Advances in Inverse Electromagnetic Scattering Theory for Complex Periodic Media

Photonic and chiral photonic crystals—materials engineered with nanometric to micrometric periodic structures—are of great significance to science, industry, and national defense. Their ability to control and manipulate light enables various technological applications, including optical communication devices, lasers, sensors, solar cells, and thermal management systems. In materials science, some defects can be intentionally added to these periodic structures to create new materials with advanced properties, such as precisely controlling light in specific areas and accurately routing light signals. The ability to control waves in a small space can allow for more compact device designs. Studying how light interacts with periodic structures that contain defects will offer a cost-effective method for optical testing of designs—a crucial step in developing these new materials. This project will contribute to the study of inverse electromagnetic scattering theory in complex periodic media. The main objective is to simulate wave–material interactions and reconstruct defects using measurements of the scattered waves at a certain distance. This research also supports quality control of optical devices fabricated from purely periodic structures. Graduate and undergraduate students will participate in and receive training as part of this research. This project investigates the direct and inverse scattering problems governed by Maxwell's equations in an infinite locally perturbed bi-periodic layer, coupled with general constitutive relations. In addition to the inherent complexity of Maxwell’s equations, the presence of defects breaks the periodicity, posing significant challenges for both theoretical analysis and numerical solvers. Various techniques—including the Floquet-Bloch transform, volume integral equations, the fast spectral method, qualitative methods, and other approaches—will be flexibly employed to develop useful tools for addressing the challenges in both problems, summarized in four major topics. Questions to be addressed for the direct problem include (i) establishing a Rellich-type identity to study the existence and uniqueness of solutions of Maxwell's equations and (ii) developing a novel numerical method that combines the Floquet-Bloch transform, volume integral equations, and a fast spectral method to solve the problem and study the convergence. Questions to be addressed for the inverse problem include (iii) designing an innovative inversion scheme to reconstruct defects using the measurements of the scattered waves, without relying on prior knowledge of the periodic structure and requiring only minimal data for accurate performance and (iv) investigating the discreteness of interior transmission eigenvalues for such scattering media. 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.