On the Termination of the Perfectly Matched Layer with Local Absorbing Boundary Conditions

Unsplit [4–5] perfectly matched layers (PML, see [1] for the original split-field formulation) have proven to be effective alternatives to local absorbing boundary conditions (ABC) for the truncation of computational domains employed in the numerical solution of electromagnetic wave propagation problems. The PML (0≤ z< d) is placed adjacent to a computational domain Ac (z< 0) and is truncated at z= d by imposing a homogeneous Dirichlet boundary condition on the tangential Electric fields there, e.g., B= 1 in Fig. 1. This is equivalent to enclosing the computational domain with a metal (perfect electric conductor, or PEC) box whose inner walls are coated with a wave absorber of depth d and loss profile σ(s), where s is the depth coordinate into the layer. The electric and magnetic conductivities in the layer are then proportional to σ(s)= σmax sn , where n= 0, 1, 2, the constants of proportionality being the permittivity 2 and permeability μ, respectively. In applications the layer is tuned by varying σmax to achieve the maximum absorption of outgoing waves for given d and discretization parameters. A properly tuned layer typically provides more than 3 orders of magnitude reduction in spurious reflection due to artificial grid truncation over that afforded by classical approaches. Therefore, it is of interest to explore the possibility of further improving such performance by altering the boundary condition used to terminate it. Previous work on terminating the Berenger PML with local ABCs [2, 3] did not take into account the presence of loss in the layer and only examined the reflection properties of the composite layer obtained with operators B derived for the lossless wave equation. Consequently, little improvement was evident with those approaches leaving the hope that more improvement can be realized if the loss is taken into account. Herein, using as a starting point the two-dimensional unsplit PML equations [4, 5] for the TM polarization in a layer that is perpendicular to the ẑ-axis [5], we derive and implement