First demonstration of machine-designed ultra-flat, low-cost directive antenna

In this paper, we present a fully automated procedure for the direct design of a novel class of single-feed flat antennas with patterning of a conductive surface. We introduce a convenient surface discretization, based on hexagonal cells, and define an appropriate objective function, including both gain and input matching requirements. The reference geometry is constituted by a very thin, single feed-point square panel. It features a backing metal plate (“ground”) and a top conductive layer, which is automatically patterned to achieve the desired radiation and input matching properties. The process employs an evolutionary algorithm combined with a boundary element electromagnetic solver. By applying this method, we designed an antenna tailored to the 2.4 GHz ISM frequency band, with a size of 24cm×24cm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$24\,\hbox {cm} \times 24\,\hbox {cm}$$\end{document}, i.e., 2×2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2 \times 2$$\end{document} wavelengths and an height of 4 mm, or 0.03 wavelengths. Measured data confirmed the expected high gain (13 dBi), with a remarkable aperture efficiency (higher than 50%, including losses), thus validating the proposed approach.

www.nature.com/scientificreports/ optical metasurfaces 19 . To the best of the authors' knowledge, no previous work has achieved, or even considered, the direct design of directive antennas (with gain above 10 dBi), with dimensions well beyond resonant size. The proposed method is then applied to the design of a broadside antenna working in the 2.4 GHz ISM frequency band, with a size of 24 cm × 24 cm ( 2 × 2 at the working frequency) and with a thickness of 4 mm.
Measurements performed on the fabricated prototype confirmed the simulated performance, with high gain and remarkable efficiency both with regard to aperture illumination and losses.
To the best of the authors' knowledge, this is the first time that the automated design of flat antennas of this size is reported in the open literature, and demonstrated experimentally. This has been possible by the innovation reported here, which can be summarized in the following items: (1) the overall architecture of the antenna, that is able to capture the relevant wave phenomena; (2) the hexapix paradigm, that leads to a directly realizable structure; (3) the theoretical work that leads to the formulation of an effective objective function for the machine design.

Results
Structure. The design of the antenna starts with the choice of its overall architecture. This step is crucial in obtaining design robustness, and must allow for direct fabrication with readily available PCB technology.
The architecture in Fig. 1a was chosen; it fulfills the requirements for low-profile and low-cost. The machine design then consists in finding the pattern of the conductive layer. The latter is printed on a thin dielectric support, suspended above a metallic ground plane. The antenna is fed at the center by a coaxial connector through the ground plane, and the conductive layer is connected to the center conductor, as in Fig. 2. The overall structure is very thin, with a height h ≪ , where is the operational wavelength. All relevant sizes are reported in Fig. 1a.
The top conductive layer was partitioned into 1230 hexapix cells (Fig. 1b), resulting in a deeply sub-wavelength resolution for the patterning, d ≈ /28 . The advantage of the hexapix over the usual square cell consists in its ability to prevent isolated corner contacts, which represent a major cause of uncertainty during the manufacturing phase 20 . Indeed, in a previous work 21 we used the square cells in the design of a similar antenna, with  www.nature.com/scientificreports/ the same specifications and size. There, we found that it was impossible to satisfy both gain and input matching requirements, as a result of the excessive manufacturing sensitivity caused by metallic corner contacts.
Design algorithm. We observe that in our approach each possible structure is described by a binary string. Therefore, it is natural to address the evolutionary design process via a Genetic Algorithm (GA). GA is a well known heuristic stochastic global optimization method based on principles derived from natural evolution. It targets non-convex and non-linear problems, such as those arising in practical electromagnetic design problems [22][23][24] .
Single elements of the search space are called individuals and consist of strings of parameters called chromosomes which encode all relevant features-in this case each representing a given patterning. These belong to sets called populations. The algorithm advances by coupling the elements within the population to produce the subsequent generation. This is done by means of operators inspired by natural processes: selection picks individuals from the population based on their fitness value, crossover creates a new individual by combining the genetic content of two parent individuals and mutation introduces variation in the genes randomly.
The design objective is expressed as the maximization of a fitness function, which gives the degree of goodness of a particular individual with respect to the set goals. The effectiveness of the objective (fitness) function is crucial in ensuring practical usefulness of any optimization result. To this end, our objective addressed both gain, and input matching of the antenna.
It appears natural to consider maximization of the gain along the desired direction of maximum radiation, here being the normal to the antenna face. However, we found that this choice was not optimal in the overall process. Analysis showed that this criterion was too strict in the early phases of the evolutionary design: individuals with high directivity but radiations somewhat off the maximum direction were excluded from further evolution, with detrimental effects. Therefore, we employed the more relaxed, gain-related solid-beam efficiency 25 . This quantity is defined as the relative fraction of power �P(��) radiated in a given solid angle around the desired beam direction. For our purpose, we used a normalized beam efficiency, where P rad is the total radiated power. The solid angle has the meaning of a "control volume": when tends to zero, the employed beam efficiency BE recovers the common definition of directivity; is one of the algorithmic design parameters, and it is apparent that it should be a fraction of the targeted beam-width (e.g. Half-Power Beam-Width, HPBW). The chosen value, after some experimentation, is reported in Table 1; the value indicated there assumes a conical region, �� = π �θ 2 , and �θ is reported. Its comparison with the actual realized HPBW can be inferred from Table 2.
The second parameter is the input matching T. It is defined in the usual way as the ratio of the accepted power P a to the incident power P inc , where |S 11 | 2 is the ratio between reflected and incident powers (with usual 50 reference). Both indicators, BE and T, are derived from full-wave simulation of the full structure for any concerned surface patterning.
It is important to note that the product of BE and T has a direct physical interpretation, being proportional to the total realized gain of the antenna 25 . This suggests a practical way of combining the two objectives into a single fitness function. If considered in logarithmic (dB) format, one would have the sum of the two goal indicators, BE and T. We thus defined the fitness function as a weighted product: www.nature.com/scientificreports/ This corresponds to a convex combination in logarithmic format, and allows more flexibility compared to the plain product. The weighting exponents α and β = 2 − α can be chosen so as to provide convergence toward a robust solution. Suitable values were found via extensive trials on the specific problem. Finally, we remark that this objective function also possesses a physical upper bound. In fact, T ≤ 1 , and where D max is the maximum directivity achievable with the considered area. This corresponds to ideal unit ( 100% ) aperture efficiency (see below for definition).
In summary, the employed fitness function has a physical meaning, it includes only two parameters to be set ( α and ), and also possesses a physical upper bound. This is important to ensure a successful outcome. With regard to the stopping criterion, it was decided to end the optimization when the best solution had not changed after a given number of generations N stag (stagnation limit). In Table 1 the values of the main algorithmic parameters are reported.
Simulation. The evaluation of the fitness function requires a full-wave electromagnetic analysis of the antenna. In principle, any simulation method and any implementation could be used. However, the binary nature of the patterning is especially suited to the use of a Boundary Element, Integral Equation approach; the related boundary value problem is called Electric Field Integral Equation (EFIE). For the numerical solution, the surface is discretized into triangular patches and the (equivalent) surface current is expressed as a linear combination of a set of local basis functions f n (Boundary Elements) 26  where Z is the system matrix, V is the vector of incident electric field and I is the vector of unknown coefficients I n to solve for. The EFIE-MoM results in a fully populated matrix Z . However, as recognized by several works 10,12 , the binary operations involved in the geometrical description of the radiating surface allow a quick build of the system matrix at each iteration. This is obtained directly from the system matrix of the mother structure consisting of full metal.
The design goal included a linear polarization along the maximum radiation direction. This requirement was incorporated by simply enforcing structural symmetry, as the feeding is at the center of the structure. This was also effective in reducing the search space, as only half of the structure was considered.
Results of design algorithm. The evolution of the design algorithm can be observed in Fig. 3, in which the fitness value of the best individual is reported for each generation. The fitness value is normalized with Once the design process ended, it provided a set of available structures. These correspond to the population of the last generation; each is characterized by a value of fitness. Fig. 4a shows the best and chosen structure. We found instructive to examine also the second-best solution, reported in Fig. 4b. It differs significantly in its geometric features, but performs comparably, as indicated by a fitness value close to the best one. This is intrinsic to the use of sub-wavelength details, and is also an indication of the robustness of the chosen architecture.
The antenna was manufactured employing the best configuration, shown in Fig. 4a. It was fabricated using standard PCB technology. The upper, patterned conductive layer was etched on a dielectric I-Tera 27 layer with a thickness of 0.254 mm (Fig. 5a). This dielectric-supported conductive layer was flipped back and then suspended above the metal ground plane by means of (eight) plastic spacers (Fig. 5b). The structure is fed by an SMA connector mounted on the back of the ground plane, with the center conductor connected to the patterned conductive layer. Due to the flip-back layout of the conductive layer, the connection to the coaxial pin was ensured via fitness = 0.6629 fitness = 0.6597  www.nature.com/scientificreports/ a plated through-hole and a small metal pad on the opposite face of the patterned layer. This choice allowed to carefully solder the pin to the latter pad, which is easily accessible (inset of Fig. 5b).

Measured results.
It is important to clarify that all measured results refer to actual parameters measured as per current standard 25 . In particular, we remark that the modified gain indicator (1) was used in the optimization process, but not in the measurements. Also, during the design process, the conductors were considered lossless, but the measured gain does of course account for the actual losses of the employed conductor materials.
In the evaluation of flat antennas performance, a relevant figure of merit is given by the aperture efficiency ε ap , which quantifies how much of a surface effectively radiates. It is defined as where A geom is the geometric area, while A eff = G max 2 /4π is the effective area. The main figures of merit are summarized in Table 2. The gain is relatively stable across the band, with a slight decrease for increasing frequency. The maximum value of 13.7 dB is obtained at 2.40 GHz, at which the aperture efficiency also reaches its maximum of 60% . At 2.44 and 2.48 GHz the peak gain is 13.3 and 12.1 dB respectively, with corresponding aperture efficiencies of 53% and 41% . Energetic (Ohmic) efficiency was estimated by comparing measured gain and directivity, resulting in losses less than 1 dB over the whole bandwidth.
Measured values of directivity in the E-and H-plane are shown in Fig. 6 for frequencies of 2.40, 2.44 and 2.48 GHz, corresponding to the lower, center and upper limit of the ISM band respectively. As is evident, the measurements compare very well with simulations, with negligible discrepancies < 0.3 dB in the main beam for all frequencies. The polarization of the radiated fields is purely linear on the symmetry plane, as enforced in the design, and therefore it is not shown here. In the entire main beam, the cross-polarization remains below −20 dB. Furthermore, design symmetry forces the radiation pattern in the H-plane to be symmetric. Sidelobe levels remain satisfactorily below −25 dB in the E-plane and −13 dB in the H-plane.
Regarding the input reflection coefficient (Fig. 7), the proposed antenna shows a satisfactory matching over the whole design bandwidth, with values of |S 11 | below −10 dB and as low as −15.6 dB at the center frequency, where the optimization was performed. A comparison with simulated data showed a slight shift in the resonance peak, which can be attributed to the difficulties in modeling the feed accurately. Regardless of this, the reference bandwidth at −10 dB remains comparable in the two cases, with 86 MHz obtained from simulations and 109 MHz from measurements.

Discussions.
We have presented a novel procedure for the direct design of flat directive antennas. Surface shaping is obtained by inclusion/removal of hexagonal cells; this makes the optimization efficient, and free of manufacturing uncertainties. The design is based on a Genetic Algorithm combined with a boundary element electromagnetic solver.
The procedure was applied to the design of an antenna for the 2.4 GHz ISM frequency band. The antenna was fabricated and measured. Measurements confirmed the satisfactory results, notably an aperture efficiency as high as 60% and a gain higher than 13 dBi.
Because of the absence of a specific beam-forming network, we expect a favourable scaling of ohmic losses at higher frequencies, as in similar single-feed antennas. Therefore, it is interesting to put the presently measured aperture efficiency (on average, about 50%) in comparison with that of similar antennas based on metasurfaces. Results reported 28 for a very high gain antenna in the X band (8.4 GHz) indicate efficiencies of the order of 35%, which is lower than the one achieved in our case, as expected at a higher frequency. Our significantly larger efficiency suggests a favourable projection for higher frequencies and larger gains.
The demonstrated robustness of the proposed method thus paves the way to machine design flat antennas with very high gain at high frequencies.

Methods
Design methodology. The electromagnetic simulation is based on our implementation of well known algorithms 14 for the Method of Moments matrix computation, while the Genetic Algorithm has been implemented in MATLAB. The code has been deployed on a cluster with 24 computational nodes, and the optimization took 21 hours to reach convergence.
The combined requirement for maximum radiation along broadside and linear polarization corresponds to a Perfect Magnetic Conductor (PMC) boundary condition on a symmetry plane perpendicular to the surface. This was directly enforced in the build phase of the system matrix. No additional care was needed for the discretization, since the basis functions already forced the current to be tangential to the symmetry plane, as required by the PMC boundary condition. www.nature.com/scientificreports/ For the purpose of reducing the computational effort, the optimization was carried out at a single frequency; we chose the centerband frequency 2.44 GHz of the targeted ISM band. The relatively stable performance over the entire band justified this choice.
The dielectric material backing the patterned metal layer was conceived as a mere mechanical support, and to allow its manufacturing with PCB technology. The flip-back layout was thus adopted to minimize field www.nature.com/scientificreports/ penetration in the dielectric. In this way, its final impact on electromagnetic phenomena was negligible, allowing the use of an efficient in-house solver which did not consider this dielectric layer. Because of the above, the performance of the actual antenna was verified by full simulation prior to fabrication. This was done with a commercially available EM solver (CST Microwave Studio 29 ); the simulated model incorporated all the details of the final design, i.e., inhomogeneity due to the dielectric substrate, thickness of the metalization, and details of the coaxial feed. A shift in the resonance frequency was observed with respect to the simplified model used during the optimization. Field analysis revealed a standing-wave behavior in one direction, and a traveling wave behavior in the orthogonal direction; this was expected because of wave physics and the imposed symmetry. Therefore, the geometry was subsequently refined by shrinking it in the direction corresponding to standing-wave resonance. This allowed to tune the resonance frequency (dotted line in Fig. 7) while preserving the radiation properties. It is important to stress that this final tweak was solely due to the neglect of the supporting dielectric layer during the design process, in turn due to the readily available in-house software. Therefore, it is not a limitation of the proposed approach. Measurement methodology. The realized antenna was measured in the Spherical Near Field (SNF) range facility installed in the anechoic chamber of the Politecnico di Torino (Turin, Italy). The size of the chamber is 5 × 5 × 4 m 3 , it is fully lined with 1800 absorbers, and it can be used for frequencies between 700 MHz and 40 GHz. The system was realized by Orbit FR (date of installation: July 2009). It is equipped for the measurements of antennas up to a maximum dimension of about 1.5 m and a weight of 30 kg.
The RF system is based on an Agilent PNA E8363B 30 coupled to a commercial measurement software (MiDAS) 31 ; the latter performs a NF to FF transformation via spherical wave expansion 32 . The field radiated from the Antenna Under Test (AUT) was sampled in amplitude and phase at appropriate angular intervals (depending on the frequency and of the radius of the sphere which encloses all sources). The scan radius, i.e., the distance between the center of rotation of the AUT and the phase center of the probes, is of about 2.5 m.

Data availability
The data that support the findings of this study are available from the corresponding author on request.