Calculate complex reflection/transmission coefficients (S-parameters) and extract the effective metamaterial parameters (refractive index, impedance, permittivity, permeability). The simulation results are compared with the published results by D. R. Smith et al.

Minimum product version: 2019a r1

Updating the model with your parameters

Understand the simulation workflow and key results

Metamaterials are typically composed of sub-wavelength, periodic structures and can exhibit very interesting material properties such as negative refraction. In this example, we will calculate the S-parameters of double-split-ring resonators and extract the effective metamaterial parameters. Then, we will run another simulation using the extracted effective material parameters and compare its results with those of the original inhomogeneous metamaterial.

Instructions for running the model and discussion of key results

1.Open the simulation file, Metamaterial_parameter_extraction_Smith.fsp, and click the “Run” button.

2.Run the script file, Metamaterial_parameter_extraction_Smith.lsf, to plot some representative plots of the s-parameters as well as the effective material parameters.

S-parameters

The S-parameter results returned by the “grating_s_parameters” analysis group correspond to the complex reflection (\(S_{11}\)) and transmission (\(S_{21}\)) coefficients of the metamaterial alone, accounting for the phase accumulation due to the spacing between the source/monitors and the metamaterial boundaries. The dip in the phase of the \(S_{21}\) indicates the presence of a negative index band, which will be confirmed from the effective parameters below. Further information about S-parameter calculation can be found here.

We use the technique described in Smith et al. to extract the effective material parameters from S-parameter measurements. The following technique assumes the device behaves symmetrically for forward and backward propagations. According to the Eq. (9) of the Smith paper, the effective refractive index and the effective impedance can be calculated as follows:

$$n_{eff}=\frac{1}{kd}{cos}^{-1}\left(\frac{1\ -{{\ S}_{11}}^2+{S_{21}}^2}{2S_{21}}\right)$$

$$z_{eff}=\sqrt{\frac{{(1+{\ S}_{11})}^2-{S_{21}}^2}{{(1-{\ S}_{11})}^2{-{\ S}_{21}}^2}}$$

The effective permittivity and the effective permeability can be easily obtained from the following relations:

$$\varepsilon_{eff}=n_{eff}/z_{eff}$$

$$\mu_{eff}=n_{eff} z_{eff}$$

The following plots show the extracted effective material parameters from the simulation. Note that the effective refractive index is indeed negative at 8~12 GHz range, corresponding to the dip in the \(S_{21}\) shown above.

Note:

Determining \(n_{eff}\) and \(z_{eff}\) (and therefore \(\varepsilon_{eff}\) and \(\mu_{eff})\) unambiguously is a challenge due to the multi-valued nature of the complex inverse cosine function. Choosing a wrong branch will lead to incorrect results. The Smith approach in this example is not particularly robust in choosing correct branches, hence should be viewed as a starting point for your effective parameter extraction work, rather than something that will work for arbitrary metamaterials. For a review of the various extraction methods, see the “Additional Resources” section below.

1.Open the simulation file, Metamaterial_parameter_extraction_Smith_effective_material.fsp, and click the “Run” button.

2.Run the script file, Metamaterial_parameter_extraction_Smith_effective_material.lsf, to plot the reflection results from both the inhomogeneous and homogeneous (effective) metamaterials.

In this example, the reflection property of the metamaterial using the extracted material parameters is compared with that of the inhomogeneous metamaterial. The bulk effective material is created by using the magnetic electric Lorentz (MEL) model, which can analytically represent the complex permittivity and permeability. Although Smith's paper mentions that, for this example, the effective method does not rigorously satisfy an effective medium limit, a reasonably good agreement can be seen between the original (inhomogeneous) and the effective (homogeneous) metamaterials. The slight discrepancy between the two results can be mainly attributed to the simplification and adjustment we made when creating the bulk MEL material as described in following section. Since the main purpose of this section is to validate the extracted data, we did not pursue highly agreed results. With careful adjustment, you may get a better agreement with your own metamaterial design.

Description of important objects and settings used in this model

The operating frequency of this simulation is in the GHz range, which is very low compared to the range where the default settings in the software would work. Below are some of the key settings to be considered:

•Simulation time might need to be increased to allow enough time for the source to inject properly and the residual field to dissipate completely. For further information about the effect of simulation time on the simulation results, please see the Simulation time and frequency domain monitors page.

•Most metals can be replaced by a perfect electric conductor (PEC) in the GHz range, allowing relatively coarse meshes around the metal without compromising much on the accuracy of the simulation. So, the PEC material was used in place of the copper in this example.

For further details on low-frequency simulation settings, please see here.

The “grating_analysis group considers the distances between the source/monitors and the metamaterial to compensate the extra phase introduced during the S-parameter calculation. For a correct S-parameter calculation for the metamaterial alone, it is crucial to set the center and the span of the metamaterial in the Analysis – Variables tab of the analysis group.

In the second step of the simulations, the inhomogeneous metamaterial was replaced with a rectangle bulk slab, which again was assigned the MEL material. The extracted permittivity was entered as a sampled data in the base material “bulk” of the “mel” material. The magnetic part of the “mel” parameters were used to represent the extracted permeability with the analytical permeability of the MEL material.

Since the imaginary part of the extracted permittivity is negative, we chose a little wider absorption of the permeability for the "mel". We also chose a simple two-coefficient fitting for permittivity, which neglects the imaginary part and the anti-resonance of the real part. This was to avoid a diverging simulation due to its large imaginary part.

Instructions for updating the model based on your device parameters

1.The inhomogeneous metamaterial is currently composed of two split-rings on top of a substrate and a wire at the bottom. The relevant parameters are shown on the right and can be customized in the Properties tab of the structure group.

2.When using your own metamaterial, disable or delete the current metamaterial and update the center and span values in the “grating_s_parameters” analysis group with those of your new metamaterial.

3.The source is a child object of the “grating_s_parameters” and its wavelength range, polarization and injection angle can be set in the Setup tab of the analysis group.

4.When changing the wavelength of the source, the simulation time might need to be adjusted. A longer wavelength generally means a longer simulation time, but other factors such as the simulation span and properties of any resonances present can affect the required simulation time. Adding a time monitor and checking its time signal is recommended to ensure the simulation time is long enough for the source to inject its pulse and for the fields decay.

5.The example uses Periodic boundaries in the y and z directions since an infinite array of metamaterials is assumed. However, if you want to consider a metamaterial which is finite in a certain direction, you need to change the corresponding boundaries to PML.

Information and tips for users that want to further customize the model

The current example deals with a symmetric metamaterial, where the S-parameters are identical for forward and backward propagations \(({\ S}_{11}={\ S}_{22}\ and\ {\ S}_{21}={\ S}_{12}\)). When simulating a nonsymmetric metamaterial or a symmetric metamaterial with different background indices on opposite sides of it, the S-parameters can differ depending the propagation direction. The “grating_s_parameters” analysis group is coded to give correct S-parameter results for nonsymmetric structures as well. However, the provided script for effective parameter extraction would not work since the formula assumes a symmetric metamaterial (and background). For further information about effective parameter extraction for nonsymmetric metamaterial, please visit here.

If you want to simulate an active metamaterial whose refractive index changes because of charge injection, you need an additional simulation using the CHARGE solver to calculate the charge distribution for a given bias. For further information about active metamaterials, please refer to the Active Terahertz Devices example.

Additional documentation, examples and training material

Related KB pages:

•General methodology for metamaterial simulations

•Effective metamaterial parameter extraction

•Negative refraction in a bulk negative index metamaterial

Related publications:

•D. R. Smith, D. C. Vier, Th. Koschny, and C. M. Soukoulis, "Electromagnetic parameter retrieval from inhomogeneous metamaterials", Phys Rev E 71, 036617 (2005)

•Arslanagić, S., Hansen, T. V., Mortensen, N. A., Gregersen, A. H., Sigmund, O., Ziolkowski, R. W., & Breinbjerg, O.,“A review of the scattering-parameter extraction method with clarification of ambiguity issues in relation to metamaterial homogenization,” IEEE Antennas and Propagation Magazine, 55(2), 2013, pp. 91-106

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