I thought that I had a good understanding of band structures in 1D photonic crystals. Yesterday I did a simple test by comparing the band structures calculated using one unit cell with periodic boundary condition (PBC) and 3 unit cells with PBC. It turned out that the results were different! In the latter case additional bands are found, which are the images of the original ones shifted by \Delta k = 2\pi/3a and 4\pi/3a (a: the period of the lattice). So what went wrong?
I did additional tests with 2,4,5 unit cells. And the images are shifted by {\pi/a}, {\pi/2a, \pi/a, 3\pi/2a}, {2\pi/5a, 4\pi/5a, 6\pi/5a,8\pi/5a}, respectively. Considering that the distances shifted are integer times of 2\pi/(whole width of simulated structure), I gradually realized that somehow I was simulating structures with a period of the whole width, instead of the intended, smaller width of the unit cell. Why would this happen?
There are two possibilities: (1) The machine error makes each of the unit cell in the simulated structures slightly different, so the true period of the simulated structures is in fact the whole width. (2) Somehow imposing the PBC on the whole structure does not guarantee that the solutions are Bloch waves \psi(x) = exp(ikx) u_k(x), with u_k(x) = u_k(x+a).
Take the simulation with 3 unit cells for example. Indeed I found that the two shifted images of the original (correct) band structures correspond to waves with u_k(x) = u_k(x+3a) only, which confirms (2) and implies (1). Interestingly, |u_k(x)| = |u_k(x+a)| does seem to hold for these two images. Why? One can understand this observation from a physical point of view, since the system is still quasi-periodic and the probability of finding photons (electrons) at the same position in different unit cells should still be roughly the same.
A variant of what I observed is the following: Simulating the same total width but reducing the period of the structures by an integer n. One would have expected only part of the band diagrams in [-\pi/a,\pi/a], since the Brillouin zone now is [-n\pi/a,n\pi/a]. But due to the slight difference of each unit cells, one ends up with the whole folded band structures.
Keep in mind that PBC only guarantees u_k(x)=u_k(x+A), where A is the length of the computational cell. If A=ma where m is an integer greater than 1 and a is the lattice constant, then the "additional bands" are duplicates with the wrong k in phase factor: psi_{n,k}(x) = e^{ikx} u_{n,k}(x), which is consistent with your (my) observations.
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