Eigenvectors of the Permittivity Tensor in Periodic Dielectrics

[sph711]

TL;DR Summary

Looking for a derivation or proof that the eigenvectors of the effective permittivity tensor of a periodic dielectric structure align with the principal axis of the crystal symmetry of said periodic strucutre.

Hi all,
(first post here :D)

I am working on periodic dielectric structures in the long-wavelength limit (wavelength much larger than the periodicity). In the long wavelength limit the periodic strucutre can be homogonized and described via an effective permittivity (or refractive index) tensor.

I think it would make sense that the eigenvectors of said homogenized permittivity tensor would correspond to the principal axis of the crystal symmetry of the periodic structures. This also corresponds well with what I am seeing in my simulations. However, I cannot think of a way to show this formally.

Could anybody point me to an existing proof or guide me in how to approach the problem mathematically?

 

[Andy Resnick]

TL;DR Summary: Looking for a derivation or proof that the eigenvectors of the effective permittivity tensor of a periodic dielectric structure align with the principal axis of the crystal symmetry of said periodic strucutre.

Interesting question!

I'm not sure this is always true, I'm thinking about acousto-optic devices (with strained crystal lattices), and also chiral phases (especially liquid crystals). I can't find a reference either though, perhaps this one will give you some ideas:

https://opg.optica.org/directpdfacc...-17-6-4442.pdf?da=1&id=177143&seq=0&mobile=no

 

[sph711]

Interesting question!

I'm not sure this is always true, I'm thinking about acousto-optic devices (with strained crystal lattices), and also chiral phases (especially liquid crystals). I can't find a reference either though, perhaps this one will give you some ideas:

https://opg.optica.org/directpdfacc...-17-6-4442.pdf?da=1&id=177143&seq=0&mobile=no

Thank you for your reply. However, the link you provide did not work for me

I did some more research today, and I think I found what I was looking for. I came across the Neumanns principle which states "The symmetry of any physical property of a crystal must include the symmetry elements of the point group of the crystal." (properties of materials, Robert E Newnham - page 35 -

[Link to PDF of copyrighted textbook deleted by the Mentors]

If I understand it correctly this should mean that yes, the principle axis of the crystal are the same as the Eigenvectors of material properties, there is also an analytical form of this principle in the link above. But I did not have time yet to go through it.

However, I think you are right, this should not always be the case. I might be wrong, but I think it is limited to mateirals that can be described with a symmetric tensor. Will update as soon as I am sure.