Accurate and rapid numerical solution of the Helmholtz and related partial differential equations in complex geometries is key to future progress in device design, in imaging, and in basic science. However, at high frequencies (many wavelengths across the system) this becomes prohibitively challenging using direct discretization, due to the multiscale nature of the problem. The investigator seeks to build upon boundary-based methods which have been uniquely successful (up to a thousand times faster than the competition) in solving eigenmode problems hundreds of wavelength in size with spectral accuracy in two dimensions, and to extend them to the scattering problem, to more general media and periodic boundary conditions, and to three dimensions. These methods are global approximation by particular solution basis sets, and the scaling method for Dirichlet eigenmodes. Proposed extensions include: 1) use of fundamental solutions basis sets, and their analysis via the role of singularities in the analytic continuation of the wave field, 2) exploiting a little-known analytic formula for the fundamental solution in linear graded-index materials, enabling non-piecewise-constant media to be solved on the boundary, 3) error analysis of a reformulation of the scaling method via the Dirichlet-to-Neumann map for the domain, 4) application of such methods to the spectrally accurate solution of dielectric photonic crystal band structure, and to `quantum chaos' (the wave and spectral properties of cavities with ergodic ray dynamics).
The impact of our technology such as radar, microwave communication (eg cellphones), optics and lasers, acoustics, medical ultrasound imaging, and miniaturized quantum devices has been, and will continue to be, profound and far-reaching. To design all such devices, one must calculate how they will reflect, guide and trap waves, and this is a time-intensive, difficult and sometimes unreliable computation. The computer algorithms proposed by the investigator will make such calculations faster and more accurate, particularly when the objects are large or complicated in shape. This is expected to lead to improvements in the design of, for example, optical signal-processing devices (which rely on microscopic periodic structures the size of the wavelength of light), promising candidates for the next generation of fast (post-silicon) computers. A deeper grasp of quantum chaos (the behavior of waves trapped in cavities which cause chaotic bouncing of rays) would impact nanoscale quantum wave devices such as quantum dots, super-fast quantum computers, as well as areas of pure mathematics and physics theory. The proposal also provides training in applied and computational mathematics at both graduate and undergraduate levels, and a course on the ``Mathematics of Music and Sound'' introducing non-majors to waves, modes, and resonance.
The work of the PI supported by this grant was to develop faster and more accurate computer algorithms for the solution of wave problems, such as the following: how precisely do waves such as light, radar, or sound, scatter when they strike an object of given shape? What set of wave frequencies will set a cavity of given shape into resonance? Applications are numerous: imaging, radar, design of microscopic optical devices for communications, lasers, quantum physics, thin-film photovoltaic designs for solar energy, etc. The PI's work involved creating algorithms of high accuracy (many digits correct). The PI created methods for the scattering of waves in two dimensions from polygons, from infinite arrays of smooth obstacles, and recently from multi-layer dielectrics with inclusions, of the type used in on-chip photonics or thin-film solar cells. The algorithms are efficient in that they require only a few seconds of computation time on a laptop. The PI also completed mathematical analysis of a new algorithm for high-frequency cavity resonances, proving that the method is robust and has errors of controlled size. This is fastest known algorithm for this challenging wave problem. Many of these algorithms are incorporated into freely-available computer code (a MATLAB toolbox) called MPSpack, enabled by the grant, and co-authored with Timo Betcke. This code lets engineers and other users try out the algorithms easily. The PI also guided related research projects for several undergraduates, developed a course for non-science majors on the Mathematics of Music and Sound, gave a public lecture to high-school students, disseminated research results at around 20 seminars and conferences, co-organized an International Conference on Spectral Geometry, and co-organized minisymposia at other conferences.
