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Reflectance greater than unity in high Q-factor metasurface

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I am simulating an unit cell of infinite array metasurfaces, aiming to achieve a high Q-factor at the resonance wavelength nearly 785 nm. Si3N4 is used for the nanoantennas and silica is for the substrate with a water medium covering the metasurfaces. I used a mesh size of 10 nm for the nanoantenna elements and extremely finer mesh for other domains. The reflectance value found here at the resonance wavelength is higher than unity. Ideally the value should be maximum of unity but never exceed the unity. I checked that different mesh sizes provides different reflectance values. I would be grateful if someone can help me providing useful suggestions avoiding such unrealistic values.


6 Replies Last Post 11.08.2021, 14:02 GMT-4
Robert Koslover Certified Consultant

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Posted: 3 years ago 10.08.2021, 17:27 GMT-4
Updated: 3 years ago 10.08.2021, 17:28 GMT-4

You said: "I checked that different mesh sizes provides different reflectance values." Well, that could be your red flag right there. if those differences appear significant, then you have a numerical issue. If you are sure there is nothing wrong physics-wise (but there may be) in your model, then consider using even finer meshes (especially in key locations) and/or higher order elements, or both. Finally, how close to unity reflection did you expect to see (before you did the Comsol model)? And how much did the reflection exceed unity? I prepare RF models all the time, and I also occasionally see unphysical results, such as reflected > incident power, or power output > input, etc. But (in most of my models) such errors are usually quite small and not representative of any serious issues. (For example, if you realistically anticipated, say, 99.98% reflection, but instead observed 100.02% reflection in the model, then it may be fair to say that the calculation worked relatively well.) So... just how far from the expected reality are your computed numbers?

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Scientific Applications & Research Associates (SARA) Inc.
www.comsol.com/partners-consultants/certified-consultants/sara
You said: "**I checked that different mesh sizes provides different reflectance values.**" Well, that could be your red flag right there. if those differences appear significant, then you have a numerical issue. If you are sure there is nothing wrong physics-wise (but there may be) in your model, then consider using even finer meshes (especially in key locations) and/or higher order elements, or both. Finally, how close to unity reflection did you expect to see (before you did the Comsol model)? And how much did the reflection exceed unity? I prepare RF models all the time, and I also occasionally see unphysical results, such as reflected > incident power, or power output > input, etc. But (in most of my models) such errors are usually quite small and not representative of any serious issues. (For example, if you *realistically* anticipated, say, 99.98% reflection, but instead observed 100.02% reflection in the model, then it may be fair to say that the calculation worked relatively well.) So... just how far from the expected reality are your computed numbers?

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Posted: 3 years ago 11.08.2021, 05:19 GMT-4

Dear Robert thank you for your reply. I beleive that there is no problem in physics setup. To validate my simulation setup, I reproduced some published results using the IDENTICAL model but at a different wavelength, ex. 1550 nm. The results were matched and the reflection never exceeds 100%. When scaling the size and wavelength, reflection goes wrong. According to the theory, my design should provide 100% reflection. It would be okay if I got slightly above 100% but here I got 153%. Strange thing is that changing 'extremely finer' mesh to 'extra fine' and 'finer mesh' results reduction of reflection from 153% to 89% and 82% respectively. Thus, for extra fine and finer meshs, the results are reasonable at least not unphysical. I expected to get a close to 100% reflection for user defined highly dense mesh. Changing the user defined mesh sizes (from dense to more denser) do not change the reflection. Can I share my model with you if you want to have a look? I am stuck.

Dear Robert thank you for your reply. I beleive that there is no problem in physics setup. To validate my simulation setup, I reproduced some published results using the IDENTICAL model but at a different wavelength, ex. 1550 nm. The results were matched and the reflection never exceeds 100%. When scaling the size and wavelength, reflection goes wrong. According to the theory, my design should provide 100% reflection. It would be okay if I got slightly above 100% but here I got 153%. Strange thing is that changing 'extremely finer' mesh to 'extra fine' and 'finer mesh' results reduction of reflection from 153% to 89% and 82% respectively. Thus, for extra fine and finer meshs, the results are reasonable at least not unphysical. I expected to get a close to 100% reflection for user defined highly dense mesh. Changing the user defined mesh sizes (from dense to more denser) do not change the reflection. Can I share my model with you if you want to have a look? I am stuck.

Robert Koslover Certified Consultant

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Posted: 3 years ago 11.08.2021, 10:59 GMT-4
Updated: 3 years ago 11.08.2021, 11:12 GMT-4

Well, I encourage you to post your model to the forum. Please select "Clear all solutions" and "Clear all meshes" to shrink the file, before uploading. There is a pretty good chance that someone at this forum will be able to find something at fault in your model. By the way, you mentioned that this is a high Q system, operating at/near resonance. Maybe that is the source of your difficulties. A sufficiently sharp resonance may sometimes be difficult to model properly, since very small changes to geometry (or to other factors that determine a resonance condition) can cause very large changes in behavior. As a result, the finite discretization that is always required in the numerical solution may introduce significant inaccuracy. Just a guess.

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Scientific Applications & Research Associates (SARA) Inc.
www.comsol.com/partners-consultants/certified-consultants/sara
Well, I encourage you to post your model to the forum. Please select "Clear all solutions" and "Clear all meshes" to shrink the file, before uploading. There is a pretty good chance that someone at this forum will be able to find something at fault in your model. By the way, you mentioned that this is a **high Q** system, operating at/near resonance. Maybe that is the source of your difficulties. A sufficiently sharp resonance may sometimes be difficult to model properly, since very small changes to geometry (or to other factors that determine a resonance condition) can cause very large changes in behavior. As a result, the finite discretization that is always required in the numerical solution may introduce significant inaccuracy. Just a guess.

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Posted: 3 years ago 11.08.2021, 11:30 GMT-4
Updated: 3 years ago 13.08.2021, 20:44 GMT-4

I have uploaded the model and this provides a sharp resonance at 785.15 nm. Regarding your guess, the resonance wavelength is extremely sensitive to the design parameters. Therefore, your assumption might be true. Surprisingly, the reflection above 100% only appears at the resonance wavelength and at other wavelengths it works fine. If this is the meshing problem, then can you please check my model and suggest how to avoid such problem?

I have uploaded the model and this provides a sharp resonance at 785.15 nm. Regarding your guess, the resonance wavelength is extremely sensitive to the design parameters. Therefore, your assumption might be true. Surprisingly, the reflection above 100% only appears at the resonance wavelength and at other wavelengths it works fine. If this is the meshing problem, then can you please check my model and suggest how to avoid such problem?

Robert Koslover Certified Consultant

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Posted: 3 years ago 11.08.2021, 13:44 GMT-4

Your model requires the Wave Optics Module to execute and I don't have a license for that, so I will only comment just a little more. I noticed that your model doesn't seem to have any lossy materials in it. So it would seem that the only place left for EM power to go is into your PMLs. If I were preparing this as an RF model, I would assign non-zero loss tangents (or, alternatively, non-zero conductivities, for dielectrics) and likewise non-infinite conductivities for conductors, which would better represent real-world materials and (inevitably) yield a reduced Q. I presume you can represent that physics in your case similarly by means of specifying non-zero values for the imaginary components of the index of refraction.

-------------------
Scientific Applications & Research Associates (SARA) Inc.
www.comsol.com/partners-consultants/certified-consultants/sara
Your model requires the Wave Optics Module to execute and I don't have a license for that, so I will only comment just a little more. I noticed that your model doesn't seem to have *any* lossy materials in it. So it would seem that the only place left for EM power to go is into your PMLs. If I were preparing this as an RF model, I would assign non-zero loss tangents (or, alternatively, non-zero conductivities, for dielectrics) and likewise non-infinite conductivities for conductors, which would better represent real-world materials and (inevitably) yield a reduced Q. I presume you can represent that physics in your case similarly by means of specifying non-zero values for the imaginary components of the index of refraction.

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Posted: 3 years ago 11.08.2021, 14:02 GMT-4
Updated: 3 years ago 13.08.2021, 20:44 GMT-4

Thank you for your suggestion and I will definitely implement that to see if there is any difference. By the way, I have attached the same model using RF module if you want to have a look.

Thank you for your suggestion and I will definitely implement that to see if there is any difference. By the way, I have attached the same model using RF module if you want to have a look.

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