On Aug 4, 2010, at 10:32 AM, Alexandr Sadovnikov wrote:
But now I suppose that I can use only the hard and unchangeable
formula of
the eps and mu dependencies on the frequency. I mean the formula
from the page
http://ab-initio.mit.edu/wiki/index.php/Material_dispersion_in_Meep
So if I want to sent the other formula of the frequency dependency
of mu
I should change the meep code and recompile it? Maybe it is only my
crazy idea...
To get any other formula, you can fit an expression of the form
supported in Meep to your desired frequency dependence, within your
bandwidth of interest. The more complicated the frequency dependence,
the more Lorentzian terms you will need in your (nonlinear) fitting
problem.
The basic issue is that Meep is a simulation in the time domain, not
in the frequency domain, so it cannot put in the frequency dependence
directly. Instead, in the time domain, a frequency-dependent
susceptibility is transformed into a convolution, and the trick is to
find a form of convolution that can be implemented efficiently with
minimal auxiliary storage. In Meep's case, the Lorentzian terms are
turned into a second-order ODE in the polarization, as explained in
the manual.
It turns out that the types of convolutions that are most efficiently
implemented in the time domain correspond to a frequency dependence
that is a ratio of polynomials in the frequency (a "rational
function"), so the trick is to fit things to this form.
This corresponds to a well-known problem in digital signal processing,
where a rational function in frequency corresponds to what is known as
a recursive filter or an IIR ("infinite impulse response") filter.
Hundreds of papers have been written on the general problem of finding
the optimal IIR-filter approximation to a desired frequency response
in a given bandwidth, what is known as a "filter design" problem.
Meep's sum-of-Lorentzians approach is actually an IIR filter in
disguise, a rational function in a particular factored form that is
motivated by the fact that a Lorentzian peak has a direct physical
interpretation in terms of coupling resonantly with a quantum
transition.
Steven
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