Yatpac - The Ultimate Open Source TLM Simulation Package

Plane wave impinging onto a lossless dielectric material

1. Example description

In this example a plane wave propagating in free-space and impinging onto a lossless dielectric material is simulated. The plane wave is generated by exiting a planar rectangular cross section with a constant electric field component.

The plane wave propagates in a rectangular region bounded by a perfect electric conductor (PEC) on top and bottom, and with a perfect magnetic conductor (PMC) on the left and right boundaries. In this case the boundary conditions play the role of a periodic extension reproducing the free-space case.

To investigate the penetration of the wave into a lossless dielectric material, a material of dielectric constant eps_r = 4.0 is inserted inside the region in such a way that the plane wave hits this material at time step 700.

2. TLM model

The different material properties for free-space and the dielectric material are defined in the "C-Block" of the I3D TLM model file I3D.PW_lossless.

The first 6 parameters describe the physical dimensions of a part of space in terms of discrete coordinates. The following parameters define the material properties.

The first 3 numbers after the coordinate parameters are related to the permeability of the material and are set to zero here. The next 3 numbers are related to the dielectric constant eps_r of the material, and are calculated by

Y = 2*(2*eps_r - 2).
In our example we have eps_r = 4, therefore Y = 12. With these three parameters different dielectric constants can be defined for the three Cartesian directions x, y and z. In this way anisotropic material properties may be modeled. We consider an isotropic material and hence the three parameters are equal.

3. Simulation Results

3.1 Electromagnetic Field Visualization

As expected, due to the dielectric the incoming wave splits into a reflected and a transmitted wave. This can be seen on the screenshots form the visualization Fig. 1-2-3.

Fig. 1: Plane wave propagation before the dielectric.

Fig. 2: Reflected and transmitted plane waves.

Fig. 3: Reflected and transmitted plane waves after some timesteps.

3.2 Computed Time-Domain Signals

In Fig. 4, the field of the plane wave is measured in constant intervals along the propagation. On the left side of the dashed line we see the undisturbed Gaussian pulse propagating in the free-space. On the boundary (indicated by the dashed line at timestep 700) a certain amount of the wave reflected back with opposite sign. The yellow and the brown signals refer to the transmitted fields inside the dielectric.

Fig. 4: Time domain signals

In Fig. 5 we display the intensity of the electric field (E) and magnetic field (H) multiplied with the characteristic impedance of the wave (Z0) in free-space. As expected they are overlapped whereas the two reflected are separated since they have different polarization.

Fig. 5: Time domain signals of the intensities of E and Z0*H.

In Fig. 6 we display the intensity of the electric field (E) and magnetic field (H) multiplied with the characteristic impedance of the wave (Z_m) of the dielectric material In this this case the amplitude of the Z_m*H is half since the dielectric permittivity is eps_r=4.0. We note that in case of the dielectric material, the pulse is disturbed due to the TLM numerical dispersion. More details will be given in Section 3.2.

Fig. 6: Time domain signals of the intensities of E and Z_m*H. In the dielectric material.

In Fig. 7 the time evolution evolution of the Gaussian pulse is shown at two different positions (of relative distance Delta x) to evaluate the phase velocity in free-space. We may note that the simulated result corresponds to the the excepted analytical one given by Delta t = Delta x / co where co is the speed of the light in the free-space, Fig. 7. of the phase velocity is displayed in the free-space.

Fig. 7: Time evolution measurement of the plan waves to compute the phase velocity in free-space.

In Fig. 8 we have repeated the same in the dielectric. Also in this case the we may note that the Gaussian pulse is disturbed due to the TLM numerical dispersion, see Section 3.2.

Fig. 7: Time evolution measurement of the plan waves to compute the phase velocity inside the dielectric material.

3.2 Numerical Dispersion of the TLM Algorithm due to the Coarseness of the Mesh

In the following we display the effect of the numerical dispersion of the TLM algorithm due to the coarseness of the mesh. The results are shown for the free-space case and for the case of the dielectrics. The results are visualized for travelling Gaussian pulses representing the plane waves at different wavelengths and for a given minimum discretization size of Delta x.

In Fig. 7 we show the free-space case. As we may see from the simulation results, no numerical dispersion in the free-space is observed even for relatively low ratio between the shortest wavelength and the discretization step Delta x .

Fig. 8: Snapshot of tree different travelling Gaussian pules taken in the same position with different wavelengths, in the free-space case.

In Fig. 8 we report case of the wave propagating in the lossless dielectric malarial. In this case, the wavelength, due to the dielectric material of eps_r = 4, is half and the effect of the dispersion can be seen in Fig. 8.

Fig. 9:Snapshot of tree different travelling Gaussian pules taken in the same position with different wavelengths, with dielectric material.