Nanodisk array, Unphysical results from scuff-transmission

[gevero]

Hi

I am using scuff-transmission to compute transmittance and reflectance of a Germanium Nanodisk Array. The geometrical parameters are:

• nd_r = 1000nm (disk radius)
• nd_h = 1000nm (disk height)
• nd_d = 12000nm (square array pitch)
• theta=0.0 (incident angle)
• everything is in vacuum, and integration plane are ad +/- 10000nm

The calculation in itself is tricky, since the particle array supports an "in plane" diffraction order at around 12000nm, i.e. at a wavelength corresponding to the array pitch, appearing in reflection and transmission spectra as a sharp feature. As it is shown in the figures below, scuff-transmission returns unphysical results. They seem to be independent from the mesh or the mesh symmetry, or the interpolation parameter. The problems in the transmission and reflection spectra are as follows:

• results are unphysical (out of the 0,1 interval)
• TE and TM spectra are different while they should be equal for the square lattice
• It is not shown here, but spectra are a strong function of the energy flux integration plane position.

Most details of the simulations are here.

PS A simple Au ND array, with weakly scattering particles, without in plane lattice modes, gave correct results.

EDIT:

• The same problematic behavior appears for an array of strongly scattering Au Nanodisks
• If The Poynting vector is integrated far from the array, the spectra display an oscillatory behavior. If it is integrated closer, the oscillatory behavior is cured, even in the spectra are physically incorrect
• Closer integration planes imply finer interpolations (I do not know why, yet it seems so...), so may be in the end results are a function the interpolation parameter.
• work in progress...

[HomerReid]

Thanks for posting this detailed report.

As of fairly recently, SCUFF-TRANSMISSION actually computes the reflection and transmission coefficients r, t in two different ways: (a) by integrating the Poynting flux over the unit cell area, as always, and (b) using an approach that computes r,t directly from the surface currents.

The results of the latter computation are reported to the .transmission file on the same lines as the results of the former computation. For example (as stated at the top of the .transmission file), column 7 is the amplitude of the upward-traveling scattered TE wave excited by a TE incident wave, while column 9 is the amplitude of the upward-traveling TM wave excited by a TM incident wave. Columns 11, 13 similarly give the amplitudes of the downward-traveling waves. (The file also contains data on the phases of these amplitudes and on the cross-coefficients, i.e. the outgoing TE wave arising from incoming TM excitation.)

Note that these are the amplitudes of the upward-traveling and downward traveling scattered waves, which is almost but not quite the same thing as the transmission and reflection coefficients. The reason for this is that in cases where the uppermost region is connected to the lowermost region of the geometry (which is true in your example, but would not be true for example in a thin-film geometry) the incident wave propagates into the uppermost region. So the transmission coefficient t is actually 1+a where a is the amplitude of the forward-scattered wave, i.e. the number reported on columns 7,8 and 9,10 of the data file.

I just took a look at your .transmission file (thanks for posting it in your report), and the data on columns 7 and 9 don't display the pathologies you report: they are the same for the TE and TM cases and they don't appear to be nonphysical. Could you take a look and see what you think? It's possible the Poynting-vector integration is badly behaved in your case, while the direct calculation from the surface currents may be better behaved.

[gevero]

Hi Homer

Thanks as always for the very quick answer. Following your suggestion, I proceeded as follows. I used the coefficients computed with the surface current approach and defined my TE (the same goes for TM) transmission and reflection as:

• 
• 

Then I compared the results with the ones computed with two different modal methods, i.e. S4 and EMUStack. For Transmission spectra I would say that we are almost there... while the reflection ones are quite different. In this case I do trust the results of the modal methods, given that they are in quantitative agreement even if they use completely different approaches. The most outstanding problem is the fact that transmittance is greater than 1 at a given point, but I wonder if it is simply some kind of normalization or phase problem, because the physics of the problem seems to be captured here. As before most details of the simulation are here.

Thanks a lot