Nonlinear optics

Nonlinear light-matter interactions have been drawing attention of physicists since the 1960's. Quantum mechanics played a significant role in their description and helped to derive important formulas showing the dependence on the intensity of the electromagnetic field. High intensity light is able to generate second and third harmonics which translates to generation of electromagnetic field with multiples of the original frequency. In comparison with the linear behaviour of light, the nonlinear interactions are smaller in scale. This makes perturbation methods well suited for obtaining solutions to equations in nonlinear optics. In particular, the method of multiple scales is deployed in paper 3, where it is used to solve nonlinear dispersive wave equations. The key difference in our multiple scale solution is the linearity of the amplitude equation and a complex valued frequency of the mode. Despite the potential ill-posedness of the amplitude equation, the multiple scale solution remained a valid approximation of the solution to the original model. The results showed great potential of this method and its promising wider applications. Other methods use pseudo-spectral methods which require an orthogonal set of eigenfunctions (modes) used to create a substitute for the usual Fourier transform. This mode transform is only useful if it succeeds to represent target functions well. Papers 1 and 2 deal with investigating such modes called resonant and leaky modes and their ability to construct a mode transform. The modes in the first paper are the eigenvalues for a quantum mechanical system where an external radiation field is used to excite an electron trapped in an electrical potential. The findings show that the resonant mode expansion converges inside the potential independently of its depth. Equivalently, leaky modes are obtained in paper 2 which are in close relation to resonant modes. Here, the modes emerge from a system where a channel is introduced with transparent boundaries for simulation of one-directional optical beam propagation. Artificial index material is introduced outside the channel which gives rise to leaky modes associated with such artificial structure. The study is showing that leaky modes are well suited for function representation and thus solving the nonlinear version of this problem. In addition, the transparent boundary method turns out to be useful for spectral propagators such as the unidirectional pulse propagation equation in contrast to a perfectly matched layer

Resurgence and hydrodynamic attractors in Gauss-Bonnet holography

We study the convergence of the hydrodynamic series in the gravity dual of Gauss-Bonnet gravity in five dimensions with negative cosmological constant via holography. By imposing boost invariance symmetry, we find a solution to the Gauss-Bonnet equation of motion in inverse powers of the proper time, from which we can extract high order corrections to Bjorken flow for different values of the Gauss-Bonnet parameter λGB. As in all other known examples the gradient expansion is, at most, an asymptotic series which can be understood through applying the techniques of Borel-Padé summation. As expected from the behaviour of the quasi-normal modes in the theory, we observe that the singularities in the Borel plane of this series show qualitative features that interpolate between the infinitely strong coupling limit of N=4 Super Yang Mills theory and the expectation from kinetic theory. We further perform the Borel resummation to constrain the behaviour of hydrodynamic attractors beyond leading order in the hydrodynamic expansion. We find that for all values of λGB considered, the convergence of different initial conditions to the resummation and its hydrodynamization occur at large and comparable values of the pressure anisotropy

Geometric manipulation of light : from nonlinear optics to invisibility cloaks

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2012.Cataloged from PDF version of thesis.Includes bibliographical references (p. 189-203).In this work, we study two different manipulations of electromagnetic waves governed by macroscopic Maxwell's equations. One is frequency conversion of such waves using small intrinsic material nonlinearities. We study conversion of an input signal at frequency w1 to frequency Wk due to second or third harmonic generation or four-wave mixing using coupled-mode theory. Using this framework, we show there is a critical input power at which maximum frequency conversion is possible. We study in depth the case of third harmonic generation, its solutions, and their stability analysis. Based on the dynamics of the system, we propose a regime of parameters that 100%- efficient frequency conversion is possible and propose a way of exciting this solution. We also look at same analysis for the case of degenerate four-wave mixing and come up with 2d and 3d designs of a device that exhibits high-efficiency second-harmonic generation. Second, we consider proposals for invisibility cloaks to change the path of electromagnetic waves in a certain way so that the object appears invisible at a certain frequency or a range of frequencies. Transformation-based invisibility cloaks make use of the coordinate invariance of Maxwell's Equations and require complex material configuration e and p in the cloak. We study the practical limitations of cloaking as a function of the size of the object being cloaked. Specifically, we study the bandwidth, loss, and scattering limitations of cloaking as the object gets larger and show that cloaking of objects many times larger than the wavelength in size becomes practically impossible.by Hila Hashemi.Ph.D

Scalable Performance Analysis of Massively Parallel Stochastic Systems

The accurate performance analysis of large-scale computer and communication systems is directly inhibited by an exponential growth in the state-space of the underlying Markovian performance model. This is particularly true when considering massively-parallel architectures such as cloud or grid computing infrastructures. Nevertheless, an ability to extract quantitative performance measures such as passage-time distributions from performance models of these systems is critical for providers of these services. Indeed, without such an ability, they remain unable to offer realistic end-to-end service level agreements (SLAs) which they can have any confidence of honouring. Additionally, this must be possible in a short enough period of time to allow many different parameter combinations in a complex system to be tested. If we can achieve this rapid performance analysis goal, it will enable service providers and engineers to determine the cost-optimal behaviour which satisfies the SLAs. In this thesis, we develop a scalable performance analysis framework for the grouped PEPA stochastic process algebra. Our approach is based on the approximation of key model quantities such as means and variances by tractable systems of ordinary differential equations (ODEs). Crucially, the size of these systems of ODEs is independent of the number of interacting entities within the model, making these analysis techniques extremely scalable. The reliability of our approach is directly supported by convergence results and, in some cases, explicit error bounds. We focus on extracting passage-time measures from performance models since these are very commonly the language in which a service level agreement is phrased. We design scalable analysis techniques which can handle passages defined both in terms of entire component populations as well as individual or tagged members of a large population. A precise and straightforward specification of a passage-time service level agreement is as important to the performance engineering process as its evaluation. This is especially true of large and complex models of industrial-scale systems. To address this, we introduce the unified stochastic probe framework. Unified stochastic probes are used to generate a model augmentation which exposes explicitly the SLA measure of interest to the analysis toolkit. In this thesis, we deploy these probes to define many detailed and derived performance measures that can be automatically and directly analysed using rapid ODE techniques. In this way, we tackle applicable problems at many levels of the performance engineering process: from specification and model representation to efficient and scalable analysis

Spectral analysis for stochastic models of large-scale complex dynamical networks

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2008.Includes bibliographical references (p. 179-196).Research on large-scale complex networks has important applications in diverse systems of current interest, including the Internet, the World-Wide Web, social, biological, and chemical networks. The growing availability of massive databases, computing facilities, and reliable data analysis tools has provided a powerful framework to explore structural properties of such real-world networks. However, one cannot efficiently retrieve and store the exact or full topology for many large-scale networks. As an alternative, several stochastic network models have been proposed that attempt to capture essential characteristics of such complex topologies. Network researchers then use these stochastic models to generate topologies similar to the complex network of interest and use these topologies to test, for example, the behavior of dynamical processes in the network. In general, the topological properties of a network are not directly evident in the behavior of dynamical processes running on it. On the other hand, the eigenvalue spectra of certain matricial representations of the network topology do relate quite directly to the behavior of many dynamical processes of interest, such as random walks, Markov processes, virus/rumor spreading, or synchronization of oscillators in a network. This thesis studies spectral properties of popular stochastic network models proposed in recent years. In particular, we develop several methods to determine or estimate the spectral moments of these models. We also present a variety of techniques to extract relevant spectral information from a finite sequence of spectral moments. A range of numerical examples throughout the thesis confirms the efficacy of our approach. Our ultimate objective is to use such results to understand and predict the behavior of dynamical processes taking place in large-scale networks.by Víctor Manuel Preciado.Ph.D

An Experimental Characterization of the Mechanical Properties of Thermal Barrier Coatings at Elevated Temperatures

This research program developed the apparatus and associated techniques to mechanically characterize the complex modulus of hard coatings across a temperature range from about 70 deg F to 900 deg F. Major effort in designing, analyzing, and experimentally validating the chamber were performed to establish that it isothermally heated a beam specimen, accomplished modal detuning, and achieved a near free-free boundary condition, and that the chamber was characterized for its forcing excitation. Novel aspects of the chamber include non-contact for the excitation, nearly non-contacted boundary conditions, and measurement of the field variables within the specimen using a hybrid experimental-numerical approach. This allowed for very low damping values to be measured. A common thermal barrier coating material, 8YSZ, was characterized in the chamber to determine its loss-factor (damping) and storage modulus (stiffness), at both a system-level, and well as, extracted bulk material properties-sense at temperatures from 70 to 900 deg F. The use of the free-decay technique using logarithmic decrement was the primary means used to characterize the coating, although some forced response was also performed and showed agreement. Some specimens that were bare titanium and bond-coat-only were studied as well. The former resulted in the discovery that the chamber is a very sensitive to slight modulus changes in classical engineering materials and the latter was shown to have fairly minimal influence on the coated beam system dynamics

MATHEMATISATION: SOCIAL PROCESS & DIDACTIC PRINCIPLE

The 69th CIEAEM conference was held from 15th to 19th July 2017 at Freie Universität Berlin, Germany. It successfully involved 100 participants from 20 countries all over the world. CIEAEM 69 was dedicated to Professor Christine Keitel, president of CIEAEM from 1997 to 2003, who tragically passed away one year before the conference. The programme of the conference started with a panel that revisited “Mathematics (Education) and Common Sense”, the theme of the 47th CIEAEM conference, which was held in Berlin in July 1995 and which was hosted by Christine. At the conference, researchers, teachers, educators, and students met to discuss, in a collaborative and inspiring environment, the most prominent problems, obstacles and resources in mathematics education; they also presented their latest research findings in the several conference activities: plenary and semi-plenary talks, two round tables, working groups, workshops, and poster presentations (forum of ideas). As in previous CIEAEM meetings, Working Groups constituted the beating heart of the conference, allowing the participants to fruitfully discuss in critical and constructive ways, in the true CIEAEM spirit, research studies and approaches from different perspectives on the conference theme: Mathematisation: social process & didactic principle. There were four Working Groups: (A) Mathematisation as a didactic principle: mathematizing and modelling of everyday contexts; (B) Mathematisation as a didactic principle: representation and generalization within mathematics; (C) Interconnecting mathematisation as a social process and as a didactic principle; and (D) Mathematisation as a didactic principle: looking at teachers of mathematics. Each Working Group discussed nine papers, and addressed the conference theme from complementary viewpoints (see the Discussion Paper), under the guidance of the group animators. The conference schedule allowed time also to deepen the plenary talks in the dedicated “Meet the plenary speaker” sessions, and to engage participants in workshops, where actual dialogue between research and practice could be fostered. This volume contains the final versions of the 53 papers presented during the conference. We thank all the contributors and the participants to the conference, because they made it such a unique experience, in which we had the good fortune to take part. We are grateful to the International Programme Committee and the Local Organizing Committee that made possible the realization of the conference in every detail with great care. Particularly, we want to thank the Working Group animators, who organized each day the sessions in inclusive as well high-quality ways. A special thanks to all the people who contributed to the realization of the conference, and to Daria Fischer, who helped in editing this volume. As a result, the CIEAEM 69 Proceedings offer a wide overview on national and international studies on the conference theme Mathematisation: social process & didactic principle. We hope that it can constitute an inspiring resource for the research community, for teachers, and for stakeholders in mathematics education. From this perspective, the possibility of free downloading offers to CIEAEM 69 participants, and also to interested people who could not take part in the Conference in Berlin, the possibility of developing a fruitful network of contacts that year after year is becoming richer and wider

ニュートン流体における粉体の二相動力学

Many scientific and technical problems which concern the dynamics of complex fluids such as multi-phase-flow and realistic flow in porous and granular media deal with the interaction between fluids and particles, rather than with the dynamics of the fluid alone. The research of how the surrounding fluid affects the dynamics of particles, or how to deal with the problem computationally for the microscopic level is still at the beginning. The aim of this study is to develop a microscopic simulation method (fluid goes around the particles) where granular particles can be simulated inside fluids to study those problems. This is done by combining the simulation method for granular particles with the simulation method for the incompressible Newtonian fluid. The granular particles are implemented via the discrete element method (DEM) where the elastic contact force between two undeformed contacting polygonal particles is proportional to the overlap area ("hard particle, soft contact"). The Gear Predictor-Corrector of 2nd-order (BDF2) is used as the time integrator to solve the equations of motion of the particles. For the fluid phase, the implementation of the incompressible Navier-Stokes equations via the Galerkin finite element method (FEM) is formulated as differential algebraic equations (DAE) with the pressures as the Lagrange parameters. The time integration is again via the BDF2 while the resulting non-linear equations are solved via the Newton-Raphson methods. The spatial discretization is via the Taylor-Hood elements from Delaunay triangulations with additional post-processing with the relaxation algorithm. The coupling of the DEM for the granular particles and the FEM for the fluid is via appropriate boundary conditions and the drag force (computed by the integration of the fluid stress tensor over the particle\u27s surface). This is being verified via the computation of wall correction factors of a sinking particle. The fluid simulation is extended to a simulation of free surfaces where the motion of the surface is integrated out according to the velocity on the surface which is obtained from the FEM-scheme. The second-order Adams-Bashforth method turns out to be the most suitable integrator for the surface motion. Compared to conventional efforts, which try to solve partial differential equations for the motion of the surface, the additional effort in our method with respect to new data structures etc. is minimal. The free surfaces code is verified by simulating the collapse of a water column. For the speed of the wavefronts, excellent agreement is obtained for large viscosity with the lubrication approximation. The agreement of the results with the experimental data for water is a further gratifying result. Two numerical experiments are conducted using the DEM-FEM code: one with a rather slow dynamics, another one relatively more "violent". The compaction simulation has shown that the addition of fluid to a granular assembly can increase the sound velocity in the system, compared to the dry case. The high viscosity slowed down the compaction, irrespective whether the system was tapped only on the ground or on the whole boundary. The granular column simulations show that for systems immersed under fluids, rolling of particles becomes less important than for the corresponding dry systems.電気通信大学201