V&V Article 5: Modeling Dispersive Materials Using EM.Tempo

Introduction

In this article, we will demonstrate EM.Tempo’s ability to simulate models and structures containing dispersive materials.  As of writing of this article, some of the features discussed below are included in the current release (13.6), while others are scheduled for an upcoming release.

An Overview of Dispersive Materials

In a dispersive material, the constitutive parameters ε_r and μ_r of a material vary with frequency.  The parameters’ dependence on frequency can be due to the natural properties of the material, or dispersion can be used as a type of macromodeling to abstract the behavior of a certain structure.  The latter is often the case in metamaterials, where a structured array of unit elements made of well-behaved material collectively appear to exhibit dispersive properties.

The dispersive properties of a material are often discussed in terms of its poles.  In EM.Tempo, materials can be modeled using multiple electric Debye poles,

<img alt="debye pole" src="http://www.emagtech.com/images/debye.png" />

multiple electric Drude poles,

<img alt="debye pole" src="http://www.emagtech.com/images/drude.png" />

or multiple electric Lorentz poles:

<img alt="debye pole" src="http://www.emagtech.com/images/lorentz.png" />

EM.Tempo can also model uniaxial materials with both electric and magnetic dispersion based on any of the pole types listed above.  This model is required for representing double negative index materials  (DNGs) and other metamaterials.

For more technical information about dispersive materials in EM.Tempo, visit the relevant section in <a href="http://wiki.emagware.com/index.php/EM.Tempo#Dispersive_Materials">EM.Cube Wiki</a>.

Simulating Dispersive Models with EM.Tempo

In this section, we will present some examples of dispersive simulations using EM.Tempo.  Throughout this section, we will try to highlight some of the finer points required to obtain an accurate simulation of a dispersive model.

Drude Plasma Slab

For this experiment, we will use EM.Tempo’s periodic boundary condition to simulate a plasma slab with infinite extents in the x and y directions, but a finite z-thickness of 1.5 cm. The Drude model often provides a good abstraction for a plasma, so a Drude pole with ω_p = 1.803e11 rad/s, and ν_c = 2e10 rad/s is used as the dispersive model.

Here is an overview of the rest of the simulation parameters:.

• Project Frequency: 40 GHz
• Project Bandwidth: 60 GHz
• Grid Spacing: 0.1mm in all directions
• Termination Criterion: -100 dB

Note that, due to the infinite nature of ε as frequency tends toward zero in the Drude model, the excitation bandwidth is set up to avoid the very bottom of the band.  Also, since energy in the low end of the band may take a very long time to dissipate (the magnitude of the conductivity is very high here), we set a relatively stringent termination criteria. Figure 1 shows the results for the reflection coefficient of the plasma slab as computed by EM.Tempo and compare them to the data given by Ref. [1], demonstrating a very good agreement.

Jerusalem-Cross Frequency Selective Surface (FSS)

In this example, we will model the sandwiched FSS structure described in [2].  A Jerusalem-cross patterned PEC layer is sandwiched between two Debye layers, where the Debye pole has parameters τ = 5.27e-10 and Δε = 1.5.  A table of key simulation parameters is shown below.

• Project Frequency: 5 GHz
• Project Bandwidth: 10 GHz
• Grid Spacing: 0.1mm in all directions
• Termination Criterion: -50 dB

The transmission coefficient of the Jerusalem cross FSS computed by EM.Tempo is shown in Figure 3. The computed data matches the result from Ref.  [2]. These data are quite different from the transmission coefficient of a structure where the sandwhich layers have a frequency-independent permittivity ε_r.

Nano-Particle-Coated Solar Cell

First, we will demonstrate a periodic solar-cell structure. A silicon-dioxide substrate is coated with an array of patterned silver-particle cuboids, which can be modeled with a single Lorentz pole having parameters ω_p = 2.20254e12 rad/s, δ_p = 1.4e13 rad/s, and Δε = 4.8e7. The silver-particles increase the absorption of the solar cell[2]. Some notable parameters for this simulation are listed below

• Project Frequency: 500 THz
• Project Bandwidth: 500 THz
• Grid Spacing: 1.5nm in all directions
• Termination Criterion: -50 dB

For this simulation, it is important to remember that ε(ω) will have a resonant peak near ω_p = 2.20254e12 rad/s. The project bandwidth should be chosen to avoid this peak, since obtaining accurate results at these frequencies can be very difficult. The transmission coefficient for this structure is shown in Figure 5, where the results computed by EM.Tempo are compared with the results of Ref. [2], and a reasonable agreement is observed.

Left-Handed Metamaterial

Finally, we will use EM.Tempo’s ability to model electric and magnetic dispersion to create a double negative material (DNG), which will demonstrate a negative index of refraction.

To model this, we will create a uniaxial left-handed slab where all six available parameters are described by the same Drude pole, with ω_p = 2.67e11  rad/s, and ν_c = 1e5 rad/s.  This material will have ε_r = -1 and μ_r = -1 (for n = -1) at ~30 GHz.  The simulation result and environment is shown at left; note that a Gaussian beam obliquely incident on the dispersive slab will be used to verify the negative index of refraction.

Figure 6 shows the electric field distribution on a vertical plane passing through the center of the GND box. The visualized data clearly verifies the negative index of refraction.

References

[1]: K.S. Kunz and R.J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics.  CRC Press, 1993.

[2]: K. ElMahgoub, F. Yang, and A. Elsherbeni, Scattering Analysis of Periodic Structures Using Finite-Difference Time-Domain Method.  Morgan & Claypool Publishers, 2012.