# Optimization strategy to design rotating nonlocal meta-deflectors for Risley-Prism antenna systems

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

Quynh Van Hoang

► To cite this version:

Ayoub Bellouch, Mahmoud Elsawy, Stéphane Lanteri, Erika Vandelle, Thi Quynh Van Hoang. Optimization strategy to design rotating nonlocal meta-deflectors for Risley-Prism antenna systems. EuCAP 2025 - 19th European Conference on Antennas and Propagation, Mar 2025, Stockholm, France. pp.1-5, 10.23919/EuCAP63536.2025.10999194 . hal-05336339

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# Optimization Strategy to Design Rotating Nonlocal Meta-Deflectors for Risley-Prism Antenna Systems

Ayoub Bellouch*, Mahmoud Elsawy*, Stéphane Lanteri*, Erika Vandelle†, Thi Quynh Van Hoang†

*Université Côte d'Azur, Inria, CNRS, LJAD, 06902 Sophia Antipolis Cedex, France, ayoub.bellouch@inria.fr

†Thales Research & Technology, 1 avenue Augustin Fresnel, 91120 Palaiseau, France, van.hoang@thalesgroup.com

**Abstract**—A new advanced optimization strategy is proposed to design high-performance rotating deflectors used in Risley-Prism concept for low-profile 2D-beam-steering-antenna system at Ka-Band. This new methodology combines two important ingredients: (1) the nonlocal near-field effect generated by the interactions among adjacent subwavelength elements and (2) a Bayesian learning-based global optimization algorithm for deriving the geometrical characteristics of the constituting dielectric pillars, driven by a design objective. This optimization strategy delivers superior performance compared to the traditional global optimization algorithms and requires fewer solver calls. In this scenario, the meta-deflector is specifically designed to efficiently operate for all rotation angles between 180° and 90°. The proposed solution provides remarkable performance across the entire Ka-band. Moreover, the numerical results show multiple advantages, mainly in enhancing transmission efficiency in the desired diffraction order and eliminating the cross-polarization.

**Index Terms**—2D-beam-steering antenna, Bayesian optimization (BO), efficient global optimization (EGO), Floquet modes, Gaussian Process (GP), Ka-Band, nonlocal metasurfaces (NLMs), SatCom on the move.

## I. INTRODUCTION

This work is concerned with the design of a 2D-beam-steering antenna systems used in Satellite-based communication systems (SATCOM) on-the-move applications. This system aims at providing internet access, facilitating information exchange with satellites, and ensuring secure communication for civil and military aircraft and vehicles on-the-move within the Ka-band. The conventional beam-steering solutions, as the parabolic reflectors, are capable of delivering good performance but their antenna architecture presents a significant challenge due to the bulky 3D design of these systems. The latest advancements focus on innovative low-profile 2D-beam-steering solutions based on the Risley-prism concept [2] that are cost-effective, low energy consumption and easy to manufacture, while maintaining superior performance levels within the Ka-band [3]. This solution typically consists of a planar radiating panel associated with two compact deflectors placed in the near-field region of the feed as shown in figure 1. The independent rotations of the deflectors ensure the steering capacity in elevation and azimuth planes. Elevation is obtained by controlling the relative rotation of the two deflectors and azimuth by rotating the two deflectors together. Leveraging this concept, recent beam-steering solutions have emerged based on the phase-gradient metasurfaces (PGM) concept [4]. In this configuration, the phase transformation is

locally controlled by varying the size of constituting elements in a unit cell. Indeed, the overall structure performance is determined using a lookup table that dictates the properties of each element to align with the desired wavefront. This PGM configuration provides strong but limited performance due to the lack of degrees of freedom to optimize the overall performance.

To address this limitation, we exploit here a new concept based on nonlocal effect of metasurfaces. It extends beyond the individual properties generated by each unit cell, considering the interactions among adjacent sub-wavelength elements to shape the overall effective far-field response. This new configuration provides a significant advantage by offering more degrees of freedom in a metasurface design, enabling new functionalities and extreme wave manipulation to obtain superior deflector performance compared to conventional PGM ones. The objective here is to leverage this nonlocal concept to the design of rotating metasurface-based deflectors well adapted for Risley-prism concept used in the 2D-beam-steering antennas. The essential challenge is to optimize the far-field transmission efficiency in the desired diffraction order and simultaneously eliminate the cross-polarization over different steering angles, each one reflecting a specific relative rotation of the two deflectors, throughout the entire Ka-band. To achieve this, we rely on 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) [5], employs a cheaper surrogate model to replace the actual objective function. It provides significant benefits for managing expensive black-box functions and ensures a considerable reduction in the number of evaluations [6], [7] compared to traditional global optimization methods. Thus, the potential to find promising solutions more rapidly is enhanced, as illustrated by the deflector design proposed in this work.

This contribution will focus on the design and optimization of the upper deflector, which always receive an oblique incidence coming from the lower deflector. A specific strategy is applied with taking into consideration the incidence angle at different azimuthal angles, representing different relative rotation angles between the two deflectors, corresponding to different steering angles.

## II. DESIGN METHODOLOGY

### A. Meta-deflector design
<figure><img src="image_2.png" /><figcaption>Fig. 1. 2D-beam-steering architecture based on Risley scanner concept, composed of a radiating panel and 2 subwavelength deflectors.</figcaption></figure>

1) Risley-prism concept: the proposed 2D-beam-steering antenna architecture is inspired by optical approaches based on Risley scanners [2], [3]. The system contains an RF feed and two diffractive meta-deflectors that rotate independently. This configuration enables 2D-beam steering within a wide angle scanning by manipulating the individual rotations of the deflectors. Figure 1 presents a general view of our proposed system architecture based on Risley concept. In our case, it is composed of a radiating source emitting a left-handed circular polarized (LHCP) electromagnetic wave in Ka-band and a pair of rotatable compact sub-wavelength all-dielectric metasurfaces deflectors positioned in the near-field region above the antenna to steer the beam to a desired angle. The direction of the 2D-scanning beam can be analytically predicted using the phase method [8]. To achieve a full coverage of the solid conical area, it is necessary to employ the same deflection angle $\alpha$ for both deflectors. Thus, the elevation and azimuth angles of the beam can be calculated using:

$$ \theta = \arcsin(\sin \alpha) \sqrt{2 [\cos(\phi_2 - \phi_1) + 1]} \quad (1) $$

$$ \phi = \arctan\left(\frac{\sin \phi_1 + \sin \phi_2}{\cos \phi_1 + \cos \phi_2}\right) \quad (2) $$

where $\alpha$ is the deflection angle of one deflector, $\phi_1$ and $\phi_2$ are the rotation angles from x-axis of the two deflectors, respectively (Figure 1). The designed deflector aims to generate the first-order mode (-1, 0) at a desired deflection angle of $\alpha = 30^{\circ}$ at 30 GHz for a normal input incidence, leading to a period of 20 mm (corresponding to 2 wavelengths at 30 GHz).

In this study case, we are focusing on the optimization of the upper deflector and we are targeting a relative rotation $\Delta\phi = |\phi_2 - \phi_1|$ from 180° to 90° between deflectors. According to (1), this adjustment enables variations in the elevation steering angle from $\theta = 0^{\circ}$ to $\theta = 47^{\circ}$. On the other side, the simultaneous rotation of both deflectors enables covering 360° in azimuth.

2) Full-wave analysis: we are aiming to optimize the design of the second deflector (the upper one) of the 2D-beam-steering antenna architecture. To achieve this, a LHCP plane wave is excited into the $(\theta_0, \phi_0)$ plane. An oblique incidence angle of $\theta_0 = 30^{\circ}$ is used to model the behavior of the (-1, 0)

transmitted mode coming from the first deflector. Moreover, we adjust the value of the angle $\phi_0 = \Delta\phi \in [180^{\circ}, 90^{\circ}]$ of the LHCP wave to mimic the behavior of the relative rotation of the second deflector relative to the first one. In this nonlocal configuration, cylindrical all-dielectric subwavelength pillars are employed, with only 5 pillars per period, compared to the 7 or 8 typically employed in the conventional local approach [3]. A composite material with high permittivity ($\epsilon_r = 8.9$) is used for the design of the pillars. A substrate support base with a low permittivity material ($\epsilon_r = 2.3$) is added to ensure the mechanical rigidity of the deflector while mitigating the reflections. The deflector is modeled as an infinite periodic structure with periodic boundary conditions (PBC). Moreover, a perfectly Matched Layer (PML), along with Absorbing Boundary Conditions (ABC), are applied to the top and bottom of the structure to prevent parasitic reflections and accurately model the far-field behavior away from the structure. Using the Floquet modal decomposition theory, the transmitted electromagnetic field can be described as a set of Floquet modes, or in other word diffraction orders. For each order, we define and compute the mode diffraction efficiencies $\eta(m, n)$ in Ka-band frequency range. The obliquely incident wave in LHCP is excited at the lower Floquet port. Hence, the transmission efficiencies of the diffraction modes are computed at the upper Floquet port.

<figure><img src="image_3.png" /><figcaption>Fig. 2. Periodic macrocell of the upper deflector with input oblique incidence simulated by the DGTD solver</figcaption></figure>

To model the periodic macrocell, a full-wave approach that involves solving the complete system of 3D time-domain Maxwell equations is used. For this purpose, we exploit a Discontinuous Galerkin Time-Domain (DGTD) solver that we have recently developed [9], [10] and which is implemented in our in-house simulation software called the DIOGENES [11].
An illustration of the periodic macrocell of the meta-deflector simulated by the DGTD method is shown in Figure 2.

## B. Optimization strategy

An inverse design problem is formulated to optimize the deflector's macro-cell structure. In the present case, the objective is to enhance the diffraction efficiency $\eta_{LHCP}(-1, 0)$ for the intended LHCP polarization in the desired diffraction order $(-1, 0)$, which corresponds to the desired angle of the beam steering, while concurrently minimizing the efficiency level for other diffraction modes. An additional aim is to maximize the ratio between the co-polarization LHCP and cross-polarization RHCP. Our challenge lies in optimizing these related objectives across all relative rotations between deflectors, $\Delta\phi \in [180^\circ, 90^\circ]$, and throughout the entire Ka-band. Solving this inverse problem requires a computationally efficient optimization strategy, which is capable of handling a several design parameters while minimizing DGTD solver calls.

To effectively resolve this problem, we consider here one of the most advanced optimization techniques based on a statistical learning approach, known as the Efficient Global Optimization (EGO) method [6], [7], which belongs to the class of Bayesian learning methods. The EGO method is a global optimization algorithm that substitutes the complex and costly iterative electromagnetic evaluation process with a simpler and cheaper model. Contrary to the traditional global optimization strategies like Genetic Algorithms, the EGO method is not based on adaptive sampling, but on a surrogate model to built on a set of available fitness observations. In our case, we use Gaussian Process (GP) which, due to its statistical nature, provide both a prediction of the cost function and a measure of the uncertainty of the prediction. This surrogate model utilizes a statistical learning criterion related to the optimization target (usually called merit function) in order to identify which design should be tested in the next iteration that would provide better results close to the predefined goal. 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 Latin Hypercube Sampling (LHS) 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. The GP mean provides the surrogate model that we use to approximate the objective function and the GP variance represents the uncertainty of the model. Next, a statistical merit function is utilized to identify what areas in the domain should be targeted for exploitation and which should be explorated. In our case, we rely on the expected improvement, which estimates the expected progress in the cost and whose maximum defines the next design parameters set to be evaluated 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.

# III. NUMERICAL RESULTS

## A. Objective function

The aim of optimization is to find the most efficient arrangement and size of the dielectric pillars to enhance the performance of the rotating deflector within the Ka-band. The desired deflection will be provided by the effective behavior of the entire macrocell due to the near-field coupling between pillars. The advantage of this nonlocal effect is increasing the number of parameters, which gives the optimization process more flexibility in choosing design dimensions.

The parameters $\mathbf{X}$ to be optimized include the radius $r_i \in [1, n_p]$, the total height of the pillars $h_i \in [1, n_p]$, the height of cones $h_c$, along with the $y_i \in [1, n_p]$ positions of each element within the macrocell. Opting for this choice rather than controlling the distance between pillars offers more flexibility. Nevertheless, to prevent elements from exceeding the macrocell boundaries, overlapping each other and to ensure compliance with fabrication conditions, constraints on their positioning are imposed. The principal route to enhance performance is to increase deflection efficiency of the mode $(-1, 0)$ and reduce the efficiencies of other modes, but more importantly to maximize the efficiency $\eta_{LHCP}(-1, 0)$ of the LHCP co-polarization. A further related objective is to minimize the Axial Ratio (AR), i.e. the ratio between the major and minor axes of an elliptical polarization. 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), which is the ratio between the co- and cross-polarizations, which allows to eliminate the unwanted polarization RHCP.

To achieve the targeted performance, the two interrelated objectives, i.e., the $\eta_{LHCP}(-1, 0)$ efficiency and the AR factor, are combined into a unified cost function to be minimized for each rotation. Given the complexities involved in optimizing over a continuous interval of relative rotations $\Delta\phi$ and the high computational cost for each value, we focused on specific key rotation values: $\Delta\phi \in \{180^\circ, 150^\circ, 120^\circ, 90^\circ\}$. The minimal fluctuation in efficiencies during deflector rotation ensures a stable high performance across the entire range. Thus, the constrained optimization problem to solve can be written as follows:

$$ \text{Minimize} \quad f(\mathbf{X}) = \frac{1}{4} \sum_{i \in \{1, 2, 3, 4\}} J_{\Delta\phi_i} + \sigma(J_{\Delta\phi}) \quad (3) $$

$$ \text{s.t.} \quad \begin{cases} y_2 - R_{print} - 2r_1 \geq \Delta p + \Delta s, \\ y_{i+1} - y_i - R_{print} - r_i \geq \Delta p, & i \in [2, n-1] \\ y_n + r_n \leq P_y - \Delta s \end{cases} $$

with $J_{\Delta\phi_i} = \frac{1}{2}(2 - \bar{\eta}_{LHCP}(-1, 0) - \frac{1}{AR^2})$ represents the objective function for a specific rotation angle $\Delta\phi_i$, $\sigma(J_{\Delta\phi})$ the standard deviation that is minimized to guarantee
similar levels of performance, $\Delta s = 0.01$ mm a fixed distance to the behavior shapes the overall characteristics observed in the farsides of the macrocell, $\Delta p = 0.02$ mm a fixed distance between two consecutive pillars, $R_{print} = 1.25$ mm (radius of the printing nozzle) and $n = 5$ (number of pillars).

## B. Optimization results

As previously described, the optimization was carried out using the EGO method. In the first phase, a DOE database consisting of 35 points was constructed. In the EGO phase, an surrogate model is built based on these DOE designs and is refined during the optimization to find the global minimum. Figure 3.a illustrates the optimization process, with the objective function evaluations from the DOE phase represented as purple points within the purple zone, and the evaluations from the EGO phase shown as blue points. The meta-deflector designs have been evaluated using the DGTD-$P_2$ method where $P_2$ refers to the second-order polynomial interpolation used for the DG discretization. The optimal design found by optimization is shown in Fig 3.b. It was reached after only 35 iterations, considering 15 parameters.

<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 3. Optimization using the EGO method (a) Evolution of the objective function with the number of full-wave solver calls (b) Best design.</figcaption></figure>

The EGO algorithm facilitates achieving convergence with much fewer iterations compared to traditional leading optimization algorithms. It is clear that the configuration of the obtained design, as shown in Fig 3.b, differs from those produced by traditional PGMs. Firstly, the periodicity between pillars is no longer maintain, and the element diameters do not follow the phase gradient required to cover the 360° cycle. Moreover, the pillar heights are non-uniform, introducing a novel aspect that, to our knowledge, has not been previously explored in beam-deflection systems.

<figure><img src="image_6.png" /><figcaption>Fig. 4. Diffraction efficiency and the AR of the macrocell</figcaption></figure>

The results are presented in Fig. 4 across all frequencies of the Ka-band for the four rotation angles $\Delta\phi \in \{180^\circ, 150^\circ, 120^\circ, 90^\circ\}$ incorporated in the optimization, as well as for selected intermediate values $\Delta\phi \in \{165^\circ, 135^\circ, 105^\circ\}$. In terms of co-polarization efficiency, the optimal design demonstrated notably high values not only for the optimized angles but also for the intermediate values, consistently exceeding 70% and maintaining a stable performance across nearly the entire band. On the other hand, the optimal design achieved very low levels of AR, remaining
below the 3 dB threshold for all values except at $\Delta\phi = 90°$, where a small region exhibits resonance near to 30.2 GHz due to the excitation of guided modes in the pillars. Overall, this design offers minimal fluctuations in LCP efficiency and the AR ratio during deflector rotation, ensuring consistently high performance and robustness across the entire range of the relative rotation, surpassing conventional designs.

<figure><img src="image_7.png" /><figcaption>Fig. 5. Field maps of $Re(E_x)$, $Re(E_y)$ and $Re(E_z)$, respectively obtained for the optimized design at $f_0 = 29$ GHz for two different cases of rotation $\Delta\phi = 180°$ and $\Delta\phi = 105°$.</figcaption></figure>

In this configuration, the near-field coupling between pillars contributes to generate the overall effective deflection in the far-field. This nonlocal effect is illustrated in the field maps presented in Fig. 5. These outcomes offer a comprehensive view of the interactions among pillars, revealing how the collective behavior shapes the overall characteristics observed in the far field pattern. The deflection observed is in agreement with the values predicted by the theoretical model (1). This confirms the reliability of the adopted optimization strategy, highlighting its success in designing high-performance rotating

metasurfaces deflectors.

## IV. CONCLUSION

In this paper, we presented a novel optimization strategy based on Bayesian learning and the nonlocal metasurface concept, paving the way for the innovation of high-performance rotating subwavelength deflectors for 2D-beam-steering antenna systems operating in the microwave regime. Moreover, the EGO algorithm used here, with a surrogate model built on a Gaussian process, shows great potential for identifying optimal solutions with a notably reasonable number of full-wave solver calls, surpassing classical global optimization methods as evolutionary algorithms. The optimized design proposed here with only 5 cylindrical pillar per macrocell provides notably high values of the LHCP efficiency of the transmitted (-1, 0) mode coupled with an important axial ratio reduction across the entire Ka-band for different obliquely incident waves representing various relative rotations of the meta-deflectors.

## ACKNOWLEDGMENT

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.

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