Numerical simulation of an arbitrarily ended asymmetrical slab waveguide by guided-mode extracted integral equations

The scattering phenomenon from an arbitrary-shaped end of a asymmetrical slab waveguide for the cases of TE and TM guided modes is simulated by means of boundary integral equations that are called guided-mode extracted integral equations. The integral equations that we derive can be solved by the conventional boundary-element method. Numerical results are presented for problems of three-layer asymmetrical waveguides with tilted ends. The reflection coefficient, reflected and scattered powers, and radiation patterns are calculated numerically for the cases of incident TE and TM guided modes.     

Analysis of mode scattering from an abruptly ended dielectric slab waveguide by an accelerated iteration technique

A new modification of the integral equation method using an iteration technique with "accelerating" parameters is presented to solve the problem of guided-mode scattering from an abruptly ended asymmetrical slab waveguide. The optimal choice of the parameters is shown to be closely connected with the variational principle. The electric-field distribution at the terminal plane, the reflection coefficient of the guided mode, and the far-field radiation pattern are computed. Numerical results are presented for several cases of abruptly ended waveguides, including the systems with constant and variable profiles of the refractive indices. The phenomenon of the radiation pattern rotation is examined in detail.     

Analysis of coupling between two slab waveguides in the presence of ring resonators

The coupling phenomena between two slab waveguides in the presence of ring resonators are investigated through a rigorous integral equation analysis. A Green's-function-theory approach is utilized to develop the integral equation formulation. The solution is obtained by applying an entire-domain Galerkin technique, using the orthogonality properties of involved wave functions in the ring resonators. The field's transmission and reflection coefficients are accurately computed and a practical directional-coupler approach is discussed. The presented results reveal strong resonant coupling phenomena between the two waveguides and the resonators.     

Metal iris influence on guided-mode diffraction

The effects of the presence of metal irises on guided-mode propagation through a symmetrical three-layer slab waveguide are examined by using the integral equation method. The aperture electric field distribution is expressed in terms of a finite series of Chebyshev polynomials. The modal reflection and transmission coefficients, the near-field structure, and the far-field radiation pattern are calculated, while numerical results are presented for several iris apertures.     

Modulation of coupling in a photonic switch by resonant interference

A novel photonic switch structure is described in which the coupling of light between two fiber waveguides is controlled by the resonant interference of a third waveguide. The switching action is controlled by a small variation of the index of refraction of the control waveguide by the application of either photo-optical (Kerr) techniques or electro-optical (Pockels) techniques. The control waveguide can be either a fiber waveguide or a slab waveguide. The equations for the waveguide coupling were obtained by analytical approximations from coupled-mode theory. A beam-propagation simulation was also used. The results of the two models were compared. Multiple resonant interferences were observed in the case of a slab waveguide.     

Reflectivity properties of an abruptly ended asymmetrical slab waveguide for the case of transverse magnetic modes

We deal with the scattering phenomenon from an abruptly terminated asymmetrical slab waveguide for the case of transverse magnetic (TM) modes. The analysis uses both the integral equation method and the variational technique. The reflection coefficient of the dominant TM guided mode and the far-field radiation pattern are computed, and the discontinuity of the electric field distribution on the core-clad interface is exhibited. Numerical results are presented for several cases of abruptly ended waveguides, including the three-layer slab guide and the structure with variable profile of the refractive index.     

Reflectivity of buried slab waveguides

The scattering properties of an abruptly ended buried slab waveguide for both TE and TM modes are examined by an improved iteration technique that is based on the integral equation method with "accelerating" parameters. The waveguide is considered a symmetrical slab, for which the weakly guiding conditions are invalid, and it is embedded in a different dielectric material. The tangential electric field distribution on the terminal plane, the reflection coefficient of the first TE and TM guided modes, and the far-field radiation pattern are computed. Numerical results are presented for several ended waveguides, while special attention is given to the far-field radiation pattern rotation and the terminal field distributions.     

Guided-mode analysis by the Lanczos-Fourier expansion

The Lanczos-Fourier series expansion is employed to analyze the guided-mode field in an asymmetrical slab waveguide, the core of which has an anisotropic and inhomogeneous dielectric permittivity. A system of linear homogeneous equations is derived by the collocation technique with consideration of the wave equation and the appropriate boundary conditions at the interfaces between the core and cladding media. The propagation constants are found from a determinant equation that ensures the existence of a nontrivial solution of the system. Numerical results are presented for several cases of dielectric permittivity, including the constant, parabolic, linear, and anisotropic cases. This approach is found to converge reasonably fast, and Richardson's extrapolation technique is applied to accelerate the convergence further. The approach can be easily generalized from the scalar to the vector equation, and, as an example, we consider the guided modes of a circular fiber.     

Vectorial modal analysis of dielectric waveguides based on a coupled transverse-mode integral equation. II. Numerical analysis

We present numerical implementation and verification of a rigorous full-vector, integral-equation formulation suitable for analyzing modal characteristics of complex, two-dimensional (2D) rectangular-like dielectric waveguides. By dividing the waveguide into vertical slices, a system of integral equations we call vector-coupled transverse-mode integral equations (VCTMIE) is derived. The entire electromagnetic mode fields are completely determined by one-dimensional unknown field functions on the slice interfaces. To further reduce numerical computation, we expand these functions in terms of the guiding modes of a slab waveguide with a large normalized frequency. Through orthogonal projection the resulting nonlinear eigenvalue and eigenvector matrix formulation enables us to obtain the effective mode index with 10(-7) precision and to compute with high resolution the 2D vectorial mode field solutions of an open dielectric waveguide. We show stable and speedy convergence of our method as well as techniques to overcome the Gibbs phenomenon in the reconstruction of the transverse fields.     

Analytical model of spatial four-waveguide coupler

Analytical model of spatial four-guide coupler is presented by utilizing Marcatili's principle, where the spatial four-guide coupler can be described by two perpendicular five-layer slab waveguides. The common expressions to calculate the effective indices and the mode field distributions are shown in this paper. Introducing perturbation corrections to take the effect of the distinct regions into account, more accurate results can be obtained. The effective refractive indices and coupling lengths of specific examples are investigated to verify the feasibility of the analytical model by comparing with the numerical approaches such as finite difference (FD) numerical calculation and beam propagation method (BPM).     

Channel waveguides of insoluble conjugated polymers

We demonstrate a fast and simple technique of patterning insoluble, nonbleachable, conjugated polymer films into channel waveguides by striploading with photoresist. Two types of coupling gratings were tested for this new system. The coupling parameters, the linear losses and the beam confinement are discussed. This technique makes it possible to explore this class of materials in the field of integrated optics. A new data evaluation method for a more precise measurement of losses in slab waveguides is introduced.     

An arbitrarily-oriented plane crack in an anisotropic elastic slab

The problem of an arbitrarily-oriented plane crack in an anisotropic elastic slab is considered. Through the use of a Fourier transform technique, the problem is reduced to a system of simultaneous Fredholm integral equations of the second kind. Once these integral equations are solved, relevant quantities such as the crack energy can be readily computed. Numerical results pertaining to the stability of a plane crack in a particular elastic slab are given.     

Guided modes in a uniaxial multilayer

An algorithm is presented for simulation of guided modes in a multilayer uniaxial structure with each layer characterized by its own ellipsoid of refractive indices and direction of optical axis. The proposed approach is based on presenting an electromagnetic field in each layer as a linear combination of ordinary and extraordinary waves coupled through the boundary conditions. The problem is reduced to two dimensions by considering the waves with a given projection of the wave vector on the plane of the waveguide. No a priori assumption about the guided-mode polarization is required in this method. Hybrid polarized modes appear naturally as solutions of a system of linear equations with respect to the amplitudes of the ordinary and extraordinary waves. The proposed approach covers a wide variety of important practical cases including isotropic waveguides, surface waves at the boundary between positive uniaxial crystal and isotropic medium, surface plasmons at metallic interfaces, uniaxial multilayers in a very general form, and leaky modes in such structures.     

Attenuation in waveguides on FR-4 boards due to periodic substrate undulations

The guided-mode attenuation associated with optical-interconnect-polymer waveguides fabricated on FR-4 printed-circuit boards is quantified. The rigorous transmission-line network approach is used and the FR-4 substrate is treated as a long-period substrate grating. A quantitative metric for an appropriate matrix truncation is presented. The peaks of attenuation are shown to occur near the Bragg conditions that characterize the leaky-wave stop bands. For a typical 400 microm period FR-4 substrate with an 8 microm corrugation depth, a buffer layer thickness of about 40 microm is found to be needed to make the attenuation negligibly small.     

Guiding mode expansion of a TE and TM transverse-mode integral equation for dielectric slab waveguides with an abrupt termination.

We propose a rigorous transverse-mode integral equation formulation for analyzing TE and TM electromagnetic radiation fields on the facet of dielectric slab waveguides with an abrupt termination in free space. Both exact waveguide guiding modes and discretized radiation modes are included in the kernels of the integral equation. To reduce the size of the matrix that approximates the exact integral equation, we expand the unknown field at the junctions in terms of guiding modes of a selected waveguide with sufficiently large normalized frequency and core thickness. By direct matrix inversion, we obtain numerical solutions of the scattered fields at the junctions. Our method can be used to study the field distribution as well as the energy reflection and transmission coefficients of dielectric waveguides with multiple step discontinuities.     

Application of time-domain finite-volume method to some radiation problems in two and three dimensions

Maxwell's curl equations in the time domain are cast into a conservation form and discretized using a finite-volume procedure. The Lax-Wendroff explicit scheme is then used to solve the discretized Maxwell's equations, resulting in second-order accuracy with respect to both time and space. This time-domain scheme does not require an interlaced mesh for electric and magnetic fields. Radiation pattern and input impedance of some typical parallel-plate waveguides and mutual coupling for arrays of parallel-plate waveguides are computed and, whenever possible, compared with closed form exact or approximate solutions. Radiation patterns for a two-dimensional horn and a rectangular waveguide are presented.<>     

Guided-mode resonance filter with an antireflective surface consisting of a buffer layer with refractive index equal to that of the grating

A type of guided-mode resonance filter (GMRF) with an antireflective surface consisting of a buffer layer with refractive index equal to that of the grating is proposed, and the approximate design approach is presented. The relation between the filter linewidth and the coupling loss is used to analyze the filter properties by using different derivation methods. It is shown that the dispersion equation of the slab waveguide may provide a reliable approximation in estimating the resonance locations of the GMRF with an antireflective surface. The buffer layer functions as an intermediate layer between the grating and waveguide layers. This reduces the coupling and out-coupling of a mode of the waveguide, which results in significant reduction of the coupling loss and the filter linewidth with the antireflection condition nearly preserved. By changing the thickness of the buffer layer, different linewidths can be obtained with spectral symmetry and sideband suppression almost kept the same. The slight shift of resonance wavelengths due to the variety of the buffer layer thickness and the etching effects can be adjusted to the design value by changing the grating period. Accurate etch depth control to avoid underetching is necessary. The electric field distributions under resonance conditions shows that the buffer layer increases the mode confinement, thus narrowing the filter linewidth.     

Analysis of an optically controlled photonic switch

The principle that the coupling of light between two fiber waveguides can be controlled by the resonant interference of a third waveguide has been developed [Attard, Appl. Opt. 37, 2296-2302 (1998)]. Here significant details concerning the operation of a photonic switch are obtained, and a more complete analysis is presented. Multiple-resonant conditions are identified for slab and fiber control waveguides at large indices of refraction. Thus a selection of materials with an appropriate refractive index and a Kerr coefficient is rendered more easily. Furthermore it is shown that the light used to control the index of refraction in the control waveguide does not enter the output of the photonic switch but remains confined to the control waveguide, for either a slab or a multimode fiber control waveguide. Spatial fluctuations of the control light beam in the control waveguide do not affect the operation of the photonic switch. Tolerances have been determined for the spacing between the control waveguide and the photonic coupler and also for the index of refraction of the control waveguide.     

Boundary element analysis of reinforced shear deformable shells

In this paper, stiffened shear‐deformable shells are analysed using the boundary element method. Coupled boundary integral equations are presented for describing curved shells under general loading conditions. The equations are based on boundary integral equations for plane stress and plate bending, with coupling terms arising from the curvature of the shell. Domain integrals are transformed into boundary integrals using the dual reciprocity technique. Stiffeners are modelled as curved beams, continuously attached to the shell. Numerical solutions calculated using the present method are compared with finite element results in two examples. Copyright © 2002 John Wiley & Sons, Ltd.