Cylindrical Waveguides for Microwave Spoof Surface Plasmon Polaritons

This work presents the theoretical and numerical design of a cylindrical waveguide that supports Spoof Surface Plasmon Polaritons (SSPP) at microwave frequencies. First, a theoretical model is proposed to assess the propagation of SSPPs in a cylindrical geometry and to estimate the surface impedance required to enable these wave modes. The reliability of the theoretical model is verified against numerical results. The proposed cylindrical waveguide embeds a cladding metasurface with a dielectric substrate covered by a lattice of square metallic patches. According to numerical estimations, the proposed design allows the propagation of SSPPs in the frequency range 2.2–5.7 GHz. Notably, the cut-off frequency is lowered by approximately 3.5 GHz with respect to conventional metallic solutions. The target application of the proposed waveguide is efficient plasma generation.


I. INTRODUCTION
Surface Plasmon Polaritons (SPP) are surface waves originally observed on metal-dieletric interfaces [1].The electromagnetic (EM) field interacts with the metal's free electrons in the optical regime, exciting a collective oscillation that propagates along the interface [2].Two notable properties of SPPs are the strong confinement of the fields to the interface and the presence of EM components along the direction of propagation [2].SPPs have been widely employed in diverse optical applications such as biosensing, microscopy, and electromagnetically induced transparency [1], [2].Nonetheless, relying on engineered structures made of subwavelength elements, it is possible to realize a medium with a negative effective permittivity at microwave frequencies, allowing the The associate editor coordinating the review of this manuscript and approving it for publication was Bilal Khawaja .
propagation of the so-called Spoof Surface Plasmon Polaritons (SSPP) [3].A largely adopted solution to implement this technology in the GHz frequency range envisions metallic gratings that emulate a negative permittivity material [4].SSPPs are particularly appealing to miniaturize circuits [5] given their low interference with the surrounding EM waves [6] and, thanks to the field confinement, the almost null cross-talk between several SSPP waveguides stacked together [7].Rectangular waveguides that support SSPP propagation in the range of millimeter-waves have been recently proposed [8], [9].Moreover, this technology allows the simple realization of bendable and conformal structures based on ultrathin corrugated metallic films [10], [11].SSPPs are also very appealing for sensing applications given that such wave modes are sensitive to the geometry of the metallic structures and the local dielectric environment [12].Sensors based on the propagation of SSPP waves have been realized for biomedical purposes [13], [14].Another intriguing application field in which SSPPs have been employed is efficient plasma generation [15].Indeed, a resonance between EM waves and free electrons is proven to be an effective way to produce and heat plasma [16], [17].
Metasurfaces are a well-known means to control surface waves [18].They are two-dimensional versions of metamaterials with subwavelength thickness [19].Notably, metasurfaces combine the capability to reconfigure EM waves massively (e.g., blocking, absorbing, concentrating, dispersing, or guiding) with lightweight and easy fabrication [20].Among their properties, metasurfaces allow manipulating the effective surface impedance emulating materials with negative permeability/permittivity [18].From one side, this enables the propagation of SSPPs-like waves [21].On the other hand, this feature has been exploited to control the EM propagation in waveguides coated with a metasurface [22].From a theoretical perspective, several works investigated the capabilities of a waveguide featuring an arbitrary surface impedance: examples are available for both rectangular [23] and cylindrical [24], [25] geometries.It is possible to stimulate the propagation of TEM modes in hollow waveguides [26] and to realize compact phase shifters [27] relying on a suitable surface impedance.Cylindrical waveguides equipped with a cladding metasurface have been proposed to enhance the operation bandwidth of a horn antenna, enabling the propagation of hybrid modes [28] or to remove the degeneracy between right-handed and lefthanded polarizations [29].Nonetheless, the main application for which cladding metasurfaces are designed is lowering the cut-off frequency of hollow waveguides with respect to conventional solutions [30].Examples with rectangular [31], [32] and cylindrical [33], [34], [35], [36] geometries are available in the literature.Notably, experimental validations of this concept proved the possibility of almost halving the cut-off frequency [31], [35].
This work aims to present a novel cylindrical waveguide designed to support SSPPs at microwave frequencies.The solution proposed envisions the use of a metasurface-cladded waveguide.The field in which this device can find its main application is plasma production [37], [38].In fact, it is well known that field-confined and resonant solutions can lead to efficient plasma generation.Examples are the Trievelpice-Gould waves in the framework of Helicon plasma sources [39] and the exploitation of the electron cyclotron resonance to design ion sources [40].Specifically, the paper addresses the design of the waveguide as a building block to realize the plasma source.Moreover, the cylindrical geometry is adopted to guarantee axial symmetry, a property usually required in several plasma applications as electric space propulsion [41].

II. METHODOLOGY
A theoretical model to assess the propagation of SSPPs in a cylindrical waveguide is proposed.Next, the numerical setup to verify the theoretical model and a realistic design of a metasurface-cladded waveguide that supports SSPPs at microwave frequencies are presented.

A. THEORETICAL MODEL
An equivalent model is introduced to assess the SSPP propagation within a metasurface-cladded waveguide (see Fig. 1).A cylinder of radius a and with an infinite extension along the z-axis is immersed in a background medium with negative relative dielectric permittivity ε r [1].The material that constitutes the cylinder is air.Following standard procedure, it is possible to solve the Maxwell's equations in the equivalent system [42,Chap. 3.4], [43,Chap. 5.3].Given that SSPP waves usually present a transverse magnetic field distribution [2], a TM 01 propagation has been investigated [21].The dependence of the EM fields from the time t and axial coordinate z is assumed in the form exp(jωt−jβz), where ω is the angular frequency in rad/s and β is the axial wave number.The following parameters are introduced to solve Maxwell's equations, where k 0 = ω/c 0 is the wavenumber in air and c 0 is the speed of light.The resultant EM fields expressed in a cylindrical reference frame (ρ,φ,z) read where E a is a complex constant in V/m, η 0 is the impedance of free space, and I 0 , I 1 , K 0 , K 1 are the modified Bessel's functions of the first and second kind, respectively.The dispersion relation which regulates the wave propagation is determined by imposing the continuity of the H φ field across the interface at ρ = a: Notably, the EM fields computed in Eqs.3-5 for ρ ≤ a are equivalent to the ones obtained substituting the interface at ρ = a with a surface whose impedance reads [36]: The definition of Z s is fundamental to link the equivalent model to the actual propagation of a SSPP in a metasurfacecladded waveguide [24].In fact, the theoretical formulation is based on the assumption (verified in the following) that the TM 01 mode described in Eqs.3-5 can propagate in a realistic cylindrical waveguide given that the cladding metasurface presents an equivalent surface impedance Z s [21].
It is worth mentioning that a similar analysis can also be done for a TE mode, assuming that the background medium is characterized by a negative relative magnetic permeability µ r [21].At the same time, the theoretical model can be generalized to handle propagation that does not enforce cylindrical symmetry, but this requires accounting for hybrid modes [30].

B. NUMERICAL MODEL
The commercial software CST Studio Suite ®has been adopted to verify the theoretical model and to design a cylindrical waveguide that supports the propagation of SSPP waves.Maxwell's equations are solved in the frequency domain, and an unstructured tetrahedral grid is adopted to mesh the computational domain.The numerical setup depicted in Fig. 2 is employed to verify the theoretical model.The native material type ''Surface Impedance'' has been attributed to the component labeled as ''Background''.In this way, a given value of Z s is imposed at the edge of the waveguide filled with air.To stimulate the propagation of a TM 01 mode, a back-plate in a perfect electric conductor (PEC) has been connected to the waveguide via a discrete port.Open boundary conditions are imposed along all the directions apart from the assumption of a null tangential electric field in correspondence of the plane z = 0 where the backplate is located.The design of a realistic metasurface-cladded waveguide that supports SSPP waves is depicted in Fig. 3.The strategy to stimulate the propagation of the TM 01 mode and the boundary conditions are the same as in the previous setup.The major difference is that the cladding metasurface is simulated, and no surface impedance is imposed.The cladding metasurface consists of a lattice of square patches located inside a dielectric tube, which, in turn, is enclosed within a cylindrical PEC surface.

III. THEORETICAL ANALYSIS
First, the theoretical model has been verified against numerical results.Second, the theoretical model has been exploited to assess the propagation of SSPP waves in a metasurfacecladded waveguide.

A. VERIFICATION
The numerical setup to validate the theoretical model consists of a waveguide of radius a = 10 mm and length L = 100 mm.The backplate is separated from the waveguide by a distance s = 2 mm.The frequency has been varied in the range f = 2-3 GHz.In the numerical simulations, the TM 01 mode propagation is triggered by imposing at the ''Background''   component the values of Z s reported in Table 1.Notably, Z s is purely imaginary [21] and its amplitude is estimated via Eqs.6-7 assuming ε r = −0.5.Fourier-transforming the E z field with respect to the z coordinate, it is possible to estimate the axial wavenumber β from the numerical results [24].At the same time, for a fixed set of the a, f , and ε r parameters, β can be estimated via the dispersion relation in Eq. 6.The results of these two approaches, numerical and theoretical, respectively, are depicted in Fig. 4 showing a remarkable agreement (errors < 1%).To further corroborate the validity of the theoretical model, the radial profile of E z is reported in Fig. 5. Also, in this case, mismatches between numerical and theoretical results are below a few percent points.

B. EM PROPAGATION
The theoretical model is exploited to assess the propagation of SSPP waves in a metasurface-cladded waveguide.First, the dispersion relation in Eq. 6 is visually represented in Fig. 6a.Eq. 6 allows determining one among the three adimensional parameters ak 0 , ε r , and β/k 0 given that the other two are specified.From a physical standpoint, ak 0 depends on the ratio between the wavelength in air λ 0 and the radius of the waveguide a.In the framework of metasurface-cladded waveguides, ε r can be considered an auxiliary parameter that describes the wave dynamics at the edge of the waveguide (i.e., it is related to Z s ).Finally, β/k 0 describes the wave propagation within the waveguide.As expected for SSPPs,

TABLE 2.
Relative dielectric permittivity ε r associated to the values of surface impedance Z s reported in Fig. 7 for ak 0 = 0.5.
the axial wavelength λ z can be shorter than λ 0 given that the dispersion relation envisages values of β/k 0 > 1 [2].Moreover, a resonance behavior (i.e., β/k 0 ≫ 1) is encountered for ε r → −1 [2], [21].A feature specific to SSPPs propagating in a cylindrical geometry is that TM fields occur for ε r < −1 only if the waveguide is sufficiently ''large'' (ak 0 > 1).If the waveguide is ''small'' (ak 0 ≲ 1) propagation is still possible for 0 > ε r > −1.Namely, a properly designed cladding metasurface might avoid the occurrence of the low-frequency cut-off typical of hollow waveguides [30].Clearly, this is a hypothetical situation given that the smaller the waveguide (i.e., the smaller ak 0 ), the larger Z s to guarantee the wave propagation (see Fig. 6b).Interestingly, Im(Z s ) > 0 as expected for the propagation of TM surface waves, and |Z s | → ∞ in correspondence of the resonance [21].To sum up, a practical utilization of the theoretical model envisions the use of Fig. 6a to determine ε r that guarantees the propagation of SSPPs (i.e., fixed β/k 0 ) given a certain geometry and frequency (i.e., ak 0 ).This value is then inserted in Fig. 6b to determine Z s that the cladding metasurface shall present.
A key feature of SSPP propagation is the fieldconfinement [3].To verify this property in cylindrical geometry, the normalized amplitude of the E z field is reported in Fig. 7.The higher Im(Z s ), the more the field is confined close to the walls of the waveguide.This behavior is consistent with SSPP waves given that higher values of Im(Z s ) mean conditions that are closer to the resonance, namely ε r → −1 (see Table 2).

IV. NUMERICAL IMPLEMENTATION
The realistic design of a cladded-metasurface waveguide that supports SSPP waves at microwave frequencies is discussed (see Fig. 3).The following parameters are assumed: radius of the waveguide a = 20 mm, the thickness of the cladding metasurface h = 6 mm, the relative permittivity of the ''Dielectric'' component ε d = 10 which is compatible with the RT/duroid ®material [44].The lattice periodicity is assumed equal to D = πa/12 ≈ 5.2 mm, namely the inner side of the cladding metasurface is covered with 24 rows of square metallic patches.The spacing between adjacent patches is w = 0.05D ≈ 0.3 mm, and the waveguide length is L = 50D ≈ 262 mm.Finally, the back-plate is distanced s = 2 mm from the waveguide.The frequency is varied approximately in the range f = 2-6 GHz where SSPP waves are proved to propagate.In this frequency interval, D/λ 0 < 0.1 allows to treat the patch lattice as a metasurface [18].

A. RESULTS
A map of the axial electric field E z computed for two values of the frequency, f = 2.9 GHz and f = 4.9 GHz respectively, is depicted in Fig. 8.The pattern is the one expected for a SSPP wave; namely, the field is maximum in correspondence with the walls of the waveguide.Notably, for f = 4.9 GHz, remarkable field confinement is noticed, being the |E z | value computed on the axis of the waveguide almost two orders of magnitude smaller than the one at the walls.Fig. 8 describes a propagative TM 01 mode with an almost constant amplitude.The maxima registered in correspondence of z ≈ 0 are associated with the stimulation of evanescent modes by the wave coupler, namely the back-plate [42].Similarly, the maxima at the front side of the waveguide (z ≈ L) are due to wave reflections because of the mismatching with the free space.The latter is not an issue since this waveguide type is meant to be coupled with a plasma reactor that shall be properly designed in terms of equivalent impedance [45].
As previously mentioned, by Fourier-transforming E z , it is possible to derive the axial wavenumber β and, in turn, the axial wavelength λ z = 2π/β (see Fig. 9a).Notably, these values are real, i.e., they describe a propagative wave only for f ≳ 2.2 GHz.A cut-off frequency is then found at f ≈ 2.2 GHz; this aspect is further discussed in section IV-B.SSPP waves subsist up to f ≈ 5.7 GHz.For larger frequencies, a TM 01 field still propagates, but it presents the features observed in conventional metallic waveguides: E z maximum on the axis and β/k 0 < 1 [42,Chap. 3.4].The frequency at which this transition occurs corresponds to the cut-off of the TM 01 mode in a conventional cylindrical waveguide [42,Chap. 3.4].Thus, an additional property of the proposed metasurface-cladded waveguide is to lower the TM 01 cut-off frequency by almost 3.5 GHz with respect to the equivalent metallic solution: from f ≈ 5.7 GHz to f ≈ 2.2 GHz [30].
According to Fig. 9a, λ z decreases with f while the normalized wave number β/k 0 increases.Indeed, the larger field confinement found at higher values of f is consistent with the increment of β/k 0 (see section III-B).To better understand the dynamics of SSPP waves in a realistic metasurface-cladded waveguide, the auxiliary parameter ε r is estimated via Eq.6 (see Fig. 9b).This is possible given that ak 0 is known from the geometry and the frequency considered, and β/k 0 from the analysis of the numerical results.Similarly, the values estimated for Z s are reported in Fig. 9b.It is found that ε r → −1 increasing f , namely the system tends toward the resonance [21].Concurrently, Im(Z s ) reaches a plateau at ≈ 1.1 k .Notably, having an almost constant Im(Z s ) while ε r → −1 is consistent with theoretical predictions given that the normalized radius ak 0 grows with f (see section III-B).
The consistency of the propagating TM 01 mode with the SSPP waves is eventually performed by comparing the E z profile evaluated numerically against the one estimated with the theoretical model and the parameters reported in Fig. 9. Results sampled along the radial direction for a fixed axial position are depicted in Fig. 10.Curves agree reasonably well for ρ < a, proving further that the waveguide supports SSPP waves.Marginal inconsistencies close to ρ ≈ a are due to fringing effects, given that homogenization is no longer valid near the metasurface [18].The E z fields within the ''Dielectric'' component (green background in Fig. 10) decrease to 0 in correspondence of the outer PEC layer at ρ = a + h.

B. DISCUSSION
From a physical perspective, the propagation of SSPP waves is regulated by the surface impedance of the cladding metasurface [21], [30].The latter is driven by design parameters such as a, h, D (see Fig. 3) and the geometry of the unit cells that constitute the metasurface (e.g., square patches, loops, or Jerusalem crosses [30]).Specifically, the trends reported in Fig. 9 strongly depend on how Z s varies with f and, therefore, on the design of the metasurface.The same holds for the frequency interval in which SSPP waves propagate.A qualitative analysis is reported in Fig. 11, which supports the existence of a cut-off at f ≈ 2.2 GHz.As the theoretical model prescribes, SSPPs propagate only if Z s assumes certain values.The latter is depicted in Fig. 11 as a function of f for three different values of ε r .Namely, for a given frequency, SSPPs can propagate only if the metasurface presents a surface impedance in the interval delimited by the curves for 0 > ε r > −1.As an approximation, the cladding metasurface is assumed to present the same Z s of a planar structure made of a lattice of square metallic patches [21,Eq. 16].Relying on this assumption, the cut-off frequency, namely the lowest frequency for which Z s is sufficiently large to intercept the curve for ε r → 0, occurs at f ≈ 2.2 GHz.Moreover, the estimated value Im(Z s ) ≲ 1.0 k fairly matches the numerical one (see Fig. 9b).Despite the accurate consistency with the numerical results, it is worth stressing that this should be taken only as a qualitative analysis given that the cladding metasurface has been approximated as a planar one [46].Nonetheless, this is sufficient to confirm the governing mechanism of this waveguide.For frequencies below the cut-off, the cladding metasurface presents too low values of the modulus of Z s for sustaining SSPP waves [18].

C. PERFORMANCE ASSESSMENT
Finally, the performance of the proposed metasurface-cladded waveguide is compared against other solutions presented in 23196 VOLUME 12, 2024 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
the literature (see Table 3).Hollow waveguides relying on SSPP propagation usually operate in the range of millimiterwaves [8], [9].Instead, the present design works at markedly lower frequencies consistently with usual plasma production technology [37], [38].At the same time, other solutions that operate in the microwave range target applications as stimulating the propagation of TEM waves [22], [26], [31] or reducing the cut-off frequency [30], [33], [34], [35], [36].The latter, in particular, rely on hybrid (e.g., HE 11 ) or TE waves.Thus, the specificity of this work depends on the analysis of SSPP propagation in hollow waveguides at microwave frequencies.

V. CONCLUSION
This work presents a novel cylindrical waveguide that supports SSPPs at microwave frequencies.The design relies on a cladding metasurface consisting of a lattice of square metallic patches placed on the inner surface of a substrate tube.A theoretical model has been proposed to assess the propagation of SSPPs in this cylindrical geometry, and verification against numerical results has proven its reliability.According to numerical predictions, the proposed design allows the propagation of SSPPs in the frequency range f ≈ 2.2-5.7 GHz.For higher frequencies, the propagating wave mode is no longer confined to the waveguide's wall, i.e., it is no longer a SSPP.The low-frequency cut-off occurs since the modulus of the surface impedance of the cladding metasurface is not large enough to sustain SSPPs.Notably, the proposed design reduced the cut-off frequency by almost 3.5 GHz with respect to a conventional metallic cylindrical waveguide of similar size [42].
The main application of such a cylindrical waveguide that supports SSPPs at microwave frequencies is plasma production.Strong field confinement and electron resonance are, indeed, desired characteristics of efficient plasma sources [37], [38].The coupling of this waveguide with a plasma reactor will be left to future work along with the production of a working prototype.About this point, manufacturing a lattice of square patches on the inner surface of a substrate tube is feasible yet nontrivial [31], [35].Possible strategies to realize a proof of concept are discussed in Appendix.

APPENDIX VIABLE MANUFACTURING PROCESSES
Several strategies can be adopted to fabricate the proposed metasurface-cladded waveguide.The first option exploits conventional techniques for printed circuit boards, such as inkjet or screen printing.In this case, in order to produce the cylindrical shape of the waveguide, a flexible substrate should be used that can be printed flat and later be bent and locked in shape by an external dielectric/metal sheet.So far, ultrathin copper strips have been deposited on a dielectric film to produce SSPP devices operating in the range of millimiterwaves [4].The growing demand for flexible circuits (e.g., sensors and wearables) has fostered the adoption of several dielectric materials such as polyethylene (PET or PEN) and polydimethylsiloxane (PDMS) [47], [48], [49].Notably, substrates with a dielectric permittivity larger than ten times the vacuum one have been proposed [50].Conductive inks based on metal nanoparticles are commonly employed for the realization of the conductive pattern [50].
A second option involves novel technologies that have the potential of producing conductive patterns directly on curved surfaces [51].With intense pulsed light-induced mass transfer, patterned circuits based on zinc nanoparticles were fabricated on complex, convex and concave surfaces [52].A micromolding-based method (MIMiC) has been recently reported which enables scalable printing of complex conductive patterns on curvilinear substrates [53].In this case, the waveguide shall be produced in two parts, namely two hemicylinders, where the substrate does not need to be flexible but can be produced with any suitable dielectric material.The external metal sheet would ensure tightness and geometrical accuracy in the final assembly.

FIGURE 1 .
FIGURE 1. Schematic of the equivalent system used to investigate the propagation of SSPPs in a cylindrical waveguide.

FIGURE 2 .
FIGURE 2. Cut view of the numerical setup used to verify the theoretical model.

FIGURE 6 .
FIGURE 6.(a) Axial wavenumber β normalized with respect to k 0 , and (b) imaginary part of the surface impedance Im(Z s ) vs. relative dielectric permittivity ε r .Results of the theoretical model for different values of the adimensional parameter ak 0 .

FIGURE 7 .
FIGURE 7. Modulus of the normalized axial electric field |E z /E a | vs. normalized radial position ρ/a.Results are associated with different values of the surface impedance Z s .

FIGURE 8 .
FIGURE 8. Modulus of the axial electric field |E z | vs. axial z and radial ρ coordinates for (a) f = 2.9 GHz, and (b) f = 4.9 GHz.

FIGURE 9 .
FIGURE 9. (a) Estimated axial wavelength λ z and wavenumber β.(b) Equivalent relative dielectric permittivity ε r and surface impedance Z s vs. frequency f .

FIGURE 11 .
FIGURE 11.Surface impedance Z s as a function of the frequency f for three values of the parameter ε r .Stars indicate the intercepts of these curves with the analytical estimation of Z s for a planar metasurface (''Planar Metasurface'').

TABLE 1 .
Imaginary part of the surface impedance Im(Z s ) vs. wave frequency f according to the theoretical model.

TABLE 3 .
Performance of the proposed metasurface-cladded waveguide against other solutions presented in the literature.The symbol ∅ indicates diameter.