High Frequency Cavity Eigenmodes: Rapid Computational Methods, Applications and Asymptotics

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The investigator seeks to develop efficient and robust numerical tools for calculating the resonant modes of cavities in two or more dimensions. These modes describe behavior of a wide variety of linear wave systems of technological importance, such as electromagnetic, optical, and acoustic cavities, elastic membranes, transverse modes of metallic waveguides, and bound energy states of quantum particles. It is hard to overestimate the impact that mathematical understanding of waves has had on society over the last century, particularly in communications technology. At high frequencies the wavelength becomes much smaller than the cavity size, and conventional numerical discretization methods become impractically slow, making the problem computationally challenging. Boundary integral equation methods, by using the known propagation across a uniform medium, solve the same problem with a much smaller number of degrees of freedom (scaling like the boundary area rather than the cavity volume), but suffer from the need to locate each resonant frequency (eigenvalue) individually.Building on exciting developments which emerged from the quantum physics community, the investigator has pioneered methods which combine this small number of degrees of freedom with the ability to compute a long sequence of modes simultaneously. This is based upon the `scaling method', a little-known variant of the Method of Particular Solutions. The remarkable efficiency gain compared to other boundary methods scales like the number of wavelengths on the boundary (its `electrical area'), and has been shown to be a thousand times faster than any other known method at very high frequencies. However, the mechanism controlling the size of errors intrinsic to the method is unknown. The investigator proposes to apply the tools of numerical analysis to understand these errors, to seek to widen the range of cavity geometries, and boundary conditions, to which it can be applied (for instance, cavities with re-entrant corners, or in three dimensions), and to write a robust software package that can allow other researchers to benefit from these new efficient methods.The investigator will continue to apply the method in two exciting areas to which it is well adapted: i) the design and modeling of micro-cavity dielectric lasers, which can give higher powers for optical fiber communications and could enable fabrication of integrated optics `on a chip', and ii) `quantum chaos', the semiclassical (high frequency) behavior of resonances of cavities whose shape leads to chaotic ray motion. Understanding the distribution and statistics of wave intensities in such modes has impact in atomic physics and chemistry, as well as for mathematical questions arising in automorphic forms and number theory. Quantum chaos can also model electronic transport and dissipation in quantum dots, nanoscale laboratories for coherent wave electron effects, which are promising candidates for quantum computers. Finally, the development of rapid solvers for resonant modes, especially for the Maxwell equations, would have far-reaching benefits for the design and shape optimization of optical and microwave resonators used throughout the engineering world.