# 2D Maxwell complex coordinate stretching PML

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1. Apr 27, 2015

### 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

Last edited: Apr 27, 2015
2. May 2, 2015

### Greg Bernhardt

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

3. May 2, 2015

### 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 [Broken]

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

Last edited by a moderator: May 7, 2017