Application of Novel High Order Time Domain Vector Finite Element Method to Photonic Band-Gap Waveguides

摘要

In this paper we motivate the use of a novel high order time domain vector finite element method that is of arbitrary order accuracy in space and up to 5th order accurate in time$ and in particular, we apply it to the case of photonic band-gap (PBG) structures. Such structures have been extensively studied in the literature with several practical applications$ in particular, for the low loss transmission of electromagnetic energy around sharp 90 degree bends (1). Typically, such structures are simulated via a numerical solution of Maxwell's equations either in the frequency domain or directly in the time domain over a computational grid. The majority of numerical simulations performed for such structures make use of the widely popular finite difference time domain (FDTD) method (2), where the time dependent electric and magnetic fields are discretized over a 'dual' grid to second order accuracy in space and time. However, such methods do not generalize to unstructured, non-orthogonal grids or to higher order spatial discretization schemes. To simulate more complicated structures with curved boundaries, such as the structure of (3), a cell based finite element method with curvilinear elements is preferred over standard stair-stepped Cartesian meshes$ and to more efficiently reduce the effects of numerical dispersion, a higher order method is highly desirable (4). In this paper, the high order basis functions of (5) are used in conjunction with the high order energy conserving symplectic time integration algorithms of (6) resulting in a high order, fully mimetic, mixed vector finite element method.