  # Normalized transmission for TFSF

1. What device are you trying to simulate? Include diagrams if available.
Optical filter.

2. What results are you trying to obtain? Be as specific as possible.
Transmission spectrum.

3. Description of the problem or issue.
How I can interpret the transmission response normalized to source intensity? The transmission response normalized to source intensity has similar, if not the same, shape of graph as transmission response normalized to source power. However, the unit is different. The transmission response normalized to source power is unitless while the transmission response normalized to source intensity is area. Here are my questions:
(1) How can I interpret the transmission response normalized to source intensity? For instance, what does it mean by 1.4 m^2 (cross-section) at 300nm? The green line and red line represent the cross-section areas of the sphere and TFSF source. Is there a meaning to the interception between the green line and cross-section?

(2) Another question is about the Lumerical software calculation. Is there a difference between the calculation done by Lumerical software between source power and power flowing through a monitor? The user can set the area for both. According to Lumerical, source power is proportional to the area of the source. In Lumerical, is the calculation of the power flowing through a monitor related to the area of the monitor?

(3) I am simulating an optical filter with a cavity where the light will be going through. If I adjust the area of the power source to be similar to the size of the cavity where the light will be going through, will I get a relatively accurate result of the transmission graph when I normalize the transmission response to the source power?

4. Lumerical product and software version.
Lumerical 2019b

1 Like

Hello @emh711,

Because plane waves have infinite extent the power of a plane wave is infinite as well, so it does not really make sense to normalize with respect to power when using plane wave or TFSF sources. In this case I would say the best way to think about the scattering cross-section \sigma would be as a proportionality constant relating the power scattered P_{scat} to the intensity of the incident light I_0: P_{scat} = \sigma I_0

I don’t think there is any significance to the intersection between the physical cross-section and the scattering cross-section.

The calculation of the power from a source and the power transmitted through a monitor is basically the same: integrating the Poynting vector across the area of the source/monitor. The exact formulas are given on these pages:

For plane wave or TFSF sources, increasing the area will increase the amount of light generated by the source, so the source power will increase. For finite sized sources, like mode sources or Gaussian sources, as long as the source area is large enough such that the field profile is low at the boundaries of the source, increasing the area of the source will not change the source power by much, because more light will not be produced by the source.

If you expand a monitor into a region where there is light, the total power passing through the monitor will change. In this sense, the power flowing through the monitor is related to the area of the monitor. However, if you expand the monitor into an area with no light, the power transmission of the monitor will not change.

The transmission result is related to the geometry of the source, so if you change the source the transmission will change. With more information I would be able to give more specific advice, can you please give more details on the cavity structure, and the type of source you are using?

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Thank you for answering my questions.
I am simulating nanohole. One set of simulations will be about a single nanohole while another set will be about a nanohole array.

I have a few more questions about my simulations:
(1) In the case of a single nanohole, I used TFSF as power source. So, for the transmission plot (800nm from filter), I should normalize it to source intensity. Is it okay if I normalize the transmission plot from nanohole array (this one, I used plane wave as power source) to source intensity as well?

(2) I did convergence tests by changing the number of PML layers, different mesh settings (i.e., mesh accuracy, mesh cell per wavelength), and inner mesh setting (dx, dy, dz). I got different results between mesh accuracy = 8 and mesh cell per wavelength = 60nm. If my interpretation is correct, the result from mesh cell per wavelength = 60nm should be more accurate. Is that correct? Another question is: What is the meaning of mesh cell per wavelength?

mesh accuracy 8. fsp
mesh accuracy 8. log

(3) What values of the x and y span of the nanohole array should I choose if I want to simulate the nanohole array as a single nanohole? Will an x and y span greater than 2000nm be large enough distance between the nanoholes?

Hello @emh711,

You could normalize the periodic results with respect to the source intensity, but I don’t think this would be a good comparison to the non-periodic results. It is difficult to compare periodic and non-periodic results as different quantities are usually being measured in each case. With the plane wave source in a periodic simulation, the total transmitted field in one unit cell is measured and normalized with respect to the incident power in one unit cell.

The TFSF source measures the scattered field, which is the difference between the total field in the simulation and the field that would be transmitted through the unpatterned substrate. This is usually normalized with respect to the source intensity to give the scattering or absorption cross-section. There is more information on the TFSF source on these pages:

“Mesh cells per wavelength” specifies the targeted mesh spacing for the automatic meshing algorithm with the following formula:
dx = (minimum wavelength)*(refractive index)/(mesh cells per wavelength)

A mesh accuracy of 8 targets 34 mesh cells per wavelength, so 60 mesh cells per wavelength would be expected to be more accurate.

This will depend on which source you are using and which results you are measuring. If you are measuring the power transmission with the plane wave source, as you increase the span of the unit cell the contribution of the single nanohole will become negligible, and the result will converge to the transmission of the unpatterned substrate. If you are measuring the scattering cross-section of the single nanohole with the TFSF source, then this result will not depend on the source span.

I hope this helps. Let me know if you have any questions.

I have a couple more questions.
(1) I am using a plane-wave source measuring power transmission. I increased the x span and y span from 2000nm to 3000nm. You can find the transmission plots below:

x span and y span = 2000nm

x span and y span = 3000nm

The peak positions seem to be at similar locations. However, the values of power transmission vary.

• How can I interpret the results that I get?
• Can I assume the result from x span and y span = 3000nm to be the same as the unpatterned substrate?

(2) I am trying to simulate this nanohole array example from Lumerical:
Nanohole Array • What is the meaning of normalizing transmission to the area of the hole divided by the unit cell area? • What is the meaning of R+T? • Is the result posted on the Lumerical website for the nanohole array (sp_array) accurate?

For my simulation, I was simulating only the nanohole array without the substrate.
I used a different setting from the original file. I used mesh accuracy = 8 instead of 2. Simulation time = 2000nm instead of 500nm. PML layer = standard 64 instead of a steep angle. The result that I got is slightly different from the original sp_array file. The peak shifted to the right.

Transmission plot from Lumerical setting (without substrate)

Transmission plot from the user-defined setting (without substrate) (mesh accuracy =8)

• Was the change a result of increasing mesh accuracy from 2 to 8?
• Can I see the result (from changing the setting) that I get as an accurate result?

(3)

• In terms of convergence test, does it need to be done for every single simulation? If I am varying a parameter of the structure (for example, nanohole cavity diameter), then will I need to do a convergence test for every time that I modify the diameter? Or, a single simulation setting that is found with a convergence test using nanohole with 200nm diameter can be applied to others (diameter = 50, 100, 300nm) as well?

• If I modify the structure from nanohole to nanohole with thin film, can the simulation setting that is found with convergence testing done on nanohole used on nanohole with thin film?