1 Dec 2006 20:20

## Re: Gap position vs effective index

On Fri, 1 Dec 2006, Kaushik Dayal wrote:
> Do you have a reference for these theorems? I'm particularly
> interested in the case when the material is anisotropic, ie, in what
> norm should \delta\epsilon be small?

The first theorem, about multiplying the index everywhere by a constant,
is proved in e.g. Photonic Crystals: Molding the Flow of Light, by
Joannopoulos et al. (see chapter 2, the section on scaling).

Perturbation theory for Maxwell's equations can be found in a number of
places.  You can find some references in my paper, Phys. Rev. E 65, p.
066611 (2002), which points out an important case in which standard
perturbative methods fail.  [It's hard to find a reference that works it
all out completely generally, partly because important generalizations are
still being made.  I'd like to write a textbook on such things at some
point, but probably not for a couple of years.]

I'm not sure of a good reference for perturbation theory in the
anisotropic case, but it is easy to derive from the generalized
eigenproblem for the electric field E.  Instead of \int \Delta\epsilon
|\vec{E}|^2, you more generally get
\int \vec{E}^* \cdot \Delta\epsilon \cdot \vec{E}
If this integral is positive, and \Delta\epsilon is small enough that
perturbation theory is accurate, then the frequency shift is negative. It
should be sufficient for the spectral norm of \Delta\epsilon to be small,
although in general rigorous proofs of convergence for perturbative
methods with finite perturbations tend to be difficult and rarely done.

Steven


Gmane