Gino Biondini - US grants

NSF Award Abstract - DMS-0101476

Mathematical Sciences: FRG: Mathematical and Computational Methods for High-Data-Rate Optical Fiber Communications

Abstract
DMS-0101476
Kath


The goal of this research project is to develop new methods that can be used to determine the behavior of optical transmission systems under realistic circumstances. This will be accomplished by a combination of various techniques. One approach will exploit the mathematical structure of fiber transmission models in order to eliminate unessential degrees of freedom. The reduced models that will result will be more tractable mathematically and also much more computationally efficient. Another approach that will be used is the application of linearization and importance sampling techniques to enable the simulation of systems at realistic data error rates. These methods will be combined to study the main sources of impairment in optical fibers in order to achieve an accurate evaluation of system performance. All the techniques to be developed will be carefully validated by comparison to more computationally time-consuming models and to experiments.

The development of high-data-rate optical fiber communications is one of the great technological achievements of the late 20th century; in the last decade alone, data rates have increased by four orders of magnitude. This enormous increase has made possible the growth of the global Internet that promises to continue to revolutionize day-to-day communications. Because demand for further growth continues unabated, however, system capacity is becoming limited by fiber transmission effects. It has therefore become crucial to accurately model and calculate the impairments due to non-ideal fiber properties when designing systems. Due to the tremendous data capacity that will be required of future transmission systems (terabits per second of aggregate capacity) and the need for extremely small transmission error rates (less than one error per trillion bits), realistic attempts to model and predict the effects of these impairments as they appear in practical systems present a number of difficult mathematical and computational challenges. The techniques that will be developed in this collaborative research project are expected to yield large reductions in the computational time required to model optical communication systems, and at the same time produce new insights into system behavior. Because these methods will be capable of providing detailed information about system performance at realistic data error rates, we believe they will lead to significant changes in the way in which optical transmission systems are modeled, and, ultimately, in the way that they are built.

Novel transmission formats hold the potential for large increases in
the total system capacity of optical fiber transmission systems. At
high bit-rates, however, nonlinear and stochastic physical effects
contribute to limit the overall system performance. The deterministic
impairments are mainly due to the combination of nonlinearity and
dispersion. The stochastic impairments occur due to amplifier noise
(which induces amplitude, timing and phase fluctuations) as well as
polarization-mode dispersion. All of these effects produce impairments
which lead to unacceptable rates of transmission errors. However,
the large scale and complexity of these systems, the variety of effects
involved, and the extremely low bit-error-ratios required of these
systems (which requires studying the occurrence of extremely rare
events) all contribute to make the modeling of optical fiber
communications a challenging task. Recent work has demonstrated that
careful mathematical and computational modeling can be very effective
in describing the behavior of realistic optical fiber transmission
systems. This research project aims at evaluating the potential of
new transmission formats and assessing how each of them is affected
by the various transmission impairments. The methods that will be
developed in this project are expected to make an impact on how
these systems are modeled and designed. In addition, because
new ultra-short pulse lasers share many similarities with
dispersion-managed optical transmission systems, the mathematical
techniques that will be developed as part of this research project
will help researchers understand the behavior of these lasers and
their fundamental limits.

The development of high-capacity optical fiber communications has
been a major technological advance that enabled the widespread use
of the internet and the world-wide-web which revolutionized our
day-to-day interactions. The demand for further increase in the
total transmission capacity remains unabated, however, fueled by
emerging applications such as video-on-demand, video-conferencing
and others requiring very large bandwidths. A key feature of this
collaborative research project is the combined use of sophisticated
mathematical and computational methods to model the behavior of
realistic lightwave systems, with the aim of developing the accurate
tools which are needed to efficiently study the behavior of optical
fiber transmission systems and to design the next generation of
systems. As such, the outcome of this project will contribute to
strengthening the national infrastructure and maintaining U.S.
competitiveness in an area which is of great national interest,
thus benefiting not just researchers, but also the community at
large.

A group of applied mathematicians in the Department of Mathematics at
the University at Buffalo proposes to purchase a distributed-shared memory multiprocessor cluster,workstations, and supplementary software. The new equipment will be used to support ongoing mathematical research projects in modeling and simulations. These research projects include the study of nonlinear and stochastic effects in fiber optics systems, analysis of coupling and uncertainty in renal hemodynamics, and an investigation of the dynamics and ill-posedness in models of crystalline surfaces especially under conditions of strong anisotropy. These projects share several features. Becauses of the interdisciplinary collaboration of each project, the co-investigators require a computing environment that allows them to rapidly develop initial model systems, usually systems of partial or ordinary differential equations, and explore solutions to these equations without extensive coding. As a model matures, the governing equations may then be solved by moderate-sized numerical simulations. Efficient numerical simulation of these dynamic models requires a computing infrastructure with large memory capacity and high bandwidth. Again because of the interdisciplinary nature of the research projects under study, parameter exploration and model validation require significant computational effort. Moreover, stochastic effects can play an important role in the model dynamics, adding to the computing burden. Comparison of model results with experiments, and communication of these results, is facilitated by advanced visualization software. The cluster to be purchased will serve these requirements, and will significantly enhance the co-investigators? research capabilities.

A group of applied mathematicians in the Department of Mathematics at the University at Buffalo proposes to purchase a distributed-shared memory multiprocessor cluster, workstations, and supplementary software. The new equipment will be used to support ongoing modeling and simulation research in fields of engineering and bio-medical application that are computationally intensive. Because the co-investigators work in highly interdisciplinary fields of study, the mathematical research advances made in these projects quickly impact fields of applied science. The cross-disciplinary nature of the co-investigators provides a stimulating environment for the training of graduate students. The co-investigators also provide opportunities for undergraduates to experience research in mathematics. The proposed equipment will benefit these students - the graduate students in their research careers, and the undergraduate students as they leave the university for graduate study or enter the workforce.

One of the most important recent advances in nonlinear optics has been the development of a new generation of very stable short-pulse lasers, with typical pulse durations of just a few femtoseconds. Such lasers have numerous applications. Among others, they provide precise sources for spectroscopy and frequency metrology, and are an essential component in optical atomic clocks. Because of the many physical effects present and the vastly different timescales involved, however, making quantitative predictions about the behavior of these systems and their fundamental performance limits is a challenging task. This project will address these challenges by developing new mathematical models aimed at describing these lasers' behavior and an accompanying set of analytical and computational tools. The new models, which will explicitly take into account the inherent complexity of these systems and the multiple time scales present, will then be used to analyze the mode-locking and nonlinear dynamics of pulses in the laser cavity. Moreover, these new models will guide the development of statistical computational algorithms that can accurately quantify the impact of noise and calculate the probability of the rare events that limit the accuracy and ultimate reliability of these systems.

Femtosecond lasers have a wide range of applicability. Beyond metrology and atomic clocks, already mentioned, important applications are a new generation of global positioning systems, the probing of transients occurring during chemical reactions and the ablation of tumors. Thus, this project will impact not only the scientific community, but also society at large. The new mathematical models and computational methods that will be derived as part of this project will substantially advance the theory of femtosecond lasers, and they will make it possible to accurately and efficiently predict the performance of these lasers. As a result, they will provide useful tools for the scientists and engineers who design and build these systems. An important part of the project will also be the training of students and junior researchers: through this interdisciplinary effort, students will be trained to use concepts, methods and techniques outside their main discipline, thus greatly enriching their educational and professional experience.

This project concerns a class of nonlinear evolution equations (NLEEs) usually referred to as soliton equations or integrable systems. Over the past fifty years, a large body of knowledge has been accumulated on these systems, which continue to be extensively studied worldwide. Nonetheless, problems in which the boundary conditions play a crucial role still pose significant challenges. This project is aimed at undertaking a wide-ranging investigation of initial-boundary value problems for integrable NLEEs, both continuous and discrete, in both one and two spatial dimensions, including their applications in several concrete scientific and technological settings. This study will be carried out by developing and using exact methods, such as the inverse scattering transform and direct methods, in combination with appropriate asymptotic and numerical techniques. Specific problems that are being studied include: (i) nonlinear Schrodinger systems (scalar and vector, continuous and discrete) with non-zero boundary conditions at space infinity; (ii) boundary value problems for discrete and continuous NLEEs; (iii) characterization of soliton interactions for all of the above systems; (iv) classification of solitary wave structures in (2+1)-dimensional integrable systems such as the Davey-Stewartson system.

Nonlinear wave equations are well-known to model a variety of physically interesting phenomena arising in areas ranging from fluid dynamics and nonlinear optics, to plasmas, cosmology and quantum field theory. The study of these equations is especially attractive because it offers a unique combination of interesting mathematics and concrete physical/technological applications. In particular, the systems that will be studied in this project model wave propagation in water waves, optical fibers, lasers and Bose-Einstein condensates. Hence the outcomes of this project will not only advance our mathematical understanding, but they will also provide practical information that will help scientists and engineers. The training of undergraduate and graduate students is also an integral component of this project. One of the PIs has co-authored a monograph on NLS systems that has been referenced extensively and is helping the dissemination of knowledge in the field of nonlinear waves. Both PIs have a long record of working with both graduate and undergraduate students, and will work both with undergraduate students and with Ph.D. students as part of this project.

This project involves theoretical and applied investigations of certain ubiquitous nonlinear wave phenomena. More specifically, the systems that will be studied belong to a class of nonlinear evolution equations referred to as soliton equations or integrable systems, which arise as mathematical models in various areas of applications, ranging from fluid dynamics and nonlinear optics, to low-temperature physics and Bose-Einstein condensates, to name a few. In particular, one of the model equations that will be studied describes the formation of rogue waves in the open ocean. These extreme events, which are characterized by wave crests up to four times bigger than the average, have been known to cause significant damages to vessels and other equipment. Effects similar to rogue waves have also been recently observed in optical fibers. A precise mathematical description of these model equations is also a key component in the design of optical fiber transmission systems. The project outcomes will help to better characterize the behavior of these systems and the properties of their solutions, they will elucidate the role that such solutions play in the generation of rogue waves, and they will contribute to paving the way for a deeper understanding of these important nonlinear phenomena. The project will provide a rich educational experience through research for graduate students at the State University of New York at Buffalo and undergraduate students at the University of Colorado at Colorado Springs.

Mathematically, the overarching goal of the project is a wide-ranging investigation of initial and initial-boundary value problems for integrable nonlinear evolution equations, thereby contributing to closing the gap between our knowledge of linear and nonlinear integrable systems. More precisely, the project comprises problems where the boundary conditions play a key role, as well as various kinds of singular limits. Specific problems that will be studied include: (i) the development of the inverse scattering transform for coupled systems of equations of nonlinear Schrodinger (NLS) type with non-zero boundary conditions at infinity; (ii) the study of initial-boundary value problems for NLS equations on a finite interval with linearizable, periodic or nearly periodic boundary conditions; (iii) the study of the long-time asymptotics of NLS systems affected by modulational instability and of semiclassical and dispersionless limits for NLS equations and the Korteweg-deVries equation with periodic boundary conditions.

This collaborative project expands the research programs of the Principal Investigators on mathematical models of optical phenomena. It comes in response to the NSF initiative on "Optics and Photonics". The interaction between light and optical media is one of the most fruitful areas of study in applied physics and provides the basic mechanism underlying devices such as lasers and optical amplifiers. For decades, it has been providing a rich source of new physical phenomena, among the latest being "slow light", the recently-observed slowing-down of light pulses to the speed of a bicycle. Slow light can potentially be used in devices such as optical memory. This project is aimed at understanding the physical mechanisms underlying the slow light phenomenon by using a remarkable, highly accurate mathematical model that can be solved with explicit formulas. The validity of this model and its explicit solutions will be verified using numerical simulations of more realistic models and careful comparisons with experiments. Interdisciplinary training in applied mathematics and nonlinear optics will be provided to graduate and undergraduate students, and a lively, challenging research and training environment for both student groups will be established.

The slowdown of light pulses is modeled as the interaction between an optical pulse and an active medium with two or three working levels, the latter being a prototypical case known as the Lambda configuration. This interaction is described by completely integrable Maxwell-Bloch equations with non-vanishing boundary conditions, a new twist. This project is a mathematical study of novel dynamics generated by the interaction of light with two-level media and the Lambda-configuration medium, and includes: (i) developing a systematic, completely integrable theory of the dynamics for the two-level and Lambda-configuration Maxwell-Bloch equations with non-zero boundary conditions, (ii) using the analytical results of step (i) to describe phenomena related to slow light, (iii) numerical studies of dynamical phenomena in more general cases in which the two-level and Lambda-configuration Maxwell-Bloch equations are not integrable. The completely-integrable description of slow light involves two new aspects: (1) non-zero boundary conditions, (2) non-trivial evolution of the spectral data. The understanding of the first aspect will be extended from the Nonlinear Schroedinger equation to the Maxwell-Bloch equations by studying scattering and inverse-scattering problems with the spectral parameter on a Riemann surface. The second aspect is complicated by the presence of the former and involves a careful derivation of how spectral data evolves from the initial state of the medium and finding correct cancellations of highly oscillatory terms. In addition to generating new models and descriptions of the dynamics exhibited by light interacting with active optical media, the project will advance the theory of completely integrable systems.

The most familiar manifestation of nonlinear dispersive waves is perhaps that of a breaking ocean wave. However, nonlinear dispersive waves are ubiquitous in nature, appearing in many fields ranging from water waves to optics, acoustics, ferromagnetics, condensed matter, cosmology and beyond. When dissipation and nonlinearity are the dominant physical effects in a system, the regime of small dissipation often gives rise to shock waves (the most familiar manifestation being a sonic boom). The analogous phenomenon when dissipation is replaced by dispersion is that of dispersive shock waves. These are non-stationary coherent, multiscale oscillatory structures. Physical media giving rise to dispersive shock waves range from water waves to superfluids, nonlinear photonics, and magnetic spin systems. For example, in fluid dynamics, dispersive shocks are known as undular bores. Even though much work has been done over the past fifty years to understand these phenomena, many fundamental questions remain. A first component of this project involves the development of mathematical tools for studying the behavior of solutions of certain nonlinear evolution equations describing nonlinear wave phenomena in more than one spatial dimension. The second component of the project involves the application of these tools to study a variety of areas of interest, ranging from water waves to optics, networks, and statistical physics. Finally, the project will also serve as a vehicle for training several Ph.D. students.

The mathematical study of nonlinear media giving rise to dispersive shocks often leads to certain systems of hyperbolic conservation laws. Over the last fifty years, various methods have been applied with success to study these kinds of systems. However, the behavior of solutions of dispersive nonlinear wave equations in more than one spatial dimension is not as well understood. In particular, a mathematical characterization of formation and propagation of dispersive shocks in two spatial dimensions is still largely an open problem. Recent work by the PI has opened up new avenues to study some long-standing open problems in this regard. Specifically, this project comprises four classes of problems: (i) Use of the Whitham modulation equations for the Kadomtsev-Petviashvili equation (the so-called KP-Whitham equations, which were recently derived by the PI) to study the temporal evolution of piecewise-constant soliton initial data and the formation and dynamics of dispersive shock waves in 2+1 dimensions. (ii) Study of the integrability structure of the above-mentioned KP-Whitham equations and their exact solutions. (iii) Derivation and application of Whitham modulation equations for various (2+1)-dimensional evolution equations of nonlinear Schrodinger type. (iv) Application of small dispersion limits and Whitham modulation theory to characterize phase state diagrams of p-star networks and certain random matrix models. The overarching theme of the project is to advance our understanding of dispersive wave phenomena in more than one spatial dimension.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.