M Del Mastro

M Ettorre

Fellow, IEEE Anthony Grbic
email: agrbic@umich.edu

Michele Del Mastro
email: michele.delmastro@univ-rennes1.fr

Dual-Band, Orthogonally-Polarized LP-to-CP Converter for SatCom Applications

Keywords: Circular polarization, dual-band polarizer, periodic structures, LP-to-CP conversion, SatCom, metasurfaces. I

Dual-Band, Orthogonally-Polarized LP-to-CP Converter for SatCom Applications

The design of highly-efficient dual-band, orthogonallypolarized SatCom antennas is not straightforward. Combining linear-to-circular polarization (LP-to-CP) converters with linearly-polarized (LP) antennas represents a simpler solution that meets the above-mentioned SatCom specifications. Typically, the up-and down-link are covered by combining two distinct radiating apertures with LP-to-CP converters [START_REF] Stankovsky | Spatial polarizers for CTS structure-based antenna arrays[END_REF]. In this paper, we propose a novel dual-band circular polarizer to attain orthogonal CPs in the K/Ka-band. This solution may be used by a single aperture SatCom terminal covering both bands.

Polarization converters usually consist of cascaded frequency selective surfaces (FSSs) [START_REF] Munk | Frequency selective surfaces: theory and design[END_REF]. Multi-layer meander lines or more complex geometries have been designed to achieve broadband, single-band LP-to-CP converters [START_REF] Joyal | Analysis and design of thin circular polarizers based on meander lines[END_REF]- [START_REF] Hosseini | A circuit-driven design methodology for a circular polarizer based on modified Jerusalem cross grids[END_REF]. Multi-parameter optimizations are usually employed in their design. A semi-analytic method to design single-band LP-to-CP converters has recently been proposed in [START_REF] Manzillo | Systematic design of a class of wideband circular polarizers using dispersion engineering[END_REF]. As in the design of negative-refractive index (NRI) meta-materials [START_REF] Antoniades | Compact linear lead/lag metamaterial phase shifters for broadband applications[END_REF], [START_REF] Antoniades | A broadband Wilkinson balun using microstrip metamaterial lines[END_REF], the design procedure in [START_REF] Manzillo | Systematic design of a class of wideband circular polarizers using dispersion engineering[END_REF] is based on engineering the dispersion of the considered unit cell's phase-shifter. In [START_REF] Manzillo | Systematic design of a class of wideband circular polarizers using dispersion engineering[END_REF], the unit cell consists of three cascaded metasurfaces. By modeling each metasurface with a given circuit topology, the axial ratio (AR) bandwidth can be broadened, thus reducing insertion losses. However, the works to date [START_REF] Joyal | Analysis and design of thin circular polarizers based on meander lines[END_REF]- [START_REF] Manzillo | Systematic design of a class of wideband circular polarizers using dispersion engineering[END_REF] do not provide a systematic procedure to address the design of dual-band LPto-CP converters with orthogonally-polarized transmission.

The design of dual-band, orthogonally-polarized LP-to-CP converters remains an open area of research. Only a few works can be found in literature. The elements of equivalent circuit models (ECMs) are usually obtained using multi-parameter optimizations. The goal is to maximize the transmission at two frequencies and shape the AR response according to the project specifications. Multi-layer FSS-based structures have been proposed operating either in reflection [START_REF] Naseri | A dual-band dual-circularly polarized reflectarray for K/Ka-band space applications[END_REF]- [START_REF] Tang | Low-profile compact dual-band unit cell for polarizing surfaces operating in orthogonal polarizations[END_REF] or transmission [START_REF] Naseri | Dual-band dual-linear-to-circular polarization converter in transmission mode application to K/Ka-band satellite communications[END_REF], [START_REF] Wang | Single-layer dual-band linear-to-circular polarization converter with wide axial ratio bandwidth and different polarization modes[END_REF]. In [START_REF] Wang | Single-layer dual-band linear-to-circular polarization converter with wide axial ratio bandwidth and different polarization modes[END_REF], the entries of ECMs are obtained by performing optimizations in Keysight Advanced Design System (ADS) [START_REF][END_REF]. The ECMs of [START_REF] Naseri | Dual-band dual-linear-to-circular polarization converter in transmission mode application to K/Ka-band satellite communications[END_REF] have been recently presented in [START_REF] Naseri | Equivalent circuit modeling to design a dual-band dual linear-to-circular polarizer surface[END_REF], but they refer to a given unit cell's geometry. Generic ECMs irrespective of unit cell's geometry have not been presented yet. Alternatively, all-dielectric cascaded metasurfaces can achieve multi-band, multi-functional polarization controllers [START_REF] Ranjbar | Analysis and synthesis of cascaded metasurfaces using wave matrices[END_REF], [START_REF] Ranjbar | Broadband, multiband, and multifunctional all-dielectric metasurfaces[END_REF]. Such low-loss topologies are very attractive but in general thicker than cascades of patterned metallic surfaces. The solution proposed in [START_REF] Arnieri | A SIW-based polarization rotator with an application to linear-to-circular dual band polarizers at K/Ka band[END_REF] combines a four layer meander line polarizer with a polarization rotator, but the final system is bulky. Finally, exploiting the real-time re-configurable properties offered by time-modulated metasurfaces would be a further way to realize dual-band, orthogonally-polarized LP-to-CP converters, as was done for polarization rotators in [START_REF] Wu | Tunable metasurfaces: a polarization rotator design[END_REF]. However, no prototype has been presented in the open literature yet.

The goal of this contribution is to provide a fully analytic design procedure for dual-band, orthogonally-polarized LP-to-CP metasurface-based converters. A dual-band quarter-wave plate is realized by cascading three anisotropic electric sheet admittances, separated by two isotropic dielectric slabs. As shown in Fig. 1, 45 • -slanted LP plane-waves are converted to left-and right-hand circularly-polarized waves in the downand up-link channels of K/Ka-band SatCom link. The problem is studied by considering two equivalent shunt-loaded transmission line (TL) problems for each transverse field component (i.e., xand y-polarized waves). The desired polarization conversion is achieved by enforcing 100% transmission at two design frequencies in the two considered bands. A ±90 • phase delay is also enforced between the two transmitted transverse components to achieve orthogonal CPs to the two bands. The proposed ECMs are fully characterized using analytic formulae.

The paper is organized as follows. The design procedure is outlined in Section II. Section III proposes the design of a dual-band, orthogonally-polarized LP-to-CP converter in the K/Ka-band. The design is validated by a prototype in Section IV and its performance is compared with the state-of-the-art. Section V discusses the capability and the limitations of the presented analytic model. Finally, Section VI concludes the paper.

II. ANALYTIC MODEL

A LP-to-CP converter is also referred to as a quarter-wave plate [START_REF] Born | Principles of optics[END_REF]. The transverse field components are phased ±90 • with respect to each other while maintaining equal amplitudes. Fig. 2 shows the geometry of the considered unit cell's phaseshifter. It consists of three sheet admittances cascaded along zaxis. Two dielectric slabs of thickness d and relative dielectric constant r are used to separate the sheets. The transverse extent of the structure is assumed infinite. The outer sheets Ȳs1 are diagonal and equal to each other. The inner admittance Ȳs2 is also diagonal but distinct. The electric admittance of each sheet is given by

Ȳsn = Y xx sn 0 0 Y yy sn , n = {1, 2} (1)
Given that the admittances are diagonal, the structure in Fig. 2 can be analyzed by considering each orthogonal transverse component separately. To streamline the mathematical formulation, we will consider scalar admittances hereinafter.

A. ABCD matrix formulation

The structure in Fig. 2 can be seen as the unit cell of periodic structure consisting of a transmission line (TL) loaded with reactive elements. Therefore, the theory of two-port microwave networks can be used in the analysis [START_REF] Pozar | Microwave engineering -3th ed[END_REF]. Let us assume that the admittance matrix of such a network is symmetric (reciprocal) and its entries are purely imaginary quantities (lossless). Each sheet admittance can be expressed in terms of the Bloch phase delay φ of the unit cell. To this end, the transmission matrix of the unit can be computed [START_REF] Pozar | Microwave engineering -3th ed[END_REF]

A B C D = A 1 B 1 C 1 D 1 1 0 Y s2 1 1 ξ D 1 B 1 C 1 A 1 (2)
where

A 1 B 1 C 1 D 1 = 1 0 Y s1 1 cos θ iη d sin θ i sin θ η d cos θ ( 3
)
where θ is the electrical length of each dielectric slab and

η d = η 0 / √ r , where η 0 is the characteristic impedance of free- space. Moreover, ξ = det A 1 B 1 C 1 D 1 = A 1 D 1 -B 1 C 1 ,
where det{•} indicates the determinant of a matrix. Finally, the inner and outer sheet admittances can both be expressed in closed formulae as function of Bloch impedance Z B and Bloch phase delay φ of the unit cell in Fig. 2. After some algebraic manipulations and exploiting the symmetry of the network [START_REF] Grbic | Super-resolving negative-refractive-index transmission-line lenses[END_REF], we obtain

Y s1 = i η d cot θ - η d Z B cot(φ/2) (4)
Y s2 = i η d 2 cot θ - Z B η d csc 2 θ sin φ (5)
A detailed derivation of equations ( 4) and ( 5) is reported in Appendix A.

B. Maximum transmission condition

The transmission coefficient S 21 of the unit cell in Fig. 2 is given by

S 21 = cos φ + i sin φ 2 Z B η 0 + η 0 Z B -1 (6)
Transmission is maximized by setting

|S 21 | = 1 in (6), yielding Z B η 0 + η 0 Z B = 2 ⇒ Z B = η 0 (7)
Equation [START_REF] Martinez-Lopez | A multilayer circular polarizer based on bisected splitring frequency selective surfaces[END_REF] shows that the network exhibits maximum transmission when impedance is matched to free-space. This result is easy to understand being equivalent to the zero reflection condition in TL theory [START_REF] Pozar | Microwave engineering -3th ed[END_REF], [START_REF] Marcuvitz | Waveguide handbook[END_REF]. It is important to note that equation [START_REF] Martinez-Lopez | A multilayer circular polarizer based on bisected splitring frequency selective surfaces[END_REF] does not necessarily lead to maximum efficiency [START_REF] Marcuvitz | Waveguide handbook[END_REF]. Finally, substituting (7) into ( 4) and ( 5) provides the sheet admittance values for 100% transmission regime

B s1 (φ, f ) = {Y s1 } = 1 η d cot θ - cot(φ/2) √ r (8)
B s2 (φ, f ) = {Y s2 } = 1 η d 2 cot θ - √ r csc 2 θ sin φ (9)
C. Design procedure

The design procedure for dual-band, orthogonally-polarized LP-to-CP converters starts by specifying an appropriate frequency behavior for each electric sheet admittance. Once this is done, the design work-flow consists of enforcing the maximum transmission condition [START_REF] Martinez-Lopez | A multilayer circular polarizer based on bisected splitring frequency selective surfaces[END_REF] at two distinct design frequencies: f 01 and f 02 . The systematic method depends on the variable φ xx 01 , i.e. the Bloch phase delay of the unit cell in Fig. 2

Y xx s1 =    iωC xx s1 , if B xx s1,01 > 0 (iωL xx s1 ) -1 , if B xx s1,01 < 0 (10)
where

C xx s1 = B xx s1,01 /ω 01 (11)
L xx s1 = ω 01 |B xx s1,01 | -1 ( 12
)
where ω is the angular frequency and ω 01 = ω(f 01 ).

Next, the phase delay for x-polarized fields φ xx 02 is obtained by inverting (8) at the frequency f 02 , resulting in

φ xx 02 = 2 arctan 1 √ r cot θ 02 -η 0 B xx s1,02 (13)
where θ 02 = θ(f 02 ) and B xx s1,02 = B xx s1 (φ xx 02 , f 02 ) is calculated by inverting [START_REF] Antoniades | Compact linear lead/lag metamaterial phase shifters for broadband applications[END_REF] at the frequency f 02 . It is worth noticing that this procedure may be repeated assuming the frequency response of Y xx s2 is that of a single lumped reactive element. This calculation is straightforward and not reported here for the sake of brevity.

Once φ xx 01 and φ xx 02 are known, we enforce the phase delay for y-polarized waves to be ±90 • and ∓90 • shifted with respect to x-polarized fields at the two design frequencies f 01 and f 02 , respectively. It follows

φ yy 01 = φ xx 01 ± 90 • (14)
φ yy 02 = φ xx 02 ∓ 90 • (15)
Enforcing ( 14) and ( 15) guarantees orthogonal LP-to-CP conversions at the two design frequencies. At this point, equation ( 13) is inserted into (9) and B xx s2 (φ xx 02 , f 02 ) is calculated at the frequency f 02 . Likewise, B yy s1 and B yy s2 are computed at the two design frequencies, by substituting both ( 14) and ( 15) into ( 8) and ( 9), respectively. Thus, the frequency responses of Y xx s2 , Y yy s1 , and Y yy s2 are assumed to be those of series-LC resonators, yielding

Y yy s1 = iωL yy s1 + 1 iωC yy s1 -1 (16)
Y xx s2 = iωL xx s2 + 1 iωC xx s2 -1 (17)
Y yy s2 = iωL yy s2 + 1 iωC yy s2 -1 [START_REF] Munk | Frequency selective surfaces: theory and design[END_REF] where To summarize, this model only requires the phase delay for x-polarized waves φ xx 01 as an input parameter. The bandwidths are maximized by sweeping φ xx 01 and observing its impact on the AR and the transmission. This is a direct consequence of having modeled Y xx s1 as a single shunt inductor or capacitor. It is worth noting that a LC-series topology could be chosen for Y xx s1 as well. The latter would provide two design parameters, namely φ xx 01 and φ xx 02 . This scenario would require a more complex approach (e.g., optimizations) to search for the best values for φ xx 01 and φ xx 02 . Besides, inductors and capacitors are easier to design than resonators using patterned metallic claddings.

C ψψ sm =                      |B ψψ sm,01 |(1-∆f 2 ) ω01(1+∆f |∆B ψψ sm |) , if B ψψ sm,01 > 0 B ψψ sm,02 < 0 |B ψψ sm,01 |(1-∆f 2 ) ω01(1-∆f |∆B ψψ sm |) , if 0 < B ψψ sm,01 < B ψψ sm,02 - |B ψψ sm,01 |(1-∆f 2 ) ω01(1-∆f |∆B ψψ sm |) , if B ψψ sm,01 < B ψψ sm,02 < 0 ( 19
)
L ψψ sm =                              |B ψψ sm,01 |-ω01C ψψ sm ω 2 01 C ψψ sm |B ψψ sm,01 | , if B ψψ sm,01 > 0 B ψψ sm,02 < 0 C ψψ sm < |B ψψ sm,
-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -3.5 -3 -2.5 -2 -1.5 -1 (b)

III. DUAL-BAND, ORTHOGONALLY-POLARIZED LP-TO-CP

CONVERTER FOR SATCOM APPLICATIONS Here, a design example is reported and discussed. The goal is to design a dual-band LP-to-CP converter for next generation SatCom terminal antennas. Such device is required to attain polarization diversity between the down-link (17.7-21.2 GHz) and the up-link (27.5-31 GHz) communication channels in the K/Ka-band. We enforce a LP-to-LHCP conversion in the lower frequency band, as well as a LP-to-RHCP conversion in the upper band. In the following, the bandwidth of the polarizer will be referred to as the frequency range over which the AR < 3 dB and the transmission is higher than -1 dB.

Using the model proposed in Section II, we will first focus on retrieving all the design parameters. Afterwards, a physical realization of the sheets is proposed and discussed. Each sheet is realized as a sub-wavelength textured metallic cladding: a metasurface [START_REF] Pfeiffer | Millimeter-wave transmitarrays for wavefront and polarization control[END_REF], [START_REF] Pfeiffer | Bianisotropic metasurfaces for optimal polarization control: analysis and synthesis[END_REF].

A. Numerical results based on ECMs

As detailed in Section II, we will focus on a class of polarizers consisting of three metasurfaces, separated by two isotropic dielectric slabs [START_REF] Pfeiffer | Bianisotropic metasurfaces for optimal polarization control: analysis and synthesis[END_REF]. Two Rogers RO3003 TM panels ( r = 3.00, tan δ = 0.0010 @ 10 GHz) of thickness d = 1.524 mm are used as dielectric substrates. They are bonded together using Taconic FastRise TM FR-27-0030-25 (F) ( r = 2.72, tan δ = 0.0014 @ 10 GHz) of ∼ 80µm pressed thickness. The overall thickness of the polarizer is approximately 3.145 mm. The design frequencies are f 01 = 19.5 GHz and f 02 = 29 GHz. As mentioned earlier, the only parameter the designer sets is φ xx 01 . In fact, a single reactive element is employed to describe the frequency response of the admittance sheet Y xx s1 . Thus, the parameter φ xx 02 is not a degree of freedom in the model, but is computed from [START_REF] Pfeiffer | Millimeter-wave transmitarrays for wavefront and polarization control[END_REF]. Given an incident LP plane-wave with Efield polarized in the direction (x ± ŷ) / √ 2, the impact of the choice of φ xx 01 on AR and transmission bandwidths is observed. By enforcing LHCP in [START_REF] Pfeiffer | Bianisotropic metasurfaces for optimal polarization control: analysis and synthesis[END_REF] and RHCP in [START_REF] Pozar | Microwave engineering -3th ed[END_REF], contour plots of the AR and the transmission are shown in Fig. 3. Values of φ xx 01 within the range [75 • , 90.5 • ] lead to dual-band LPto-CP conversions with orthogonal CP functionality. Unlike in [START_REF] Pfeiffer | Millimeter-wave transmitarrays for wavefront and polarization control[END_REF], a 360 • -phase coverage is not achieved while designing multi-band polarization converters. Furthermore, different bandwidths result from distinct values of φ xx 01 . Considering the trade off, a phase delay of φ xx 01 = 82.5 • is chosen. This operating point is in the middle of the range of solutions [75 • , 90.5 • ], as graphically depicted by a dashed red line in Fig. 3. It offers large bandwidths and robustness of the system to tolerances.

The corresponding AR and transmission coefficients are plotted in Fig. 4. Specifically, the polarizer conveys LHCP and RHCP fields in the frequency bands 17.6-21 GHz and 28.5-29.7 GHz, respectively. The model of Section II has been used to predict the frequency behavior of each sheet admittance. All sheet admittances can thus be studied by means of ECMs, as depicted in Fig. 5 for xand y-polarized waves. Y xx s1 is modeled as a shunt inductor, whereas series-LC resonators model the remaining sheet admittances. For the sake of completeness, the values of the lumped circuit elements are listed in Table I.

B. Physical design and full-wave results

Each sheet admittance is realized with periodic subwavelength patterned metallic cladding [START_REF] Munk | Frequency selective surfaces: theory and design[END_REF]. The unit cell's and geometrical sizes designed through full-wave simulations in ANSYS HFSS 2018.2 [START_REF]HFSS High-frequency structure simulation[END_REF]. The simulation setup can be found in [START_REF] Pfeiffer | Bianisotropic metasurfaces for optimal polarization control: analysis and synthesis[END_REF]. The first two fundamental Floquet modes excite the unit cell, whose sides are surrounded by periodic boundary conditions. The frequency responses are then extracted from the reflection coefficient [START_REF] Pfeiffer | Bianisotropic metasurfaces for optimal polarization control: analysis and synthesis[END_REF] for both xand y-polarized fields and matched to those of the ECMs in Fig. 5. The choice of the unit cell's geometry is inspired by standard Jerusalem cross shapes, whose simplified ECM consists of series-LC resonators [START_REF] Langley | Improved empirical model for the Jerusalem cross[END_REF], [START_REF] Costa | Efficient analysis of frequencyselective surfaces by a simple equivalent-circuit model[END_REF]. noting that in Fig. 6(c) the large inductance L xx s1 is realized as a very small capacitor. In terms of physical design, there is no metal pattern along x-axis for the outer sheets (see Fig. 6(a)). This is only possible when the sheet admittance approaches the open condition, as follows

Y xx s1 C xx s1 →0 or L xx s1 →+∞ ----------------→ 0 (22)
Finally, the full-wave simulation setup is depicted in Fig. 7(a). The stack-up of the polarizer is simulated as a structure that is transversely periodic. The illumination consists of a LP plane-wave with the E-field polarized in the direction α = (x ± ŷ)/ √ 2. Normal incidence is considered in the simulations. Fig. 7(b) shows that a LP-to-LHCP conversion is performed in the lower band and LP-to-RHCP in the upper band. The simulated AR is plotted in Fig. 7(c). The 3-dB-AR bandwidths are 22% (17.2-21.6 GHz) and 6% (28.8-30.6 GHz). The transmission coefficient is plotted in Fig. 7(d). The designed polarizer exhibits larger 3-dB-AR bandwidths than its ECM. The transmission is 0.7 dB lower than the ECM results in the upper band (see Fig. 7(d)). This is due to a difference between ECMs and physical design. In fact, the outer metasurface's unit cell and ECM have a different circuit topology for x-polarized waves, see Fig. 6(c). However, the transmission is acceptable since it is above -1 dB everywhere in the bands of interest.

IV. PROTOTYPE AND EXPERIMENTAL RESULTS

Several prototypes were fabricated, using standard printed circuit board (PCB) technology. A schematic representation of the converter cross section is depicted in Fig. 8(a). The patterned metallic surfaces are etched on the faces of the substrates, with a 45 • -rotated orientation with respect to the circuit board edges. A photograph of the fabricated board is shown in Fig. 8(b). The transverse size of the converter is ∼ 7λ max , where λ max is the free-space wavelength at 17 GHz. The overall dimension of the polarizer is 123.74 × 123.74 × 3.14 mm 3 . As shown in Fig. 8(b), the inner textured metal cladding is not visible.

A. Measurement setup

The dual-band, orthogonally-polarized LP-to-CP converter was experimentally characterized using a quasi-optical measurement system (Thomas Keating Ltd) [39]. A schematic drawing of the experimental setup is shown in Fig. 9. The fabricated boards are illuminated by a collimated Gaussian beam, under normal and oblique incidence. Specifically, two corrugated circular-horn antennas are employed for illumination and reception. The first horn (TX in Fig. 9) feeds a pair of reflectors, whose purpose is to focus the incident Gaussian beam on the device under test (DUT). A second pair of reflectors focuses the field coming from the sample to a receiving corrugated circular-horn antenna (RX in Fig. 9). The horns are connected to a PNA network analyzer (Keysight N5227A) with coaxial cables. The experiments in the band 17-31 GHz were carried out considering two separate measurement setups to cover the two bands 18-26 GHz and 26-40 GHz, respectively.

A free-space thru, reflect, line (TRL) calibration procedure is performed to calibrate the input ports located at the interfaces of the coaxial cables. The effects of waveguide transitions and horn-antennas are not calibrated out, but a timegating procedure is applied to isolate the DUT. The transmission coefficients are measured by rotating the receiving horn in the direction of the xand y-axis (see Fig. 8(a)). The DUT is characterized in terms of its scattering matrix. Finally, the AR and the transmitted power are calculated by post-processing the measured data.

B. Measured results for normal incidence

The measured results are plotted in Fig. 10, when the DUT is illuminated under normal incidence. As shown in Fig. 10(a), the dual-band and the orthogonally-polarized transmission of the polarizer have been demonstrated. A 45 • -slanted LP planewave is transformed into LHCP and RHCP plane-waves in two separate bands. The CP's purity is characterized in terms of AR, whose plot is shown in Fig. 10(b). Also, the total efficiency of the polarizer is plotted in Fig. 10(c). Several experiments were performed on four fabricated boards. The measured results were in excellent agreement with full-wave simulations, thus demonstrating the reliability of fabrication and measurement processes. A slight frequency shift exists between the measured and simulated results at lower frequencies. This is likely due to a phase variation resulting from fabrication tolerances. From Fig. 10(c) it is evident that the measured transmitted power matches very well in amplitude with simulations. This points to the fact that there is a phase mismatch at lower frequencies between measurement and simulation. Fig. 11(a) compares the phase difference ∆φ = φ yy -φ xx in measurement and simulation. A percentage variation of more than 10% occurs between the measured and simulated phase difference ∆φ at lower frequencies, whereas such a difference is lower than 5% in the upper band, as shown in Figs. 11(b) and(c). This difference is likely due to fabrication tolerances. Indeed, tolerances in the substrate's thickness d impact the electrical paths of the propagating waves. Likewise, variations in the dielectric constant r result in different refraction behaviors. In both cases, the phase delay φ may vary with respect to the design value. These effects can be observed using ECMs in Section II. Figs. 12(a The AR response remains nearly unvaried in the upper band. Therefore, the AR response exhibits a more robust stability to tolerances at higher frequencies than at lower. This behavior at lower frequencies is attributed to the presence two close resonances in the transmission coefficient response (see Fig. 10(a)). Small variations of the phase delay in the structure (see Fig. 2) have a stronger impact on shifting the resonances. On the other hand, only one resonance appears in the upper band, as shown in Fig. 10(a). A more robust frequency behavior is then expected in this case. This effect is confirmed by measurements shown in Fig. 10(b).

To conclude, the performance of the converter is summarized in Table II. The polarizer performs a LP-to-LHCP conversion over a 21% fractional bandwidth. Furthermore, a LP-to-RHCP conversion is performed over a 6% fractional bandwidth. These bandwidths are calculated considering a transmission higher than -1 dB.

C. Measured results for oblique incidence

The fabricated boards were also tested under oblique incidence. The DUT is mechanically-rotated on the quasi-optical test-bench. The impinging waves illuminate the DUT at different angles θ 0 , as shown in Fig. 9. Specifically, two types of illuminations were considered. In the first scenario, the impinging plane-wave presents the E-field polarized along the α direction (TE illumination). Secondly, a plane-wave with the The AR performance is generally stable for incident angles up to ±45 • . In particular, the 3-dB-AR bandwidths remain roughly unperturbed with respect to normal incidence, when a TE illumination is considered (see Fig. 13(a)). However, Fig. 13(b) shows that the -1-dB-transmission bandwidth slightly narrows with respect to normal incidence.

In general, the measured results differ for TE and TM illuminations. In particular, the TM case is more sensitive to oblique incidences. This is likely due to the asymmetry of the polarizer's unit cell, as depicted in Fig. 6(a)-(b). The 3-dB-AR bandwidth is very stable for angles of incidence up to ±45 • , within the upper frequency band. On the contrary, the lower bandwidth is slightly reduced, as shown in Fig. 13(c). The total efficiency of the polarization converter remains nearly unvaried for all incident angles, as plotted in Figs. 13(d). A detailed and complete summary of the polarizer's performance is reported in Table III for both TE and TM illuminations. The total bandwidth in Table III is calculated as the frequency range where AR < 3 dB and the total transmission is higher than -1 dB. The TM illumination results in narrower bandwidths at lower frequencies, whereas the opposite behavior is observed at higher frequencies. Moreover, the difference between TE and TM illuminations is mitigated at upper frequencies. The coupling effect between the polarizer sheets is, indeed, reduced at those frequencies and a quasi-plane-wave field propagation can be assumed through the metasurfaces. * The total bandwidth refers to the frequency range for which AR < 3 dB and the total transmission is better than -1 dB. * The total bandwidth refers to the frequency range for which AR < 3 dB and the total transmission is better than -1 dB.

TM illumination

θ 0 = 15 • θ 0 = 30 • θ 0 = 45

*

The total bandwidths refer to the frequency ranges for which AR < 3 dB and the total transmission is better than -1 dB. ** λ is the free-space wavelength at 25 GHz. *** This result refers to full-wave simulations only. **** This result is reported in terms of the free-space wavelength at 15.2 GHz.

D. State-of-the-art comparison

A comparison with state-of-the-art dual-band, orthogonallypolarized LP-to-CP converters for SatCom applications is presented here. Table IV lists some prior articles that proposed a solution to this problem. Our converter represents a good trade-off between performance and size. Polarization conversion is efficiently performed over larger frequency bands with respect to the state-of-the-art. The bandwidths in Table IV to frequency bands for which AR < 3 dB and transmission is above -1 dB. As detailed in Section IV-C, the angular stability is also highly improved. Angular stability is dependent on the choice of geometrical shapes for the metasurfaces' unit cells. In fact, the geometries in Fig. 6(a)-(b) mitigate the mutual coupling between the transmitted orthogonal transverse components. Jerusalem's cross-like FSSs are well known for ensuring low sensitivity to oblique incidences [START_REF] Li | A novel FSS structure with high selectivity and excellent angular stability for 5G communication radome[END_REF], [START_REF] Hosseini | A novel AMC with little sensitivity to the angle of incidence using 2-layer jerusalem cross FSS[END_REF]. Moreover, the overall thickness of the converter also represents a good trade-off. Ultra-thin polarizers [START_REF] Naseri | Dual-band dual-linear-to-circular polarization converter in transmission mode application to K/Ka-band satellite communications[END_REF], [START_REF] Cao | A novel chiral metamaterial circular polarizer based on e-shaped structure[END_REF] exhibit narrower bandwidths. On the contrary, more complex systems [START_REF] Arnieri | A SIW-based polarization rotator with an application to linear-to-circular dual band polarizers at K/Ka band[END_REF], [START_REF] Molero | All-metal 3-D frequency selective surface with versatile dual-band polarization conversion[END_REF] are too bulky. Our polarizer's thickness is electricallysmall. The impact of dielectric losses is therefore reduced and high transmission is observed over larger bands. Also, this paper proposes a generic analytic design procedure for dualband polarizers. In contrast to [START_REF] Naseri | Dual-band dual-linear-to-circular polarization converter in transmission mode application to K/Ka-band satellite communications[END_REF], [START_REF] Naseri | Equivalent circuit modeling to design a dual-band dual linear-to-circular polarizer surface[END_REF], our method is not dependent on the geometry of the metasurfaces' unit cells.

The proposed model allows for the synthesis of dual-band polarizers in terms of generic ECMs. Once the ECMs are obtained, the metasurfaces' unit cells can be designed using various geometries. Analytic formulae also provide insights to overcome the use of multi-parameter optimization processes.

V. IMPACT OF d AND r ON AR BANDWIDTHS

In this section, we discuss the impact of the substrate thickness d and dielectric constant r on AR bandwidths. The ECMs introduced in Section II avoid using optimizations since φ xx 01 is the only parameter controlled by the designer. This choice enables having an overall view of the maximum achievable performance of the polarizer. For instance, it is sufficient to sweep the value of φ xx 01 and observe the corresponding AR response. However, the simplicity of the model may exclude potential solutions. For instance, other parameters could be used to improve the performance of the design. Here, we consider the substrate's thickness d and dielectric constant r as representative example.

In Figs. 14 and 15, the impact of a 20% variation in d and r with respect to their design values (d nom = 1.524 mm and r,nom = 3.00) is plotted in terms of AR, respectively. These parameters clearly represent key assets in engineering the 3-dB-AR bandwidth. For instance, Fig. 14(a) shows that larger bandwidths are achieved at higher frequen- cies when d = 1.2d nom . This operating point resembles achieved in previous contributions [START_REF] Naseri | Dual-band dual-linear-to-circular polarization converter in transmission mode application to K/Ka-band satellite communications[END_REF], [START_REF] Wang | Single-layer dual-band linear-to-circular polarization converter with wide axial ratio bandwidth and different polarization modes[END_REF]. Likewise, broader 3-dB-AR bandwidths are obtained at lower frequencies when r = 0.8 r,nom or r = 1.2 r,nom (see Fig. 15(a)).

Using d and r as design parameters may impact the overall design of the structure. One issue is the physical implementation of ECM's elements. The range of values of ECM's entries is dependent on the geometry of the metasurfaces' unit cells. Assuming a given unit cell geometry, only a limited range of circuit element values can be realized. To give an example, the geometry in Fig. 6(a) and (b) can realize capacitance and inductance in the range of [6.5, 25] (fF) and [2.5,15] (nH), respectively. As shown in Fig. 14(b) and 15(b), the design complexity fairly increases for d = 0.8d nom and r = 0.8 r,nom , since the capacitance C xx s1 reaches very low values. In this scenario, the geometries in Fig. 6(a) and (b) cannot be employed in the physical design. Different geometries can obviously help in overcoming this issue. Nonetheless, the set of usable geometries is limited, given the electrically-small unit cell's size at K/Ka-band. Finally, fabrication constraints represent a further limitation in the implementation of the circuit elements.

Moreover, it is worth mentioning that Figs. 14 and 15 report results around the operating point (d nom , r,nom ). In this scenario, (d, r ) = (d nom , r,nom ) represents the optimal choice for a variation of d and r up to 20%. This does not mean that a better operating point cannot be achieved by optimizing d, r and φ xx 01 over larger ranges. One could possibly trade off a better performance for increased computational complexity. This is something that will be explored in future work.

VI. CONCLUSION

A novel design method for dual-band, orthogonallypolarized linear-to-circular polarization converters is proposed in this contribution. The polarization converter consists of three cascaded sheet admittances, separated by two dielectric slabs. Each sheet admittance is modeled using equivalent circuit models based on periodically-loaded transmission lines. By enforcing 100% transmission at two separate frequencies, the equivalent circuit models of the polarization converter are described by means of closed-form analytic expressions. The bandwidths of the polarizer are thus engineered according to specifications. The proposed method avoids the use of multi-parameter optimizations. A dual-band, orthogonallypolarized linear-to-circular polarization converter is designed in the K/Ka-band for satellite communication applications. A prototype was fabricated in printed circuit board technology. Measurements were performed to validate the numerical results. The measured results are in very good agreement with full-wave simulations. Under normal incidence, the polarizer efficiently performs linear-to-left-hand-circular polarization conversion over the band 18-22.2 GHz (∼ 21%). In addition, linear-to-right-hand-circular polarization conversion is demonstrated in the band 28.7-30.4 GHz (∼ 6%). These bandwidths are calculated as the frequency ranges over which the axial ratio is below 3 dB and the transmission is above -1 dB. The polarizer performance remains stable under oblique incidence for angles up to ±45 • . The axial ratio remains lower than 3 dB over the bands 17-22 GHz (∼ 25.6%) and 28.6-30 GHz (∼ 4.7%) when the plane-wave's incident angle is 45 • . Furthermore, the insertion loss remains lower than 1.2 dB in this scenario. Such a device may find application in satellite communication terminals for polarization control based on its low profile and stable performances with scanning. For instance, it can be integrated with linearly-polarized satellite communication terminal antennas to enhance robustness and isolation between transmitted and received signal. This appendix details the derivation of equations ( 4) and [START_REF] Joyal | Analysis and design of thin circular polarizers based on meander lines[END_REF]. A scalar formulation is used to streamline the notation. The network is assumed to be symmetric, reciprocal and lossless. Under this assumption, the following relation holds [START_REF] Pozar | Microwave engineering -3th ed[END_REF]

ξ = det A 1 B 1 C 1 D 1 = A 1 D 1 -B 1 C 1 = 1 (IA.1)
where A 1 , B 1 , C 1 and D 1 are defined in [START_REF] Pearson | Next generation mobile SATCOM terminal antennas for a transformed world[END_REF]. Their extended form is given by A 1 = cos θ (IA. The unit cell in Fig. 2 is assumed infinitely-periodic along z-axis. This allows a Bloch phase delay φ and the Bloch impedance Z B to be defined. The Bloch impedance Z B can be expressed in terms of an ABCD matrix formulation [START_REF] Pozar | Microwave engineering -3th ed[END_REF], yielding

Z B = -iB D tan φ (IA.6)
where A, B, C and D are defined in [START_REF] Bayer | Ka-band user terminal antennas for satellite communications[END_REF]. Using the matrix multiplication, they can be written as

A = 1 ξ (A 1 D 1 + B 1 C 1 + B 1 D 1 Y s2 ) B = 1 ξ (2A 1 B 1 + B 2 1 Y s2 ) (IA.8) C = 1 ξ (2C 1 D 1 + D 2 1 Y s2 ) (IA.9) D = A (IA.10)
The Bloch phase delay can be found as function of the ABCD matrix entries [START_REF] Pozar | Microwave engineering -3th ed[END_REF], yielding

cos φ = A = A 1 D 1 + B 1 C 1 + B 1 D 1 Y s2 (IA.11)
Exploiting (IA.1), the relation (IA.11) reduces to

cos φ = 2A 1 D 1 + B 1 D 1 Y s2 -1 (IA.12)
and can be arranged in the following form:

1 + cos φ D 1 = 2A 1 + B 1 Y s2 (IA.13)
Now, we can focus on finding the expression of the Bloch impedance Z B as function of the Bloch phase delay φ, the ABCD matrix and Y s2 , by substituting (IA.8) and (IA.10) into (IA.6). In formulae [START_REF] Grbic | Super-resolving negative-refractive-index transmission-line lenses[END_REF]: Y s2 is found by equating (IA.17) and (IA.18), yielding

Z B = -iB 1 (2A 1 + B 1 Y s2 ) (A 1 D 1 + B 1 C 1 + B 1 D 1 Y s2 ) tan φ (IA.
Y s2 = i η d sin(2θ) sin 2 θ - Z B η d csc 2 θ sin φ (IA.19)
Finally, equation ( 5) is obtained using the following trigonometric identity: sin(2θ) sin 2 θ = 2 cot θ.

Fig. 1 .

 1 Fig. 1. Exploded view of the dual-band, orthogonally-polarized LP-to-CP converter. The incident waves are linearly polarized at 45 • with respect to the orientation of the metal patterns. They are converted into right-and left-hand circularly-polarized waves in two separate bands. The polarizer is a symmetric structure consisting of three patterned metallic claddings (metasurfaces).

Fig. 2 .

 2 Fig. 2. Geometry of the problem under analysis. Three anisotropic sheet admittances are cascaded along the z-axis. The external sheet admittances are equal.

17

 17

Fig. 3 .

 3 Fig. 3. Contour plots of the absolute value of (a) axial ratio and (b) transmission coefficient as a function of frequency and φ xx 01 . The dashed red line indicates the chosen operating point for the design.

17

 17

  Down-link of K/Ka-band.

  Up-link of K/Ka-band.

Fig. 4 .

 4 Fig. 4. Numerical results for the dual-band, orthogonally-polarized LP-to-CP converter with φ xx 01 = 82.5 • . Axial ratio and transmission coefficients for both RHCP and LHCP are plotted in two bands: (a) 17-21 GHz and (b) 27-31 GHz.

Fig. 5 .

 5 Fig. 5. Equivalent circuit models of the structure. Networks for xand ypolarized waves are analyzed separately.

  (a) Outer sheets' unit cell. (b) Inner sheet's unit cell. Outer sheets: x-pol fields.

17

 17

  Inner sheet: x-pol fields.

17

 17

  Outer sheets: y-pol fields.

17

 17

  Inner sheet: y-pol fields.

Fig. 6 .

 6 Fig. 6. Patterned metallic geometries of the metasurfaces' unit cell: (a) outer and (b) inner sheets. All dimensions are given in millimeters. The simulated susceptances for the patterned metallic geometries are shown in (c)-(e) for xand (d)-(f) for y-polarized fields. The black and grey dot markers indicate the desired design values at the frequency f 01 and f 02 , respectively.

Fig. 7 .

 7 Fig. 7. (a) Simulation setup of the polarizer's unit cell. A comparison between the circuit network and full-wave simulations [36] is reported. (b) Transmitted LHCP and RHCP. (c) Axial ratio. (d) Total transmission. The fullwave simulations considers a normal incidence onto the polarizer. Besides, the incident E-field is linearly-polarized in the direction α = (x ± ŷ)/ √ 2.

  (a) Cross section of the polarizer. (b) Fabricated prototype.

Fig. 8 .

 8 Fig. 8. (a) PCB stack-up of the polarizer. (b) Fabricated prototype. The overall size of the board is 123.74 × 123.74 × 3.14 mm 3 . The outer metal pattern is highlighted by a zoomed-in photograph's detail.

Fig. 9 .

 9 Fig. 9. (a) Schematic drawing of the experimental setup used to measure the polarizer: quasi-optical test-bench [39]. (b) Photograph of the test setup.

Fig. 10 .

 10 Fig. 10. Comparison between measured and full-wave performance of the polarizer. (a) Transmitted LHCP and RHCP. (b) Axial ratio. (c) Total transmission of the polarization converter. The measurements refer to a normal illumination of the DUT.

  Phase difference ∆φ = φ yy -φ xx . Uplink of K/Ka-band.

Fig. 11 .

 11 Fig. 11. (a) Phase difference ∆φ = φ yy -φ xx comparison between measurements and simulations. Percentage error of measured data in the (a) downlink (b) uplink of the K/Ka-band.

  ) and (b) show the AR variation when fabrication tolerances are considered for d and r , respectively. The tolerances δ toll = 0.0762 mm and δ r,toll = 0.04 are provided by Rogers Corporation [40]. It is clear from Fig. 12 that tolerances in d and r cause a frequency shift at lower frequencies.

  Tolerances of the substrate's thickness.

Fig. 12 .

 12 Fig. 12. Tolerances effects of (a) d and (b) r on AR using ECMs. The nominal values of d and r are varied by fabrication tolerances provided by Rogers Corporation [40].

Fig. 13 .

 13 Fig. 13. Measured performance of the polarization converter under different incident angles in the βz-plane where β = α × ẑ, see Fig. 9. (a)-(c) Axial ratio. (b)-(d) Total transmission. Both vertical (TE) and horizontal (TM) Efield illuminations are considered.

Fig. 14 .

 14 Fig. 14. (a) AR versus frequency for different substrate's thicknesses d. (b) Related circuit parameters.

17

 17

Fig. 15 .

 15 Fig. 15. (a) AR versus frequency for different substrate's dielectric constants r . (b) Related circuit parameters.

ACKNOWLEDGMENT

  The authors would like to thank dr. Laurent Le Coq and Frederic Boutet for their help in performing the the measurements.APPENDIX A DERIVATION OF Y s1 AND Y s2 IN CLOSED FORM

2 )B 1 =C 1 = Y s1 cos θ + i sin θ η d (IA. 4 )D 1 =

 21141 iη d sin θ (IA.3) iY s1 η d sin θ + cos θ (IA.5) A. Closed form of Y s1

14 )

 14 By substituting (IA.11) and (IA.[START_REF] Pfeiffer | Millimeter-wave transmitarrays for wavefront and polarization control[END_REF]) into (IA.[START_REF] Pfeiffer | Bianisotropic metasurfaces for optimal polarization control: analysis and synthesis[END_REF], we obtainZ B = -iB 1 (1 + cos φ) D 1 cos φ tan φ = -iB 1 D 1 tan(φ/2) (IA.15)where the trigonometric identity 1+cos φ cos φ tan φ = 1 tan(φ/2) was used.By inserting (IA.3) and (IA.5) into (IA.15), the following relation is foundZ B = η d sin θ (iY s1 η d sin θ + cos θ) tan(φ/2) (IA.16)Finally, (4) is obtained by inverting (IA.[START_REF] Grbic | Super-resolving negative-refractive-index transmission-line lenses[END_REF]) and applying some algebraic manipulations.B. Closed form of Y s2By inserting (IA.2) and (IA.3) into (IA.8) and using the relation (IA.1), the second entry B of the overall ABCD matrix takes the following formB = iη d sin(2θ) -Y s2 η 2 d sin 2 θ (IA.17)Furthermore, B can be expressed in terms of the Bloch unit cell's phase delay φ and Bloch impedance Z B[START_REF] Pozar | Microwave engineering -3th ed[END_REF], yielding B = iZ B sin φ (IA.18)

for x-polarized waves at frequency f 01 . φ xx 01 is the only required input parameter to the model

	. After
	assuming a value for φ xx 01 , equations (8) and (9) are used to compute the sheet susceptances B xx s1,01 = B xx s1 (φ xx 01 , f 01 ) and B xx s2,01 = B xx s2 (φ xx 01 , f 01 ) at frequency f 01 . The frequency response of Y xx s1 is assumed to be that of an inductor or capacitor. According to the sign of B xx s1,01 , the outer sheet admittance for x-polarized waves Y xx s1 is expressed as a
	function of the frequency, as follows

  01 |/ω 01

	|B ψψ sm,01 |-ω01C ψψ sm ω 2 01 C ψψ sm |B ψψ sm,01 | ,	if	0 < B ψψ sm,01 < B ψψ sm,02 C ψψ sm < |B ψψ sm,01 |/ω 01
	|B ψψ sm,01 |+ω01C ψψ sm ω 2 01 C ψψ sm |B ψψ sm,01 | ,	if	B ψψ sm,01 < B ψψ sm,02 < 0
					(20)
	where
	m =	2, {1, 2},	if if	ψ = x ψ = y	(21)

and ∆f = f 01 /f 02 , ∆B ψψ sm = B ψψ sm,01 /B ψψ sm,02 where B ψψ sm,01 = B ψψ sm (φ ψψ 01 , f 01 ) and B ψψ sm,02 = B ψψ sm (φ ψψ 02 , f 02 ).

TABLE I CIRCUIT

 I ELEMENTS OBTAINED BY USING THE ANALYTIC METHOD PROPOSED IN SECTION II

		Outer sheets (n = 1)	Inner sheet (n = 2)
	C xx sn C yy sn	n.a. 6.97	20.5 12.3	(fF)
	L xx sn L yy sn	14.5 5.98	4.80 3.10	(nH)

  The proposed geometries are shown in Figs. 6(a) and 6(b). The metasurface's transverse cell dimensions are 0.28λ min × 0.28λ min , where λ min is the free-space wavelength at 31 GHz. Gap capacitors are designed in combination with meandered lines, to attain the circuit elements in TableI. Their susceptances are plotted

in Figs.

6(c

) to 6(f) for normal incidence in the frequency band 17-31 GHz. The desired susceptance values ((

8

) and (

9

)) are also plotted using black and grey dot markers at the frequencies f 01 and f 02 , respectively. Very good agreement between ECMs and full-wave simulations is observed. No optimization process was performed in the design. It is worth

TABLE III MEASURED

 III FRACTIONAL BANDWIDTHS OF THE PROPOSED DUAL-BAND, ORTHOGONALLY-POLARIZED LP-TO-CP CONVERTER UNDER OBLIQUE

			PLANE-WAVE INCIDENCE
	TE illumination	θ 0 = 15 • LHCP RHCP	θ 0 = 30 • LHCP RHCP	θ 0 = 45 • LHCP RHCP
	Axial ratio (< 3 dB) Transmission (> -1 dB)	[17, 22] [18, 21.9]	[28.6, 30] [28.3, 31]	[17, 22] [18, 21.6]	[28.6, 30] [28.3, 31]	[17, 22] [18.6, 21.4]	[28.6, 30] [28.6, 31]	(GHz)
	Total relative bandwidth	19.5	4.8	18.2	4.8	14.0	4.8	(%)

TABLE IV COMPARISON

 IV WITH STATE-OF-THE-ART DUAL-BAND, ORTHOGONALLY-POLARIZED LP-TO-CP CONVERTERS

	Ref.	Total bandwidths (%)	Thickness (λ)	Unit cell size (λ)	Analytic design	Angular stability ( • )
	[24], [26]	2.5 and 1.7	0.08	0.37	Quasi	±30 ***
	[25]	14.1 and 8.1	0.13	0.33	No	±20
	[27]	n.a.	0.06	0.41	No	n.a.
	[31]	8 and 2.3	1.05	n.a.	No	n.a.
	[32]	3.5 and 2.6	1.3 ****	0.57 ****	No	±5 ***
	This work	21.3 and 6	0.26	0.23	Yes	±45

Manuscript received XX/XXwork was supported by the Direction Générale de l'Armement and by Brittany Region, France.