Dr. Johnson,
Thank you for your response. I believe my simulation is going unstable
not because there is gain, but because of it's magnitude, and the fact
that it doesn't die down as the source is turned off. I only have gain
for a very small piece of the simulation and when I launch a pulse at
this piece, the fields grow during the entire simulation, even after the
source is off. Considering the gain section is a low index material
sandwiched between two higher index slabs, the power should very quickly
go into the transparent higher index slabs. Also, the magnitude of the
gains is very unreasonable. I have seen gain on the order of 10^205.
That's a tad high, even for my toy problem.
I did a numerical experiment based on your suggestion for stability
though. I used ctl file I uploaded earlier and did three separate runs
of the same physical situation at three different resolutions. I
modeled a Gaussian source launched at 45 degrees from a slab n=3.22 onto
a trench 140nm thick filled with eps=1.89 eps"=-1 and I recorded the
transmitted and reflected flux. I recalculated the reference field for
each resolution befor doing the actual run. (That bit me once a few
years ago)
At resolution = 50 T-flux = 1.43E18 R-flux = 5.73E11;
at resolution = 100 T-flux = 9.02E11 R-flux = 1.10 E12;
at resolution = 200 T-flux = 3.8 E12 R-flux = 4.86 E12.
You notice that these numbers vary and don't seem to be converging in
any way. I've done some research and there seems to be a wide variety of
thoughts on numerical stability for FDTD and complex / dispersive
materials. I would like to look into this further with meep. Could you
please tell me specifically which numerical method is implemented in
meep to calculate the response to complex materials? I read through
"Materials In Meep" and it seems like you are using the Axillary
Differential Equation Method but there are a few variations on that that
will behave differently for stability. If you could please point me to
a paper that discusses the method implemented I would greatly appreciate
it. I'll make sure to update this thread with any results I get.
Best Regards,
Nathan Huntoon
Steven G. Johnson wrote:
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
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