Bloch Functions: Explaining the Bloch-Floquet Theorem

[danja347]

For propagation in a periodic dielectric crystal i can by combining Maxwells equations under certain conditions get:

$$\bold{\nabla}\times{1\over\epsilon(\bold{x})}\bold{\nabla}\times\bold{H}=\left({\omega\over{c}}\right)^2\bold{H}$$

I can apply Bloch-Floquet theorem and then draw a lot of conclusion.
Where does Bloch-Floquet theorem come from, when can I apply it and how can it be explained?
Please help or give references to where i can read about it.

Thanks!

/Daniel

 

[wenty]

Why not try google.For example the first search resualt I got is:

http://www.elettra.trieste.it/experiments/beamlines/lilit/htdocs/people/luca/tesihtml/node7.html

 

[Claude Bile]

The essence of the theorem is, that any solution to a periodic variation in the refractive index, must itself be periodic (To me, this is a fairly logical conclusion).

There is a wealth of information about this theorem, as it is not only used in photonic crystals, but also to study bandgaps in metals and semi-conductors.

As for the google, I thought the 3rd one down was pretty good (A little more layman for those that don't have a large mathematics background).

Claude.

 

[Gokul43201]

Also, if you have access to a library, find 'Solid State Physics' by Ashcroft & Mermin or Kittel. Both have derivations of Bloch's Theorem.

 

[rainbowings]

Here's an insight into Bloch's theorem that most texts do not mention:

The idea is that in a period potential, the probablilty of finding an electron at some location should be equal to the probability of finding the electron at all other places which are identical due to periodicity- and this makes sense. Here's the punchline - since the lwavefunctionl^2 gives the probability, this means that at all those places the wave function can differ only by a phase. The not so obvious thing is that this is not just any phase, but a phase whose argument is a function of the lattice vector.

 

[danja347]

Thank you all for you replies... I think I am getting a better and better understandning about how things work!

So, thanks again!

/Daniel