Mathematical framework for hyperbolic metamaterial

This project is mainly concerned with building a comprehensive mathematical framework for hyperbolic metamaterial, for electromagnetic wave and elastic wave. In recent years there were some relevant studies in the literature by this PI and other researchers on metamaterial in electromagnetic and elastic setting, as well as their exotic applications like super-focusing, subwavelength resonance, etc. In this project, we will focus our research efforts in a new emerging and promising field, hyperbolic metamaterial, and aim to establish a rigorous and complete theoretical framework for it in both electromagnetic and elastic setting. First, we concentrate on electromagnetic hyperbolic metamaterial. We will use potential theory to reach an effective medium theory, and to this end a generalized version of layer potential operators will be defined and studied. The effective medium theory shall fully unveil the far-field behaviour of hyperbolic metamaterial. Secondly, when taking near-field into consideration, discussion on scattering theory would be required, and we will again generalize the existing scattering theory to fit the highly anisotropic medium case. Through detailed discussion on scattering coefficient from hyperbolic metamaterial, we expect to draw a clear picture on the mathematical theory lies behind those exotic properties like super-resolution and super-collimation. We would even expect some predictions on properties yet to be discovered in experiments. Thirdly we will turn our focus to elastic hyperbolic metamaterial. The near-field and far-field behaviour, like what we have done to optic case, will be studied for elastic case, and for this reason a generalized version of potential operators and scattering coefficients should be studied for Lame system. With those tools we shall be able to clearly look through the nature of elastic hyperbolic metamaterial.