On the optimal design of the PML absorbing boundary condition for the FDTD code

An analytical method to predict and optimize the performance of Berenger´s (see J. Computat. Phys., vol.114, p.185-200, 1994) perfectly matched layer (PML) absorbing boundary condition (ABC) for finite-difference time-domain (FDTD) simulations is described. The shape of the conductivity in the PML layers has to be chosen carefully to obtain the best performance for a given number of layers. The relative error is shown to be the composite of two distinct effects: the theoretical reflection coefficient given by the PML layers backed by a metal plane and the second-order error in the differential intrinsic in the FDTD formulation. A theoretical expression to evaluate this error as a function of the number of PML layers and the shape of the conductivity is given, and the result is compared to that obtained for several FDTD test cases. The good agreement of the shapes of the theoretical and numerically derived curves allows the use of the theoretical formulation to optimize the PML region as a function of the shape of the conductivity, resolution, and number of layers