Re: MPB problem: refraction of EM wave between two photonic crystals

On Tue, 2 May 2006, Huazhong Wang wrote:
> I am studying the refraction of EM wave between two photonic crystals. In 
> general, the refraction of light obeys Snell's law when light travels from 
> one medium to another one. what about the refraction in phtonic crystal?

In a photonic crystal, unless you are working in the limit where the 
wavelength is much longer than the period, it is not described by a single 
well-defined "index" of refraction.

At an interface with a crystal, you can have multiple reflected and 
refracted beams and other complicated behaviors.

> My first question is if MPB can apply in such a photonic crystal which 
> is made up two same kind structures (for example, both of them are 
> triangle structures) except the different lattice constant (for example, 
> one structure has the lattice constant of a and another has 1.2a.).
>
> My second question is how to analysis the refraction propeties if MPB gives 
> me the simulation results. For example, how to know bandwidth of refraction 
> light from the band diagram? this question likes Q factor question in laser.

MPB computes eigenmodes.  This is a useful tool and an important piece of 
the puzzle in understanding refraction phenomenon, but it solves only part 
of the problem.

In general, I would recommend doing something like the following:

1) In MPB, analyze each crystal individual and get the wavevector diagram 
(the contour plot of frequency vs. wavevector).

2) Use the wavevector diagram to analyze what refracted/reflected beams 
are possible and in what directions.  (See e.g. Luo et al., Phys Rev. B 
65, 201104 (2002) for an example of this sort of analysis.)  This does not 
tell you the amplitude or phase of the refracted beams, however.

3) Use some other method, e.g. a time-domain code (see jdj.mit.edu/meep) 
to simulate the refraction/reflection.  This both verifies that you did 
step (2) correctly and gives you the actual amplitudes etcetera.

(You can do (3) to start with, of course, but the wavevector diagram gives 
you a "map" to understand and interpret the reflection/refraction results, 
which may be difficult to interpret otherwise.)

Steven