2D Maxwell complex coordinate stretching PML

[Knaapje]

Hello, I'm trying to derive the perfectly matched layer for the TM mode Maxwell's equations using a complex coordinate stretching. As seen in http://math.mit.edu/~stevenj/18.369/pml.pdf . But I'm running in a bit of trouble somehow.

$\partial_t H_x =-\mu^{-1} \partial_y E_z\\ \partial_t H_y =\mu^{-1} \partial_x E_z\\ \partial_t E_z =\epsilon^{-1} \left(-\partial_y H_x + \partial_x H_y \right)$​

After applying the transformations in the x- and y-direction, the equations look like this:

$i\omega H_x = -\mu^{-1} \partial_y E_z + \sigma_y(y) H_x\\ i\omega H_y = \mu^{-1} \partial_x E_z + \sigma_x(x) H_y\\ i\omega E_z = \epsilon^{-1} \left(-\partial_y H_x + \partial_x H_y\right) +\left(\sigma_x(x) + \sigma_y(y)\right) E_z + \psi + \phi + \frac{i\sigma_x(x) \sigma_y(y) E_z}{\omega}\\ i\omega\psi = \epsilon^{-1}\sigma_x(x)\partial_y H_x\\ i\omega\phi =-\epsilon^{-1}\sigma_y(y)\partial_x H_y$​

Where two auxiliary differential equations have appeared due to integration terms in the Ez differential equation. However, it appears as though I should have another one, as there is still an integration term left in the Ez differential equation. This is contrary to what I've heard/read should happen. Is there any reason why this term should be absent (or why it does not contribute a lot to the solution in the physical domain)?

Any help would be greatly appreciated, as I've been trying to figure this out for some time.

EDIT: fixed the blank space between the equations

 

[Wxfsa]

I am no expert but you can look into "2D absorbing boundary conditions."
I got that part correct, right? You have a Z polarized E field, xy polarized H field traveling in xy plane?
http://www.engr.uky.edu/~gedney/courses/ee624/notes/EE624_Notes6.pdf


I have implemented this in 1D, you need to search for 2D cases.