Photonics – Chinese

Photonics – Chinese

之前在Lumerical 論壇上討論的帖子 Simulation of Modal Volume in plasmonic cavity

    • mwyu.pt06

      Dear Lumerical support team


      討論Purcell factor and mode volume 的計算在PLASMONIC CAVITY上

      該帖子也有討論下面這篇論文上的MODE VOLUME 以及 PURCELL FACTOR 的計算

      我想問一下 現在還找的到嗎?



    • Guilin Sun
      Ansys Employee



    • mwyu.pt06

      Dear 孫老師


      再點進 support registration



    • mwyu.pt06


      當我register for support

      出現support code

    • Guilin Sun
      Ansys Employee

      没有support code说明你不合格获得Premium support。我试一下能不能吧原来的帖子发这里。如果发不了,只好请你再重新开帖子提问了。


    • Guilin Sun
      Ansys Employee

      Simulation of Modal Volume in plasmonic cavity

      Hi everyone,

      I have a question about the simulation of confined mode volumes using the modal volume monitor in the FDTD solver. Although I get a reasonable curve for the Re(V), the values are odd. First of all, there is no hint in what dimension they are plottet. I assume it’s m^3 since it’s a volume, but if that was the case, the values would be several orders of magintude higher than what I’d expect from another modal volume calculation using the Purcell factor. On the other hand, I’m not too sure how big the monitor needs to be and what source parameters etc. I need to set in order to get valid results. The system consists of a spherical gold nanoantenna coupled to a gold mirror.

      cheers and thanks in advance

      Dear @mayer-martin

      Modal volume can be calculated using the equations described in the link below:

      As you can see, to calculate modal volume we need index and filed profile within each mesh cell. The results for modal volume are in um^3. Based on my understanding, modal volume results should be independent of source intensity or type of illuminations.

      Regarding the size of monitors, it depends on where you want to calculate the modal volume. If you have a cavity, I think that the modal volume has to be calculated over the cavity region (unless mentioned otherwise in your reference or paper).

      If your geometry is complicated, you can filter the cavity area using the index monitor results (this technique relies on index differences). Please see the link below on how we used this approach to calculate absorption only over a specific region:

      I hope this was helpful.

      Dear @bkhanaliloo

      Thanks for the response. I still have doubts that I’m doing the simulation right. When I use the Purcell Factor of a dipole in the cavity, it gives me a calculated effective modal volume of around 670 nm³. When I use the modal volume simulation, it gives me a modal volume of roughly 3.97e-13 nm³ at 539 nm. If I now calculate the effective modal volume by dividing the result through (lambda/refractive index)³, I get 6.95e6, so 6 orders of magnitude higher. I’m not sure what’s the problem, maybe I chose wrong symmetries etc… I attached the file with the system how I simulated it.

      CoreShell_4nm_gold_Modal_Volume_support.fsp (338.3 KB)

      Dear @mayer-martin

      Thanks for the clarification and simulation file.
      As you mentioned, there will be two approaches to calculate mode volume: directly using analysis group and Purcell factor .

      1. I am not quite sure why you include only half the sphere inside the mode volume analysis group? Also, since your structure contains areas outside the cavity, you need to limit your calculation only in the cavity region. Please see my previous post for more information.

      Moreover, when you use analysis group, did you use any apodization ? The idea here is to wait until injected pulse is dissipated and record only cavity modes.

      Please note that you need to watch for hot spots that may be created due to meshing in metal objects.

      1. Regarding Purcell factor calculations, what was the value for the cavity Q? Here is a good link on how to calculate quality factor:


      Dear @bkhanaliloo
      Thanks for the answer!

      1. I was not entirely sure how much spacing around the cavity I need to have to record a proper modal volume. I did not use apodization so far. I have no experience with setting up simulations for these mode volumes, so I guess I’m doing a lot fo things rather wrong. What would be an appripiate approach to setup the right monitor size etc?
      2. I used a rather simple approach calculating the FWHM of the Purcell Factor and used that for Q calculation. Not sure how exact this method was, but at least it gave a rather reasonable value in the right order of magnitude while I’m still struggling to get a proper result with the modal simulation itself. But I’d like to verify the Q factor calculation by using the latter simulation and since the values are so completely different I’m kind of stuck on this.

      Thanks for your support!

      Dear @mayer-martin

      Thanks for the clarification.

      I think it would be a good idea to start with a simple case. I do not have an example handy, but I was thinking to study a dielectric sphere in free space and compare the results from two different approaches. If this sounds good to you, can you please prepare the simulation file?

      Here are a few things to bare in mind regarding these simulations:

      1. To calculate mode volume by integrating over the field intensity, you need to capture only the field distribution at resonant mode. Thus, to exclude the incoming light intensity, you need to modify apodization settings of the monitors .
      2. To exclude the surrounding area from the calculation, you can filter the free space results. Please see the link below for more information:

      FDE Solver - Modal Analysis - Power Integration over a Triangle

      1. Purcell enhancement factor that you are referring from the KB page is the simplified version, and assumes that dipole is polarized along the electric field and is in resonant with the cavity. Please see this paper (Eq. 1) for more information:

      Resonant enhancement of the zero-phonon emission from a colour centre in a diamond cavity

      The benefit of using a dielectric sphere is that we do not need to worry about hot spots as well. It would be very helpful if you could find a reference that has the mode volume and Q factor reported.

      Please let me know of your thoughts.

      Dear @bkhanaliloo
      Thanks for the feedback.

      1. In the example on the kb, they use dipoles as sources and frequency domain monitors instead of the modal volume monitor. How should I adapt this to my setup since I’m using a normal incident beam and a modal volume monitor?
      2. Shouldn’t be the surrounding area be excluded by the way the modal volume is calculated? As far as I understand it, it integrates over the field and than divides by the maximum value?
      3. Both, the Pucell Factor as well as the modal volume with different approaches are shown here:
        They extract the modal volume from the Purcell Factor and also with a field integratin method and yield the same results.


      Dear @mayer-martin

      Thank you very much for sharing the paper link. It was quite useful.

      I got some initial results, which are in the same ballpark as the paper.

      First I tried to reproduce the Fig. S6 (supplementary material). I created a new simulation file, and added a dye layer in the gap region. From paper, I think the material with n=1.4 is covering the gap region rather than enclosing sphere. Does this make sense?

      This is the plot that I am getting for the Purcell factor (in the attached simulation file, disable TFSF and mesh-TFSF , and enable dipole, meshNP, and set y-min bc to PML) as a function of wavelength running a simple script below:

      lambda = linspace(0.55,0.75,200)*1e-6;


      purcell = dipolepower(f)/sourcepower(f);



      Plot is slightly different, but results are in the same order of magnitude. This might be due to dipole polarization or the geometry. You may need to perform convergence testing as well.

      Checking the supplementary material, I think their approach is more similar to Calculation Type 1 in this link:

      By setting the calc type to 1 from Analysis tab of mode-volume, here is the plot that I am getting:

      As you can see, I am getting a mode volume ~10-20 nm^3 at 640nm from both method.

      The mode volume box in my calculation is enclosing only the gold sphere, while a box of 82.5 x 82.5 x 68.4nm is used in the paper. It will be a good idea to check the calculations in the paper and also the box size for more precise results.

      Here is the simulation file attached for your review:

      GoldNP_Vmode.fsp (291.9 KB)

      I hope this was helpful.

      Dear @bkhanaliloo

      Thank you very much, this helps a lot! I just got some last general questions about the general setup:

      Our system actually consists from a goldsphere covered with dye material, but I guess for the mode volume it won’t make much different if you assume a layer on the film instead, what do you think?

      Why do you only enclose the goldsphere for the mode volume? The mode is occuring between goldsphere and film so I thought the actual cavity needed to be in the box. I guess I have a wrong idea of how this simulation works in detail.

      I saw you made an extra mesh with small mesh size for the dye layer. Was this for the purcell factor calculation? If I do a Purcell Calculation for the system with the CoreShell-Particle, I see additional peaks in the curve that should not be there. Might this be due to artefacts because I used to small mesh size (0.5nm)?

      Once again, thank you so much for your help!

      Best Regards!

      Dear @mayer-martin


      Our system actually consists from a goldsphere covered with dye material, but I guess for the mode volume it won’t make much different if you assume a layer on the film instead, what do you think?

      That’s what I expect, but I think it is a good idea to run the simulation to make sure that your intuition is correct.


      Why do you only enclose the goldsphere for the mode volume? The mode is occuring between goldsphere and film so I thought the actual cavity needed to be in the box. I guess I have a wrong idea of how this simulation works in detail.

      I think you are right and box has to enclose the gap region as well. In the paper, they are using even a larger box. It will be a good idea to run simulations with different sizes for mode volume analysis group and compare the results. They also point out some concern regarding the imaginary part of gold permittivity, and you may consider even modifying the calculations.


      I saw you made an extra mesh with small mesh size for the dye layer. Was this for the purcell factor calculation?

      When you have small features like this, you need to use a mesh override region to make sure that it is resolved properly. Please see the link below for more information:

      Generally, you expect the results to converge using finer mesh. Since gap is 0.9 nm, I recommend you to use finer mesh of (at least) 0.3nm to have a minimum of three mesh cells in the gap region.

      Okay, thank you once again.
      Seems like we’re getting there. I attached one last file for modal volume / Purcell simulations where I hopefully did everything right for the system as we actually wanna show it. Maybe you can give some last feedback/hints.

      For the Purcell Factor, I chose a mesh size for the NP of 0.1 nm while the mode volume are done with 2 nm mesh size since the simulation size would become extremely large otherwise. Does this make sense?


      CoreShell_4nm_gold_Modal_Volume_4.fsp (398.2 KB)

      Dear @mayer-martin

      I think you are on the right track. Since your gap is larger (4nm instead of 0.9nm in the paper), you can start with a mesh of 1nm in the gap region (only in the z-direction), and then use refined mesh over Silica (I think 0.1 nm will be super fine and may take a very long time to finish). You can also use conformal variant 1 when you have metal in your structure. Using symmetries would be quite helpful to speed up the simulations. Please compare your simulation file with the one that I sent you before for further considerations.

      Regarding the results, I expect you to obtain a lower Purcell factor and higher mode volume.

      Dear @bkhanaliloo

      Thanks for your support. The simulations are running now. I guess it’s simply given due to the rather weak definition of the mode volume that the results vary extremely. For calculation method 1 for the mode volume analysis, I get a rather sensible value (173 nm³), although also this value varies a lot if I change for example the auto shutoff. The other calculation methods give me values that are 5-6 orders of magnitude above. Is there any specific reason I could bring up why method 1 is most suitable for this system except for “It gives the most convinient values”?


      Dear @mayer-martin

      I went over the supplementary material of the paper, and I think none of the available calculation methods match the paper. I have made a mistake claiming that approach 1 is similar to the paper approach! I might have got excited when I saw results from approach 1 are matching the paper ones, and did not pay attention into the details.

      Here is a snap of the paper:

      As you can see, the authors had to use a more general equation to calculate mode volume. This equation takes into account the lossy dispersive gold material. The energy density (W(r,w)) has both the E and H fields, and mode volume is complex. This means that you need to start a fourth approach and introduce a new energy density identical to the paper ones, and do the calculation in the Analysis tab of the mode volume.

      Authors also mentioned that they have used a mesh accuracy of 0.3 nm. It is possible that they have used a mesh override over the entire simulation region to obtain a homogeneous mesh.

      Please take a second look at the paper and make sure that we are not missing any important details. For example, I did not know that the second approach in KB application example is valid only for a lossless and dispersion-less material.

      I hope this was helpful.

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