Cross-Polarization Control in FSSs by Means of an Equivalent Circuit Approach

Received June 18, 2021, accepted July 4, 2021, date of publication July 12, 2021, date of current version July 20, 2021. Digital Object Identifier 10.1109/ACCESS.2021.3096715 Cross-Polarization Control in FSSs by Means of an Equivalent Circuit Approach CARLOS MOLERO 1, (Member, IEEE), ANTONIO ALEX-AMOR 1, FRANCISCO MESA 2, (Fellow, IEEE), ÁNGEL PALOMARES-CABALLERO 1, AND PABLO PADILLA 1 1Departamento de Teoría de la Señal, Telemática y Comunicaciones, Universidad de Granada, 18071 Granada, Spain 2Department of Applied Physics 1, Escuela Técnica Superior de Ingenieria Informática, Universidad de Sevilla, 41012 Sevilla, Spain Corresponding author: Antonio Alex-Amor ([email protected]) This work was supported in part by Spanish Research and Development National Program under Project TIN2016-75097-P, Project RTI2018-102002-A-I00, Project TEC2017-84724-P, Project PID2020-112545RB-C54, and Project EQC2018-004988-P; in part by the Predoctoral under Grant FPU18/01965; and in part by Junta de Andalucia under Project P18-RT-4830 and Project B-TIC-402-UGR18. ABSTRACT This paper presents an efficient equivalent circuit approach (ECA), based on a Floquet modal expansion, for the study of the co- and cross-polarization in frequency selective surfaces (FSS) formed by periodic arrays of patches/apertures in either single or stacked configurations. The ECA makes it possible the derivation of analytical expressions for the generalized scattering parameters associated with the proposed circuit networks. Furthermore, the proposed circuit approach is an efficient surrogate model that can be combined with optimization techniques and artificial intelligence algorithms for the efficient design of FSS structures, saving efforts in the computation compared to time-consuming full-wave simulators and tedious synthesis (simulation-assisted) techniques. Due to the simplicity of the topology of the involved networks, the ECA can also be advantageously used to gain physical insight. The proposed approach is applied and validated in different FSS configurations where the cross-pol component plays a fundamental role in the design, as in circular polarizers, polarization rotators, and reflectarray cells. INDEX TERMS Frequency selective surface, equivalent circuit approach, co- and cross-polarization terms, metamaterials. I. INTRODUCTION Controlling the behavior of the electromagnetic (EM) waves is a traditional challenge of the microwave and antenna engineering community [1], [2], which still raises a lot of attention [3], [4]. The modification of the transmission and reflection responses of an impinging EM wave on a periodic structure has been studied for different frequency ranges, with some examples from microwave to terahertz regimes found in [5]–[13]. In these works, the selected structure to provide the desired control of the incident EM is the so-called frequency selective surface (FSS) [14], [15]. The FSS can be classified into two categories depending of the implemented geometry of the unit cell; namely, FSS based on either apertures or patches [14]. One of the most appreciated functionalities in the design of FSS is the control of the cross-polarization. This capability The associate editor coordinating the review of this manuscript and approving it for publication was Gerardo Di Martino . enables the design of polarization-converter devices [16], [17] because both the co-polarized and cross-polarized components of the EM wave have to be tuned in order to obtain the desired polarization (horizontal, vertical, circular or elliptical). Polarization converters based on apertures [18]–[22] and based on patches [23]–[25] are reported in the literature. In most of the previous works, the incident wave is transmitted along the FSS structure, although there are also FSSs where the incident wave is totally reflected. This fact is taken into advantage in radar cross section (RCS) applications in order to rotate the polarization of the incoming wave towards its orthogonal polarization and, thus, achieve a RCS reduction [26]–[29]. Given the importance of controlling the cross polarization in transmission and/or in reflection by means of FSS, an efficient design of these structures is imperative. The optimization of the FSS design can be carried out by using an EM simulator. However, its computational cost is very high because hundreds of full-wave simulations are generally VOLUME 9, 2021 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ 99513 C. Molero et al.: Cross-Polarization Control in FSSs by Means of ECA needed to reach the desired goals. An alternative strategy to substantially reduce the computational cost is replacing the full-wave simulations by equivalent circuit models that represent the performance of the FSS in a cost-effective way [30]–[35]. These circuit models can then be combined with optimization and artificial intelligence algorithms for an efficient design of the device, especially when a large number of iterations are required in the process. Among the different techniques to obtain a circuit model of periodic structures [25], [36]–[41], the equivalent circuit models based on Floquet analysis [42]–[45] have shown remarkable outcomes even for intricate structures [43], [46]–[50]. One of the most relevant characteristics of this method comes from its analytical (or quasi-analytical) nature. It means that no previous full-wave frequency-sweeping simulations are needed in order to accurately take into account the effects of higher order harmonics and couplings in single and stacked FSS structures in a very extended frequency band [42], [43], [49]. In this work, we present analytical expressions for both copol and cross-pol scattering parameters of single and stacked aperture- and patched-based FSSs. In order to have complete independence of full-wave simulators, the formulation is based on Floquet modes in contrast with other state-of- the-art synthesis techniques [25], [32], [37]. Furthermore, as discussed in our previous work [49], the analytical nature of the approach makes that linear transformations of the scatterers (such as displacement, rotation, and/or scaling) can easily be introduced to model in a simple and efficient manner complex FSS stacks. The present work goes one step further in that direction and includes the treatment of the cross-pol component in both aperture- and patch-based FSS structures. This provides a great advantage by allowing the aperture/patch scatterers can be different between layers. It represents an improvement with respect to [50], where only patch-based FSSs with the same base geometry are studied. The paper is organized as follows. Section II presents the derivations and the analytical expressions for the co- and cross-polarization in aperture-based FSS. In Section III, the calculation of the generalized scattering parameters for patched FSS is given. Both sections discuss the application of the formulation for both single and stacked FSS. Section IV shows several examples of FSS based on either apertures or patches that are characterized using the developed analytical expressions. Finally, in Section V, the main conclusions of the work are drawn. II. CROSS-POLARIZATION IN APERTURE ARRAYS A. SINGLE APERTURE PROBLEM The unit cell for the problem under consideration in the present section is depicted in Fig.1, where the scattering of an impinging time-harmonic (ejωt) plane wave on an infinite 2-D periodic structure has been reduced to a classical discontinuity problem in waveguides. For simplicity, we will consider but suppress the ejωtdependence in the following derivation. FIGURE 1. (a): Unit cell in perspective. (b): Unit cell in the yz-plane. The planar discontinuity consists of a infinitesimally thin metallic sheet with an arbitrary-shaped aperture. This discontinuity is located at z=0 and embedded between two semi-infinite homogeneous and isotropic dielectric media with permittivity ε(L/R) r, as shown in Fig.1(b). The unit cell is bounded by periodic boundary conditions (PBC) and the incident plane wave is arbitrarily oriented. The electric field magnitude of the incident wave is assumed to be unity, thus letting its magnetic-field magnitude identical to the value of admittance associated with the wave. Next we follow the general procedure reported in [42], [43] although here the necessary derivations to obtain closed-form expressions associated with the cross-pol components will be explicitly shown. Since the interaction of the incident wave with the discontinuity excites an infinite number of harmonics/modes in the unit cell, the transverse electric field at both sides of the discontinuity plane can be described in terms of the following Floquet expansion: E(L) t(x,y;ω) =(1 +R)ecp 00(x,y)+Vxp,(L) 00 exp 00(x,y) +X0 n,m Vcp,(L) nm ecp nm(x,y)+X0 n,m Vxp,(L) nm exp nm(x,y) (1) E(R) t(x,y;ω) =Tecp 00(x,y)+Vxp,(R) 00 exp 00(x,y) +X0 n,m Vcp,(R) nm ecp nm(x,y)+X0 n,m Vxp,(R) nm exp nm(x,y) (2) where the superscripts ‘‘cp’’ and ‘‘xp’’ denotes co- and crosspolarization terms with respect to the polarization of the incident wave, respectively. For TE incidence, the crosspolarization superscript becomes TM and vice versa. The superscript ‘‘L/R’’ denotes the left/right-side dielectric medium. The factor 1 in (1) accounts for the unit-amplitude of the incident electric field, here characterized by the harmonic of order m,n=0,0. The coefficients Rand Tare the corresponding copolar reflection and transmission coefficients, respectively. The coefficient Vxp,(L/R) 00 is the amplitude of the ’cross-pol’ harmonic of order m,n=0,0 at both sides of the discontinuity, excited thanks to the interaction between the incident harmonic and the discontinuity. The coefficients Vcp/xp,(L)/(R) nm are the amplitudes associated with harmonics of order n,mat both sides of the discontinuity as well. All the above coefficients are function of the angular frequency (ω) of the impinging wave. The prime symbol in the series stands for summation ∀n,mexcept n,m=0,0. 99514 VOLUME 9, 2021 C. Molero et al.: Cross-Polarization Control in FSSs by Means of ECA The spatial profile of the transverse electric field associated with each harmonic is represented by ecp/xp nm and defined in terms of TM/TE harmonics as eTM nm (x,y)=1 √pxpy exp[−j(kxnx+kymy)] √pxpyˆ eTM nm (3) eTE nm(x,y)=1 √pxpy exp[−j(kxnx+kymy)] √pxpyˆ eTE nm (4) with ˆ eTM nm =kxn ˆ x+kym ˆ y qk2 xn +k2 ym (5) ˆ eTE nm =kym ˆ x−kxn ˆ y qk2 xn +k2 ym h=ˆ eTM nm ׈ zi(6) kxn =qε(L) rk0sin θcos φ+2nπ px (7) kym =qε(L) rk0sin θsin φ+2mπ py (8) where pxand pyare the periodicity along xand yaxis, k0the free-space wavenumber, and θ, φ are the elevation and azimuth incident angles. At the discontinuity plane, the transverse electric field is different from zero only at the aperture region . A key assumption of the present approach is that the spatial profile, E, of this transverse electric field at the aperture is considered to be frequency-independent; namely, the total transverse field, Eap, can be factorized in the following way Eap(x,y;ω)=A(ω)E(x,y)∀x,y∈(9) where A(ω) can be regarded as the amplitude which does depend on frequency whereas E(x,y) describes the spatial distribution which does not depend on frequency. This approximation has been found to be valid up to the second excitable resonance of the considered scatterer. Nonetheless, this usually leads to a large operation bandwidth for the ECA, even within the grating-lobe regime [34], [49]. In fact, the onset of the grating-lobe regime corresponds to the cutoff frequency of high-order propagating harmonics. In this sense, the ECA is capable of reproducing very exotic behaviors where several diffraction orders are present, regardless of the electrical size of the unit cell under the assumptions described above. Since the electric field is continuous in the aperture region, we can write E(1) t(x,y;ω)=E(2) t(x,y;ω)≡Eap ∀x,y∈. (10) If this continuity of the electric field is forced for the projection of the fields onto the modal functions eTM/TE nm , it allows us to calculate the amplitudes of the harmonics Vcp/xp, (L/R) nm by means of the following expressions: (1 +R)=A(ω)˜ E(kx0,ky0)·ˆ ecp 00 (11) Vxp,(L/R) 00 =A(ω)˜ E(kx0,ky0)·ˆ exp 00 (12) Vcp/xp,(L/R) nm =A(ω)˜ E(kxn,kym)·ˆ ecp/xp nm .(13) It should be noted that the expressions for Vcp/xp,(L/R) nm at the left- and right-hand side coincide, which allows us to remove the superscript (L/R) in the following. In the above expressions we have made use of the fact that Z E(x,y)·[ecp/xp nm (x,y)]∗d=˜ E(kxn,kym)·ˆ ecp/xp nm (14) where ˜ E(kxn,kym)=1 √pxpyZ E(x,y) ej(kxnx+kymy)d(15) can be recognized as the 2-D Fourier transform of the spatial profile E(x,y) at (kxn,kym). Thus, if we now define a frequency-independent parameter Ncp/xp nm given by Ncp/xp nm =˜ E(kxn,kym)·ˆ ecp/xp nm (16) the expressions (11) – (13) can be rewritten in general as Vcp/xp nm Ncp/xp nm =(1 +R) Ncp 00 (17) what indicates that each amplitude is actually proportional to the voltage factor (1 +R). As already discussed in [43], the factors relating each amplitude and (1 +R) can be interpreted, from a circuit point of view, as the value of the turns ratio corresponding to transformers. It is worth noting here that these transformer ratios turn out to be frequency independent, which will be very relevant from an operational standpoint. If we now force the continuity of the power through the aperture: Z Eap ×[H(L) t]∗d=Z Eap ×[H(R) t]∗d(18) it is then possible to obtain the reflection coefficient Rafter considering that the transverse magnetic fields at both sides of the discontinuities are given by the following Floquet expansions: H(L) t(x,y;ω) =(1 −R)Ycp,(L) 00 hcp 00(x,y)−Vxp 00 Yxp,(L) 00 hxp 00(x,y) −X0 n,m Vcp nmYcp,(L) nm hcp nm(x,y)−X0 n,m Vxp nmYxp,(L) nm hxp nm(x,y) (19) H(R) t(x,y;ω) =TY cp,(R) 00 hcp 00(x,y)+Vxp 00 Yxp,(R) 00 hxp 00(x,y) +X0 n,m Vcp nmYcp,(R) nm hcp nm(x,y)+X0 n,m Vxp nmYxp,(R) nm hxp nm(x,y) (20) where hcp/xp nm (x,y)=ˆ z×ecp/xp nm (x,y) and Ycp/xp,(L/R) nm is the admittance associated with the harmonics of order (n,m): YTE,(L/R) nm =β(L/R) nm ωµ0(21) YTM,(L/R) nm =ωε0ε(L/R) r β(L/R) nm (22) VOLUME 9, 2021 99515 C. Molero et al.: Cross-Polarization Control in FSSs by Means of ECA with the longitudinal wavenumber, β(L/R) nm , given by β(L/R) nm =qε(L/R) rk2 0−k2 xn −k2 ym.(23) Thus, introducing (19) and (20) into (18), the reflection coefficient is found to be R=|Ncp 00 |2(Ycp,(L) 00 −Ycp,(R) 00 )−|Nxp 00 |2(Yxp,(L) 00 +Yxp,(R) 00 )−Yeq |Ncp 00 |2(Ycp,(L) 00 +Ycp,(R) 00 )+|Nxp 00 |2(Yxp,(L) 00 +Yxp,(R) 00 )+Yeq (24) with Yeq =X0 nm |Ncp nm|2(Ycp,(L) nm +Ycp,(R) nm ) +X0 n,m|Nxp nm|2(Yxp,(L) nm +Yxp,(R) nm ).(25) Notice that Yeq corresponds to an infinite summation of admittances connected in parallel. Each individual admittance, Ycp/xp,(L/R) nm , is associated with a semi-infinite transmission line section that is connected to the input line (corresponding to the incident harmonic) through a transformer of turns ratio Ncp/xp nm . Electromagnetic coupling between adjacent cells is taken into consideration by the multi-modal nature of the present equivalent approach; namely, Floquet harmonics of lower and higher order are coupled together through the parallel connections appreciated in Fig.2. For the sake of simplicity, Yeq will be considered as a single admittance (it includes the contribution of all the harmonics except the ones corresponding to m,n=0,0). The corresponding 4-port equivalent circuit can therefore be schematically represented as the one in Fig. 2. Four transmission lines connected in parallel are distinguished: two lines in green that describe the propagation of the incident harmonic through the media (L/R) and two lines in pink that describe the propagation of the (0,0) cross-polarized harmonic in both dielectric media. All these lines are connected to Yeq via transformers with turns ratio Ncp/xp 00 . Standard circuit rules can now be employed to calculate the scattering parameters of the above network. As shown in Fig. 2, the nodes n1and n2share the same voltage, given by (1 +R)=T(coming from the superposition of the incident and reflected waves at the discontinuity plane). The voltages in the rest of the lines are then Vcp 00 =(1 +R)=T(26) Vxp 00 =(1 +R)Nxp 00 Ncp 00 (27) with associated currents given by Icp,(R) 00 =TY cp,(R) 00 (28) Ixp,(L) 00 =(1 +R)Nxp 00 Ncp 00 Yxp,(L) 00 (29) Ixp,(R) 00 =TNxp 00 Ncp 00 Yxp,(R) 00 .(30) FIGURE 2. (a): Multi-modal equivalent circuit for a single 2-D slot-like periodic array when assuming the incidence of the harmonic n,m=0,0. The interaction of the incident harmonic and the discontinuity excites all the higher-order harmonics, represented by their corresponding transmission lines. (b): Same equivalent circuit, with the higher-order harmonics grouped in the term Yeq. Defining now the corresponding arrays of power wave amplitudes [51], the following generalized scattering parameters are obtained: Scp→cp 11 =R(31) Scp→cp 21 =Tv u u t Ycp,(R) 00 Ycp,(L) 00 (32) Scp→xp 11 =(1 +R)Nxp 00 Ncp 00 v u u t Yxp,(L) 00 Ycp,(L) 00 (33) Scp→xp 21 =TNxp 00 Ncp 00 v u u t Yxp,(R) 00 Ycp,(L) 00 .(34) B. STACKS OF APERTURES Stacks of two (or more) identical or nonidentical coupled apertures can also be analyzed via the equivalent circuit approach, provided that they have the same px×py dimensions (namely, the size of the unit cell should be 99516 VOLUME 9, 2021 C. Molero et al.: Cross-Polarization Control in FSSs by Means of ECA FIGURE 3. (a): Unit cell in perspective. (b): Unit cell in the YZ-plane. the same). An example of this scenario is shown in Fig.3(a). Two metallic perforated screens are separated by a dielectric material with relative permittivity ε(M) rand thickness d. The unit cell now contains two semi-infinite dielectric slabs of relative permittivity ε(L/R) rand the finite slab with ε(M) r, as illustrated in Fig. 3(b). Following a similar rationale as in [52], the obtaining of the circuit network for this structure is based on the singlearray case. Each individual aperture taking part in the stack admits to be represented by the circuit in Fig. 2. Both individual circuits can be joined by connecting the transmission lines associated with each one of the harmonics inside the finite dielectric slab (namely, connecting all the lines associated with harmonics of the same order). By means of this operation and after several manipulations, the final equivalent circuit can be represented by the classical π-network shown in Fig.4. Again, we have separated the co- and crosspolarized terms in the leftmost and rightmost dielectric media by its corresponding transmission lines. The admittances forming the π-block can be split into two contributions. On the one hand, the external contribution given by the harmonics n,m6= 0,0 in the leftmost and rightmost dielectric media, Y(L) p,1=X0 n,m|Ncp,(1) nm |2Ycp,(L) nm +X0 n,m|Nxp,(1) nm |2Yxp,(L) nm (35) Y(R) p,2=X0 n,m|Ncp,(2) nm |2Ycp,(R) nm +X0 n,m|Nxp,(2) nm |2Yxp,(R) nm (36) where the transformers now include the superscript (1/2), referring to the leftmost or rightmost aperture, respectively. On the other hand, the internal block includes the information about all the possible electromagnetic couplings between the two aperture arrays. This is due to the multi-modal nature of the elements that form both the internal and external blocks. In this case, the internal block can be expressed as a πnetwork topology whose elements are given by Y(M) p,1=jX ∀n,m|Ncp,(1) nm |2Ycp,(M) nm +X ∀n,m|Nxp,(1) nm |2Yxp,(M) nm  ×tan(β(M) nm d/2) (37) Y(M) p,2=jX ∀n,m|Ncp,(2) nm |2Ycp,(M) nm +X ∀n,m|Nxp,(2) nm |2Yxp,(M) nm  ×tan(β(M) nm d/2) (38) FIGURE 4. Equivalent circuit for stacked aperture arrays. Y(M) s= −jX ∀n,m Ncp,(1) nm [Ncp,(2) nm ]∗Ycp,(M) nm +X ∀n,m Nxp,(1) nm [Nxp,(2) nm ]∗Yxp,(M) nm csc(β(M) nm d).(39) The topology of the circuit for the stack is slightly more complex than the circuit for the single case. The expressions for the scattering parameters are those in (32), (33) and (34). However, it can be inferred from Fig. 4that the nodes n1and n2do not share now the same voltage, that is, (1 +R)6= T. A simple circuit analysis reveals that the voltages (1 +R) and Tare related by the following expression: T=(1 +R)Y(M) s Y(M) s+Y(R) in (40) with Y(R) in =Y(M) p,2+Y(R) p,2+|Ncp,(2) 00 |2Ycp,(R) 00 +|Nxp,(2) 00 |2Yxp,(R) 00 . (41) This new value of Tcan be introduced in (32) and (34) in order to achieve the final expressions for the generalized scattering parameters. FIGURE 5. Unit cell for (a): a single-patch array, (b): a pair of coupled arrays of patches. III. PERIODIC PATCH ARRAYS SINGLE AND STACK Problems involving patches arrays can be conceived as complementary to the aperture case and, thus, its treatment can follow the general guidelines given above. Figs. 5(a) and (b) illustrate the unit-cell problem for single and coupled patch arrays, respectively. Focusing on the single-array case, the derivation of the reflection coefficient is achieved after imposing the following two conditions VOLUME 9, 2021 99517 C. Molero et al.: Cross-Polarization Control in FSSs by Means of ECA at the discontinuity plane, specifically in the area of the metallic patch : ˆ z×[H(R) t(x,y;ω)−H(L) t(x,y;ω)] =Jpa(x,y;ω) (42) E(L) t(x,y;ω)=0=E(R) t(x,y;ω) (43) where Jpa is the surface current density on the patch. The transverse magnetic and electric fields are described in terms of Floquet harmonics as in (1), (2), (19) and (20). The surface current is assumed to be given by a spatial distribution independent from frequency, namely, Jpa(x,y;ω)=B(ω)J(x,y).(44) After manipulating (42) and (43), taking into account (44), the following expression relating the amplitudes of the harmonics and the reflection and transmission coefficients of the incident one is found: Vcp/xp nm Ycp/xp, (L) nm +Ycp/xp, (R) nm Ncp/xp nm =−(1 −R)Ycp,(L) 00 +TY cp,(R) 00 Ncp 00 (45) where T=(1 +R), and Ncp/xp nm =˜ J(kxn,kym)·ˆ ecp/xp nm (46) with ˜ J(kxn,kym) being the 2-D Fourier transform of J(x,y) at (kxn,kym). The reflection coefficient Rcan be written as R= Ycp,(L) 00 −Ycp,(R) 00 |Ncp 00 |2−|Nxp 00 |2 Yxp,(L) 00 +Yxp,(R) 00 +Zeq−1 Ycp,(L) 00 +Ycp,(R) 00 |Ncp 00 |2+|Nxp 00 |2 Yxp,(L) 00 +Yxp,(R) 00 +Zeq−1(47) with Zeq =X0 n,m |Ncp nm|2 Ycp,(L) nm +Ycp,(R) nm +X0 n,m |Nxp nm|2 Yxp,(L) nm +Yxp,(R) nm .(48) As in the aperture-array case, Zeq includes all the harmonics of orders n,mexcept those with n,m=0,0. Similarly, electromagnetic coupling between adjacent cells is taken into consideration by the multi-modal nature of the circuit in Fig.6(a). However, at the light of (48), the connection among all these elements is now in series in contrast to the parallel connections observed in the aperture case. The crosspolarization elements of order n,m=0,0 also take part in the group of elements connected in series. Following the theory in [51] and using the expression for Vcp/xp nm obtained after manipulating (45), the generalized scattering parameters for the case of a single patch-based FSS are found to be Scp→cp 11 =R(49) Scp→cp 21 =(1 +R)v u u t Ycp,(R) 00 Ycp,(L) 00 (50) FIGURE 6. (a): Equivalent circuit for the single array. (b): Equivalent circuit including voltages when assuming the incidence of an harmonic (labelled as the co-polarized one). Scp→xp 11 =Nxp 00 Ncp 00 (R−1)Ycp,(L) 00 +TY cp,(R) 00 Yxp, (L) 00 +Yxp, (R) 00 v u u t Yxp,(L) 00 Ycp,(L) 00 (51) Scp→xp 21 =Nxp 00 Ncp 00 (R−1)Ycp,(L) 00 +TY cp,(R) 00 Yxp, (L) 00 +Yxp, (R) 00 v u u t Yxp,(R) 00 Ycp,(L) 00 (52) The couple of patch-arrays separated by a distance d depicted in Fig.5(b) also admits to be represented by an equivalent circuit. Similarly as in the aperture-case, we proceed by connecting transmission lines associated with individual harmonics. Fig. 6(b) shows the final topology. The transmission lines related to the cross-pol term of order n=m=0 in inner medium (M) are connected. This procedure is also realized by the rest of higher order harmonics (all of them included in Zeq in Fig. 6(a)). The resulting topology is not appropriate to construct individual blocks and derive a final π-network. Furthermore, the complexity of calculating the scattering parameters for the cross-pol term increases considerably. A possible way to address this calculation appeals to transfer-matrix notation. In particular, the composition of scattering (S) matrices has been the method employed. Each S-matrix describes and represents the coupling between all the harmonics excited in an individual patch array. The final S-matrix of the system is the result of the composition of all the individual matrices. The transmission/reflection coefficients of the cp- and xp-components are directly extracted from this final scattering matrix. It is worth remarking that the calculation is still cumbersome and no physical insight is inferred from the process. 99518 VOLUME 9, 2021 C. Molero et al.: Cross-Polarization Control in FSSs by Means of ECA IV. NUMERICAL EXAMPLES In this section, some numerical examples of FSS structures are given in order to validate the analytical circuit approach and show its accuracy and computational efficiency. We compare the results with commercial full-wave simulators as well as with previously published works in the literature. Examples of aperture and patch arrays (single and stacked layers) are reported. FIGURE 7. (a) Linear-to-circular polarization converter. (b) Single-layer aperture-based FSS presented in [22] acting as a circular polarizer. (c) Spatial profile considered at the aperture, computed with (53). (d) Phase and amplitude of the transmission parameters (x-pol and y-pol waves). The shadowed area indicates the frequency range where the transmitted wave is circularly polarized (AR≤3 dB). Losses are considered in the dielectric, ˆ εr=εr(1 −jtanδ), εr=2.95, tanδ=0.025. A. APERTURE-BASED CIRCULAR POLARIZER The first structure under analysis is the aperture-based linear- to-circular polarizer presented in [22]. The operation of the structure as well as its forming layers can be visualized in Figs. 7(a) and (b), respectively. The FSS is formed by a periodic array of cross-shaped apertures lying on a lossy dielectric substrate. When a 45◦linearly-polarized plane wave impinges the structure, the transmitted x-pol and y-pol components are delayed by π/2 radians while maintaining the same level of amplitude. This is due to the different lengths that the two magnetic dipoles forming the cross have, which leads to a circularly-polarized wave at the output. The analytical spatial profile of the electric field assumed in the cross-shaped aperture, displayed in Fig.7(c), is mathematically modeled as [53] E(x,y)=cosπy/ly q1−(2x/wx)2ˆ x+cos(πx/lx) q1−2y/wy2ˆ y.(53) In the above expression, the horizontal (y-oriented E-field) and vertical (x-oriented E-field) magnetic dipoles are independently modeled. The phase and amplitude coefficients of the transmitted x-pol and y-pol waves are plotted in Fig. 7(d) and compared with the measurements shown in [22]. As observed, there is a good agreement between the ECA results and the experimental data. For the computation of 501 frequency points, the circuit approach took only 2.28 seconds. This demonstrates the efficiency of the present model. Furthermore, it should be remarked the ability of the equivalent circuit to easily account for dielectric losses in the substrate. Slight differences are seen in the amplitude coefficient at the end of the displayed frequency range. This is attributed to the large size of the cross-shaped apertures as well as the field coupling between horizontal and vertical dipoles, not considered in expression (53). The shadowed area in Fig. 7(d) highlights the frequency range where the axial ratio is below 3dB; namely, where the transmitted wave is circularly polarized from a practical point of view. B. ASYMMETRIC LOADED MAGNETIC DIPOLES The second example is inspired by the synthesis method of anisotropic FSS reported in [37], based on the use of equivalent circuits. The FSS structure is formed by asymmetric loaded magnetic dipoles, as can be seen in Fig. 8(a). As discussed in [54] for meander-line gratings, a frequencyindependent sine-shaped (half sine) spatial profile can be assumed to be excited at the aperture. The considered profile is constant along the slot width wand is mathematically modeled as E(x,y)=Esec(x,y)+Ecor(x,y) (54) where Esec(x,y)=             sin(πx L)ˆ y,in region S1 sin(π[y+L1] L)ˆ x,in region S2 sin(π[x+L2] L)ˆ y,in region S3 (55) takes into account the sinusoidal variation along the different longitudinal sections [see Fig. 8(a)] of the loaded dipole (L1=lS1+w,L2=lS1+2w+lS2), and Ecor(x,y)=(Ecor(C1)(x,y),in corner C1 Ecor(C2)(x,y),in corner C2 (56) VOLUME 9, 2021 99519 C. Molero et al.: Cross-Polarization Control in FSSs by Means of ECA FIGURE 8. (a) Front view of the unit cell containing the asymmetrical loaded magnetic dipoles. (b) Spatial profile considered at the aperture, computed with (54), and a detail of a corner section. (c) Single array. (d) Two-layer stack whose plates are rotated 20 degree. TE normal incidence is assumed. Geometrical parameters: px=py=10 mm, lS1=2mm, lS2=6mm, lS3=1mm, w=0.5mm, α=20◦, and h=2.5mm. includes the boundary conditions at the corner sections, with Ecor(C1)(x,y)=1−y wsinπL1 Lˆ x +1−x wsinπlS1 Lˆ y(57) Ecor(C2)(x,y)=1−y wsinπL0 2 Lˆ x +1−x wsinπL2 Lˆ y(58) and L0 2=lS1+w+lS2. Note that the total length of the loaded dipole is L=lS1+lS2+lS3+2w. The magnitude of the spatial profile can be observed in Fig.8(b). As appreciated in (57),(58), a linear decay that maintains the continuity of the fields has been assumed at the corner sections. The same consideration is valid when dealing with currents in patch structures, as in Fig. 9. Although this approach may seem simple, it has proven to give accurate results [54]. Fig. 8(c) illustrates the transmission parameters (amplitude and phase) of the co-pol (cp→cp) and cross-pol (cp→xp) terms for a single, free-standing layer when a TE-polarized plane wave illuminates the FSS. An excellent agreement FIGURE 9. (a): Unit cell in perspective and under front view. (b): |S11|data obtained by the ECA and CST for TE normal incidence. Structure parameters (lengths in mm): p=4.5, d=1.2, w(L) =w(R) =0.2, x(L) 0= −2.225, x(L) F= −0.625, y(L) 0= −0.7, y(L) F=0.9, x(R) 0=0.625, x(R) F=2.225, y(R) 0= −0.9, y(R) F=0.7, εr=10. is observed between our ECA results and the commercial software CST. It should be noted that the length of the horizontal arms of the dipoles control the cross-pol level; namely, the larger the horizontal arms are, the higher the cross-pol term is. A variant of the above structure is formed when the previous layer is stacked together with a second rotated layer of equal dimensions (angle of rotation α=20◦). In our equivalent circuit approach, the incorporation of this second layer to account for the spatial profile of the rotated aperture simply demands a linear transformation of the original spatial profile (54), as pointed out in [49]. The transmission parameters (amplitude and phase) of the stack are plotted in Fig.8(d), where a good agreement of our ECA results is observed with CST data, despite the complexity of the FSS under analysis. Using the same computer, the circuit model took 2.02 and 3.18 seconds to compute the scattering parameters at 501 frequency points of the single array and the two-layer stack, respectively, while CST (default mesh size, 60 Floquet harmonics) took 6 and 8 minutes in these calculations. It is then apparent that, within its validity range, the ECA is a very efficient procedure to study the scattering properties of FSSs as well as an excellent tool for optimization tasks where many computations are required. C. DOUBLE L-SHAPED DIPOLE A first example of a patch-array periodic structure to be analyzed with the circuit model is inspired in the structure 99520 VOLUME 9, 2021 C. Molero et al.: Cross-Polarization Control in FSSs by Means of ECA reported in [55]. The unit cell, shown in Fig. 9(a), comprises two identical L-shaped metallic dipoles printed on a grounded dielectric slab. Each individual dipole will be labelled as (L) or (R), referring to the left- or right-side position in the cell. The (L)-dipole is rotated 180◦with respect to the (R)-dipole, thus creating an asymmetric unit-cell pattern. The presence of the ground plane ensures full-reflection operation. In the frame of the circuit model approach described in Sec. III, the ground plane is included in the circuit in Fig.6 just by short-circuiting the lines associated with the rightmost medium with permittivity ε(R) r. The length of all these shortcircuited transmission lines coincides with the dielectric-slab thickness. As required by the ECA, the spatial profile of the surface current distribution, J(x,y), in each of the L-shaped dipoles is assumed to be frequency independent. The longitudinal component of the current is taken to have the form of a half-sine function, distributed along the corresponding strip, and constant along the strip width. Likewise the example in the previous section, the surface-current function is mathematically expressed in two parts: J(L/R)(x,y)=J(L/R) sec (x,y)+J(L/R) cor (x,y) (59) where J(L) sec(x,y)=       sinπx L(L) ˆ x,in S1 sin π(d(L) x+w(L) +y−y(L) 0) L(L) !ˆ y,in S2 (60) J(R) sec(x,y)=       sinπy L(R) ˆ y,in S1 sin π(d(R) y+w(R) +x−x(R) 0) L(R) !ˆ x,in S2 (61) and J(L) cor(x,y)=sin πd(L) x L(L) !x−x(L) F w(L) ˆ x +sin π(d(L) x+w(L)) L(L) !y−(y(L) F−w(L)) w(L) ˆ y (62) J(R) cor(x,y)=sin πd(R) y L(R) !y−y(R) F w(R) ˆ y +sin π(d(R) y+w(R)) L(R) !x−(x(R) F−w(R)) w(R) ˆ x (63) with d(L/R) x=x(L/R) F−x(L/R) 0 d(L/R) y=y(L/R) F−y(L/R) 0 L(L/R) =d(L/R) x+d(L/R) y+w(L/R). The corresponding transformer ratios Ncp/xp nm are given in terms of the 2-D Fourier Transform of J(L/R); namely, according to (46) Ncp/xp nm =h˜ J(L)(kxn,kym)+˜ J(R)(kxn,kym)i·ˆ ecp/xp nm .(64) It is worth remarking that if the dipoles are very close each other, some degree of mutual coupling is expected, affecting the excitation of individual surface-current distribution in each dipole. Therefore, the individual currents in the left-and right patches are not expected to be identical to the ones of each patch acting separately. Since the proposed current profiles assume no mutual coupling, the accuracy of the surfacecurrent profiles deteriorates as both dipoles get closer. In order to validate the circuit-model approach as well as the surface-current functions, Fig.9reproduces one of the results reported in [55, Fig. 7] where a wideband polarizer is presented. Both the left and right dipoles are identical (dimension in the caption of Fig. 9). The dielectric substrate is 1.2 mm thick with relative permittivity ε(R) r=10. The structure is excited by TE normal incidence (electric field directed along the y-axis). The reflection of this field component (cp) and the orthogonal field component (xp) are both plotted in Fig. 9(c). The cell configuration is such that wideband field rotation is achieved, as shown by the high reflection magnitude of the xp-component from 14.5 GHz to 23 GHz in contrast with the small reflection magnitude of the cpcomponent. The agreement between results provided by CST and those given by the circuit model is quite acceptable, with some slight differences caused by the approximated nature of the surface-current functions employed for this example. Possible sources of inaccuracies can be associated with the modelling of the current in corners, realized by including a simple linear decay in the sine function. The neglecting of mutual-coupling effects in the surface current functions is also expected to yield some inaccuracies. In turn, the computation time required by the ECA is very short in comparison to the CPU time of CST. The ECA employed 1.13 seconds to reproduce the solution, whereas CST needed 5 minutes approximately. Hence, ECA stands out as a very quick and efficient analysis tool if a small degree of inaccuracy is acceptable. D. SPLIT-RING DIPOLE A second case study of the circuit model for patch arrays is the structure whose unit cell is shown in Fig. 10(a)-(b). The patch is a circular ring section, defined by the arc comprised between the angles φ0and φF. The width of the patch is defined between the inner and outer radii R1and R2. As the ring section is printed on a grounded dielectric slab, the corresponding equivalent circuit is topologically identical to the one employed in the previous example. The surface current is now assumed to be directed along the ring section; namely, the longitudinal component of the current takes the direction of the polar unit vector ˆ φ(the current is null at the ends of the patch). A half-since function, adapted to the geometry of VOLUME 9, 2021 99521