Simulation of Electromagnetic Waves with Sinusoidal Pulses on Metal through the Finite Difference Time Domain (FDTD) Method

Understanding electromagnetic wave physical phenomena in three dimensions can be challenging. This research aims to create a model for how electromagnetic waves travel through a three-dimensional medium. One of the best ways to visualize this type of electromagnetic wave is Finite-Difference Time-Domain (FDTD). The simulation results revealed that the properties of the dielectric material and conductivity affect the shape of the sinusoidal pulse propagation, where the electromagnetic wave experiences reflection and transmission. The properties of the material influence the electric and magnetic fields in various ways, affecting the interaction between the transmitted and reflected waves. This interaction is influenced by several factors, such as permeability, and leads to either constructive or destructive interference.


Introduction
James Clark Maxwell presented the theory that described how waves propagate in both a vacuum and metallic materials in 1865.Electromagnetic waves are a type of wave where their propagation direction is perpendicular to both the electric and magnetic fields.Faraday explained his theory that the flow of electricity changes with changes in the magnetic field.The dielectric constant of the medium in a vacuum or conductive material determines the magnitude of the electric field.Meanwhile, the magnetic field depends on the permeability of both the vacuum and the material.
Electromagnetic waves are a type of transverse wave that is invisible to the unaided eye, making them difficult to envision.In 1966, Kane Yee introduced the FDTD method to explain electromagnetic wave propagation in free space and became the first to provide visual representation [1].The FDTD method has a number of advantages, including that the algorithm is easy to understand and can be used in various domains, such as time, space, and frequency.Additionally, FDTD can be used to display electromagnetic waves in vacuum or conducting materials.The FDTD iteration approach employs discrete differential and integral equations.
The application of the FDTD method requires basic and in-depth knowledge regarding the physical phenomena to be visualized [2].Since 1920, research related to the FDTD method has been applied in various fields, including the propagation of electromagnetic waves, seismology, scattering and radiation, as well as acoustic waves in various materials [3].
Several previous studies related to FDTD have explored various aspects of FDTD, including plasma [4], crosstalk analysis [5], and electromagnetic waves on biconcave lenses [6].Additionally, the electromagnetic waves of organic solar cells can be analyzed using FDTD [7].Another application of FDTD is for scattering effects on gold and microwaves on hyperthermia media [8,9].Meanwhile, FDTD can also work in antennas [10] and the telecommunications industry for waveguides and cavities [11,12].
The application of 3-dimensional FDTD is required to increase understanding in studying the physical phenomena of electromagnetic waves in the medium.For this reason, implementing the Perfect Matched Layer (PML) theory [13] is required.In previous studies, most researchers used the FDTD method to evaluate one-and two-dimensional material properties.The use of FDTD to understand variations in conductivity and permittivity in three dimensions has not been applied in observing electric field phenomena in dielectric mediums.The research proposed in this study can be used to determine materials and identify characteristics according to the utilization of electromagnetic waves.

Method
In this research, the proposed electromagnetic wave simulation involved a visual presentation of Maxwell's equations in three dimensions, particularly focusing on absorbing boundary conditions (ABC).Maxwell originally formulated the differential equation in the form of a time-dependent electromagnetic radiation field [13], and the differential equation of a wave propagating in three dimensions can be derived and reduced as described in Equations ( 1) to (6). ) Equations ( 1) to (3) illustrate Gaussian divergence propagation concerning the relationship between the magnetic and electric fields in the Cartesian coordinate plane system.Meanwhile, equations ( 4) to (6) explain the divergence equation of the magnetic field intensity concerning time within Cartesian coordinates.The FDTD model requires selecting a wave propagation mode, such as transverse magnetic (TM) or transverse electric (TE), to create a three-dimensional simulation.
A similar thing related to discrete construction for time is written t = nΔt.Function equations, in general, for position and time discrete and the curl operator can be written with the Equation ( 7) & ( 8).

(
) ( ) On the other hand, the discrete representation of the differential equation of Maxwell's equations depends on position and time by using the curl operator, resulting in the simplification of Equation (8) to Equation (9).
The same is true for second-order equations on the y and z axes, where the magnetic and electric field patterns can be written down From equations (3) to ( 6), if written discretely, they are as follows: Note that the relationship between D and H in equation 10 is equivalent to the relationship in the spatial and time domains.The influence of material properties in the form of conductivity can be shown in equations (12) and (13).
In the FDTD simulation, it is essential to adjust the initial conditions, especially to ensure that the speed of electromagnetic waves in a vacuum does not exceed the speed of light.Determining the time domain interval uses the FDTD stability principle for n-dimensions which can be seen in equation (15), can be applied to overcome this problem [14].
In order to prevent the reflection of electromagnetic, it is necessary to implement boundary conditions for both electric (E) and magnetic (H) fields in n-dimensional.Mathematically, the electric field E FDTD technique is determined by considering the value of the magnetic field [15].The theory that states the boundary conditions is known as the absorbing boundary condition theory (ABC theory).
The computer's ability to determine the FDTD simulation speed needs to be established from the outset.The waves generated by the source are spread evenly in empty space, and this phenomenon has led previous researchers to develop the Perfect Match Layer (PML) theory to better understand the propagation of electronic waves [16].

Results and Discussion
Figure 1 presents a 3D visualization of electromagnetic waves as suggested by the researchers.Electromagnetic waves travel sinusoidally in both dielectric and conductive media.The initial procedure taken by the researcher involved configuring input parameters, including initial time, permittivity, conductivity, NStep, and pulse width.The initial parameter values for input are as follows: conductivity of 0.04 C/m 2 , dielectric constant of 4.2, NStep 100, initial time 20, and pulse width of 6.
The visualization of electromagnetic waves in Figure 1 shows that differences in the height of electromagnetic wave intensity are shown by color gradations.Electromagnetic waves are depicted in color contours of spatial propagation.This contour-shaped electromagnetic wave programming language can be written as follows: The sinusoidal pulses used are electromagnetic waves with an electric field amplitude of 200 N/C.Writing a programming language for pulse propagation in the electric and magnetic field loop process can be written as follows: For k = 1 to KE -1 For j = 1 to JE -1 For i = 1 to IE -1 dx(i, j, k) = dx (i, j, k) + 0.5 * (Hz (i, j, k) -Hz (i, j -1, k) -Hy (i, j, k) + Hy (i, j, k -1)) dy(i, j, k) = dy (i, j, k) + 0.5 * (Hx (i, j, k) -Hx (i, j, k -1) -Hz (i, j, k) + Hz (i -1, j, k)) dz(i, j, k) = dz (i, j, k) + 0.5 * (Hy (i, j, k) -Hy (i -1, j, k) -Hx (i, j, k) + Hx (i, j -1, k)) dx2(i, j, k) = dx2 (i, j, k) + 0.5 * (Hz2 (i, j, k) -Hz2 (i, j -1, k) -Hy2 (i, j, k) + Hy2 (i, j, k -1)) dy2(i, j, k) = dy2 (i, j, k) + 0.5 * (Hx2 (i, j, k) -Hx2 (i, j, k -1) -Hz2 (i, j, k) + Hz2 (i -1, j, k)) dz2(i, j, k) = dz2 (i, j, k) + 0.5 * (Hy2 (i, j, k) -Hy2 (i -1, j, k) -Hx2 (i, j, k) + Hx2 (i, j -1, k)) Figure 2 illustrates the sinusoidal propagation of electromagnetic wave pulses within the material, spreading uniformly in all directions.The propagation speed of electromagnetic waves in free space is comparable to the speed of light [17].When electromagnetic waves pass through a dielectric and conductive medium, the pulses exhibit a decaying pattern.In addition, there is also a wave superposition in the medium.The calculation results based on the input parameters provided show that the propagation attenuation value is around 29.54 rad/m and the attenuation is around 3.73 Np/m.This attenuation phenomenon occurs after electromagnetic waves pass through the surface of the dielectric medium, aligning with findings from previous studies [8].  Figure 3 visualizes the propagation of a sinusoidal pulse formed in an electromagnetic wave that is decaying along the direction of propagation.Figure 3 illustrates the occurrence of reflection and transmission, which is a consequence of changing phases in the Ex and Hy field components begin to change phase, and wave superposition occurs.The simulation results reveal that the distribution of sinusoidal pulses in the medium shows a decrease in the wave amplitude, consistent with the previous studies [16].The decrease in amplitude or intensity leads to the decay of the electric and magnetic fields due to the unequal and uniform superposition of waves in the medium.It is important to note that this specific phenomenon had not been previously documented in the previous studies [18].
Figure 4 describes the reflection of electromagnetic waves in the form of sinusoidal pulses which experience sufficient reflection and attenuation after NStep 57.The phenomenon of reflection of electromagnetic waves is influenced by the propagation and attenuation values of the waves as well as the superposition of waves due to the material on the wall.

Conclusion
This research involves the simulation of 3D wave propagation in a dielectric and conductive medium.
The simulation results reveal that the dielectric material properties and conductivity affect the shape of the sinusoidal wave pulse propagation, including both transmitted and reflected waves.The dielectric value affects the components of the magnetic and electric fields components in diverse directions as they interact between the transmitted and reflected waves.Consequently, due to permeability effects, the superposition of waves can vary, leading to both destructive and constructive interference patterns along the way.

Figure 2 .
Figure 2. Electromagnetic waves enter the medium.