Waveguide Problems Research Articles

Forced guided waves in linearly elastic plates (I) – An examination of the normal-mode expansion method

Guided waves in a plate can be generated by external loads such as body forces and surface tractions. The region of the plate where the external loads are applied is called the loading zone. Guided waves inside the loading zone are called forced guided waves. The classical normal-mode expansion method developed by Auld and Kino in 1973 has been used widely by numerous researchers and practitioners to study forced guided wave problems in plates, including both elastic and electromagnetic waves. As Part I of this investigation, the current paper shows that the classical normal-mode expansion method of Auld and Kino does not yield the exact elastodynamic solution when applied to forced Lamb waves, because its solution does not satisfy the Hooke’s law inside the loading zone. The classical normal-mode expansion method, however, does yield the exact elastodynamic solution when applied to forced horizontally polarized shear waves in that its solution satisfies all the governing equations including the equation of motion, the Hooke’s law and the prescribed traction boundary conditions. In Part II, a modified normal mode expansion method will be developed to mitigate the above limitations of the classical normal mode expansion method.

Lambert W function methods in double square well and waveguide problems

Using methods related to the Lambert W function, we present solutions of two apparently different problems: (1) The one-dimensional double square well potential in quantum mechanics and (2) The transverse electric and magnetic modes for a step-index electromagnetic waveguide. The solution techniques provide insight into the bound energy states for the single and double square well problems and the allowed modes of propagation in a waveguide with varying refractive indices. The solutions can be viewed in either of two related complex plane representations. Comparison of the solution geometries suggests that interesting applications may be possible in nanostructures and devices which are designed to be sensitive to small changes in their environment.

Sapphire Single-Crystal Waveguides and Fibers for Thz Frequency Range

The paper is devoted to problem of waveguide designing for the THz frequency range of electromagnetic spectra. We proposed different types of THz waveguides on the base of the sapphire shaped crystals. Due to the advantages of the shaped crystal growth technique and unique combination of sapphire physical properties, the sapphire waveguides with relatively low losses, near-zero dispersion and inertness of the waveguide characteristics with respect to external conditions can be used effectively for guiding of the THz radiation in the harsh environments.

A numerical study of the pollution error and DPG adaptivity for long waveguide simulations

High-frequency wave propagation has many important applications in acoustics, elastodynamics, and electromagnetics. Unfortunately, the finite element discretization for these problems suffers from significant numerical pollution errors that increase with the wavenumber. It is critical to control these errors to obtain a stable and accurate method. We study the effect of pollution for very long waveguide problems in the context of robust discontinuous Petrov–Galerkin (DPG) finite element discretizations. Our numerical experiments show that the pollution primarily has a diffusive effect causing energy loss in the DPG method while phase errors appear less significant. We report results for 3D vectorial time-harmonic Maxwell problems in waveguides with more than 8000 wavelengths. Our results corroborate previous analysis for the Galerkin discretization of the Helmholtz operator by Melenk and Sauter (2011). Additionally, we discuss adaptive refinement strategies for multi-mode fiber waveguides where the propagating transverse modes must be resolved sufficiently. Our study shows the applicability of the DPG error indicator to this class of problems. Finally, we illustrate the importance of load balancing in these simulations for distributed-memory parallel computing.

Mode transformation for a Schrödinger type equation: Avoided and unavoidable level crossings

Methods elaborated in quantum mechanics for the Landau–Zener problem are generalized to study the non-adiabatic transitions in a wide class of problems of wave propagation, in particular in the waveguide problems. If the properties of the waveguide slowly vary along its axis and the phase velocities of two modes have a degeneracy point or are almost degenerate near some point, the transformation of modes may occur. The conditions are formulated under which we can find formal asymptotic expansions of modes outside the vicinity of the degeneracy point and write out explicitly the transition matrix. The starting point is rewriting the governing equations in the form of the Schrödinger type equation. The Hamiltonian is assumed to be the result of a small perturbation of an operator with a degeneracy point of the crossing types of two eigenvalues. The perturbation of the Hamiltonian produces a close pair of simple degeneracy points. Two regimes of mode transformation for the Schrödinger type equation are identified: avoided crossing of eigenvalues (corresponding to complex degeneracy points) and an explicit unavoidable crossing (with real degeneracy points).

EXACT NON-REFLECTING BOUNDARY CONDITIONS WITH AN FDTD SCHEME FOR THE SCALAR WAVE EQUATION IN WAVEGUIDE PROBLEMS

EXACT NON-REFLECTING BOUNDARY CONDITIONS WITH AN FDTD SCHEME FOR THE SCALAR WAVE EQUATION IN WAVEGUIDE PROBLEMS

Physical problems of diffraction waveguide used in large field of view

Diffraction gratings have been widely used in waveguides. They can transmit light beams or images from the in-coupling end to the out-coupling end at predetermined positions. However, when they are applied to augmented reality and virtual reality with large field of view and color light sources, there will arise some problems such as mismatch and missing field of view, non-uniform emission, and others. Therefore, starting from these physical problems, the upper limit of the field of view for diffractive waveguide and the complete theoretical boundary formula of the field of view are derived, and on this basis, in-depth research is conducted on monochromatic waves and multicolor waves, respectively. It is concluded that the single-layer diffractive waveguide supports the theoretical upper limit of the monochromatic wave field angle of about 48° under normal high refractive index of <i>n</i> = 1.75, and supports the theoretical upper limit of the multicolor wave field angle of 26.4° for coefficient <i>q</i> = 1.3. Clearly, a larger field of view requires a higher refractive index <i>n</i> and a smaller <i>q</i> value. The boundary conditions of field integrity indicate that reducing the maximum diffraction angle of the long wave and thinning the thickness of the waveguide layer can solve the problem of missing field of view. The practical maximum diffraction angle generally does not exceed 75°, and the thickness of the waveguide layer is about 0.5 to 1.0 mm generally based on the incident field of view. Finally, a method of expanding each total internal reflection field of view into a ray tracing diagram and a distribution function of pupils to receive light energy at various angles are obtained. In this way, the optimal position of the out-coupling grating region can be achieved, and the inverse of the distribution function is used to constrain the angular distribution of the projected light or the grating efficiency, and then receiving uniform exit image at any position becomes possible. The uniformity of the monochromatic waveguide increases from 0.27 to 0.15, and the uniformity of the long wave in the single grating multicolor waveguide rises from 0.4 to 0.28. The results of these studies will undoubtedly help to solve the problem in the diffractive waveguides used in large field of view and multicolor light.

A boundary element method for waveguide problems and its application for topology optimisation

A boundary element method for waveguide problems and its application for topology optimisation

PROPAGATION OF ELECTROMAGNETIC WAVES IN STRUCTURES WITH BIAXIAL ANISOTROPIC MEDIA OF MONOCLINIC SYNGONY. PART 2: WAVEGUIDE STRUCTURES

The processes of propagation of electromagnetic waves in closed and open waveguides filled with biaxial anisotropic media of monoclinic syngony have been sufficiently fully and strictly modeled for the first time. The media with anisotropy of dielectric permittivity and magnetic permeability and relevant key waveguide problems, which accurate solution has to provide the successful simulation of more complicated problems that are of undoubted interest for both theory and practice, are considered.

Leaky waves in planar dielectric waveguide

A new analytical and numerical solution of the electrodynamic waveguide problem for leaky modes of a planar dielectric symmetric waveguide is proposed. The conditions of leaky modes, corresponding to the Gamow-Siegert model, were used as asymptotic boundary conditions. The resulting initial-boundary problem allows the separation of variables. The emerging problem of the eigen-modes of open three-layer waveguides is formulated as the Sturm-Liouville problem with the corresponding boundary and asymptotic conditions. In the case of guided and radiation modes, the Sturm-Liouville problem is self-adjoint and the corresponding eigenvalues are real quantities for dielectric media. The search for eigenvalues and eigenfunctions corresponding to the leaky modes involves a number of difficulties: the problem for leaky modes is not self-adjoint, so the eigenvalues are complex quantities. The problem of finding eigenvalues and eigenfunctions is associated with finding the complex roots of the nonlinear dispersion equation. To solve this problem, we used the method of minimizing the zero order. An analysis of the calculated distributions of the electric field strength of the first three leaky modes is given, showing the possibilities and advantages of our approach to the study of leaky modes.

Integral Equation Method for Calculating Irregular Waveguides Using the Generalized Lorentz Lemma

A method for solving boundary value problems for irregular waveguides that combines the generalized Lorentz lemma with a collocation technique is proposed. The method allows one to reduce boundary value problems to systems of integro-differential equations, which are algebraized by identifying collocation points with the points of localization of auxiliary sources.

Mixed Spectral-Element Method for the Waveguide Problem With Bloch Periodic Boundary Conditions

The mixed spectral-element method (MSEM) is applied to solve the waveguide problem with the Bloch periodic boundary condition (BPBC). Based on the BPBC for the original Helmholtz equation and the periodic boundary condition (PBC) for the equivalent but modified Helmholtz equation, two equivalent mixed variational formulations are applied for the MSEM. Unlike the traditional finite-element method and spectral-element method (SEM), both these mixed SEM schemes are completely free of spurious modes because of their use of the Gauss’ law and the curl-conforming vector basis functions structured by the Gauss–Legendre–Lobatto points. A simple implementation method is used to deal with the BPBC and the PBC for the mixed variational formulations so that both schemes can save computational costs over the traditional methods. Several numerical results are also provided to verify that both schemes are free of spurious modes and have high accuracy with the propagation constants.

Three-dimensional boundary fitted parabolic-equation model of underwater sound propagation.

A three-dimensional (3-D) parabolic-equation (PE) method utilizing a higher-order split-step Padé algorithm and a boundary fitted grid has been developed to accurately solve 3-D underwater sound propagation problems with non-planar or tilted boundaries. At each PE marching step, the split-step Padé algorithm enables the method of alternating directions to implement the square-root Helmholtz operator by carrying out its one-dimensional (1-D) derivative components alternately, and it also allows a straightforward application of the 1-D non-uniform Galerkin method to discretize the solution mesh. The advantage of the boundary fitted grid to improve PE solution accuracy is most profound in the case of fitting to a pressure release surface as its boundary condition is a scalar and has no direction. This method can also be applied to a sloping interface by rotating the grid to align with the interface. Numerical problems of semi-circular waveguide and tilted wedge were solved using this boundary fitted PE method, and benchmark reference solutions were used to examine and confirm the accuracy of the PE solutions. Future applications include modeling 3-D acoustic scattering from a rough sea surface and 3-D sound propagation in beach environments.

A novel approach to solve eigenvalue and forced response problems in waveguide theory by means of bi-orthogonality relations

In this paper recent advances on solving challenging problems in dynamics of multimodal symmetric waveguides (having identical properties of positive/negative going waves) using bi-orthogonality are summarised. It has already been shown by the authors that bi-orthogonality, valid for arbitrarily complicated unbounded symmetric waveguides, greatly facilitates analysis of their forced response in any excitation conditions. Recent findings have shown that bi-orthogonality is a powerful tool also for bounded waveguides and by direct use it resolves the classical Boundary Integral Equations (BIE) into a set of individual relations between modal amplitudes at the boundaries. Moreover, bi-orthogonality permits to uniquely identify special sets of boundary conditions for which the eigenfrequency spectra may be found directly from the dispersion diagrams. These solutions are available simply by drawing horizontal lines specifying an inverse of the wave length of interest in the (Ω, k)-dispersion diagram and so the eigenfrequency spectrum may be read directly as intersections of these lines with dispersion curves. This paper therefore serves to promote this method, discuss applications, new possibilities, physical interpretation and its generalisation to a broader class of waveguide problems.

A new technique for the analysis of the physical discontinuity in a hollow rectangular waveguide with dielectric inserts of varying profiles

ABSTRACTThe main objective of this research is to present an effective technique to solve the problem of hollow rectangular waveguide. The second objective is to examine the influence of the hollow rectangular waveguide with dielectric material between the hollow rectangle and the metal on the output field. The proposed technique is very effective in relation to the analytical methods because it allows to calculate any discontinuous problem in the cross section. The contribution is important in that it is an improvement over the earlier method that was based on the Laplace and Fourier transforms and the inverse Laplace and Fourier transforms and also that it is capable to analyze the physical discontinuity in a hollow rectangular waveguide. A comparison with the known transcendental equation will show in order to examine the validity of the theoretical model. We can predict the waveguide parameters for obtaining the Gaussian behavior of the output field. The application is useful for straight hollow rectangular waveguides with dielectric material between the hollow rectangle and the metal in the microwave regime. The application can be practical in the laboratory by connecting the hollow waveguide to the analyzer spectrum in order to analyze the effect of the dielectric profile on the behavior of the waveguide.

Powerful relativistic generators on rectilinear electron beams with electrodynamic filters at the ends

This paper presents the results of the calculations of high-power generators of microwave radiation on the rectilinear electron beam on resonators with electrodynamic filters at the ends of the resonators. The results of calculations of high-power generators of microwave radiation on rectilinear electron beam on resonators with electrodynamic filters at the ends of the resonators are presented. The methodology for calculating processes in the irregular waveguide is based on the ideas of A. G. Sveshnikov, based on the coordinate transformation method, which allows searching for a solution of the excitation problem of the irregular waveguide by expanding this solution into the system of basic functions of the regular waveguide. allows you to search for a solution to the excitation problem of an irregular waveguide by expanding this solution into a system of basic functions of a regular waveguide. This allows reducing the 3-dimensional problem to the one-dimensional one allows us to reduce a 3-dimensional problem to a one-dimensional one, which significantly reduces the amount of computation and increases the accuracy of the solution. The calculations are carried out using the computer program Gyro-K. The presence of electrodynamic filters at the ends of the waveguide makes it possible to increase the quality factor of the resonator and reduce microwave radiation to the cathode region. We demonstrate that the power of such generators may reach 21 MW, which may provide effective suppression of the electronic devices of the alleged enemy. Kolosov S. V., Batura M. P., Shatilova O. O. Powerful relativistic genera- tors on rectilinear electron beams with electrodynamic filters at the ends. Ural Radio Engineering Journal . 2019;3(4):369–377. DOI: 10.15826/ urej.2019.3.4.003

Coupled Mode Analysis for 3D Stress-Free Elastic Acoustic Waveguide

Acoustic sensors and acoustic measurements receive much attention in various applications. Because waveguides are commonly used in sensor design, theoretical means to study acoustic propagation and interaction in waveguides are necessary. However, current methods for elastic wave coupling, including the transfer matrix method and coupled mode theory in planar 2D waveguides, are not satisfactory. In this work, a coupled mode analysis for acoustic waves in 3D stress-free elastic waveguides is proposed. Similar to the coupled mode theory in optical waveguides, the analysis is presented by the evolution of modal amplitudes. It can solve various modal conversion and scattering problems in elastic waveguides with small changes of cross sections and stress-free boundaries. To demonstrate the practicability, the coupled mode analysis is used to calculate the reflection spectrum of the newly proposed structure, the acoustic fiber Bragg grating. In a notch-based grating fabricated on a thin cylindrical waveguide, the results from coupled mode analysis are in good agreement with those from the transfer matrix method, which has been already validated experimentally. The coupled mode analysis is a promising method to solve various scattering problems.

Theory for anomalous refraction of visible light by the plane array of metallic waveguides

The effect of anomalous or negative angle refraction in the periodic systems of silver waveguides is investigated. This effect arises solely due to the waveguide interaction and requires the system of specific geometrical configuration. It is also shown that the propagation constant provides a flexible mechanism to manipulate the anomalous refraction. We employ the multiple scattering formalism based on the exact solution of the Maxwell equations for a single waveguide problem. Also the method based on the lattice sum transformation into a fast convergent series is involved. This technique dramatically decreases the time necessary for performing the numerical simulations and increases the accuracy of the computational process. It is shown that all the directions, which the periodic system is possible to radiate, can be obtained beforehand, i.e. analytically, without numerical simulations. The latter ones are needed to determine energy the flux going into each of those directions.

A Nonlocal Operator Method for Partial Differential Equations with Application to Electromagnetic Waveguide Problem

A Nonlocal Operator Method for Partial Differential Equations with Application to Electromagnetic Waveguide Problem

Residual distribution schemes for Maxwell’s equations

This paper aims to deliver another approach of solving hyperbolic system of equation into the field of computational electromagnetics, known as the residual distribution (RD) or fluctuation splitting method. The RD scheme fills the interstice between finite-element method (FEM) and finite-volume method (FVM), where its idea of calculating flux residual imitates the FVM, but the numerical solutions within a discretized triangular mesh is interpolated with similitudes to the FEM. It was originally designed as a remedy to capture shock problem in Euler system under compact stencil, but its extension to linear-preserving scheme which is the main cause of this work, capable of giving second-order-accurate results and is congenial to the FEM framework. The RD scheme is always vertex-centered therefore has all the conserved variables E and H located at the same nodal point. This topology evades having both vertex-centered and cell-centered coordination in one single mesh, and permits time-discretization other than the staggered time marching scheme, such as the common backward-time discretization which is unconditionally stable. Another contribution of this work is to suggest that row-mass-lumping of the consistent mass-matrix would not mar too much the second-order-accuracy of LDA-RD scheme, which is an upwind scheme where the mass-matrix an impediment for time-updating during the last decade. A prior reconstruction of the upwind mass-matrix is done before lumping up the mass-matrix so that the time-marching nodal update is consistent with the distribution of mass-matrix. The current work covers 3 basic exercises of electromagnetics, they are 2D radiation problem, 2D scattering problem and 3D waveguide propagation. For the 2D scattering and 3D waveguide problems, the perfect electric conductor (PEC) boundary condition is successfully applied using RD construction.