Change in the permittivity tensor of a material in 3D and 2D simulations

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#1

Hello,
I have a permittivity tensor as a material definition, in which z direction is the direction of magnetization and light propagation. E=[e11, -ig, 0; ig, e22, 0; 0, 0, e33]
This means it is suitable for 3D simulations, but for 2D simulations, y direction is the direction of interest. So I wonder how should I change the permittivity matrix to adopt with 2D simulations.


#2

Dear @skharratian15

Can you please elaborate more on why you need to change the matrix permittivity? My understanding is that you need to define your material with permittivity tensor and then use it in 2D or 3D simulations, and I do not think that you will need to modify the permittivity.

The only concern is that since permittivity tensor is anisotropic, you need to diagonalize it first. This is explained in the link below:
https://kb.lumerical.com/en/index.html?materials_anisotropic.html

Please let of your thoughts and I will be glad to be of a help.

Thanks


#3

Thank you so much for the reply.
As I have mentioned, the permittivity matrix that I have is compatible with a configuration in which the magnetization and anisotropic direction is in Z direction, which suits with the 3D simulations in Lumerical. But in 2D simulations, the Y direction is the important direction (for my case, direction of magnetization and light propagation); so I think I should change the matrix accordingly before diagonalizing it.


#4

Dear @skharratian15

If I understood your question, you want to rotate your axises along x-axis by 90 degrees. As a result, you need to transform the permittivity tensor into new axes. For a rotation along the x-axis, you will need to unitary rotation matrix in the form of:

ux=(1, 0, 0
        0, cos(θ), -sin(θ)
        0, sin(θ),  cos(θ)
                          )

where θ= π/2. Thus you need to calculate:

ϵ_new = ux_dagger * ϵ_old * ux

Doing so, the new permittivity tensor would be:

ϵ_new = (ϵ11, 0, ig
            0, ϵ33, 0
            -ig, 0,  ϵ22
                        )

Now you can can diagonalize this matrix using grid attributes and eig command.

I hope I could answer your question, but please do let me know if you had any other questions.

Thanks