or above the threshold power. The results are compared with analytical solutions for these structures. The onset and the end of the physically unstable solutions regime coincides with the two power thresholds for the increasing and decreasing powers     

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