Helical antennas are commonly used to launch waves in ICRF. One example of the helical antennas is shown in Figure 1, where two rings on both sides are connected by two helical straps displaced by 180 degrees with respect to each other.

A pair of terminal points is located at the center of one ring or one strap in order to feed the RF current. The pitch of the strap is in the right hand sense with respect to the external magnetic field B₀, and therefore the antenna in Figure 1 is called the right helical antenna. RF electric fields in the cylindrical coordinate are generally represented as

where E₀ is the complex amplitude, ω is the wave angular frequency, m is the azimuthal mode number, and k is the axial wave number. It is well known that the right helical antenna excites electric fields given by Equation (1) with m = −1, and the left helical antenna with m = +1 in the direction of B₀. Since the m = −1 fields are left-hand polarized (E_L), it couples to ICW, of which wave fields rotate in left-hand sense as they propagate in the direction of B₀. The m = +1 fields, which are right-hand polarized (E_R), are suitable to launch HW in an appropriate frequency range. In VASIMR [2] [3] two helical antennas are equipped: the left helical antenna is used to generate plasmas by HW, and the right helical antenna to heat ions by ICW at the ion cyclotron resonance.

We note that the right helical antenna shown in Figure 1, which excites the m = −1 fields in the direction of B₀, excites m = +1 fields simultaneously in the direction opposite to B₀. This means bi-directional excitation of the electric fields with different modes from a single helical antenna, and it can be used both to generate the plasma and to heat ions.

We calculate propagation, damping and power absorption of waves when the antenna excites m = −1 or +1 fields in a non-uniform plasma. Maxwell equations with a source term representing the antenna current density are solved in an arbitrary non-uniform axisymmetric magnetic field and plasma density in the two-dimensional r-z plane. The wave field E satisfies the equation:

where ω is the wave angular frequency, c is the velocity of light, μ₀ is the permittivity in vacuum, is the antenna current density, and is the dielectric tensor. For an axisymmetric external magnetic field B₀ with, is given by

with

Here, ω_pe is the plasma frequency, v_i and v_e are the thermal velocity of ions and electrons, respectively, and k_|| is the parallel wave number determined by the local cold plasma dispersion relation [9] [10] .

Equation (2) is solved numerically in a model device depicted in Figure 2. The device diameter is 100 mm for −400 mm < z < 0, and 350 mm for 0 < z < 550 mm. The applied magnetic field strength on axis is schematically given in the lower part of the figure. It has a peak at some point and decays in both sides. RF waves are

launched from an antenna consisting of three axisymmetric current loops L₁, L₂, and L₃ located near z = −200 - −100 mm.

It should be noted that the helical type antenna cannot be directly implemented since the calculation is in two- dimensional. We approximate the current in the helical antenna by plural loop currents with azimuthal phase difference along the axial direction. Approximate positions of a Langmuir probe and a Faraday cup, which are to be used in the following experiments, are illustrated in the device.

The current density in the helical antenna is approximated by

where J_k, r_k, z_k, and ϕ_k are the amplitude of the current, the radial position, the axial position, and the angle of the azimuthal position of the k-th loop of the antenna, respectively [11] , and ω/2p = 1.8 MHz. We represent the helical variation by the discrete step change in the azimuthal angle of each loop, and consider only the azimuthal component of the current. In this expression, when B₀ is in the +z direction, m = +1 represents the right-rotating field and m = −1 the left. The summation in m is taken for m = −1 and m = +1, which gives azimuthal variation of J_θ as cosθ (apart from ϕ) that is just the fundamental component of the current in the ring conductors at both ends of the helical antenna and also the helical straps (neglecting axial current) shown in Figure 1. A collection of appropriately phased circular antennas might also be employed in the real magnetoplasma thruster design.

We first use one loop (J₁ = 1, r₁ = 25 mm, z₁ = −130 mm, J₂ = J₃ = 0) only in order to focus on propagation and damping of fields of a specific azimuthal mode number in non-uniform plasma. The present two-dimen- sional calculation gives profiles of physical quantities in the r-z plane represented by the shaded region in Figure 2 for the plasma composed of electrons, H⁺ ions, and H₂ with the electron-neutral collision frequency ν divided by ω being 0.05. Figure 3 shows the calculation results of (a) the assumed magnetic field strength, (b) the assumed electron density n_e of the hydrogen plasma, (c) the left circularly polarized field E_L, (d) the right circularly polarized field E_R, (e) the power absorption for ions P_ion, and (f) the power absorption for electrons P_elec

in the case of m = −1 mode with the maximum magnetic field strength B_m = 1.5 kG. Here, the electric field amplitude is a complex number and its real part (Re) is plotted. Each frame represents the color contour map in the r-z plane with the legend on the right. The profile of n_e follows that of in the direction parallel to B₀ and parabolic profile in the radial direction. For this magnetic field profile, the ion cyclotron resonance is located at z = 10 mm. It is found that E_L is large near the axis while E_R is small and exists around the plasma edge. RF power is absorbed by ions at the ion cyclotron resonance point as shown in (e), and much smaller power is absorbed by electrons.

It is clearly seen in (c) and (e) that E_L extends both sides from the antenna and strongly damped near z ~ 0 at resonance, yielding the power absorption by ions. Since no resonance point exists in the left side of the antenna, the wave field exhibits standing wave profile.

We use three loops of the antenna (z₁ = −250, z₂ = −200, z₃ = −150 mm, and r_k = 25 mm), and set ϕ₁ = −p/2, ϕ₂ = 0, ϕ₃ = p/2 to approximate the right helical antenna. In Figure 4 are plotted assumed, Re and Im parts of E_L (solid and broken blue lines), Im(E_R) (broken green line), P_ion (red line), and P_elec (purple line) in the case of (a) B₀//+z and (b) B₀//−z with the maximum magnetic field strength B_m = 1.3 kG at z = −200 mm. We plot m = −1 component for E_L and P_ion, and m = +1 component for E_R and P_elec, since E_L (E_R) is the main component of the electric fields of m = −1 (+1) mode. All quantities are of on axis value except P_elec that is at r = 1 cm.

The ion cyclotron resonance is located at the axial position of crossing point of B₀ and horizontal dotted line marked as ω/ω_ci = 1. We see in Figure 4(a) that the amplitude of E_L is large on the right side of the antenna (indicated as ANT) and it propagates towards the positive-z direction because the time dependence of E_L is E_L(t) = Re(E_L)cosωt + Im(E_L) sinωt. The amplitudes of Re(E_L) and Im(E_L) are damped at one of the cyclotron resonance points (positive side of the antenna) and, as a consequence, the wave power goes to ions as shown by large increase in P_ion at the resonance point of positive side of the antenna. The amplitude of Im(E_R) is large at the negative-z side of the antenna. The propagation direction of E_R is not clear since the standing wave is formed between the metal walls of both sides. The value of P_elec is slightly larger at the negative-z side than at the positive-z side.

When the direction of B₀ is reversed so that B₀//−z, completely opposite phenomena are observed in Figure 4(b); the amplitudes of Re(E_L) and Im(E_L) are damped at the cyclotron resonance point on the negative-z side, where P_ion is particularly large, while the amplitude of Im(E_R) is large at the positive-z side of the antenna and P_elec is asymmetrically larger at the positive-z side with respect to the center of the antenna. Figure 4 reveals that the right helical-like excitation of the antenna generates m = −1 mode fields to the positive-z direction and m = −1 mode fields to the negative-z direction in the case of B₀//+z, and vice versa in the case of B₀//−z. This bi-directionality of selective excitation of the m = −1 and +1 modes leads to controlling the ion heating by E_L and the plasma production through electron acceleration by E_z associated with E_R.

It is found in the previous section that the RF field of m = −1 and +1 mode provides ion and electron heating, respectively. It is usual to use the helical antenna with positive or negative pitch for each mode excitation. We try to launch both the m = −1 and +1 modes from a single helical antenna and perform ion heating and plasma production simultaneously.

The configuration of the experimental apparatus [12] is the same as shown in Figure 2 with the loop antenna replaced by the helical antenna of right helical pitch in the +z direction that is shown in Figure 1. The ion saturation current I_is corresponding to the plasma density is measured by a Langmuir probe (LP) located at z = −320 mm and the parallel ion temperature T_i by a Faraday cup (FC) at z = 260 mm (see Figure 2). The electron temperature T_e can be measured by the Langmuir probe and the Faraday cup. The helical antenna is 200 mm long and fed by a 2-kW RF amplifier at 1.8 MHz. Low density plasmas are first produced by using RF at 13.56 MHz with hydrogen gas and introduced to the chamber from the left side as a base plasma. Then the 1.8 MHz RF power is energized and time-dependent plasma parameters are measured.

Hydrogen gas is supplied to the device at a pressure of 1.6 × 10⁻⁴ Torr, and the current in the magnetic coils set to I_AC = 50 A, which amounts to B_m = 1.18 kG. Figure 5 shows the time variation of I_is as measured by LP after switching on the base plasma at 0.2 ms and the 1.8-MHz RF at 1.2 ms for the magnetic field directed to (a) the +z direction and (b) the −z direction. Timing of each pulse is illustrated at the top of the figure by the black line for the base RF and the red line for the 1.8-MHz RF. It is seen that I_is increases by the RF pulse in the case (a) but not in (b). The present right helical antenna is expected to excite the left (right) rotating RF field in the +z (−z) direction. The RF field propagating to the −z direction (toward LP) is of m = +1 mode for the case (a) and of m = −1 mode for the case (b). This explains larger increase in I_is with the RF pulse for the case (a) since the m = +1 mode wave is expected to couple with electrons to enhance ionization.

Data from FC is given in Figure 6, where T_i is plotted versus the magnetic coil current I_AC with and without the RF pulse for the magnetic field directed to (a) the +z direction and (b) the −z direction. The ion cyclotron resonance (B_m = 1.18 kG) locates near the antenna for I_AC = 50 A. It is evident that only in the case (a) T_i increases over the values for the base plasma with larger increment for the magnetic field closer to the ion cyclotron resonance.

Ion energy distribution function is obtained from the relation;

where V_C and I_C are the retarding voltage and the current of the collector of FC, respectively, and V_P is the plasma potential measured by LP [13] . The ion energy distribution function, i.e., the value of Equation (5) is plotted as a function of V_C in Figure 7 for (a) B₀//+z and (b) B₀//−z with I_AC = 50 A. In our device V_P is negative with respect to a reference point, and the abscissa in Figure 7 that corresponds to the energy in eV spans from negative to positive. It is shown that with the application of 1.8-MHz RF f shifts toward higher energy side by 3 eV or so and becomes broader in width in the case of (a), but not in (b). The ions are heated by the RF only when the magnetic field direction is from the antenna to FC.

This is again consistent with the characteristics of field excitation of the right helical antenna. The RF field propagating to the +z direction (toward FC) is of m = −1 mode for the case (a) and of m = +1 mode for the case (b), and the m = −1 mode RF that is left-circularly polarized can heat ions near the ion cyclotron resonance in the perpendicular direction. The perpendicular energy is converted to the parallel energy in the diverging magnetic field and is observed by the parallel-aligned FC.

T_e can be measured both by LP and FC. In Figure 8 left two graphs, (a) and (c), show T_e measured by LP and right two graphs, (b) and (d), measured by FC. Also, upper two, (a) and (b), are for the magnetic field directed to

the +z direction and lower two, (c) and (d), for the −z direction. In all four graphs T_e is plotted as a function of I_AC. If we compare the increment in T_e by the application of 1.8-MHz RF (the difference in the data values with and without RF), the incremental value is relatively large in the cases of negative-z side with B₀//+z (upper left) and positive-z side with B₀//−z (lower right). The m = +1 mode RF field that couples with electrons propagates toward the negative-z direction (toward LP) for B₀//+z and positive-z direction (toward FC) for B₀//−z as discussed in Section 2.3. This explains larger increase in T_e in the two cases in Figure 8.

We see from Figures 5-8 that the right helical antenna with B₀ in the +z direction can enhance the plasma production in one direction and can heat ions in the other direction simultaneously. This is the consequence of bi-directional mode-selective wave excitation of the helical antenna. A straight antenna has no capability of mode-selection, and bi-directional production and heating would not be obtained.

These results lead to the possibility that a magnetoplasma thruster with variable specific impulse is composed of only one helical antenna and one RF power source. The plasma production takes place at one side (say upstream side) of the antenna where the magnetic field is lower to allow HW propagation, and the ion heating takes place at downstream side where the magnetic field is higher to fulfill ion cyclotron resonance condition. We have confirmed in the experiment that the negative-z (upstream) side plasma with densities increased by the 1.8 MHz RF flows positive-z (downstream) side and exhausted through the magnetic nozzle. In reality Figure 8(a) and Figure 6(a) show the dependence on magnetic field strength of electron temperature at the negative-z side of the antenna and ion temperature at the positive-z side, respectively. This is the direct indication that by changing the magnetic field strengths in both sides of the antenna relatively, we are able to control the ratio of T_e to T_i, namely, the plasma production to the ion heating. It is necessary to confirm the present findings with higher RF power, which is left for future experiments.