Rigorous, full-vectorial source-type integral equation analysis ofcircularly curved channel waveguides

A source-type integral equation method is presented to determine the propagation constants, the radiation losses, and the electromagnetic field distributions of the discrete (“guided”) modes in circularly curved, integrated optical channel waveguides embedded in a homogeneous background. The method can be extended to the case of a multilayered background, e.g. a ridge waveguide. The source-type integral equation forms an eigenvalue problem, where the electric field strength represents the eigenvector. This problem is solved numerically by applying the method of moments. Numerical results are presented for various rectangular channel waveguides situated in a homogeneous embedding and compared with those of other modeling methods     

Full-wave analysis of dielectric frequency-selective surfaces using a vectorial modal method

A novel vectorial modal method is presented for studying guidance and scattering of frequency-selective structures based on lossy all-dielectric multilayered waveguide gratings for both TE and TM polarizations. The wave equation for the transverse magnetic field is written in terms of a linear differential operator satisfying an eigenvalue equation. The definition of an auxiliary problem whose eigenvectors satisfy an orthogonality relationship allows for a matrix representation of the eigenvalue equation. Our proposed technique has been applied to the study of lossy all-dielectric periodic guiding media with periodicity in one dimension. This method yields the propagation constants and field distributions in such media. The reflection and transmission coefficients of a single layer under a plane-wave excitation can be obtained by imposing the boundary conditions. Study of the scattering parameters of the whole multilayered structure is accomplished by the cascade connection of components as characterized by their scattering parameters. Results obtained with this method for the propagation characteristics of a one-dimensional periodic dielectric medium are compared with those presented by other authors, and results for the scattering of several dielectric frequency-selective surfaces (DFSS) are compared with both theoretical and experimental results presented in the literature, finding a very good agreement. A symmetrical band-stop filter with a single waveguide grating is also demonstrated theoretically.     

A K-space method of moments solution for the apertureelectromagnetic fields of a circular cylindrical waveguide radiatinginto an anisotropic dielectric half-space

A method of moments solution is presented for the problem of determining the electromagnetic aperture fields that occur when a circular cylindrical waveguide radiates into an anisotropic dielectric-covered ground plane. The method of moments solution is derived by using a reaction matching technique and by using K-space theory to develop the aperture matrix elements of the system. An example of radiation into an anisotropic dielectric is given. The anisotropic dielectric for this example is assumed to be formed by the presence of a static magnetic field placed within a plasma     

Rigorous boundary integral equation solution for general isotropicand uniaxial anisotropic dielectric waveguides in multilayered mediaincluding losses, gain and leakage

Bounded and leaky eigenmodes of arbitrary shaped polygonal dielectric waveguides embedded in a multilayered medium are determined based on a rigorous full-wave analysis. The dielectric waveguides consist of isotropic or uniaxial anisotropic material. Losses and gain inside the layers and the waveguides are allowed. The eigenmodes are determined with a boundary integral equation technique in conjunction with the method of moments. Results for the propagation constants are presented for a number of waveguides and, where possible, compared with published data. Special attention is devoted to the transition from a dielectric waveguide to a perfectly conducting waveguide when the loss tangent of the waveguide material changes from zero to infinity     

Analysis of waveguide antenna arrays with protruding dielectricelements

The two-dimensional problem of radiation of TE and TM waves from a waveguide array with protruding smooth dielectric elements of arbitrary shape is considered, and solution algorithms are suggested. The algorithms are based on applying the method of auxiliary sources for the representation of electromagnetic fields outside and inside the protrusions in combination with the method of integral equations for the electric field at the waveguide aperture. The point matching of the field tangential components on the protrusion-to-free-space boundary and at the waveguide aperture is used to reduce the problem to a system of linear algebraic equations for the amplitudes of the auxiliary filamentary currents and of the waveguide aperture electric field, which is assumed to be piecewise constant. The amplitudes obtained from the solution of the system are used for computing the array reflection coefficient and element pattern, which are shown in some cases to be significantly dependent on the protrusion shape. Examples of arrays with flat-topped element patterns resulting from array geometry numerical optimization are also presented     

The Method of Calculation of Open and Closed Waveguides,Which Is Based on the Modified Method of Discrete Sources

The problem of calculation of eigenmode characteristics is considered for 3D open dielectric waveguides and waveguides with the anisotropic impedance boundary condition. An efficient algorithm is proposed for finding eigenmodes of waveguides having different cross-sections. It is based on the solution of the auxiliary problem of diffraction of the field of a filamentous source located inside the waveguide. The method is tested using weakly directing dielectric waveguides having circular and elliptical sections and a waveguide with the two-sheeted section. The results of calculation of dielectric waveguides obtained with the help of the proposed method, the method of finite elements, and the method of the integral equation over the waveguide section are compared. The circular and elliptical waveguides and the waveguide having a two-sheeted section with an anisotropic impedance are also investigated. The calculation results are compared with the results obtained with the use of the method of separation of variables and the Galerkin method.     

Domain integral equation analysis of integrated optical channel andridge waveguides in stratified media

A domain integral equation approach to computing both the propagation constants and the corresponding electromagnetic field distributions of guided waves in an integrated optical waveguide is discussed. The waveguide is embedded in a stratified medium. The refractive index of the waveguide may be graded, but the refractive indices of the layers of the stratified medium are assumed to be piecewise homogeneous. The waveguide is regarded as a perturbation of its embedding, so the electric field strength can be expressed in terms of domain integral representation. The kernel of this integral consists of a dyadic Green's function, which is constructed using an operator approach. By investigating the electric field strength within the waveguide, it is possible to derive an integral equation that represents an eigenvalue problem that is solved numerically by applying the method of moments. The application of the domain integral equation approach in combination with a numerically stable evaluation of the Green's kernel functions provides a new and valuable tool for the characterization of integrated optical waveguides embedded in stratified media. Numerical results for various channel and ridge waveguides are presented and are compared with those of other methods where possible     

Finite-Element Analysis of Dielectric-Loaded Waveguides

A finite-element analysis in which nonphysical spurious solutions do not appear has been established to solve the electromagnetic field problem of the closed waveguide filled with various anisotropic media. This method is based on the approximate extremization of a functional, whose Euler equation is the three-component curlcurl equation derived from the Maxwell equations, with a new conforming element. Specific examples are given and the results are compared with those obtained by exact solutions and longitudinal two-component finite-element solutions. Very close agreement was found and all nonzero eigenvalues have been proved to have one-to-one correspondence to the propagating modes of the waveguide.     

Modification of Marcatili's method for the calculation ofanisotropic rectangular dielectric waveguides

Mareatili's method for the calculation of rectangular dielectric rod waveguides has been modified for uniaxial anisotropic dielectric materials. The averaging method without guessing the field distribution has been used to derive the equations. Experimental results for an anisotropic sapphire waveguide show a good agreement with those calculated using the approach developed here     

Moment method solution of double step discontinuities in waveguides

The problem of scattering of a double step waveguide discontinuity is investigated. Two coupled sets of equations in terms of the unknown magnetic current sheets on the apertures are obtained and solved for the electric field distribution using the moment method. The scattering matrix coefficients describing reflection, transmission, and mode conversions at a symmetric double step waveguide discontinuity are obtained and presented graphically for specific junctions.     

The discontinuity problem of a rectangular dielectric post in arectangular waveguide

A simple numerical technique for the solution of the discontinuity problem of a symmetric loaded rectangular dielectric post centered in a rectangular waveguide is presented. The waveguide is divided into three regions where the field is expressed in suitable waveguide modes. By applying the continuity condition at the common surfaces of the regions, a system of linear equations determining the reflection and transmission coefficients is formed. Several examples are compared with experimental results and show the validity of the method     

Propagation characteristics of microstrip transmission line on ananisotropic material ridge

A microstrip transmission line residing on an electrically anisotropic material ridge embedded in a multilayered environment is studied using a coupled set of integral equations (IE's). The full-wave IE formulation easily accommodates arbitrary material anisotropy and inhomogeneity in the finite ridge region using equivalent polarization currents residing in a multilayered isotropic background. New results are presented for uniaxially anisotropic ridge structures which show that the transmission line propagation constant is sensitive to anisotropy for certain ridge structures and insensitive for others, compared to the conventional line on an infinite substrate. Results are also presented for a transmission line printed on a nonreciprocal solid-state magnetoplasma ridge. The current distribution associated with the dominant microstrip mode is investigated, where it is found that the transverse component of current is much larger for the ridge geometry than for the infinite substrate case, although the transverse component is still small compared to the longitudinal component     

Improved coupled-mode theory for anisotropic waveguide modulators

An improved coupled-mode theory for the propagation of modulated light waves in anisotropic dielectric waveguides is presented. Starting from Maxwell's equations, a partial differential equation is derived to describe mode-coupling between two normal modes which may propagate in anisotropic waveguide systems under modulation. The theory is applied to the analysis of typical waveguide modulators; examples for LiNbO₃ phase modulator, Mach-Zehnder, and directional coupler modulator are presented. The theory is applicable to both bulk and waveguide modulators/switches from DC to the high-frequency band. The only limitation is that the modulating wave has to propagate collinearly to the light waves     

Modes of a symmetric slab optical waveguide in birefringent media--Part 1: Optical axis not in plane of slab

We derive the eigenvalue equation and the expressions for the field distribution for the TM modes of an anisotropic symmetric slab waveguide; the TE modes are identical to the isotropic case. It is assumed that the field components do not depend on theycoordinate and that thecaxis of the uniaxial crystal lies in thex-zplane. We find that, in spite of the symmetry of the waveguide, even and odd modes do not exist because the phase fronts of the modes are tilted with respect to the waveguide axis. However, the absolute magnitude of the Hycomponent of the field has even or odd symmetry. The interesting case of an anisotropic slab waveguide with the optical axis located in the plane of the slab will be presented in Part II of this paper.     

Circularly Symmetric Optical Waveguide with Strong Anisotropy

A solution to the problem of wave propagation in an anisotropic, circularly symmetric optical waveguide is presented. Exact solutions are given for the step-index case when the core and the cladding consist of uniaxial materials with their optical axes parallel with the axis of the cylindrical waveguide.     

Moment method with isoparametric elements for three-dimensionalanisotropic scatterers

A new method for computing the frequency-domain electromagnetic fields scattered from, and penetrating into, arbitrarily shaped, three-dimensional, lossy, inhomogeneous anisotropic scatterers is presented. The method is based on a general volume integro-differential formulation of the scattering problem, and consists of the numerical solution of the coupled integral equations by the moment method and point matching. A particularly powerful feature of this method is that the numerical model of the scatterer is obtained by parametric volume elements and the basis functions used to represent the field within each element are the same used in the finite-element method. Element integration problems due to the singular kernel of the integral equations are treated in some detail. Numerical results for both the isotropic and the anisotropic spherical scatterer are presented, including comparisons with results obtained by different numerical methods for the isotropic cases considered. The capability of the numerical code presented here to deal with cases where the material parameters of the scatterer are given by singular matrices is discussed for two particular examples     

A Planar transversal junction of two planar waveguides

A rigorous analytic method is used to solve the wave-propagation problem for a planar transversal junction of two planar waveguides. The problem is reduced to an iteratively solved system of integral equations of the second kind. Dependences of the transmission and reflection coefficients of guided waves, radiation patterns, and the power of the radiation field on the ratios of the waveguide thicknesses and the waveguide permittivities are obtained.     

Low-loss optical branching waveguides consisting of anisotropicmaterials

Low-loss branching waveguides of the mode-conversion type consisting of anisotropic materials are proposed, and their basic wave-guiding characteristics are studied by means of coupled-mode theory. Two mode-conversion sections are introduced on both input and output sides of a conventional symmetric branching waveguide. Each arm of the branching waveguides is assumed to be a single-mode slab waveguide except for the tapered section. A coupled-mode system of equations describing mode-conversion phenomena with respect to the transverse magnetic (TM) mode in the branching waveguides is derived from the field expansion in terms of local normal modes. A Runge-Kutta-Gill method is used to numerically solve the coupled-mode equations. It is found that the proposed branching waveguides suffer mode-conversion losses to a much lesser extent than conventional branching waveguides     

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     

A computation of the reflection and transmission coefficients for dielectric coated waveguides

A previously developed variational technique for finding the approximate solution to the electromagnetic field inside waveguides of varying shapes and containing non-uniform dielectric and magnetic materials is applied to the specific case of an axisymmetric waveguide containing three layers of different dielectric materials. The magnetic permeability is taken equal to unity. The method enables the problem to be reduced to a two point boundary value problem for a pair of second order, linear, ordinary differential equations. The values of the reflection and transmission coefficients obtained by this method are in good agreement with those derived from solving the partial differential equations for the field.