Complex coordinate system as a generalized absorbing boundary condition

We have demonstrated that the perfectly matched layer (PML) in Cartesian coordinates is equivalent to solving the wave equation or Maxwell´s equations in a complex coordinate system. Therefore, closed form solutions that exist in the real coordinate system map to solutions in the complex coordinate system. In the complex coordinate system, the boundaries exist in a complex space, providing absorbing boundary conditions. Hence, this transformation provides a new view of PML in the Cartesian coordinates, clearly showing that a mapping to a complex coordinate system does not induce reflections, explaining why PML works near the corner of a simulation region, and when a dielectric interface, or a metallic surface, extends to the edge of a simulation region.