effective refractive index

There is probably no "simple" way for you to do this, but there are some relatively straightforward methods. The question you are asking is referred to as "effective medium theory" or "homogenization" in the literature.

If your meta-material is a random collection of dielectric particles, you could look into Maxwell-Garnett or Bruggemann theories. These are very easy equations to use to compute effective refractive index (actually effective dielectric constant). I provide a simple summary of these in Chapter 13 of my dissertation. See pages 276 to 279.

If your meta-material is better described as a photonic crystal, there are several approaches you can take. The simplest essentially calculates a weighted average of the refractive index. These are accurate when the wavelength is much longer than the period of your lattice, but cannot always for the contribution of resonance to the effective refractive index. I do not discuss these approaches in my dissertation, but searching the literature for "effective medium theory" will turn up lots of papers for you.

The next level of complexity involves using a numerical model. For dielectric structures, the plane wave expansion method works well and there are some codes available online. I devote all of chapter 5 in my dissertation to this method. For metallic structures, the transfer-matrix-method will work well for you. In either case, your effective refractive index is determined by solving an eigen-value problem. I discuss this somewhat indirectly in my dissertation in section 2.4.3 (pp. 47-50) when I describe how to determine group and phase indices of refraction. The problem with this approach is that there is a built in assumption that the photonic crystal is of infinite extent. It turns out that the effective index of finite size devices changes a bit from the infinitely periodic case.

Perhaps the most complicated, but most accurate, is to apply a scattering numerical model such as rigorous coupled-wave analysis (RCWA) for dielectric structures, or the transfer-matrix-method for metallic structures. You model scattering from your device, then through equations or more modeling, determine what the effective index was. Think of it this way, keep replacing your structure with a homogeneous material and adjust the refractive index until you get the same reflectance and transmittance as you did with your original meta-material. I devote all of Chapter 6 to RCWA, but unfortunately do not discuss transfer-matrices or how to calculate effective index of refraction as I described here.

This discussion should get you started and hopefully doing a literature search with the right key words will turn up some good papers for you.

Good luck!
-Tip

P.S. If you are interested, my dissertation can be downloaded at:

https://www.edaboard.com/download.php?id=124081

Oh, that is very detailed answer.

In fact, my structures are rather simple. They are split ring resonators with rods coupled to transmission line. That's it.

I tried to implement the algorithm in this paper:
"Robust method to retrieve the constitutive effective parameters of metamaterials"
PHYSICAL REVIEW E 70, 016608 (2004)

without step D, becasue I didn't understand it very well.

Anyway, I'm getting very good results for the structures in free space (SRR-rod in free space). Yet, I'm getting questionable behavior for microstrip case (SRR-rod coupled to microstrip line).

Any hint about that?

After a quick skim of part D in this paper, I think by "branch" they mean there are multiple possible answers. I am guessing this because that is a problem I encountered some time ago when computing effective indices of dielectric photonic crystals and diffraction gratings.

It makes sense there are multiple possible answers for effective refractive index in thick slabs. the method described in this paper may be more efficient than what I did, but my approach was very intuitive for me and worked well. I started with a very thin slab where there would only be one solution. I then slowly increased thickness of the slab and tracked my original solution as I did this. When thickness was final, I had my answer.

You may want to start by calculating the transfer matrix (or scatting matrix) assuming your device is infinitely periodic in z. This, of course, will not account for finite slab thickness but it will give you an answer that is close and get you started.

-Tip

Added after 4 minutes:

I suppose I should ask: Are you modeling this device to compute effective indices, or characterizing in the lab with a network analyzer?

In fact, I simulated one cell of the structure to find out all the eigen modes.
So, I know already how should I get the refractive index for infinite structure.

Then, I simulated different number of cells. Furthermore, I fabricated all of them and measured them as well.

The S-parameters agreement is excellent between simulation and measurment.

I started to have problem in the retrieving the permibility and permittivity (the effective one). I see some inconsistency when number of cells are increase!

So, I started to doubt the retrieving algorithm, this is the complete story may be.

Is there any hint that might help ?

Hmmm...

What numerical method are you using to compute your eigen-modes and scattering analysis? Does the inconsistency get worse at you add cells? Are you using MATLAB? If it is a method I am familiar with, perhaps I could run your codes on my machine.

I'm using Microwave studio to compute the eigen-modes, and HFSS for the S-parameters.
MATLAB compute the parameters as mentioned in the paper I cited.
It soesn't matter if MATLAB or anyother program.
I compared the results with some published results and my code gave very accurate results.

That's why, I wanted to see if there is any other code to compare with mine for multiple cells. It seems, the problem starts with multiple cells choosing the appropriate root.

I see. Perhaps if you saved your structure to a data file, other users of HFSS could compute the s-parameters with that same tool to verify all is correct with your method. From there you can focus on processing the s-parameters. Perhaps doing a frequency sweep of the s-parameters could help you determine which root, or branch, is correct.

Good luck!
-Tip

Hi everybody

I am now working on the same problem. Did anyone find the refractive index, permittivity and permeability using HFSS for a metamaterial structure (SRR+wire)? If yes, please help me.

Thanks..

hi abuantenna
i simulate the paper:
"Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients"
D. R. Smith, S. Schultz, P. Markos and C. M. Soukoulis
Physical Review B, 65, 195104 (2002)

but i think there are some problem in my Matlab code or in paper dscription. if you extract the permitivity and permiability from the S-parameters, upload your Matlab code, plz.
regards

Can you post your examples in here?
Thank you!

Hmn...this topic is very interesting..I am new..nice to know u guys..

hi abuantenna

you mention u used microwave studio to compute the eigen modes

can you please let us know what is the procedure to that in microwave studio

and it would be great if you post your model or explanation of procedure to do it


regards

hi all,

it's great discussion.

i am also studying same prob. As abuantenna said, I also read "Robust method to retrieve the constitutive effective parameters of metamaterials" paper and "Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients".

i have already writen C-codes to retrieve effective parameters using Robust method. I also verified this method by some data from published paper elsewhere. It's work quite well. In my understanding, abuantenna obtained S-parameters from CST microwave studio (or Transfer matrix method or anything), and then from these results he can extract the effective parameters. It 's normal manner. Now i also using TMM code and CST to obtain S-parameters, but i met some problems in setting up the initial configuration as shahid78 asked. So you can help us by some explainaions about this procedure.

thanks rrumpf because of ur interesting discussion.

hi tungsin

im realy interrested that if you simulate :

"Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients"
D. R. Smith, S. Schultz, P. Markos and C. M. Soukoulis
Physical Review B, 65, 195104 (2002)

i do that and after writing the matlab code. there are some difference between mine and smith output!

Hi all,
I have redo the same work reported in this article "Electromagnetic parameter retrieval from inhomogeneous metamaterials" published in PRL by Smith in 2005. The results are the same. but the probleme is when I change the excitation (the wave propagation now is perpendicualr the particle plane under study) this method doesn't work anymore.So I moved to another method.
For people who used CST I would like to mention that they have to deembed Sij in order to get correct results. if the results are not deembed the retrieval method gives wrong results.

best wishes.

as I told, I used Robust method to extract effective parameters. I think it better than the method Smith mentioned in his paper (PRB, 65, 195104). I guess you used Sij for different thickness (L) to choose the correct branch of Re(n) (linear fitting of nkL versus L). Am I correct? So if I'm correct, we can discuss more.

in addition, you should think about problem helio1972 suggested. I remembered in PRB, 65, 195104 you cannot get any data to extract effective parameters. If you did Sij by CST with input parameters from his paper, you have to be careful to deembed Sij in order to get correct results (phase). If not, you result can be wrong. The best way is getting Sij amplitude and phase also from papers and using our code to retrieve. But I think you can not because you using Smith's method, you need many data with different thicknesses. Because the phase is very important. If you use robust method, you can. I also using Robust method to verify some Smith's results in PHYSICAL REVIEW E 71, 036617 (2005). Next time I can show you my reproducing to compare the output data in above paper.

hi all,

for everybody to be interested in retrieving effective eps and muy, i found that if you obtain eps and muy from CST data (Sij amplitude and phases), the results could be wrong because of phases. It should be inversed in some cases but actually I dont understand clearly.

I'm a newbie.
Can anyone explain for me?

thanks alot

hi tungsin
whould you please upload the robust method paper. i whould like to exmine that for my problem.
regards

ok i am going to upload some papers which i think to be important

first is robust method paper, second one is another paper that mentioned about the different calculation in homo and inhomogeneous structure. I think you can verify ur code by using data from second paper (both cases: homo and inhomogeneous structures).