Take a look at Lectures 7 and 8 here:

http://emlab.utep.edu/ee5390em21.htm

These lectures do a pretty good job of explaining all of this.   Some of the electronic notes have been revised and improved since the videos were recorded, so be sure to look at both.

Plane waves do not exist inside periodic structures.  The modes that do exist are called Bloch waves and they look like bumpy plane waves.  There are good pictures of these in the Lectures I have identified.

Think of the Brillouin Zone (BZ) as a map of how similar a lattice looks in each direction.  The more spherical the BZ, the higher the symmetry of the lattice.  Other than the BZ being derived from the same lattice as the Bloch wave, I do not think there is a very direct connection between the two.  The BZ applies to the reciprocal lattice, whereas the Bloch wave applies to the direct lattice.  All of this is discussed in Lectures 7 and 8.

Hope this helps!

Thank you!

Hi

The connection between Bloch waves and the Brillouin Zone is the following. Bloch waves are uniquely defined only by wavevectors inside the BZ. If you take a Bloch wave characterized by a wavevector K and then another characterized by a wavevector K + G, where G belongs to the reciprocal lattice, they are in all respects the same Bloch wave. Look up the excellent  book from Joannopulos.

Best

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