Re: How to calculate the surface bands of photonic crystal slab?

On Mon, 20 Nov 2006, zhang hao wrote:
> Dear Steven, in your 2003 paper [PRB 68/045115], you used MPB to calculate
> the surface bands of a photonic crystal slab. Could you please tell me
> whether the left figure of Fig. 3(a) is the supercell you used, and whether
> you look on the photonic crystal slab as a one-dimensional photonic crystal
> constructing by the supercell? Thanks a lot!

First of all, that paper was looking at two-dimensional photonic crystals, 
*not* photonic-crystal slabs (although of course one could do an analogous 
calculation for a slab).

Although I don't have the input file that was used for that figure in the 
paper, I don't think you can infer the supercell from figure 3(a), for two 
reasons.

First of all, for a surface-state calculation, you want to use only a 
single unit cell in the periodic direction(s) parallel to the surface.  In 
figure 3(a), several unit cells are shown for illustration purposes.

Second, in the direction perpendicular to the surface, you want a 
supercell, with enough air (on one side of the surface) and photonic 
crystal (on the other side of the surface) that the exponentially 
localized surface mode(s) don't "see" the boundary of the supercell.  How 
big your supercell has to be therefore depends on the structure and on 
what wavevector you are looking at, which determines how localized your 
states are.

I'm not sure exactly what you mean by "one-dimensional photonic crystal". 
It is true that a surface in a 2d photonic crystal is periodic only along 
one dimension, as I mentioned above.  But the dielectric function is not 
univariate, so the system is not "one-dimensional" per se.

Cordially,
Steven G. Johnson

PS. Because of the periodic boundary conditions of the supercell, your 
surface-state calculation will have two surfaces of the photonic crystal 
(there is no way to make the crystal semi-infinite).  There are two 
approaches to dealing with this.  One approach is to make the surfaces 
symmetric, as in figure 3 of the PRB, and look at say the even modes (or 
the odd modes, it doesn't matter), which will have identical surface 
states on both surfaces, with the same dispersion relation if the 
supercell is large enough.  Another approach is to terminate *one* of the 
surfaces with a termination that doesn't support surface states; this 
requires some experimentation if you don't know what termination you need, 
however.