This example has been updated. Find the latest version at Metamaterial Parameter Extraction - Smith.
We will calculate and plot the magnitude and phase of the the scattering (S) parameters for a negative index metamaterial. The metamaterial is comprised of a split ring resonator (SRR) and a wire and yields a band of negative refractive index at microwave frequencies. Simulation results will be compared to the results published by D.R. Smith et al.
The file s_parameters_effective_eps_mu.fsp contains the simulation of the structure shown in Figure 2 from Smith et al. As shown in the image above, the simulation contains a cubic unit cell of length 2.5mm. Periodic boundary conditions are used to extend the structure in the y and z directions, and a plane wave source operating at 5-20GHz is injected in the x direction.
The substrate is 0.25 mm thick and composed of FR4 which has a permittivity of 4.4 and a loss tangent of 0.02. A copper split ring resonator (SRR) and wire are positioned on opposite sides of the substrate. The width of the wire is 0.14 mm, and it runs the length of the unit cell. The outer ring length of the SRR is 2.2 mm and both rings have a linewidth of 0.2 mm. The gap in each ring is 0.3 mm, and the gap between the inner and outer rings is 0.15 mm. The thickness of the copper is given as 0.017mm in Smith, but since this is much smaller than the wavelength, we use a 2D sheet to represent it.
Since in the GHz range most metals act like perfect electrical conductors (PEC), the PEC material model is used for the copper elements. Also, note that the material fit for FR4 deviates from a straight line as given by the permittivity and loss tangent. This is due to the fact that materials cannot be fit by a straight line over the whole frequency range. However, this does not noticeably affect the frequency dependence of the S parameters. In addition, a straight line model for the material properties is not completely accurate since in practice FR4 material properties are frequency dependent.
The conformal variant 1 mesh refinement option is used in this example to take full advantage of the conformal meshing technology to accurately represent the ring widths. In addition, the autoshutoff min in the advanced tab of the FDTD region is reduced to 1e-7. This ensures that the fields decay sufficiently at the resonance frequencies before the simulation automatically shuts off. Relatively coarse mesh settings are used for demonstration purpose due to the properties of PEC. Results with finer mesh are presented in a later section on this page.
Using the parameter extraction techniques described in the Parameter extraction page, we will calculate the effective refractive index, and related properties, for this structure. First, let us check if the transmitted wave can be regarded as plane wave, as required by the parameter extraction analysis. The intensities of Ey field component from the T Monitor are shown below at two wavelengths:
Plane wave verification - abs(Ey)^2 from T monitor at wavelength 60 mm
Plane wave verification - abs(Ey)^2 from T monitor at wavelength 15 mm
It can be seen that the Intensity variation in the near field is only on the order to 1e-3, which can be considered as uniform, thus justifies the use of near field point monitor. The uniformity of the intensity can be further increased if the finer override mesh is extended.
When the simulation is finished, run the script in the S parameters analysis group to calculate the S parameters and reproduce figures 3a-3f from Smith et al.
S-parameters, index, impedence, permittivity, permeability plots from the s_params analysis group. These plots are obtained with finer mesh (dx=dy=0.03mm, dz=0.025mm in the mesh override).
Advanced note: Calculating the phase of the S parameters
The S parameter are defined assuming that the incident phase is 0 at the left edge of the substrate, and the reflected and transmitted phases are measured at the left and right sides of the substrate respectively. In the simulation setup, the source is placed at approximately -4.2mm, the monitor measuring reflection is at -5mm and the monitor measuring transmission is measured at +5mm. This results in a phase offset of the measured field compared to the desired result for the S parameters, however this can be easily corrected as long as the positions of the source, the substrate and the monitors are known. The script makes this phase correction but the user must enter some of the parameters to define the source position, background index and the position of the substrate.
To test both the amplitude and phase, the simulation file s_parameters_test.fsp can be used. in this file, we simply have a planar substrate with no metallic components so the S parameters can be easily calculated theoretically. After running this fsp file, the script file s_parameter_test.lsf can be used to compare the S parameters with the theoretical results. It will display the following figures.
We can simulate the electromagnetic response of the metamaterial using the extracted material parameters. Although Smith's paper mentions that for this example the effective method does not rigorously satisfy an effective medium limit, we can still get some results reasonably agreed with the original metamaterial which is inherently inhomogeneous.
Before setting up the simulation file, we need some analytical parameters from the extracted data in order to use the magnetic electric Lorentz (MEL) model since it has dispersive and lossy permeability. By some analysis, we can use the permeability in MEL model, which is relatively easy to have analytical expression, whereas for the extracted permittivity, we can import it into the material database, which is used as the base material for the MEL model. Some estimated parameters for the analytical permeability are listed below:
delta_mu = 0.6 (H/m)
wm = 6.1e10 (rad Hz)
delta_m = 1.43e9 (rad Hz)
To set up the simulation file with minimum effort, we can modify the structure in the original file to be a bulk slab with material of MEL created from above description. Since now it is a bulk material, we can simulate only in 2D. What we want to compare is its transmission and reflection, thus in the analysis group s_params, only R and T are kept and all others are deleted. You can run the file s_using_extracted_parameters.fsp which is ready to use.
After simulation, we can plot the results with the original transmission/reflection for comparison as follows:
From the above figure, we can see that the results using the extracted/simplified effective data agree well in certain degree with those using the original metamaterial.
Since the imaginary part of the extracted permittivity is negative, we chose a little wider absorption of the permeability for the MEL model named "mel".
Once the extracted permittivity is imported as the base material (named "bulk") of the MEL model, the default setting of the material fitting can lead to a good fitting. However, due to its large imaginary part, the simulation can diverge at late time. To avoid this, we simply chose the simplest two-coefficient fitting, which neglects the imaginary part and the anti-resonance of the real part. Even with such simplification, the result is reasonably good with the original transmission and reflection. Since the main purpose of this section is to validate the extracted data, we do not pursue highly agreed results. With careful adjustment, you may get a better agreement with your own metamaterial design.
For the magnetic electric Lorentz model, please refer the Material Database section in the Reference Guide to get more information.