Automatic Design of a CP Fan-Beam Linear Slotted Array in SIW Technology

In this paper, we present a technique for the automatic optimization of a right-hand circular-polarized (RHCP) linear slotted array in substrate integrated waveguide (SIW) technology for the so-called “last-mile” communications, in which a narrow (broad) beam is required in the elevation (azimuthal) plane. The procedure can be divided into two steps; i) the optimization of isolated single slot pairs, by means of two physics-based functionals able to minimize both slot pair back-scattering and axial-ratio, ii) the optimization of the entire linear array of slot pairs, by means of a global functional able to match the co-polar component of the synthesized aperture distribution to the target one, i.e., a uniform magnetic current distribution to radiate a narrow beam in the elevation plane. A prototype of array was optimized at 24.3GHz and measured in the bandwidth [24.1, 24.5] GHz, showing a fair agreement against full-wave simulations, thus validating the exposed techniques.


I. INTRODUCTION
Substrate integrated waveguides (SIWs) are a promising technology for the design of low-cost and low-profile planar electromagnetic (EM) devices at microwaves/millimeter waves, for the control of guided waves [1]- [5] and for the design of low-cost planar antennas [6]- [10]. They consist of two conveniently spaced rows of metallic pins able to behave as equivalent electric walls. The main advantage of SIWs is undoubtedly their flexibility, i.e. several kinds of passive devices such as filters, power combiners, splitters, cavities, and junctions can be easily designed, and their reduced size, thus allowing scalability at microwaves/millimeter waves. Such kind of devices can be profitably analyzed by adopting ad hoc numerical techniques such as the method of moments (MoM) [11], [12] or hybrid methods [13], able to efficiently compute coupling between pins and radiating slots.
On the other hand, in recent years physics-based optimization techniques were successfully developed to design The associate editor coordinating the review of this manuscript and approving it for publication was Wanchen Yang . radial-line slot-arrays (RLSA), i.e. radial waveguides loaded by several slots on their top plate, in order to shape an arbitrary desired aperture field distribution, both for antenna [14]- [18] and focusing [19]- [23] applications. In [24], the Authors introduced the technique here investigated to design and optimize a circular polarized (CP) fan-beam linear array in SIW, but neither experimental validation of the proposed technique nor the detailed synthesis of the entire array was presented at that time. Instead, in this paper we extend the automatic techniques developed for the optimization of CP RLSA antennas [15] to design and optimize a sectoral beam CP linear array in SIW technology, by considering not only the slot pair relative positions and lengths are modified during the optimization loop, but also their mutual orientations, in order to improve the polarization purity of the radiated field.
The EM analysis of such structures is performed by using an in-house MoM [11], [12], whereas the optimization loop is based on suitable functionals and geometry-based update rules.
The optimized CP linear slotted array we propose can profitably find application in so-called ''last-mile'' communications (for instance, in rural or sparsely populated areas), in which the antenna is required to radiate a fan beam, i.e., characterized by a broad (narrow) pattern in the azimuthal (elevation) plane.
Section II presents a simple analytical model that allows the definition of two complex functionals and suitable update rules for the geometrical optimization of single right-hand circular polarized (RHCP) slot pair in isolation (i.e., without considering the mutual coupling between adjacent pairs). Such optimized slot pairs are stored in a database (DB) for different lengths and then used as building blocks for synthesizing the starting array configuration. Then, a further optimization loop is developed to take into account also the mutual coupling between different slot pairs, based on a global functional F and associated update rules. To validate the proposed approach, in Section III we present the design of a sectoral pattern RHCP linear array in SIW at 24.3GHz, and a prototype of the array together with measurements. Finally, conclusions are drawn.

II. FORMULATION OF PHYSICS-BASED FUNCTIONALS FOR THE OPTIMIZATION OF CP LINEAR ARRAY IN SIW
In this Section, a simple analytical model is developed in order to define a suitable set of parameters for the optimization of the array unit cell, namely a RHCP pair of slots with centers in P 1,2 = (∓ X , ∓ Z ) and directed alonĝ u 1,2 = ± sin θ sx + cos θ sẑ (Fig. 1). As extensively done in [15], [25], narrow slots can be modeled as equivalent magnetic dipoles, whose moment is M 1,2 = M 1,2û1,2 . The SIW is fed by the dominant mode, which can be assumed as a TE z 10 mode in an equivalent rectangular waveguide [1]. Its electric and magnetic field distributions can be written as [26] A cos π a e x e ∓jβ g z , in which the minus (plus) sign applies to waves propagating along the positive (negative) z-direction, β g is the propagation constant along the waveguide, a e is the equivalent rectangular waveguide width fitting the fundamental SIW propagating mode, whereas A is an arbitrary modal amplitude.
To optimize slot pair positions and orientations for the linear array design, it is required i) to reduce their backscattering to the feeding point, that is equivalent to minimize the S 11 parameter, and ii) to ensure their RHCP polarization purity, i.e. the magnitude of the axial ratio AR should be also minimized. With reference to Fig. 1, before the optimization, the two slots of the pair are spaced 2 Z = λ g /4 (being λ g = 2π/β g the guided wavelength inside the SIW) to be fed in quadrature and oriented with an angle θ s = π/4 to radiate a circular polarization [15]. Moreover, a fixed displacement along x-direction of X = (a r − D p )/6 to avoid slot superposition, and a slot width always linked to its length by the aspect-ratio W i = L i /5 will be considered throughout the paper.
For the sake of clarity, in Fig. 2 a flowchart of the overall procedure adopted for the linear array optimization is presented, in which also the pertinent sections of the paper associated with each optimization loop are highlighted (orange rectangles).

A. MINIMIZATION OF BACK-SCATTERING BY A SINGLE SLOT PAIR
To minimize slot pair back-scattering, we require the modal amplitude A − 10 of the back-reflected wave to vanish. By resorting to the reciprocity theorem [26] and by identifying with M tot = M 1û1 + M 2û2 the total magnetic dipole moment of the slot pair, one can easily calculate A − 10 as Since the first term in square brackets cannot be zero, only the second one can be set to zero in order to cancel out single slot pair back-scattering, hence we find that in order to have |S 11 | = 0 it should be M 1 exp(jβ g Z ) = M 2 exp(−jβ g Z ), or equivalently Therefore, a complex functional G can be defined in order to minimize the slot pair back-reflection, namely that should be optimized in order to obtain |G| = 1, G = 0. Since the equivalent magnetic dipole moment M i of the i-th slot is proportional to its length L i , for a given average length L c of the pair one can adjust |G| by setting The correction L to the slot length is then updated at each iteration step s according to the rule in which a suitable damping factor χ L < 1 was introduced to prevent undesired oscillations of the algorithm. Indeed, (5) is based on the fact that if |G| > 1, then |M 1 | > |M 2 |, therefore to adjust |G| → 1, L is increased so that L 1 is reduced and L 2 is increased and in turn, |M 1 | decreases and |M 2 | increases, so that |G| is finally reduced. Analogous reasoning shows that if |G| < 1, L is reduced, accordingly to (4), to increase |G| → 1.
In addition, the displacement between slots can be used to compensate the phase difference between their equivalent magnetic dipole moments M 1,2 , in order to achieve G = 0. Hence, according to (4) and by invoking a simple model for slots based on their magnetic polarizability [15], [25], for which M 1,2 ≈ ±β g Z , then G ≈ ±4β g Z , the distance between slots of a single pair is updated at each iteration step s according to the rule in which another damping factor χ z < 1 was introduced.

B. MINIMIZATION OF SINGLE SLOT PAIR AR
In order to have RHCP slot pair such to ensure polarization purity, the cross-polarized LHCP of M tot has to be set equal to zero, namely being θ s the slot orientation, andp RH ,LH = (ẑ ∓ jx)/ √ 2 the RHCP/LHCP polarization unit vectors. Then, by separating the real and imaginary parts in (7), one finds the relations Hence, a proper functional H can be defined to reject the cross-polarized component of the total magnetic dipole moment, i.e., Since by definition |H | = |G|, the condition to satisfy |H | = 1 is already embedded in (5), then only the condition VOLUME 7, 2019 H = 0 has to be added. It can be realized by defining one more update rule for slot orientations at each iteration step, namely in which a further damping factor χ θ < 1 was introduced to control algorithm convergence.

C. CREATION OF THE OPTIMIZED SLOT PAIR DB
The functionals (4)-(9) and the update rules (5)- (6) and (10) defined in Sects. IIA and IIB can be used to optimize a set of slot pairs for the synthesis of a suitable starting design of the linear array in SIW. Such an approach, although suboptimal because it inherently neglects the mutual coupling between different slot pairs, allows for a faster full optimization (i.e., a smaller number of iterations) of the entire array with respect to that based on an arbitrary choice of the starting aperture distribution. The set of optimized slot pairs in isolation is parameterized with respect to their lengths in a certain range of interest L i ∈ [L min , L max ], and then stored in a DB whose entries are the slot geometrical parameters, the achieved axial ratio and the spill-over efficiency, i.e., the ratio of the radiated power by the single slot pair and the incident power on it, as shown in Fig. 3. The spill-over efficiency is of primary importance since it allows the synthesis of the starting aperture distribution to be used for the optimization of the entire linear array, as it will be detailed in the next subsection. The final value of the functionals after the optimization is G ≈ H ≈ 1 for any slot lengths.

D. FULL AUTOMATIC DESIGN OF RHCP LINEAR ARRAY IN SIW
Once developed an automatic technique for single slot pair optimization in terms of back-scattering reduction and polarization purity, a suitable starting design for entire linear array optimization is defined in order to radiate a sectoral pattern in the azimuthal plane, that is a uniform aperture distribution A = const. along the array. In a traveling-wave linear array composed by N slot pairs, a uniform magnitude of the current distribution can be synthesized by properly choosing the spill-over efficiency η n of each slot pair (i.e., the ratio of radiated power by slot pairs and accepted by the SIW), whereas the constant phase distribution required for radiating a broadside beam is achieved by spacing each slot pair of a uniform amount Z (0) c = λ g (Fig. 1), being λ g the wavelength inside the SIW and the superscript (0) is due to the fact that slot pair spacing is imposed only at the beginning of the optimization loop, and then corrected at each iteration step.
With reference to Fig. 4, if a unitary incident power P inc = 1W is supposed for simplicity, the spill-over efficiency of each pair should be chosen in such a way to satisfy simultaneously the relations from which one finds by substitution a general formula for calculating the required efficiency of the i-th slot pair, namely This solution also ensures a vanishing residual power in the waveguide beyond the last slot.
Once the target spill-over efficiency distribution of the array is determined, each slot pair composing the array is synthesized as it was in isolation by resorting to the DB, to find a suitable starting point for the subsequent global optimization. To this aim, a global functional F is then introduced to match the desired magnetic current distribution to the actual co-polar component of the total magnetic dipole moment as in [15], that is in which the average parametersĀ = 1/N N n=1 |A n | and M = 1/N N n=1 |M n ·p RH |, being N the number of slot pairs, were used.
Finally, to close the optimization loop, the following rules were developed to update slot pair lengths L c and positions Z c , namely in which other two damping factors χ l , χ z < 1 were introduced to avoid oscillations of the algorithm. In the next Section, an example of the automatic design of a linear array in SIW is presented, together with measurements of the realized prototype.

III. PROTOTYPE AND MEASUREMENTS
To prove the effectiveness of the above-described automatic technique, we propose here the design and realization of a sectoral pattern, RHCP SIW linear array, at the central frequency f = 24.3GHz. The SIW is realized in a standard ROGERS 6002 substrate of thickness h d = 1.524mm, electrical permittivity r = 2.94 and loss tangent tan δ = 0.0012. According to the design criteria for SIWs described in [1] to prevent higher-order modes to propagate and to minimize leakage effects due to the discretization of continuous electric walls by metallic pins, we found the following geometrical parameters for the SIW: width a r = 6.5mm, pin diameter of D p = 0.5mm, and spacing between adjacent pins of T p = 2D p = 1mm (Fig. 1).
By calculating the wavenumber dispersion with CST Microwave Studio (Eigenmode Solver), we found the guided wavelength in the SIW structure to be λ g ≈ 8.63mm at the central frequency. Moreover, we assumed a centrally-fed linear array of length L = 220mm composed by N = 28 slot pairs, hence the spill-over efficiency distribution of the slot pairs able to synthesize a uniform magnetic current along the linear array was calculated by using (12). A DB of optimized slot pairs with lengths in the range L c ∈ [3, 4]mm was then created by optimizing each slot pair in isolation according to (4)-(9). In Fig. 3, a summary of the information contained in the DB is presented: indeed in correspondence of each slot pair length belonging to the prescribed range of interest, (a) the pair spill-over efficiency, (b) the slot back-scattering (achieved by minimizing |S 11 |), and the (c) CP polarization purity (achieved by minimizing AR) are optimized, thus obtaining (d) |G| = |H | → 1, G → 0 and also H → 0.
By using the slot pair DB to generate the starting array configuration (Fig. 2), the entire array global optimization was activated to properly take into account the effects of mutual coupling between adjacent slot pairs, according to the functional F in (13) and to the update rules for slot pair lengths L c and positions Z c given by (14). In Fig. 5 (a), the complex functional F is shown in magnitude and phase when the optimization process has reached the convergence (35 iterations) and no more significant changes occur in the geometrical parameters if the number of iterations is increased. As it is apparent, the slot pairs in correspondence of the end of the antenna cannot match the requirement |F| → 1/ √ η, due to geometrical unavoidable constraints on their lengths, in order to not overlap the pins of the SIW structure. Moreover, in Fig. 5 (b), the co-polar |M co | and cross-polar   |M cx | magnitudes of the magnetic current distribution synthesized by the linear array are shown, thus demonstrating the efficient rejection of the LH circular polarization. Instead, in Fig. 5 (c), the axial ratio magnitude |AR| is also shown in correspondence of each slot pair.
Finally, a prototype of RHCP linear slotted array in SIW was automatically synthesized at the central frequency and realized by using a standard PCB technology, as shown in Fig. 6. The linear array is fed at its center by the rear pin of a standard SMA flanged panel connector. The coaxial  feeding has an inner (outer) radius of 0.325mm (0.52mm) and it is filled by a dielectric of permittivity r = 2.2. The rear pin of the connector has a smaller radius (0.15mm) than inner conductor of the coaxial feed; it is inserted inside a hole drilled in the dielectric substrate and then soldered to the circular patch on the top copper face of the antenna, as shown in the inset of Fig. 7. An annular slot of inner (outer) radius equal to 0.69mm (0.82mm) is etched on the top metallic plate of the SIW to match the antenna to Z 0 = 50 at the design frequency, as shown in Fig. 7. The simulated and measured reflection coefficients at the coaxial input port are shown versus frequency in Fig. 8, clearly demonstrating the antenna matching.
Measurements of the prototype at central frequency (f = 24.3GHz) revealed a nice agreement with the fullwave simulations of the optimized device made by using CST   Microwave Studio, as it is apparent from Figs. 9-10: in the plane φ = 0 • (φ = 90 • ) a narrow (broad) pattern is successfully radiated, as required. Moreover, in Fig. 11 the measured maximum directivity and gain are compared versus frequency in the bandwidth [24.1, 24.5]GHz. At 24.3GHz, we found  a directivity (gain) in the direction of maximum radiation θ = 0 • equal to D max = 22dBi (G max = 21.5dBi), hence the amount of losses due to metal finite conductivity and to the dielectric loss tangent contributes to only 0.5dB. Moreover, in the considered bandwidth [24.1, 24.5]GHz, the antenna gain decreases of approximately 1dB, thus demonstrating also its stability when frequency varies. In Fig. 12, instead, the measured field plot of the antenna directive plane (φ = 0 • ) is shown at different frequencies in the considered bandwidth, clearly showing a negligible pointing angle tilt versus frequency, but a degradation in terms of side lobe level, as provided by the antenna central feeding.
It is worth noting that in Fig. 10, an oscillation in the measured co-polar (RHCP) component of the radiated field occurs, due to edge effects of the metallic plane on which the antenna is mounted (Fig. 6).
The axial ratio achieved in the direction of maximum radiation θ = 0 • is |AR| ≈ 1.3dB, thus demonstrating the RHCP polarization purity, as required. For the sake of completeness, in Figs. 13-14 also the comparison between the measured (solid lines) and simulated (dotted lines) axial ratios is provided at the design frequency versus the elevation angle θ, both in the φ = 0 • and φ = 90 • planes. Moreover, VOLUME 7, 2019 to demonstrate the antenna polarization purity over the entire bandwidth of interest [24.1, 24.5]GHz, the measured axial ratio versus frequency is shown in Fig. 15.
To conclude, in Fig. 16 the measured maps in the (a)-(b) φ − θ and in the (c)-(d) u − v planes of the co-polar (RHCP) and cross-polar (LHCP) components of the radiated electric field are shown at the central frequency.

IV. CONCLUSION
We proposed a fully automatic technique to design a RHCP linear slotted array in SIW technology to radiate a fan-beam. The design technique was based on two subsequent optimization steps: i) the first, based on local functionals able to minimize single slot pair back-scattering to the feeding point and its axial ratio, and ii) the second based on a global functional that matches the starting aperture distribution to the target uniform one. A prototype of linear array in SIW at f = 24.3GHz was successfully optimized, realized and measured in order to validate the overall algorithm.