Forward and Inverse Scattering problems in Topological Insulators and paraxial wave models

This research project first concerns the field of topological insulators, whose topological features lead to unusual transport properties with potential applications in communication and quantum computers. Topological characteristics afford these materials some protection against manufacturing defects and loss of transmission resulting from scattering effects. Topological quantum computers offer a potentially transformative approach to a central obstacle in quantum computing, namely the fragility of quantum information against decoherence. A second component of the research project involves free-space optical communication using laser light and its limitations due to blurring effects caused by turbulent fluctuations in the atmosphere. The two sides of the proposal are related by the promise of novel structured laser light generated by topological cavities. A central element of the project will be the mentoring of early career researchers and training of master's students and PhD students. While the topological classification of non-interacting discrete models is mostly understood, many surprises arise in continuous descriptions of, e.g., photonics and geophysical models. One main thrust of this proposal will be to elucidate this issue and develop a stable classification of such materials. A more quantitative understanding of their transport properties requires the introduction of novel scattering theories. With those in place, the proposal will also focus on the inverse problems theory of such materials and assess what role the non-trivial topology may have on coefficient estimations. In the field of open space communication, modeled as a problem of wave propagation in a random turbulent medium, the main thrust of the proposal will be to devise structured light incident fields that are as immune to atmospheric turbulence as possible. 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.