A basic question about Band Diagram Calculation for PCs

[kaps_nit]

energy band diagram calculation

Dear all,

After reading number of papers on photonic band gap calculation, I found that the calculation is always done for a particular direction. Say for example, if we consider a square lattice, we only consider the perimeter, rather than the whole area of the reduced Brillouin zone. Some argues that ω(k) for all other points within the reduced Brillouin will lie within the boundary set by the high symmetric points. But the answer still remained vague to me.

Can anybody explain to me the physics behind this???

Thanks

 

[rrumpf]

calculation of pcs

I have thought about this same thing. The argument is that the band extremes almost always occurs at the key points of symmetry. So if you want to calculate the width of a photonic band gap, checking the key points of symmetry will give you the right answer 99.99% of the time and is much faster than calculating every point in the irreducible Brillouin zone.

But you are asking about the 0.01% of the time and an explanation of what is happening. Qualitatively, as your wave "looks" in any particular direction, it sees a Bragg grating. Each direction through the lattice has a slightly different Bragg grating. Imagine a lattice constructed of an array of spheres. There is a smooth and consistent transition from "Bragg grating" to "Bragg grating" as you scan directions through the lattice. This ensures no surprises in terms of how the band gaps shift around. The shortest and longest period gratings will always occur in the directions you would expect. This same geometrical argument holds for most shapes your lattice may be composed of.

I suppose if your lattice were composed of something oddly shaped (maybe like a donut or snowman), it could effectively produce additional Bragg planes in your lattice, above and beyond what that lattice itself would give you by default. I "think" perhaps this would give you a lattice where a band extreme could fall somewhere other than a key point of symmetry.

This could, perhaps, make an interesting paper. I know there is some interest in indirect photon transitions and this could play a role in that.

If you are looking for more on photonic band diagrams and qualitative descriptions of this kind of thing, you might be interested in chapter 2 of my dissertation. You can download for free from here:

**broken link removed**

Hope this helps!
-Tip

 

[kaps_nit]

Thnak you, it is in deed a nice explanation.
And for your information, I have been following your thesis for quite a some time for my masters project which is also on PCs.
I had one doubt regarding the band diagram, what does zero value of k vector signify?

 

[rrumpf]

That is a good question!! K=0 is an anomalous point where the fields at opposite sides of the unit cell are perfectly matched and not out of phase in any way. Zero frequency with an infinite wavelength is one solution, but there are others as you can see in the band diagrams.

It is a mistake to think of a K=0 solution as a Bloch mode without any variations in field amplitude. Remember through the Bloch-Floquet theroem that the field is the product of a periodic envelope and a plane wave. When K=0, the plane wave component is unity so you are just left with the periodic envelope.

-Tip

 

[kaps_nit]

your replies clearly shows your indepth knowledge in this field.

As I have understood from your explanasion, if a plane wave of frequency ω2 propagates though the PC slab, it will be decomposed into bloch modes with different k-vectors and one among them will be K=0, which will have only variation similar to the crystal variation. right?

 

[rrumpf]

Correct!