Re: Propagation constant

On Oct 30, 2008, at 5:35 AM, Yiling Qi wrote:
> Dear Steven,
>     My simulated structure is a 2D periodic photonic lattice (the  
> contrast of refractive index is very small,  the order is only  
> around exp(-4). Then kx and ky are meaningful for the bandstructure,  
> but apparently there will be no bandgap with respect to frequency(w)  
> along kx/ky. But it is supposed to have bandgaps of propagation  
> constant(beta) to kx/ky. The solution form can be deduced to be  
> &(wavefunction)=exp(-iqz)u(x,y), and following the Bloch's theorem,  
> the eigenfunction u(x,y) can be written as u(x,y)=Uk(x,y)*exp(iK*r),  
> where Uk(x,y) is periodic have the period of the lattice. From this  
> solution form, it seems exp(-iqz) replaces exp(iwt) in the ordinary  
> plane wave form. So do you think MPB can still solve it?


MPB can certainly solve any problem of that form.  However, you still  
need a frequency in order to define your mode; given K (the in-plane  
wavevector with components kx and ky) and q (the z wavevector,  
although have switched the sign convention so that q = -kz) there are  
infinitely many eigenmodes with different frequencies.  Without a  
frequency your problem seems to be ill-posed.

MPB solves the problem where, given (kx,ky,kz) you find the  
frequencies.  Alternatively, with find-k you can give (kx,ky) and the  
frequency and then solve for kz (q).

Steven