Lecture -- Dispersion Relation & Index Ellipsoids

[@eastofthegreenline3324]

Intuitive and clear. Great animation and accessible/helpful lecture!

[@halbarad7932]

5:25 Could you refer me to a source or tell me how it can be shown? Although I have seen this formulation of the eigenpolarization in textbooks such as Born, Wolf Principles of Optics, I've never really understood where it came from. Funnily enough, I tried solving the eigenvector problem symbolically in MATLAB and I did NOT get the same expression. Another problem I have with it is that it doesn't actually check out unitwise because you get m/rad which is not unitless (if your goal is normalized polarization), nor is it the unit which the electric field have. Also, in dielectric slabs you need two polarizations, TM and TE. Apparantelly, here you get just one, my question is, why? I guess I need to be wrong when textbooks have it the way you do, but to me, this expression just comes out of nowhere.

[@empossible1577]

I have been waiting for many years for somebody to ask this. I have simply written here what I had in my personal notes from graduate school and always wanted to perform the derivation myself but have not. It has always just been low enough on my priorities that I never got around to it. If I was to attempt it, my first attempt would be exactly as you have done using the symbolic toolbox in MATLAB. As for polarization, in anisotropic media away from a principal axis, the dispersion surfaces are separate. This means for a given k vector, there is only one possible polarization. The other polarization for a wave in the same direction would have a different magnitude k vector. You put this on my mind again. Maybe I will give it a try to fill in the steps and hopefully correct the units. Thank you and sorry I don’t have a better answer for you!

[@halbarad7932]

@@empossible1577 I understand that in anisotropic media you have two surfaces which correspond to two waves and each one has just one polarization. Again, I will refer to the dielectric slab as an example, where we will analyze the situation in xz plane (x is the direction of propagation and y is invariant). In there, considering diagonal permittivity tensor, you get two polarizations, TE for one wave and TM for the other wave, you can find that in lot of papers. We work in the xz plane, meaning that ky in the formula for eigenpolarization (as in the video) drops out to 0. And herein lies my problem with this formula. Even though the kx, kz are conditioned on what wave we are considering, once you place ky to 0, there is no way to get TE polarization, an eigenpolarization, that should exist there in theory, but the formula doesn’t actually reflect that. Also just so you know, I'm asking this because part of the goal for my bachelor's thesis is to get a dispersion relation for normal modes in an infinite slab with rotated anisotropic tensor, meaning I can't simply assume a polarization as others have done. I have done the simulations in COMSOL and currently, I'm trying to get the dispersion analytically to fit the COMSOL data. Btw. this goes without saying, but I really appreciate the work you do Dr. Rumpf. Your videos are amazing. I watch some of them just out of curiosity. For instance, I don't even need transformation optics, but it's just fun to learn about it :D.

[@empossible1577]

@@halbarad7932 I think I am at a dead end for you. I will have to work through the math in order to help you. This may take some time because I also have a lot of other things going on. It did take a quick stab it. For the dispersion relation I got ka^2*kb^2*kc^2 + ka^6 + kb^6 + kc^6 = 0. I need to figure out of this is correct and then figure out how to morph this into something that resembles what I have in the notes. Then I will use this to derive the equation for polarization. I am wondering if this is what you got as well?

[@halbarad7932]

@@empossible1577 Not sure if you mean the free space dispersion, but if yes then that equation should never higher than of order 4, since the surface generated by it is a quartic one. In any way, you don't need to do that, although I'm glad that you do, I'm sure you are very busy, so don't feel the need to help out. I was just thinking that maybe you know the answer off the top of your head, but it's okay. In the end, I might figure it out and then I can get back to you on that, if you are interested of course :).

[@empossible1577]

@@halbarad7932 I am absolutely interested!

[@antonknse]

Thanks a lot for the simple and clear explanations !!! Please keep going!!!

[@empossible1577]

I work on the lectures every day! Thank you!

[@annankldun4040]

Thank you so much!

[@empossible1577]

You are welcome!

[@alikmirzoyan5422]

Amazing thank u so much

[@terrywright6156]

Fab! I wonder can you in principle have a complex k in one direction and real in another since the dispersion relations are quadratic. In which case can one wave be evanescent and the other propagating.

[@keeqianling]

This is great -- thank you!

[@yehezkieladhi1388]

Amazing lecture

[@empossible1577]

Thank you!