Inhomogeneous Waveguide Research Articles

Parabolic propagation in a weakly range-dependent duct: Approaching higher orders systematically

This work illustrates a technique that exploits energy-conserving transformations to split a CW field into a pair of components that propagate via uncoupled parabolic equations in opposite directions along the axis of a weakly inhomogeneous waveguide. A systematic series of these transformations produces this splitting at successively higher orders while avoiding backscatter. In the interest of clarity, the simplest possible nontrivial case is considered: waves of vertical displacement on a stretched membrane with a smooth density inhomogeneity along the y direction that forms a duct in the x direction. (This case is truly two-dimensional and its only environmental variable, density, is continuous.) This transformation technique provides an efficient means of incorporating the effects of weak environmental inhomogeneity in higher-order parabolic propagation. [Work supported by ONR.]

Obtaining optimal time-delay, source localization and tracking estimates in free space and in an ocean waveguide

Asymptotic methods for obtaining optimal time-delay, source position and track estimates are derived for measurements in free-space and in an ocean waveguide using nonlinear statistical inference theory. The methods are useful for determining necessary conditions on controllable variables such as signal-to-noise ratio and sample size to attain desired design error thresholds. For free-space time-delay estimation, it will be shown both analytically and intuitively that the matched filter cannot be unbiased and attain minimum variance unless the kurtosis of the signal’s energy spectrum exceeds the signal-to-noise ratio [Naftali and Makris, J. Acoust. Soc. Am. 110, 1917–1930 (2001)]. In an ocean waveguide, it will be shown that signal-to-noise ratios and sample sizes necessary to attain practical localization error thresholds can become prohibitively large when random waveguide inhomogeneities such as internal waves degrade intermodal coherence.

An asymptotic theory for internal reflection in weakly inhomogeneous elastic waveguides

An asymptotic theory for internal reflection in the plane elastic waveguide, slowly varying along one of the longitudinal directions, is developed by the method of matched asymptotic expansions. The reflection takes place within cross-sections in which the vibration frequencies coincide with the cut-off frequencies. After transforming the original elasticity equations into a Schrödinger type equation, the results are derived in a general form. These results may be applied to other areas of mathematical physics, provided the governing equations allow such a transformation.

Efficiency of Spatial Matched Filtering of Shallow-Water Sound Waveguide Modes

Peculiarities of the mode shadow formation in a randomly inhomogeneous oceanic waveguide are studied using vertical radiating and receiving arrays. It is shown that the maximum mode shadow contrast can be reached for optimal parameters of the radiating and receiving arrays if characteristics of the waveguide at the points of location of the source and receiver are known. A decrease in the mode shadow contrast due to scattering of few-mode signals by random inhomogeneities in oceanic waveguides is analyzed.

On The Existence of Backward Waves In Metallic Waveguides

A proof is provided for the fact that whenever there is a frequency region of a backward wave in metallic waveguides filled with media which do not induce coupling between transverse and longitudinal fields, then there exists an adjacent region with a complex propagation constant. This proof is based on the Method of Moments approach used in converting the problem of calculating the modal fields of the inhomogeneous and/or anisotropic waveguide, to a matrix eigenvalue problem. It is also shown that for those structures which are additionally frequency independent, a necessary condition for the existence of backward waves is that TE-TM modes of the empty waveguide whose eigenfunctions are used in the representation of the fields in application of the Method of Moments, be coupled. In other words the waveguide must be inhomogeneous or anisotropic or both for a backward wave mode to exist.

Narrowband signals propagation in randomly inhomogeneous shallow water waveguide

In the presented work results of experiment and theoretical modeling are considered for the narrowband (ratio of frequency band to frequency ∼0.1) sound propagation at the long range (up to 200 km) acoustic track in Barents sea. Modes selection for these conditions can be used for acoustical tomography of large-scale perturbations. Characteristics of the sound signals passing through this area (and in turn feasibility of tomographic methods) are determined by mutually competing mechanisms: waveguide dispersion, providing modes filtering, modes attenuation and modes coupling due to random inhomogeneities, masking separation of modes. Experimental results presented in this work show significant influence of noise (or modes coupling). For interpretation of experimental results (frequency and time dependencies of arriving pulses) background internal waves are taken as a main reason of the modes coupling. It is shown that for the modes separation it is necessary to use vertical array along with the measuring difference in arrival time, and correlation processing of received signals. [Work supported by RFBR, grants Nos. 03-05-64568 and 02-02-16509.]

Wave motion in infinite inhomogeneous waveguides

The analysis of wave motion in infinite homogeneous waveguides, having a complicated cross-section and/or an irregular inclusion, is a rather difficult task for the majority of available methods, especially when striving for accurate results. In contrast, this presented procedure performed in the frequency domain, is simple to apply. It yields correct results because the radiation conditions are considerably accurately satisfied, and it offers a clear parametric insight into wave motion. This procedure uses the FE modelling of an analysed section of the waveguide. It is based on the decomposition of wave motion, distinguishing propagating and non-propagating wavemodes by solving the eigenvalue problem. The presented examples demonstrate the effectiveness of this procedure, whilst a comparison between computed and analytical results demonstrates its accuracy.

Exact treatments of light propagation in inhomogeneous slab waveguides using supersymmetric quantum mechanics

Investigation of the inhomogeneous slab waveguide characteristics are basic demands for developing all-optical systems. Traditionally, numerical methods are used for evaluating the waveguide parameters. In this paper, thestandard techniques of supersymmetric quantum mechanics are developed in order to obtain a large class of exactly solvable inhomogeneous media with position-dependent index of refraction.

Discrete solitons in inhomogeneous waveguide arrays.

The existence and dynamical properties of discrete solitons in inhomogeneous waveguide arrays with a Kerr nonlinearity are studied in two different configurations. First we investigate the effect of a longitudinal periodic modulation of the coupling strength on the dynamics of discrete solitons. It is shown that resonances of internal modes of the soliton with the longitudinal structure may lead to soliton oscillations and decay. Second we study the existence and stability of discrete solitons in arrays exhibiting a linear variation of the waveguide effective index in the transverse direction. We find that resonant coupling between conventional discrete solitons and linear Wannier-Stark states leads to the formation of so-called hybrid discrete solitons.

TM mode in inhomogeneous slab waveguide as an exactly solvable oscillator-like Hamiltonian

In all practical waveguides, the index of refraction produced by a diffusion process (e.g. ion exchange) is inhomogeneous. For determining the optical characteristics, the numerical approach is suitable in the general case. In this paper we will obtain a mathematical analogy between a quantum mechanical harmonic oscillator-like Hamiltonian and the Helmholtz equation for TM-mode electromagnetic fields propagating in the slab waveguide. Using this approach, oscillator-like index of refraction profiles for exact solutions of the Helmholtz equation are proposed.

Various numerical techniques for analysis of longitudinal wave propagation in inhomogeneous one-dimensional waveguides

This paper presents various numerical techniques for the analysis of the one dimensional wave equation with variable coefficients, for governing materials with spatially varying thermal, inertial and elastic properties. Approximate time domain Finite Elements (FE) and frequency domain Spectral Finite Elements (SFE) are formulated, being computationally efficient, accurate and fast convergent compared to any other existing tool. The Sturm-Liouville problem corresponding to this wave equation is discussed and numerically solved using Prufer transformation, and the effect of inhomogeneity on the natural frequencies and mode shapes is elicited. As an application, Functionally Graded Materials (FGM), where the material properties are tailored to vary in spatial directions, are chosen, and the longitudinal wave propagation in an FGM rod due to high frequency impact load is studied. As a possible application of FGM, attenuation of waves reflection at a joint of dissimilar materials is investigated. The notion of inhomogeneous waves, used in dissipative media of more than one dimension, is retained in this one-dimensional approximation.

Scattering losses from planar waveguides with material inhomogeneity

We consider the propagation of guided waves in a planar waveguide that has a finite-length segment of inhomogeneous media. Except for the inhomogeneous segment, which varies in length from 0 to 160λ, the waveguide is piecewise homogeneous everywhere. The inhomogeneity is modelled by two-dimensional random permittivity fluctuations that are numerically generated from an assumed Gaussian correlation function. In 2D, the Maxwell equations are solved in the frequency domain for both TE and TM polarization by using modal expansion methods, perfectly matched absorbing boundary layers and the R -matrix transfer matrix algorithm. The guided waves are excited by a Gaussian beam incident on the waveguide aperture. For various waveguide design parameters, numerical results are given for waveguide power loss per unit length of waveguide inhomogeneity. The power loss curves are calculated as the average from numerous realizations of the random permittivity and the coefficient of variation is also given.

The scattering of guided waves in partly embedded cylindrical structures.

The scattering of ultrasonic guided waves at a point where a free cylindrical waveguide enters an embedding material is investigated. A modal solution that is valid when the guided waves are incident from the free section of the waveguide is developed. It is shown that in this case it is valid to consider only the modal fields over the cross section of the waveguide, neglecting the fields in the embedding material. As an application, the scattering of the lowest-order longitudinal mode in a cylindrical waveguide, L(0,1), is examined in detail. As well as considering epoxy resin as an embedding material, the case where the embedding material is replaced by a perfectly rigid boundary is discussed. The latter gives some insight into the role of nonpropagating and inhomogeneous waveguide modes in the scattering process. The results from the modal solution are validated using Finite Element modeling, very good agreement being obtained.

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.

Green's Function Expansions in Dyadic Root Functions for Shielded Layered Waveguides — Abstract

Dyadic Green's functions for inhomogeneous parallel-plate waveguides are considered. The usual residue series form of the Green's function is examined in the case of modal degeneracies, where secondorder poles are encountered. The corresponding second-order residue contributions are properly interpreted as representing "associated functions" of the structure by constructing a new dyadic root function representation of the Hertzian potential Green's dyadic. The dyadic root functions include both eigenfunctions (corresponding to first-order residues) and associated functions, analogous to the idea of Jordan chains in finite-dimensional spaces. Numerical results are presented for the case of a two-layer parallel-plate waveguide.

Method of integral equation in the theory of weak-guiding inhomogeneous optical waveguides

A method of integral equation is stated and applied in the analysis of the characteristic and noncharacteristic modes in 3D weak-guiding inhomogeneous optical waveguides. The dispersion curves for the modes of diffused channel waveguides are studied near the critical conditions in absorbing media. The accuracy of recovering the permittivity profile in a channel waveguide from the far-field radiation of the fundamental mode is estimated.

Analytical solution of non-homogeneous anisotropic wave equations based on differential transfer matrices

We present a new analytical method for solution of non-homogeneous anisotropic optical systems, based on the extension of transfer matrices into differential form. This approach can be used for exact calculation of various functions including reflection and transmission coefficients, band structures and bound states. We discuss improvements of the method for infinite periodic structures and bounded modes. Then simplifications of the approach are considered for the TE and TM polarized isotropic structures, pseudo-isotropic and uniaxial non-magnetic media. Moreover a general variational representation of bound states is introduced. As application examples, we consider the reflection from inhomogeneous structures, the band structure of infinite periodic media and bounded modes in inhomogeneous anisotropic waveguides. Excellent agreement between the results from our differential transfer matrix method and other methods is observed where comparison is possible.

A fast reduced-order model for the full-wave FEM analysis of lossy inhomogeneous anisotropic waveguides

The evaluation of the frequency response of waveguiding structures by means of the full-wave finite-element method requires solving a large generalized eigenvalue problem for each frequency. This paper describes a novel approach, based on the singular-value decomposition, which drastically reduces the order of the eigenvalue problem. By inspection of the singular values, the accuracy level of the procedure may be controlled. The technique is applied to the analysis of open and closed waveguides with arbitrary cross section, lossy conductors, and anisotropic dielectric layers, by means of vector elements of generic order; higher order elements are shown to allow the accurate evaluation of fields inside lossy conductors with fewer unknowns, besides exactly modeling normal field discontinuities at material interfaces. Examples of application of the reduced-order technique are shown concerning both non-TEM and quasi-TEM structures.

Lamb wave propagation in inhomogeneous elastic waveguides

The problem of Lamb wave propagation in an axially multilayered waveguide is treated by a multimodal approach. A general formalism is proposed that avoids the numerical divergence due to evanescent...

Acoustic field of a vertical cylindrical antenna in an inhomogeneous waveguide with an impedance lower boundary

An expansion of the field of a vertical antenna located in an inhomogeneous waveguide in terms of the normal waves of a homogeneous reference waveguide is obtained. The frequency dependence of the radiation resistance is analyzed numerically for various antenna depths and sound velocity profiles. Variations in the radiation resistance are correlated with the variations in the sound velocity.