## Photonics

Topics related to Lumerical and more

#### Speed of Light in FDTD

• Mikhailo Galinsky
Subscriber

I would like to draw your attention that the speed of light in Lumerical does not seem to be equal to the speed of light in textbooks. The latter is equal to 0.2997925 microns per femtosecond whereas the speed of light in Lumerical is equal to 0.2846045 μm/fs.

It can be checked, for example, using the following program parameters:

·         FDTD area x: [-20, 20], y: [-20, 20], z: [-2, 160], filled with vacuum

·         plane wave with wavelength 1 micron, pulse length 30 fs, offset 60 fs

·         boundary conditions are periodic (on all three axis)

·         simulation time is 1200 fs, steps 0.2 micron on all three axis.

·         Wavevector is directed along OZ axis. The source is deployed at z = 0.

The propagation speed can then be evaluated by computing the time needed for the pulse to cover the whole distance (162 micron).

In addition, I noticed that using the function “material etch” causes unexpected absorption, which may affect on speed measurement.

I would be highly grateful for your assistance on this issue. I am missing something here? Or is the speed of light really different in Lumerical? Is there a way to correct it to the right value?

Sincerely,

Mikhailo Galinsky.

• Guilin Sun
Ansys Employee

Hi Mikhailo, you have asked a very good and subtle question. I happend to be in developing FDTD algorithms years ago and would be happy to share some insights for you.

First, your test result that the simulated speed is slower than the physical speed is correct ! This is because, after discretization, it creates numerical dispersion, meaning v_simulated is not equal to v_theory. Please refer to this link: getnumericalpermittivity - Script command

You can read some FDTD books, or publications regarding to this.

The numerical dispersion is the theoretical error of the method, similar to other source of errors for other algorithm. It does not mean FDTD is bad. Every algorithm has numerical errors. The best part for FDTD is, as long as other settings are proper such as PML, the numerical disperison error can tend to be zero.

As for the pulse amplitude, because of the numerical dispersion which also causes group dispersion, the pulse is spreading wider at longger distance. So the amplitude becomes smaller, which is correct, and does not violates the energy conservation. If you check the field amplitude or power transmission, it will be correct, since the temporal part is normalized out. https://optics.ansys.com/hc/en-us/articles/360034394234-Understanding-frequency-domain-CW-normalization

I hope this helps.

• Mikhailo Galinsky
Subscriber

Hello!

I checked that the pulse speed indeed tends to the speed of light when dz tends to zero. Unfortunately, making the mesh fine enough to reach the correct propagation speed would lead to an estimated simulation time of 14 months for the relatively simple problem that I am trying to simulate. It would be appreciated if the algorithm could be improved to yield the correct speed for coarser meshes. Also, I suggest that this issue should be better documented, as I failed to find any helpful information in tutorials and forums.

Finally, my other question remains unanswered. If I understand correctly, the material “etch” is supposed to imitate the vacuum; the results obtained with “etch” and without any material should thus be equal. But they are not. Did I misunderstand something? Your help on this other point would be most welcome.

Thank you for your time and attention,

Sincerely,

Mikhailo Galinsky.

• Guilin Sun
Ansys Employee

Hi Mikhailo , This is a known source of error similar to other errors in other algorithm. Although there are many different suggestions for the improvement, the current algorithm is the only one that is robustic. Most importantly we are interested in the fields not the speed. If you want to correct it, you can do a post correction using the numerical dispersion relation, which is usually unnecessary. Many papers have been published to address this numerical dispersion. You can search "FDTD numerical dispersion" for more details. As far as I can tell, FDTD has been the most popular simulation method in computational electromagnetics for more than two decades with reasons.

I tested with the regular backgound material of n=1 and etch and found that they gave exactly the same result. Basically the "etch" is to use the material of n=1. Please use time monitor to compare the signals.