On Thu, 9 Nov 2006, Minghui Han wrote:
In your 2002 paper on the perturbation theory with shifting material boundaries, you formulate the problem in terms of parallel and normal fields. I am hoping that I can follow that procedure to do some perturbation analysis based on MPB result. Then the question is how I can determine the normal vector as well as epsilon_parallel and epsilon_normal for each grid point. The most direct way is of course, to use the same spherical-quadrature method as implemented in MPB.
I've applied this perturbation method in MPB lots of times. Normally when I want to look at the frequency shift due to a shifted material boundary, however, I have an analytical description of the boundary that gets shifted (e.g. a cylinder, or a flat interface, or...) so finding the normal vector isn't an issue.
Note that you shouldn't apply this method to analyze surface roughness, at least not in the limit where the correlation length is comparable to the amplitude of the roughness. In that limit, additional corrections are required, as described in our 2005 paper [ Appl. Phys. B. 81, p. 283 ].
But I am wondering if there is some simpler way for me to extract such information. I have tried to do the svd decompostion on the output epsilon tensor at each grid point, but the problem is that I don't know how to automatically decide which diagonal element corresponds to the parallel epsilon and which corresponds to the normal epsilon. Do you have any suggestions on this?
Yes. When you do the eigen-decomposition of the dielectric tensor around the interface in MPB, it will have two equal eigenvalues and one different eigenvalue. The eigenvector for the different eigenvalue corresponds to the surface-normal direction, and the eigenvectors for the equal eigenvalues correspond to the surface-parallel directions.
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