Plasma-Based Reflecting and Transmitting Surfaces

A theoretical model is proposed to preliminary assess the performance of plasma-based reflecting and transmitting surfaces. The model has been verified and, subsequently, exploited in a feasibility study to implement beam steering and polarization control. Theoretical findings laid the basis for the numerical design of a reconfigurable plasma-aided horn antenna. Notably, the solution proposed is capable of both beam steering and polarization control relying on a plasma-based transmitting surface. Specifically, the main lobe can be steered in the range 0° – 50° maintaining the gain >10 dBi, the relative side lobe level < −10 dB, and the reflection coefficient < −10 dB. The bandwidth is 1 GHz for an operation frequency of 10 GHz. Polarization conversion is feasible maintaining values of the plasma parameters and magnetic induction field compatible with the technology at the state of the art.


I. INTRODUCTION
Gaseous Plasma Antennas (GPAs) are devices in which an ionized gas, namely plasma, is used to transmit and receive electromagnetic (EM) signals [1]. GPAs offer several advantages over metallic antennas [2]. First, it is possible to reconfigure the antenna performance (e.g., radiation pattern and operation frequency) by electronically controlling the plasma properties (e.g., density) [3], [4]. Second, when the plasma is switched off, the antenna almost stops interacting with the EM signal [5]. Thus, GPAs are particularly suitable to form arrays or to be applied in scenarios where stealth is required [6], [7]. Third, the EM response of a GPA is frequency dependent; thus, antennas with different operation frequencies can be stacked together [8].
Among GPAs, plasma-based reflecting surfaces are particularly suitable to implement beam steering [9], polarization The associate editor coordinating the review of this manuscript and approving it for publication was Ali Karami Horestani . control [10], and to combine these two features [11]. Numerical designs have been proposed to implement plasma-based reflecting surfaces via Cold Cathode Fluorescence Lamps (CCFL), which is a plasma production technology at the state of the art [12], [13]. At the same time, plasma-based transmitting surfaces are put forward to enable beam steering operations [14], [15], [16] or polarization control [17]. The possibility to exploit plasma discharges in a transmitarray [18], [19] or a plasma lens [20] has been investigated too.
The scope of the present paper is twofold. First, a theoretical model is proposed to assess the feasibility of a reconfigurable plasma-based transmitting and reflecting surface. Second, the numerical design of a plasma-aided horn antenna is presented. The latter relies on a reconfigurable plasma-based transmitting surface placed in front of a metallic horn antenna. Theoretical models to handle the propagation of EM waves through a plasma slab are available for both reflecting [9], [10] and transmitting [21], [22] surfaces. The novelty of the proposed approach lies in the adoption of the formalism of ABCD matrices [23] to quickly evaluate the performance of both reflecting and transmitting surfaces in terms of beam steering and polarization control. Regarding the plasma-aided horn antenna, this is the first design ever proposed that enables both beam steering and polarization control via a plasma-based transmitting surface. Similar features have been demonstrated relying on conventional metasurfaces [24], [25], [26], [27] but never in the framework of GPAs.

II. METHODOLOGY
The relative permittivity of plasma ε pl is a tensor that accounts for the anisotropy triggered by a magnetic induction field B 0 [28]. It is convenient to express ε pl in a Cartesian reference frame where the z-axis is aligned with B 0 , The adimensional parameters S, D, and P read [28] where the coefficients X , Y , and U are strictly related to the plasma properties. The following relations apply [28] where ω p is the plasma frequency, ω c the cyclotron frequency, ν the collision frequency, and ω the wave frequency. Notably, ω p , ω c , and ν are driven by macroscopic plasma parameters [28] ω p = q 2 n e mε 0 , where q is the elementary charge, m the electron mass, ε 0 the vacuum permittivity, n e the plasma density, B 0 the intensity of the magnetic induction, n 0 the neutral density, and K a rate constant which depends on the electrons' temperature T e . Specifically, ω p depends on n e , which, in practice, can be controlled by varying the electrical power used to sustain the plasma discharge. Similarly, ω c depends on B 0 , which can be reconfigured relying on electromagnets. The parameter ν is associated with Ohmic losses and has a major dependence on n 0 , which, according to the plasma discharges at the state of the art [29], is related to the maximum value of n e required.
The following dispersion relation describes the wave propagation in plasma [28] where u is the index of refraction and the adimensional parameter Q reads The angle θ forms between the vector B 0 and the direction in which the EM waves propagate. Eq. 5 has two solutions: u + and u − , respectively (the notation refers to the sign adopted in Eq. 6). Namely, two wave modes characterized by the index of refraction u + and u − coexist in a magnetized plasma [10]. This holds unless B 0 = 0, since an unmagnetized plasma is an isotropic material in which just one degenerate wave mode, associated with the index of refraction u, propagates [9]. Wave modes are propagative if their index of refraction is almost real [10]. This occurs in an unmagnetized plasma if n e ≤ n p e , where n p e identifies the value of plasma density for which the wave frequency equals ω p [9]. On the other hand, if B 0 ̸ = 0, intervals of propagation are delimited by the thresholds n R e , n 0 e , n p e , and n L e [10]. These values of plasma density trigger specific resonances or cut-offs [28]. For example, the condition n e > n L e in a magnetized plasma implies that EM waves are evanescent [10].

A. THEORETICAL MODEL
A theoretical model is proposed to evaluate the feasibility of plasma-based transmitting and reflecting surfaces. Unlike other approaches proposed in the literature [9], [10], the current model relies on the formalism of ABCD matrices [23], which allows treating configurations both in transmission and reflection. A plane wave propagating along the ξ -axis (see Fig. 1) is assumed to impinge normally on a uniform plasma slab of thickness ξ pl . Both magnetized and non-magnetized plasmas can be handled. If present, the magnetic induction field B 0 , aligned with the z-axis, forms an angle θ with the ξ -axis and, in turn, with the direction of propagation of the plane wave. In a transmitting configuration, air extends to the infinite on both sides of the plasma slab. The transmitted and reflected fields read VOLUME 11, 2023 where E i , E t , and E r are the complex vectors that describe the incident, transmitted, and reflected fields, respectively. T is the transmission coefficient evaluated at the second airplasma interface, while is the reflection coefficient at the first air-plasma interface (see Fig. 1). On the other hand, in a reflecting configuration, the plasma slab is assumed to be placed on top of an infinite ground plane made from a Perfect Electric Conductor (PEC). The reflected field reads where 0 is the reflection coefficient at the air-plasma interface (see Fig. 1). The coefficients T , , and 0 can be computed according to the formalism of the ABCD matrices. The components of the ABCD matrix associated to the plasma slab read [23] where λ is the wavelength in air and Z 0 is the intrinsic impedance of vacuum. In case of a transmitting surface, the coefficients T and read [23] While, for a reflecting surface, 0 reads [23] where T and are computed according to Eq. 10. It is worth specifying that, to avoid a cumbersome notation, the relations presented in this section apply to each wave mode, namely, they hold straightforwardly for a nonmagnetized plasma. General results can be easily obtained by combining the contributions of multiple wave modes according to the procedure described in [10]. At the same time, the proposed model holds if the intensity of the signal is not high enough to affect plasma properties. Considering the technology at the state of the art [29], this condition applies if the signal power is less than several Watts [3].
Finally, once E r and E t are known, the capability of a plasma surface to exploit polarization conversion is quantified in terms of Polarization Conversion Ratio (PCR) and Axial Ratio (AR) [10], [11]. The former describes the cross-polarization conversion while the latter conventionally defines a circularly polarized signal if AR < 3 dB [30].

B. NUMERICAL MODEL
Numerical simulations are performed with the commercial software CST Studio Suite ®. Analyses are accomplished to verify the theoretical model and to preliminary design a reconfigurable plasma-aided horn antenna. The former simulations aim at reproducing the plasma slab depicted in Fig. 1. Thus, a rectangular plasma block of thickness ξ pl is analysed assuming Floquet boundary conditions along the η-axis and ζ -axis while open boundary conditions are imposed along the direction of propagation of the plane wave (i.e., the ξaxis) [9], [10], [11]. The horn antenna is simulated assuming open boundary conditions in all directions. An exhaustive description of the antenna design is reported in section IV.

III. THEORETICAL RESULTS
The theoretical model has been verified against numerical results to assess its reliability. Subsequently, it has been exploited to compare the performance of plasma-based transmitting and reflecting surfaces.

A. VERIFICATION
The theoretical model has been verified against numerical results in the case of both non-magnetized and magnetized plasmas (see Fig. 2 and Fig. 3, respectively). Given a wave frequency f = 10 GHz, a non-magnetized plasma slab of thickness ξ pl = λ = 30 mm is considered. A plane wave linearly polarized along the η-axis is assumed to impinge on the slab. Comparison against numerical results is reported in terms of the amplitude of the T and coefficients, which describe a transmitting surface (see Fig. 2(a)), and phase of 0 , which is associated with a reflecting surface (see Fig. 2(b)). Results show a good agreement between theoretical and numerical results, with differences within 1%. The same slab is adopted to verify the theoretical model in the case of a magnetized plasma with B 0 = 100 mT, and θ = 45 • . The amplitude of T and the phase of , associated with a transmitting surface, are depicted in Fig. 3. Provided the capability of a magnetized plasma to affect polarization, it is necessary to refer T and to the η and ζ components of the transmitted and reflected fields, respectively. For example, T ζ η indicates the ratio between the ζ component of E t and the η component of E i . Also, in this case, theoretical results match well the numerical ones, with minor differences within 1%. This confirms the reliability of the proposed model.

B. TRANSMITTING AND REFLECTING SURFACES
The theoretical model has been exploited to compare the performance of a plasma-based surface either in transmitting or reflecting mode, with the aim to assess the pros and cons of the two configurations. A non-magnetized plasma slab of thickness ξ pl = λ = 30 mm (f = 10 GHz) is analysed in transmitting and reflecting configuration in Fig. 4. Consistently with the literature [9], the amplitude of the reflection coefficient 0 is close to the unit for each value of n e considered. This is associated with a mild Ohmic loss occurring in the plasma medium [9]. On the other hand, in the transmitting configuration T drops for n e > n p e which is consistent with the absence of waves propagating for this range of densities. This aspect imposes a limit on the possibility of steering a transmitted signal. Indeed, even if its phase can be reconfigured over 360 • varying n e , only in the range n e ≤ n p e the intensity of the transmitted signal is non-negligible. In the analyzed configuration, the phase of T can be controlled over approximately 180 • if |T | ≈ 1 is required. This limitation can be overcome by enlarging ξ pl , in fact, the phase of 0 varies more rapidly than for T given that the path travelled by a wave in a reflecting configuration is doubled with respect to the equivalent transmitting configuration. The theoretical model predicts a value of ξ pl ≈ 2λ to guarantee a control of the phase of T over 360 • ensuring |T | ≈ 1.
Similar considerations hold if a reflecting surface is meant to implement polarization conversion. In Fig. 5 two arrange-ments are analyzed to implement cross-polarization and linear to circular polarization (LP-to-CP) conversion, respectively. Consistently with the literature [10], the assumption θ = 0 • is adopted to implement cross-polarization conversion, which is triggered by the Faraday rotation associated with the propagation of R-and L-waves [28]. As proven in Fig. 5(a), the condition PCR ≈ 1 is achieved both with transmitting and reflecting surfaces. Nonetheless, the former case requires higher values of both B 0 and ξ pl . LP-to-CP conversion is obtained assuming θ = 90 • [10]. In this case, polarization is controlled by relying on the propagation of Oand X-waves [28]. The condition AR < 3 dB is achieved both for transmitting and reflecting configurations, but the former impose more demanding requirements in terms of B 0 and ξ pl (see Fig. 5(b)).
To sum up, transmitting plasma surfaces are proven to be an appealing solution given that the values of n e , B 0 , and ξ pl identified to implement beam steering and polarization control are consistent with the plasma technology at the state of the art [29]. Clearly, this solution requires a larger value of ξ pl with respect to a plasma reflecting surface [9], [10], [11] to ensure a propagation path long enough to implement the desired features while maintaining |T | ≈ 1. Similarly, higher, but still realistic, values of B 0 are needed to exploit polarization conversion.

IV. PLASMA-AIDED HORN ANTENNA
Based on the results presented in section III, a numerical model has been developed for the design of a reconfigurable plasma-aided horn antenna. The latter consists of a plasma-based transmitting surface placed in front of a metallic horn antenna (see Fig. 6). The operation frequency is Where s is the thickness of a dielectric vessel that confines each plasma element. The EM response of the dielectric is computed assuming the relative permittivity ε r = 2, and loss tangent tan δ = 5.4 × 10 −3 . The signal is fed via a waveguide port placed on the back of the horn antenna. It is worth noting that to enhance the reliability of the proposed design, the electrodes to sustain the discharge and the dielectric vessel to confine the plasma have been included. Moreover, ν = 3.1 × 10 8 Hz consistently with the plasma technology at the state of the art [29].

A. BEAM STEERING
The proposed design is analysed in terms of beam steering capabilities. The value of n e in each of the ten plasma  Table 1. elements that constitute the transmitting surface is varied to impose a certain phase profile and, in turn, to tilt the direction of the main lobe in the azimuth plane (ξ -η). A steering angle φ max = −50 • is achievable assuming n e according to Table 1. Notably, the prescribed values have been derived by applying the array factor rule [30] to the results of the theoretical model described in section II. Reconfiguring n e , it is possible to vary |φ max | in the range 0 • -50 • maintaining the gain > 10 dBi, the relative Side Lobe Level (SLL) < −10 dB, the reflection coefficient |S 11 | < −10 dB, and the total radiation efficiency η > 0.85 (see Fig. 7 and Table 2).
The results obtained regarding beam steering can be analysed according to the theoretical model described in section II. It provides qualitative indications given that some of its hypotheses are not used in the design at hand. For example, the EM field radiated by a horn antenna in the near field is appreciably different from that of a plane wave [30] and, according to the present theoretical formulation, the dielec- tric vessels that confine the plasma have been neglected. Nevertheless, the theoretical model provides a preliminary estimation of n e . This is done by selecting the n e values that produce the phase shift prescribed by the array factor rule to induce a certain steering angle (see Table 1). Moreover, some values of n e reported in Table 1    require higher values of n e [9]. This is the main limitation of the steering range since imposing |φ max | > 50 • causes the increase of the elements in which n e > n p e and, in turn, |T | < 1. Thus, the radiation pattern is distorted and steering angles larger than 50 • are not achievable with the proposed horn antenna. From a technological standpoint, the maximum value n e = 2.41 × 10 18 m −3 is compatible with plasma discharges at the state of the art [29].
The proposed design is further analysed as the operation frequency varies (see Fig. 8 and Table 3). Remarkably, the  conditions gain > 10 dBi, SLL < −10 dB, |S 11 | < −10 dB, η > 0.85, and φ max within ±1 • of the nominal value are maintained in the range 9.1-10.1 GHz. This has been obtained for the specific case φ max = −40 • , but comparable results hold for different steering angles. It is worth mentioning that the aperture efficiency η A of the proposed plasma-aided horn antenna varies in the range 0.65 − 0.8 following the same trend of the gain [30]. These values are compatible with standard horn antennas [30]. Finally, the bandwidth of the plasma-based systems is generally large compared with more standard solutions [24], [27], consistently with the reflecting surface technology [9], [10], [11].

B. POLARIZATION CONVERSION
The proposed plasma-aided horn antenna is reconfigurable also in terms of polarization, provided that a suitable induction magnetic field is generated by electromagnets in the nearby. To this end, a uniform value of n e and B 0 is imposed on all the plasma elements (see Table 4). Cross-polarization conversion can be achieved assuming θ = 0 • [10]. According to Fig. 9, PCR > −1 dB in correspondence of an angle of aperture ≈ 60 • centered in correspondence of the main lobe. LP-to-CP conversion is feasible too if θ = 90 • [10] and ψ = 25 • ; the latter indicates the angle between B 0 and the η-axis. In this case, AR < 3 dB within an angle of aperture ≈ 30 • centered in the broadside direction. Notably, CP is not achieved assuming ψ = 45 • , as prescribed by the theoretical model (i.e., E i rotated of 45 • with respect to B 0 [10], [11]). This result depends on the EM field generated by a horn antenna which, in the near field, is appreciably different from that of a plane wave [30].
The possibility to implement polarization conversion comes at the cost of a gain < 10 dBi, even though the condition |S 11 | < −10 dB is maintained (see Table 4).
Consequently, the aperture efficiency decreases to η A = 0.6 − 0.45, while η > 0.85. From a technological standpoint, the values of n e and B 0 required to implement polarization conversion are feasible at the state of the art [29].

C. SENSITIVITY ANALYSIS
The robustness of the proposed plasma-based horn antenna is assessed via a sensitivity analysis. The effect on the antenna performance of the dielectric vessels' thickness s and relative permittivity ε r as well as plasma collision frequency ν has been investigated. The performance when the antenna is exploited for LP-to-CP conversion is depicted in Fig. 10. The values s = 0.2 mm and ε r = 2, selected for the design, allow minimizing AR. Specifically, a tolerance of ±50% on both parameters ensures AR < 3 dB. At the same time, increasing the collision frequency above the design value ν = 3.1 × 10 8 GHz causes a decrease in the gain and degradation of the AR. Nonetheless, AR < 3 dB if ν ≈ 1.5 × 10 8 GHz, which is 5 times larger than the design value. So, there is a sufficient safety margin in selecting these design parameters.
The AR is the indicator more affected by s, ε r , and ν. Thus, the operation margins are larger if the plasma-based horn antenna is intended for beam-steering only. For example, increasing ν of one order of magnitude causes a decrease of the gain of ≈ 1 dB (see Table 5). At the same time, other performance indicators as φ max , SLL and |S 11 | are marginally affected by s, ε r , and ν. These trends can be explained from a physical standpoint since Ohmic losses occurring within plasma correlate with ν [9]. Instead, s and ε r can modify reflections occurring at the plasma interfaces with a consequent effect on the transmission coefficient T [9]. This might be studied with an upgraded version of the theoretical model, including the propagation through the dielectric slab [23]. Nonetheless, such an effect seems not critical for the present design, so this upgrade of the theoretical model is left for future work.

V. CONCLUSION
A theoretical model that relies on the formalism of the ABCD matrices has been proposed to preliminary design plasma-based reflecting and transmitting surfaces. It has been verified and exploited to prove the feasibility of plasma-based surfaces for beam steering and polarization control. The findings of the theoretical model laid the basis for the design of a reconfigurable plasma-aided horn antenna. The latter relies on a plasma-based transmitting surface placed in front of a metallic horn antenna. The numerical design accounts for practical constraints as the presence of a dielectric vessel that confines the plasma and the interaction between the transmitting surface a realistic EM field. For the first time, the capability to perform both beam steering and polarization control has been demonstrated for a GPA operated in transmission. The possibility to steer the main lobe in the range 0 • − 50 • with a gain > 10 dBi, a relative SLL < −10 dB, |S 11 | < −10 dB, and η > 0.85 has been proven over a bandwidth of 1 GHz. Moreover, both cross-polarization and LP-to-CP conversion are feasible assuming values of the plasma parameters and magnetic induction field compatible with the technology at the state of the art [29].
The realization and testing of the plasma-aided horn antenna will be the subject of future work. In fact, the proposed design is technologically feasible, but several challenges shall be overcome before its practical implementation. First, the numerical design relies on rectangular plasma elements. One technological solution to implement this design envisions the use of Dielectric Barrier Discharges (DBD), which have been widely employed in the framework of plasma display panels [31]. Nonetheless, advances in terms of electrode design and power consumption are needed for their exploitation in the field of GPAs. It is worth specifying that this is not a killing factor for the technology at hand since, according to recent measures, the assumed values of plasma density and magnetic induction field are feasible with plasma discharges commonly employed for GPAs [11], [12]. Second, the design of the electronics to sustain the plasma shall be optimized and miniaturized according to the solutions recently proposed for space electric propulsion [32], [33]. To conclude, the path toward realizing a reconfigurable plasma-aided horn antenna is challenging but feasible.