Re: About the lightcone and projected slab mode

Dear Prof. Steven,

        OKey! I got it.
        I have used a non-orthogonal supercell for calvulating
triangular lattice. That may be the problem.
        I chose the supercell like this:
                                       o
                                     o
                                  X
                               o
                            o
        I will change it into orthogonal supercell (rectangle) when
calculating slab mode.
        Professor, Thank you very much for your kind help~

Best wishes

                      Yours,
                        Ryan

On Nov 16, 2007 7:19 AM, Steven G. Johnson <[email protected]> wrote:
> On Thu, 15 Nov 2007, Ryan Hao wrote:
> >     I know sent the supercell with no defect to calculate slab mode.
> > But how to project it to the waveguide direction?
>
> Just plot it as a function of the waveguide-parallel k vector.
>
> The only thing you need to be careful of is to arrange your supercell so
> that one lattice vector points along the waveguide direction and one
> lattice vector is perpendicular.  e.g. in a triangualr lattice you would
> use a supercell something like
>
>    o   o   o   o   o   o   o   o   o   o    x
>      o   o   o   o   x   o   o   o   o      |
>    o   o   o   o   o   o   o   o   o   o    |____ y
>
> (where "o" is a hole and "x" is the defect, for example, and the x
> direction is parallel to the line defect and y is perpendicular).
>
> >     Just a large supercell is enough? why? Could you please talk into
> > it a little bit more in details?
>
> There are two ways to proceed.
>
> One is just to use a large supercell with no defect, in which the slab
> bands are folded many times...in the limit as the supercell becomes
> infinitely large, the slab bands will form a continuous region when
> plotted vs. k_x.
>
> More efficiently, use a small supercell (even just a single unit in the y
> direction), and compute the bands for many k_y values from 0 to 0.5.  When
> plotted as a function of k_x, this will outline your continuum slab-band
> regions.
>
> In principle, you could also take the bands in the primitive
> non-orthogonal unit cell (for a triangular lattice) of the defect-free
> system and project them.  This requires some care, however, as described
> in my paper "Linear waveguides in photonic-crystal slabs," in Phys. Rev.
> B. 62, p. 8212 (2000).  It is easier to use the supercell technique above
> to get an orthogonal unit cell.
>
> Regards,
> Steven G. Johnson
>
>
>