Scattering field formulation used in DG-FEM

[discworld]

Hello!

Reading up on simulations of electromagnetic scattering with DG-FEM and trying some myself, I got stuck.
In some of papers I have read, a scattering field formulation is used, in which the total field is linearly decomposed in incident field and scattering field:

$E^{T}=E^{S}+E^{I}$

And, the 2D equations for the scattering field in a lossless, isotropic medium are:

$\epsilon_{r} \frac{\partial E^{S}}{\partial t} = \nabla \times H^{S} - (\epsilon_{r} - \epsilon_{r}^{I}) \frac{\partial E^{i}}{\partial t}$
$\mu_{r} \frac{\partial H^{S}}{\partial t} = -\nabla \times E^{S} - (\mu_{r} - \mu_{r}^{I}) \frac{\partial H^{i}}{\partial t}$

My problem is in the interpretation of the "scattering field" and "incident field" in this context. In every use I see of this formulation $\epsilon_{r}$ is space dependent, while $\epsilon_{r}^{I}$ is a constant - specifically, the incident medium's permittivity (same for the permeability). How can this work for multi-substrate cases, where, if I am thinking correctly, the medium considered incident should change?

(I am quite confused with the affair in general, so any clarifications are quite welcome)

 

[eljose79]

Hello there,

Thank you for sharing your thoughts and questions about the scattering field formulation and its interpretation. As a fellow scientist in the field of electromagnetic scattering, I can understand your confusion and would be happy to provide some clarifications.

Firstly, the scattering field can be defined as the part of the total field that is scattered by the medium. It is the result of the interaction between the incident field and the medium, and it carries information about the properties of the medium. On the other hand, the incident field is the part of the total field that is not affected by the medium and is known or controlled by the experimenter.

In the context of the equations you mentioned, the incident field is assumed to be a known, constant field with a certain permittivity and permeability, while the scattering field is dependent on the medium's properties (represented by the space-dependent permittivity and permeability). This formulation is valid for cases where the incident medium remains constant, and the medium being studied is the only one that changes.

However, for multi-substrate cases where the incident medium also changes, the scattering field formulation would need to be modified to account for this. This can be done by considering the incident field as a superposition of multiple known fields with different permittivities and permeabilities, and then solving for the scattering field using the appropriate equations.

I hope this helps to clarify the concept of scattering and incident fields in this context. If you have any further questions or concerns, please don't hesitate to reach out. Keep up the great work with your simulations!