calculate velocity

Hi, everyone! I want to get the group velocity of a pulse in FDTD calculation. I input a gaussian pulse into a waveguide and record the incident and output pulse in time domain. If the waveguide length is L, and the peaks of the two pulse is seperated by Δt, can the group velocity be calculated by vg=L/Δt?

If this method is right, should the incident pulse be narrow enough at frequency domain to garantee the accuracy of group velocity?

Hope someone can help me, thanks!

in principle yes.

mostly yes. It depends on the geometry/# of cells per wavelength lambda.

If you have a very fine mesh then pretty much every thing is ok.
But since FDTD suffers from numerical dispersion you have two problems.
1) the group velocity depends on the direction
2) the group velocity depends on lambda/cell size

So a pulse with narrow spectrum will lessen the effect of 2.
If the wave guide is straight then 1) will also not be too much of a problem.

Ideally you compute the group velocity for one frequency at a time using one of the
low-dispersion FDTDs

See for example

K. L. Shlager and J. B. Schneider, "Comparison of the Dispersion Properties of Several Low-Dispersion Finite-Difference Time-Domain Algorithms," IEEE Trans. Antennas Propagat., vol. 51, no. 3, pp. 642--653, 2003.

at

http://www.eecs.wsu.edu/~schneidj/journal.html

So the conclusion is that it depends on what you want to measure. If you want to get an accurate
group velocity measurement low-dispersion FDTDs are needed.

If you are interested in measuring the accuracy of standard FDTD then I would at least use pulses
with a very narrow spectrum

Thanks, iyami! Yes, you have given valuable suggestions. I'm simulating a slow light photonic crystal waveguide now. Because the group velocity dispersion is very strong in my case, I have to narrow the pulse spectra to get accurate vg. But this will longthen the computation time.
And I have read some paper which suggest to calculate the transmission spectra at first, and then make fourier transform to get the temperal light pulse. So the vg can also be calculated. What do you think about this method?

Actually there is one more problem. I assumed a non-dispersive material.
If you use a dispersive material then

1) group velocity != phase velocity

2) the method you use to deal with dispersion also matters.

How do you get the transmission spectrum without Fourier transforms?
If you excite a single frequency then amplitudes are ok, but if you use a pulse then
how do you get the spectrum of the transmitted/reflected signal without Fourier transforms?