The goal of the proposed project is to develop novel mathematical and simulation tools for studying electromagnetic wave interacting with arbitrarily curved dispersive interface. Great challenges exist in developing efficient and reliable numerical methods for such interactions. Physically, jumps in wave solution and its derivatives across the dispersive interface are time dependent. Numerically, the existing algorithms suffer a serious accuracy reduction due to their incapability to handle such time variant jumps. Computationally, this interface error will be significantly amplified when coupling with the staircasing approximation in treating curved interface. Due to these challenges, an extremely expensive mesh resolution of about 100 grid points per wavelength was commonly practiced in the metamaterial simulations. In this project, the investigator will rigorously analyze the time dependence and cross coupling of electromagnetic field components at the dispersive interface. Novel formulations will be derived for commonly used dispersive material and metamaterial models to convert time dependent jump conditions into time independent ones and to minimize the cross coupling. Building on these mathematical modeling, a second order accurate interface algorithm will be developed to deal with arbitrarily curved dispersive interface, by using only a simple Cartesian grid. This higher order of accuracy will promise a higher numerical resolution, so that the computational burden of the existing simulations can be significantly relieved.
Dispersive media are ubiquitous in nature, such as in biological tissues, rocks, soils, and plasma. The numerical simulation of dispersive media is crucial to a wide range of electromagnetic and optical applications, such as microwave imaging for early detection of breast cancer, double negative metamaterial based subwavelength imaging system, and cloaking devices. The proposed mathematical modeling, algorithm development, and numerical computations will address key scientific challenges in an interdisciplinary filed lying at the interface of computational mathematics, physics, and electric engineering. The planned research activities will bring new advances to computational mathematics and lead to reliable simulation tools for the characterization, analysis, and design of various practical engineering devices and systems. These tools in turn may offer a better means for analyzing or calibrating some basic physical laws, such as the one governing the resolution limit of the sub-diffraction imaging system. In addition, this project will provide an interdisciplinary research training environment which could inspire and promote more students to purse careers in science and engineering.
We have constructed new mathematical and simulation tools for studying electromagnetic wave interacting with arbitrarily curved dielectric and dispersive interface. Physically, when both permittivity and permeability are discontinuous across the dielectric interface, the electromagnetic wave solution will be discontinuous. Moreover, for dispersive interface, jumps in wave solution and its derivatives across the dispersive interface are time dependent. Without proper interface treatments, the usual numerical methods will converge slowly or even fail to converge. To capture such jumps in discontinuous dielectric interface problems, we have constructed new jump conditions which relate the field components on both hand sides of the interface and can be decomposed from two dimensions (2D) to one dimension (1D). To capture time dependent jumps in dispersive interface problems, we have proposed novel transverse magnetic (TM) Maxwell systems in treating both Debye and Drude dispersive materials, which couple the wave equation for the electric component with Maxwell’s equations for the magnetic components. Such hybrid formulation enables us to track the transient changes in the regularities of the electromagnetic fields across the interface. Building on these mathematical modeling, we have developed matched interface and boundary (MIB) methods to enforce the jump conditions in the finite difference time domain (FDTD) discretization in the vicinity of the dielectric and dispersive interfaces. A versatile grid scheme is considered so that the MIB methods can accommodate arbitrarily complex geometry by using only simple Cartesian grids. Benchmark computations have been conducted to validate the proposed MIB-FDTD algorithms. For dielectric interface problem, the proposed MIB method can secure a second order of accuracy in handling any 2D discontinuous Maxwell interface problem, including both transverse electric (TE) and TM modes. For dispersive interface problem, the MIB enhanced FDTD algorithm attains a higher order of convergence that the plain FDTD algorithm in treating both Debye and Drude materials, so that the computational burden of the existing FDTD simulations can be significantly relieved. In addition, this project also provides an interdisciplinary research training opportunity to two math graduate students and one math undergraduate student. This will inspire and promote these students to purse careers in science and engineering.
