Experimental study of electromagnetic wave scattering from a gyrotropic gaseous plasma column

The control of electromagnetic waves has long been the interest of researchers in physics and engineering. Alternative methods of redirecting, filtering, modulating, and attenuating waves have led to various applications in wireless sensing and communication. In recent years, the development of engineered metamaterials has led to properties such as negative refraction and near-zero permittivity that are not attainable in natural or previously synthesized materials. Negative refraction enables cloaking^1–3 as well as the ability to focus waves to below the diffraction limit.⁴ 

The research on engineered photonic and plasmonic structures, some of which behave as optical analogs to topological electrical insulators,^5–7 have led to more exotic electromagnetic wave control. Most of these previous studies have been focused on the use of metallic and dielectric materials as the constituent building blocks, althoughgases and liquids have recently been used for constructing the metamaterial or photonic domains.

One interesting medium that has only recently been explored for incorporation into such structures is a gaseous plasma. Gaseous plasmas possess unique features not present in other material phases such as low collisional damping and continuously variable physical properties that can be tuned in time and space. Often modeled to have a Drude dielectric response where the tunable parameter is plasma density, the relative permittivity of plasma $εp$ can be adjusted to achieve metallic (⁠$εp<0$⁠) or dielectric (⁠$0<εp<1$⁠) behavior with a temporal response limited by rates of plasma recombination or ionization.

A magnetized plasma has an anisotropic permittivity that results in a gyrotropic response. As a result, it is a topologically non-trivial bulk material that can possess chiral, unidirectional interfacial edge states that are immune to scattering.^7–9 This behavior leads to non-reciprocity, enabling potentially reconfigurable or tunable one-way waveguides that may be of interest in various applications.

However, experimental plasmas exhibit many non-ideal factors that represent a disconnect between what is experimentally plausible and the exciting theoretical results previously demonstrated for gyrotropic materials. As such, producing a bulk, uniform magnetized plasma medium with geometry tailored to specific applications is not feasible in practice. To combat this, devices composed of smaller gyrotropic elements, such as the cylindrical plasma discharges in this study, can be designed to exhibit the same physics as a larger bulk gyrotropic medium. Before using such smaller components, however, it is important that their gyrotropic response is confirmed and well-characterized.

Leveraging the understanding gathered from past simulations and theoretical studies that describe the scattering from a gyrotropic plasma column,^10–13 this paper demonstrates a low-pressure magnetized plasma discharge with an electromagnetic gyrotropic response. Characterization of this phenomenon allows it to be utilized as an experimentally viable component for photonic devices such as magnetized plasma metamaterials and photonic crystals, non-reciprocal waveguide structures, and other devices that require a tunable gyrotropic response.

An experimental platform is designed and fabricated to study the scattered wave from a magnetized plasma rod. Figure 1 shows the experimental setup, which consists of a broadband microwave horn antenna (A INFO LB-20180 2—18 GHz) that is used to send incident transverse-electric polarized (TE) electromagnetic waves from 3 to 8 GHz in frequency along the x-axis defined in Fig. 1(b). In this study, we define the TE polarization such that the electric field of the source is perpendicular to the plasma rod. The gaseous plasma discharge is positioned in the path of the wave and placed in the center of a Helmholtz coil setup to allow for axial magnetization of the plasma discharge. The scattered wave from the plasma rod is measured by the open-source electromagnetic field robot OpenEM.¹⁴ The robot consists of two orthogonal stepper motors positioned 6 cm away from the rod, allowing it to map a 15 × 30 cm² area with a 0.75 cm step size. A computer controls the linear stages using OpenEM Labview software, and the amplitude and the phase of the wave are measured at each mapping grid point via a vector network analyzer (HP 8722D). θ is defined as the angle at the rod's center between each point of the mapped area and the x-axis shown in Fig. 1(b).

The plasma rod is a custom-fabricated discharge source consisting of a 15 mm diameter quartz tube with an inner wall thickness of 1 mm and a length of 290 mm, filled to approximately 250 Pa with argon and added mercury, the vapor pressure of which is determined by the operating temperature. The gas temperature during nominal operation is approximately 330 K, resulting in a mercury vapor partial pressure of about 3.5 Pa. The discharge is driven by an alternating-current (AC) ballast at 33.1 kHz frequency with 144 V peak-to-peak voltage and 298 mA root mean square (RMS) discharge current at the nominal operating condition. These types of plasma discharge tubes have been used in our past studies as non-magnetized inclusions in dielectric photonic crystals^15,16 and as the basic building block for 2D and 3D non-magnetized plasma photonic crystals.^17,18 The plasma density is tuned by using a variable AC transformer to vary the AC voltage delivered to the ballast, resulting in a typical achievable plasma frequency range of about 2–9 GHz. These plasma frequencies, as well as other plasma properties such as mean electron energy and electron momentum scattering frequency, can be estimated from discharge operating conditions such as the RMS voltage and current, fill pressure, and discharge geometry (area and length of the positive column), via BOLSIG+,^15,19 which is an online tool that solves the zero-dimensional Boltzmann equation for the electron energy distribution function in the local field approximation. From an estimate of the local reduced electric field, BOLSIG+ returns the mean electron energy, electron mobility, and electron momentum collision frequency. For the conditions in which the electromagnetic scattering is studied in this paper, the BOLSIG+ suggests that the plasma and collision frequencies are approximately 9 and 2.3 GHz, respectively, and the electron temperature is approximately 1.7 eV, assuming a current-carrying plasma diameter of 13 mm.²⁰ 

The Helmholtz arrangement for generating the external magnetic field consists of two water-cooled coils driven by a DC power supply (10 A, 100 V). The setup provides a quasi-uniform 47.5 mT magnetic field for magnetically biasing the plasma placed at the center of the coils. The upper and lower coils are conduction cooled using lines that circulate water at 2.5 °C to eight aluminum cooling pods in contact with the coils. A block diagram depicting functional elements of the entire setup is shown in Fig. 2.

We investigate the scattered field for two different plasma configurations: magnetized with an upwards oriented magnetic field (B+) and a downwards oriented magnetic field (B−). Experimental results are complemented by corresponding electromagnetic simulations to better understand the dynamics of the scattering fields from the magnetized plasma column. Figure 3(a) plots the scattered field's normalized Poynting vector S/S_max obtained using the frequency-domain solver from Computer Simulation Technology (CST) Microwave Studio 2022.²¹ The plasma rod is 3D modeled as a smaller gyrotropic plasma core (4.5 mm in radius) separated from a 1 mm thick quartz envelope (inner radius of 6.5 mm) by a thin, 2 mm non-ionized gap (with vacuum permittivity) to account for the non-uniform density profile in the bulb. This three layer model is a simple representation of the actual distribution of plasma density within a discharge tube,²² and the results are in acceptable agreement with experiments involving plasma photonic crystals.^17,18 In the simulations, we use periodic boundary conditions along the z-axis to mimic an infinitely long plasma column. The computed scattered fields are also in the z = 0 plane consistent with the experimental measurements. The plasma frequency, electron momentum collision frequency, and cyclotron frequency are $ωp=$ 5.28 $×1010$ rad/s, $νc=$ 1.34 GHz, and $ωce=$ 8.4 $×109$ rad/s, respectively. The values for ω_p and ν_c used to obtain optimal agreement between measured and computed scattering, while slightly different than those estimated from previous non-magnetized studies, are consistent with estimates derived from the BOLSIG+ simulations.

The plasma's relative anisotropic permittivity (⁠$ε̿p$⁠), which is responsible for its gyrotropic scattering, assuming a time dependence of the form $e+jωt$⁠, is given by

where

The permittivity tensor, non-symmetric due to the applied magnetic bias, defines the interaction of the electromagnetic wave with electrons within the plasma. Governing the off-diagonal terms, $εg$ characterizes the strength of the gyrotropy of the magnetoplasma. In a non-magnetized plasma, permittivity is isotropic and the radiated wave from the plasma rod is oriented in the direction of propagation of the incident wave. As a result of the interaction with a gyrotropic plasma rod, the radiated field from the excited plasma is rotated.

Figure 3(b) presents the normalized Poynting vector S/S_max of the scattered electromagnetic field experimentally measured at 4.42 GHz in the area delimited by the black box on the simulation results. Experimentally, a scattered wave is obtained by performing two wavefronts scans. For the first scan, the plasma discharge is removed, and the incident wave emitted by the horn is mapped out. For the second scan, the plasma discharge is inserted, the magnetic coils are activated, and the incident and scattered waves are collected by the robot with amplitude and phase recorded at the exact same locations as for the first scan. The scattered wave is calculated by subtracting these two scans. We see that the electromagnetic simulations and experimental measurements are in good agreement. There is a clear asymmetry in the scattering induced by the magnetized plasma, a confirmation of the active gyrotropic response. For an upwards oriented (B+) field, the scattered field is gyrotropically bent to the left of the x-axis along which the incident wave is propagating. For a downwards oriented (B−) field, the scattered field is gyrotropically bent to the right of the x-axis. This experimentally demonstrates the ability to bend electromagnetic waves using a magnetized plasma rod by changing the direction of the axially applied magnetic field on the plasma.

In the black boxed region of Fig. 3(a), the scattered power P_s is computed at every angle θ of the experimental and simulated wavefronts by integrating the Poynting vector along a given θ. Figure 3(c) presents the scattered power in the (B−) and the (B+) configurations, respectively, denoted by $Ps−$ and $Ps+$⁠. Good quantitative agreement is obtained between simulated and experimental results, and peak scattered power is notably reached at equal angles.

Figure 4 shows the difference, $ΔP=Ps−−Ps+$⁠, of experimentally obtained scattered powers at every angle θ between the (B+) and (B−) configurations over the 3.50–5.65 GHz range, normalized by the maximum scattered power difference, $ΔPmax$⁠, detected over this range. In the (B−) configuration, scattering into negative angles is favored, and in the (B+) configuration, scattering into positive angles is favored. The peak scattered power is approximately situated between 20° and 40° over this range of frequencies. As expected from previous theoretical studies of a collisionless magnetized plasma,¹⁰ some frequencies seem to display stronger degrees of gyrotropic response. In our experimental study, such a frequency band is located around 5.15 GHz. Furthermore, we expect the maximum scattered power's angle to change with variations in the magnitude of the external magnetic field.¹⁰ For example, CST simulations at 4.42 GHz suggest that doubling the external magnetic field keeping all other parameters constant could double this angle derived from the black-boxed region in Fig. 3(a).

The fact that this magnetically biased plasma is indeed gyrotropic is not unexpected as the anticipated electron Hall parameter with collisionality estimated from these scattering measurements suggests that for these discharges, $ωce/νc≈6$⁠, consistent for conditions of modestly magnetized electrons. Furthermore, assuming an electron temperature of $Te≈$ 1.7 eV as estimated from the BOLSIG+ simulations (mean electron speed, $ce≈×106$ m/s), the electron gyroradius, $rg≈×10−4$ m, is much smaller than the plasma discharge diameter, and so the plasma is not expected to be significantly demagnetized by electron collisions with the quartz wall.

In conclusion, we demonstrated a magnetically biased low-pressure plasma discharge with substantial gyrotropic response despite non-ideal experimental factors such as plasma collisionality. This response is detected in the 3.5–5.6 GHz frequency regime by measuring the rotation in the forward-scattered wave phase fronts. Our experimental investigation is in good quantitative and qualitative agreement with simulations. As such, this study paves the way toward the use of these plasma elements as building blocks for complex devices such as magnetized plasma photonic crystals, metamaterials, topological insulators, and non-reciprocal waveguides.

This research is partially supported by the Air Force Office of Scientific Research through a Multi-University Research Initiative (MURI) with Dr. Mitat Birkan as Program Manager. J.A.R. acknowledges support by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Department of Energy Computational Science Graduate Fellowship under Award No. DE-SC0019323. L.S.H. acknowledges support of the Stanford France Center for Interdisciplinary Studies.