Numerical Solution of Coupling Between Two Collinear Parallel-Plate Waveguides

The problem of coupling between two collinear parallel-plate waveguides is investigated numerically using moment methods. The exciting mode in the waveguide is assumed as the incident field, and the integral equation for the induced current is expressed in terms of the reflected, transmitted, and evanescent currents on the waveguides. The integral equation is then solved numerically by a point-matching method and the reflection and the transmission coefficients and the radiated fields are obtained. To examine the accuracy of the results, the special case of a semi-infinite exciting waveguide coupled to a finite coupled waveguide is also considered and is solved numerically by treating the singularities of the induced currents using a transformation method. For a TE/sub 0,1/ excitation of the exciting waveguide, the results of both numerical methods are compared with the analytical results obtained previously using the Wiener-Hopf technique, and are found to be in good agreement. The methods are then used to study the effect of the coupled waveguide on the radiation field.     

Computation of TM and TE modes in waveguides based on a surfaceintegral formulation

The surface integral formulation is used for the computation of TM and TE modes propagating in dielectric loaded waveguides. This formulation makes use of the surface equivalence principle whereby the field at any point internal or external to the waveguide can be expressed in terms of equivalent surface currents. This procedure reduces the original problem into a set of integro-differential equations which is then reduced to a matrix equation using the method of moments. The solution of this matrix equation provides the propagation characteristics of the waveguide and the equivalent surface currents existing on the waveguide walls. The equivalent surface currents can be used to compute the fields at all points, both inside and outside the waveguide. The surface integral method has been used to compute the propagation characteristics of waves propagating in dielectric loaded waveguides. The computed results agree very well with analytical and published data. A method that can be used to remove spurious modes is illustrated     

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     

Propagation and coupling properties of integrated opticalwaveguides-an integral equation formulation

The propagation and coupling properties of integrated optical waveguides are analyzed by means of the electric field integral equation approach. The kernel of the integral equation is the Green's function of a two-layered medium. The Galerkin's method is then employed to solve the integral equation numerically. The set of basis and test functions consists of entire domain plane wave functions. Fast convergence and superior accuracy are the advantages of the chosen set of basis and test functions. The method is used to compute the propagation and coupling properties of several structures. Very good agreement is observed with previously published results. Field distributions of several coupled mode structures, such as the symmetrical and asymmetrical coupler are also investigated and presented. Finally, the same method is used to produce the field distribution of waveguides having more complex cross section like the trapezoidal waveguide     

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     

Eigenmodes of sectoral coaxial ridged waveguides

The results of numerical investigation of sectoral coaxial ridged waveguides eigenmodes of two configurations (with a ridge on inner or outer wall) for different cross-section dimensions are presented. In particular, dependences of cutoff wave numbers on geometrical dimensions ratios for first four modes are investigated, electric field components distributions for these modes have been obtained and the optimization of sectoral coaxial ridged waveguides has been carried out to provide maximal single-mode operation frequency band. Two optimal configurations of waveguides with single-mode operation bandwidth ratio 5.6:1 are obtained. It is shown that smaller cross-section dimensions at the fixed single-mode operation frequency band has the waveguide with the ridge at the inner round wall. The size of the gap between the ridge and the round wall of optimal waveguide is identical for both configurations and is determined by the required ratio of cutoff frequencies of two lower TE modes. Calculations are conducted utilizing the mathematical model obtained in [1] by the integral equation technique with the correct account of singular behavior of the field at the ridge.     

Aperture Coupling and Dipole Excitation in Planar Waveguide Partially Filled With Left-Handed Material

We present physical concepts and formulations for a parallel-plate waveguide which is partially filled with stratified right-handed and left-handed media and fed by apertures. Based on an exact analysis, high power transmissions can be obtained if the medium parameters and layer thicknesses are properly chosen. Such a structure is called a super waveguide since the transmitted power is extremely larger than that in a conventional air-filled waveguide. The equivalence principle and stratified medium theory are used to set up the integral equation in terms of the magnetic currents on apertures. We have applied the method of moments to discretize the integral equation and solved it numerically. The impact of such magnetic currents to the high-power transmission and their interaction to the original dipole source are investigated. From numerical results, we notice that the transmission power is not as high as we anticipated if the source is outside the waveguide. If the source is placed inside the waveguide through apertures, however, the super waveguide will be realized.     

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 planar arrays of shunt slots in ridged waveguides

Planar infinite arrays, comprising longitudinal slots cut in ridged waveguides, are analysed using a Galerkin formulation for the tangential electric field at both slot interfaces, combined with the finite element method for generating the waveguide Green function used in the integral equation. Computations agree very well with previously documented results     

Domain-integral analysis of channel waveguides in anisotropicmultilayered media

A domain-integral equation method is presented to determine both propagation constants and the electromagnetic field distributions of guided surface wave modes in integrated optical waveguides. Both the waveguide and its multilayered embedding are anisotropic. The permittivity tensor of the embedding is assumed to be piecewise homogeneous. The kernels of the domain-integral equations consist of Green's tensors. The integral equations form 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 an anisotropic ridge waveguide, embedded in an anisotropic multilayered medium     

Solutions for Some Waveguide Discontinuities by the Method of Moments (Short Papers)

The electromagnetic boundary value problem of two waveguides coupled by an aperture or an aperture in a waveguide radiating into free space may be described by an integral equation. An analytical solution to this integral equation cannot be readily found due to the complexity of the kernel. However, extremely useful results may be obtained if the method of moments is employed to reduce the integral equation to a matrix equation which can be solved by known methods. In this short paper, series and shunt slots in a rectangular waveguide are analyzed using this technique.     

Mutual Coupling Between Parallel-Plate Waveguides

The radiation field and mutual coupling between two identical parallel-plate waveguides having the same axis of symmetry are investigated. Jones' method of formulation is applied and a modified Wiener-Hopf equation is obtained. Expressions for the radiated field in free space, reflected field in the exciting waveguide, and transmitted field in the coupled waveguide are obtained and the reflected and transmitted fields are expressed in terms of waveguide modes. The reflection coefficient for each mode is represented by three terms, two of which are due to reflections at the open end of the exciting waveguide and are constant along the waveguide. The third term is the contribution from the field scattered by the open end of the coupled waveguide and decays along the waveguide according to the radiation condition. Similarly, the transmission coefficient of each mode is represented by three terms, two of which decay along the coupled waveguide and the third one is constant. The radiation field is also divided into three terms. One of them is due to the radiation from the open end of the exciting waveguide and the other two are the contribution of multiple interactions between the two waveguides. Computed results for the reflection and transmission coefficients and the radiation field are shown for TE/sub 0,1/ excitation and various separation distance of the waveguides. The results for the reflection and transmission coefficients are oscillating functions of period /spl pi/, and approach gradually the well-known final values of a single excited wavegnide.     

Compound coupling slots for arbitrary excitation of waveguide-fedplanar slot arrays

A resonant coupling slot cut in the common broad wall of two crossed rectangular waveguides is analyzed. The slot is offset from the center line and tilted with respect to the longitudinal axis of the main waveguide, whereas it is centered-tiled in the branch waveguide. It is shown that the slot offset and tilt control the branch waveguide excitation amplitude with a phase variability of 360°. Pertinent integral equations are developed, taking into account finite wall thickness. The integral equations are solved for the slot aperture electric field using the method of moments. Dominant mode scattering by the slot in both waveguides is obtained     

Overlap integral analysis for second-harmonic generation within inverted waveguide using mode dispersion phase match

We have discussed the design of a multilayer waveguide toward an efficient nonlinear interaction. We have investigated five- and seven-layer inverted waveguides. The effect of the interlayer thickness and refractive index on the waveguide overlap integral and phase match thickness has been described. A simple approach for improving the nonlinear interaction within an inverted waveguide through maximization of the overlap integral is presented.     

Waveguides of Arbitrary Cross Section by Solution of a Nonlinear Integral Eigenvalue Equation

The problem of electromagnetic wave propagation in hollow conducting waveguides of arbitrary cross section is formulated as an integro-differential equation in terms of fields at the waveguide boundary. Cutoff wave numbers and wall currents appear as eigenvalues and eigenfunctions of a nonlinear eigenvalue problem involving an integro-differential operator. A variational solution is effected by reducing the problem to matrix form using the method of moments. A specific solution of the problem is developed using triangle expansion functions in the method of moments. The solution is simplified by symmetry considerations and is implemented by two digital computer programs. Listings and full documentation of these programs are available. This solution yields accurate determinations of cutoff wave numbers, wall currents, and distributions of both longitudinal and transverse modal field components for the first several modes. Illustrative computations are presented for the single-ridge waveguide, which has a complicated boundary shape that does not lend itself to exact solution.     

Test structures for characterization of electrooptic waveguidemodulators in lithium niobate

A method of determining the critical parameters of waveguide modulators, using a set of test devices fabricated on a single chip, is presented. The five parameters are the depth and lateral Ti diffusion lengths, the peak index change in the waveguides, the electrooptic coefficient, and the buffer layer dielectric constant. The finite element method is used for calculation of optical modes in waveguides with graded refractive index profiles. The integral equation method is used for calculation of the static electric field due to electrodes in a three-layer structure of air, buffer layer, and LiNbO₃. The test set includes a planar waveguide, Mach-Zehnder modulators, symmetrically perturbed directional couplers, and widened X modulators. Several test chips have been fabricated using different fabrication conditions. The parameter values determined using this method are compared with those reported by other authors     

Conductor loss in hollow waveguides using surface integralformulation

The power-loss method along with a surface integral formulation is used to compute the attenuation constant in hollow waveguides of arbitrary cross-section. An E-field integral equation is developed for the surface electric currents which is transformed into a matrix equation using the method of moments. An iterative technique, i.e. Muller's method, is used to obtain the relation between the propagation constant and frequency. The attenuation constants have been calculated and formulated for various waveguides and are in good agreement with published data     

Three-dimensional analysis of electromagnetic fields in rectangularwaveguides by the boundary integral equation method

A rapidly convergent expression of electromagnetic fields in rectangular waveguides is proposed for three-dimensional electromagnetic field analysis by using the integral equation method. The new method is an improved image expansion method utilizing the rapid convergence of the orthogonal expansion method. By this new method, the slow convergence of the orthogonal method with currents near an observation point can be removed completely. In order to investigate the adequacy of the new expression, the fields produced by the line electric and magnetic current segments are calculated and compared with the values obtained by the orthogonal expansion method. This confirms that the new expression gives accurate numerical values with a short computing time. Electromagnetic fields in a rectangular waveguide with circular metallic and dielectric posts are analyzed by using the new expression. From computed values, equivalent circuits of the metallic and dielectric post are obtained and compared with values obtained by N. Marcuvitz (1951). Reasonably good agreement is obtained     

Computation of cutoff wavenumbers of TE and TM modes in waveguidesof arbitrary cross sections using a surface integral formulation

A procedure is described for obtaining the cutoff wave numbers of transverse electric (TE) and transverse magnetic (TM) modes in waveguides of arbitrary cross section. A surface integral equation approach is used in which the E-field equation has been transformed into a matrix equation using the method of moments. An iterative technique is used to pick the eigenvalues of the solution matrix which corresponds to the waveguide cutoff wave numbers. The salient features of this technique are its speed, its simplicity, and the absence of any spurious modes when waveguides of arbitrary cross section are treated. The first four modes are tabulated for various waveguides, and the results are in very good agreement with published data