On Jun 2, 2008, at 8:36 PM, YC, Liu wrote:
> Hello, everybody:
>
> Nice to see this forum.
> I have question about the waveguide simulation using Meep, hope you
> can help me this out.
> I am seeking how to get the effective index (beta/ko) of the eigenmode
> of lossy waveguide by using
> Meep. For lossy waveguide, the effective index is a complex number,
> and how I can get both the
> real part and imaginary part of it by using Meep?
The best way is probably to get the complex frequency (the loss in
time) for a real beta and then convert this into a loss in space (the
complex propagation constant beta at a real frequency) by dividing by
the group velocity.
To get the loss in time, you make your computational cell a cross-
section of your waveguide (i.e. 2d for a waveguide with constant cross-
section), and set Bloch-periodic boundary conditions (via the k-point
input variable) -- this specifies your (real) beta.
To get the loss in time, you then treat it exactly the same as a
resonant-cavity problem (like the ones in the Meep tutorial): you hit
the system with a short pulse source, monitor the field at some point,
and then analyze the result with harminv ... all of which is done if
you call run-kpoints. This will give you the complex frequency omega
at the given beta, where imaginary part is the loss rate in time.
That is, you have omega(beta_r) = omega_r + i omega_i, where _r and _i
denote real and imaginary parts. Now, what you want to do is to get
the complex beta at the real frequency omega_r. This is given by:
beta(omega_r) = beta_r - i omega_i / v_g + O(omega_i ^2)
That is, to first order in the loss, the imaginary part of beta (the
propagation loss) at the real frequency omega_r is given just by
dividing omega_i by the group velocity v_g = domega/dbeta, which you
can get from the dispersion relation in the absence of loss. This
relationship is just a consequence of the first-order Taylor expansion
of the dispersion relation omega(beta) in the complex plane.
The above is only valid if the loss is small, i.e. omega_i <<
omega_r. This should always be the case in any reasonable waveguide,
where the light can travel for many wavelengths before dissipating/
escaping. If you have extremely large losses so that it only
propagates for a few wavelengths or less, then you would have to treat
the problem differently -- but in this case, the whole concept of a
"waveguide mode" is problematic, so I'm assuming that's not the case
for you.
Steven
_______________________________________________
meep-discuss mailing list
[email protected]
http://ab-initio.mit.edu/cgi-bin/mailman/listinfo/meep-discuss
