OP: Collaborative research: Nonlinear theory of slow light

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.