A Monolayer Dually Polarized High-Order Space Harmonic Leaky-Wave SIW Antennas With Fixed and Scanning Beam Capability

In this work, the concept of high-order space harmonics (HSHs) applied to a hybrid form of nonradiative dielectric (NRD) waveguide and substrate-integrated square waveguide (SIW) involving both TE10 and TE01 modes is investigated with aim to develop multi-functional dually polarized leaky-wave antennas (DP-LWAs). A polarization-selective coupling (PSC) mechanism is used to guide the two orthogonally oriented modes in the hybrid structure to achieve a polarization diversity feature. The polarization effect on unit-cell geometrical structures is examined for different characteristics of LWA such as propagation constant behavior, leakage ratio, and scanning range variation. In addition, we demonstrate how the extrinsic characteristics of HSHs in a one-dimensional periodic LWA and its unit-cell analysis can provide design flexibility through two case studies. In the first case of demonstration, we show a dually polarized multi-beam LWA design with wide scanning capability in which we use both even and odd HSHs of n $\mathrm {\epsilon }$ [−1, −2]). The second antenna design case is concerned with a DP-LWA having the capability of fixed and scanning beams operation by using a combinatory set of PSC and four HSHs of n $\mathrm {\epsilon }$ [−1, −2, −3, −4]. Two DP-LWA prototypes are demonstrated as proofs of concept, and the obtained experimental results show a very good agreement with their simulation and analysis counterparts.

them hard to integrate with other parts. Planar OMT structures were investigated in [23], [24] at the expense of using multi-layers. Nevertheless, a low-profile monolayer planar OMT with integration capability should be highly attractive for 5G applications and beyond.
Additionally, considerable efforts have been made in realizing dually polarized antennas (DPAs) due to the spectrum limitation and high data demand in cellular systems. DPAs judging from their mechanisms and radiation properties are divided into two different categories. The first category belongs to the DPA with the capability of fixed beam operations [25], [26], [27], [28], [29], [30], [31], where the antenna beam of each polarization targets a similar angle. These types of DPAs are more suitable for channel capacity improvement especially in dense areas where we encounter user plane (UE) traffic [32], [33]. The second category is related to DPAs featuring a different beam direction for each polarization [34], [35]. These types of DPAs provide a wider coverage compared to the first category since the beam of each polarization targets different angles. It should be mentioned that besides this categorization, the features of beam steering and polarization diversity can also be beneficial for multi-user streaming and broadcasting where the use of a multiple input multiple output (MIMO) technology can improve the cell throughput (base station) performances [36], [37].
Several methods were proposed to realize a dually polarized LWA (DP-LWA), such as the dual-mode transmission line [38], symmetrical spoof surface plasmon polariton lines [39], back-to-back slot array [40], OMT based LWA [35], partially reflected surface (PRS) LWA [41], metamaterial LWA [42], and lens LWA [43]. However, all those methodologies can only provide a single beam operation for each polarization which makes them unsuitable for multi-beam operations, which are essential in 5G new radio (NR) [44]. A multi-beam mechanism can be used to enhance the spectral efficiency, power delay profile, reduce recovery times [45], and also to provide reduction in computational complexity and access times [46]. Furthermore, a code book technique can be applied to simultaneous multi-beam radiators to achieve faster initial access times [47]. Some methods were also used to achieve common source multi-beam LWAs, such as dual-mode transmission line [48], defected ground structure (DGS) [49] and aperture coupled feeding [50], [51]. However, applying all these methodologies [48], [49], [50], [51] to a DPA for achieving a multi-beam operation is challenging, and to the authors' best knowledge, do not yet exist in literature. In addition to multi-beam operation, dual-polarization, and beam steering features, it is also beneficial for the antenna to have a fixed beam operation, especially for pointto-point communications [52]. However, the beam in most LWAs usually steers with frequency which makes them unamenable for fixed beam operations. Besides, using single beam operation there will be a high possibility of link failure due to the blockage effect. This issue can be overcome by using multi-beam operation and assigning a code book to the radiated signals. In this case, if one of the beams fails then the alternative one can still provide a reliable connection [53].
In this work, we introduce a method for designing and analyzing 1D-periodic LWA as a common source dually polarized multi-beam antenna. We try to benefit from the hybrid form of the nonradiative dielectric (NRD) waveguide and the substrate-integrated square waveguide (SIW) to achieve the polarization diversity for our LWA design. The authors in [54], [55] demonstrated a planar OMT which has the potential integration capacity with antenna. However, they did not investigate this feature in their work since the focus was on developing the concept and theory of OMT operation. Hence, in this work, we demonstrate a comprehensive analysis in this regard; we analyze the behavior of two OMT modes (TE 10 and TE 01 ), along with an LWA, for different higher-order space harmonic (HSH) of each OMT operation mode. This work differs from [56] where the authors discussed the HSH behavior for a single polarization LWA and their focus was on the broadside condition of n = −2 space harmonic. Here, however, we tackle the HSH behavior with the presence of two orthogonal modes in which the geometrical effect of the LWA's unit cell on the propagation constant, leakage ratio, and stop-band for single and dual-mode operations is investigated and their differences are highlighted. Furthermore, the design flexibility in terms of multi-beam alignment of the HSH for two OMT modes is also investigated.
In the first design reported in this work, we demonstrate a dually polarized multi-beam steering LWA where we use a combination of one even and one odd HSH of n [−1, −2] of each OMT mode. It should be mentioned that a combination of fixed beam operation along with beam steering can sufficiently reduce the volume and size of wireless communication systems and present an alternative solution for future broadcasting designs. This work also demonstrates a dually polarized multi-beam LWA with the capability of providing simultaneous fixed and scanning beam operation. In the second design, we use a combination of polarization-selective coupling (PSC) mechanism and four HSHs belonging to n [−1, −2, −3, −4]. This design is different compared to [57] where two reflector antennas with 180 • phase shift between the reflected waves were used to achieve a single fixed-beam operation for their LWA. This contribution differs from [52] and [58] where the fixed single beam LWA (single polarization) is achieved at the expense of adding a mirror and glide-symmetric prisms to the structure. While, in this work, we achieve fixed multi-beam operation by utilizing the HSH concept along with beam scanning feature. To the best of our knowledge, this is the first time that an antenna with the capability of simultaneous fixed and scanning beams operation is proposed and demonstrated. Experimental results show a good agreement between the simulation, analysis, and measurements (CST STUDIO SUITE used in all simulations and analyses).
This work is organized as follows. Section II investigates LWA's unit-cell for two oriented modes and shows the analysis and experimental validation of a dually polarized multi-beam steering LWA. Section III gives the modeling, analysis and experimental validation of a DP-LWA with simultaneous fixed and scanning multi-beam capabilities. Finally, Section IV concludes the work and discusses future perspectives.

A. EFFECT OF POLARIZATION ON UNIT-CELL RADIATED POWER
In this section, we study and analyze the effects of the TE 10 and TE 01 polarization diversity in connection to our antenna radiating element for a DP-LWA design based on the PSC concept. The performances of the periodic LWAs are predictable thanks to careful analysis of their unit-cell radiating element [4]. The geometrical shape of the radiating element is usually chosen based on the current distribution of the transmission line propagating mode. Most of the uni-polarized periodic LWAs operate with a single mode to prevent additional modes from any hybrid-mode coupling in their operation band [59]. However, for dual-polarization (if two modes are used for the antenna operation), it is necessary to examine the coupling effect between the two operating modes and its impact on the antenna performances. In addition, for each polarization, we also need to investigate the geometrical shape of the LWA's unit-cell to make sure we can get an acceptable radiating performance from it.
As a first step, we study and analyze the effect of the SIW-LWA unit-cell for two modes, namely TE 10 and TE 01 . Here we are interested in examining the unit-cell in the Ka-band. Therefore, the square section of the SIW-LWA's unit-cell is considered to be 3mm, using an RT6002 substrate. Based on these dimensions, the cutoffs of the TE 10 and TE 01 modes are 29 GHz, and the higher mode cutoff starts from 41.2GHz. The behavior of most periodic LWAs is usually predictable thanks to a careful analysis of their Brillouin diagram, where the relation between different space harmonics can be written as β n = β 0 + 2nπ/p. In this work (DP-LWA), we want to study the behavior of the propagation constant (β 0 ) of the two modes TE 10 and TE 01 . Here the focus is on the n = −1 space harmonic. By choosing the periodicity of the LWA's unit-cell as p = 6.9mm and the square section's length as 3mm, the contribution of the n = −1 space harmonic is a single fast wave (β −1 /k 0 ≤ 1) for the frequencies above 36 GHz where the fundamental and n = −2 space harmonics remain as a slow wave due to the condition of β 0 /k 0 ≤ 1 and β −2 /k 0 ≥ 1, respectively. Fig. 1(a) shows the Brillouin diagram analysis in this case (the propagation model is theoretically calculated). Here we achieve the same Brillouin diagram for the TE 10 and TE 01 modes because of the square section unit-cell. It should be mentioned that besides the analysis of the Brillouin diagram, it is also necessary to investigate the radiating element's effect on the performance of the LWA.   1(b) shows different radiating geometries that can be used for our LWA's unit-cell. Here the focus is on the upper frequency of the Ka-band, so the length of the cross-slot, X-slot, square-slot and diameter of circleslot are considered to be λg/2 (2.2mm) and the width of cross-slot, and X-slot are λg/10 (0.44mm). The response of the impedance/admittance behavior of these LWA's unitcells is demonstrated in Figs. 1(c)-(f). In this analysis, we consider modes 1 and 2 of the LWA's unit-cell as TE 10 and TE 01 , respectively, where the relation for impedance and admittance behavior based on the Z and Y matrix analysis [60] can be written as: (1) (1) TE 01 →B y = −1/Y 2(2)1 (2) (2) where By and Cz are the parameters of the ABCD matrix of the LWA's unit-cell. The analysis of Figs Another point that needs to be considered here is related to the radiated power response of LWA's unit-cell. The analysis of Figs. 1(c)-(f) shows that the radiating element with a wider geometry (square-slot in this case) radiates more power compared to other LWA's unit-cells. However, the slope variation of the impedance/admittance behavior of the square-slot is sharper compared to other LWA's unit-cells in the frequency range of interest. Thus, we expect to have more gain variation for our LWA, which is not desirable. The analysis of Figs. 1(c)-(f) shows that the cross-slot can provide a suitable tradeoff between the slope variation and the real part of unit-cell's impedance/admittance behavior for both polarizations (TE 10 and TE 01 modes) within the frequency range of interest. Thus, we choose the cross slot as the radiating element for the design of the DP-LWA.

B. EFFECT OF POLARIZATION ON UNIT-CELL PROPAGATION CONSTANT AND LEAKAGE RATIO
It should be mentioned that most of the periodic LWAs operate with a single polarization. However, adding a second polarization to the LWA will highly affect the performance of the primary one. To address this issue, we investigate the effect of single-and dual-polarization on the performance of our periodic LWA unit-cells. Figs. 2(a)-(b) show the propagation constant and leakage ratio behavior for different LWA's unit-cells for the case of single-and dual-polarization where the phase and attenuation constants can be expressed as [60]: Here A and D are two components of the ABCD matrix and they can be found from the Z and Y matrices. In the analysis of Figs. 2(a)-(b), we consider having transverse and longitudinal slots with single polarization and crossslots for dual-polarization. The periodicity in the unit-cells of Fig. 2(a) is chosen based on the analysis of Fig. 1(a) and is equal to 6.9mm and the length and width of all slots are λg/2 (2.2mm) and λg/10 (0.44mm), respectively. Besides, the square section of the SIW-LWA's unit-cell is set to 3mm with the substrate RT6002. Based on these dimensions, the same cutoff is obtained for the two modes as in Section II-A. At first, we compare the operation of the TE 10 mode between the single and dual-polarization of our LWA's unit-cells. The analysis of Fig. 2(b) shows that the propagating TE 10 mode can provide a wider stop-band frequency range for the case of a single polarization (transverse slot) compared to the dual-polarization (cross-slot). In addition, the leakage ratio of propagating TE 10 mode for the case of a single polarization (transverse slot) has a higher value at the pass-band and stop-band frequencies compared to the dual-polarization (cross-slot). Hence, we should expect to have more radiated power in the single polarization (transverse slot). Furthermore, the propagation constant behavior of the propagating TE 10 mode also has different behavior for each case of the single and dual polarization at the passband frequency. It is expected that the propagating TE 10 mode for the dual-polarization provides more scanning range compared to the single polarization. This is attributed to the higher value and increased variation of its propagation constant (β −1 ) (over the pass-band). On the contrary, the propagation constant behavior and leakage ratio variation of the propagating TE 01 mode has approximately similar behavior in both the single polarization (longitudinal slot) and dual-polarization (cross-slot). Hence, their radiated power and scanning angle variations must be similar. Another notable point from the analysis of Fig. 2(b) is that we also have different responses of propagating TE 01 and TE 10 modes for dual-polarization cases (cross-slot). Here the TE 10 mode can provide a wider stop-band frequency compared to the TE 01 mode scenario while the leakage ratio of these modes has an approximately similar response in their common passband frequency range. In addition, the propagation constant (β −1 ) of the TE 01 mode shows more variation as compared to that of the TE 10 mode (in the common passband) which results in a wider scanning range and angular misalignment.
To overcome the issue of misalignment between propagation constant variations in each polarization, we investigate the geometrical effect of the cross-slot LWA's unit-cell for the two polarizations of our DP-LWA. Figs. 2(c)-(d) show the effect of slot length variation of the cross-slot for the propagating TE 10 and TE 01 modes, respectively. The periodicity of unit-cells in Figs. 2(c)-(d) is chosen based on the analysis of Fig. 1(a). The analysis of Figs. 2(c)-(d) shows that using a longer slot not only provides a wider stop band, but it also shifts down the lower edge of the stop band to lower frequencies while slightly affecting the upper stop band edge as well. In addition, using the longer slot decreases  Fig. 2(e) shows that a wider slot not only provides a wider stop band, but also it shifts up the upper edge of the stop band to higher frequencies while slightly affecting the lower edge of the stop band. Also, it slightly affects the propagation constant (β −1 ) in the passband frequency range. While for the TE 01 mode, using a wider slot decreases the variation and values of the propagation constant (β −1 ), which results in a shorter scanning range. Besides, using a wider slot for the propagating TE 01 mode also increases the stop band and shifts down the lower edge of stop band to lower frequencies, while slightly affecting the upper edge of stop band.
The analysis of Figs. 2(c)-(f) suggests that a symmetric cross slot could provide an approximate similar response of leakage ratio for both TE 10 and TE 01 modes in the common passband frequency range. However, the propagation constant (β −1 ) of these modes has different responses in their common passband frequency ranges. Interestingly, choosing the cross-slot as our LWA's radiating element gives us the flexibility of controlling the propagation constant for each propagating modes of the TE 10 and TE 01 since it can be approximately considered as a combination set of a transverse slot and a longitudinal slot. From the analysis of Figs. 2(c)-(f), it is expected that an unsymmetrical crossslot can decrease the difference between the propagation constants of each propagating mode, TE 10 and TE 01 . This is shown by plotting the β −1TE10 -β −1TE01 of the LWA's unit-cell. In addition, if the condition of β −1TE10 − β −1TE01 ≈ 0 is satisfied, we should expect that the phase difference between the two modes also approaches zero ( ϕTE 01 -ϕTE 10 ≈ 0) and that their related beams must be in alignment. However, the β −1TE10 -β −1TE01 condition is only valid and applicable for the single space harmonic (this study case n = −1) operation for our DP-LWA. The beam's alignment for a dual-polarized operation (two modes operation in this study) is more complex when we are simultaneously dealing with more than one space harmonic (as fast waves) at the same frequency, since the distinction between the phase operation of each mode and their related HSH becomes more challenging and complicated.

C. BEAM ALIGNMENT OF HSH
In order to address this issue in a comprehensive manner, first, we analyzed and studied the behavior of phase variation of different unit-cells with the capability of supporting two space harmonics as fast waves at the same frequency. Based on the analysis of β n = β 0 + 2nπ/p, we expect to get the contribution of n [−1, −2] space harmonics as fast waves by choosing the permittivity, effective width and periodicity of our SIW-LWA's unit-cell as 2.94, 3mm, and 8.5mm, respectively, in the frequency range of 35 < f GHz < 39. Fig. 3(a) shows its Brillouin diagram analysis, where the n = 0 space harmonic remains as slow waves and n [−1, −2] space harmonics behave as fast waves for frequencies above 35GHz. The Brillouin diagram of Fig. 3(a) is also applicable to both TE 10 and TE 01 modes since the SIW-LWA's unit-cell is designed to have a square section dimension (3mm). Figs. 3(b)-(c) show symmetric and asymmetric cross-slot type radiating elements with their phase shift difference response of two propagating modes of TE 10 and TE 01 ( ϕTE 01 -ϕTE 10 ), respectively. In the analysis of Figs. 3(b)-(c) at first, we try to investigate the phase response behavior for two symmetric cross-slots (type A), where the length and width dimensions are set to be λg/2 and λg/10 for f = 35.5GHz (black line) and f = 38.5 GHz (blue line), respectively. We chose these two frequencies since they can be set as lower and upper frequencies for the passband of the Brillouin diagram of Fig. 3(a). The analysis of Fig. 3(c) shows that using a symmetric cross-slots (blue and black lines) cannot provide a flat and low response for the phase shift difference of two propagating modes (TE 10 and TE 01 ). However, based on the analysis of Fig. 2, we should expect to have less difference between two propagating modes' phase shift variations by using an asymmetric cross-slot in which the transverse slot has a shorter length compared to the longitudinal slot (type B). Fig. 3(c) shows that a transverse slot with dimensions of 2.2mm, and 0.5mm along with a longitudinal slot with dimensions of 2.5mm, and 0.5mm in a cross slot can provide a low value and flat response for the phase shift difference variations between the two propagating modes (red line). Contrarily, an asymmetric cross-slot with a longer geometrical length for its transverse slot (type C) cannot provide such a flat response (green line). Besides, our analysis also shows that choosing a longer dimension for the transverse slot compared to the longitudinal slot, in addition to shifting up/down the longitudinal slot in an asymmetric cross slot (type D), not only can give us a flat response for the two propagating modes' phase shift difference but also gives us the feasibility of shifting up/down the phase shift difference between them (brown and purple lines).
Comparing the A-D responses of the radiating element in Fig. 3 indicates that a better beam alignment must be achieved for type B since it realizes a lower value of phase shift difference between the two propagating modes. It should be mentioned that the beam angle alignment between the two propagating modes in Fig. 3 does not comply with the value of the phase shift difference between the two propagating modes. This non-compliance occurs as it shows the total phase response difference between the two propagating modes and their related HSH's phases. In order to find an accurate beam angle difference between the two propagating modes and their related HSH, we investigate the behavior of the propagation constant of each mode along with the propagation constant response of each mode's HSH. Here we are interested in analyzing the unit-cells of Fig. 3 for 35 < f GHz < 39. Based on β n = β 0 + 2nπ/p, the total phase response for each propagating mode in the frequency range of our interest must be the summation of βTE 10 (n = −1) + βTE 10 (n = −2) for TE 10 mode and βTE 01 (n = −1) + βTE 01 (n = −2) for TE 01 mode. They can be written as: where l z is the length of periodic unit-cell of our LWA. The difference between the two propagating modes is: It should be mentioned that the factor of 2 in (9) is only applicable when we are dealing with 2 space harmonics simultaneously (this case n [−1, −2]). However, the conditions in (9) can also be extendable for any set combination of different space harmonics as:  Fig. 3 (types B and D).
where m is the number of fast wave space harmonics in the frequency range of interest. One point that needs to be considered here is that the beam angle difference between similarly involved space harmonics of each propagating mode must comply with propagation constant difference value. The difference between their arcsine function can be written as: To investigate the beam angle difference between the two modes of TE 10 and TE 01 and their related HSH, in Fig. 4 we plot the propagation constant difference of TE 10 and TE 01 modes of the two SIW-LWAs' unit-cells of Fig. 3 (types B  and D). This analysis shows that the value of βTE 01 -βTE 01 in type B of the SIW-LWAs' unit-cell has a lower value compared to type D, which shows a good agreement with the geometrical unit-cell analysis of Figs. 2-3. To examine the validity of our analysis we choose a random frequency, i.e., 38GHz, to check the beam behavior of the two propagating modes in comparison. Based on Fig. 3, at f = 38GHz we have two space harmonics as fast waves, so m in (11) must be set to 2, and the value of k 0 is set to be 2.35. The value of βTE 01 -βTE 01 in Fig. 4 at f = 38GHz for unit-cell types B and D are 0.15 and 0.6, respectively. Dividing these values by 2 and putting them in the arcsine function of (11) give us the angle of 1.82 • and 7.73 • for SIW-LWAs' unit-cells type B and D, respectively. So, for type B of SIW-LWA's unit-cell in Fig. 3, we should expect that the beam angle difference between two n = −1 space harmonics related to the TE 10 and TE 01 modes must be 1.82 • at f = 38GHz. Similar angle difference must be expected for two beams of n = −2 space harmonics at f = 38GHz in a type B unit-cell. On the other hand, for type D unit-cell, the expected angle difference is 7.73 • at f = 38GHz for each pair of n [−1, −2] space harmonics. Fig. 5 shows the simulated radiation pattern of two SIW-LWAs based on type B and D of SIW-LWA's unitcells in Fig. 3 (the length of the antennas considered to be more than 10λ 0 in this example). Fig. 5 shows a 2 • and 7 • difference between each pair of beams for our LWAs based on type B and D unit-cells, respectively. This shows a good agreement with our unit-cell analysis of Figs. 3-4.  Fig. 3.

SIW-LWA's unit-cells of
The analysis of Figs. 3-5 also show that we can feasibly shift the beams of each mode by choosing a suitable unit-cell (Type D) for our LWA with a careful analysis of the propagation constant behavior of the LWA's unit-cell. It should be mentioned that besides the study of the DP-LWA's unit-cell behavior, we also need to analyze its effect on the bandwidth for our antenna to make sure that the two propagating modes of our DP-LWA have roughly the same bandwidth operation. Fig. 6 shows the simulated S-parameter responses of two LWAs based on type B and D unit-cells of the previous example (the antenna length is considered to be 10λ 0 in this analysis). The analysis of Fig. 6 shows that the type B DP-LWA can provide a better alignment between the S-parameter responses compared to the type D for the two propagating modes of TE 10 and TE 01 . Thus, it is chosen in the fabricated prototype.

D. EXPERIMENTAL VALIDATION AND CONFIGURATION OF SIW-DP-LWA
The geometry of the designed SIW-DP-LWA is shown in Fig. 7. Here we utilized PSC-OMT as a compact feeding network since it provides two orthogonal modes of TE 10 and TE 01 . The dimensions of the PSC-OMT were calculated based on the analysis of [61] and applied to the proposed design. In addition, a transition between the PSC-OMT and the WR28 waveguide is designed to support the operation over the frequency range of interest (35 < f GHz < 39). Here we use a triangular shape for our transition section (taper section) with the length of 9mm where we removed the top and bottom copper layer of the substrate. This transition is used on both ports of the PSC-OMT along its vertical and horizontal axes for the excitation of TE 10 and TE 01 modes, respectively. For the LWA section, we used a type B cross slot radiating element. The length of the LWA's section is considered to be more than 10λ 0 in this analysis. In addition, due to the reciprocal behavior of HSH in the Brillouin diagram analysis of Fig. 3, we used two sets of SIW-DP-LWAs in this design to have a symmetric multi-beam radiation for our antenna. Fig. 8 shows the fabricated prototype of the SIW-DP-LWA with the PCS-OMT and transition section to WR28. The dielectric material used in this design is RT6002 with a dielectric constant of ε r = 2.94 and thickness of 3mm. In addition, we also applied a metal epoxy in the via section of the PSC-OMT in order to ensure the via's wall is well covered with metal to keep the antenna performance acceptable. The type B cross slot has transverse and longitudinal slots lengths of 2.2mm and 2.5mm, respectively, with a width of 0.5mm. Figs. 9 (a)-(b) show simulated and measured S-parameters of the proposed SIW-DP-LWA for ports 1-2 and ports 7-8, respectively. The measured reflection coefficients of TE 10 (port 1 and 7) and TE 01 (port 2 and 8) modes for the antenna are below −10dB for the frequency ranges of 35.3 < f GHz < 38.9 and 35.7 < f GHz < 38.7, respectively. This shows that the two propagating modes have an approximately similar bandwidth behavior for our DP-LWA. In addition, the simulated and measured leakage ratio of the antenna for the two propagating modes has approximately similar behavior. There is a small discrepancy between the measured and simulated results due to fabrication tolerances. According to the analysis of Fig. 3(a), it is expected to have one proper beam (n = −2) and one improper beam (n = −1) for our antenna. Fig. 10 shows the antenna radiation patterns at two frequencies; f = 36.5GHz and f = 38.4GHz for the two propagating modes of TE 10 and TE 01 where we have one backward beam and one forward  Fig. 3(a), we expect that for both modes the beam related to the n = −2 space harmonics scans faster than the n = −1 space harmonic beam since the arcsine function variation of β −2 /k 0 is closer to 1 compared to the β −1 /k 0 . This behavior is also validated in Fig. 10, in which the beam of n = −2 space harmonic covers more angles than the beam of n = −1 space harmonic for the same frequency variation. Here the measured scanning ranges of n [−1, −2] space harmonics are 4 • to 22 • and −67 • to −33, respectively, which shows a good agreement with the ratio analysis of arcsine β −n /k 0 of Fig. 3(a). In addition, based on the analysis of Fig. 4, a misalignment is expected between the beams of TE 10 and TE 01 modes for our antenna. Fig. 10 also shows that we have a 2 • mismatch alignment between the beams of TE 10 and TE 01 modes. Thus, a good agreement is observed between the analysis, simulations, and measurement results of the proposed DP-SIW-LWA in which a wide scanning range can be achieved by using the HSH (n [−1, −2]) concept.

A. ANALYSIS AND MODELING
A DPA based on its radiation mechanism can be divided into two categories: a DPA with fix beam operation and  a DPA with scanning beam operation. To the best of the authors' knowledge, a DPA with the capability of both scanning and fixed beam operations does not exist in literature. Therefore, in this work, we demonstrate an DPA with fixed and scanning beam capabilities by using the HSH concept. The core idea of the proposed design is to assign a specific task (scan or fix) to each polarization of the proposed DP-LWA. However, several aspects need to be analyzed and their trade off needs to be investigated. In the proposed design, first, we need to choose the topology of each polarization and decide about their function and operation task. In addition, other antenna characteristics such as gain level alignment and gain stability between each polarization need to be analyzed. In the proposed design and as a first step, we chose to assign the scanning and fixed beam operations to the TE 01 and TE 10 modes, respectively.
In the second step, we analyze the Brillouin diagram with the unit-cell geometrical effect (periodicity) on the antenna performances. From that, we can predict the HSH behavior in the fast wave region and evaluate the scanning range variation. The total behavior of the scanning range in periodic LWAs is predictable based on the arcsine function analysis. This function has two specific features when applied in the analysis of periodic LWAs. First, as long as we keep the ratio of β n /k 0 close to zero in the frequency range of the antenna, we expect the scanning range variation related to the involved space harmonic not to vary too much due to the specific behavior of the arcsine function. However, periodic LWAs usually encounter the stopband issue when the ratio of β n /k 0 approaches zero. Therefore, using this feature of the arcsine function to overcome the wide scanning range requires applying methods to suppress the open stopband problem, such as a phase reversal technique [62], impedance matched unit cells [56], [63], [64], and reflection cancelation [65]. The second feature of the arcsine function, which can be applied to overcome the limitation of a wide scanning range in periodic LWAs is the possibility of decreasing the variation of β n /k 0 in the passband frequency range operation of the antenna, leading to the decrease in the scanning range variation. Since the scope of this work is not focused on the band edge issue of periodic LWAs, we chose to investigate the possibility of applying the second feature to our antenna.
From the analysis of Fig. 3(a), the scanning range variation of n = −1 space harmonic decrease at the third passband frequencies (f ≥ 35GHz) compared to lower ones. This is due to the higher value of k 0 which decrease the variation of β −1 /k 0 . On the other hand, the n = −2 space harmonic has a wide scanning range variation compared to the n = −1 space harmonic in the third passband as the variation of β −2 /k 0 is close to 1. Thus, there is a possibility of decreasing the scanning range variation of the n = −2 space harmonic by approaching the fourth or higher passband since the ratio of β −2 /k 0 will be decreased. Fig. 11 shows the Brillouin diagram analysis of a one-dimensional periodic SIW-LWA with 5 passbands (periodicity, dielectric propagating medium, and effective width are 17mm, 2.94, and 3mm, respectively). Here we are interested in setting our analysis for the upper frequency range of Ka-band (f ≥ 36GHz). Therefore, based on the analysis of Fig. 11, the fundamental space harmonic  Fig. 11 also shows that the n = −2 space harmonic has the lowest scanning range variation in the fifth passband since its propagation constant is located near the center of the Brillouin diagram, resulting in the lowest value and variation ratio of β n /k 0 . Thus, we expect to have very low scanning range variation (around 5 • ) for our antenna in the frequency range of interest (36 < f GHz < 38).
A point that needs to be considered here is that the topology of applying scanning and fixed beam operation requires different periodicities (based on Brillouin diagram analysis of Fig. 3 and Fig. 11). Thus, to avoid modifying the geometrical shape of the antenna radiating element, we should choose the periodicity ratio between both polarizations as PTE 01 /PTE 10 = 0.5 or PTE 10 /PTE 01 = 2. Besides, taking the periodicity into the consideration for the proposed DP-LWA has a direct effect on the number of radiating elements for each polarization. Since one polarization has a higher number of radiating elements. Therefore, as a first estimation, we should expect to have a higher gain for one polarization compared to the other (if only we judge based on the radiating element number). A point that needs to be considered is that we also need to evaluate the effect and influence of the radiating elements on the antenna performances for both operating modes of TE 01 and TE 10 especially when we are dealing with the dually polarized operation.
To investigate this effect and as a third step, we analyze the geometrical effect of the radiating elements on the radiated power response of each polarization (TE 10 and TE 01 modes). First, we consider a symmetric cross slot and its longitudinal slot as our radiating element where they are assigned to the TE 10 and TE 01 modes, respectively (TE 01 and TE 10 modes are assigned to the Brillouin diagram of Fig. 3 and Fig. 11, respectively). Fig. 12 shows the radiated power response of these unit-cells for the frequency range of interest (36 < f GHz < 38). This analysis shows that the TE 01 and TE 10 radiated power responses have different behavior for each of these unit-cells. The TE 10 mode has a higher radiated power compared to the TE 01 mode for the case of a symmetric cross slot, while for the case of a longitudinal slot it has much less radiated power compared to the TE 01 mode. A point that needs to be considered here is that the TE 01 mode has an approximate similar response for both radiating elements (symmetric cross slot and longitudinal slot) and its slope variation is not sharp, while the slope variation for the TE 10 has a sharp variation for the case of a symmetric cross slot which results in more gain variation for this antenna polarization (TE 10 mode). To overcome this issue, we investigated the geometrical effect of our radiating elements on their radiated power response for two polarizations of our DP-LWA. In this regard, we analyzed the effect of slot width variation on the radiated power responses of TE 10 and TE 01 modes for the case of a crossslot and a longitudinal slot. The analysis of Fig. 12 shows that the radiated power response of TE 01 mode is increased by increasing the longitudinal slot width for both cases of cross slot and longitudinal slot. However, for the case of TE 10 mode, the radiated power response will decrease for the case of the asymmetric cross slot while it remains close to zero for the case of a longitudinal slot and it is much less compared to the TE 01 mode. It should be mentioned that in addition to the LWA unit-cell analysis, we also need to check the S-parameter response of each antenna polarization to ensure that the DPA has an acceptable performance. Fig. 13 (a)-(b) show the S-parameters responses for the case of DP-LWA with symmetric and asymmetric unit-cells, respectively (dimensions of LWA unit-cells specified in the capture of Fig. 13). The analysis of Fig. 13 shows that the antenna polarization related to the TE 10 mode has a narrower bandwidth compared to the TE 01 mode since it has a narrower passband frequency range. This shows a good agreement with the Brillouin diagram analysis of Fig. 11 and Fig. 3. In addition, S-parameters responses (S21) related to the TE 10 mode for the case of LWA with symmetric unit-cell have a sharper variation compared to the TE 01 mode, while both modes have smooth variation for the case of LWA with asymmetric unit-cell. In this regard, we should expect to have a mismatch between the gain level of each antenna polarization for the case of LWA with symmetric unit-cell. Besides, From Fig. 13 (b) it is expected that the beam levels of TE 10 and TE 01 modes have approximately similar levels since the radiated power of TE 10 mode (with four beams) is around 3 dB higher than TE 01 mode (with two beams).  for the TE 10 mode. This shows a good agreement with the Brillouin diagram analysis of Figs. 11 and 3. Besides, these analyses show that we have a mismatch between the gain level of each polarization for the case of LWA with symmetric unit-cell (4.2dB difference). However, a good gain alignment is observed between the antenna polarization for the case of LWA with asymmetric unit-cell (0.1dB difference). This shows a good agreement with the analysis of Figs. 12-13. Thus, with a careful analysis of Brillouin diagrams and their related unit-cells, we are able to achieve a DPA with fixed and scanning beam capabilities. We could also assign the features of scanning and fixed beam operations to the TE 10 and TE 01 modes, respectively. However, the use different radiation elements will be required for our DPA to find a suitable tradeoff between the antenna characteristics of the two propagating modes. In Section III-B we validate our analysis by showing a prototype of the proposed DP-SIW-LWA. Such a dual operation can be beneficial in next-generation access points. Fig. 15 shows the design prototype of the proposed SIW-DP-LWA with simultaneous fixed and scanning beam capabilities. Here, we again use a PSC-OMT for our DPA. The dielectric material used in this design is RT6002 with a dielectric constant of ε r = 2.94 and thickness of 3mm. In the proposed DPA's prototype and the SIW-LWA's section, we use two sets of radiating elements in which we assign the periodicity of 17mm and 8.5mm to each. The first radiating element is an asymmetric cross slot in which the length of the transverse and longitudinal slot dimensions are set to 2.2mm, and their widths are set to 0.7mm and 1mm, respectively. The second radiating element is a longitudinal slot with length and width of 2.2mm and 1mm, respectively.

B. VALIDATION AND PROTOTYPE
Figs. 16 (a)-(b) show the simulated and measured S-parameters response of the proposed SIW-DP-LWA for ports 1-2 and ports 7-8, respectively. The measured reflection coefficients of the TE 10 (port 1 and 7) and TE 01 (port 2 and 8) modes for the antenna are below −10dB for the frequency ranges of 36.35 < f GHz < 38 and 36 < f GHz < 38, respectively. There is a small discrepancy between the measured and simulated results due to fabrication tolerances. According to Fig. 3(a) and Fig. 11, it is expected to have two improper beams (n [−1, −2]) and two proper beams (n [−1, −2]) for the TE 10 mode and one proper beam (n = −2) with one improper beam (n = −1) for the TE 01 mode of our DP-LWA. In addition, we used two sets of DP-LWA in this design to have a symmetric multibeam radiation for our antenna. Fig. 17 shows the antenna radiation patterns at the two frequencies of f = 36.5GHz and f = 37.5GHz for two propagating modes of TE 10 and TE 01 . Here we have two forward beams and two backward beams for the TE 10 mode and one backward beam with one forward beam for the TE 01 mode of our antenna. Besides, based on Fig. 3(a), we should expect that the beam related to the n = −2 space harmonic of the TE 01 mode scans faster than the n = −1 space harmonic beam of this mode since the arcsine function variation of β −2 /k 0 is closer to 1 compared to the β −1 /k 0 . In addition, Fig. 11 predicts to have a very low scanning range variation for each of HSH's beams of the TE 10 mode since the variation of β n /k 0 of this mode is very low, and they are not close to 1. Fig. 17  , respectively. This shows a good agreement with the ratio analysis of the arcsine β n /k 0 in Fig. 3(a) and Fig. 11. In addition, the scanning range variation for the fixed beam operation in this work is in an acceptable range compared to the scanning range variation in [57] and [66], which have around 12 • variation. Thus, an agreement observed between the analysis, simulation, and measurement results of the proposed DP-LWA in which the simultaneous fixed and scanning features are achieved by using the PSC mechanism and the HSH concept. Fig. 18(a) also shows the antenna gain performance for each polarization of the proposed DP-LWA. The peak gains for both TE 10 and TE 01 modes are 14.8dBi and 14.6dBi, respectively. An acceptable gain alignment is observed between those polarizations. This shows a good agreement with the unit-cell radiated power analyses of Fig. 12. The maximum gain difference between the beams related to the TE 10 and TE 01 modes are 0.2dB and 0.3dB, respectively (the gain values presented in Fig. 18(a) are the average values of the beams of TE 10 and TE 01 modes). Fig. 18(b) provides the side lobe level (SLL) of each polarization where they are operating in an acceptable range better than −12dB over the whole operating frequency range. Fig. 18(c) gives the radiation efficiency where a good agreement is observed for both polarizations. A maximum radiation efficiency of 84% is obtained for each polarization. Besides, a high level of coupling between antenna ports could be highly beneficial to reduce the effects of interference. Fig. 18(d) shows the port coupling level of the proposed antenna. A coupling level better than -40dB is observed for the whole operating frequency range. Table 1 shows the comparison between proposed work and others LWAs. The work presented in Section II (LWA with scanning feature) achieves more bandwidth and lower scanning range compared to [40] while these features improved in [39] by using a plasmon polariton LWA. The works of [39], [40] generated a single beam while the work presented here can provide multi-beam operation from each antenna polarization which can improve the link reliability and availability [53]. Besides, the antenna demonstrated in Section III takes the benefit of shrinking the size and achieving a lower scanning range variation compared to the reflector based LWAs in [57], [66] for the case of a fixed beam operation. On the other hand, using dispersive symmetric prism [58] and glide symmetric prism [52] can provide a lower scanning range and wider bandwidth at the expense of a larger geometry for the antenna to achieve a fixed beam operation. Finally, this work demonstrates a combination of fixed and scanning beam operations which can reduce the volume and size of wireless communication systems, thereby presenting an alternative solution for future broadcasting and wireless developments.

IV. CONCLUSION
The concept of applying HSHs in a hybrid form of the NRD waveguide and the square SIW structure for the TE 10 and TE 01 modes is proposed and applied to several theoretical and experimental prototypes in this work. The properties of polarization diversity related to the unit-cell geometrical structures for different features of an LWA are investigated. It is shown that the propagation constant of the TE 10 and TE 01 modes can be manipulated by the geometrical structure of the LWA's radiating element for achieving the design flexibility of the proposed antenna. In addition, the effect of the difference between the two propagation constants for the two modes TE 10 and TE 01 on the beam alignment of their related HSHs is investigated. Besides, the feasibility of even and odd HSHs (n [−1, −2, −3, −4]) of the two propagating modes (TE 10 and TE 01 ) in different passbands is studied, where we show that the features of scanning and fixed beam operations are achievable with a careful analysis of the Brillouin diagram and propagation constant variation of each HSH. In this regard, the PSC mechanism along with the HSH concept is used in this work to demonstrate a multifunctional SIW-DP-LWA with the capability of simultaneous fixed and scanning multi-beam operations. In this work, two experimental prototypes of the proposed antennas are shown as proofs of concept to validate the analysis method formulated in this work. The measured results show a good agreement with simulated counterparts, thus validating the proposed LWAs.