Re: Relationship between primitive cell and reciprocal lattice vectors

On Apr 21, 2008, at 12:37 AM, Bruce Burckel wrote:
>
> As a prelude to studying a larger problem, I have been knocking  
> around a toy
> problem with MPB and have confirmed some cloudy areas in my  
> understanding of
> the program. I would like to study the behavior of a "nominally" 1-D  
> bragg
> stack with variable angle with respect to the x-axis in the unit  
> cell. I say
> nominally 1-D because with either 0 degrees of tilt (horizontal  
> slabs) or 90
> degrees of tilt (vertical slabs) the problem is truly 1-D, but with  
> angles
> between these two, the problem becomes 2-D in the sense that the  
> unit cell
> requires finite dimensions in both x and y. To keep things consistent
> regardless of the tilt angle, I would like to study the problem as a  
> 2-D
> problem for all angles.

No it doesn't.  If the dielectric function is a multilayer film and  
hence is univariate, the problem can always be reduced to a 1d one  
because the other dimensions are separable.  In MPB, you would still  
specify a 1d unit-cell with two no-size dimensions, and off-axis  
propagation is specified simply by giving a k vector that has a  
component along one or both of the no-size directions.

In fact, you *really* don't want to give a non-zero cell size along  
the invariant directions, because doing so will result in artificially  
folded bands.  (This is a common point of confusion.  I actually posed  
this as a question on the mid-term exam for my nanophotonics course  
last year.  See question 2 in: http://www-math.mit.edu/~stevenj/18.369/spring07/midterm.pdf 
  and the solution in http://www-math.mit.edu/~stevenj/18.369/spring07/midterm-sol.pdf)

Your unit cell should be dictated by the periodicity of the structure,  
not by the propagation direction.

Regards,
Steven G. Johnson