Conventional Realization in Cartesian Coordinates

The split-field forms of time-dependent Maxwell s equations along the -direction are 

As an example, consider the original PML absorber in a 2-D computational space. Regarding the propagation of the TE case, the fourth-order electric field expression for Ex in the interior of the layer, becomes 

For the 3-D case, the study yields more complicated expressions. Thus, for reasons of brevity, only one of the twelve split-field PML subcomponents will be provided here. The remaining update equations can be then likewise extracted. For illustrative purposes, the electric field Exy subcomponent is given by 

Another point of notification is that both electric and magnetic field quantities are required on the right-hand side of the previous expressions at a certain time-step. However, most of them are not assumed to be stored in the computer s memory at this particular moment. This problem is overcome in two different ways. The first and the simpler one uses the most recent temporal value of the component, available via the time-advancing procedure, while the second way applies the semi-implicit approximation, assuming that the unknown quantity can be interpolated by averaging two of its already calculated values. 

Due to the generalized structure of (4.5)-(4.12), the previous technique can be profitably applied to the unsplit-field PML as well. For example, a potential scheme for further enhancement is the robust reflectionless sponge layer, developed in [48], As expected, the use of fourth-order spatial and temporal operators increases the absorber s versatility, while maintaining its well-posed and stable profile. 

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