On Mar 5, 2009, at 11:35 PM, Nathan wrote:
Is this true when just the complex part of epsilon is negative? I was aware of the instability for non-dispersive negative epsilon, but thought it was only for the real part.
When I said "negative epsilon", I was referring to a negative real epsilon; sorry, I didn't realize that this wasn't what you are doing.
A non-dispersive complex epsilon, regardless of the sign of the imaginary part, also leads to exponential growth regardless of the sign of the imaginary part. Analytically, if the product of the frequency [using the sign convention exp(-i omega t)] and the imaginary part of epsilon is negative, you have gain. In any time domain simulation, you cannot avoid having both positive and negative frequencies (even if you have a positive-frequency complex-fields source, you will get negative frequencies from numerical noise if nothing else).
Is this true when just the complex part of epsilon is negative? I was aware of the instability for non-dispersive negative epsilon, but thought it was only for the real part. I know the FD-TD method can model non-dispersive gain situations, but is there a limit on the value of the gain? I have found some work that suggests the time step must be reduced when modeling complex epsilon to ensure stability, but that didn't seem to fix my issue. I tried cutting the Courant number in half and the results were the same in my simulation.
I see that in your case, you actually want the gain. What I don't understand is this: you put gain into your simulation, so why do you think that exponentially-growing fields are incorrect? That's the usual consequence of having gain.
(In a physical system, the exponential growth is truncated because any real gain will saturate. It's possible to model such materials in FDTD: see http://link.aps.org/abstract/PRB/v73/e165125)
Technically, the question of whether the simulation is stable is separate from whether the solutions are exponentially growing once you have a gain medium, but this also means that you can't conclude that the simulation is unstable just because you see exponential growth in a gain medium. Rather, to check whether the simulation is numerically stable, you need to keep doubling the resolution and see if the solution blows up more and more quickly (= unstable) or blows up at a fixed rate (= stable, blow-up rate determined by the physical gain). I haven't really thought about this case myself, so I can't say offhand whether there is a Courant stability condition for FDTD in the case where epsilon has gain.
Steven _______________________________________________ meep-discuss mailing list meep-discuss@ab-initio.mit.edu http://ab-initio.mit.edu/cgi-bin/mailman/listinfo/meep-discuss
