Showing papers in "Waves in Random and Complex Media in 2007"

Introduction to Wave Scattering, Localization and Mesoscopic Phenomena. Second edition

Abstract: For any researcher working with waves in disordered media, it is crucial to have a solid reference to elementary principles with the details worked out in a complete and transparent way. This is ve...

Optimal synthesis of 2D phononic crystals for broadband frequency isolation

Abstract: The spatial distribution of material phases within a periodic composite can be engineered to produce band gaps in its frequency spectrum. Applications for such composite materials include vibration and sound isolation. Previous research focused on utilizing topology optimization techniques to design two-dimensional (2D) periodic materials with a maximized band gap around a particular frequency or between two particular dispersion branches. While sizable band gaps can be realized, the possibility remains that the frequency bandwidth of the load that is to be isolated might exceed the size of the band gap. In this paper, genetic algorithms are used to design squared bi-material unit cells with a maximized sum of band-gap widths, with or without normalization relative to the central frequency of each band gap, over a prescribed total frequency range of interest. The optimized unit cells therefore exhibit broadband frequency isolation characteristics. The effects of the ratios of contrasting material properti...

Leaky modes in twisted microstructured optical fibers

Abstract: The purpose of this paper is to investigate the effect of a twist on the losses of leaky modes in microstructured optical fibers (MOF). A helicoidal system of coordinates is introduced to define the structure and to set up the problem. These coordinates, albeit non-orthogonal, preserve the translational invariance in a way that allows a two-dimensional finite element model similar to that of classical straight waveguides. The Perfectly Matched Layer (PML) technique is used to compute the leaky modes in the fibers. Helicoidal coordinates and PML are unified under the point of view of geometrical transformations which allows the design of ‘twisted PML’ that provides the right tool to determine the leaky modes in the twisted structures.

A simplified asymptotic theory for ocean surface electromagnetic wave scattering

Abstract: The normalized radar cross-section (NRCS) expression of the Local Curvature Approximation (LCA-1) is derived to first order. The polarization sensitivity of this model is compared to the Kirchhoff Approximation (KA), Two-Scale Model (TSM), Small Slope Approximation (SSA-1) and Small Perturbation Method (SPM-1) to first order in the backscattering configuration. Analytical comparisons and numerical simulations show that LCA-1 and TSM could be rewritten with the same formulation and that their polarization sensitivities are comparable. Comparisons with experimental data acquired in C- and Ku-band reveal that the polarization sensitivities of these models are not adequate. However, the NRCS azimuth modulation predicted by LCA-1 is found to be dependent on polarization and sea surface roughness. This property of the LCA-1 model yields to an azimuth modulation for the polarization ratio. Based on the surface curvature correction concept, a simplified electromagnetic model is proposed. The curvature correction ...

Waves and fracture in an inhomogeneous lattice structure

Abstract: We analyze a crack propagating in an inhomogeneous rectangular lattice in the state of anti-plane shear. The filtering properties of such a lattice are linked to the energy dissipation due to waves initiated by the crack. The influence of the inhomogeneities within the lattice on the lattice trapping is investigated.

Homogenization of 3D finite photonic crystals with heterogeneous permittivity and permeability

Abstract: We consider a heterogeneous magneto-dielectric photonic crystal and derive the so-called ‘homogenized Maxwell system’ via the multi-scale method and provide ad hoc proofs for the convergence of the electromagnetic field towards the homogeneous one using the notion of two-scale convergence. The homogenized medium is described by anisotropic matrices of permittivity and permeability, deduced from the resolution of two annex problems of electrostatic type on a periodic cell. Noteworthily, this asymptotic analysis also covers the case of photonic crystals with non-cuboidal periodic cells. We solve numerically the associated system of partial differential equations with a method of fictitious charges and a finite element method (FEM) in order to exhibit the matrices of effective permittivity and permeability for given magneto-dielectric periodic composites. We then compare our results in the 2D case against some Fourier expansion approach and provide duality relations in the case of magneto-dielectric checkerb...

Approximations to wave propagation through doubly-periodic arrays of scatterers

Abstract: The propagation of waves through a doubly-periodic array of identical rigid scatterers is considered in the case that the field equation is the two-dimensional Helmholtz equation. The method of matched asymptotic expansions is used to obtain the dispersion relation corresponding to wave propagation through an array of scatterers of arbitrary shape that are each small relative to both the wavelength and the array periodicity. The results obtained differ from those obtained from homogenization in that there is no requirement that the wavelength be much smaller than the array periodicity, and hence it is possible to examine phenomena, such as band gaps, that are associated with the array periodicity.

Scattering of elastic waves by multiple elastic circular cylinders with imperfect interface

Abstract: In the current paper a general method is presented for the rigorous solution for the scattering of elastic waves by a cluster of elastic circular cylinders of infinite length. The interface separating the cylinder from the surrounding media is considered to be homogeneous imperfect. Specifically, the tractions are continuous but the displacements are discontinuous and proportional in terms of interface stiffness parameters to their respective traction components. Using the exact theory of multipole expansion, analytic solutions for the scattered and internal fields excited by an incident plane P-wave, an incident cylindrical P-wave and an incident plane SV-wave are derived. Numerical results for directivity patterns and scattering cross-sections are presented for a finite hexagonal array of elastic circular inclusions with imperfect interface. The results show that the sequence of maxima and minima in the curves of scattered cross-sections becomes more undistinguishable as the interface becomes more imper...

Three-dimensional electromagnetic nonlinear inversion in layered media by a hybrid diagonal tensor approximation: Stabilized biconjugate gradient fast Fourier transform method

Abstract: This paper presents an efficient three-dimensional nonlinear electromagnetic inversion method in a multilayered medium for radar applications where the object size is comparable to the wavelength In the first step of this two-step inversion algorithm, the diagonal tensor approximation is used in the Born iterative method The solution of this approximate inversion is used as an initial guess for the second step in which further inversion is carried out using a distorted Born iterative method Since the aim of the second step is to improve the accuracy of the inversion, a full-wave solver, the stabilized biconjugate-gradient fast Fourier transform algorithm, is used for forward modelling The conjugate-gradient method is applied at each inversion iteration to minimize the functional cost The usage of an iterative solver based on the FFT algorithm and the developed recursive matrix method combined with an interpolation technique to evaluate the layered medium Green's functions rapidly, makes this method h

Tailoring dispersion properties of photonic crystal waveguides by topology optimization

Abstract: The paper describes a systematic method for the tailoring of dispersion properties of slab-based photonic crystal waveguides. The method is based on the topology optimization method which consists in repeated finite element frequency domain analyzes, analytical sensitivity analyzes and gradient based design updates. The goal of the optimization process is to come up with slow light, zero group velocity dispersion photonic waveguides or photonic waveguides with tailored dispersion properties for dispersion compensation purposes. Two examples concerning reproduction of a specific dispersion curve and design of a wide bandwidth, constant low group velocity waveguide demonstrate the efficiency of the method.

Bistatic scattering from one-dimensional random rough homogeneous layers in the high-frequency limit with shadowing effect

Abstract: Many fast asymptotic models of electromagnetic scattering from a single rough interface have been developed over the last few years, but only a few have been developed on stacks of rough interfaces. The specific case of very rough surfaces, compared to the incident wavelength, has not been treated before, which is the context of this paper. The model starts from the iteration of the Kirchhoff approximation to calculate the fields scattered by a rough layer, and is reduced to the high-frequency limit in order to rapidly obtain numerical results. The shadowing effect, important under grazing angles, is taken into account. The model can be applied to any given slope statistics. Then, the model is compared with a reference numerical method based on the method of moments, which validates the model in the high-frequency limit for lossless and lossy inner media.

Depolarization regions of nonzero volume in bianisotropic homogenized composites

Abstract: In conventional approaches to the homogenization of random particulate composites, the component phase particles are often treated mathematically as vanishingly small, point-like entities. The electromagnetic responses of these component phase particles are provided by depolarization dyadics which derive from the singularity of the corresponding dyadic Green functions. Through neglecting the spatial extent of the depolarization region, important information may be lost, particularly relating to coherent scattering losses. We present an extension to the strong-property-fluctuation theory in which depolarization regions of nonzero volume and ellipsoidal geometry are accommodated. Therein, the size, shape and spatial distribution of the component phase particles are taken into account. The analysis is developed within the most general linear setting of bianisotropic homogenized composite mediums (HCMs). Numerical results are presented for a representative example of a bianisotropic HCM, namely a Faraday chir...

Scattering of ice coupled waves by a sea-ice sheet with random thickness

Abstract: Arctic sea-ice contains imperfections such as cracks, leads and pressure ridges that scatter flexural-gravity waves. Models for predicting scattering have been described in the literature, concentrating mainly on singular isolated features with simplified shapes or on arrays of such features. In reality ridges are seldom simple and leads are rarely entirely free of ice. Here we describe a model in which the scattering by a sheet of arbitrary thickness can be simulated. Linear wave theory and Green's functions are used to derive the governing equations for a numerical model of a two-dimensional (in the vertical) system. We examine wave scattering by random ice sheets, identifying trends in behavior as the wave period and the length, median thickness and variance of the sheet are changed. It has been suggested that wave scattering could be used to identify sea-ice thickness, a task which is difficult or expensive by other methods, and here we examine a technique by which this could potentially be achieved. ...

Effect of the illumination length on the statistical distribution of the field scattered from one-dimensional random rough surfaces: analytical formulae derived from the small perturbation method

Abstract: We consider a dielectric plane surface with a local cylindrical perturbation illuminated by a monochromatic plane wave. The perturbation is represented by a random function assuming values with a Gaussian probability density with zero mean value. Outside the perturbation zone, the scattered field can be represented by a superposition of a continuous spectrum of outgoing plane waves. The stationary phase method leads to the asymptotic field, the angular dependence of which is given by the scattering amplitudes of the propagating plane waves. The small perturbation method applied to the Rayleigh integral and the boundary conditions gives a first-order approximation of the scattering amplitudes. We show that the real part and the imaginary part of the scattering amplitudes are Gaussian stochastic variables with zero mean values and unequal variances. The values of variances depend on the length of the perturbation zone. In most cases, the probability density function for the amplitude is a Hoyt distribution ...

Bloch-wave homogenization for spectral asymptotic analysis of the periodic Maxwell operator

Abstract: This paper is devoted to the asymptotic behavior of the spectrum of the three-dimensional Maxwell operator in a bounded periodic heterogeneous dielectric medium T = [−T,T]3, T > 0, as the structure period η, such that η−1 T is a positive integer, tends to 0. The domain T is extended periodically to the whole of ℝ 3, so that the original operator is understood as acting in a space of T-periodic functions. We use the so-called Bloch-wave homogenization technique which, unlike the classical homogenization method, is capable of characterizing a renormalized limit of the spectrum (called the Bloch spectrum) [6]. The related procedure is concerned with sequences of eigenvalues Λη of the resolvent of the order of the square of the medium period, which correspond to the oscillations of high-frequencies of order η−1. The Bloch-wave description is obtained via the notion of two-scale convergence for bounded self-adjoint operators, and a proof of the ‘completeness’ of the limiting spectrum is provided. The results o...

The role of surface tension in elastic wave scattering in an inhomogeneous poroelastic medium

Abstract: It is well known that many porous media such as rocks have heterogeneities at nearly all scales. We applied Biot's poroelastic theory to study the propagation of elastic waves in isotropic porous matrix with spherical inclusions. It is assumed that the heterogeneity dimension exceeds significantly the pore size. Modified boundary conditions on poroelastic interface are used to take into account the surface tension effects. The effective wavenumber is calculated using the Waterman and Truell multiple scattering theory, which relates the effective wave number to the amplitude of the wave field scattered by a single inclusion. The calculations were performed for a medium containing fluid-filled cavities or porous inclusions contrasting in saturating fluid elastic properties. The results obtained show that when we consider elastic wave propagation in poroelastic medium containing soft inclusions, it is necessary to take into account the capillary pressure. The influence of the surface tension depends on the d...

A stochastic model for acoustic attenuation

Abstract: Propagation of waves in gases or liquids has been observed for a long time in homogeneous or non-homogeneous media In acoustics, attenuation is a significant problem which is studied mainly through the ‘equations of change’ of fluid mechanics These equations are based only on the macroscopic characteristics of the medium Microscopic variations, related to other phenomena like Brownian motion or critical opalescence, are not taken into account This paper provides a random one-ray model This model explains the proportionality between the standard attenuation and the product between length and frequency squared in a logarithmic scale The wave is shown to be necessarily associated with noise, even if this noise cannot be observed by devices Furthermore, the ‘coefficient of variation’ defined in turbulent environments can be explained as a random version of the usual coefficient of attenuation

Transfer matrix method for point sources radiating in classes of negative refractive index materials with 2n-fold antisymmetry

Abstract: We introduce a transfer matrix algorithm well-suited for negative refractive index materials. We achieve a clean mathematical derivation of the electromagnetic field radiated by finite and countable sets of harmonic point sources within a class of perfect lenses [1] which present some periodicity along one axis. In the case of a periodic set of point sources, combining a coordinate transformation [2] with the transfer matrix method enables a rigorous calculation of the vector field within perfect corner lenses consisting of intersecting planes delimiting positive and negative index media, in the spirit of [3]. In contrast to [4] where two negative corners sharing the same vertex led to spatial oscillations of surface plasmons being inversely as ln (σ), we observe that 2n negative corners (related by 2n-fold antisymmetry) lead to linearly decreasing absorption σ, for large n. This may result in a better trap for light in a practical device such as a poor man's corner reflector made out of thin sectors alte...

The spatial focusing of a leaky time reversal chaotic cavity

Abstract: Elastic waves propagating inside a solid chaotic cavity create a diffusive random field that contains both longitudinal and shear waves. In the current paper, we are interested in the field radiated in a fluid in contact with such cavity. The goal of this paper is to predict the spatial focusing properties that can be obtained in the fluid using a time-reversal piezoelectric transducer in contact with the cavity. We present a statistical approach that supposes a fully diffused wavefield inside the cavity with an equipartition of energy between longitudinal and shear waves. We show that the critical angles of transmission in the solid–fluid interface generate a cut-off of the spatial frequencies and then a degradation in the spatial focusing. This limitation can be overcome using a rough surface. A set of experiments conducted in the MHz range confirm the theoretical model.

Dynamic behaviour of speckle cluster formation

Abstract: In this work, we analyse the speckle cluster structure generated when a coherently illuminated diffuser is imaged by introducing a multiple aperture pupil mask in front of the lens plane. We demonstrate that the speckle cluster originates from the complex speckle modulation generated by multiple interferences among the wavefront passing through each aperture. The auto-correlation function of the intensity distribution when using a multiple aperture pupil arrangement is calculated. Besides, we demonstrate that the autocorrelation function and the intensity corresponding to a single scattering element of the input are coincident. This result allows interpretation of the dynamics behaviour of the speckle cluster formation by considering the result obtained by a single scattering element. Then, we determine the pupil mask geometrical parameters that control the cluster behaviour and therefore the condition for obtaining a highly repetitive cluster structure that we define as a ‘regular cluster’. The theoretic...

C-band scattering simulation of a Scots pine shoot

Abstract: Scots pine (Pinus sylvestris L.) has the widest distribution of any pine and is one of the most important timber trees in Europe and one of the main species of boreal forest, which have an important role in climate change studies. The leaf area index (LAI) is one of the key input parameters for the climate change models. Recently a relationship between C-band backscattering and the LAI has been detected for Scots pine. To understand the C-band microwave characteristics of Scots pine shoots the backscattering is simulated using the so-called discrete dipole approximation (DDA), which is the only possible, nearly exact method for this problem. The backscattering of the shoot is dominated by the needles. The VV/HH backscattering ratio of closely spaced parallel and perpendicular shoots averages to that of a single shoot. For a simulated whole Scots pine crown the VV/HH backscattering ratio is related to the total orientation distribution of the needles. The variation range of the VV/HH backscattering ratio w...

A new method for measuring the outer scale of atmospheric turbulence

Abstract: A method, based on the correlation functions of the wandering (lateral fluctuations) of two separated thin parallel laser beams propagating through the atmosphere, is proposed for measuring the outer scale of atmospheric turbulence. The method utilizes the ratio between the correlation functions of the wandering in two perpendicular planes. A simple relationship to obtain the outer scale from the measured correlation functions is established for a particular model of turbulence, the modified von Karman model. The method is validated by experimental data both in the laboratory and in the open air. To this purpose special measurements, which were conducted during an outdoor measurements campaign in 2004, are analysed. An instantaneous value of the outer scale can be obtained accurately. The results show that the size of outer scale of atmospheric turbulence changes very rapidly with time and that the proposed method can be applied in real atmospheric conditions.

Improved diagonal tensor approximation (DTA) and hybrid DTA/BCGS–FFT method for accurate simulation of 3D inhomogeneous objects in layered media

Abstract: This paper presents an improved diagonal tensor approximation (DTA) and its hybridization with the stabilized biconjugate-gradient fast Fourier transform (BCGS–FFT) algorithm to solve a volume integral equation for three-dimensional (3D) objects in layered media. The improvement in DTA is obtained for lossy media through a higher-order approximation. The interaction between the dyadic Green's function and the contrast source is efficiently evaluated by the (FFT) algorithm through the convolution and correlation theorems. For the hybrid implementation, the DTA solution is used as an initial estimate and a preconditioner in the BCGS–FFT algorithm in order to solve the forwards modelling problem accurately with fewer iterations than the conventional BCGS–FFT algorithm. The accuracy and convergence of the DTA, BCGS–FFT and hybrid DTA/BCGS–FFT methods are compared extensively with several numerical examples. Numerical results show that (a) the improved DTA formulation enhances the accuracy and (b) the DTA/BCGS...

Asymptotic analysis of Bloch-Floquet waves in a thin bi-material strip with a periodic array of finite-length cracks

Abstract: We present an asymptotic algorithm to solve a problem of wave propagation in a thin bi-material strip with an array of cracks situated at the interface between two materials. For small frequencies we construct an asymptotic solution which takes into account the singular behavior near the crack tips and the smooth nature of the oscillation far away from them. We construct the boundary layer solutions near the crack tips. The boundary layers are harmonic solutions in scaled domains. Dispersion equations are derived and solved within the frame of the asymptotic model.

Bloch–Floquet waves and controlled stop bands in periodic thermo-elastic structures

Abstract: An algorithm for controlling the stop bands for elastic Bloch–Floquet waves within a periodic structure is proposed. Explicit asymptotic estimates of frequencies of translational and rotational standing waves, together with the numerical estimates of the stop band frequencies, are given. Thermal pre-stress is introduced and used to control the position of the stop bands on the dispersion diagram.

Wave propagation in a one-dimensional randomly perturbed periodic medium

Abstract: We study the variance of the solution of a periodic randomly perturbed one-dimensional Schrodinger operator after propagation through N periods. It is shown that if the frequency of propagation lies inside the band, then the total variance is proportional to Nσ2, where σ is the intensity of the white noise. However, if the wave frequency is close to the band edge (where the transfer matrix has a Jordan block structure), the resulting variance is proportional to Nσ2/3. Thus, propagation becomes highly sensitive to random perturbations. Numerical simulations reveal that even low noise in a periodic potential can suppress transmission near the band edges and make it strongly irregular inside the band. Further increase of the noise amplitude leads to intermittent behaviour of the transmission coefficient, and makes transmission possible only for a few random frequencies in the band.

Fresnel formulation for multi-element lamellar diffraction gratings in conical mountings

Abstract: A formulation of the modal method for multi-element dielectric lamellar diffraction gratings is presented. It combines the conventional semi-analytic Kronig–Penny formulation with a Fresnel scattering matrix approach to the solution of the diffraction problem. The theory is intuitive, applicable to complex geometries, and provides insight into diffraction grating physics. With the Fresnel matrix extensions the method potentially has greater generality than other formulations, and is applicable to non-trivial waveguide problems, such as characterizing the coupling between 1D photonic crystal waveguides. A numerically stable way of using the modal method for multilayer stacks of lamellar gratings is described.

Propagation of p-polarized electromagnetic waves obliquely incident on stratified random media: Random phase approximation

Abstract: We consider the propagation of p-polarized electromagnetic waves obliquely incident on stratified random dielectric media. Using the invariant imbedding method generalized to random media and applying the random phase approximation, we derive a simple analytical expression of the localization length and calculate the disorder-averaged reflectance and transmittance and the fluctuations of the localization length and the reflectance as functions of the incident angle. We also calculate the disorder-averaged intensity profile of the magnetic field inside the random medium. We find that within the random phase approximation, the p wave can be delocalized and transmitted completely at a certain critical incident angle, which is bigger than the Brewster angle in the uniform case.

A Review Of “Modeling Fluctuations in Scattered Waves”

Abstract: Almost everything we see around us – blue sky, clouds, material objects, people – is attributable to the scattering of light by matter. Objects that are not due to scatter are mirror-like, or smoot...

Creation of bulk elementary excitations in superfluid helium-II by helium atomic beams at low temperatures

Abstract: The experimental results of creating bulk elementary excitations (BEEs) in isotopically pure liquid helium-II by helium atomic beams at low temperatures ∼ 60 mK are presented. In the present experiment, BEE signals generated by 4He-atomic beams incident on the liquid free surface were detected by a bolometer positioned in the liquid helium-II. Some detected signals were very weak and depended on the heater power. Some examples of BEE detected signals are shown. Also, group velocities of the detected BEEs are evaluated and the threshold velocities of the helium atoms are discussed. The present experimental results demonstrate BEE creation, such as the third non-dispersive Zakharenko waves (supra-thermal phonons), with energies ∼17 K (the Cooper pairing phenomenon doubles the supra-thermal phonon energy E k ∼ 2 × 17 K∼34 K in order to fulfil the energy conservation law) in the positive roton branch of the BEE energy spectra by helium atomic beams with suitable energies ∼ 35 K, which perturb the liquid surfa...