Why does first photonic band go to zero at the centre of the Brillouin zone? I have been plotting photonic band diagrams of various geometries recently and I identify if it is correct by looking if it goes to zero at the Brillouin zone centre, $\Gamma$. I realised early on that I don't understand the physical reason why it goes to zero at $\Gamma$ and still haven't learnt properly why it is so. I feel like this is probably something obvious. Why does the first band of TE and TM mode always go to zero at $\Gamma$?

 A: 
Why does the first band of TE and TM mode always go to zero at $\Gamma$?

It's not always. $\Gamma$ point is a point where quasiwavevector $\vec k=0$. The fact that the dispersion curve goes to zero frequency there means that Maxwell's equations with periodic boundary conditions around a single Wigner–Seitz cell have a "standing wave" solution, which actually represents a static electromagnetic field.
Now, not all the periodic media will have such a static-field eigenmode. E.g. if we consider a grid of fully closed cavities with conducting borders, the lowest-frequency mode in such a system will be a periodic repetition of the ground mode of the cavity, which due to the combination of interface conditions on the borders and uniformity of $\varepsilon$ and $\mu$ inside can't have zero frequency.
The example above is, of course, extreme: the bands there all get flattened into single frequency levels. But it doesn't have to be that extreme. This open access paper discusses some 2D photonic crystal structures with zero-frequency stop band and normal, not too flattened, pass bands (and some more extreme structures, closer to the one discussed above).
