Resonance Phenomena in Wave Scattering

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ShipmanDMS-1411393 Optical and electronic devices, as varied and complex as they come, are based on a common underlying physical principle called resonance. This principle lies behind lasers, filters, antennas, and light-emitting diodes, as well as devices whose components operate at the nanoscale. The investigator studies efficient and robust ways to exploit the phenomena of resonance in device design and how to control variations in device performance resulting from inevitable manufacturing imperfections. The theory of resonance has deep mathematical roots, which are developed to gain a better working understanding of new physical phenomena for the design of modern-day devices. The project trains graduate and undergraduate students to develop skills at the interfaces of mathematics, physics, and engineering, paying special attention to bridging communication gaps between the disciplines. Undergraduate students are involved in creating an online computational tool for virtual experimentation and design of optical structures. Four problems of resonance are studied. (1) An electromagnetic resonance of a doubly periodic waveguide is associated with a complex eigenvalue of the Maxwell equations, which encodes the frequency and spectral width of the resonance. The investigator considers new waveguide structures that allow both characteristics of a resonance to be tuned by varying the angle of incidence of a source wave. This ability to tune is precluded in existing constructions of resonant waveguide modes due to the symmetry that they utilize. This symmetry is broken by exploiting non-reciprocal materials, special geometries, and bifurcations of symmetric guided modes into antisymmetric modes, to construct asymmetric tunable waveguide resonances. (2) Embedded eigenvalues in locally perturbed periodic quantum graphs are responsible for resonance phenomena in two-dimensional molecular structures such as bi-layer graphene. Building on a recent result of his on embedded eigenvalues, the investigator employs asymmetry to control resonant scattering characteristics in quantum graphs. (3) Even in materials with small coefficients of nonlinearity, field amplification due to resonance causes strongly nonlinear effects, such as bistability and hysteresis. The investigator quantifies these effects in a tractable model in which a photon transmission line is coupled to a local exciton field, producing a nonlinear field of photon-exciton quasiparticles known as polaritons. (4) Random fluctuations of a waveguide's material properties and geometry introduce variability into features of resonance, such as their central frequency and spectral width. The investigator calculates this variability using perturbation analysis of the scattering matrix where the perturbative parameters are random variables.