Re: Gap position vs effective index

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?

Thanks,
Kaushik

-- 
__________________________________
Kaushik Dayal
Aerospace Engineering and Mechanics
University of Minnesota

On 11/30/06, Steven G. Johnson <[email protected]> wrote:
> On Thu, 30 Nov 2006, [email protected] wrote:
> > Is it *necessarily* true that all the bands move to lower frequency values
> > as the effective index of the photonic crystal is increased ? I thought
> > this was, in general, true. But now I have some calculations that show
> > otherwise, and am a little confused.
>
> I don't know what you mean by "effective index" in this context.  If you
> mean "average" index, with the usual definition of "average", then this
> statement is not correct in general.
>
> There are a few statements that you can prove rigorously.
>
> First, if you multiply the index everywhere by a constant C (i.e. for
> *all* the materials by the *same* constant), then the eigenfrequencies
> decrease by the same factor of C.
>
> Second, if you change all of the dielectric constants everywhere by
> \Delta\epsilon << 1, where \Delta\epsilon can be any arbitrary function of
> position as long as it is small everywhere, then the frequencies decrease
> (or increase) if the integral of \Delta\epsilon |E|^2 is positive (or
> negative).  That is, the frequencies decrease if you slightly increase the
> *average* dielectric constant *weighted* by |E|^2.  (This is proved from
> perturbation theory.)
>
> Third, as a consequence of the second theorem, if you change the
> refractive index *everywhere* by a non-negative amount, the frequencies
> must decrease (or at least cannot increase).
>
> What does *not* follow is that increasing the index "on average", where
> the average is not weighted by n|E|^2, will necessarily decrease the
> frequencies.  In particular, if you increase the index in some regions and
> decrease it in other regions, the frequency could either increase or
> decrease depending on where the electric field is concentrated.
>
> Steven
>
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