# Nonlocal metasurfaces based on advanced Bayesian learning for satellite communications

Ayoub Bellouch, Mahmoud Elsawy, Stéphane Lanteri, Erika Vandelle, Thi

Quynh Van Hoang

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Ayoub Bellouch, Mahmoud Elsawy, Stéphane Lanteri, Erika Vandelle, Thi Quynh Van Hoang. Non-local metasurfaces based on advanced Bayesian learning for satellite communications. 2025. [hal-04850081v2](https://hal.science/hal-04850081v2)

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Preprint submitted on 4 Mar 2025 (v2), last revised 20 Nov 2025 (v3)

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Nonlocal metasurfaces based on advanced Bayesian learning
for satellite communications

Ayoub Bellouch¹, Mahmoud Elsawy¹*, Stephane Lanteri¹, Erika Vandelle²,
Thi Quynh Van Hoang²

¹*Inria, Côte d'Azur University, CNRS, LJAD, 2004 route des Lucioles, Sophia Antipolis, 06902, Provence-Alpes-Côte d'Azur, France.

²Thales Research & Technology, 1 avenue Augustin Fresnel, City, 91120, Palaiseau, France.

*Corresponding author(s). E-mail(s): mahmoud.elsawy@inria.fr;

Contributing authors: ayoub.bellouch@inria.fr; stephane.lanteri@inria.fr;

erika.vandelle@thalesgroup.com; van.hoang@thalesgroup.com;

Abstract

Beam-steering antennas, based on sub-wavelength deflectors, aim to facilitate information exchange with satellite communication (SatCom) on-the-move systems. Recent beam-deflection solutions are often based on the phase-gradient metasurfaces (PGM) concept to generate the desired wavefront. However, the performance remains inadequate for advanced communication technologies due to band-width limitations and insufficient efficiency. Here, a highly-efficient broadband beam-deflection system is designed, fabricate, and characterize experimentally for SatCom applications. To achieve that, an advanced Bayesian learning strategy, accompanied by an innovative nonlocal metasurface (NLM) configuration, is employed. This non-trivial broadband design leverages a sophisticated coupling between adjacent sub-wavelength elements, together with varying thicknesses along the propagation direction to manipulate the incoming beam. The novel beam-deflector is illuminated with a circularly polarized wave and is designed to operate in the Ka-Tx band from 27.5 to 31 GHz. The optimized structure demonstrates exceptionally high performance compared to the conventional phase-gradient synthesizing counterparts. The experimental validation exhibits an excellent agreement with the numerical simulation, highlighting the potential of this optimization-assisted design to advance metasurface technology for various beam-steering applications.

**Keywords:** Bayesian optimization (BO), Beam-steering antenna, Efficient Global Optimization (EGO), Gaussian Process (GP), Ka-band, NonLocal Metasurfaces (NLM), Phase Gradient Metasurface (PGM), Satellite Communication (SatCom)

# 1 Introduction

Beam-steering antennas, embedded in SatCom on-the-move systems, aim to offer better connectivity and ensure secure information exchange [1–3]. This innovative advancement is critical in the context of the rapidly growing demand for advanced communication systems, driving progress in civil aviation, defense, and space exploration technologies [4]. In general, these antenna systems are characterized by their ability to adjust their pointing direction, enabling large angular coverage and scanning. Conventional solutions of mechanical beam-steering, such as parabolic reflectors [5], are capable of delivering high performance, as evidenced by their widespread use in various applications, particularly in satellite TV reception. However, their cumbersome 3D architecture creates challenges regarding space constraints and aerodynamic issues [4, 6]. To circumvent these constraints, the objective is to innovate and develop alternative low-profile beam-steering antennas that are cost-effective, low energy consumption, and easy to manufacture, while
ensuring robust performance standards. To this end, a recent antenna architecture has been proposed, inspired by optical approaches based on the Risley prism concept [7–10]. It consists of a radiating feeder associated with two deflectors, whose rotations ensure the 2D-beam-steering capacity in elevation and azimuth planes.

To design beam-steering antennas with high-performance deflectors, recent approaches have turned to technologies based on metamaterials and metasurfaces due to their attractive features. These structures are characterized as inhomogeneous sub-wavelength elements arranged in a periodic pattern [11, 12]. This configuration allows to produce peculiar EM properties [13, 14], enabling a better control of phase, amplitude, and polarization [15–18]. In this regard, all-dielectric metasurfaces (ADMs) made from high-refractive-index materials have been suggested to design high-performance optical and millimeter-wave beam-steering systems [19–23]. Their all-dielectric nature ensures high transmission efficiency over a wider frequency bandwidth compared to other concepts such as impedance-sheet metasurfaces [24]. In addition, the fabrication of ADMs is facilitated by additive manufacturing [25], which offers many degrees of freedom for the design and a moderate cost.

Recent metasurface beam-deflection systems primarily rely on the PGM concept. The conventional PGM are constructed by arranging a linear array of sub-wavelength elements with a spatially-varying phase. This arrangement aims to create a phase gradient response in the transmitted far-field, enabling deflection control of the EM wave [26–28]. In this configuration, the phase transformation is locally controlled by varying the dimensions of the motif through each unit-cell. This type of design is known as local metasurfaces, which incorporates quasi-periodic sub-wavelength elements. Each element generates independently its own scatterer response, considering a periodic environment and neglecting the real interactions between the adjacent elements. Indeed, the overall performance is determined using a lookup table that dictates the properties of each element to align with the desired wavefront. On the other hand, the nonlocal approach has recently emerged in the field of photonics [29–31]. It extends beyond the individual effects generated by each unit-cell, considering the interactions among adjacent sub-wavelength elements to shape the overall effective far-field response at the output of the structure. This NLM configuration provides a significant advantage by offering more degrees of freedom in the metasurfaces design, enabling extreme wave manipulation to optimize the overall response.

In this paper, we leverage on the recent advances in NLM concept in photonics to design a highly-efficient beam-deflection system, composed of sub-wavelength all-dielectric cylindrical pillars, in microwave regime. In this regard, we exploit the significant nonlocal interactions to improve overall performance by maximizing the diffraction efficiency in the desired direction, further discriminating the cross-polarization, simultaneously. In addition, we overcome the bandwidth limitations encountered in conventional designs (including nonlocal ones [29, 31]), achieving high performance across the entire Ka-Tx band for SatCom applications. To realize this task, we rely on a novel pioneering approach by introducing a non-uniform distribution of the pillars' heights, further exploiting the sophisticated near-field coupling by optimizing the distances between the adjacent elements. To achieve this, we leverage recent progress in statistical learning-based optimization techniques to develop a methodology for optimizing the beam-deflection design. This type of method, called Bayesian optimization (BO) [32–34], employs a cheaper probabilistic surrogate model to replace the actual objective function, providing significant benefits for managing expensive and black-box functions, which is crucial due to the numerical complexity involved in accurately modeling NLM designs. To take advantage of this statistical approach, we have opted for one of the most advanced techniques belonging to the class of BO methods, known as Efficient Global Optimization (EGO) [33–39]. The EGO method makes use of a surrogate model, which is derived from a Gaussian Process (GP) model, to reduce the computational cost of optimizing expensive objective functions based on EM full-wave simulations, whereas traditional global optimization strategies like Genetic Algorithms [29, 39–41] rely on adaptive sampling and require considerably more evaluations, making them impractical for costly simulations [38]. In addition, the EGO algorithm employs a statistical learning criterion, called an acquisition function [42], related to the optimization target, to identify which design should be tested in the next iteration that would provide better results close to the predefined goal.

Based on our novel methodology, we demonstrate that the NLM deflector can achieve superior performance compared to conventional PGMs. Furthermore, the adopted statistical learning strategy ensures a significant reduction in the number of EM full-wave evaluations, enhancing the potential to find promising solutions more rapidly. The optimized design is fabricated and characterized experimentally to confirm the remarkable broadband response of our NLM design in terms of radiation gain in the desired direction and the discrimination of the unwanted polarization component.
<figure><img src="image_1.png" /><figcaption>**Fig. 1:** The first row represents a conceptual illustration of a beam-steering antenna system with metasurface deflector for SatCom on-the-move applications. The inset of the periodic deflector configuration is depicted in the second row (see blue arrows). The structure is illuminated with a left-handed circularly polarized (LHCP) wave from the substrate side. The objective of the metasurface configuration is to steer the incoming LHCP wave to a desired direction dictated by an angle $\theta$ as illustrated by the blue arrow direction. Meanwhile, decriminalizing the right-handed circularly polarized (RHCP) cross-polarization as illustrated by the red arrows direction. Further, the description of the single macro-cell is shown in the bottom-right, including the simulations details. The configuration is surrounded by periodic conditions (PBC) along x and y directions. Perfectly matched layers (PMLs) is applied to the top and bottom to prevent parasitic reflections and accurately model the far-field. Two different metasurface configurations are exhibited as indicated by the two black arrows.</figcaption></figure>

# 2 Results

## 2.1 Metasurface deflector design

For our target application, we seek to design a metasurface deflector to steer the incoming beam with a deflection angle in the range of $\theta \in [29°, 33°]$ at the Ka-Tx band for SatCom. Relying on the first-order diffraction mode, and considering normal incidence, we have chosen the period of the macro-cell to be $P_x = 20$ mm along x-axis. We considered a sub-wavelength period $P_y = 2.6$ mm along y-axis to limit the diffraction in the out-of-plane direction. The schematic representation of the macro-cell configuration is shown in Fig. 1. The macro-cell is composed of sub-wavelength all-dielectric cylindrical pillars. A composite material with high permittivity ($\epsilon_r = 8.5$) is used for the pillars. We consider
a non-trivial distribution of cylindrical pillars with pyramidal tips to break the symmetry along the propagation direction and further manipulate the circularly polarized incoming beam. In addition, a substrate support base is used to ensure the mechanical rigidity of the deflector, with a low permittivity material ($\epsilon_r = 2.3$) to minimize the reflection. The macro-cell configuration is simulated by our full-wave Discontinuous Galerkin Time-Domain (DGTD) solver [43–45]. The Floquet modal decomposition theory is used to describe how the far-field is split into diffraction orders, which in the present case amounts to define and compute the mode diffraction efficiencies $\eta(m, n)$.

It is worth mentioning that in our NLM design, we consider solely 5 pillars per macro-cell to allow more freedom for the pillars during the optimization and further manipulate the interactions between them. As a reference, in the traditional PGM-based deflector, where the phase is locally controlled by each single pillar, 8 elements were required to cover the total phase shift range, and to reduce the phase quantization error. We refer to the supplementary information section, for more details regarding the design of the conventional PGM. Within our nonlocal scenario, the pillars interact not only with the external waves directly incident upon them but also with the waves coming from adjacent sub-wavelength pillars, resulting in a strong near-field coupling effect. These interactions form the overall behavior of the macro-cell, referred to as the effective field. To tailor this response and maximize the potential to find broadband solutions, we grant the pillars the freedom to move within the grid. In addition, we introduce variations in thickness along the propagation direction, a parameter traditionally dictated by the permittivity in conventional designs. This novel configuration, to the best of our knowledge, has not been previously introduced in the literature.

## 2.2 Optimization strategy

An inverse design problem is formulated to optimize the NLM deflector’s macro-cell structure. In the present case, the objective is to enhance the transmission efficiency $\eta(-1, 0)$ of the diffraction mode order $(-1, 0)$, which corresponds to the desired deflection angle of the beam, while concurrently minimizing the transmission efficiency level of the other diffraction modes. An additional objective is to minimize the ratio between the co- and cross- polarizations. Hence, this optimization problem seeks to determine the positions and dimensions of cylindrical pillars that achieve the best performance by minimizing a sophisticated objective. Here, we take advantage from the nonlocal effect to offer the optimization process more degrees of freedom in the choice of the design dimensions. The parameters $\mathbf{X}$ to be optimized include the radius $r_i \in [1, n_p]$, total height of pillars $h_i \in [1, n_p]$, height of pyramidal tips $h_c$ and distance between consecutive pillars $d_i \in [1, n_p - 1]$, leading to $3n_p$ optimized parameters, with $n_p$ is the number of pillars. To construct the objective function, it is necessary to incorporate the two key performance elements mentioned previously. The principal objective is to increase transmission efficiency of the mode $(-1, 0)$, but more importantly to maximize the efficiency $\eta_{LHCP}(-1, 0)$ of the LHCP polarization to maintain the same polarization as the incident wave. The second objective is to minimize the Axial Ratio (AR), i.e. the ratio between the major and minor axes of an elliptical polarization (EP). The idea is to ensure a CP nature of the transmitted wave, because the configuration of the pillars does not necessary preserve the nature of the incident wave polarization. By minimizing the AR factor, we will simultaneously raise the cross-polarization discrimination (XPD), defined as the ratio between the co- and cross-polarization. In our case, this means mitigating the unwanted polarization RHCP. The relation between the AR and XDP ratios is given by the following formula :

$$AR = \frac{\text{Major axis}}{\text{Minor axis}} = \frac{XPD + 1}{XPD - 1} \in [1, +\infty[, \quad (1)$$

$$\text{with } XPD = \sqrt{\frac{\bar{\eta}_{LHCP}(-1, 0)}{\bar{\eta}_{RHCP}(-1, 0)}}.$$

In order to achieve the desired performance, the two related objectives, i.e. the $\eta_{LHCP}(-1, 0)$ diffraction efficiency and the AR factor, can be combined into the same cost function to be minimized. Thus, the optimization problem to solve can be written as:

$$\text{Minimize } f_n(\mathbf{X}) = \frac{1}{2} \left( 2 - \bar{\eta}_{LHCP}(-1, 0) - \frac{1}{AR^n} \right) \in [0, 1] \quad (2)$$

where $n$ is the exponent order, adjusting the weighting of the AR in the objective function. Solving this inverse design requires a rigorous and computationally efficient optimization strategy, which is capable
of treating several design parameters while minimizing the number of EM full-wave solver calls. For this purpose, we consider here the EGO method, introduced in section I, which belongs to the class of BO statistical learning approach. This choice is made to significantly reduce the number of function evaluations needed compared to conventional optimization techniques, which is crucial to handle with our sophisticated 3D structure. In practice, the EGO is based on two phases. The first one is the design of experiment (DOE), in which an initial database is generated. In essence, a uniform sampling strategy is deployed in order to generate different designs in which the cost function is evaluated using our electromagnetic DGTD solver. In the second phase, a GP model is constructed to fit the data obtained from DOE. This GP model allows us to predict the values of the cost function in the parameter space without the need to perform additional simulations. Therefore, one can estimate at any point of the design space, the objective function (mean of the GP model) and an uncertainty value (variance of the GP model). The mean and the variance are used together to determine a statistical merit function. In our case, we rely on the expected improvement, which is a function whose maximum defines the next design parameters set to be evaluated. That is to say, in the search parameter space where this function is maximized, we extract the corresponding parameter values, and the corresponding design will be simulated using the DGTD solver. Then the database is updated accounting for this new observation (construction of a new GP model based on the updated database). We repeat this process until a predefined convergence criterion is reached, or when the expected improvement is sufficiently small.

## 2.3 Macro-cell optimization results

In this optimization scenario, the objective function $f_n(\mathbf{X})$, described in equation (2), is optimized with a value $n = 2$ to provide an adequate weighting between the two related objectives. The number of pillars used here is $n_p = 5$ per period, leading to 15 optimization parameters. As explained previously, the optimization has been performed using the EGO method. During the first phase, we build a DOE database with 75 points. It represents five times the number of parameters, which is sufficient to construct an initial GP surrogate model. In the second phase, i.e., the EGO phase, this surrogate model is refined during the optimization process to find a global minimum.

Figure 2a summarizes the optimization process, in which the objective function evaluations of the DOE phase are displayed with purple points and the evaluations obtained during the EGO phase are displayed with blue points. The deflector designs have been evaluated using the a DGTD-$\mathbb{P}_2$ method where $\mathbb{P}_2$ refers to the second-order polynomial interpolation used for the DG discretization. We set our simulation budget to 250 iterations. Interestingly, the optimization process identified several global minima. The results confirm that the EGO algorithm achieves convergence with significantly fewer iterations, despite the considerable number of parameters, unlike traditional optimization algorithms. The best design satisfying our objective function is shown in Fig. 2b. It is worth mentioning that this optimal design differs from traditional PGM designs (see Fig. 1 and the supplementary information for more details). First, the distance between the adjacent pillars' centers is no longer constant, and the evolution of the pillars' diameters does not conform to the phase-gradient approach. Additionally, the heights of the pillars are not uniform, which introduces a novel aspect that has not been previously explored in the literature on beam-deflection systems. The field maps of the optimized structure is exhibited in Fig. 2c. Each field map is presented with an individually normalized scale to enhance clarity, particularly for the $E_z$ component, which is considerably weaker than the others. As can be seen, the near-field coupling between pillars significantly enhances the overall effective deflection observed in the far-field.

The performance of the macro-cell configuration is depicted in Fig. 2d across all frequencies of the Ka-Tx band. A comparison was made between the optimal design, represented by the blue curve, and the conventional design based on phase-gradient approach, represented by the dashed red curve. In terms of co-polarization efficiency, the optimal design achieves notably high values, reaching approximately 80% with a stable level across the entire band, compared to around 65% for the conventional design with fluctuations in some areas. On the other hand, this NLM design achieved very low levels of AR, falling below the imposed threshold of 1.5 dB, whereas the PGM design only adhered to this threshold within a limited frequency range around 29 GHz. These outcomes offer a comprehensive view of the non-intuitive interactions among pillars, revealing how the nonlocal behavior exceptionally improves the overall performance compared to the classical conventional PGM design strategy.

## 2.4 Finite structure performance

The next step involves validating the results for the finite structure without periodic boundary conditions to carefully consider the influence of edge effects. We built a finite structure with a 260 mm diameter,
<figure><img src="image_2.png" /><figcaption>**Fig. 2:** Optimization results for the nonlocal metasurface configuration. **a** Evolution of the desired objective function as a function of the full-wave solver calls. The purple zone represents the DOE phase, while the blue one refers to the second EGO phase. **b** A schematic view of the best optimal design during the optimization process. **c** Field maps of Re($E_x$), Re($E_y$) and Re($E_z$) obtained for the optimized design at $f_0 = 29$ GHz. **d** Comparison between a conventional design (red curves) and an optimized design (blue curves): performance of the macro-cell in efficiency and Axial Ratio versus frequency (the dashed gray lines correspond to 1.5 dB and 3 dB level).</figcaption></figure>

corresponding to 13 macro-cell periods, a dimension deemed sufficiently large to approximate the behavior of an infinite deflector in a periodic environment. The total diameter of the deflectors is 280 mm, consisting of a 260 mm effective surface and a 20 mm fixation section.

The designs for both conventional and optimized deflectors are illustrated in Fig. 3a,b. Deflector functionality was evaluated by illuminating each one with a LHCP Gaussian source at normal incidence. The LHCP co-polarization efficiency and the AR ratio of both designs were evaluated using the Transient solver of the CST microwave suite. The efficiency calculation is detailed in Methods.

Figures 3c,d,e present a comparative analysis of the far-field radiation patterns in the deflection plane for the two deflectors at 28 GHz, 29 GHz and 30 GHz, respectively. The results indicate that the optimal design given by the EGO achieves enhanced directivity in the desired direction. Furthermore, the optimized design also exhibits a notable reduction in side lobes levels (SLL) and a significant minimization of the RHCP cross-polarization when compared to the PGM conventional deflector.

Figures 3f,g depict the performance of the finite structures in terms of LHCP co-polarization transmission efficiency and AR, respectively, across the frequency range of interest from 27.5 to 31 GHz. The optimized deflector demonstrates a notably high efficiency, exceeding 80%, while maintaining a robust circular polarization with an AR less than 1 dB throughout the entire Ka-Tx band. Overall, the optimized deflector features superior performance relatively to its conventional counterpart. The results obtained with the macro-cell configuration are also given in the graphs, showing that the finite structure achieves similar results, thereby validating the adopted modeling approach. The AR results confirm that the finite structure performance aligns closely with the macro-cell, even though AR levels are highly sensitive to
<figure><img src="image_3.png" /><figcaption>**Fig. 3:** Finite structures of 260 mm diameter: **a** Conventional deflector, **b** Optimized deflector. Comparison of radiation patterns of the co- and cross-polarization in the deflection plane for the conventional deflector (red curves) and the optimized deflector (blue curves) illuminated by an LHCP Gaussian source: **c** at 28 GHz, **d** at 29 GHz, **e** at 30 GHz. Performance of the macro-cell and the finite 260mm-diammeter structure: conventional deflector (red curves) and optimized deflector (blue curves) **f** Efficiency of the LHCP co-polarization component, **g** Axial Ratio versus frequency (the dashed gray lines correspond to 1.5 dB and 3 dB level).</figcaption></figure>

various factors such as the strong near-field coupling between pillars and edge effects. Despite these sensitivities, the optimized deflector maintains values below the 1.5 dB-threshold across the entire frequency range of interest, outperforming the conventional design in circular polarization quality.

These findings highlight the advantages of the NLM design in achieving enhanced and reliable radiation performance. They also validate the modeling approach, demonstrating that macro-cell simulations can reliably predict the performance of sufficiently large finite structures (from approximately ten periods). This insight is crucial, as it enables efficient macro-cell optimization with high confidence in the expected performance of the corresponding finite structure, significantly saving time and computational resources during the design and optimization stage.

## 2.5 Prototyping and measurement

For experimental validation, only the optimized deflector was manufactured. This component was fabricated using Zetamix Epsilon filament [46] and an advanced FDM-based 3D printing technique tailored
for metasurfaces based on sub-wavelength cylindrical pillars [47]. The optimized deflector weighs approximately 450 g, compared to 530 g estimated for the conventional counterpart. A photograph of the fabricated deflector appears in Fig. 4a.

<figure><img src="image_4.png" /><figcaption>**Fig. 4:** Measurements of the RLSA combined with the optimized deflector: a) Photograph of the experimental setup in the anechoic chamber, b) Results of radiation patterns: Simulation (dashed lines) and measurement (solid lines) results of cut planes at 28, 29, 30 GHz.</figcaption></figure>

In the previous section, the optimized deflector was evaluated using an ideal Gaussian source as illuminator. However, in practice, the optimized deflector was illuminated by a circularly polarized radial line slot antenna (RLSA) [48]. The gap between the RLSA and the deflector was set to 5 mm, corresponding to half the wavelength at 30 GHz. It is important to note that the RLSA possesses a narrower frequency bandwidth than the Gaussian source, which limits our ability to evaluate the wideband behavior of the optimized deflector. Additionally, the circular polarization of the RLSA is not as ideal as that of the Gaussian source, resulting in a slightly higher axial ratio when the optimized deflector is combined with this source compared to the performance obtained with the Gaussian source. The radiation pattern measurements were conducted by mounting the entire system on a multi-axis positioner in an anechoic chamber, as shown in Fig. 4a. The measurement protocol is described in the Methods. Figure 4b compares the simulated and measured radiation patterns in the deflection plane at 28, 29 and 30 GHz. The main beam direction and grating lobes positions show good agreement between simulation and measurement. These directions correspond to the diffraction orders angles of the periodic deflector and can be analytically calculated using the grating law. The angle dispersion in frequency is 1°/GHz, derived from the relationship between deflection angle and frequency for a given deflector period.

Figures 5a-f present the 3D radiation patterns at 28, 29 and 30 GHz, revealing a close alignment between simulation and measurement across the entire 3D pattern. The energy is concentrated in the main beam at a deflection angle of 31° at 29 GHz, with a 3dB-beamwidth of approximately 2.75° in both
<figure><img src="image_5.png" /><figcaption>**Fig. 5:** 3D radiation patterns: simulation results at **a** 28 GHz, **b** 29 GHz, **c** 30 GHz, and measurement results at **d** 28 GHz, **e** 29 GHz, **f** 30 GHz.</figcaption></figure>

simulation and measurement. While the measured diffused level is slightly higher than in the simulation, it remains below -10 dB relative to the main lobe.

Figure 6 illustrates the measured realized gain of the LHCP co-polarization and the AR across frequencies, alongside the standalone RLSA results. The 3 dB frequency bandwidth is 6.4%, covering the range from 27.9 to 29.75 GHz. This bandwidth primarily depends on the radiating panel that illuminates the dielectric deflector. Recent advancements in corporate-feeding slot arrays and open stub arrays have shown great potential to widen the frequency bandwidth [49, 50].

<figure><img src="image_6.png" /><figcaption>**Fig. 6:** RLSA combined with the optimized deflector – Simulation (green curves) and measurement (blue curves) results: **a** Realized gain and **b** Axial Ratio versus frequency (the dashed gray lines correspond to 1.5 dB and 3 dB level). The measured results of the RLSA alone are shown in gray curves.</figcaption></figure>

Realized gain measurements match the simulation well up to 30 GHz; beyond this frequency, the measured gain drops and diverges by 3 dB from the simulated gain at 31 GHz. Ongoing investigations aim to identify the cause of this discrepancy. The AR shows similar behavior in both simulation and measurement, with level less than 2 dB over almost the whole frequency range of interest. Generally speaking, the experimental results align quite well with the numerical simulations. Yet, the discrepancy observed
above 30 GHz may stem from fabrication imperfections and variation in the measured permittivity of the fabricated pillars. Nonetheless, the overall performance confirms the fidelity of our NLM concept.

## 3 Discussion

In this paper, we introduced the NLM concept within beam-deflection systems operating in Ka-Tx band for SatCom, paving the way for the development of high-performance deflector designs. As discussed in this study, employing the near-field interactions unlocks new systems functionalities and extreme wave manipulation. This advantage was combined with an advanced optimization strategy based on statistical learning to offer the potential to determine optimal solutions with a reasonable number of full-wave solver iterations. The EGO algorithm, based on Bayesian learning, enables fast and robust predictions of the objective function, facilitating the determination of highly efficient designs within a reasonable computation time. As highlighted, this optimization strategy outperforms evolutionary algorithms and local gradient techniques by minimizing the number of iterations needed to reach convergence.

The optimized design proposed here with only 5 cylindrical pillars per macro-cell offers a significant efficiency increase of the (-1, 0) transmitted mode of the circularly polarized incident wave, coupled with an important axial ratio reduction across the Ka-Tx band compared to the conventional phase-gradient-based design without optimization. Furthermore, the radiation pattern results of the complete deflector demonstrate remarkable performance gain with reduced unwanted radiation and overall improved efficiency. The measurement results confirm the simulated performance of the NLM deflector, demonstrating the advantages of this design approach in achieving lightweight, low-cost, low-profile beam-deflection systems. The proposed nonlocal metasurface approach together with the optimization strategy facilitate the development of antenna systems equipped with efficient deflectors capable of 2D-beam-steering with wide field-of-view. This makes these systems perfectly suited for SatCom on-the-move applications, as they guarantee reliable wave transmission and ensure strong connectivity.

## 4 Methods

### Macro-cell simulation

An EM full-wave approach was used to simulate the deflector's macro-cell in the optimization process by solving the complete system of time-domain Maxwell equations. For this purpose, we exploit the Discontinuous Galerkin Time-Domain (DGTD) solver that we have recently developed and which is implemented in our in-house simulation software called the DIOGENES.

### Finite structure simulation

The performance of the complete antenna, consisting of the RLSA and the optimized deflector, was evaluated using the Finite Integration Technique (FIT) Transient Solver in CST Microwave Studio 2021. The efficiency calculation involved three parameters: the directivity of the standalone LHCP Gaussian source $D_{Gaus}$, the directivity of the system comprising the LHCP Gaussian source and a deflector $D_{Gaus+def}$, and the cosine of the deflection angle $\theta$ (with $\theta = \arcsin(\lambda_0/P_x)$) as follows:

$$ \eta = \frac{D_{Gaus+def}}{D_{Gaus} \cdot \cos \theta} \qquad (3) $$

### Measurement protocol

Measurements were performed over the frequency range of 27 GHz to 31 GHz, using a dual-polarized circular waveguide antenna as the probe and a SATIMO QR18000 as the gain reference antenna. Angular measurements were taken in steps of 0.25° for elevation and 0.5° for azimuth. The measurement was performed in the near-field region at a distance of 3.0 meters (see Fig. 4a). The collected data were subsequently transformed to the far-field to obtain the radiation patterns.

**Data availability.** All data are available from the corresponding author upon request.

**Acknowledgements.** This work was co-funded by the Defense Innovation Agency (AID) of the French Ministry of the Armed Forces as part of the RAPID AEROCOM project No. 2022 29 0954. In addition, the authors would like to thank their project partner, Nanoe company, for its contribution in providing the materials and manufacturing the deflector. Numerical experiments presented in this paper were carried out using the PlaFRIM (see http://www.plafrim.fr/) and Azzurra (https://calculs.univ-cotedazur.fr)
HPC clusters. PlaFRIM cluster is supported by Inria, CNRS (LABRI and IMB), Université de Bordeaux, Bordeaux INP and Conseil Régional d'Aquitaine. The Azzurra cluster supported by the French government through the France 2030 investment plan managed by the National Research Agency (ANR), as part of the Initiative of Excellence Université Côte d'Azur under reference number ANR-15-IDEX-01. The authors are grateful to the Université Côte d'Azur's Center for High-Performance Computing (OPAL infrastructure) for providing resources and support to Azzurra.

**Author contribution.** A.B. and M.E. suggested the NLM design configuration. A.B. performed the simulations and the optimization of the structure. A.B. and M.E and S.L. discussed the numerical and optimization results. S.L. and T.Q.V.H. coordinated the project. T.Q.V.H. and E.V. conceived the experiments and carried out the measurements. All authors participated in writing and reviewing the manuscript.

**Conflict of interest.** The authors declare no competing interests.

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