Unbounded Waveguide Research Articles

Analysis of guided and leaky modes of planar optical waveguides using transparent boundary conditions

We present a numerical approach to compute and characterize both guided and leaky modes in a multilayer planar optical waveguide made of any lossy and dispersive materials. Usually, in numerical calculations based on finite element methods, perfectly matched layers (PMLs) are used to truncate the unbounded substrate and cover layers. However, it is difficult to make such PMLs transparent for both guided and leaky modes at the same time, and often, different or even contradictory PML parameters are required for these two types of modes. In contrast, the transparent boundary conditions (TBCs) that we use in this work can terminate the unbounded waveguide and, simultaneously, provide perfect transparency for the modes. In addition, this type of boundary condition does not contaminate the solutions with non-existent modes, such as PML modes. More importantly, the TBC approach yields the nonlinear eigenvalue solutions that can be intrinsically mapped to the parameter space of transverse wavenumbers in the claddings. This allows us to uniquely determine the power flow properties of all the calculated modes. A finite element Python package is developed to treat a variety of planar waveguides in a robust and systematic way.

Model order reduction of layered waveguides via rational Krylov fitting

Rational approximation recently emerged as an efficient numerical tool for the solution of exterior wave propagation problems. Currently, this technique is limited to wave media which are invariant along the main propagation direction. We propose a new model order reduction-based approach for compressing unbounded waveguides with layered inclusions. It is based on the solution of a nonlinear rational least squares problem using the RKFIT method. We show that approximants can be converted into an accurate finite difference representation within a rational Krylov framework. Numerical experiments indicate that RKFIT computes more accurate grids than previous analytic approaches and even works in the presence of pronounced scattering resonances. Spectral adaptation effects allow for finite difference grids with dimensions near or even below the Nyquist limit.

Transparent boundary condition for the calculation of eigenmodes in transversely infinite waveguides

Abstract. Waveguides play one of the key figures in today's electronics and optics for signal transmission. Corresponding simulations of electromagnetic wave transportation along these waveguides are accomplished by discretization methods such as the Finite Integration Technique (FIT) or the Finite Element Method (FEM). For longitudinally homogeneous and transversely unbounded waveguides these simulations can be approximated by closed boundaries. However, this distorts the original physical model and unnecessarily increases the size of the computational domain size. In this article we present a boundary condition for transversely open waveguides based on the Kirchhoff integral which has been implemented within the framework of FIT. The presented solution is compared with selected conventional methods in terms of computational effort and memory consumption.

Analytic solution to a waveguide featuring caustics and shadow zones

Exact solutions to waveguides are useful tools for validating numerical waveguide models. In this paper, a modal solution to an unbounded waveguide with a range-independent but depth-dependent profile is described. The sound speed profile utilized is a symmetric, piecewise n2-linear profile matched above and below to a homogenous half-layer with density and sound speeds that prevent any reflections at the interface. In this environment, caustics and shadow zones are formed. A nearly exact solution to this environment could help improve other numerical methods' accuracy near caustics and shadow zones. In this mathematical analysis, both proper (trapped) and improper (leaky) modes are included. Arbitrarily high frequencies are permitted through the use of numerically well-conditioned approximations to Airy functions, particularly for the dispersion relations and mode-shape evaluations. The proper (real) and improper (complex) modal eigenvalues are found approximately, with root-finding algorithms used for eigenvalues near cut-off to improve the approximation. Mathematical details are presented along with a brief discussion of how well a few common numerical solvers agree with the analytic solution derived.

Adaptive mode solver for varying refractive index profile’s waveguides with modified spectral element method

A mode solver based on the spectral element method with mesh adaptation is proposed to calculate the modal characteristics of a semivectorial field in open varying-index optical waveguides. General optical waveguides with varying refractive indices are studied for the transverse electric and transverse magnetic cases, where perfectly matched layers (PMLs) are used to truncate the unbounded waveguides. The optical field expanded by a suitable set of orthogonal basis points (Gauss–Lobatto–Legendre or Gauss–Chebyshev collocation points) through the spectral element method are all represented by processing of characterization of composite material through mesh adaptation. By this combined solver, a large number of accurate PML modes can be easily calculated. Our results are still found to be in good agreement with those produced by other numerical methods, but with more efficiency. Additionally, the relations of the PML modes’ distribution to the parameters of PMLs are analyzed through the pitchfork phenomenon in the spectrum.

Dispersion relations of the modes for open nonhomogeneous waveguides terminated by perfectly matched layers

The perfectly matched layer (PML) is a widely used tool to truncate the infinite domain in modal analysis for optical waveguides. Since the PML mimics the unbounded domain, propagation modes and leaky modes of the original unbounded waveguide can be derived. However, the presence of PML will introduce a series of new modes, which depend on the parameters of PML, and they are named as Berenger modes. For two-dimensional step-index waveguides, the eigenmode problem is usually transformed into an algebraic equation by the transfer matrix method (TMM). When the waveguide is nonhomogeneous, in which the refractive index in the core is varied, TMM is not available. In this paper, we use the differential TMM to derive the dispersion relation. We also deduced the asymptotic formulas for leaky modes and Berenger modes separately, which are accurate for large modes.

An alternative to Dirichlet-to-Neumann maps for waveguides

We are interested by the treatment of the radiation condition at infinity for the numerical solution of a problem set in an unbounded waveguide. We propose an alternative to the classical approach involving a modal expression of Dirichlet-to-Neumann (DtN) operators. This method is particularly easy to implement since it only requires the solution of boundary value problems with local boundary conditions. The corresponding approximate solution is comparable in accuracy to the one obtained by truncating the infinite series in the DtN maps.

Explicit finite element artificial boundary scheme for transient scalar waves in two‐dimensional unbounded waveguide

AbstractTo simulate the transient scalar wave propagation in a two‐dimensional unbounded waveguide, an explicit finite element artificial boundary scheme is proposed, which couples the standard dynamic finite element method for complex near field and a high‐order accurate artificial boundary condition (ABC) for simple far field. An exact dynamic‐stiffness ABC that is global in space and time is constructed. A temporal localization method is developed, which consists of the rational function approximation in the frequency domain and the auxiliary variable realization into time domain. This method is applied to the dynamic‐stiffness ABC to result in a high‐order accurate ABC that is local in time but global in space. By discretizing the high‐order accurate ABC along artificial boundary and coupling the result with the standard lumped‐mass finite element equation of near field, a coupled dynamic equation is obtained, which is a symmetric system of purely second‐order ordinary differential equations in time with the diagonal mass and non‐diagonal damping matrices. A new explicit time integration algorithm in structural dynamics is used to solve this equation. Numerical examples are given to demonstrate the effectiveness of the proposed scheme. Copyright © 2011 John Wiley & Sons, Ltd.

A Unified Mode Solver for Optical Waveguides Based on Mapped Barycentric Rational Chebyshev Differentiation Matrix

A unified high-accuracy mode solver is proposed to compute the modes of planar optical waveguides. The advantage of this solver is composed of three aspects. The first is the high accuracy of mapped barycentric rational Chebyshev differentiation matrix (MBRCDM) for approximating differential operator. The second is the high efficiency of multidirectional Newton iteration for root finding on complex plane. The third is the outstanding performance of perfectly matched layer (PML) as an absorbing boundary condition (BC). Specifically, MBRCDM method is accurate enough to compute as many modes as necessary for waveguides bounded with regular Robin BCs. MBRCDM-PML method, where PML is used to truncate an unbounded waveguide followed by the MBRCDM method, performs very well. MBRCDM-Newton method, where multidirectional Newton iteration for a dispersion equation is implemented with some initial values obtained by MBRCDM method, is more efficient. By this combined solver, a large number of accurate modes can be easily calculated. This solver is particularly essential for the computing of high index modes.

Mechanism of azimuthal mode selection in two-dimensional coaxial Bragg resonators

The analysis of electrodynamic properties of two-dimensional (2D) Bragg resonators of coaxial geometry realizing 2D distributed feedback was carried out using a quasioptical approach of coupled-wave theory and three-dimensional (3D) simulations. It is shown that the high selectivity of a 2D Bragg resonator over the azimuthal index originates from the topological difference in the dispersion diagrams of the normal symmetrical and nonsymmetrical waves near the Bragg resonance frequency in a double-periodic corrugated unbounded waveguide. For a symmetrical mode near the Bragg frequency it was found that the group velocity tends to zero as well as its first derivative. This peculiarity of the dispersion characteristic provides the conditions for the formation of an eigenmode with a Q-factor essentially exceeding the Q-factors of other modes. The results of the theoretical analysis coincide well with results of 3D simulations using the CST code “MICROWAVE STUDIO” and confirm the high azimuthal selectivity of the system.

The method of waveform images for finite waveguides with resistive terminations subject to arbitrary initial conditions

Reflection at normal incidence of a plane wave can be described by imaging the incident wave profile on the opposite side of the boundary. This concept has been introduced in a few texts, but only for nondissipative conditions. In this paper the procedure to describe a purely resistive boundary is generalized, and then the concept is extended to describe the transient response of a one-dimensional, finite length waveguide. The field generated by arbitrary initial conditions is characterized by an infinite number of images, which leads to a representation of the acoustic field as oppositely propagating waves in an unbounded waveguide. Both graphical and mathematical descriptions of these waves are derived, with the former shown to provide significant insights. Mathematical analysis of the image construction leads to identification of several fundamental acoustic phenomena, including acoustic modes and reverberation time. From an instructional viewpoint the ability to explore fundamental acoustic phenomena without recourse to solving differential equations makes the waveform image concept especially useful as an introductory tool.

On the use of waveform images to describe the initial response of finite‐length waveguides

The d’Alembert solution of the wave equation can be adapted to describe reflection from planar boundaries. One technique for doing so images the incident wave on the opposite side of the boundary. This concept has been introduced in a few texts, most extensively by Morse and Ingard [Theoretical Acoustics, McGraw‐Hill, New York (1964), pp. 106–115], but only for nondissipative ends (infinite or zero impedance.) This paper formalizes the procedure for the case where the boundary has a resistive impedance that is independent of frequency, and then extends it to treat waveguides of finite length. It is shown that the field that results from arbitrary initial conditions can be represented by an infinite number of images. This leads to a representation of the acoustic field as oppositely propagating wave in an unbounded waveguide, with only a limited number of images overlapping at any instant. Both mathematical and graphical descriptions of these waves are derived. In addition to assisting the student to understand the evolution of the field, mathematical analysis of the image construction leads to a number of physical and mathematical insights to fundamental acoustic phenomena. These include the fact that the field in the dissipationless case can be represented as a modal series with associated natural frequencies, and a quantitative understanding of the manner in which the field decays when either end is dissipative. A corollary of the latter analysis is an expression for reverberation time that is remarkably similar to the Norris–Eyring formula. From an instructional viewpoint, the fact that all results are derived without recourse to solving differential equations makes the image waveform concept especially useful as a way of introducing new students to fundamental acoustic phenomena.

Interaction Mechanism of the Parametrically Excited Solitons

The parametrically excited solitons manifest themselves with many distinct particle-like behaviors. Here the internal dynamics of the soliton-soliton interactions is explored both analytically and numerically. The results confirm the attraction between polarity-like solitons and the repulsion between opposite polarity ones. The attraction can make a pair of like polarity solitons form a standing (or oscillating) bound state. Therefore, multisolitons in an unbounded waveguide would asymptotically develop into an array of mutually independent standing soliton(s) or/and the bound state(s). It is shown that the bound state has two unidirectional momenta flowing toward the symmetric center, and thus the solitons do not perform the collisions in the classical sense. It is further demonstrated that the observed periodic "collisions" are actually due to the dissipative-induced collapses and the parametric recreations of solitons.

Dirichlet-to-Neumann Maps for Unbounded Wave Guides

Dirichlet-to-Neumann (DtN) boundary conditions for unbounded wave guides in two and three dimensions are derived and analyzed, defining problems that are suitable for finite element analysis. In the most general cases considered wave numbers may vary in arbitrary cross sections. The full DtN operator, in the form of an infinite series, is exact. Nonunique solutions may occur when this operator is truncated. Simple criteria for the number of terms in the truncated operator that guarantee unique solutions are presented. A simple modification of the truncated operator leads to uniqueness for any number of terms. Numerical results validate the performance of DtN formulations for wave guides and confirm the criteria for uniqueness.

A simple example of a trapped mode in an unbounded waveguide

This letter presents concerns on analysis of a trapped mode in a simple one-dimensional waveguide. The chosen system is an unbounded elastic string lying on a Winkler elastic foundation. It is shown that in such a system there is the single eigenfrequency corresponding to a trapped mode.

Finite‐element analysis of dielectric slab waveguide with finite periodic corrugation

AbstractTo calculate the reflection/transmission characteristics of a dielectric slab waveguide with finite periodic corrugation, a numerical approach based on the finite‐element method is proposed. Although the finite‐element method has already been applied to discontinuity problems in dielectric waveguides, it was virtually impossible from the viewpoints of computation time and computer capacity to apply the finite‐element method directly to a waveguide with many discontinuities such as a finite periodic waveguide. In the present article, a “substructure” has been introduced to the finite‐element formulation to treat a large number of discontinuities. Consequently, the interaction of surface waves and radiated waves among all discontinuities is automatically taken into account. Also, an unbounded waveguide is replaced by a corresponding bounded one; this creates a problem where an artificial boundary should be set. It was found that in a finite periodic waveguide, a good approximate solution can be obtained by choosing the position of the artificial boundary with an adequate distance from the waveguide, since the dependence of the solution on the position of the artificial boundary is not significant. The validity and usefulness of this method was confirmed by comparing the numerical results with the values obtained from other theoretical methods and the experimental values.

Scattering by Abrupt Discontinuities on Planar Dielectric Waveguides

Two unifying aspects of the problem of scattering by an abrupt discontinuity on a planar dielectric waveguide are considered. The first aspect is concerned with the numerical solution of the scattering problem through consideration of a corresponding bounded waveguide problem in which perfect electric or magnetic conductor bounds with variable locations are introduced. It is shown that the solutions to the bounded problem, when numerically integrated over a range of bound locations defined within half a wavelength, allow the complete mode spectra of the unbounded waveguide to be accurately accounted for. Scattering solutions for both TE- and TM-modes are presented for a wide range of discontinuities and, in the TE-mode case, are in agreement with results obtained using the method due to Rozzi [5]. The second aspect is concerned with the relationship between scattering and inter-waveguide mode orthogonality. Based on a simple iterative scheme, a meaningful physical interpretation of the scattering process is developed. This allows the scattering to be classified as being of first or higher order, to be explained in terms of the physical characteristics of the mode fields.