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Motivated by the emerging field of plasma antennas, electromagnetic wave propagation in and scattering by inhomogeneous plasma structures are studied through finite-difference time domain (FDTD) techniques. These techniques have been widely used in the past to study propagation near or through the ionosphere, and their extension to plasma devices such as antenna elements is a natural development. Simulation results in this work are validated with comparisons to solutions obtained by eigenfunction expansion techniques well supported by the literature and are shown to have an excellent agreement. The advantages of using FDTD simulations for this type of investigation are also outlined; in particular, FDTD simulations allow for field solutions to be developed at lower computational cost and greater resolution than equivalent eigenfunction methods for inhomogeneous plasmas and are applicable to arbitrary plasma properties such as spatially or time-varying inhomogeneities and collision frequencies, as well as allowing transient effects to be studied as the field solutions are obtained in the time domain.

A plasma dipole is an antenna with a radiating structure based on a plasma element instead of a metallic conductor [

These advantages can be employed in solutions to one of the great challenges in the synthesis of antennas for telecommunication links, which is the lack of flexibility in changing parameters, like, for example, the radiation pattern, once the antenna is deployed. This problem can be avoided by application of antenna arrays, where control over the excitation of each element allows the radiation pattern to be conformed. These antenna arrays are particularly interesting for use in reconfigurable antennas in mobile communications and satellite link applications [

Despite the advantages in utilizing plasmas as conducting elements in antennas, there are still several difficulties and obstacles for this technology. Obtaining the real characteristics of a plasma antenna, in a generic situation, requires the complete description of the plasma configuration, which creates theoretical and numerical difficulties when trying to analyze such systems, as outlined in [

The theoretical investigation of electromagnetic field behavior within a cylindrical inhomogeneous plasma structure is usually carried out through eigenfunction expansions [

In this paper, a simulation scheme based on FDTD techniques is proposed to investigate the problem of electromagnetic propagation within arbitrary inhomogeneous plasma structures appropriate for telecommunication applications and the waves scattered by such structures. Such simulations have already been successfully applied to the study of electromagnetic propagation through or near the ionosphere [

For the ease of comparison with results from the literature, the plasma under consideration is inhomogeneous, cold, unmagnetized, and collisional. The plasma is also assumed to be confined to a cylindrical structure and under illumination from a transverse magnetic (TM) plane electromagnetic wave with electric field parallel to the cylinder’s axis and in a situation of normal incidence, as shown in Figure

Plane wave incidence on the plasma cylinder.

The remaining of this work is organized as follows. In Section

The electromagnetic wave propagation in a cold, collisional, isotropic plasma with no background magnetic field can be characterized by the relative permittivity
_{p}_{0} is the permittivity of free space.

Equating (_{ef} and associated plasma current density in the frequency domain to be found as

Applying an inverse Fourier transform and some algebraic manipulations allows (

The relevant equations that govern the evolution of the electromagnetic fields within the plasma are then given, in the time domain, by

The FDTD technique consists of discretizing Maxwell’s equations through the use of finite difference approximations for the derivatives. This section will provide the basic characteristics of the FDTD method utilized in this work.

The basic scheme laid out by Yee in his seminal paper [_{x}_{y}_{t}_{t}_{s}_{x}_{y}_{t}_{x}_{y}

Electromagnetic fields are spatially discretized over a finite Cartesian grid as per Figure

Generic Cartesian spatial cell (

This term can be efficiently handled by means of an auxiliary differential equation (ADE) formulation [

The update equation for the electric field requires knowledge of the plasma current at time step

These update equations naturally take into account any spatial variations in both the electron density and the collision frequency, as the values for each are different at each computational cell.

The incident wave under consideration for the FDTD technique is a plane wave propagating towards the right (positive

Total-field scattered-field technique used to generate the incident fields and calculate the scattered far fields.

Region separation for the TFSF technique. Region 1 consists of total fields while region 2 consists only of scattered fields

Detailed view of the field components adjacent to the TFSF boundary. A computational cell at the corner of the TFSF boundary is shown

Absorbing boundary conditions (ABCs) based on one-way wave equations [

The FDTD technique provides a time domain solution for the electromagnetic fields, but the frequency domain solution can be obtained by an efficient implementation of in-step Fourier transforms on the fields [

With the steady-state fields calculated as described in Section

Equivalence principle.

Fictitious boundary for the problem at hand. Fields inside and outside the boundary are continuous across it

Equivalent problem with fields inside the fictitious boundary set to zero and material properties set to that of free space

With the equivalent electric and magnetic currents at the fictitious boundary, the electromagnetic potentials for a two-dimensional problem can be written in terms of Green’s functions as

Additionally, by considering an observation point in the far field, the asymptotic expression for the Hankel function can be used, resulting in, for the

To validate the FDTD technique, it is applied to a homogeneous plasma and the solution compared with solutions obtained from eigenfunction expansions, which can be efficiently calculated for homogeneous cases. The simulation parameters are as follows: time discretization ∆_{t}^{−12} seconds, spatial discretization ∆_{s}^{−3} meters, maximum temporal step _{t}_{x}_{y}

Figure

Time evolution of the intensity of the electric field, in volts/meter, across the computational domain, for the homogeneous case where incident wave frequency is set to _{in} = 10 GHz, electron density is set to ^{17} m^{−3}, and electron collision frequency is set to ^{6} Hz.

Figures

Comparison between the magnitude of the electric field, in volts/meter, within the cylinder obtained by the eigenfunction method and the FDTD simulation for different incident frequencies; electron density is set to _{0} = 5 × 10^{17} m^{−3}, and electron collision frequency is set to ^{6} Hz.

Eigenfunction, _{in} = 5 GHz

FDTD, _{in} = 5 GHz

Eigenfunction, _{in} = 10 GHz

FDTD, _{in} = 10 GHz

Eigenfunction, _{in} = 15 GHz

FDTD, _{in} = 15 GHz

Comparison between the magnitude of the electric field, in volts/meter, within the cylinder obtained by the eigenfunction method and the FDTD simulation for different electron densities; incident wave frequency is set to _{in} = 10 GHz, and electron collision frequency is set to ^{6} Hz.

Eigenfunction, _{0} = 5 × 10^{17} m − 3

FDTD, _{0} = 5 × 10^{17} m^{−3}

Eigenfunction, _{0} = 1 × 10^{18} m^{−3}

FDTD, _{0} = 1 × 10^{18} m^{−3}

Eigenfunction, _{0} = 5 × 10^{18} m^{−3}

FDTD, _{0} = 5 × 10^{18} m^{−3}

Comparison between the magnitude of the electric field, in volts/meter, within the cylinder obtained by the eigenfunction method and the FDTD simulation for different collision frequencies; incident wave frequency is set to _{in} = 10 GHz, and electron density is set to _{0} = 5 × 10^{17} m^{−3}.

Eigenfunction, ^{6} Hz

FDTD, ^{6} Hz

Eigenfunction, ^{7} Hz

FDTD, ^{7} Hz

Eigenfunction, ^{8} Hz

FDTD, ^{8} Hz

Comparison between the magnitude of the electric field, in volts/meter, inside the plasma for linear cuts through the cylinder obtained by the eigenfunction method and the FDTD simulation for different incident frequencies; electron density is set to _{0} = 5 × 10^{17} m^{−3}, and electron collision frequency is set to ^{6} Hz.

Variables

Variables

Comparison between the magnitude of the electric field, in volts/meter, inside the plasma for linear cuts through the cylinder obtained by the eigenfunction method and the FDTD simulation for different electron densities; incident wave frequency is set to _{in} = 10 GHz, and electron collision frequency is set to ^{6} Hz.

Variables

Variables

Comparison between the magnitude of the electric field, in volts/meter, inside the plasma for linear cuts through the cylinder obtained by the eigenfunction method and the FDTD simulation for different collision frequencies; incident wave frequency is set to _{in} = 10 GHz, and electron density is set to _{0} = 5 × 10^{17} m^{−3}.

Variables

Variables

Figure

Comparison between the scattering amplitudes obtained by the eigenfunction method and the FDTD simulation for different variable parameters of the homogeneous plasma.

Variable incident frequency _{in}

Variable collision frequency

Variable electron density

These results show that the FDTD simulations provide solutions that differ less than 1% from those obtained by the eigenfunction method that is widely used in the literature in most cases and less than 5% in the worst case. It can also be seen that the FDTD technique allows the study of the time domain evolution of the wave propagation, so transient effects can be analyzed. Additionally, it can be seen that the algorithm is still precise even for the plasma that is overly dense in relation to the incident frequency (Figures

An inhomogeneous plasma is now investigated, with the same simulation parameters as the homogeneous case. The plasma inhomogeneity is defined by the quadratic density profile
_{0} is the plasma density at the center of the plasma,

Figure

Time evolution of the intensity of the electric field, in volts/meter, across the computational domain, for the inhomogeneous case where incident wave frequency is set to _{in} = 10 GHz, central electron density is set to _{0} = 5 × 10^{17} m^{−3}, and electron collision frequency is set to ^{6} Hz.

Figures

Magnitude of the electric field, in volts/meter, within the inhomogeneous plasma cylinder obtained by the FDTD simulation for different incident frequencies _{in}. Central electron density is set to _{0} = 5 × 10^{17} m^{−3}, and electron collision frequency is set to ^{6} Hz.

_{in} = 5 GHz

_{in} = 10 GHz

_{in} = 15 GHz

Magnitude of the electric field, in volts/meter, within the inhomogeneous plasma cylinder obtained by the FDTD simulation for different central electron densities _{0}. Incident wave frequency is set to _{in} = 10 GHz, and electron collision frequency is set to ^{6} Hz.

_{0} = 1 × 10^{17} m^{−3}

_{0} = 5 × 10^{17} m^{−3}

_{0} = 1 × 10^{18} m^{−3}

_{0} = 1^{18} m^{−3}

_{0} = 2 × 10^{18} m^{−3}

_{0} = 5 × 10^{18} m^{−3}

Magnitude of the electric field, in volts/meter, within the inhomogeneous plasma cylinder obtained by the FDTD simulation for different collision frequencies _{in} = 10 GHz, and central electron density is set to _{0} = 5 × 10^{17} m^{−3}.

^{6} Hz

^{7} Hz

^{8} Hz

Magnitude of the electric field, in volts/meter, inside the plasma for linear cuts through the cylinder obtained by the FDTD simulation for different incident frequencies. Central electron density is set to _{0} = 5 × 10^{17} m^{−3}, and electron collision frequency is set to ^{6} Hz.

Variables

Variables

Magnitude of the electric field, in volts/meter, inside the plasma for linear cuts through the cylinder obtained by the FDTD simulation for different central electron densities _{0}. Incident wave frequency is set to _{in} = 10 GHz, and electron collision frequency is set to ^{6} Hz.

Variables

Variables

Magnitude of the electric field, in volts/meter, inside the plasma for linear cuts through the cylinder obtained by the FDTD simulation for different collision frequencies _{in} = 10 GHz, and central electron density is set to _{0} = 5 × 10^{17} m^{−3}.

Variables

Variables

Comparison between the scattering amplitudes obtained by the eigenfunction method and the FDTD simulation for different variable parameters of the inhomogeneous plasma.

Variable incident frequency _{in}

Variable collision frequency

Variable central electron density _{0}

One qualitative analysis is straightforward: as expected from inspecting (_{p}_{in} = 5 GHz in Figure _{0} = 5 × 10^{18} m^{−3} in Figure

Additionally, another qualitative analysis of the results shows a phenomenon of wave path deflection when the propagation is through inhomogeneous plasmas. This is due to the spatial variation in the electron density, which in turn causes a spatial variation in the refraction index of the plasma medium. Continuous spatial variations in refraction indexes, in turn, are well-known to cause ray deflection.

In broad terms, two different behaviors can be observed from the presented results: (1) electromagnetic waves penetrating the plasma and propagating while being conditioned by the plasma, that is, suffering dispersion, attenuation, and deflection, when appropriate to each situation’s characteristics, and (2) electromagnetic waves being reflected from the plasma and exhibiting very low penetration (or, for the inhomogeneous cases, very low penetration after a certain point in the inhomogeneous cylinder). These two different types of behavior are related to the real part of the plasma’s dielectric permittivity, with penetration possible for

Behavior shift for the plasma dielectric permittivity as a function of the local plasma density _{in}. Red region represents

_{in}

_{in}

These results are consistent with previous qualitative results in the literature obtained by the application of the eigenfunction expansion technique, but solutions are obtained at lower computational cost and with greater resolution when applying the FDTD technique.

A limitation that comes with the FDTD technique for inhomogeneous plasmas, however, is the staircasing effect on the local plasma density

For the cases presented herein, the difference between the actual plasma density function and its staircase approximation in the FDTD grid is shown in Figure

Comparison between the actual inhomogeneous density function for the plasmas considered herein and its staircase approximation in the FDTD grid.

FDTD simulations are of great value in identifying different behaviors within the plasma structure and exploring the effects of the plasma parameters on the electromagnetic propagation as well as studying transient effects. In particular, for inhomogeneous cases or any case with other kinds of spatial complexity (e.g., complicated geometries for the structure), the computational cost of running analytical methods based on eigenfunction expansions is prohibitively high, and the results obtained lack good enough resolution to be precisely analyzed, two limitations that do not exist for FDTD simulations. The FDTD technique described herein allows for arbitrary time or spatially varying parameters to be incorporated in the simulation, as well as providing step-by-step transient solutions, two features that eigenfunction expansions lack.

Understanding the effects of plasma inhomogeneities in electromagnetic wave propagation and scattering and being able to correctly simulate those effects are important steps in the design of plasma devices such as telecommunication antennas, especially when the inhomogeneities are time-dependent or present sharp spatial variations. The algorithm presented herein is an efficient solution that, to the authors’ knowledge, has not been previously applied to 2D plasma systems in the context of telecommunication devices.

Future perspectives for this work include extending the numerical algorithm for a TEz-polarized incident wave. Due to the nature of TMz-polarized waves, the electric field was restricted to having only a

Another perspective is including ionization processes in the algorithm, which would allow the simulation of the start up and turn off of a device; so far, the plasma has been considered to be in a steady state of ionization; that is, the source responsible for ionization is considered to be active for a long time and recombination processes are ignored.

With these extensions, the algorithm would be able to simulate fully self-consistent plasma systems in three spatial dimensions, thus allowing for the full simulation of an entire device like a plasma antenna or even the interaction between multiple devices operating simultaneously.

The authors declare that there is no conflict of interest regarding the publication of this paper.

This work was supported by the Brazilian Agency CNPq under Grant 131640/2015-1.