Higher Order diffraction artifact in spectrum for Nanowire Arrays

Hello everyone,

I encountered some problems when simulating the reflectance/transmittance of ordered GaN Nanowire (NW) arrays. The NWs have a fixed diameter and length and are arranged in a periodic hexagonal lattice. Here i´m simulating over a unitcell containing 2 NWs with periodic BCs in the x and y direction and PML on the z axis.
The NWs are illuminated with a plane wave source (375-800nm) at normal incidence.
Now because the NWs behave like a grating there are some artifacts occuring at certain wavelengths, in dependence on the distance between the NWs, and the location of this artefact can be calculated before the simulation as shown here (DOI: 10.1021/nl200201b) which occours in my case at around 500nm Wavelength for a distance of 1000nm between the NWs.

Now to my Question:
Is there a way to reduce the error (which is exagerated by the FDTD algorithm) or in the best case remove it completly?

The problem is that it seems that the spectrum obtained before or after the artifact is not a continous function if you exclude the artifact in many cases. It looks more like a step which can extend over a range of ~100nm wavelength.

Thank you in advance for your time!

Best regards

(Exemplary Graphs)

Reflection from above the NWs inside with the Source above the NWs. Here one can see the step from ~400-500nm.

Transmission from below the NWs inside with the Source inside the Substrate to the top. Here no step is visible and the function appears to be continous.

Hi Richard,

Are you simulating P-polarized light? If so, the discontinuities could be physical effects known as Wood’s anomalies.

It may not be possible to completely get rid of the discontinuities if they are physical, but you can improve the performance of the PML boundaries above and below the structure by using the “steep angle” PML profile and making sure that the PML is at least half a wavelength away from the grating structure to ensure that there is no artificial coupling between the structure and PML.

Hopefully this helps!

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The PML boundaries are at least half a wavelength away from the structure and i already tried all different kinds of PML settings and layer constelations but with no success.

Here are two images of the setup:

From the top the FDTD volume spans over on hexagonal unit cell with.

and the setup from the side:

The source is neither s nor p-polarized because the angle of incidence is 0.
The graphs only change slightly when sweeping the polarization of the source.

Is it correct to use Periodic BCs in this case? Or do i have to use Bloch BCs?

I have some early measurements for the reflection which don´t show these anomalies mentioned in my last post.

The strength of the artifact also depends heavily on the substrate i use: If the substrate and the nanowires are made out of the same material the artifact is sometimes less visble in comparison to fo example a sapphire substrate under the GaN nanowires.

Thank you for your answer!

Best Regards,

Dear Richard,

From the screenshots I don’t see any problems with the setup.

Periodic boundary conditions can be used for the source at normal incidence, and Bloch boundaries only need to be used if the source injection angle is non-zero.

One thing you may want to try is to see if the results change when you run the simulation for a longer time. You may need to lower the auto shutoff min threshold which is set in the Advanced options tab of the FDTD solver region object to do this.

In your previous post you mentioned that the location of the anomaly can be calculated before the simulation, so I am assuming that it is due to a physical effect. I wasn’t able to access the article that you referred to - do you know if the spectrum is supposed to be continuous before and after the anomaly?

I would suspect that if you have tried increasing the number of PML layers and running the simulation for a longer time, and the results are not changing, then the result might be physical at the frequencies other than the points where the anomalies occur and the discontinuity before and after the anomaly might be expected. Please let me know what you think.

Hello nlui,

I tried it with a way longer autoshutoff and PML boundaries atleast 2um away from the structure and nothing changed.
I allready thought about it beeing a physical effect due to grating orders but i´m currently looking into it and it looks promising at the moment.

I sadly don´t know if the spectrum is supposed to be continous… It was just a guess.

But i have two other question if you don´t mind:

If i want to simulate the inclination angle dependence, do i have to use Bloch BCs on all 4 sides?
I tried using a BFAST source but my computer is pretty slow so it would take “forever” to finish.
I decied to only look at certain frequencies of interest without a broad band source (so only a single wavelength). In that case the normal source should be sufficient?

I added anti-symmetric BCs to the x axis and symmteric BCs to the y axis while allowing symmetry on al boundaries.
This should drastically reduce the simmulation time?
The graphs are unchanged so there should be no problem with that aslong as i dont change the polarization angle and inclination angle?

Thank you very much for your Help!
I really appreaceate it!

Best Regards,

Hi Richard,

Bloch boundaries only need to be used if the source is angled in the direction of the boundary. For example on the image of the setup at the top of the following page, the source is only angled with respect to the x boundaries and not the y boundaries so Bloch boundaries can be used in the x-direction and periodic boundaries (or periodic symmetric boundaries) can be used in the y-direction:

Running single wavelength simulations using the Bloch/periodic type plane wave instead of BFAST will give accurate results.

Regarding your second question, you’re correct. The boundaries that you mentioned will give equivalent results and reduce simulation time for the normal incidence source with the source polarization that you showed in the previous screenshots of your setup.