Investigation of elastic modes propagating in multi-wire helical waveguides

Elastic guided waves have some potential for non-destructive inspection of civil engineering multi-wire steel cables. However, wave propagation inside such structures is not yet fully understood. This paper investigates multi-wire helical waveguides with special attention to the common seven-wire strand configuration (one straight core surrounded by one layer of six helical wires). A helical coordinate system is first proposed. Though non-orthogonal, this system preserves translational invariance along the helix centreline to explicitly perform a spatial Fourier transform. Then, it is shown that for the analysis of multi-wire helical strands a twisting system—which is a special case of helical systems—is translationally invariant. A semi-analytical finite element method is developed reducing the problem on the cross-section only. A straightforward computation of energy velocity is proposed. Dispersion curves for a single straight wire and a helical wire are first computed to verify the adequacy of the twisting system. Finally the seven-wire strand is analysed using simplified contact conditions. Theoretical dispersion curves are compared to low-frequency magnetostrictive measurements. Good agreement is found for the first compressional-like mode and its associated veering central frequency (‘notch frequency’).     

Numerical investigation of dispersion relations for helical waveguides using the Scaled Boundary Finite Element method

In this paper, the dispersion properties of elastic waves in helical waveguides are investigated. The formulation is based on the Scaled Boundary Finite Element method (SBEFM). With a set of orthogonal unit basis introduced as the contravariant basis, the helical coordinate is firstly considered, where components of tensor retain the dimension of original quantity. Based on the strain–displacement relation, the eigenvalue matrix is obtained about wavenumbers and frequencies. The cross section of the waveguides is discretized by using high-order spectral elements. Moreover, the formulated linear matrix is utilized to design efficient and accurate algorithms to compute the eigenvalues of helical waveguides. Compared with the Pochhammer–Chree curves, the convergence and accuracy of the SBFEM are discussed. Finally, we give some dispersion curves for a wide range of lay angles and analyze in detail properties of cut-off frequency, mode separation and mode transition for elastic wave propagation in the helical waveguides.     

Critical ray-like propagation modes along graded-index planar optical waveguides

Graded-index planar optical waveguides and surface compression layers were formed simultaneously in the surface layer of glass plates by ion exchange. A change in the refractometer patterns was observed. Before ion exchange, only a critical ray fringe was observed, but with ion exchange a guided wave fringe appeared on the high effective refractive index side of the ‘critical ray’ fringe, and the number of guided wave fringes increased. The guided wave fringe or fringes were birefringent, whereas the ‘critical ray’ fringe was kept nonbirefringent. It was concluded that the ‘critical ray’ propagated along the bottom of a waveguide, ie at the foot of the refractive index distribution.     

Numerical modelling of TE and TM modes in lossy nonlinear optical waveguides

The intensity-dependent characteristics of nonlinear TE and TM waves in a thin film bounded by lossy nonlinear media are analysed by the finite-element method. In this approach, the power-dependent complex propagation constants and local complex electromagnetic field distributions of lossy nonlinear TE and TM waves are obtained directly from the given lossy nonlinear waveguides, without any approximations. Numerical results for these modes in a lossy nonlinear waveguiding system with different absorption coefficients are given. It is shown that the complex dispersion relations, for both TE and TM modes, are strongly power dependent.     

Equal-phase-velocity propagation for the TE and TM modes of planar metal-clad optical waveguides

An asymmetrical four-layered metal-clad dielectric optical waveguide can support the TE_N and TM_N modes forN0 with the same phase-velocity if the core thickness,t ₃, and the buffer layer thickness,t ₂, are suitably chosen. Only metals that have a negative real part ₁ of their complex permittivity at optical frequencies are considered as the cladding material. It is shown that the application of a lossless approximation which set ₁=0 enables the development of a real expression which may be solved directly for the values oft ₂ andt ₃ that are required to achieve the equal-phase-velocity characteristic. Parametric plots for the guide dimensions are presented for representative waveguides and it is shown that an appropriate choice oft ₂ andt ₃ will reduce the absorption losses of the TE and TM modes to negligible levels. This property of the waveguide will find application in thin film optical devices such as polarization rotation modulators that require the simultaneous propagation of TE and TM polarized waves.     

Numerical algorithm for waveguide and leaky modes determination in multilayer optical waveguides

A new algorithm for waveguide and leaky modes spectrum calculation in multilayer planar waveguides is proposed. The method is of high computational efficiency due to the preliminary search for propagation constants approximation. Numerical efficiency of this algorithm is demonstrated for several waveguide structures. Waveguide and leaky modes propagation constants calculated for these structures using proposed method are in good agreement with previously published results.     

Principal modes in multimode waveguides

We generalize the concept of principal states of polarization and prove the existence of principal modes in multimode waveguides. Principal modes do not suffer from modal dispersion to first order of frequency variation and form orthogonal bases at both the input and the output ends of the waveguide. We show that principal modes are generally different from eigenmodes, even in uniform waveguides, unlike the special case of a single-mode fiber with uniform birefringence. The difference is most pronounced when different eigenmodes possess similar group velocities and when their field patterns vary as a function of frequency. This work may provide a new basis for analysis and control of dispersion in multimode fiber systems.     

Leaky modes in optical waveguides

Solutions of the scalar wave equation are presented giving, explicitly, the excitation and subsequent attenuation of leaky modes in optical waveguides.     

Guided modes in left-handed waveguides

We investigate basic properties of linear guided modes in a waveguide slab with the core of a material with simultaneously negative effective permittivity and permeability surrounded by a normal dielectric cladding. The existence and dispersion of all possible types of guided modes have been studied in detail. We demonstrate that the two orthogonally polarized, TE and TM slow guided modes, with the same frequency and the same wave number can exist simultaneously. Their group velocities, however, can be parallel or anti-parallel.     

Mode propagation in curved waveguides and scattering by inhomogeneities: application to the elastodynamics of helical structures

This paper reports on an investigation into the propagation of guided modes in curved waveguides and their scattering by inhomogeneities. In a general framework, the existence of propagation modes traveling in curved waveguides is discussed. The concept of translational invariance, intuitively used for the analysis of straight waveguides, is highlighted for curvilinear coordinate systems. Provided that the cross-section shape and medium properties do not vary along the waveguide axis, it is shown that a sufficient condition for invariance is the independence on the axial coordinate of the metric tensor. Such a condition is indeed checked by helical coordinate systems. This study then focuses on the elastodynamics of helical waveguides. Given the difficulty in achieving analytical solutions, a purely numerical approach is chosen based on the so-called semi-analytical finite element method. This method allows the computation of eigenmodes propagating in infinite waveguides. For the investigation of modal scattering by inhomogeneities, a hybrid finite element method is developed for curved waveguides. The technique consists in applying modal expansions at cross-section boundaries of the finite element model, yielding transparent boundary conditions. The final part of this paper deals with scattering results obtained in free-end helical waveguides. Two validation tests are also performed.     

Correlation between helical surface waves and guided modes of an infinite immersed elastic cylinder

Scattering of obliquely incident plane acoustic waves from immersed infinite solid elastic cylinders is a complex phenomenon that involves generation of various types of surface waves on the body of the cylinder. Mitri [F.G. Mitri, Acoustic backscattering enhancement resulting from the interaction of an obliquely incident plane wave with an infinite cylinder, Ultrasonics 50 (2010) 675-682] recently showed that for a solid aluminum cylinder, there exist acoustic backscattering enhancements at a normalized frequency of ka?0.1. The incidence angle α_c at which these enhancements are observed lies between the first (longitudinal) and second (shear) coupling angles of the cylinder. He also confirmed the observations previously reported by the authors that there exist backscattering enhancements of the dipole mode at large angles of incidence where no wave penetration into the cylinder is expected. In this paper, physical explanations are provided for the aforementioned observations by establishing a correlation between helical surface waves generated by oblique insonification of an immersed infinite solid elastic cylinder and the longitudinal and flexural guided modes that can propagate along the cylinder. In particular, it is shown that the backscattering enhancement observed at ka?0.1 is due to the excitation of the first longitudinal guided mode travelling at the bar velocity along the cylinder. It is also demonstrated that the dipole resonance mode observed at incidence angles larger than the Rayleigh coupling angle is associated with the first flexural guided mode of the cylinder. The correlation established between the scattering and propagation problems can be used in both numerical and experimental studies of interaction of mechanical waves with cylinders.     

An orthogonality relation for the modes of wave propagation in an elastic circular cylinder

In this note it is shown that the orthogonality relations obtained by Fama for the eigenfunctions for the elastostatic deformation of a circular cylinder also apply for the elastodynamic case.     

Trapped modes in finite quantum waveguides

The eigenstates of an electron in an infinite quantum waveguide (e.g., a bent strip or a twisted tube) are often trapped or localized in a bounded region that prohibits the electron transmission through the waveguide at the corresponding energies. We revisit this statement for resonators with long but finite branches that we call ??finite waveguides??. Although the Laplace operator in bounded domains has no continuous spectrum and all eigenfunctions have finite L ₂ norm, the trapping of an eigenfunction can be understood as its exponential decay inside the branches. We describe a general variational formalism for detecting trapped modes in such resonators. For finite waveguides with general cylindrical branches, we obtain a sufficient condition which determines the minimal length of branches for getting a trapped eigenmode. Varying the branch lengths may switch certain eigenmodes from non-trapped to trapped or, equivalently, the waveguide state from conducting to insulating. These concepts are illustrated for several typical waveguides (L-shape, bent strip, crossing of two strips, etc.). We conclude that the well-established theory of trapping in infinite waveguides may be incomplete and require further development for applications to finite-size microscopic quantum devices.     

Waveguide modes in coupled-resonator optical waveguides

Coupling between cavities within coupled-resonator optical waveguides (CROWs) affects transmission properties. Our results show that, in addition to the number of the waveguide modes, the group velocity of electromagnetic waves transmitted in the CROWs can be controlled by changing the stiffness of coupling between cavities. This is significant for the applications.     

Guided modes in asymmetric graphene waveguides

An asymmetric quantum well in graphene can act as a slab waveguide for electron waves in a manner analogous to the electromagnetic waves in dielectrics. Guided modes and the probability current density are analyzed in the graphene electron waveguide induced by asymmetric electrostatic potential. The modes in an asymmetric graphene waveguide include guided modes, “cover modes”, “substrate modes” and “radiation modes”. The conditions for a guided mode are quantified. It is found that the fundamental mode is absent when both the Klein tunneling and classical motion are present. The confinement of electrons for lower order mode is stronger than for higher order mode. We hope that these characteristics in asymmetric graphene waveguide can provide potential applications in graphene-based waveguide devices.     

Wave propagation in deep-subwavelength mode waveguides

This Letter proposes a dielectric waveguide with deep-subwavelength mode sizes. Results of both frequency domain and time domain analysis show that the effective mode area is below λ(0)(2)/400 and can even reach λ(0)(2)/1000 (λ(0) is the wavelength in vacuum). The effective electrical mode area can be comparable to that of a hybrid plasmonic subwavelength confinement waveguide, with reduced optical absorption. In contrast to slot waveguides, which guide light in low-index materials, the proposed structure guides light in high-index materials. Results obtained in this Letter show that the losses are sensitive to the surface roughness on the tens of nanometers scale. The structure can be used to design ring resonators with a quality factor comparable to that of a diffraction-limited dielectric ring resonator with the same standing wave numbers. The property can be applied in nonlinear effect enhancement or laser design with ultralow threshold.     

Disappearance of modes in planar Bragg waveguides

We show how the forbidden bands of a Bragg reflector may shrink to points and how some classes of the guided modes of a planar Bragg waveguide may disappear altogether by shrinking with the forbidden bands. We derive the general conditions to determine the missing modes and explain these conditions with examples. It is possible, for example, to design a Bragg waveguide that rejects all antisymmetric modes and supports only symmetric modes for the TE polarization. We also highlight the effect of Brewster incidence on the interpretation of the missing modes for the TM polarization.     

Low-loss surface modes and lossy hybrid modes in metamaterial waveguides

We show that waveguides with a dielectric core and a lossy metamaterial cladding (metamaterial-dielectric guides) can support hybrid ordinary-surface modes previously only known for metal-dielectric waveguides. These hybrid modes are potentially useful for frequency filtering applications as sharp changes in field attenuation occur at tailorable frequencies. Our results also show that the surface modes of a metamaterial-dielectric waveguide with comparable electric and magnetic losses can be less lossy than the surface modes of an analogous metal-dielectric waveguide with electric losses alone. Through a characterization of both slab and cylindrical metamaterial-dielectric guides, we find that the surface modes of the cylindrical guides show promise as candidates for all-optical control of low-intensity pulses.     

Mode propagation of ultrasound in hollow waveguides

It has previously been shown that there is close agreement between theoretical and experimental behaviour of pulses of ultrasound propagating in solid cylindrical waveguides. Waveguides are used in a number of areas of medical ultrasonics and it is therefore important to be able to model sound propagation in them accurately. This paper extends the analysis of guided wave propagation to hollow waveguides. In particular, frequency spectra of modes of progagation are given and theoretical group velocity curves are compared with experimental results. Signal strengths of modes propagating in both solid and hollow stainless steel waveguides of similar cross-sectional area are also compared.