This work will address mathematical descriptions of the dynamics arising in three novel effects generated by the interaction of light with an optical medium in the lambda- configuration. The first is random polarization switching of light pulses propagating through a lambda-configuration medium, whose description gives rise to the rare combination of a simultaneously completely integrable and stochastic partial differential equation. The second is the "light stopping" phenomenon, which requires the development of a new inverse-scattering-transform technique and soliton theory for the Maxwell-Bloch equations with nonvanishing boundary conditions at infinity. The third is light propagating through a combined, lambda-configuration metamaterial, which requires the derivation of a new model to be studied through a combination of exact solutions, asymptotic, and numerical techniques, and may give rise to descriptions of novel phenomena such as simultaneous color and direction switching and a related nonlinear version of Anderson localization, as well as mechanisms for loss compensation in metamaterials. More broadly, the work will advance the theory of completely-integrable systems, and especially a successful description of loss compensation in metamaterials, may have impact on practical nonlinear optics. Interdisciplinary training in applied mathematics and nonlinear optics will be provided to graduate and undergraduate students.
Interaction between light and active optical media (these are media for which such interaction is particularly strong) is one of the most fruitful areas in applied physics and provides the basic mechanism underlying devices such as lasers and optical amplifiers. It has continued to be a rich source of new physical phenomena, among the latest being "light stopping," during which specially prepared optical pulses are slowed down to a fraction of the light speed, and which could be used in designing optical memory. In addition, this interaction could also be used to reduce losses in optical metamaterials (artificial nano-composites with previously unattainable optical properties such as perfect lens focusing) and thus help advance their development from the current proof-of-concept experiments to eventual practical optical devices. Due to the great variety of physical phenomena it exhibits, light interaction with active media has also given rise to a broad range of mathematical descriptions of their dynamics. Three novel such mathematical descriptions will be developed in the course of this work, including those of "stopped light," a certain potential type of loss compensation in a metamaterial, and finally a rare exact description of light propagating through a highly disordered medium.
The project developed mathematical descriptions of a number of wave-like phenomena associated with light propagation through materials. In the process, three PhD students and three undergraduates were trained. Intellectual Merit Three main mathematical descriptions and/or techniques were obtained: The first is a rare, exact, explicit mathematical description of light pulses propagating through a highly disordered medium. The second concerns propagation of light through an artificial metamaterial, i.e., a transparent material containing nano-size metallic inclusions. First, propagation through a metamaterial which deflects a certain color of light so that it makes objects of that color invisible was computed. Then, we investigated propagation of two-color light through a metamaterial in which one color propagates in the normal regime and the other in the regime of negative refraction, which leads to unusual effects such as perfect lensing. Color and direction switching of light pulses through such a metamaterial was computed. The same physical mechanism is used in experiments to pump the negative-refraction regime and thus compensate for the typically large losses in it, and thus enable such metamaterials to begin approaching device size and so become applicable. The third was the development of mathematical tools for describing slowdown of light pulses to velocities of a car or even a pedestrian. Slowing down and stopping light has potential uses in computing, such as in optical memory. The same techniques can also be used to understand why certain waves can interact to form unusually large new waves, such as rogue waves in the ocean. Broader Impact Six articles describing the accomplished results were published in refereed journals, and 18 conference presentations were given. The PI organized three special sessions at SIAM and AIMS conferences. Three undergraduate students were trained and produced a refereed publication. All three went on to PhD studies in (applied) mathematics at Brown University, Duke University, and the University of Arizona. Three PhD students were trained on the projects. Two are still in training, but the first already completed her PhD, went on to a prestigious instructorship at New York University's Courant Institute, and is now an Assistant Professor at the University of North Carolina, Chapel Hill. All three contributed to current and forthcoming refereed publications, and presented lectures and posters at conferences. At our home institution, Rensselaer, we were all active participants of a seminar on Nonlinear Science.
