Fast and well-conditioned boundary integral equation solvers for the numerical simulation of graphene plasmon

Abstract

Surface plasmonic polaritons (SPPs) are evanescent electromagnetic waves coupled to free electron plasma oscillations. It was first observed in metallic gratings in 1902 by Wood. Modern plasmonics gained renewed interest since the discovery of the extraordinary optical transmission through a periodic array of subwavelength hole arrays. The strong confinement of SPPs in the subwavelength scales and their huge electromagnetic field enhancements have led to significant applications of plasmonics structures in near-field imaging, spectroscopy and bio-sensing, solar cells, nano-photonics, etc. Graphene is rapidly emerging as a powerful plasmonic material recently by combining the appealing features of SPPs and easiness to tune electrically. It also opens applications of SPPs in lower frequency regimes such as the terahertz to mid-infrared frequencies.

This thesis is concerned with computational modeling of plasmonic phenomenon in graphene. Two main difficulties arise in solving the associated mathematical models numerically. First, surface plasmonic modes are strongly confined with subwavelength scales, and at the same time, they are highly oscillatory along the graphene surface. Hence numerical schemes have to be designed so as to resolve the oscillatory nature of plasmonic waves. Second, the two-dimensional graphene has sharp edges. Such edge effect may give rise to additional difficulties, such as ill-conditioning of the discretized linear system.

In this thesis, we develop an integral-equation solver for numerical simulations of graphene plasmon. The integral equation is formulated along the graphene surface, which reduces the degree of freedom significantly compared with volumetric methods. Another advantage is that the radiation condition at infinity is enforced automatically. Due to the edge effect mentioned above, the classical Calderon formula does not lead to a well-conditioned integral formulation anymore. To alleviate the ill-conditioning, following the ideas by Prof. O. Bruno, we regularize the integral equation using scaled integral operators and generalized Calderon formula. The Nystrom scheme is then applied to discretize the integral operators and the Generalized minimal residual (GMRES) iterative method is employed to solve the linear system. Numerical examples are demonstrated to illustrate the effectiveness of the proposed integral-equation solver. Finally, we carry out numerical analysis to study the errors arising in numerical approximation of the integral equation and its solution.