Modeling Double Negative Materials

There has been growing interest in the electromagnetic behavior of materials that have both permittivity and permeability with negative real parts. This article describes how so-called Double Negative Materials (DNG) can be simulated in CST MICROWAVE STUDIO® (CST MWS) using built-in dispersive material models. As a test vehicle we use a dielectric slab illuminated from an angle of 20 deg by a Horn Antenna. The electromagnetic fields have been monitored at different frequencies from a single simulation. The problem has been reduced to a 2D problem by using electric boundary conditions on the top and bottom of the assembly.

Figure 1: Test vehicle for DNG simuations: DNG dielectric slab illuminated by a Horn Antenna

Time domain simulations are generally broadband simulations. All meta- and DNG-materials show frequency dispersion. To consider this frequency-dependent material behavior in broadband simulations, the most common models up to second-order dispersion are available in CST MWS. This includes relaxation and resonance effects as well as plasma or even gyrotropic media. In each case the microscopic material behavior is represented by a macroscopic description of the permittivity or permeability in the frequency domain. For the modeling of a negative real part in the dielectric function, the Drude (or plasma dispersion) can be used. The governing equations can be seen below. Hereby, the high frequency parameter limit is indicated by the infinity symbol. The plasma frequency (ω...

F) and the collision frequency (νc) are the so-called Drude Parameters.

Figure 2: Drude Dispersion Formulae

Fig. 3 shows the dispersion curve for this artificial material for a certain set of parameters. In this model, the magnetic and dielectric losses - influenced by the collision frequency - are very small. The real part for both, the permittivity and permeability, are negative for frequencies below 25 GHz. For the further discussion, we will observe the propagation of an electric wave at 3 distinct frequencies.

Figure 3: Dispersion curve for Drude Material

At 20 GHz, permittivity and permeability is -1. The wave impedance of such a material will be equal to the vacuum wave impedance of 377 Ω. As a result of this perfect match, any wave hitting the slab can propagate through the boundary without any reflection. However, due to the negative real parts, the phase velocity becomes negative inside the slab. Consequently, the phase will travel “backwards” in direction of the source. It should be noted that there is still energy transfer away from the source as can easily been seen by observing the power flow at this frequency.

Figure 4: E-Field for slab permittivity and permeability equal to -1

Figure 5: Powerflow for DNG Slab

At 24.56 GHz, the permittivity and permeability become zero. In such a medium the phase velocity becomes infinite. There is no “wave” visible inside then media. For a infinite thick slab total reflection would occur. As a result of the finite thickness of the slab, there is still an energy transfer through the media. It is interesting to observe that the excited wave resembles a plane wave.

Figure 6: E-Field for slab permittivity and permeability equal to 0

Above 24.56 GHz, the permittivity and permeability become positive but are still smaller than 1. Consequently the phase velocity in such media is faster then the speed of light. The visible wavelength becomes larger than the vacuum wavelength as can be clearly seen below.

Figure 7: E-Field for permittivity and permeability smaller than 1

This article shows the possibilities to model DNG Material with the time domain solver of CST MWS by using dispersive material properties. Even if the material used is purely artificial, simulations gain a valuable insight into the propagation behavior of electromagnetic wave through such a media.

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