# Bloch functions

1. Apr 22, 2005

### 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?

Thanks!

/Daniel

2. Apr 22, 2005

### 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 [Broken]

Last edited by a moderator: May 2, 2017
3. Apr 25, 2005

### 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.

4. Apr 25, 2005

### Gokul43201

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

5. Apr 25, 2005

### 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 probablity 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.

6. Apr 26, 2005

### 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