Synthesis of 3-D Polarization Rotation Transmissive Surface by Arbitrary Angle Based on Orthogonally Inserted Parallel-Couple Slotlines

In this article, a polarization rotation transmissive surface (PRTS) of arbitrary angle based on a three-dimensional (3-D) frequency selective structure is proposed. The PRTS unit cell is composed of three parallel-coupled slotline sections with open-short ends etched on vertically and horizontally inserted single-layer printed circuit boards (PCBs). Under the incidence of vertically polarized waves, the spatial waves are converted into guided waves by the open-circuited end of one slotline section, and then the guided waves are coupled to two other slotline sections that can act as two independent coupling transmission paths in the orthogonal planes. This can be observed as a power divider, and the power division ratio can be fully controlled by virtue of the coupling strengths of the two coupling transmission paths. Thus, the polarization of incidence waves can be rotated at an arbitrary angle at the output port. To simplify the design procedure, a filtering power division theory and an equivalent circuit model are first used to obtain the theoretical response. Subsequently, a coupling matrix is employed to synthesize and design the proposed PRTS. To confirm the effectiveness of the proposed PRTS, two examples with different polarization rotation angles and fractional bandwidths at the same center frequency of 5 GHz and return loss of 20 dB are designed, fabricated, and measured. The measured results confirm our theoretical predictions and agree with the simulated results, thus demonstrating that the proposed PRTS exhibits stable frequency response under a large oblique incidence, in addition to their easiness in fabrication.

Over the past few decades, many studies have been carried out for PRTSs with 90 • polarization rotation (PR). Multilayers, including wire grids, have been reported [16], [17], which have the advantages of broadband operation, low insertion, and easy fabrication. Design approaches based on the substrate-integrated waveguide (SIW) cavity technology have been presented [18], [19], [20], which can obtain good frequency selectivity and sharp-roll off skirts at passband edges. However, the large profile of these structures and the fabrication cost consequently limit their practical applications. Recently, structures with double-sided slot resonant [21], [22], or orthogonal capacitive surfaces [23] have been reported and they are typically composed of back-toback slots or patch resonators coupled via a narrow aperture. These types of PRTSs can provide thin thickness, wideband, and low inserted loss (IL). Whereas these designs are limited because of the large size of the periodic elements, thus resulting in poor angular stability. Moreover, they can only provide 90 • PR.
Arbitrary polarization rotators have received considerable attention because they can meet the requirements of different PR angles [24], [25], [26], [27]. A polarization rotator with meander-line grids that allows for the arbitrary rotation of linear polarization has been described [24]. In [25], a polarization rotator of arbitrary angle has been reported, where each unit cell contains double metallic layers etched with a slot. By adjusting the rotation direction of the slot on each layer, the electric field direction can be rotated when penetrating the structure. A waveguide polarization rotator has been used to demonstrate that an arbitrary angle of polarization plane rotation can be obtained by EM-coupling two conjugated quadruple-slot planar-chiral irises [26]. Broadband polarization rotators for rotating arbitrary polarization angles by improved SIW cavities are presented in [27], [28], where the PR can be achieved by controlling the rotation angle of input and output slots etched on two sides of the SIW cavities. Although these above-mentioned PRTSs have the same advantages in terms of design, they suffer from their sensitivity to oblique incidence because they are designed by two-dimensional frequency selective surfaces  with high periodicity in transversal cross-section.
In recent years, a class of three-dimensional (3-D) FSSs based on a periodic array of cavity elements has been proposed [29], [30], [31]. Because of the special construction of unit cells, 3-D FSSs can provide multi-mode resonators in the longitudinal direction in the FSS element, thus resulting in high-order filtering response or wideband performance [30]. Furthermore, 3-D FSSs exhibit good angular stability because of the miniaturized transversal size of these 3-D unit cells. More recently, ultra-wideband bandpass 3-D FSSs based on direct synthesis methods have been investigated and designed [32]. Therefore, owing to the operation characteristics of 3-D FSSs, many designs have been proposed [33], [34], [35].
In this article, a novel concept of synthesizing and designing PRTS with arbitrary angles based on 3-D FSSs is presented, which comprises three straight slotline sections with open short ends etched on an orthogonally inserted printed circuit board (PCB). Under the incidence of linearly polarized incident waves in the vertical direction, the spatial waves are converted into guided waves at the open end of one slotline and then coupled to the other two parallel slotline sections etched on the orthogonal PCB. Thus, the combination of the two perpendicular electric fields will be rotated by the two coupling paths at the output port. To reveal this procedure, a filtering power divider concept with an equivalent circuit model is developed to analyze the operating characteristics of the proposed PRTS. Meanwhile, a synthesis method with a coupling matrix is used to design the proposed PRTS. The synthesis and design procedure are then discussed in detail. Two examples with different PR angles and fractional bandwidths at the same center frequency and return loss (RL) are designed. In final, two examples with fractional bandwidths of 20% and 13% with PR angles of 30 • and 45 • are fabricated and measured, respectively. Good agreement is obtained between simulated and measured results, showing that the proposed PRTS has stable PR in the operation band. Furthermore, owing to the small transversal size of the unit cell compared with the wavelength in the air, the proposed PRTS exhibits angular stability. Fig. 1(a) shows the perspective view of one unit cell of the proposed PRTS, which comprises orthogonal PCB pieces with two parallel slotlines arrayed in the vertical direction (x-axis) and another slotline arrayed in the horizontal direction (y-axis). The dielectric substrate of PCB is selected by Rogers 4350B with a relative permittivity of 3.66 and a loss tangent of 0.0037. Fig. 1(b) and Fig. 1(c) show the detailed layouts of the vertical and horizontal slotline structures. It is observed that a pair of parallel-coupled slotline sections (named as Slot1 and Slot2) with short-open ends are etched on the vertical PCB piece and another slotline section (named as Slot3) is etched on the horizontal PCB piece. The three slotline sections can work together to develop the desired co-and cross-polarized transmission paths by coupling Slot1 and Slot2, as well as Slot1 and Slot3, respectively.

B. WORKING PRINCIPLE AND ANALYSIS
The relationship between the electric field and the PR angle is illustrated in Fig. 1(a). The PR angle can be calculated using the following simple equation: From (1), the PR angle θ can be determined by controlling the amplitude ratio of the electric field vectors in the two orthogonal polarizations. Fig. 1(a) also shows a diagrammatic sketch of the simulation setup and electric field distributions of the proposed PRTS, where the red and blue arrows denote the electric and magnetic fields, respectively. When the direction of the electric field is oriented along the x-axis, the incident waves are converted into guided waves by the interface between the free space and the open end of Slot1, and then the guided waves are coupled to the two other slotline sections Slot2 and Slot3 by dominant magnetic coupling, respectively. Finally, the guided waves are converted into spatial waves again at the open ends of Slot2 and Slot3. Owing to the controllability of the two orthogonal electric fields along the Slot2 and Slot3, the outgoing electric field can be rotated by a certain angle with respect to those of incidence. To demonstrate the performance of the PRTS, we define R xx = |E rx |/|E ix | and R yx = |E ry |/|E ix | to represent the co-and cross-polarized reflection coefficients, respectively. Similarly, the co-and cross-polarized transmission coefficients are expressed as T xx = |E tx |/|E ix | and T yx = |E ty |/|E ix |, respectively. Herein, E rx , E ry , E tx , E ty , and E ix are corresponding to the amplitudes of the coand cross-polarized reflection waves, co-and cross-polarized transmission waves, and co-polarized incident waves, respectively. Fig. 2 shows the simulated S-parameter results of the proposed PRTS obtained using CST Microwave Studio (CST-MWS) with periodic boundary conditions and Floquet ports. The magnitude of the co-polarized reflection coefficient R xx is less than −20 dB in the operation band. The insertion losses of the co-and cross-polarized transmission coefficients

FIGURE 3. Coupling topology of the proposed PRTS element.
T xx and T yx are approximately −1.27 dB and −6.42 dB, respectively. Therefore, a PR angle of 30 • with a slight phase difference can be obtained from 4.5 to 5.5 GHz. It should be mentioned that the cross-polarized reflection coefficient R yx is always kept below −30 dB over the entire frequency band in the simulation, thus it is neglected in the discussion.
To examine the operation principle of the proposed PRTS, a schematic topology can be established, as depicted in Fig. 3, where R 1 , R 2 , and R 3 denote three resonators with respect to the slotline sections Slot1, Slot2, and Slot3. The spatial waves at both sides of PRS are represented by S and L, respectively. Besides, the cross-and co-polarized coupling coefficients are denoted by M 12 and M 13 . The external quality factors are represented by Q S1 , Q 2L , and Q 3L . It is noted that each transmission path is arranged linearly to form a direct coupling path without cross-coupling in selection. This coupling topology can be viewed as a power divider, i.e., a merger of two second-order Chebyshev bandpass filters with a common resonator R 1 . To achieve the same RL performance as a real two-port second-order Chebyshev bandpass filter, the coupling coefficients M 12 and M 13 should satisfy the following equations [36] where M 12 and M 13 are the coupling coefficients between resonators R 1 and R 2 , as well as R 1 and R 3 of an ideal two-port second-order Chebyshev bandpass filter, respectively. The coefficients α and β are needed to satisfy the condition of α 2 + β 2 = 1. Moreover, the three external quality factors of Q S1 and Q 2L , as well as Q 3L , should be equal. Thus, they can be set as Q e . Fig. 4 depicts the corresponding equivalent circuit model of the proposed PRTS, where the source and load impedances are denoted by Zs and Z L , respectively. Herein, Zs and Z L are equal and thus can be substituted by Z 0 . Each resonator developed by the open-short-ended slotline section can be considered a λ/4 resonator. As per the operation mechanism of a λ/4 resonator, it acts either as a shunt LC resonator based on its open-circuit end or a series LC resonator by its short-circuit end [37]. Therefore, the transmission network can be represented by the alternative J and K inverters, as well as λ/4 resonators, which can be calculated as follows: where Z r represents the characteristic impedance of the λ/4 resonator, and the corresponding admittance Y r = 1/Z r . Y 0 represents wave admittance. It is mentioned that J S1 , J 2L , and J 3L are equal. When the external equation factor Q e and the coupling coefficients (M 12 and M 13 ) are determined, the theoretical bandpass filter response of the proposed PRTS can be obtained by using (4)-(6).

III. SYNTHESIS PROCEDURE A. COUPLING MATRIX EXTRACTION
In the coupling topology of Fig. 3, each coupling branch only comprises the main coupling, without cross-coupling or stray coupling. According to the coupled-filter theory, each coupling branch can be easily represented by a coupling matrix, which can be extracted using known specifications, such as center frequency, bandwidth, and RL in the passband [38]. The coupling matrix can be given as follow: where m S1 and m 2L are the coupling coefficients of the source and load to the corresponding resonators, respectively. m 12 is the normalization-coupling coefficient between the two coupling resonators. The coupling coefficients between nonadjacent resonators are set to zero because of no crosscouplings. Once the coupling matrix is allotted by the given specifications, the initial values of external quality factor Q e and coupling coefficient M 12 can be extracted using the well-known formulas: where FBW is the fraction bandwidth. Note that the coupling coefficient M 12 is an ideal two-port second-order Chebyshev bandpass filter. The coupling coefficient M 13 is calculated in the same way as that of M 12 . Based on the above analysis, the parameters Q e , M 12 , and M 13 can be finally achieved by the known specifications. In the meantime, the theoretical response of the proposed PRTS can be obtained as well. The next step in the design procedure is to establish a relationship between the calculated parameters (Q e , M 12 , and M 13 ) and the physical dimensions of the proposed PRTS.

B. EXTERNAL QUALITY FACTOR Q E
To extract the external quality factors Q e , an open-shortended slotline section is etched on the vertical PCB with periodic boundary conditions under the incidence of x-polarized EM waves, as shown in Fig. 5(a). The incident spatial waves are converted into guided waves by the open end of the slotline section and will be blocked by the short end. Hence, the incident waves will be completely reflected such that the reflection coefficient R xx is almost equal to 0 dB. However, the phase response Rxx distinctly decreases with respect to frequency, as shown in Fig. 5(b). Furthermore, the simulated phase Rxx is equal to zero at the resonant frequency f 0 , and the absolute bandwidth between the phase shifts of ±90 • is represented by f. As per the coupled-resonator theory [39], the extraction of quality factors can be calculated as To examine the sensitivity and effect of the physical dimensions on the external quality factor Q e and the resonant frequency f 0 , a set of parametric studies has been conducted. Fig. 6 shows the external quality Q e and resonant frequency f 0 versus different values of w, l 1 , and g. It can be observed from Fig. 6(a) that the curves of the external quality factor Q e get reduced by increasing the slotline section width w or the space g. However, the resonant frequency f 0 is slightly affected by the parameters w and g. As shown in Fig. 6(b), the resonant frequency f 0 can be significantly decreased with increasing l 1 , whereas the external quality factor Q e is slightly decreased with increasing l 1 . Hence, it can be well figured out that the desired Q e can be calculated by the parameters w and g, and the resonant frequency f 0 primarily depends on the slotline section length.

C. COUPLING COEFFICIENTS M 12 AND M 13
To obtain the coupling coefficient of the two coupling slotline sections Slot1 and Slot2, the geometry and simulation setup with two slotline sections etched on the orthogonal PCB under x-polarized incidence, as shown in Fig. 7(a). It is observed that two additional metallic layers with rectangular apertures are employed between the PRTS and the free space, respectively. This configuration aims to achieve extremely weak coupling between the source (or load) and the resonators. To eliminate additional effects between the metallic layers and PRTS interfaces, a certain distance (denoted as s) away from the PRTS is required. In this case, the extraction of the coupling coefficient between the two slotline sections is almost unaffected by the port excitation. Note that the rectangular apertures in both metallic layers are selected with a relatively small size to ensure minimal influence on the resonant properties of the resonators. The simulated S-parameter results versus the length of the rectangular aperture l s are shown in Fig. 7(b), where f 1 and f 2 are the lower and upper resonant frequencies for the two coupling slotline sections. It is seen that a weak port excitation is achieved. The coupling coefficient of the two coupling slotline sections can be calculated using the following formula: To study the relationship between the coupling coefficient M 12 and physical dimensions, the variation of the coupling coefficient M 12 and resonant frequencies f 1 and f 2 versus the slotline width w, coupling length l c , and space g are explored. It is worth pointing out that the dimensions of the two slotline sections are the same. As depicted in Fig. 8(a), the two resonant frequencies are almost constant by modifying the slotline width w under each g. Thus, the coupling coefficient M 12 is not sensitive to the variation of w. As shown in Fig. 8(b), the two resonant frequencies f 1 and f 2 move away from each other by increasing the coupling length l c or decreasing the space g, resulting in strong coupling. Therefore, the desired coupling coefficient  = 17.69, d = 0.508, l1 = 11.01, w = 0.81, g = 1.1, s = 13, and ws = 0.8).
M 12 can be primarily realized by adjusting the two parameters l c and g. Fig. 9 shows the simulation setup to extract the coupling coefficient M 13 . The extraction procedure of M 13 is the same as that of M 12 . The distributions of the coupling coefficient M 13 versus the coupling length l c and slotline width w is shown in Fig. 10. It is seen that M 13 increases as increasing the coupling length l c , while slotline width w has negligible impact on it.
Until now, the theoretical value of the external quality factor Q e and the ideal coupling coefficients M 12 and M 13 have been calculated using the coupled matrix. The practical coupling coefficients M 12 and M 13 can be calculated and extracted by utilizing (2) and (3). The initial dimensions are then determined by extracting the external quality factor and coupling coefficients. To implement the proposed PRTS as designed by using the above synthesis approach, the entire design procedure can be summarized as follows: Step 1: Determine the PR angle and the ratio of coupling coefficients, and prescribe the second-order bandpass response, center frequency, bandwidth, and RL.
Step 2: Calculate the coupling matrix using the prescribed parameters [38].  Step 3: Compute the theoretical values of the external quality factor Q e and the ideal coupling coefficients M 12 and M 13 .
Step 4: Determine the physical dimensions as per the extracted external quality factor Q e and the two practical coupling coefficients M 12 and M 13 . Finally, it is necessary to compensate for unexpected effects by slight adjustments,   so as to ensure that the desired PR angle and the frequency response meet the design requirements.

A. THEORETICAL DESIGN EXAMPLES
Two examples with second-order Chebyshev filtering response and different FBWs (20% and 13%) at the same center frequency of f 0 = 5 GHz and RL of 20 dB are designed to confirm the synthesis procedure of the proposed PRTS. Example A is designed to realize the PR angles of 30 • and 60 • , and example B can achieve the PR angles of 30 • and 45 • . The calculated parameters are listed in Table 1, and the corresponding physical dimensions are presented in Table 2.
It is worth mentioning that the coefficients of the coupling matrix of the two examples are equal to each other, i.e., m S1 = 1.225, m 12 = 1.658, m 2L = 1.225. Besides,  Fig. 1(a). (a) Fig. 1(a). (a)  the port impedance is Z 0 = 377 . The S-parameters and PR comparisons between the theoretical results achieved by the equivalent circuit model and the full-wave EM simulation results obtained in CST-MWS of example A is shown in Fig. 11. As is observed in Fig. 11(a) and Fig. 11(b), two PR angles of 30 • and 60 • can be realized over the frequency range of 4.5 -5.5 GHz, and the simulated phase differences, = Txx − Tyx , of the two transmission paths are less than 0 • ± 2.3 • and 0 • ± 6 • compared to theoretical ones, respectively. The small discrepancies in the phase difference are possible because the two transmission paths are not exactly symmetrical in the unit cell. For design example B, a PR angle of 30 • with a maximum phase deviation of 0 • ± 3.5 • is obtained, as shown in Fig. 12(a). Fig. 12(b) shows that the phase difference under the PR angle is set to 45 • . It is seen that is less than 0 • ± 4.5 • . The reflection coefficients R xx of the two examples are below −20 dB over the operation band. The simulation results are displayed to confirm the feasibility of the design concept.

B. FABRICATION AND MEASUREMENT
Prototypes A and B with PR angles of 30 • and 45 • are then fabricated and measured, respectively. Fig. 13 shows the assembly details of our design, where two narrow slits  are cut half away along vertical and horizontal PCBs, respectively. In this case, the vertical and horizontal PCBs can be cross-joined to realize an entire structure using the cut slits. To ensure all the PCBs are fixed together, several vias and solder pads are added at the two sides of the PCBs. It should be noted that the performance of the PRTS is not affected by the added vias and solder pads. The PRTS prototypes were fabricated using Rogers 4350B substrate with a thickness of 0.508 mm. Fig. 14 shows the structural details of prototype A, where each PCB piece has 23 periodic unit cells, i.e., 23 × 23 elements are established and the total dimension is 250 mm × 250 mm, which corresponds to the electrical dimensions of 4.2λ 0 × 4.2λ 0 , where λ 0 represents the free-space wavelength at the center frequency. The magnified views of the four elements and the side view of the proposed PRTS show the relative location of the three coupling slotline sections and the details of the structure. Fig. 15 shows an image of the measurement scheme. The measurement system is set up in a free-space microwave anechoic chamber, which is composed of transmitting and receiving horn antennas, the measured prototype, and a vector network analyzer. In this measurement system, the two horn antennas should be kept at a certain distance (d) away from the test screen to satisfy the far-field distance requirement of the antennas, i.e., d > 2D 2 /λ, where D is the largest dimension of the antenna and λ is the wavelength of the highest operation frequency. The measured prototype was centrally positioned on a foam block. Except for minor differences in assembly, the measurement procedures of these two prototypes are similar. To obtain accurate measurement results, the measurement procedure of the PRTS is conducted in two primary steps: First, the air box and the perfect electric conductor (PEC) with the same dimension as the proposed PRTS structure are measured without the proposed PRTS. These measurements are used to calibrate the measured PRTS data. Then, the PRTS structure is measured. Finally, by normalizing the measured results to the calibration data with the help of MATLAB, the frequency response of PRTS can be obtained.

C. RESULTS AND DISCUSSIONS
The simulation and measurement results of these two designed prototypes at incident angles of 0 • , 20 • , and 40 • are compared in Fig. 16 and Fig. 17, respectively. As shown in Fig. 16(a) and Fig. 17(a), the measured T yx , T xx , and R xx are in good agreement with the simulated ones. Moreover, under the normal incidence, co-polarized reflection coefficients R xx of the two prototypes are less than −18 dB over the operation band ranges of 4.5-5.5 and 4.675-5.325 GHz, respectively. Meanwhile, the measured PR angles are within 30 • ± 1.3 • and 45 • ± 1.5 • , respectively. The differences in PR angles between the simulation and measurement results are less than 3 • for both examples. The differences are possibly attributed to fabrication tolerance, dielectric substrate deformation, and uncertainties in the measurement setup unaccounted for in the simulation model. The measured PR angles are within 30 • ± 1.8 • and 45 • ± 3 • compared to simulated results when the oblique incident angle is increased to 20 • , as shown in Fig. 16(b) and Fig. 17(b). As shown in Fig. 16(c) and 17(c), the PR angles are still relatively stable with oblique incidence up to 40 • . In order to show the total energy transmission efficiency (ETE) of the two orthogonal branches, a simple formula is given as follows: The simulated and measured ETE of the two orthogonal branches are shown in Fig. 18(a) and Fig. 18(b). It is observed that the measured ETE of the two examples A and B can be up to 98.8% and 99% under the normal incidence, respectively. As the oblique incident angle ϕ = 40 • , the total of the measured ETE of both examples is still above 98% in the operation band. It implies that almost all of the energy can be transmitted through the PRTS by the orthogonal coupling slotlines. A comparison between our designs with other PR surfaces in the literature is presented in Table 3. It reveals that although the profile of the proposed PRTS is thicker than other designs, our proposed design can achieve a smaller size of each unit cell in transversal cross-section than others and thus it has a larger oblique incident angle than others. More importantly, our proposed design can be synthesized using the coupling matrix, which can remarkably improve design efficiency and provide greater flexibility.

V. CONCLUSION
In this article, a novel PRTS of arbitrary angle based on 3-D FSSs is proposed, which is developed using three parallelcoupled slotline sections with open-short ends etched on two orthogonal dielectric substrates. The three slotline sections can form two orthogonal transmission paths. The polarization of the outgoing EM waves can be determined by the ratio of the coupling coefficient of each path. To reveal the operating mechanism, a filtering power divider concept and an equivalent circuit model are employed to analyze first, and a synthesis scheme based on the coupling matrix is utilized to synthesize and design the proposed PRTS. To validate the design concept, two examples with different FBWs and PR angles at the same center frequency and RL are developed. Finally, two PRTS prototypes with PR angles of 30 • and 45 • are fabricated and measured, respectively. The measured results agree well with the theoretical and simulated results, thus verifying that the proposed PRTS exhibits good stability under a large incident angle.