Plasma-Based Reflective Surface for Polarization Conversion

This study analyses, for the first time, the use of reflective surfaces based on magnetized plasmas for polarization conversion. The feasibility of this concept has been assessed via a theoretical model. Moreover, the numerical design of a plasma-based reflective surface is presented. The latter enables linear-to-linear (LP-to-LP) and linear-to-circular polarization (LP-to-CP) conversion over a broad frequency range, from 7.5 to 13 GHz. To this end, the applied magnetic field intensity has to be tunable over 55–140 mT and its direction steerable toward three mutual orthogonal axes. At the same time, the plasma density has to be controlled up to ${2\times 10^{18}}\,\,\text{m}{^{-3}}$ . These requirements are consistent with the plasma technology at the state-of-the-art.

conductor [15] elements, in both the transmission and reflection modes [16]. Notably, LP-to-LP [17] and LP-to-CP [18] ultrawideband conversions have been demonstrated relying on anisotropic metasurfaces. Reconfigurable metasurfaces have been proposed to integrate various polarization conversions in a single device [19]. Switching between linear polarization (LP) and circular polarization (CP) [20] as well as between right-handed and left-handed CP [21] is feasible by integrating diodes on the elements that constitute the metasurface.
An alternative approach to controlling an EM wave's polarization envisions using magnetized plasmas [22], [23]. Theoretical investigations proved that by varying the plasma properties (e.g., density), it is possible to selectively control the transmission through a magnetized plasma layer of the right-handed or left-handed CP waves [22], as well as transverse electric (TE) or transverse magnetic (TM) waves [23]. At the same time, EM signals can be controlled by relying on plasma-based reflective surfaces [24], [25]. In these devices, the unit cells consist of plasma elements placed on top of a ground plane. For the first time, the present study explores the use of reflective surfaces based on magnetized plasmas for polarization conversion. Specifically, an analytical model and a preliminary design are proposed to prove that this concept enables significant improvements in terms of polarization conversion with respect to the state-of-the-art. Indeed, a single plasma-based reflective surface might allow generic LP-to-LP conversion and LP-to-CP conversion with the possibility of selecting between the right-handed and left-handed CP. To the best of the authors' knowledge, no other solution proposed in the literature enables all these features in the GHz frequency range and for a relative bandwidth larger than 50%. Moreover, the significance of the numerical results presented in this work is strengthened by the assumption of realistic plasma properties compatible with the technology at the-state-of-the-art [3], [26].

II. METHODOLOGY
The EM response of the plasma is described via its relative permittivity ε r . Given the usual operation frequencies (i.e., in the GHz range) and plasma properties of GPAs [2], ε r is formulated according to the cold plasma model where the ions' motion is neglected [27]. ε r takes the form of a dyadic tensor to account for the anisotropy induced by the magnetostatic field B 0 [28]. Without loss of generality, it is practical to express ε r in a reference frame where the z-axis is aligned with B 0 where ȷ is the imaginary unit, and the adimensional parameters S, D, and P read [27] The relation between plasma properties and the adimensional parameters X , Y , and U follows [27]: where ω, ω p , and ω c are the operation angular frequency, the plasma frequency, and the cyclotron frequency, respectively, in rad/s. Instead, the parameter ν is the collision frequency in Hz. Here, ω p , ω c , and ν depend on the macroscopic plasma parameters according to the following relations [27]: where q is the elementary charge, m is the electron mass, ε 0 is the vacuum permittivity, n e is the plasma density in m −3 , B 0 is the intensity of the magnetostatic field in T, n 0 is the neutral background density in m −3 , and K is a rate constant that depends on the electron temperature T e . If plasma is produced from argon gas, K reads [29] where T e is expressed in eV. It is worth stressing that ω p depends on n e , and ω c depends on B 0 ; this means that in a magnetized plasma, the EM response (i.e., the reflection of an incident wave) can be controlled electronically by acting on two macroscopic parameters. Indeed, the plasma density n e depends on the electric power used to sustain the discharge [10]. At the same time, the intensity of the magnetostatic field B 0 can be varied using electromagnets (e.g., by current-controlled solenoids) [30]. The collision frequency ν is associated with Ohmic losses occurring within the plasma [24] that are proportional to n 0 . With this regard, it is worth introducing the neutral gas pressure p 0 , which represents an auxiliary parameter more often used to characterize plasma discharges rather than n 0 where k B is Boltzmann's constant and T 0 is the temperature of the neutral gas. According to the state-of-the-art for plasma discharges, the maximum value of n e is proportional to p 0 [6], [7].
In an anisotropic medium where ε r is given by (1), the relation between the components of the electric field reads [27] where E = (E x ; E y ; E z ) is the complex electric field vector, θ is the angle between B 0 and the wave vector k, and u is the refractive index. From (7), the following dispersion relation derives [27]: where the adimensional parameter Q reads For a given direction of the wave vector (i.e., for a fixed θ ), two values of the refractive index, namely, u + and u − , satisfy the dispersion relation reported in (8), where the subscripts are consistent with the signs used in (9). Thus, two types of waves may exist simultaneously, from now on called modes [27]. Specifically, it is possible to identify cutoff and resonance frequencies that delimit each mode's propagation interval (i.e., u almost real). Outside the interval within the cutoff and resonance frequency, the dispersion relation describes an evanescent wave [31]. For clarity, the cutoff occurs at u = 0, while the resonance occurs at u → ∞ [27]. For what mode + is concerned, ω p is a cutoff. Instead, mode − shows two cutoffs and one resonance frequency. The cutoffs occur at the following frequencies: and the resonance frequency occurs at For what follows, it is more practical to define the propagation intervals in terms of the corresponding plasma density values n e rather than the operation frequency ω [24]. Specifically, the critical densities n p e , n R e , n L e , and n 0 e identify the values of n e for which ω is equal to ω p , ω R , ω L , and ω 0 , respectively. Specifically, the mode + propagates for n e < n p e , while the mode − propagates for n e < n R e and n 0 e < n e < n L e . Instead, if n e ≥ n L e , the plasma behaves as a conductor, namely, only evanescent waves may exist [27].

A. Theoretical Model
A theoretical model has been developed to assess the use of a magnetized plasma as a polarization converter. A schematic of the considered setup is depicted in Fig. 1. A homogeneous plasma slab of thickness z pl is located on top of an infinite perfect electric conductor (PEC), constituting the ground plane [24]. A uniform magnetostatic field B 0 is assumed to be aligned with the z-axis of the reference frame depicted in Fig. 1. A linearly polarized plane wave impinges the plasma slab normally. The wave propagates along the ξ -axis, forming an angle θ with B 0 . The reflected wave is described via the following relation: where E i is the complex electric field vector associated with the incident wave, E r corresponds to the reflected wave, and is the reflection coefficient [24]. As done above, the superscripts refer to mode + and mode −, respectively. In particular, E + i and E − i describe the component of the incident electric field which excites the propagation of the mode + and mode − within the plasma, respectively. According to the classical transmission line model [31], the reflection coefficient reads where ρ is Fresnel's reflection coefficient at the air-plasma interface, pl is the reflection coefficient within the plasma medium, and the superscripts refer to the mode + and mode −, respectively. Fresnel's reflection coefficient reads [31] where λ = c/ f is the wavelength in air, and c is the speed of light in the vacuum. To separate the E + i and E − i components, the first step consists of expressing the incident field into the (x y z) reference frame. Therefore, for geometric construction (see Fig. 1), the following relation holds at the air-plasma interface: It is worth noting that E ξ = 0 provided that the incident field is a plane wave propagating along the ξ -axis [31]. Using (7), from which E + x and E − x can be calculated. Once E + x and E − x are known, the vectors E + i and E − i are determined via (7) and, thus, E r is calculated via (12).
Given the value of E r , we can analyze the capability of a magnetized plasma slab to act as a polarization converter. The following reflection coefficients: help quantify LP-to-LP conversion, as they are sufficient to describe the behavior of an η-axis polarized incident wave. In this case, the polarization conversion ratio (PCR) is defined as The PCR quantifies the amount of power transferred from the incident wave to its cross-polarized direction (i.e., the ζ -axis). At the same time, the parameter used to describe the LP-to-CP conversion is the axial ratio (AR), namely the ratio between the major and the minor axes of a generic polarization ellipse [32]. Conventionally, CP is defined for AR ≤ 3 dB [32].

B. Numerical Model
Numerical analyses have been accomplished with the commercial software CST microwave Studio 1 . The schematic of a simulated 1 Registered trademark. plasma element is depicted in Fig. 2. Its thickness is z pl , the square plasma block measures L pl × L pl , and the square ground plane is L × L. The parameter L also represents the lattice periodicity [24]. The plasma is handled in CST via the native dispersion model called "Gyrotropic", conceived for media described by the anisotropic permittivity reported in (1); moreover, a generic direction of the magnetostatic field is allowed. Simulations rely on an unstructured tetrahedral mesh, where Maxwell's equations are solved in the frequency domain. As for the theoretical model, a linearly polarized plane wave is assumed to normally impinge the plasma element along the ξ -axis. Two simulations have been performed: 1) singleelement analysis to verify the theoretical model and 2) array design to evaluate the capability of a plasma-based reflective surface to act as a polarization converter. The single element analysis is accomplished assuming Floquet boundary conditions along the η-axis and the ζ -axis; an open condition is imposed along the ξ -axis [24]. Moreover, L = L pl for consistency with the theoretical model, which refers to a slab geometry. Instead, open boundary conditions have been adopted along all directions to simulate the plasma-based reflective surface.

III. ELEMENT ANALYSIS
The theoretical model has been exploited to investigate the use of a magnetized plasma as a polarization converter, both LP-to-LP and LP-to-CP conversion have been considered. Numerical simulations have been performed to verify the theoretical model.

A. LP-to-LP Conversion
The considered plasma slab has thickness z pl = 21 mm, i.e., 0.7λ at the central frequency f = 10 GHz. The LP-to-LP conversion is  analyzed assuming the magnetostatic field with intensity B 0 = 95 mT and aligned with the ξ -axis, i.e., θ = 0 • . A neutral gas pressure p 0 = 0.4 mbar has been assumed for consistency with the plasma generation technology at the current state-of-the-art [2]. Without loss of generality, when the incident electric field is linearly polarized along the η-axis, the LP-to-LP conversion can be characterized via the ηη , ζ η , and PCR parameters defined in Section II-A.
The corresponding ηη and ζ η are depicted in Fig. 3 as a function of n e for L = L pl = 15 mm [24]. First, we can observe an excellent agreement between theoretical and numerical results (within an error < 1%). Regarding the modulus of the reflection coefficient, | ηη | is minimum and | ζ η | is maximum in correspondence of n e = 8.0 × 10 17 m −3 . Here, PCR ≈ 1, meaning an almost complete cross-polarization conversion occurs. Instead, for n e → 0 or n e ≈ 10 19 m −3 , PCR → 0, meaning the plasma does not affect the polarization. As a result, it is possible to rotate the LP in an arbitrary direction controlling n e . The phase of the reflected wave is almost unaffected by the magnetostatic field, being practically the same as that produced by a non-magnetized plasma element [24]. Indeed, if n e → 0, the wave is almost not influenced by the presence of the plasma, and it is correspondingly reflected by the ground plane. If n e ≈ 10 19 m −3 , the plasma behaves itself as a good conductor, and the wave is reflected at the air-plasma interface with ang( ) ≈ −180 • . Intermediate situations are associated with different propagation paths within the plasma before the wave is reflected. Specifically, if n e ≥ n L e only evanescent waves occur within the plasma, thus the EM propagation is almost entirely determined by the reflection at the air-plasma interface [24].
From a physical standpoint, the proposed LP-to-LP conversion relies on the Faraday rotation effect [27]. Namely, the two propagative modes occurring for θ = 0 • are identified as R-waves and L-waves. The former is a right-handed CP wave propagating along the direction of the magnetostatic field, whilst the latter is a left-handed CP wave [27]. Being u + ̸ = u − , the longer the path traveled by these two waves within the plasma before being reflected, the larger the rotation induced in the polarization direction. This is consistent with data reported in Fig. 3 since, for n e ≲ n R e , the Faraday rotation reaches its maximum before the R-wave becomes evanescent [27].
The capability of the proposed design to perform LP-to-LP conversion is analyzed in Fig. 4, where the operation frequency f  varies. As one can note, cross-polarization conversion can take place over a remarkably large operation bandwidth (approximately from 7 to 13 GHz) by simply reconfiguring the plasma properties. Specifically, the relative bandwidth in which PCR ≥ 0.98 is 55% of the central frequency f = 10 GHz. It is worth noting that: 1) the condition n e ≲ n R e is enforced by all the n e and B 0 pairs reported in Fig. 4 and 2) the intensity of both n e and B 0 are consistent with the plasma production technology at the current state-of-the-art [3], [26]. This confirms that the LP-to-LP conversion via plasma-based reflective surfaces is a feasible and appealing technology. Finally, Table I reports the plasma properties to achieve cross-polarization conversion at different values of z pl . It is worth noting that the values of B 0 reported here are the minimum ones that allow achieving PCR ≈ 1.0 for at least one interval of n e . In fact, the higher B 0 , the higher the Faraday rotation effect [27]. By increasing too much B 0 , the polarization direction might vary by more than 90 • , which is out of the present scope. As one can note, the required intensity of B 0 is inverse proportional to z pl . In particular, a slab thickness of z pl = 21 mm is a trade-off between the compactness and the technological feasibility of the proposed design.

B. LP-to-CP Conversion
The same plasma slab analyzed in Section III-A is investigated in terms of LP-to-CP conversion. In this case, the magnetostatic field has intensity B 0 = 110 mT, and it is aligned with the ζ -axis, i.e., θ = 90 • . Moreover, the incident electric field is linearly polarized at 45 • in the η-ζ -plane. The reflected wave is then analyzed in terms of circular polarization components through the parameter AR. Fig. 7. Schematic of the plasma-based reflective surface. Each element is characterized by L = 15 mm, L pl = 12 mm, and z pl = 21 mm. Fig. 5 shows the parameter AR as a function of n e for the same plasma element described in Section III-A. Again, an excellent agreement is found between the theoretical and numerical results (within an error <1%). The condition AR ≤ 3 dB, which corresponds to an efficient LP-to-CP conversion, is achieved for n e in the range 7.0×10 17 -1.1×10 18 m −3 . From a physical standpoint, this achievement is related to the wave modes excited by the incident electric field E i . The E ζ,i component, parallel to B 0 , couples to an ordinary wave (O-wave). In contrast, the E η,i component, perpendicular to B 0 , couples to an extraordinary wave (X-wave) [27]. According to (8), O-the waves are associated with the mode + and X-waves with mode − [27]. Provided O-waves and X-waves are orthogonal [31], the CP condition is obtained for a 90 • phase shift between + and − and, in turn, between E ζ,r and E η,r . Moreover, it is possible to switch between the right-handed and left-handed CP by rotating E i of 90 • in the η-ζ plane. A more practical implementation of this feature (not envisioned by the theoretical model in its current form) is obtained by orienting B 0 along the η-axis instead of the ζ -axis.
The proposed design is analyzed as the operation frequency f varies; see Fig. 6. By reconfiguring the plasma properties, a circularly polarized reflected wave is achievable in a large operation bandwidth (approximately from 7 to 14 GHz). Remarkably, the relative bandwidth for which AR ≤ 3 dB is 65% of the central frequency f = 10 GHz. The intensities of n e and B 0 are consistent with the plasma production technology at the state-of-the-art [3], [26]. Therefore, this confirms that LP-to-CP conversion via a plasma-based reflective surface is feasible and appealing.

IV. ARRAY DESIGN
A design closer to practical implementation has been analyzed numerically to investigate further the feasibility of a plasma-based reflective surface with polarization conversion capabilities The setup is depicted in Fig. 7: it is composed of 10×10 plasma elements, each characterized by L = 15 mm, L pl = 12 mm, and z pl = 21 mm. The lattice periodicity is L = 0.5λ at the central frequency f = 10 GHz [32]. The condition L pl = 0.8 L has been imposed to account for the additional equipment required to operate the array ensuring homogeneity and controllability of the plasma properties (e.g., vessels to confine the neutral gas and electrodes to ignite the discharge) [3].
The performance of the plasma-based reflective surface, in terms of PCR, is shown in Fig. 8 as the operating frequency is varied; here, θ = 0 • . It is confirmed that PCR ≥ 0.98 over a broad operation bandwidth, approximately 7-13 GHz. Concerning the theoretical model's results shown in Fig. 4, higher values of n e are required to obtain similar outcomes consistently to what happens in nonmagnetized systems due to the presence of propagation paths outside the plasma medium [24]. Nonetheless, this is only a minor issue since the values of B 0 and n e are still within feasible ranges for the plasma technology at the state-of-the-art [3], [26]. Similar considerations hold for AR, here computed for θ = 90 • . In this case, the operation bandwidth where AR ≤ 3 dB spans between 7.5 and 13 GHz. It is worth mentioning that the performance depicted in Fig. 8 is very close to the one derived from the reflection coefficients of each element of the reflective surface: differences are <0.01 in terms of PCR, and <0.1 dB in terms of AR. Similar results have also been obtained in the case of nonmagnetized systems [24], highlighting a generally mild cross-talking among plasma elements.
To sum up, given that 1) n e is varied up to 2.0 × 10 18 m −3 with a resolution of 1.0×10 17 m −3 , 2) B 0 is varied in the range 55-140 mT, and 3) the direction of the magnetostatic field is changed over the ξ -axis, η-axis, and the ζ -axis, then the proposed plasma-based reflective surface enables: 1) the cross-polarization conversion; 2) LP-to-LP conversion for a generic polarization direction; 3) LP-to-CP conversion with the possibility to select between the right-handed and left-handed CP. All these features are achievable from 7.5 to 13 GHz, namely, with a 55% relative bandwidth with respect to the central frequency f = 10 GHz, and for a fixed incident electric field. No other design proposed in the literature enables all these features in the GHz frequency range and for such a large operation bandwidth. Moreover, the requirements listed above are feasible with the plasma technology at state-of-the-art and assuming the magnetostatic field is generated with a suitable set of electromagnets [30].

V. CONCLUSION
A feasibility study has been accomplished to assess, for the first time, the use of reflective surfaces based on magnetized plasmas for polarization conversion. A theoretical model has been formulated to analyze LP-to-LP and LP-to-CP converters relying on this concept. Afterward, the numerical design of a plasma-based reflective surface is presented. The latter enables 1) cross-polarization conversion, 2) generic LP-to-LP conversion, and 3) LP-to-CP conversion with the possibility to select between the right-handed and left-handed CP. To the best of the authors' knowledge, this is the first concept enabling all these features in the GHz range with a relative bandwidth larger than 50%. From a technological standpoint, it requires the plasma density to be controlled up to 2.0 × 10 18 m −3 , and the magnetic field shall be varied in the range 55-140 mT with the possibility to change its direction along three perpendicular axes. The practical implementation of these constraints, while challenging, is entirely feasible. In fact, the controllability of the plasma density has been demonstrated in the field of GPAs [2], [10], and electromagnets providing variable magnetostatic fields up to 200 mT are commonly used in electric space propulsion [30].
In conclusion, the proposed solution is extremely appealing regarding polarization conversion capabilities. Thus, the realization and testing of a proof of concept will be the subject of future works. At the same time, a drawback of plasma-based reflective surfaces is their relative thickness, almost doubled with respect to other systems presented in the literature [18], [20]. Nonetheless, the most critical aspect that might hinder the applicability of these devices in realistic scenarios (e.g., intelligent reflecting surfaces-IRSs [33]) is the nonnegligible power consumption, in the range of several watts [3]. This drawback can be substantially mitigated by taking advantage of the recent technological advances made in the field of electric space propulsion in terms of optimization and miniaturization of the electronics to manage the plasma [26].