How to get the transmission phase shift at plasmonic waveguide-resonator systems?

Hello everyone!
I have a problem when I use FDTD Solutions to get the simulation of the SPPs structure presented in this paper: Plasmonic analog of electromagnetically induced transparency in multi-nanoresonator-coupled waveguide systems

the structure as shown in th picture:

The insulator is chosen to be Air. The metal is chosen to be silver whose frequency-dependent complex relative permittivity can be described by the well-known Drude model:

In the section II part D, the author realize the slow-light effect using the structure they designed. They described it by the group index ng. And in order to get the group index ,they use the equation:

Here vg stands for the group velocity in the plasmonic waveguide systems. τg and ψ(ω) are the optical delay time and transmission phase shift, respectively.
And they are shown in the follow picture:

Since I am a beginner of the FDTD Solutions,I don’t know how to get these results especially the transmission phase shift(because we can get another results from the equation if we know the transmission phase shift).
How can I get the transmission phase shift at plasmonic waveguide-resonator systems like this?
Thanks for your help!

Best wishes.
Yichen Ye

Dear @ycye0603

Thanks for reaching out. We have a section on the plasmonics in our KB page here. You can start from our simple examples there, and then you can start preparing the simulation file. Then we can discuss your results and hopefully we can reproduce these results.

To begin, maybe start with the supplementary file (link is provided on the caption of Fig.7 in the paper), and then prepare a simple file with only a straight waveguide (ignore the microdisks for now) on 2D. I hope we can get comparable results.

Dear @bkhanaliloo

Thanks for your reply. I have finished a simulation file to get the transmission and field distribution of |Hz| when the geometric parameters are set to be : d1 = 392nm, d2 = 398 nm, L1 = 280 nm, and D = 1 μm.
multi-nanoresonator-coupled waveguide systems.fsp (671.0 KB)

The results are shown as follow and are similar to the results obtained in the paper:

My problem is that I don’t know how to get the transmission phase shift like in Fig. 7. According to another topic:How to find phase shift at two sides of the rectangular resonator in FDTD solutions?, I know that we can get the phase shift by measuring the S-parameters through the mode monitor. So I want to know:
1, How to set the mode monitor to find the S-parameters in this plasmonic structure to get the phase shift? Since I couldn’t find the correct way.
2, How to obtain the relationship between the phase shift(units of π) and the wavelength just like in the paper( Fig. 7 (a)) based on the S parameters?
I know These are likely to be simple and basic questions. I wil also try my best to search for the answers through books and papers. And I sincerely look forward to your help and reply.

P.S. May I ask for your help to check out the .fsp file I uploaded to find out the mistakes and help me to optimize it? For example, in order to get the good results, the numbers of PML layers are set to be 200, will it be too thick?

Best wishes.
Yichen Ye

Dear @ycye0603

I am glad that you are getting similar results as the paper, that’s a good sign.

I think you can find your phase difference in two ways which would answer your questions (1 and 2):

a) set a Mode Expansion Monitor and two _frequency domain field and power monitor_s on the waveguide section of your structure. Mode Expansion Monitor can be placed anywhere on the waveguide section, and filed monitors are set at the location where you want to measure the relative phase (considered as input and output). Select the fundamental mode on the mode selection of Mode Expansion Monitor edit page and add these two monitors as is shown in the figure below:

Then what you need is to divide the output field to the input field. You can learn more about this technique here.

b) Second approach would be to use a line frequency domain field and power monitor along the x-axis (linear axis). This is similar to as profile monitor but records the electric and magnetic field along a line. You can choose the field component of interest (in your case Hz) and plot the angle. From this monitor you can simply extract the angle difference between two different position and for different wavelengths (choose a line plot and you can set distance or wavelength as your x-axis as is shown in the figure below). The Angle has the unit of radian in the visualizer.

  1. I think since your fields are attenuated enough before they reach the PML, you shouldn’t see much of divergence problem. I lowered the layer to default stabilized (42 layers) and simulation was still OK. Generally if you are not getting divergence problem, you should be safe in this case.

I hope this could help you to get your results.


Dear @bkhanaliloo

Hello, Thanks again for your painstaking reply and I know the way to calculate the S-parameters and how to find the phase difference using a line frequency domain field and power monitor. It’s all thanks to your help.

But I am so sorry to disturb you again. I still have problems about the results I got. Firstly, I find the S-parameters follow your advance and the example from the Component Tools Reference Guide. And the phase of S21 represents the phase shift of the electromagnetic wave.

My results are shown as follow:

This is quite different from the result in the literature. So then I tried the second method. I set a line frequency domain field and power monitor in the bus waveguide. And since the author said the length of the plasmonic system is 1000nm, so the x span of the monitor is 1000nm. The result is shown as follow:

It’s also different from the paper. I’m confused since I don’t know how to solve these problems. And when I check out my simulation file I was puzzled by the transmission of the power monitors Input and Output. Since the length of the system is set to be 1000nm, so X axis coordinates of the power monitors are set -500nm and 500nm, respectively.

We can see that the trasmission of the monitor Input is higher then the Output. In my opnion, it’s unreasonable. Is that the reason why I can’t get the correct results?

Can I beg for your help again to show me the correct results? Please have a look at my file and help me to do this simulation correctly.
multi-nanoresonator-coupled waveguide systems.fsp (5.5 MB)

Thank you very much!

Yichen Ye

I think the difference between the S-parameter method and the longitudinal field monitor occurs because when you used the monitor you didn’t subtract the phase of the input from the phase of the output.

Does this help?

Regarding the input and output transmission, I can see no problem. The power at the input should be higher than that at the ouput. The propagating plasmonic MIM mode is pretty much lossy. However, some of the input power is reflected back to the input. We can use the expansion monitor for the input port to investigate this. The forward transmission is almost 1 at the input. The backward, however, shows a resonance behaviour.

I plotted both the output spectrum and the reflected backward transmission below. I can see it makes sense.

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I read this post. But it does not exist a valid simulation result for phase.
I tested two ways but it is not similar to paper results.
Check it, please.

Yours sincerely.

What do you think about problems?


I’m interested in using FDTD to track the phase of a waveguide mode. This thread looked promising so I read through it and ran some simulations.

I took the ring resonator tutorial and only simulated a straight waveguide section to see how the S-matrix extraction technique matches up with the expected (calculated phase); that is, kL = 2pineffL/lambda. However, I can’t seem to get the two curves to match up. I thought this could have to do with the definition of “a” (the forward complex transmission coefficient), but I can’t find an analytical expression for “a” in Lumerical’s documentation.

I’ve attached the modified ring resonator code, along with a script I wrote to run the calculations and plot the curves for comparison. Please let me know if you can figure out why the simulated mode’s phase is different from the calculated mode’s phase, or if there’s an error with how I’m running this simulation.

(I know I have the option to do this with ports. However, I eventually want to simulate bent structures where the input and output are not in line with one another, in which case I won’t be able to use ports.)


ring_resonator (2)_strWG.fsp (545.8 KB)
PhaseCode3.lsf (694 Bytes)

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The difference between the calculated and simulated phase is usually attributed to the dispersion of neff. Did u consider in your calculations the fact that neff varies across wavelength?

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Yes I did. Unless the way I scripted it is incorrect, but I don’t think it


Update on this:

As Aya pointed out, dispersion of neff needs to be accounted for. The code I posted earlier takes this into account, but I hadn’t changed the expansion monitor frequency points to account for dispersion. The previous files I submitted are correct, except the number of frequency points needs to be increased (I like to use 100 points).

The “Simulated” and “Calculated” curves now match up.

Thanks Aya!



You are welcome, Peter. Let me know if you need any help.

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