Photonic band gaps: Incomplete vs complete

[Gnomie]

Hi all,

I'm currently trying to learn more about photonic band gaps. The material I am looking at is an FCC crystal of spheres with an (111) surface, and this structure is theoretically predicted to display an incomplete band gap from the gamma to L points. (Please see attachment)

Essentially I am trying to figure out how an incomplete band gap actually behaves compared to a complete band gap. A complete band gap of course reflects light of all angles of incidence while an incomplete does not. But which parts of the light actually gets reflected? For example, is it such that all light within a "cone" gets reflected (much like diffracted orders)? How can I translate the band structure information into actual real-space angles?

Thank you for your time,

Gnomie

 

[Dr Lots-o'watts]

If there is no gap at point L for example, it means a beam with a wavevector in the L direction will not reflect. You then need to take the reciprocal vector to find the direction in real space.

The range of directions depends on the specific band structure. It can be somewhat like a cone, but I imagine a cone where the base is not necessarily a circle.

And remember the L direction includes its symmetries, so in that attached diagram, there are 8 symmetrical L-points, so 8 "cones" (each one at 180deg from another, with the 8 summits meeting at the center.)

 

[Gnomie]

Thank you for your reply! :) I have now figured out some more details to help solve this:

My structure has a (111) surface. The incomplete band gap goes from (000) to (111) points in reciprocal space. The (111) point in reciprocal space corresponds to light traveling along the <111> vector in real space (that is, perpendicular to my surface). So far so good.

The challenge now is to understand which other directions fall within the incomplete band gap. That is, which real space vectors are found in the interval between the (000) and (111) points in reciprocal space? I think my problem is that I never quite understood how the x-axis in band diagrams is actually constructed, with the critical points and all. Mapping something three dimensional onto a one-dimensional axis is not easy to visualize..

I found a reference claiming that the incomplete band gap only corresponds to light perpendicular to the surface (see attachment and read subtext). Does this seem reasonable?

Once again thanks for your time.

 

[Dr Lots-o'watts]

The graph on the right is for a single direction, but the graph on the left is for different wavevectors, which include both direction and wavength information.

Of course, to relate reciprocal space to real space, we need these formulas:
http://en.wikipedia.org/wiki/Reciprocal_lattice

You can simplify the concept by a considering a 1D photonic band gap (instead of your 3D crystal), which is nothing more than a periodic multi-layer mirror (a 1D crystal right?). What happens if you tilt the mirror?

Now for a 3D crystal, you get the same, except that the phenomena happens at more than one direction, and most likely at more than one different wavelengths.