A second order dispersive FDTD algorithm for transverse electric Maxwell’s equations with complex interfaces

This work overcomes the difficulty of the finite-difference time-domain (FDTD) algorithm in solving the transverse electric (TE) Maxwell’s equations with inhomogeneous dispersive media. For such TE problems, the electric fields are discontinuous across the dispersive interfaces. Moreover, such discontinuities are time variant. A novel matched interface and boundary time-domain (MIBTD) method is proposed to solve such problems through new developments in both mathematical formulations and numerical algorithms. Mathematically, instead of handling all zeroth and first order jump conditions in a local coordinate, we directly construct the TE jump conditions which are needed in the FDTD computations in the Cartesian coordinate. Such Cartesian direction conditions depend on the time, as well as tangential and normal components of the electric flux. Driven by the jump condition modeling, we adopt the standard Maxwell’s equations in coupling with the Debye auxiliary differential equations for the electric flux as the governing equations. Computationally, the leapfrog scheme is employed for integrating the Maxwell system and time dependent jump conditions. Sophisticated interface treatments are developed in both producing the TE jump conditions and enforcing them in the FDTD algorithm, based on a staggered Yee lattice. The numerical accuracy, stability, and efficiency of the proposed scheme are investigated by considering dispersive interfaces of various shapes. The MIBTD method achieves a spatially second order of convergence in all tests. To the best of our knowledge, the present MIBTD scheme is the first FDTD method in the literature that can restore second order accuracy in treating curved dispersive interfaces for the TE Maxwell system.