OP: Mathematical Analysis of Nonlinear Optics in Periodic and Complex Media

University Of Vermont & State Agricultural College, Burlington VT

Investigators

Abstract

Light propagation in periodic media and media with balanced gain and loss is at the frontier of research in optics and applied mathematics. Periodic media, such as photonic crystals or media with periodically changing refractive index, can be manipulated to control light's propagation and achieve, for example, light trapping and routing. Progress in this area of research has promising applications for on-chip data processing at microscopic scales. By judiciously balancing the gain and loss in the so-called parity-time-symmetric configurations, one can induce the light to behave in novel ways, giving rise to a possibility of "optical diodes." Such an optical diode that is capable of transmitting a signal in only one direction would be an elemental base of optical computers. In this project, novel nonlinear behavior of light in periodic media and gain-loss-balanced media are theoretically investigated. By developing new mathematical methodologies, new insight will be gained on the nonlinear propagation of light in such media, so that their application for data processing can be assessed. From a broader perspective, this project will facilitate training of a graduate student in this important interdisciplinary area. The problems undertaken in this project are at the cutting edge of applied mathematics and optics. On the mathematical side, this project will develop novel mathematical methodologies for the treatment of contemporary nonlinear optics problems, such as a sophisticated exponential asymptotics technique for the bifurcation and linear stability of lump solitons in two-dimensional periodic media. On the physical side, the mathematical results from this project will allow us to assess the potential of optical solitons for various physical applications such as data processing on microscopic periodic structures. In addition, the theoretical investigation of actual parity-time-symmetric single-mode lasers could directly impact the nonlinear operation of those lasers and may lead to more advanced laser devices.