Electromagnetic Particle-In-Cell simulation on the impedance of a dipole antenna surrounded by an ion sheath



[1] We have newly developed a numerical tool for the analysis of antenna impedance in plasma environment by making use of electromagnetic Particle-In-Cell (PIC) plasma simulations. To validate the developed tool, we first examined the antenna impedance in a homogeneous kinetic plasma and confirmed that the obtained results basically agree with the conventional theories. We next applied the tool to examine an ion-sheathed dipole antenna. The results confirmed that the inclusion of the ion-sheath effects reduces the capacitance below the electron plasma frequency. The results also revealed that the signature of impedance resonance observed at the plasma frequency is modified by the presence of the sheath. Since the sheath dynamics can be solved by the PIC scheme throughout the antenna analysis in a self-consistent manner, the developed tool has feasibility to perform more practical and complicated antenna analyses that will be necessary in real space missions.

1. Introduction

[2] For several decades, the impedance of antennas immersed in plasmas has received a great deal of attention. The precise knowledge of the impedance of antennas aboard scientific spacecraft is essential, e.g., for the data calibration required in plasma wave observations and some plasma diagnostic techniques such as the impedance probe. The precise impedance knowledge is also useful for the circuit matching of antenna systems used in space missions.

[3] Various methods for the evaluation of antenna impedance in plasma have been developed by using theoretical approaches. Because of the complexity of the plasma dynamics around the antenna, most of the methods have introduced certain assumptions and approximations to simplify the antenna modeling and calculation of the antenna impedance. As a pioneering work in this field, Balmain [1964] theoretically derived a formulation of the input impedance of short dipole antennas in magnetized plasma with an assumption of cold plasma. Analyses of short antenna impedance in kinetic plasma have also been performed for some limited models [e.g., Kuehl, 1966, 1967; Schiff, 1970; Nakatani and Kuehl, 1976]. In those theories, an assumption of a triangular current on the antenna surface was used in order to avoid the complexity of deriving the real form of the current distribution. Although it has been considered that the assumption is valid for antennas with length sufficiently smaller than applicable wavelength, a more self-consistent method without any assumptions on the current distribution is necessary for the precise evaluation of antenna impedance. Recently, several studies have been performed to derive the real form of the current distribution in cold plasma [e.g., Bell et al., 2006]. However, there are no studies that derived a self-consistent form of the current distribution by kinetic plasma approaches.

[4] Another important point to be considered in the antenna analysis is the inhomogeneous plasma environment around the antenna. In absence of any effects of particle emission from an antenna surface, an electron-sparse region called an ion sheath is created around the surface with a floating potential. The dynamics and the detailed properties of the ion sheath have been exhaustively studied particularly in the field of active experiments [e.g., Calder et al., 1993]. In aspect of the ion sheath effect on antenna impedance, however, it has been simply regarded as a vacuum layer in a frequency range in which ions are assumed to be immobile. It has been reported that such a vacuum layer may contribute prominently to antenna impedance, and rocket and satellite observations have indicated that the the sheath impedance is important [Oya and Obayashi, 1966]. However, the inclusion of the inhomogeneous plasma effect caused by the ion sheath leads to complication in theoretical derivation of antenna impedance. Therefore, several theoretical analyses of the sheath impedance have been conducted for much simplified sheath configuration such as planar [Oya, 1965; Balmain and Oksiutik, 1969] and cylindrical [Aso, 1973] structures. Béghin and Kolesnikova [1998] proposed a numerical approach using the surface-charge distribution (SCD) method, which can consider all of the boundary surfaces involving ion-sheath interfaces around the antenna and satellite bodies with complex geometry. In the SCD method, the ion-sheath interfaces were given as parameters of the numerical tool.

[5] Recently, numerical simulations have been recognized as a powerful tool as the theoretical and experimental approaches. In the field of antenna characteristics, extensive analyses have been conducted using numerical simulations via the Finite-Difference-Time-Domain (FDTD) method [Taflove, 1995] in free-space cases. The FDTD method was also applied to plasma simulations by treating the plasma as an anisotropic and dispersive dielectric [e.g., Cummer, 1997]. The advantage of the FDTD simulations lies in the ability to treat realistic antenna geometries without too simplified approximations of the antenna current distribution. Using the FDTD simulation with the fluid-plasma description, the nontriangular antenna current distribution was suggested to have caused the deviation of impedance value from that obtained by the assumption of triangular current distribution [Ward et al., 2005]. However, in order to analyze the impedance including the plasma kinetic effects, the plasma must be modeled as particles in the FDTD simulations.

[6] In the present study, by applying the three-dimensional Electromagnetic Particle-In-Cell (EM-PIC) simulation, we developed a numerical tool for the antenna analysis in kinetic plasma environment. The developed tool enables us to perform simulation analysis including plasma kinetic effects on the antenna impedance, e.g., the existence of finite resistance below the electron plasma frequency and a change of an impedance resonance signature due to damping of plasma kinetic waves [Kuehl, 1967; Meyer-Vernet and Perche, 1989]. In addition, we incorporated the numerical model of the conducting surfaces of an antenna as inner boundaries and a boundary treatment for plasma particles on the surfaces in the simulation tool. With these treatments, we can simulate sheath dynamics in a self-consistent manner throughout the antenna analysis and evaluate antenna impedance without any assumptions on the sheath structure.

[7] The present paper presents simulation results obtained for the impedance of an electrically short dipole antenna covered with an electron-sparse region. The major motivation of this work is to demonstrate the application of PIC simulation techniques to the analysis of the antenna characteristics. We particularly focus on the impedance of a low-power transmitting antenna. The impedance calculation is fundamental and useful for the validation of the EM-PIC method. The transmitted power is small enough not to disturb the boundary environment of the simulation box so that numerical errors caused by the boundary effects are minimized. We consider a very simple situation in which a set of dipole antenna is immersed in Maxwellian, unmagnetized, and collisionless plasma. The plasma is so dense and low-temperature that the Debye length becomes smaller than the antenna length. First, we validate the developed EM-PIC simulation tool by examining the impedance without considering any effects of an ion sheath and comparing obtained results to the conventional kinetic theories [e.g., Schiff, 1970; Meyer-Vernet and Perche, 1989]. After that, we analyze the impedance characteristics of antennas covered with an ion sheath, which is created under the condition that an antenna has a floating potential. We focus on the impedance dependence on the ratio of the antenna length to the Debye length. We also discuss the dependence of sheath capacitance on the sheath thickness by the simulations with different bias potentials.

2. EM-PIC Simulation Tool for Antenna Analysis

2.1. Simulation Model

[8] The simulation tool utilized in the present paper has been developed based on the electromagnetic particle simulation code called KEMPO [e.g., Omura and Matsumoto, 1993]. In the KEMPO Maxwell's equations for electromagnetic field and the equations of motion for charged particles are solved simultaneously. Therefore, plasma kinetic effects are reflected in the field evolution in a self-consistent manner. In addition to the above EM field solver, the tool optionally has an electrostatic (ES) field solver, i.e., Poisson's equation. We call the two kinds of field solvers EM-PIC and ES-PIC modes, respectively. One can switch between these two modes in one simulation run. For one scenario of an antenna analysis, we performed one long PIC simulation in which the ES-PIC mode was used at the beginning to obtain the steady state of the plasma environment around the antenna. Then, in the middle of the simulation run after the sheath formation, we switched to the EM-PIC mode to analyze the antenna impedance. The merit of using the ES-PIC mode is that its calculation speed for the sheath formation is much faster than EM-PIC mode because the time step in the ES-PIC mode is not restricted by the Courant condition for the light-wave mode [Birdsall and Langdon, 1985]. This implies that we can set the time step to much larger value than in the EM-PIC mode and reduce the computational cost drastically.

[9] The simulation system is shown in Figure 1. We consider a three-dimensional simulation box and place a dipole antenna at its center. The simulation box is uniformly filled with mobile electrons and ions with finite thermal velocities at the initial state of a simulation run. Since our interest in the present study is in antenna impedance in a frequency range near the electron plasma frequency, the motion of ions has little effects on the antenna impedance itself. However, ion dynamics cannot be neglected to achieve a steady-state profile of the plasma environment around the antenna. For instance, the ion current is necessary to balance the electron current at the antenna surface for a floating potential of the antenna body. We assumed that the ion species is a proton and employed the real mass ratio of the protons to the electrons, i.e., 1836, in the present analysis.

Figure 1.

Model of the numerical simulation. (a) The dipole antenna is placed at the center of the simulation box. (b) The electric field Ezgap = −Vir is provided at the gap between two antenna elements.

[10] The boundary condition (BC) of the simulation box should be carefully selected in order to realize an isolated system. In the present analyses, two types of BCs are utilized: the BCs for EM and ES components. For EM component, field absorbing region for the outgoing wave is necessary to realize an isolated system. We set the field absorbing region based on Masking method [Tajima and Lee, 1981] consisting of 8 grids from the edge of the box in order to prevent the field reflection at the boundary. When we solve Poisson's equation for ES component, the Neumann condition is used. The particles which reach the edge of the simulation region are reflected back into the region. In the current analysis, we set the edge of the simulation box sufficiently far from the sheath region. This indicates that the perturbation of plasma density around the antenna never reaches the outer boundaries of the simulation box. In this condition, since the flux escaping from the simulation box is equal to the particle flux in the unperturbed background plasma, the reflecting boundary condition for particles can be substituted for a particle-loading scheme that is known as a rigorous open boundary condition for escaping particles. By combining the above treatments, we realized the isolated system of the simulation.

2.2. Antenna Treatment

[11] One of the important modifications of KEMPO is the introduction of the antenna body to the simulation model. For simplicity, we assumed that the antenna bodies are made of perfect conductors, in which the electric field component tangential to the antenna surface is zero. In the present analysis, we set the values of electric field Ez in the antenna body to zero, except for the gap between the two antenna elements, as shown in Figure 1. One should note that the antenna surface current is not artificially given but obtained by calculating the rotation of the magnetic field around the antenna body. The profile of the magnetic field around the antenna body is self-consistently solved so that the electric field satisfies the appropriate boundary conditions in the antenna body as explained above. As a result, we can evaluate the antenna impedance without any assumptions on the current distribution on the antenna surface.

[12] Another important issue that should be carefully considered is the treatments for particles which impinge into the antenna bodies. The treatments can be categorized in two models. In the first model, the antenna surfaces are perfectly transparent with respect to the plasma particles, which can pass through the antenna location. This model corresponds to a mesh-like antenna [Schiff and Fejer, 1970] that was widely used in previous related studies. Note that, if this model is used in the particle simulation, inhomogeneous plasma environment such as an ion sheath is not naturally created. We, therefore, used the model for antenna analyses in uniform plasma.

[13] In the second model, the physical existence of the antenna body is taken into consideration by introducing the internal nonplasma boundaries in the simulation system. The most important feature of this model is that a sheath is created as the result of plasma-body interactions, and thus this model is more practical than the first model. We, therefore, applied the second concept to cases of the ion-sheathed antenna. Practically, since the minimum spatial unit is one cubic cell with Δr3 volume, the cross-section of antenna is assumed to be one zone squared with Δr2 area, where Δr is a grid spacing in a three-dimensional EM-PIC method. In the present simulation model, the antenna line is composed of a series of the cubic cells, and particles whose centers move into the cell boundary are absorbed in the antenna. The charge collected by the antenna is redistributed on the grid line in the antenna body on which the z-components of the electric field Ez are defined (see Figure 1), so that an equipotential solution on the antenna body is realized. For this purpose, we use the Capacity Matrix method [Hockney and Eastwood, 1981], which can also be applied to the multi-body case. After we redistribute the surface charge, we correct the electrostatic field by solving Poisson's equation considering the modified surface charge. By this treatment, the contribution of collected particles on the charging of the antenna body can be precisely evaluated. For the outside of the antenna territory, the particle motion is advanced by linearly interpolating the field values at the particle position from the adjacent grid points, which is the scheme commonly used in PIC plasma simulations [e.g., Omura and Matsumoto, 1993].

[14] We analyzed the impedance characteristics of the transmitting antenna with a small applied signal. To simulate the transmitting antenna, we used the Delta-Gap feeding method [e.g., Luebbers et al., 1993]. In this method, the dipole antenna is fed with an input voltage Vi, which is realized by providing an electric field Ezgap at the gap between two antenna elements as follows:

equation image

To obtain the input impedance of the antenna, we need to know the current Ii at the antenna feeding point. Ii is obtained by the rotation of the magnetic field around the feeding point. Numerically, Ii is computed with

equation image

where μ0 represents the permeability in vacuum, and Bxlower, Bxupper, Byright, and Byleft are the magnetic fields which are defined at the adjacent grids to the feeding point, as shown in the right panel of Figure 1. Vi and Ii are first obtained in the time domain and are then transformed to the frequency domain by Discrete Fourier Transform (DFT). The input impedance Z of the antenna is obtained from

equation image

where equation imagei and equation imagei represent the voltage and current, respectively, at the feeding point in the frequency domain.

[15] In order to obtain the antenna characteristics over a large frequency range in a single run of the simulation, we utilized a broad spectrum pulse given as Vi = Va(d/dt)[(t/T)4exp(−t/T)] where Va and T are parameters of the pulse, and t/T represents the normalized time. The dominant spectral frequency ωd of the pulse is given as ωd = 0.152 × 2π/T and was set close to the electron plasma frequency.

2.3. Common Parameters of the Analysis

[16] Table 1 shows common parameters used in the present simulations. A grid spacing and a time step are determined appropriately so that the Courant condition for the light-wave mode is safely satisfied. In the present analysis, we have 64 × 64 × 64 cells and 512 particles per cell; namely 64 × 64 × 64 × 512 = 134,217,728 particles in the entire system.

Table 1. Simulation Parameters for the Analysis of Antenna Impedancea
  • a

    The values are given in a normalized unit system used in the simulation tool.

   Grid spacingΔr1
   Time stepΔt0.02
   Speed of lightc25
   System lengthLs64
   Number of superparticles per cell 512
Dipole antenna
   Frequency at which the antenna operates as the half-wavelength dipoleωhalf∼3.3
   Antenna half lengthLa12
   Antenna width in x and y directions (See Figure 1 for an antenna configuration.) 1
Background plasma electrons
   Plasma frequencyΠe1
   Inertial lengthce25
   Thermal velocity (variable)v01–2
   Debye length (variable)λD1–2

[17] In the present paper, we set our goal to examine the impedance characteristics in collisionless-isotropic plasma environment. The parameters listed in Table 1 are given in a renormalized unit system used in the simulation tool. In this case, the outputs are obtained as the ratio of the antenna impedance to the characteristic impedance of free space equation image, where equation image0 represents the permittivity in vacuum. The impedance values in the real physical unit are calculated using the relation equation image = 120πΩ. Hence, all the results for the impedance are given in the unit of Ω in the present paper.

[18] One of the important parameters is the ratio of the antenna length to the free-space wavelength in the frequency range of interest. In the present study, the frequency range of our interest is near the electron plasma frequency and is located well below the frequency at which the antenna operates as a well-known half-wave dipole. From this point of view, we treat the “electrically short antenna” in comparison with the free-space wavelength. In practice, the “electrically short antenna” regime is valid in most of solar-terrestrial regions except in very dense plasmas (105–106/cm3) in ionosphere, where the electron plasma frequency is so large that the free-space wavelength at the frequency becomes in the order of 10–100 m.

[19] The ratio of the antenna length to the Debye length of the background plasma is also important. The impedance resonance in plasma becomes remarkable when the antenna length is significantly larger than the Debye length, as was predicted by the previous theory [Meyer-Vernet and Perche, 1989]. We, therefore, chose the plasma parameters so that the antenna has a length greater than the Debye length in the present study.

3. Antenna Impedance in Uniform Plasma

3.1. Comparison With the Conventional Theory

[20] In order to validate the developed EM-PIC simulation tool, we examined the antenna impedance by using the transparent-antenna modeling, with which an ion sheath is not created around the antenna as described in section 2.2. The results are compared with the conventional kinetic theory which was developed by e.g., Schiff [1970]. In the theory the impedance is formulated based on the induced Electro-Motive-Force (EMF) method using Maxwell's equations and the linearized Vlasov equation as basic equations in the quasi-static limit. The formula for the antenna impedance Z in kinetic plasma is then written [Schiff, 1970] as

equation image

where Js, I0, ω, and k are the antenna current distribution, the antenna current evaluated at the antenna feeding point, the frequency, and the wave number vector, respectively. The asterisk denotes the complex conjugate. In order to adopt the normalized parameters listed in Table 1, we used the normalized form of equation (4) that is given as

equation image

where Z0 = equation image and c represent the characteristic impedance of free space and the speed of light, respectively. In equations (4) and (5), equation imageL is the plasma longitudinal permittivity and is given in the normalized form by using the kinetic theory as

equation image

where v0, Πe, and Zp represent the electron thermal velocity, the electron plasma frequency, and the plasma dispersion function, respectively, as discussed by Fried and Conte [1961]. Note that k · (equation imageL · k) = 0 gives the dispersion relation for plasma longitudinal waves.

[21] For the theoretical comparisons with the simulation results, we adopted the assumption of the triangular current distribution on the antenna surface in analytically evaluating equation (4). In the parameters used in the present analysis, the antenna length is smaller than the electron inertial length ce. Physically, ce means the skin depth of an evanescent wave mode below Πe. When ce is much larger than the antenna length, the triangular current approximation is known to be appropriate in the computation of the conventional theory for the antenna impedance [Bell et al., 2006]. In upper panels of Figure 2, we plot the theoretical curves in solid lines. Panels I-(Re) and I-(Im) show the resistance and reactance, which are the real and imaginary parts, respectively, of the impedance. In Panel I-(Im), we also superimpose the theoretical value of free-space antenna reactance, which is evaluated by the formula (−1/πωequation image0La)[ln(La/a) − 1] where La and a represent the half length and the radius of the dipole antenna, respectively [Schelkunoff and Friis, 1952]. The impedance value is plotted as a function of a normalized frequency ωe.

Figure 2.

Comparison of the antenna impedance in uniform plasma obtained by the conventional kinetic theory and the developed EM-PIC tool. Upper panels (I-(Re) and I-(Im)) represent the theoretical results of antenna resistance and reactance, respectively, for the case of λD = La/12. Lower panels (II-(Re) and II-(Im)) represent the EM-PIC results of antenna resistance and reactance, respectively. In the lower panels, the solid and dashed lines correspond to the cases of λD = La/12 and λD = La/6, respectively. The dotted line represents the free-space reactance.

[22] Meanwhile, we run EM-PIC simulations using the parameters listed in Table 1 and computed the antenna impedance by the method presented in section 2. Note that the form of the current distribution was never assumed but evaluated self-consistently in the simulations. We examined a case with the Debye length of background plasma: λD = La/12. The obtained simulation results are shown in solid lines in Panels II-(Re) and II-(Im) of Figure 2 in the same manner as the theoretical curves. Also in Panel II-(Im), we superimpose the free-space value of antenna reactance that is obtained by the simulation of the free-space case.

[23] As clearly shown in comparison between the solid lines in the upper and lower panels, the impedance profiles basically show agreement between the theoretical and simulation results. The major points of the agreement are, (1) the resistance has a finite and constant value below ω = Πe, (2) the reactance value is larger than the free-space value below ω = Πe, (3) the drastic variation of the impedance values is observed near ω = Πe, and (4) the impedance tends to the free-space value well above ω = Πe. The interpretations of these effects will be described briefly in the next subsection. On the other hand, a discrepancy is clearly seen between the theoretical and EM-PIC results near ω = Πe. The intensity of the impedance resonance is greater in the theoretical results than those of EM-PIC simulation, which is seen in both real and imaginary parts but more remarkable for the imaginary part. The possible reason causing this discrepancy will be discussed in section 5. Although the above disagreement is found, we basically confirm that the physical behavior of the antenna impedance in the plasma can be qualitatively evaluated by the developed tool.

3.2. Dependence of Antenna Impedance on Debye Length

[24] In order to examine the dependence of the antenna impedance on the Debye length of the surrounding plasma, we performed an additional EM-PIC simulation for the case of λD = La/6. The obtained result for λD = La/6 case is superimposed as dashed lines in addition to λD = La/12 case in Panels II-(Re) and II-(Im) of Figure 2. One should note that we did not change the plasma density in these two cases. In this situation, doubling the Debye length indicates quadrupling the temperature at the constant density.

[25] As shown in Panel II-(Re), the resistance has a finite and almost constant value for each case in the frequency range lower than Πe. In free space, the resistance should be less than 5 Ω for ω < Πe and the given antenna length [Stutzman and Thiele, 1997] since there is few radiation of the electromagnetic wave from the electrically short antenna. In kinetic plasma, however, the conversion of field energy excited by the antenna into the kinetic energy of the plasma electrons causes the dissipation, which leads to the equivalent resistance for ω < Πe [Kuehl, 1967]. The result confirms that wave-plasma interactions around the antenna are correctly evaluated in the present simulation. We can also see that the resistance is larger in the case of λD = La/12 corresponding to the smaller Debye length case. This dependence was also confirmed by the conventional kinetic theory although not displayed.

[26] Near ω = Πe, the large peak of the resistance value is observed, which is particularly remarkable in the case of λD = La/12. In the case of λD = La/6, the similar signature is recognized, but the peak value is lower than the case of λD = La/12. This characteristic variation of the impedance value has been referred as the impedance resonance. The enhancement of the impedance value corresponds to the presence of the poles k · (equation imageL · k) = 0 in the analytic expression of equation (4), which also gives the dispersion relation of the plasma wave mode. The impedance resonance, therefore, is considered to have much relevance to the strong interactions between the antenna and the plasma wave mode. In the present case the corresponding plasma wave is the Langmuir wave. The reduction of the peak value due to a high temperature, which corresponds to the case of the larger Debye length, was also confirmed by the theory. Another feature we can find near ω = Πe is that the peak frequency of the resonance is shifted to higher frequency for the case of λD = La/6 in comparison with the case of λD = La/12. This resonance shift was not shown in the conventional theory. There are several possible reasons for this frequency shift, which will be discussed in section 5.

[27] As to the reactance shown in Panel II-(Im) of Figure 2, the absolute value of the reactance below ω = Πe is smaller than its free-space value. This means that the antenna capacitance, defined as C = −1/(ωX), where X is the reactance, becomes greater in the plasma than in free space. The simulation results show that the antenna capacitance is larger for λD = La/12 case than for λD = La/6 case. This feature can be explained by an analogy with a capacitor separated by dielectric material with a large permittivity. In equation (6), the real part of the derivative of the plasma dispersion function Zp takes a negative value in a low-frequency limit [Fried and Conte, 1961]. Therefore, the value of the dielectric function equation imageL in the finite-temperature plasma is larger than unity at the low-frequency range. If we apply to an analogy that an antenna consists of two elements separated by a dielectric with a permittivity larger than equation image0, it makes sense that the antenna capacitance is larger in the plasma in the low-frequency range. When we consider large v0 which implies a situation of high temperature of plasma, the corresponding equation imageL approaches to unity, and the capacitance tends to its free-space value.

[28] Near ω = Πe, the reactance also shows the signature of the impedance resonance, at which the reactance is maximum. The remarkable feature found in the simulation results is that the intensity of the impedance resonance is much weakened in the case of λD = La/6. As mentioned in the interpretation of the resistance peak near ω = Πe, the impedance resonance is considered to be caused by the strong wave-antenna interactions. The plasma wave component that has a wavelength smaller than the local Debye length is readily damped, and thus large λD/La leads to the reduction of wave components which can interact with the antenna. Therefore, in the case of λD = La/6, the impedance resonance becomes weaker than the λD = La/12 case. The same tendency was shown in the theoretical calculations.

[29] In the frequency range above Πe, the antenna impedance should recover its free-space value simply because the plasma dielectric function recovers its free-space value in the frequency range well above Πe. This signature is confirmed in the theoretical results (see Panels I-(Re) and I-(Im)). Also in the EM-PIC results, it is confirmed that the resistance and the reactance tend to approach their values in free space in both cases of the Debye length. Therefore, the impedance behavior above Πe is correctly evaluated by the developed EM-PIC tool.

4. Analysis on the Antenna Covered by an Ion Sheath

4.1. Structure of an Ion Sheath

[30] In previous studies on antenna impedance [e.g., Oya, 1965], simplified models were commonly used for the plasma environments around antennas; e.g., an ion sheath created around the antenna surface was assumed to have an abrupt jump in electron density at the interface between the sheath and the uniform plasma. However, for higher accuracy and applicability to complex plasma environments which will be encountered in real space missions, it is important to establish a method of including the ion sheath of which the structure is solved by self-consistent analysis in consideration of antenna-plasma interactions. By taking advantages of the PIC simulation, we performed the impedance analysis simultaneously solving the dynamics of an ion sheath created around the antenna body. In the present section, we present the ion sheath structure obtained as a steady state during the ES-PIC mode.

[31] As a steady state in the ES-PIC mode simulation, we obtained floating potential values ϕF = −3.4kBTe/e and −2.9kBTe/e for the cases of λD = La/12 and La/6, respectively. Here, kB, Te, and e represent Boltzmann's constant, the electron temperature, and the electric unit charge, respectively. In an isothermal plasma, i.e., Te = Ti where Ti represents the ion temperature, Fahleson [1967] theoretically evaluated the floating potential as ϕF = −(kBTe/e)lnequation image ∼ −3.8kBTe/e in a condition that conductor dimensions are sufficiently larger than λD. Here, me and mi = 1836me represent the mass of electrons and protons, respectively. In the present case, however, the antenna radius is small and comparable to λD. Therefore, Fahleson's theory may not be applicable. Although the floating potential of a cylindrical conductor with a comparable radius to λD is generally difficult to formulate, its magnitude becomes smaller than that obtained with Fahleson's theory and should decrease with the ratio of the conductor's radius to λD [Mott-Smith and Langmuir, 1926]. These tendencies basically agree with those obtained in the current simulations stated above.

[32] Figure 3 shows the spatial profile of electron number density for the case of λD = La/12 in the xz plane, which includes the center of the antenna. We depict white lines at the location of the dipole antenna in the figure. An electron sparse region, shown in black, is clearly found around the dipole antenna. On the other hand, ion density was confirmed to increase around the antenna due to the attraction by the negative charged antenna but less perturbed than electron density. Since charge neutrality is broken and ions become relatively rich in this region, we call it an ion sheath. In order to examine the spatial variation of the electron density in the ion sheath region in detail, the one-dimensional density distribution is shown in Figure 4. The density is measured along the direction perpendicular to the antenna from its surface at the midpoint of the upper antenna element. The solid and dashed lines correspond to the cases of λD = La/12 and La/6, respectively. Unlike the simplified models of the ion sheath commonly used in previous studies, the electron density varies gradually in the sheath region between 0 and 1 of the normalized distance. Note that the Debye length affects the spatial gradient of the electron density at the interface between the sheath and the uniform plasma. Comparing the solid and dashed lines, we find that the spatial gradient is steeper for the case of the smaller Debye length with the lower temperature.

Figure 3.

Spatial profile of the normalized number density of electrons in the xz plane, which includes the center of the antenna in the case of λD = La/12. n0 represents the background level of the electron number density.

Figure 4.

Profiles of the number density of electron measured along the direction perpendicular to the antenna from its surface (on the axis of z = La/2 shown in Figure 3). The solid and dashed lines correspond to the cases of λD = La/12 and λD = La/6, respectively.

4.2. Impedance of an Ion-Sheathed Antenna

[33] The antenna impedance in the ion-sheath environment was computed by the developed tool by adopting the method described in section 2. Note that we kept solving the plasma dynamics in the antenna-impedance analysis with the EM-PIC mode after obtaining the steady-state structure of the sheath with the ES-PIC mode. During the antenna-impedance analysis with EM-PIC mode, however, the ion-sheath environment was hardly perturbed since the applied signal at the antenna feeding point was sufficiently small.

[34] Figure 5 shows the sheath effects on the antenna impedance in the case of λD = La/12. The solid, dashed, and dotted lines indicate the results for the ion-sheathed, uniform plasma, and free-space cases, respectively. As in the uniform plasma case, the signature of the impedance resonance is seen around ω = Πe for the ion-sheathed antenna. There are, however, some differences between the solid and dashed lines in the figure. As the frequency increases from the resonance frequency, i.e., ω ∼ Πe, the resistance value decays to its free-space value, which is found in both the uniform plasma and the ion-sheathed cases. However, as shown in Figure 5a, the resistance decays faster in the ion-sheathed case than in the uniform plasma case.

Figure 5.

Antenna resistances (left) and reactances (right) including and not including ion sheath effects for the case of λD = La/12. The dotted line in the right panel represents the free-space reactance.

[35] Below ω = Πe, as shown in Figure 5b, the absolute value of reactance is large for the ion-sheath environment in comparison with the uniform plasma case. In order to interpret these results, we show the results in terms of the antenna capacitance C = −1/(ωX) in Figure 6, in which the values are given as a product of C and Πe, so that they have the units of 1/Ω. As clearly shown in Figure 6, the capacitance C has almost a constant value in the frequency range well below ω = Πe in all cases. Particularly, the value of C is smaller for the ion-sheath case than that of the uniform plasma case. This reduction of C is caused by the presence of the ion sheath formed around the antenna and is an important effect which has been reported in previous antenna studies [e.g., Oya, 1965]. Since mobile electrons are extremely scarce in the ion sheath compared to the background plasma, the ion sheath behaves as a vacuum gap that separates the antenna surface from the background plasma. Therefore, as a simple model, the ion sheath can be considered as a capacitance between two coaxial conductors. In analogy, the inner and outer conductors correspond to the antenna body and the background plasma. The reactance caused by the sheath is added to the antenna impedance and clearly affects the total capacitance value of the antenna. In other words, the capacitance of the coaxial conductors is connected to the plasma capacitance in series so that the total capacitance in the case of the ion sheath is smaller. A discussion of the relation between the sheath structure and the antenna capacitance will be presented in the next section.

Figure 6.

Antenna capacitance in 1/Ω (see text). The solid, dashed, and dotted lines represent the antenna capacitance in the ion sheath, uniform plasma, and free space, respectively.

[36] It has been considered that the ion-sheath effects described above become less significant as the Debye length becomes larger in comparison with the antenna length, as mentioned in the work of e.g., Béghin et al. [2005]. We also examined the ion-sheath effects on the antenna impedance for the case of larger Debye length λD = La/6 and confirmed that the ion-sheath effects were correctly weakened compared to λD = La/12 case. However, the fact, that the impedance modification due to the ion-sheath effects can be observed in λD = La/6 case, shows the importance of the precise modeling of an ion sheath even in situations of the Debye length in the same order of the antenna length.

4.3. Dependence of Antenna Impedance on the Sheath Structure

[37] Several previous studies [Balmain and Oksiutik, 1969; Aso, 1973] formulated the impedance of ion-sheathed antennas by assuming that the total antenna impedance was represented by the impedance of the sheath plus that of the plasma connected in series. In these formulations, the impedance of the sheath region was obtained as a function of the sheath thickness. In this section, we examine the effects of the sheath thickness on the antenna impedance by performing additional simulations. For this purpose, we applied a DC bias potential to the antenna. By changing the bias potential as a simulation parameter, the sheath structure around the antenna changes, and thus we can examine various sheath environments without any changes in the background plasma parameters. In the present section, we examined two cases with different bias potentials: (a) ϕa = 4ϕF and (b) ϕa = 16ϕF, where ϕF = −3.4kBTe/e is a floating potential obtained in the analysis described in section 4.1 for λD = La/12. In both cases, λD was fixed to La/12, and the other parameters were set as listed in Table 1. Note that the condition of the current balance between electrons and ions at the antenna surface is not necessary in the present analysis. In this situation, the motion of ions has little effects on the analysis. We, therefore, uniformly distributed immobile ions as a background charge in order to reduce the computational memory and time required for the analysis.

[38] Figure 7 shows the electron density distribution measured along the direction perpendicular to the antenna. The solid and dashed lines correspond to the cases of (a) ϕa = 4ϕF and (b) ϕa = 16ϕF, respectively. Ion sheaths are created for both cases, but their sizes are different. The electron-free region expands in the case (b) compared to the case (a) due to the electron evacuation by the antenna potential. Note that the spatial gradient of density at the interface between the sheath and the uniform plasma is almost the same in these two cases. In the previous section, we found that the spatial gradient of the density is affected by λD. In the present analysis, λD is common between the two cases. Therefore, it is reasonable that the thickness of the electron-free region increases for the larger antenna potential without the change in the spatial gradient of the density.

Figure 7.

Profiles of the number density of electrons measured along the direction perpendicular to the antenna from its surface for the cases of ϕa = 4ϕF (solid line) and ϕa = 16ϕF (dashed line).

[39] The antenna capacitance C = −1/(ωX) is shown in Figure 8. The signature of impedance resonance is observed in the capacitance value near ω = Πe. One can find in Figure 8 that the intensity of the resonance depends on the sheath thickness; it is larger for the case (a) ϕa = 4ϕF than for the case (b) ϕa = 16ϕF. As described in section 3, the impedance resonance is due to the interaction between the antenna and the plasma wave. Therefore, the observed dependence of the resonance intensity suggests that the thick sheath separates the antenna from the plasma and then can weaken the interaction between the antenna and the plasma wave. The sheath thickness also affects the impedance well below Πe. As shown in Figure 8, the capacitance curves have nearly-plateau parts. The plateau value is larger for the thin sheath and tends to approach the free-space value as the sheath expands. This can be explained by a simple analogy with the two coaxial conductors: the larger the gap between the conductors, the smaller the capacitance.

Figure 8.

Antenna capacitance in 1/Ω. The solid, dashed, and dotted lines represent the capacitance in the cases of ϕa = 4ϕF, ϕa = 16ϕF, and free space, respectively.

[40] The dependence of the low-frequency capacitance on the sheath thickness as described above was reported in the previous studies [e.g., Balmain and Oksiutik, 1969; Aso, 1973]. They modeled the ion sheath which was divided into a vacuum region and a transition region in which the electron density increased linearly with respect to its ambient plasma level. The total impedance was calculated as a summation of the local impedances corresponding to each region. We confirmed that the theory basically agrees well with the present simulation outputs for the case of the thin sheath. However, as the sheath width becomes larger, the theoretical result doesn't approach to the free-space antenna impedance although the antenna capacitance should recover its free-space characteristic in the limit of wide sheath. Therefore, the theory is not applicable to the large sheath in comparison with the antenna dimensions. Furthermore, since the formulation of the local impedances was performed using the cold plasma approximation, any effects of a finite temperature on the sheath impedance cannot be treated in the theory. The present numerical method, therefore, has advantages in obtaining the complex characteristics of antenna impedance in inhomogeneous, kinetic plasma environments.

5. Discussion

[41] In section 3, we presented the EM-PIC simulations of the antenna impedance in uniform plasma and compared the results with those theoretically obtained. It was confirmed that the EM-PIC simulation results overall agree with the conventional theory. However, we found that the intensity of the impedance resonance, particularly for the imaginary part, is greater in the theoretical results than those obtained in the EM-PIC simulations.

[42] The difference found in the impedance resonance may be caused by the difference of the modeling of the current distribution on the antenna surface. In the developed EM-PIC tool, the form of the current distribution is not assumed unlike the theory but evaluated as a result of the self-consistent computation of the antenna near-field as mentioned in section 2.1. Figure 9 shows the antenna surface current distributions observed in the EM-PIC simulation results for the case of λD = La/12. The solid and dashed lines correspond to the profiles at the observation frequencies of ω = 1.0Πe and ω = 0.5Πe, respectively. The current distribution at ω = 1.0Πe, at which the strong impedance resonance was confirmed to occur, is clearly different from the triangular form. On the other hand, in absence of the impedance resonance, i.e., at ω = 0.5Πe, the triangular-like distribution is recovered. This implies that the strong resonance between the antenna and the surrounding plasma can affect the form of the current distribution. The nontriangular current distribution can be a possible reason for the impedance difference between the EM-PIC and theoretical results at the resonance frequency although the detailed mechanism of the formation of the nontriangular current distribution has not been sufficiently resolved yet. The observed nontriangular form is very important issue since the current distribution on the antenna surface affects not only the impedance but also other important antenna characteristics such as the effective length. However, the behavior of the resonance is quite complex, and further investigation of this issue is beyond scope of the present paper. The detailed analysis on this issue will be described in another paper.

Figure 9.

The normalized amplitude of the antenna surface current observed in the case of λD = La/12. The solid and dashed lines correspond to the observation frequencies ω = 1.0Πe and ω = 0.5Πe, respectively.

[43] In the results of the EM-PIC simulations the resonance frequency observed for λD = La/6 case is shifted to higher frequency in comparison with λD = La/12 case. This shift may be caused by the limited size of simulation box even though we realized an isolated system. We briefly discuss this issue here. The signature of the impedance resonance is resulted from the antenna-wave interactions as described in section 3. In the present plasma environment, the longitudinal plasma wave mode that can exist in the simulation system is only the Langmuir mode, of which the dispersion relation is given as ω2 = Πe2 + 3k2v02. Therefore, the resonance signature at very near ω = Πe should reflect the contribution of the interactions between the antenna and the Langmuir wave with large wavelength. However, in the present analysis, the size of the physical region in the simulation box is limited to 48λD due to the high computational cost of the EM-PIC simulation, and the plasma wave components that have wavelength larger than the size of the physical region cannot be supported in the simulation system. In addition, even for the wave components that can be supported in the system, wave components of wavelength much larger than the thickness of the absorbing layer are difficult to be completely absorbed by the absorbing layer. In this case, there is possibility that some wave components near the electron plasma frequency are reflected into the physical region from the simulation box edge. These limitations may have an influence on the EM-PIC results at the resonance frequency particularly for λD = La/6 case. The larger physical space in the simulation box is desirable in the future analysis to obtain the impedance value in a greater accuracy at frequencies close to the electron plasma frequency.

[44] Another point we should pay careful attention to is that we utilized a broad spectrum pulse emission from the antenna feeding point in order to compute the antenna impedance. Feeding too large energy can cause destruction of the electron density distribution in the sheath region in equilibrium. This effect is undesirable because we focus on the antenna impedance under the steady state of the sheath environment in the present study. We confirmed that the sheath structure obtained as the steady state of the plasma environment was not corrupted by the pulse emission. This is because the electric energy of the applied signal was set to 0.11 Esh, where Esh is calculated as an integral of the electrostatic energy in the sheath region, and was sufficiently small. We also checked several simulation results as a simple test by changing the amplitude of the applied signal and confirmed that almost the same results of impedance value were obtained in all cases except near the impedance-resonance frequency. Even near the resonance frequency, the difference of the impedance value was less than 3% when we doubled the signal amplitude. This implies the linear voltage-current characteristic is overall maintained in the wide frequency range. For the detailed analyses at the resonance frequency, this effect should be minimized by using a pulse with smaller amplitude in future studies.

[45] Finally, we mention about the application of the EM-PIC tool to the study of receiving-antenna characteristics. In the presence of the reciprocity relation between transmitting and receiving antennas, the present Delta-Gap feeding method can be directly applied also to the analysis of receiving antennas. However, in plasma environment, the reciprocity has been strictly proved only in very limited simple situations in past theories [e.g., Ishizone et al., 1976]. For environment of unknown reciprocity such as an ion-sheathed antenna, the use of the Delta-Gap feeding method should be limited to an analysis of transmitting antennas as shown in the present study. For the analysis of receiving antennas in such an environment, we should use different techniques from the Delta-Gap feeding method. As one of possible solutions, we may set up wave fields propagating in the simulation region and directly simulate the process of receiving the wave fields by the antenna. For the application of the present tool for simulating the receiving antennas in general plasma environments, we have been developing a plug-in routine with this technique. The details and the validity of the technique for the receiving antenna will be discussed in future publications.

[46] In the present section, we presented several key points of the EM-PIC simulation tool that should be carefully considered to improve the present state of the accuracy of the impedance analysis. However, the limitations described in the present section except regarding the reciprocity problem can be basically resolved by using larger computational resources. The larger memory enables us to take a larger size of the simulation box. The amplitude of the emitted broad spectrum signal can also be reduced by introducing a larger number of the superparticles. This is because a numerical noise, which is originated from the smaller number of macro-particles used in the simulation than that of real electrons, can be reduced, and thus a better signal-noise ratio can be realized by increasing the number of the superparticles. We believe that by performing large-scale computations, we can minimize the artificial effects on the EM-PIC results and analyze the antenna characteristics in greater accuracy.

6. Conclusion

[47] In order to investigate the antenna characteristics including the plasma kinetic effects in a self-consistent manner, we have developed a new antenna analysis tool by making use of EM-PIC plasma simulations. In the present study, we focused on the impedance of a low-power transmitting antenna because this basic property is useful for the validation of the EM-PIC tool. The developed tool was first validated by examining the wire-antenna characteristics in a homogeneous kinetic plasma. The obtained antenna impedance showed good agreement with the analytic results based on the conventional theory at frequencies below and above the electron plasma frequency. Near the electron plasma frequency, the dependence of the impedance-resonance intensity on the plasma temperature was qualitatively consistent with that expected analytically, although the peak values of the impedance resonance showed a discrepancy between the EM-PIC and theoretical results.

[48] The present tool was next applied to the analyses of the ion sheath effects on the antenna impedance. Since the sheath dynamics were simultaneously solved during the analyses, the effects on the antenna impedance was included in a more self-consistent manner than the previous works that assumed simplified structure of the ion sheath. As was predicted by the previous theories, the low-frequency capacitance was confirmed to be decreased by the presence of the ion sheath. The signature of the impedance resonance is also modified by the ion sheath. Particularly, it was revealed that the resonance is weakened when one applies a negative large bias potential, which leads to a thick ion sheath around the antenna. To understand the more detailed physical mechanism of the present results, further analyses of the energy distribution of electrons and the wave propagation properties in the sheath region are required. The PIC simulation method is effective for such detailed diagnosis [Usui et al., 2004]. The larger scale analyses will enable us to investigate the physical properties of the ion sheath in greater detail in the future.

[49] By examining the antenna impedance in the simple situations with and without the ion-sheath effects, we successfully demonstrated the present state of the validity and the effectiveness of the EM-PIC antenna-analysis tool. On the other hand, the present test revealed some limitations of the developed tool, which showed several important factors that we should improve in further development of the tool. Although the proposed approach is costly in the respect that it requires large computational resources and time, we believe that realistic and practical modeling is effective for obtaining the complex antenna characteristics in plasmas as well as for evaluating the validity of other low-cost methods.


[50] Computation in the present study was performed with the KDK system of Research Institute for Sustainable Humanosphere (RISH) at Kyoto University as a collaborative research project. Some calculation was performed with FUJITSU HPC2500 of ACCMS (Academic Center for Computing and Media Studies) at Kyoto University. The work was supported by Grant-in-Aid for Research Fellows of the Japan Society for the Promotion of Science (JSPS).