Dynamics of Light Interacting with Active Media

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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.