Open waveguides in a thin Dirichlet ladder: I. Asymptotic structure of the spectrum

The spectra of open angular waveguides obtained by thickening or thinning the links of a thin square lattice of quantum waveguides (the Dirichlet problem for the Helmholtz equation) are investigated. Asymptotics of spectral bands and spectral gaps (i.e., zones of wave transmission and wave stopping, respectively) for waveguides with variously shaped periodicity cells are found. It is shown that there exist eigenfunctions of two types: localized around nodes of a waveguide and on its links. Points of the discrete spectrum of a perturbed lattice with eigenfunctions concentrated about corners of the waveguide are found.     

Multiplicity of eigenvalues in certain boundary value problems for the Helmholtz equation

Boundary value problems for two-layer cylindrical waveguides (namely, for a circular two-layer closed waveguide and a dielectric waveguide in an unbounded medium) are considered. It is shown that the same eigenvalues determining the wave numbers of waves in the waveguides can correspond to different solutions of the boundary value problem.     

Second kind integral equation formulation for the mode calculation of optical waveguides

We present a second kind integral equation (SKIE) formulation for calculating the electromagnetic modes of optical waveguides, where the unknowns are only on material interfaces. The resulting numerical algorithm can handle optical waveguides with a large number of inclusions of arbitrary irregular cross section. It is capable of finding the bound, leaky, and complex modes for optical fibers and waveguides including photonic crystal fibers (PCF), dielectric fibers and waveguides. Most importantly, the formulation is well conditioned even in the case of nonsmooth geometries. Our method is highly accurate and thus can be used to calculate the propagation loss of the electromagnetic modes accurately, which provides the photonics industry a reliable tool for the design of more compact and efficient photonic devices. We illustrate and validate the performance of our method through extensive numerical studies and by comparison with semi-analytical results and previously published results.     

The interaction of electron beams and electromagnetic fields in relativistic Cerenkov generators

A method of numerical analysis of linear and nonlinear nonstationary processes in relativistic Cerenkov generators based on periodic superdimensional waveguides is proposed. The main idea lies in considering a nonregular waveguide as a sequence of wave transformers and using a cross-section method. Configurations of eigenwave fields of periodic waveguides with a high-current relativistic electron beam are considered. The processes of generation development in a section of a relativistic multiwave generator are studied. We show that the system frequency is proved to be determined by the longitudinal resonances of surface waves and internal feedback.     

Discrete spectrum of cranked quantum and elastic waveguides

The spectrum of quantum and elastic waveguides in the form of a cranked strip is studied. In the Dirichlet spectral problem for the Laplacian (quantum waveguide), in addition to well-known results on the existence of isolated eigenvalues for any angle α at the corner, a priori lower bounds are established for these eigenvalues. It is explained why methods developed in the scalar case are frequently inapplicable to vector problems. For an elastic isotropic waveguide with a clamped boundary, the discrete spectrum is proved to be nonempty only for small or close-to-π angles α. The asymptotics of some eigenvalues are constructed. Elastic waveguides of other shapes are discussed.     

Resonances for multistratified acoustic waveguides

The equation of acoustic oscillations in multistratified waveguides is considered. It is assumed that the properties of the medium do not depend on the longitudinal coordinate in a neighbourhood of infinity and may be different at different ends of the waveguides. It is proved that the truncated resolvent of the corresponding operator admits an analytical continuation through the continuous spectrum. The singularities (poles, branching points) of the truncated resolvent on the continuous spectrum are investigated. The large time asymptotic behavior of the compulsory oscillations due to periodic forces is obtained.     

Frequency shifting for solitons based on transformations in the Fourier domain and applications

We develop the theoretical procedures for shifting the frequency of a single soliton and of a sequence of solitons of the nonlinear Schrödinger equation. The procedures are based on simple transformations of the soliton pattern in the Fourier domain and on the shape-preserving property of solitons. These theoretical frequency shifting procedures are verified by numerical simulations with the nonlinear Schrödinger equation using the split-step Fourier method. In order to demonstrate the use of the frequency shifting procedures, two important applications are presented: (1) stabilization of the propagation of solitons in waveguides with frequency dependent linear gain-loss; (2) induction of repeated soliton collisions in waveguides with weak cubic loss. The results of numerical simulations with the nonlinear Schrödinger model are in very good agreement with the theoretical predictions.     

Gap opening and split band edges in waveguides coupled by a periodic system of small windows

Using an example of two coupled waveguides, we construct a periodic second-order differential operator acting in a Euclidean domain and having spectral gaps whose edges are attained strictly inside the Brillouin zone. The waveguides are modeled by the Laplacian in two infinite strips of different width that have a common interior boundary. On this common boundary, we impose the Neumann boundary condition, but cut out a periodic system of small windows, while on the remaining exterior boundary we impose the Dirichlet boundary condition. It is shown that, by varying the widths of the strips and the distance between the windows, one can control the location of the extrema of the band functions as well as the number of the open gaps. We calculate the leading terms in the asymptotics for the gap lengths and the location of the extrema.     

Dielectric waveguides of arbitrary cross sectional shape

Numerical guided mode solutions of arbitrary cross sectional shaped waveguides are obtained using a finite difference (FD) technique. The standard FD scheme is appropriately modified to capture all discontinuities, due to the change of the refractive index, across the waveguides’ interfaces taking into account the shape of each interface at the same time. The method is applied to the vector Helmholtz equation formulated to describe the electric or magnetic fields in the waveguide (one field is retrieved from the other through Maxwell’s equations). Computational cost is kept to a minimum by exploiting sparse matrix algebra. The waveguides under study have arbitrary cross sectional shape and arbitrary refractive index profile.     

Non-spectral Field Representations in Dielectric Fibre Guides

Open waveguiding structures, such as acoustic, piezoelectric,gyrotropic and dielectric waveguides, are used for a varietyof applications. An example of such open waveguides is the dielectricfibre guide used in the construction of optical communicationsystems. An understanding of the modal properties of this guideis therefore of fundamental interest to the understanding ofthe modal properties of similar open structures. We analysein detail the transverse electric (TE) and transverse magnetic(TM) polarized electromagnetic fields within a circularly cylindricaldielectric waveguide with arbitrary non-vanishing analytic radialpermittivity variation. It is shown that the field can be representedas an infinite sum of functions, including the finite numberof spectral (surface wave) and an infinity of non-spectral (complex)eigenfunctions. Additional functions are also included. Boththe field polarization and the behaviour of the permittivityat the waveguide boundary are found to affect the form and thevalidity of the representation.     

On the completeness of the system of eigenvectors of electromagnetic waveguides

Several statements of the spectral problem in the theory of regular waveguides are considered. The completeness of the system of eigenvectors of these problems, which is a consequence of the earlier established completeness property of a certain system of eigenvectors, is proved.     

Coupled dielectric waveguides with photonic crystal properties

A system of planar waveguides coupled through a periodic set of small windows is considered. It is shown that the weak periodic coupling of the waveguides leads to an additional eigenvalue band separated by a gap from the threshold of a continuous spectrum branch. Thus, the system has photonic crystal properties, which can be used to construct optical fiber devices. This system can also play the role of a SCISSOR device for reducing the group velocity of light, in optical delay lines, etc.     

The Method of Operator Pencils for Propagation of Electromagnetic Waves in Irregular Waveguide Structures

We consider the completeness of the system of normal waves in irregular waveguides.     

Substantiation of the Spectral Method for Calculating Propagation Constants for Waveguides with Microstrip and Slit Lines

We prove the spectral method for calculating the propagation constants of normal waves in screened waveguides with an irregular cross-section and a nonhomogeneous filling. The problem is reduced to the nonlinear spectral problem for operator-functions in a Hilbert space. Existence theorems are proved for spectral points and the cutoff method is substantiated.     

A novel approach to mode-tracing in SBFEM-simulations: Higher-order Taylor- and Padé approximation

This contribution presents the study of strange phenomena in wave mode representations of waveguides. For this study the waveguides are computed by means of the Scaled Boundary Finite Element Method (SBFEM). Different approaches of mode tracing are used to identify the characteristics of the resulting wave modes. Higher order differentials of the underlying eigenvalue problem are the basis for these approaches. The main idea behind this mode tracing approach is to reduce the cubic computation time to solve the eigenvalue problem for each frequency of interest. This study identifies potentially critical frequency regions and attempts to formulate a solution process. The fascinating effects at critical frequencies are displayed and a suggestion for a stabilization for the solution process is made. This study bases its conclusion on a numerical viewpoint. Main aspects in this study include high order differentials of the eigenvalue problem and the corresponding Taylor and Padé approximations for the eigenvalue problem as a whole. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Solution of boundary value problems for waveguides with anisotropic filling

A rectangular shielded waveguide with arbitrary anisotropic filling is used to describe a spectral approach to the computation of longitudinal regular shielded waveguides filled in part with a nonreciprocal medium whose parameters have an arbitrary dependence on the transverse coordinates. Numerical results are presented that confirm the validity of the algorithms developed.     

Diffraction problem for periodic waveguides

The unique solvability of the diffraction problem on a junction of three periodic waveguides and continuous dependence of the solution on the data are established. Bibliography: 7 titles. __________ Translated from Problemy Matematicheskogo Analiza, No. 37, 2008, pp. 23–36.     

Direct Sturm–Liouville problem for surface Love waves propagating in layered viscoelastic waveguides

This paper presents theoretical model for shear-horizontal (SH) surface acoustic waves of the Love type propagating in lossy waveguides consisting of a lossy viscoelastic layer deposited on a lossless elastic half-space. To this end, a direct Sturm–Liouville problem that describes Love waves propagation in the considered viscoelastic waveguides was formulated and solved, what constitutes a novel approach to the state-of-the-art. To facilitate the solution of the complex dispersion equation, the Author employed an original approach that relies on the separation of its real and imaginary part. By separating the real and imaginary parts of the resulting complex dispersion equation for a complex wave vector k = k₀ + jα of the Love wave, a system of two real nonlinear transcendental algebraic equations for k₀ and α has been derived. The resulting set of two algebraic transcendental equations was then solved numerically. Phase velocity v_p and coefficient of attenuation α were calculated as a function of the wave frequency f, thickness of the surface layer h and its viscosity η₄₄. Dispersion curves for Love waves propagating in lossy waveguides, with a lossy surface layer deposited on a lossless substrate, were compared to those corresponding to Love surface waves propagating in lossless waveguides, i.e., with a lossless surface layer deposited on a lossless substrate. The results obtained in this paper are original and to some extent unexpected. Namely, it was found that: 1) the phase velocity v_p of Love surface waves increases as a function of viscosity η₄₄ of the lossy surface layer, and 2) the coefficient of attenuation α has a maximum as a function of thickness h of the lossy surface layer. The results obtained in this paper are novel and can be applied in geophysics, seismology and in the optimal design and development of viscosity sensors, bio and chemosensors.     

The localization for eigenfunctions of the dirichlet problem in thin polyhedra near the vertices

Under some geometric assumptions, we show that eigenfunctions of the Dirichlet problem for the Laplace operator in an n-dimensional thin polyhedron localize near one of its vertices. We construct and justify asymptotics for the eigenvalues and eigenfunctions. For waveguides, which are thin layers between periodic polyhedral surfaces, we establish the presence of gaps and find asymptotics for their geometric characteristics.     

On a Nonlinear Spectral Problem for a Dielectric Waveguide with Kerr Nonlinearity

Computational Mathematics and Mathematical Physics - The frequency dependence of the propagation constants of plane layered dielectric waveguides with the Kerr nonlinearity is considered. An...