The Simulation of the Propagation of Electromagnetic Waves through a Material with Various Thicknesses Using Finite Difference Time Domain (FDTD)

A finite difference time domain method in one dimension (1D-FDTD) is employed to study the properties of the propagation of electromagnetic waves across a medium with finite thicknesses. In this paper, we simulated the propagation of a Gaussian pulse. Since we simulated in one dimension, the absorbing boundary condition (ABC) can be applied to minimize undesirable reflection which generated at the numerical boundaries. We also implemented total field / scattering field (TF/SF) to ensure the source propagated in only one direction. The analysis performed by inserting the object with both relative permittivity and permeability values higher than one decreased the pulse’s amplitude.


Introduction
Finite difference time domain (FDTD) was the most popular method which was employed in simulating electromagnetic waves for handling various problems.This method had been used in analyzing many topics, such as designing antennas (Gao et al., 2005;Lee et al., 2004), simulation electromagnetic waves in a transducer (Xie et al., 2016), numerical modeling of ground penetration radar to detect landmines (Giannakis et al., 2016) and electromagnetic waves propagation in human tissues (Mirza, et al, 2015).This method was firstly proposed by Yee in 70's (Yee, 1966).The basic concept of FDTD was solving Maxwell's equations, especially the Ampere law and Faraday law numerically.It required spatial discretization of both Electric and magnetic fields along with the temporal discretization of time propagation.Here, the Yee's lattices were based on second order of central difference.Depended on the requirement, we were able to derive the numerical equations for FDTD in 1D, 2D and 3D.Since the purpose of this paper was to analyzing of the effect of object thickness to the propagation of the electromagnetic waves here we derived the simple 1D FDTD numerical formulations.Then, we tried to analysis the propagation properties of the electromagnetic waves when the different medium with certain values of permittivity and permeability .We also looked at effect of the inserted medium's thickness when the electromagnetic propagates through that inserted medium.

The method and formulation
In this paper, since we focused on 1D, we had to reduce the 3D Maxwell equation to become only 1D Maxwell equation, such as which was only involving field components and and pulse propagated at the ̂ direction.Based on that equations (Eq.( 1)) and (Eq.( 2)), the coding for basic simulation for updating and at the free space can be written as for i = 2:Nz Ex(i) = Ex(i) +(dt/ep)*(Hy(i) -Hy(i-1))/dz; end where represented total spatial grid along the propagation.Spatial grid and temporal grid obeyed Courant condition with .Here mu and ep were permeability and permittivity of the medium where electromagnetic waves propagated.
Since we simulate 1D electromagnetic waves propagation, it was appropriate to only implementing absorbing boundary condition (ABC).Using also a gaussian pulse as a source, the coding in Eq.( 3) should be add Ex(src)=Esrc; Hy(src)=Hsrc; for the source.Here, Esrc and Hsrc were electric and magnetic fields source with A was the amplitude of magnetic source Hsrc.Parameter delt was the delay between the electric and the magnetic pulses.Parameters t0 and tau were time delay and the width of Gaussian pulse.Here, src was the grid location of the pulse's sources.The ABC was applied by adding the code H2=0; H1=0; E2=0; E1=0; Ex(1)=E2; E2=E1; E1=Ex(2); (5) Hy(Nz)=H2; H2=H1; H1=Hy(Nz-1); Here, was the total number of spatial grids.Unnecessary reflection was minimized using scattered/total field (TF/SF).Implementing this method, resulted in only one direction of the pulse's propagation.Then, we modified the update of and fields as The simulation was performed by implementing ABC, Gaussian source and TF/SF source in FDTD as it was discussed above.

Results and Discussion
In this report, the medium surrounding was free space.Then, we inserted the object which had different values of permittivity and permeability.Here, we used and The result with object thickness was 40 grids (around 78 nm) was presented in figure 1.The source was generated at grid's number 30 in free space., that the amplitude and the width of the Gaussian pulse decreased when it propagated through the object.This was because the medium of the inserted object had the values of permittivity and permeability higher than that that of its surrounding which was vacuum (free space).It meant that the density of the object was higher than the density of the free space.When the pulse propagation was captured at 750 ps (see Fig. ) was also having similar trends.It seem that increasing the width of the object was not significantly changing the profile of the pulse.We thought that this happened since we treated the object here as lossless materials.Hence, the thickness of the object became unimportant parameter.

Conclusion
The propagation of electromagnetic waves through the rectangular object had simulated using 1D-FDTD.We also noticed that the width of the rectangular object did not significantly affect the profile of the pulse of electromagnetic waves.

Figure 1 .
Figure 1.Propagation of the pulse at several time capture.In (a), propagation at 250 ps, In (b), propagation at 500 ps, In (c), propagation at 750 ps an (d) the propagation at 1000 ps.

Figure 2 .
Figure 2. Propagation of the pulse at several time capture with the thickness of tectangular object was 60 grids.In (a), propagation at 500 ps, In (b), propagation at 750 ps, In (c), propagation at 1000 ps an (d) the propagation at 1125 ps.