Re: Determine Spatial Overlap of Optical Modes

On Jun 25, 2009, at 5:56 PM, Ian D. Hosein wrote:
>   Is there a quantitative way to determine if the fields of the  
> different
> calculated bands overlap in the unit cell, especially for  
> consecutive bands
> at which a complete bandgap occurs?


Sure, but you have to be more specific about what you mean by "overlap".

Usually, when one talks about "overlap" between modes, one is  
referring to some integral involving the two field patterns.  MPB has  
built-in field-integration routines (see the manual) that can compute  
any user-defined integral you want.

For example, if your magnetic fields are H1 and H2, you could  
integrate H1* H2.  But this gives zero for two modes at the same k  
point, because the Hermitian nature of the eigenproblem guarantees  
orthogonality.

Typically, overlap integrals are relevant when you have some  
perturbation in the system that couples the eigenmodes of the  
unperturbed system.  The precise overlap integral that is relevant,  
however, depends upon the nature of the perturbation.

For example, we have a recent paper that includes (among other  
results) the correct overlap integrals for two modes at different  
frequencies that are coupled by nonlinear harmonic generation: http://math.mit.edu/~stevenj/papers/RodriguezSo07.pdf 
  (Optics Express 15, 7303-7318, May 2007).  Many other examples can  
be found in the literature.

Regards,
Steven G. Johnson