Waveguide Equations Research Articles

Almost global existence for quasilinear wave equations in waveguides with Neumann boundary conditions

In this paper, we prove almost global existence of solutions to certain quasilinear wave equations with quadratic nonlinearities in infinite homogeneous waveguides with Neumann boundary conditions. We use a Galerkin method to expand the Laplacian of the compact base in terms of its eigenfunctions. For those terms corresponding to zero modes, we obtain decay using analogs of estimates of Klainerman and Sideris. For the nonzero modes, estimates for Klein-Gordon equations, which provide better decay, are available.

Nonstationary waveguide equations for investigation of ultrawideband pulses in irregular waveguides

Nonstationary waveguide equations for investigation of the processes of excitation and propagation of ultrawideband pulses in irregular electromagnetic waveguides are derived. The equations allow direct calculation of the electric-field intensity of the pulses as a function of the spatial coordinates of the observation point and time. The intensity of the magnetic field is determined at the next stage after additional calculations. The equations proposed are a supplement to and a generalization of the equations derived earlier.

Excitation of a waveguide transformer

Integral immitance equations for steady-state excitation of a shielded waveguide transformer with any number of arbitrarily arranged adjoint semi-infinite waveguides are derived in the general formulation. Excitation can be carried by arbitrary inner sources (extraneous electric and magnetic currents), as well as by normal modes incident from infinity. The results are extended to the case of nonideal walls for magnetodielectric and metallic inclusions. The results give a generalization of the familiar integral impedance and admittance equations for inhomogeneous waveguides.

Global existence for nonlinear Klein–Gordon equations in infinite homogeneous waveguides in two dimensions

In this paper we prove a global existence result for nonlinear Klein–Gordon equations with small data in infinite homogeneous waveguids, R 2 × M , where M = ( M , g ) is a Zoll manifold. The method is based on the normal forms, the eigenfunction expansion for M and the special distribution of eigenvalues of Laplace–Beltrami on Zoll manifold.

Global solutions for nonlinear Klein–Gordon equations in infinite homogeneous waveguides

In this paper we prove a global existence result for nonlinear Klein–Gordon equations in infinite homogeneous waveguides, R × M , with smooth small data, where M = ( M , g ) is a Zoll manifold, or a compact revolution hypersurface. The method is based on normal forms, eigenfunction expansion and the special distribution of eigenvalues of the Laplace–Beltrami on such manifolds.

Design optimization of polymer electrooptic modulators

A versatile and numerically efficient finite-element method-based approach has been developed and used to solve the associated quasi-static Laplace equation for electrodes, the full-vectorial wave equation for optical waveguides, and the evolutionary beam-propagation method for bend designs, to characterize the Mach-Zehnder-based polymer modulators incorporating ridge-type waveguide structures. The effects of the rib height and the waveguide width on a single-mode operation, the symmetry of the beam profile, the insertion loss, and the bending loss of the polymer rib waveguides are presented. Further, the effect of the rib height, the waveguide width, and the electrode width on the key modulator parameters, such as V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">pi</sub> L, N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sub> , and Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> are presented, and as a consequence, an optimized design is reported.

Introducing the Finite Element Method in Electromagnetics to Undergraduates Using MATLAB

This paper shows how electrical engineering undergraduates can acquire a working knowledge of the finite element method (FEM) within a short period of time using MATLAB. For simplicity, only first-order triangular elements are considered. The scalar wave equation for homogeneous isotropic waveguides is used for introducing the FEM. Simple waveguide problems, including a design problem, are discussed as examples. It is shown how the knowledge acquired can be extended to other electromagnetic problems.

Study of a self-consistent pulsed electron flow in an irregular waveguide

A mathematical model is presented for self-consistent interaction of pulsed electron flows with the electromagnetic waves they excite in irregular waveguides. The model is applied to axially symmetric systems and based on simultaneous solution of an inhomogeneous system of nonstationary waveguide equations and relativistic equations of motion of a system of quasi-particles that approximate the electron flow. Results of calculations are presented for the case of a pulse of photoelectrons emitted from the walls of a hyperboloid of revolution. The flow is accelerated preliminarily by means of a grid that has a high transmission factor.

Nonlinear Evolution Problems

The workshop ‘Nonlinear Evolution Problems’ focussed on three types of nonlinear evolution equations: (1) Geometric evolution equations (essentially of parabolic type) (2) Nonlinear hyperbolic equations (3) Dispersive equations The programme consisted of 22 talks presented by international specialists from Australia, France, Germany, Italy, Sweden, Switzerland and the United States.Generally, three lectures were delivered in the morning sessions and two in the late afternoon which left ample time for individual discussions. In the first group of equations, in particular the Ricci flow and conformal flows such as the Yamabe flow and the Q-curvature flow were considered. A further focus was on curvature flows for hypersurfaces such as the Gauss and harmonic mean curvature flow and on geometric flows of higher order. Julie Clutterbuck presented a direct approach to certain fundamental estimates for a general class of parabolic equations. Mete Soner pointed out important relations between fully nonlinear parabolic equations and backward stochastic differential equations. The class of hyperbolic equations was represented by wave maps, Einstein's equations of gravitation and nonlinear wave equations in waveguides and cones. Among the dispersive equations the nonlinear Schroedinger equation and the KdV equation were considered. The individual discussions covered connections between approaches to the different equations in each group but also on common techniques used across the three classes of equations. Among such techniques are for instance the careful examination of the algebraic and geometric structure of the nonlinear terms. Another common theme appears to be the phenomenon of blow-up (singularity formation) for solutions and methods to describe the solution near these. Selfsimilar solutions and blow-up rates play an important role here. The organizers decided to particularly encourage the young researchers among theparticipants to present their work, including a PhD student and several recent post-doctoral fellows.

A nonlinear wave equation and exact periodic solutions in circular-rod waveguide

Under the condition of small deformation, a new nonlinear wave equation is derived to describe nonlinear wave evolution in a nonlinear elastic circular rod by means of Hamilton principle. The nonlinear constitutive relationship proposed by Cox and transverse Possion effects are simultaneously taken into account. Nonlinear wave equation and truncated nonlinear wave equation are solved by the Jacobi elliptic cosine function expansion method. The exact periodic solutions of these nonlinear equations are obtained. The limiting conditions of these solutions are also given.

Appropriate formulation of the Characteristic equation for open nonreciprocal Layered waveguides with different upper and lower half-spaces

This paper will study the most convenient way of formulating the characteristic equation for shielded, parallel-plate, grounded, and open reciprocal/nonreciprocal planar layered waveguides, including the possibility of different upper and lower half-spaces. A detailed study of the suitable mapping for each kind of waveguide will lead to the formulation of analytic characteristic equations for shielded/parallel-plate/grounded reciprocal/nonreciprocal waveguides and also for open reciprocal ones. Although no mapping has been found to remove all the branch points of the characteristic equation for open nonreciprocal waveguides with different half-spaces, a robust approach will be proposed to overcome the main drawbacks caused by the multivalued nature of this problem. The combination of the suitable formulation of the characteristic equation with a systematic integral-nature root-searching strategy bears a reliable and efficient method. Some novel numerical results will be presented for open anisotropic reciprocal waveguides and for open magnetized ferrite slabs to illustrate the performance of the present proposal.

Transformation from a multimode interference solution to a self-guiding solution

We develop a perturbation theory as an approximate solution to the nonlinear Helmholtz equation in a multimode waveguide to examine the transition from the low-power multimode interference solution to high-power self-guiding solution. We compare the perturbation results with those obtained from finite-difference beam-propagation simulations. The results indicate that the lowest-order linear mode evolves into the self-guided mode as the power in the input beam is increased.

Theoretical Analysis of Resonant Cavities Loaded with Lossy Material

Theoretical analysis of abrupt cavities loaded with lossy material has been described in detail. Dispersion equation for waveguides with lossy material and hybrid modes (HEM) matching equations for abrupt structure were derived. Further, the equations have been solved by numerical calculation to study waveguides and abrupt cavities loaded with lossy material. The theoretical analysis is available for designing and analysis RF devices. It was used to design the RF system for Ka band gyroklystron amplifier, the calculation results accord with the cold test.

Vector coupled-mode theory of dielectric waveguides

A consistent derivation of a system of vector coupled-mode (VCM) equations for parallel dielectric waveguides is presented and compared with earlier versions of the improved coupled-mode theory (ICMT). As a validity test, it is shown that the effectively scalar transverse electric and transverse magnetic (TM) coupled-mode (CM) equations are direct limits of our full VCM formulation. In particular, our formulation does not lead to the fundamental error found with earlier coupled-mode theories (CMTs) in a case of TM fields. Functional equations of our VCMT are consistent with Maxwell's equations and lead to higher precision. They can be applied to complicated arrays of strongly coupled parallel dielectric waveguides with true vectorial behavior.

Application of the method of nonstationary waveguide equations to simulating pulsed processes in irregular transmission lines

The potentialities of the model of nonstationary waveguide equations for describing pulsed processes in irregular transmission lines are studied using planar lines as an example. The problem of high-accuracy controllable-error numerical simulation is discussed. Typical examples of simulating ultra-wide-band electromagnetic pulses with an initial TEM structure in terms of time-domain representation are presented with emphasis on the interaction of the pulses with irregularities, including their transformation into longitudinal waves. Both lumped and distributed irregularities are addressed: deep corrugations that cover 90% of the transmission line’s aperture (distributed irregularities) and these corrugations in combination with sharp kinks at the boundary surfaces and permittivity steps at the boundaries of the dielectric filling (lumped irregularities). It is shown that a relative rms error involved in the calculated field intensity of no higher than 10−4 is easy to achieve.

Stress effects on the performance of optical waveguides

Stresses can cause anisotropic and inhomogeneous distribution of the refractive index. Their effects on the performance of optical waveguides have been observed in photoelectric devices. In this paper, the photo-elastic relation and wave equations for inhomogeneous and anisotropic waveguides are reviewed. The effective refractive indexes and mode shapes of planar waveguides under different stress states are obtained analytically. It is found that stress can affect the optical performance; different stress states play different roles: high stress value can change the cutoff thickness, which may induce multimode; in-plane stress causes birefringence, which may induce polarization shift and polarization dependent loss; stress concentration can change the mode shape, which may induced large transition loss; and pure shear stress has little effects on the effective refractive index.

Generalized Tschebyshev polynomial and its application in solving the strong cou pling waveguide equations

In this paper, by means of generalized Tschebyshev polynomial, the analytical so lution is derived for (N+1)×（N+1） waveguide couplers arranged in a ring with strong coupling. As a concrete example, the solution of 5×5 waveguide couplers is calculated, and the relationship between strong and weak couplings is analyze d. The property of generalized Tschebyshev polynomial is discussed in detail.

Analysis of a ridge waveguide using overlapping T-blocks

A T-block (TB) approach is proposed to analyze the dispersion relation of a ridge waveguide. The field representations of a TB are obtained with the Green's function and mode-matching technique. Rigorous, yet simple dispersion equations for symmetric and asymmetric ridge waveguides are presented using a superposition of overlapping TBs. The rapid convergence characteristics of the dispersion equation are illustrated in terms of the cutoff wavenumbers. A closed-form dispersion relation, based on a dominant-mode approximation, is shown to be accurate for most practical applications such as couplers, filters, and polarizer designs.

High-accuracy finite-difference equations for dielectric waveguide analysis II: dielectric corners

For part I see ibid., p. 1210, 2002. We present a discussion of the behavior of the electric and magnetic fields satisfying the two-dimensional Helmholtz equation for waveguides in the vicinity of a dielectric corner. Although certain components of the electric field have long been known to be infinite at the corner, it is shown that all components of the magnetic field are finite, and that finite-difference equations may be derived for these fields that satisfy correct boundary conditions at the corner. These finite-difference equations have been combined with those derived in the previous paper to form a full-vector waveguide solution algorithm of unprecedented accuracy. This algorithm is employed to provide highly accurate solutions for the fundamental modes of a previously studied standard rib waveguide structure.

Effective equations for photonic-crystal waveguides and circuits

We suggest a novel conceptual approach to describing the properties of waveguides and circuits in photonic crystals, based on effective discrete equations that include long-range interaction effects. We demonstrate, through the example of sharp waveguide bends, that our approach is very effective and accurate for the study of bound states and transmission spectra of photonic-crystal circuits and disclose the importance of evanescent modes in these phenomena.