3D reconstruction of complex dielectric function of the glass during the femtosecond micro-fabrication

We measure complex amplitude of scattered wave in the far field, and justify theoretically and numerically solution of the inverse scattering problem. This allows single-shot reconstructing of dielectric function distribution during direct femtosecond laser micro-fabrication.

Nonlinear Wave Equation with Vanishing Potential

We study the Cauchy problem for utt − ∆u + V (x)u^5 = 0 in 3–dimensional case. The function V (x) is positive and regular, in particular we are interested in the case V (x) = 0 in some points. We look for the global classical solution of this equation under a suitable hypothesis on the initial energy.

Cauchy-Type Problem for Diffusion-Wave Equation with the Riemann-Liouville Partial Derivative

2000 Mathematics Subject Classification: 35A15, 44A15, 26A33

Pseudo-Differential Operators in a Wave Diffraction Problem with Impedance Conditions

Mathematics Subject Classification: 35J05, 35J25, 35C15, 47H50, 47G30

The Operators Aγ = γA + -γA for a Class of Nondissipative Operators A with a Limit of the Corresponding Correlation Function

2000 Mathematics Subject Classification: Primary 47A48, Secondary 60G12

Green's function for paraxial equation

The theory and experimental applications of optical Airy beams are in active development recently. The Airy beams are characterised by very special properties: they are non-diffractive and propagate along parabolic trajectories. Among the striking applications of the optical Airy beams are optical micro-manipulation implemented as the transport of small particles along the parabolic trajectory, Airy-Bessel linear light bullets, electron acceleration by the Airy beams, plasmonic energy routing. The detailed analysis of the mathematical aspects as well as physical interpretation of the electromagnetic Airy beams was done by considering the wave as a function of spatial coordinates only, related by the parabolic dependence between the transverse and the longitudinal coordinates. Their time dependence is assumed to be harmonic. Only a few papers consider a more general temporal dependence where such a relationship exists between the temporal and the spatial variables. This relationship is derived mostly by applying the Fourier transform to the expressions obtained for the harmonic time dependence or by a Fourier synthesis using the specific modulated spectrum near some central frequency. Spatial-temporal Airy pulses in the form of contour integrals is analysed near the caustic and the numerical solution of the nonlinear paraxial equation in time domain shows soliton shedding from the Airy pulse in Kerr medium. In this paper the explicitly time dependent solutions of the electromagnetic problem in the form of time-spatial pulses are derived in paraxial approximation through the Green's function for the paraxial equation. It is shown that a Gaussian and an Airy pulse can be obtained by applying the Green's function to a proper source current. We emphasize that the processes in time domain are directional, which leads to unexpected conclusions especially for the paraxial approximation.

One-dimensional optical wave turbulence:experiment and theory

We present a review of the latest developments in one-dimensional (1D) optical wave turbulence (OWT). Based on an original experimental setup that allows for the implementation of 1D OWT, we are able to show that an inverse cascade occurs through the spontaneous evolution of the nonlinear field up to the point when modulational instability leads to soliton formation. After solitons are formed, further interaction of the solitons among themselves and with incoherent waves leads to a final condensate state dominated by a single strong soliton. Motivated by the observations, we develop a theoretical description, showing that the inverse cascade develops through six-wave interaction, and that this is the basic mechanism of nonlinear wave coupling for 1D OWT. We describe theory, numerics and experimental observations while trying to incorporate all the different aspects into a consistent context. The experimental system is described by two coupled nonlinear equations, which we explore within two wave limits allowing for the expression of the evolution of the complex amplitude in a single dynamical equation. The long-wave limit corresponds to waves with wave numbers smaller than the electrical coherence length of the liquid crystal, and the opposite limit, when wave numbers are larger. We show that both of these systems are of a dual cascade type, analogous to two-dimensional (2D) turbulence, which can be described by wave turbulence (WT) theory, and conclude that the cascades are induced by a six-wave resonant interaction process. WT theory predicts several stationary solutions (non-equilibrium and thermodynamic) to both the long- and short-wave systems, and we investigate the necessary conditions required for their realization. Interestingly, the long-wave system is close to the integrable 1D nonlinear Schrödinger equation (NLSE) (which contains exact nonlinear soliton solutions), and as a result during the inverse cascade, nonlinearity of the system at low wave numbers becomes strong. Subsequently, due to the focusing nature of the nonlinearity, this leads to modulational instability (MI) of the condensate and the formation of solitons. Finally, with the aid of the probability density function (PDF) description of WT theory, we explain the coexistence and mutual interactions between solitons and the weakly nonlinear random wave background in the form of a wave turbulence life cycle (WTLC).

Performance study of the Galway Bay wave energy test site floating power system

The Galway Bay wave energy test site promises to be a vital resource for wave energy researchers and developers. As part of the development of this site, a floating power system is being developed to provide power and data acquisition capabilities, including its function as a local grid connection, allowing for the connection of up to three wave energy converter devices. This work shows results from scaled physical model testing and numerical modelling of the floating power system and an oscillating water column connected with an umbilical. Results from this study will be used to influence further scaled testing as well as the full scale design and build of the floating power system in Galway Bay.

Estimation of the Crustal Bulk Properties Beneath Mainland Portugal from P-Wave Teleseismic Receiver Functions

In this work, we present results from teleseismic P-wave receiver functions (PRFs) obtained in Portugal, Western Iberia. A dense seismic station deployment conducted between 2010 and 2012, in the scope of the WILAS project and covering the entire country, allowed the most spatially extensive probing on the bulk crustal seismic properties of Portugal up to date. The application of the H-kappa stacking algorithm to the PRFs enabled us to estimate the crustal thickness (H) and the average crustal ratio of the P- and S-waves velocities V (p)/V (s) (kappa) for the region. Observations of Moho conversions indicate that this interface is relatively smooth with the crustal thickness ranging between 24 and 34 km, with an average of 30 km. The highest V (p)/V (s) values are found on the Mesozoic-Cenozoic crust beneath the western and southern coastal domain of Portugal, whereas the lowest values correspond to Palaeozoic crust underlying the remaining part of the subject area. An average V (p)/V (s) is found to be 1.72, ranging 1.63-1.86 across the study area, indicating a predominantly felsic composition. Overall, we systematically observe a decrease of V (p)/V (s) with increasing crustal thickness. Taken as a whole, our results indicate a clear distinction between the geological zones of the Variscan Iberian Massif in Portugal, the overall shape of the anomalies conditioned by the shape of the Ibero-Armorican Arc, and associated Late Paleozoic suture zones, and the Meso-Cenozoic basin associated with Atlantic rifting stages. Thickened crust (30-34 km) across the studied region may be inherited from continental collision during the Paleozoic Variscan orogeny. An anomalous crustal thinning to around 28 km is observed beneath the central part of the Central Iberian Zone and the eastern part of South Portuguese Zone.

Evans functions for integral neural field equations with Heaviside firing rate function

In this paper we show how to construct the Evans function for traveling wave solutions of integral neural field equations when the firing rate function is a Heaviside. This allows a discussion of wave stability and bifurcation as a function of system parameters, including the speed and strength of synaptic coupling and the speed of axonal signals. The theory is illustrated with the construction and stability analysis of front solutions to a scalar neural field model and a limiting case is shown to recover recent results of L. Zhang [On stability of traveling wave solutions in synaptically coupled neuronal networks, Differential and Integral Equations, 16, (2003), pp.513-536.]. Traveling fronts and pulses are considered in more general models possessing either a linear or piecewise constant recovery variable. We establish the stability of coexisting traveling fronts beyond a front bifurcation and consider parameter regimes that support two stable traveling fronts of different speed. Such fronts may be connected and depending on their relative speed the resulting region of activity can widen or contract. The conditions for the contracting case to lead to a pulse solution are established. The stability of pulses is obtained for a variety of examples, in each case confirming a previously conjectured stability result. Finally we show how this theory may be used to describe the dynamic instability of a standing pulse that arises in a model with slow recovery. Numerical simulations show that such an instability can lead to the shedding of a pair of traveling pulses.

Wind wave prediction in the Caspian Sea

In this thesis, wind wave prediction and analysis in the Southern Caspian Sea are surveyed. Because of very much importance and application of this matter in reducing vital and financial damages or marine activities, such as monitoring marine pollution, designing marine structure, shipping, fishing, offshore industry, tourism and etc, gave attention by some marine activities. In this study are used the Caspian Sea topography data that are extracted from the Caspian Sea Hydrography map of Iran Armed Forces Geographical Organization and the I 0 meter wind field data that are extracted from the transmitted GTS synoptic data of regional centers to Forecasting Center of Iran Meteorological Organization for wave prediction and is used the 20012 wave are recorded by the oil company's buoy that was located at distance 28 Kilometers from Neka shore for wave analysis. The results of this research are as follows: - Because of disagreement between the prediction results of SMB method in the Caspian sea and wave data of the Anzali and Neka buoys. The SMB method isn't able to Predict wave characteristics in the Southern Caspian Sea. - Because of good relativity agreement between the WAM model output in the Caspian Sea and wave data of the Anzali buoy. The WAM model is able to predict wave characteristics in the southern Caspian Sea with high relativity accuracy. The extreme wave height distribution function for fitting to the Southern Caspian Sea wave data is obtained by determining free parameters of Poisson-Gumbel function through moment method. These parameters are as below: A=2.41, B=0.33. The maximum relative error between the estimated 4-year return value of the Southern Caspian Sea significant wave height by above function with the wave data of Neka buoy is about %35. The 100-year return value of the Southern Caspian Sea significant height wave is about 4.97 meter. The maximum relative error between the estimated 4-year return value of the Southern Caspian Sea significant wave height by statistical model of peak over threshold with the wave data of Neka buoy is about %2.28. The parametric relation for fitting to the Southern Caspian Sea frequency spectra is obtained by determining free parameters of the Strekalov, Massel and Krylov etal_ multipeak spectra through mathematical method. These parameters are as below: A = 2.9 B=26.26, C=0.0016 m=0.19 and n=3.69. The maximum relative error between calculated free parameters of the Southern Caspian Sea multipeak spectrum with the proposed free parameters of double-peaked spectrum by Massel and Strekalov on the experimental data from the Caspian Sea is about 36.1 % in spectrum energetic part and is about 74M% in spectrum high frequency part. The peak over threshold waverose of the Southern Caspian Sea shows that maximum occurrence probability of wave height is relevant to waves with 2-2.5 meters wave fhe error sources in the statistical analysis are mainly due to: l) the missing wave data in 2 years duration through battery discharge of Neka buoy. 2) the deportation %15 of significant height annual mean in single year than long period average value that is caused by lack of adequate measurement on oceanic waves, and the error sources in the spectral analysis are mainly due to above- mentioned items and low accurate of the proposed free parameters of double-peaked spectrum on the experimental data from the Caspian Sea.

Wave and turbulence measurements at a tidal energy site

This work presents the analysis of wave and turbulence measurements collected at a tidal energy site. A new method is introduced to produce more consistent and rigorous estimations of the velocity fluctuations power spectral densities. An analytical function is further proposed to fit the observed spectra and could be input to the numerical models predicting power production and structural loading on tidal turbines. Another new approach is developed to correct for the effect of the Doppler noise on the high frequencies power spectral densities. The analysis of velocity time series combining wave and turbulent contributions demonstrates that the turbulent motions are coherent throughout the water column, rendering the wave coherence-based methods not applicable to our dataset. To avoid this problem, an alternative approach relying on the pressure data collected by the ADCP is introduced and shows appreciable improvement in the wave-turbulence separation.

Predicting Pollicipes pollicipes (Crustacea: Cirripedia) abundance on intertidal rocky shores of SW Portugal: a multi-scale approach based on a simple fetch-based wave exposure index

Understanding and predicting patterns of distribution and abundance of marine resources is important for con- servation and management purposes in small-scale artisanal fisheries and industrial fisheries worldwide. The goose barnacle (Pollicipes pollicipes) is an important shellfish resource and its distribution is closely related to wave exposure at different spatial scales. We modelled the abundance (percent coverage) of P. pollicipes as a function of a simple wave exposure index based on fetch estimates from digitized coastlines at different spatial scales. The model accounted for 47.5% of the explained deviance and indicated that barnacle abundance increases non-linearly with wave exposure at both the smallest (metres) and largest (kilometres) spatial scales considered in this study. Distribution maps were predicted for the study region in SW Portugal. Our study suggests that the relationship between fetch-based exposure indices and P. pollicipes percent cover may be used as a simple tool for providing stakeholders with information on barnacle distribution patterns. This information may improve assessment of harvesting grounds and the dimension of exploitable areas, aiding management plans and support- ing decision making on conservation, harvesting pressure and surveillance strategies for this highly appreciated and socio- economically important marine resource.

Green's function retrieval through cross-correlations in a two -dimensional complex reverberating medium

International audience

A principal component approach to space-based gravitational wave astronomy

The current approach to data analysis for the Laser Interferometry Space Antenna (LISA) depends on the time delay interferometry observables (TDI) which have to be generated before any weak signal detection can be performed. These are linear combinations of the raw data with appropriate time shifts that lead to the cancellation of the laser frequency noises. This is possible because of the multiple occurrences of the same noises in the different raw data. Originally, these observables were manually generated starting with LISA as a simple stationary array and then adjusted to incorporate the antenna's motions. However, none of the observables survived the flexing of the arms in that they did not lead to cancellation with the same structure. The principal component approach is another way of handling these noises that was presented by Romano and Woan which simplified the data analysis by removing the need to create them before the analysis. This method also depends on the multiple occurrences of the same noises but, instead of using them for cancellation, it takes advantage of the correlations that they produce between the different readings. These correlations can be expressed in a noise (data) covariance matrix which occurs in the Bayesian likelihood function when the noises are assumed be Gaussian. Romano and Woan showed that performing an eigendecomposition of this matrix produced two distinct sets of eigenvalues that can be distinguished by the absence of laser frequency noise from one set. The transformation of the raw data using the corresponding eigenvectors also produced data that was free from the laser frequency noises. This result led to the idea that the principal components may actually be time delay interferometry observables since they produced the same outcome, that is, data that are free from laser frequency noise. The aims here were (i) to investigate the connection between the principal components and these observables, (ii) to prove that the data analysis using them is equivalent to that using the traditional observables and (ii) to determine how this method adapts to real LISA especially the flexing of the antenna. For testing the connection between the principal components and the TDI observables a 10x 10 covariance matrix containing integer values was used in order to obtain an algebraic solution for the eigendecomposition. The matrix was generated using fixed unequal arm lengths and stationary noises with equal variances for each noise type. Results confirm that all four Sagnac observables can be generated from the eigenvectors of the principal components. The observables obtained from this method however, are tied to the length of the data and are not general expressions like the traditional observables, for example, the Sagnac observables for two different time stamps were generated from different sets of eigenvectors. It was also possible to generate the frequency domain optimal AET observables from the principal components obtained from the power spectral density matrix. These results indicate that this method is another way of producing the observables therefore analysis using principal components should give the same results as that using the traditional observables. This was proven by fact that the same relative likelihoods (within 0.3%) were obtained from the Bayesian estimates of the signal amplitude of a simple sinusoidal gravitational wave using the principal components and the optimal AET observables. This method fails if the eigenvalues that are free from laser frequency noises are not generated. These are obtained from the covariance matrix and the properties of LISA that are required for its computation are the phase-locking, arm lengths and noise variances. Preliminary results of the effects of these properties on the principal components indicate that only the absence of phase-locking prevented their production. The flexing of the antenna results in time varying arm lengths which will appear in the covariance matrix and, from our toy model investigations, this did not prevent the occurrence of the principal components. The difficulty with flexing, and also non-stationary noises, is that the Toeplitz structure of the matrix will be destroyed which will affect any computation methods that take advantage of this structure. In terms of separating the two sets of data for the analysis, this was not necessary because the laser frequency noises are very large compared to the photodetector noises which resulted in a significant reduction in the data containing them after the matrix inversion. In the frequency domain the power spectral density matrices were block diagonals which simplified the computation of the eigenvalues by allowing them to be done separately for each block. The results in general showed a lack of principal components in the absence of phase-locking except for the zero bin. The major difference with the power spectral density matrix is that the time varying arm lengths and non-stationarity do not show up because of the summation in the Fourier transform.