Design and Analysis of Antenna Feeding Networks Based on the Rotman Lens Using Interval Analysis (IA)

A new method for the design of antenna array feeding networks has been proposed in this work. In particular, the considered feeding networks are based on the Rotman lens. By knowing the maximum errors on the fabrication tolerances, the arithmetic of intervals and interval analysis (IA) are used for determining the lower and upper bounds of the antenna feeding network parameters of interest. Representative and preliminary numerical results are reported to show the potentialities of the proposed method. An experimental X-band Rotman lens prototype has been designed, fabricated, and assessed. (e obtained results are quite good.


Introduction
Modern telecommunication systems such as mobile smartphones offer multimedia applications and different services that require high reconfigurability despite the limited dimensions of the devices. 5G and 4G generation devices offer, in addition to usual voice and other services commonly provided by standard 3G systems, ultra-broadband Internet connection, voice-over telephony (VOIP), mobile web access, gaming services, and other exciting multimedia applications (such as a high-definition television). e use of advanced RF techniques such as MIMO antennas, multiband antenna and feeding networks, and carrier aggregation (CA) could actively improve the performances of the standard radar or radio links. In such a framework, the design of a suitable radiating system and feeding networks can play a vital role in the design of new-generation communication devices. e antenna systems and the related feeding networks for these new devices need to be light, cheap, and able to keep device performances at a high level. is results in RF devices with high complexity and in challenging RF requirements, especially on the antenna feeding network.
In the last decades, the use of reconfigurable antenna arrays with fully adaptive properties [1][2][3][4][5][6] clearly demonstrated their effectiveness to improve the performance of a telecommunication system dramatically.
is kind of antenna, commonly used in several fields such as airport surveillance and missile detection and tracking, is unfortunately very complex, expensive, and bulky to be used for commercial portable devices such as mobile phones and tablets. Moreover, the introduction of new frequency bands that belong to the millimetric wavelength results in a dramatic increase of the feeding network components such as variable attenuators and phase shifters. For low-cost applications, the use of switched beam antennas is generally adopted instead of adaptive arrays. Recently, a new kind of reconfigurable parasitic antenna, able to electronically select different configurations of the radiation patterns [7][8][9], has been successfully adopted for various practical applications, such as Wi-Fi systems [10], wireless sensor network applications [11], and sum-difference radar [12].
ese antennas offer a good compromise between fully adaptive arrays and the switched beam solution, but the complexity of the feeding network is still too complex or expensive. e Rotman lens has proven its capability to be a useful beamformer for designers of electronically scanned arrays [13][14][15][16][17][18][19]. It is particularly helpful for the design of multibeam or reconfigurable arrays because it offers a real-time delay phase shift capability that is independent of frequency and removes the need for expensive phase shifters to steer a beam over wide angles. e Rotman lens has a long history in communication systems ranging from military radar up to standard radio links [20]. e Rotman lens is a planar structure, which can be easily implemented with microstrip technology, and it is used to feed an antenna array to obtain a multibeam behaviour; since it does not require lumped elements or other RF devices, it is simple, cheap, and mechanically robust. Despite the simplicity of the design equations, they are not able to take into account all the realworld effects, such as the material or fabrication tolerances, which can actively reduce the performance of the Rotman lens and require an expensive experimental calibration tuning [21][22][23][24][25][26][27][28][29][30][31]. e proposed method in this paper is based on the use of the interval analysis (IA) and the interval arithmetic for the design of the Rotman lens. IA was initially introduced to bound rounding errors in numerical computations [32,33], and it has been then extended to the solution of linear and nonlinear equations and functional optimisation [34,35]. Nowadays, interval analysis is widely used in many engineering applications, but its applications to electromagnetics are still limited to few applications mainly concerned with the design of robust antennas [36] or microwave imaging applications [37][38][39]. In this work, the reliability of the IA is used to define an innovative design method where the Rotman lens design formulas are firstly reformulated within the interval arithmetic, and secondly, an IA-based refinement algorithm has to be applied to identify the lens parameters better. e outline of this paper is as follows: the problem of the Rotman lens design is mathematically formulated according to the principles of the arithmetic of intervals in Section 2, while Section 3 is dedicated to present the IA-based design methodology. A set of numerical experiments considering Rotman lenses in the S, C, and K bands is reported and discussed in Section 3.1 to show the effectiveness of the approach. Moreover, in Section 3.2, an experimental X-band Rotman lens prototype has been designed, fabricated, and assessed. Concluding remarks are drawn in Section 4, where innovative features and peculiarities of the proposed design method are also pointed out to encourage possible future improvements.

Mathematical Formulation
A Rotman lens is a feeding network made with a parallel plate structure. e shape is carefully chosen to produce a wavefront across the output ports, which connect the array elements that are phased by the time delay. A standard Rotman lens layout is shown in Figure 1: as it can be noticed, it consists of a parallel plate region surrounded by a number N a of output ports, N b input ports called beam ports, and some N d dummy ports used to obtain reflection-less termination around the parallel plate region. e design of a Rotman lens starts with the definition of the number of radiating elements (N a ) and their location along the array axis, the number of beams, and the required beam steering angles. e other lens parameters are the focal angle α, the focal ratio β � f 2 /f 1 , and the expansion factor defined as c � sin θ/sin α. e length of the transmission lines that connect the array elements located on the array axis must be normalised with respect to the focal length f 1 . ese parameters are shown in Figure 2, where it can be seen that F 0 , F 1 , and F 2 are the three foci of the circular beam port arc (in these points, there are no phase errors), following the formulation reported in [19] and considering the interval arithmetic's rules [32]. e tolerances of the considered ceramic material Arlon 25N are t � 0.5 ± 0.07 (mm), thickness of the metalization t m � 10 ± 2 (μm), and ε r � 3.28 ± 0.05. e mechanical tolerance of the CNC machine for the microstrip width, lens geometrical characteristics, tapered line structure, transmission line length, and antenna element dimensions is ± 0.05 (mm). Considering the above mechanical and electrical tolerances, the lens design formulas have been reformulated. A given interval is represented with a tilde as follows: . e array ports' phase centre locations (x (ap) n , y (ap) n ), n � 1, . . . , N a , are provided by the following equations: With the knowledge on the array port positions, the solution of the following equation can provide the length of the transmission microstrips connected with the antenna elements (green lines in Figure 2): where w n � W n /f 1 n � 1, . . . , N a are the normalised lengths of the transmissions line connected with the antenna e solution of equation (2) provides the intervals of each transmission line length w n n � 1, . . . , N a . e locations of the beam ports (x (bp) m , y (bp) m ), m � 1, . . . , N b , aimed at steering the main beam towards θ m , m � 1, . . . , N b , can be estimated considering the following relations: where , e above formulas allow estimating the geometrical layout of the lens, while the following one allows determining the coupling terms between the N b and the N a beam and array ports: where k � 2π/λ is the interval of the wavenumber in the dielectric substrate, d is the interval of the interelement distance between the array elements, w n and w m are the intervals of the beam and array widths, respectively, and ϕ n and ϕ m are two angles measured considering the ports' boresight direction and the segment connecting the phase centres of the ports m and n. Equation (6) provides a prediction of fundamental antenna parameters such as the array factor, phase error, and illumination amplitudes.
Equations (1)-(6) give all information needed for the geometrical structure of the Rotman lens as intervals; to obtain a better refinement of the interval estimation, the Skelboe-Moore algorithm [32] has been used. It is worth noticing that, after the design of a Rotman lens, often a manual tuning or an optimisation phase is mandatory to obtain the required performance and move towards the fabrication of an experimental prototype. e proposed method based on the interval arithmetic permits to avoid the time-consuming tuning phase and provides the lens geometrical parameters able to take into account the tolerances of materials and to guarantee the initial design requirements.

Numerical and Experimental Assessment
e proposed design method is assessed by reporting and discussing a selected set of representative results of numerical simulations in the following sections. All the numerical simulations have been carried out by considering a ceramic substrate Arlon 25N of 0.5 mm thickness with dielectric permittivity ε r � 3.28 and tan(δ) � 10 − 3 . Numerical data from the obtained experimental prototype operating in the X band (8 − 12 GHz) have been designed, fabricated, and assessed.

Numerical Assessment.
In the first experiment, an S-band (2 − 4 GHz) Rotman lens has been considered. e lens parameters are four beam ports with a scan angle of ± 25 degrees and centre frequency f c � 2.45 GHz, sixteen array ports, and four dummy ports. e S-band lens is characterised with a focal length f 1 � 5.5λ, a focal angle α � 25 (degrees), a focal ratio β � f 2 /f 1 � 0.9, and an expansion factor c � 1. e port impedance is 50(Ω). All ports are connected with tapered transmission lines aligned towards the lens centre. e tapered line length, aimed at guaranteeing a good impedance matching and a phase error reduction, is represented as L t � 1λ, while the beam and array ports are connected with microstrip lines of length L m � 0.5λ. e S-band Rotman lens geometry is obtained  Figure 4.
To assess the S-band lens performance, the return loss S 11 at the input beam ports has been estimated. In particular, the return loss vs. frequency has been measured in the range between 2.0 GHz and 3.0 GHz. e results are reported in Figure 5; as it can be noticed, the obtained results are quite satisfactory and the return loss is always in the range − 14 dB < S 11 < − 18 dB. e isolation between the input ports is estimated in Figures 6(a) and 6(b); in particular, Figure 6(a) reports the isolation between two adjacent ports P1 and P2, while Figure 6(b) reports the isolation between two distant ports P1 and P4. As it can be noticed from the data reported in Figure 6, the isolation is very good: below − 20 dB for the adjacent ports P1 and P2 and − 30 dB for the distant ports P1 and P4. To show the focusing efficacy of the lens, the surface current flowing from the beam port P1 to the ten array ports P13, P14, . . ., P22 is reported in Figure 7. Finally, the phase errors measured at the ten array ports P13, P14, . . ., P22 are reported in Figure 8; as expected, the maximum phase error is measured at the ports P13 and P22 placed at the ends of the lens. However, the maximum phase error is below 1.5 degrees.
In the next experiment, a C-band (5 − 7 GHz) Rotman lens has been considered. Also in this experiment, four input beam ports with a scan angle of ± 25 degrees, ten array ports, and eight dummy ports have been considered. Also, in this case, the port impedance is 50(Ω). e lens centre frequency is f c � 5.8 GHz. Moreover, the other C-band geometrical parameters are the same as those in the previous example; in particular, focal length f 1 � 5.5λ, focal angle α � 25 degrees, focal ratio β � f 2 /f 1 � 0.9, and expansion factor c � 1. All ports are connected with tapered transmission lines of length L t � 1λ, aligned towards the lens centre, while the tapered microstrip lines that connect beam and array ports have a length of L m � 0.5λ. e lens geometry and the positions of the ports have been obtained after the application of the design formulas (3)-(5) and the Skelboe-Moore refinement algorithm [32]. e port positions including the upper and lower limits provided by the IA analysis are shown in Figure 9, while the complete geometry of the C-band Rotman lens is reported in Figure 10. e scattering parameters of the C-band prototype have been estimated; in particular, the return loss and the isolations at the input beam ports are reported in Figures 11 and 12. e obtained results are quite good with a return loss in the range − 12 dB > S 11 > − 22 dB, isolation between two adjacent ports P1 and P2 below − 20 dB, and isolation below − 29 dB for the more distant ports P1 and P4. e current distribution flowing from beam port P4 towards the ten array ports P13, P14, . . ., P22 is reported in Figure 13; the effects of the dummy ports are particularly evident in the left side of the lens. e phase error is quite satisfactory, from the data reported in Figure 14; the phase error is always below 1.5 degrees as in the previous experiment. In the last experiment, a Rotman lens at the K band (22-26 GHz) has been considered. e considered lens structure is characterised by six input beam ports, with a steering angle of 35 degrees, four dummy ports, and twelve output array ports. e considered central frequency is f c � 24.0 GHz, and the port impedance is 50 Ω. e other lens geometrical characteristics are focal length f 1 � 5.0λ, focal angle α � 35 degrees, focal ratio β � f 2 /f 1 � 0.9, and expansion factor c � 1. e length of the tapered line L t � 1λ, and the length of microstrip lines connected with the beam and array ports L m � 0.5λ. e width of the beam, array, and dummy tapered sections is w tb � 26 mm, w ta � 8.5mm, and w td � 25.32 mm, respectively. e length of the tapered lines and of the 50Ω transmission lines which connect the ports is 1.5λ and 0.5λ, respectively. e port positions and the geometry of the K-band Rotman lens, obtained after the IA procedure and the refinement algorithm, are reported in Figures     International Journal of Antennas and Propagation and 16, respectively. e return loss at the six input beam ports P1-P6 is quite satisfactory and always within the limits, − 13 dB >S 11 > − 24 dB, predicted by the IA procedure, as reported in Figure 17. Also, the isolation between the input beam ports is quite good; in particular, the nearest ports P1 and P2 show isolation below − 20dB, while the isolation of about − 28 dB is measured between ports P1 and P6, as reported in Figures 18(a) and 18(b). e current distribution between each input beam port and the array ports is reported in Figures 19(a)-19(f ). As it can be noticed from the data reported in Figure 19, the focusing effect of the Rotman lens is quite evident. Concerning the phase error at the array ports, the maximum measured phase error at the ports placed far away from the lens centre was below 2.5 degrees, while the error at the mechanical boresight of the lens was below 1.5 degrees. e phase errors are higher concerning the other lens structures considered in the previous experiments, as expected since the wavelength is lower.

Experimental Assessment.
In this section, an experimental X-band Rotman lens prototype has been designed, fabricated, and assessed. e considered Rotman lens parameters are the following: six input beam ports, scan angle of ± 30 degrees, and centre frequency of 10 GHz with a 4 GHz bandwidth to cover the whole X-band range. e high bandwidth provided by the Rotman lens feeding network is strongly limited to 500 MHz, by the radiators (rectangular patch antennas equipped with a quarter wavelengthmatching transformer). Sixteen array ports are connected to sixteen antennas with an interelement spacing of 0.4λ.   International Journal of Antennas and Propagation Summarizing the feeding network is characterised by N a � 16 array ports, N b � 6 beam ports, and N d � 4 dummy ports. Other lens parameters are as follows: a focal length f 1 � 6, a focal angle α � 35 degrees, a focal ratio β � f 2 /f 1 � 0.9, an expansion factor c � 1, and a port impedance � 50 Ω. e lens is realised considering a ceramic substrate Arlon 25N of 0.5 mm thickness with a dielectric permittivity ε r � 3.28 and tan(δ) � 10 − 3 . A circular curvature lens on the beam and array ports has been considered, and four dummy ports are aimed at absorbing energy reflections at the top and bottom sides of the lens structure. All ports are connected with tapered transmission lines which are aligned towards the lens centre, and the tapered lines are aimed at guaranteeing a good impedance matching, good beams, and a phase error reduction. To apply formulas (1)-(4) for the geometrical estimation of geometrical lens parameters with the IA, the following interval tolerances have been considered: ε r � 3.23; 3.33 { }(µm), and L � L − 0.05; L + 0.05 { }(mm), with L being a geometrical parameter indicating a given length. After the application of design formulas and the Skelboe-Moore refinement algorithm [32], beam, array, and dummy port positions are estimated. Figure 20 shows the positions of the ports with the intervals of confidences provided by the upper and lower limits. e exact port coordinates are reported in Tables 1-3 which report the positions of the array, beam, and dummy ports, respectively. e width of the beam, array, and dummy tapered sections is w tb � 26mm, w ta � 8.5mm, and w td � 25.32 mm. e length of the tapered lines and of the 50 Ω transmission lines which connect the ports is 1.5λ and 0.5λ, respectively. e geometry of the obtained Rotman lens is shown in Figure 21. International Journal of Antennas and Propagation e phase errors at the array ports have been estimated by considering relation (6), and the results are shown in Figure 22. As expected, the higher phase errors can be found where the ports are placed far away from the lens boresight, and it can be noticed from the data in Figure 22 that the maximum phase error is about ± 2 degrees, while the amplitude errors at the outer ports are about 1.5 dB. e correspondent computed lens array factor for all the beam ports is reported in Figure 23.

X-Band Prototype Description.
After the IA design procedure and the preliminary numerical estimation of the lens characteristics, an antenna array prototype, with a Rotman lens as a feeding network, has been fabricated and assessed. In particular, sixteen rectangular patch antennas equipped with a quarter wavelength transformer have been connected to the N a � 16 array ports with sixteen microstrip lines; each length has been estimated by solving equation (2). e lens geometrical parameters are retrieved from Tables 1-3.  International Journal of Antennas and Propagation e prototype has been obtained with a CNC milling machine with a mechanical resolution of 0.05 mm. e prototype is shown in Figure 24: it has been equipped with ten subminiature type A (SMA) coaxial connectors to measure the prototype scattering parameters. An experimental setup has been arranged in an anechoic chamber. e first set of measurements concerns the estimation of the return loss at the beam ports. For the sake of comparisons, the measurements have been compared with the simulation obtained with commercial software, namely, ADS 2019. In particular, the insertion loss measures are reported in Figure 25, which also states the synthetic results obtained with ADS and considering the upper and lower limits of the geometrical lens parameters provided by the IA approach. It can be noticed from the data in Figure 25 that the measured insertion loss is quite satisfying: below − 12 dB for all the considered frequency range. e second set of measures is related to the estimation of the isolation between the beam ports. In particular, the insertion loss parameter has been estimated for each couple of ports. In particular, Figures 26(a)and 26(b) report the measured insertion loss between two adjacent ports (ports P1 and P2) and between two distant ports (ports P1 and P6). It can be noticed from the data in Figure 26 that the isolation between the beam ports is very good: about − 22 dB and − 32 dB for the adjacent and distant ports, respectively.
Also, the measured data have been compared with the simulated ones and by considering the upper and lower IA limits. e predicted and measured values are very similar. Figure 27 shows the measured array pattern rosettes at 10 GHz for beams 1-6. Moreover, it can be noticed that the main beam is correctly steered towards the angular direction of the ports. In particular, Table 4 reports the required and the measured direction of the main beam. As it can be noticed, the higher measured error is +3 degrees. e shape of the main beam is quite regular, and the secondary side lobe (SSL) level is satisfactory and comparable with the simulated results obtained with ADS.      Figures 28 and 29 report the comparisons of predicted and measured beam patterns related to beam ports p − 1 and p 3 , respectively. e numerical values have been obtained considering the upper and lower geometrical parameters provided by the IA rules and the refinement algorithm. It can be noticed that the measured main beam is precisely located between the beams obtained by considering the upper and lower IA limits, demonstrating the effectiveness of the proposed method.

Conclusion
An innovative approach based on interval analysis and the arithmetic of intervals has been proposed for the design of Rotman lens feeding networks when manufacturing errors are present. e problem of the Rotman lens feeding network design has been first recast into the IA-based framework by introducing the interval numbers aimed at describing the tolerances on the lens parameters and the arising lens illumination function. e proposed method is able to (i) manage the uncertainties and the tolerance errors due to the materials and fabrication process and (ii) define exact and analytical bounds of the lens parameters and the corresponding illumination function by using the rules of interval arithmetic. e obtained results have shown the potential of the proposed strategy; in particular, a prototype working in the X band has been designed, fabricated, and experimentally assessed. Further advances will consider the extension of the proposed method to the analysis of waveguide-based Rotman lenses.     In order to design and calculate the parameters of a microstrip lens, a mathematical model is formulated, as shown in Figure 30. However, most of it has been explained in Section 2. In the figure, the beam ports and array ports are denoted as Σ 1 and Σ 2 , respectively, and  D). In the following, the key part of the microstrip Rotman lens arithmetic is summarized. �� ε r √ ⟶ where ε eff and ε r are the dielectric constant of the microstrip line and relative dielectric constant of the substrate, respectively.