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 _______________________________________________ mpb-discuss mailing list [email protected] http://ab-initio.mit.edu/cgi-bin/mailman/listinfo/mpb-discuss
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