PARISO: pixelated antennas and reconfigurable intelligent surface optimizer

1 Introduction

Planar Printed-Circuit-Board (PCB) technologies provide a low-profile, low-cost, and fabrication-friendly platform for implementing antennas, antenna arrays, and periodic electromagnetic structures such as metasurfaces and RISs (Iqbal et al., 2025). Due to their compatibility with standard manufacturing processes, their ability to realize multi-band-multi-polarized, and multifunctional structures, PCB-based solutions have become central to modern wireless systems. While much of this development has been driven by emerging 5G and 6G networks (Nwufo et al., 2025), (Yu and Ren, 2025), the same capabilities are equally critical for next-generation Wi-Fi systems, especially Wi-Fi 6E and Wi-Fi 7, which require compact, efficient, and wideband electromagnetic components (Heinrichs et al., 2023; Tewes et al., 2023; Al-Asaslih et al., 2025).

Although RIS technology has matured rapidly in recent years and is now viewed as a key enabler for future wireless networks due to its ability to manipulate the propagation environment using passive, electrically controllable unit cells (Lu et al., 2025), it still faces multiple challenges. At mmWave frequencies (e.g., 24–100 GHz), RIS and metasurface implementations face stringent challenges, such as increased conductor and dielectric losses, higher sensitivity to fabrication tolerances, and reduced unit-cell dimensions. Enabling dense integration of electrically small unit cells on PCB platforms is a major challenge at any frequency. In 6G scenarios, where operation may extend into sub-THz frequencies and require ultra-wideband, beam-steerable, and reconfigurable environments, the ability to maintain consistent electromagnetic performance over wide spectral ranges becomes critical.

Although the use of RIS in Wi-Fi remains unclear, Wi-Fi may also benefit from the introduction of RIS into the system. Wi-Fi access points must operate across three distinct and widely separated frequency bands: 2.4 GHz (2.40–2.48 GHz), 5 GHz (5.15–5.85 GHz), and the new 6 GHz band (5.95–7.125 GHz). Supporting these bands simultaneously imposes stringent requirements on antennas and RIS elements, particularly in maintaining stable phase and amplitude responses over a very broad bandwidth. For RISs, achieving consistent phase compensation across such a spectral span is one of the most difficult barriers to practical adoption in Wi-Fi environments where channel variations and multipath propagation are significant (Monemi et al., 2025; Zhang et al., 2021).

In addition to the ElectroMagnetic (EM) challenges, the design of PCB-based antennas, metasurfaces, and RIS unit cells is highly complex and computationally demanding. As the number of degrees of freedom increases, whether in miniaturized antenna patches, pixelated metasurface layouts, or multi-state RIS unit cells, the design space becomes extremely high-dimensional and nonlinear. This complexity makes it difficult to achieve multiple objectives simultaneously, such as physical miniaturization, bandwidth enhancement, high radiation efficiency, or broadband phase stability. Recent studies in RIS synthesis and optimization (Marzban et al., 2026), (Ji et al., 2023) highlight that even moderately sized surfaces can require advanced optimization strategies due to the intricate electromagnetic coupling among elements. Similar challenges arise in antenna miniaturization, where geometric modifications intended to reduce physical size often degrade bandwidth or efficiency, demanding careful trade-off exploration.

To navigate these challenges, various optimization methods have been applied in antenna and metasurface design. Evolutionary algorithms, particularly GA, have proven effective for exploring unconventional or non-intuitive geometries. Early microstrip antenna research demonstrated that GA-based shape optimization, slot introduction, and fractal-inspired modifications can successfully improve bandwidth and achieve significant miniaturization. Several studies have combined GA with full-wave solvers such as Finite Difference Time Domain (FDTD) to automatically evaluate the electromagnetic performance of candidate geometries (Johnson and Rahmat-Samii, 1999), enabling the discovery of optimized patch shapes (Villegas et al., 2004) and slot-loaded miniaturization techniques (Sabouni et al., 2005), (Sabouni and Thesis, 2005). Pixel-based methodologies, where the conducting surface is discretized into binary metallization states, have emerged as especially powerful for generating irregular, fractal-like antenna topologies (Noghanian et al., 2025), (Khanam et al., 2025).

Motivated by these prior successes, this work adopts a pixel-based optimization framework to address the dual challenges of antenna miniaturization and RIS phase stability. The conducting patch is discretized into metallized “pixels,” and their binary on/off states form the optimization variables within a GA. All other parameters, including substrate thickness, feed location, and ground plane size or unit cell size, remain fixed. This approach enables efficient exploration of a large design space, targeting compact antennas and RIS elements capable of supporting robust performance across a frequency band. The design approach presented in this work is independent of frequency and can be applied to Wi-Fi and mmWave systems bands by appropriately defining the geometry and cost function used in the optimization process.

2 Methodology

In this article, we address two major classes of electromagnetic optimization problems relevant to modern antenna and metasurface engineering:

• Microstrip antenna miniaturization for both single-element and infinite array configurations, and,
• RIS optimization, targeting reflection phase bandwidth improvement.

To tackle these intensive computational, nonlinear problems, we employ two categories of global optimization techniques: GA and SADEA optimization, both of which are well supported within MATLAB’s optimization Toolbox (MathWorks, 2025). A brief technical background on each category is summarized in this section, while Section 3 provides detailed implementation guidelines and some examples. It is important to note that the performance of various optimizers, such as GA and SADEA, is problem dependent. The goal of this work was to provide a framework and show how the optimization tools can be utilized for the intended problems.

2.1 Simulation tools and environment

The simulation code is developed using MATLAB, which provides an integrated environment well-suited for electromagnetic optimization, combining geometry generation, PCB modeling, full-wave simulation, and advanced numerical optimization within a single workflow. These capabilities were used to implement an automated framework along with a Graphical User Interface (GUI) App developed with MATLAB App Designer named “Pixelated Antenna and RIS Optimizer–PARISO”.

The RF PCB Toolbox provides the foundation for constructing physical geometry. It enables precise definition of multilayer PCB stack-ups, including dielectric substrates, copper layers, feed structures, and slot features. Using Boolean operations and parametric control, pixelized metallization patterns and conventional patch shapes were generated programmatically.

Electromagnetic evaluation was performed using the Antenna Toolbox, which employs a hybrid Method of Moments (MoM) formulation tailored for planar and layered structures. This allows accurate simulation of PCB-based radiators and periodic unit cells within a unified solver environment.

For optimization, MATLAB’s Global Optimization Toolbox was used, which offers efficient exploration of the large, highly nonlinear design space associated with pixel-based geometries. Evolutionary algorithms such as GA were used to manipulate the binary pixel states, while surrogate models help reduce computational cost when appropriate.

Together, these methods support a customizable framework capable of targeting diverse design goals, including antenna miniaturization, bandwidth enhancement, and RIS phase response.

2.2 Microstrip antennas and arrays

Microstrip antennas are widely used in consumer electronics, wearable systems, Wi-Fi devices, radar sensors, and general wireless communication platforms due to their low-profile geometry, lightweight construction, and seamless integration with RF front ends and PCB-based circuitry. Their simple fabrication and ability to conform to planar or mildly curved surfaces make them particularly attractive for compact system designs. Despite these advantages, the relatively large physical size required for resonance remains a key limitation, especially in applications requiring multiband operation, dense integration, or ultra-thin form factors.

A broad range of miniaturization strategies has been investigated to address these constraints. Techniques such as slot loading, high-permittivity substrates, metamaterial inclusions, fractal geometries, and shorting structures have been widely reported. Review articles on microstrip miniaturization and electrically small antennas further document the evolution of these approaches and their trade-offs (Khan et al., 2015). More recent developments show that with careful geometric engineering, it is possible to achieve significant size reduction while maintaining acceptable bandwidth and radiation efficiency. For instance, optimized feeds, slot architectures, and array-level techniques have enabled compact high-gain microstrip arrays at millimeter-wave radar frequencies, achieving controlled sidelobes and low cross-polarization performance (Wang et al., 2025), (Chen et al., 2017).

In addition to optimizing individual antenna elements, this work also considers array applications by evaluating unit-cell behavior under infinite array conditions. Using periodic boundary conditions, a single unit cell can be simulated as part of an infinite two-dimensional lattice, enabling efficient analysis of array impedance, scan-dependent behavior, mutual coupling effects, and reflection/transmission phase responses. This approximation is effective for optimizing elements intended for large arrays or RIS surfaces.

Once a square patch is generated, the GA-based optimization creates a binary code to add or remove metal pixels. The PCB stack-up is generated, and the antenna is formed. The patch geometry is pixelized by discretizing the surface into an n x × n r grid determined by the user. Each pixel corresponds to a binary metallization state, and GA manipulates these states to generate new geometries. Care was taken to maintain the current continuity and protect the feed location. Although protecting some cells and removing them from the search space reduces the search size, it is required to maintain the surface current continuity and physical feasibility of creating the antenna. The symmetry is required to ensure symmetric far-field radiation. These points are explained through an example in Section 3.1.

The GA evaluates each candidate geometry with a custom cost function that queries MATLAB’s Antenna Toolbox for key metrics S₁₁ over the design band, resonant frequency, −10 dB bandwidth, and peak realized gain. The objective aggregates three aims: (i) miniaturization by driving the resonance to a lower frequency, (ii) bandwidth preservation, and (iii) radiation robustness by penalizing excessive gain loss. Each design is simulated, scored, and ranked before generating the next population. For both single-element patches and array unit cells, the overarching goal is to shrink size by shifting the resonance frequency of S₁₁ downward while keeping bandwidth and gain within acceptable limits. Because these requirements conflict during miniaturization, the framework supports multi-objective weighting of resonance shift, bandwidth, and gain, enabling designers to navigate trade-offs according to application needs.

For infinite-array optimization, MATLAB’s Antenna Toolbox models the pixelized unit cell within an infinite periodic environment using Floquet-type boundary conditions. This formulation captures scan-dependent reflection coefficient, bandwidth, and gain. The optimization process is similar to a single-element antenna, and a similar cost function is evaluated for each solution proposed by the GA. The realized gain is calculated using the array factor.

2.3 RIS

RISs offer programmable control of the propagation environment by adjusting the phase of many passive elements across a large planar surface. Tutorials and surveys identify RIS as a key enabler for beyond 5G/6G due to its low power, low-cost potential for coverage and interference management, but they also underscore the high-dimensional, quantized, and coupling-aware nature of the underlying optimization tasks (Feng et al., 2023). In this work, we targeted frequency robust reflection phase bandwidth. In the context of Wi-Fi, the major concern is to use the sam structure over the required Wi-Fi band with a stable phase. Therefore, we used the optimization goal of minimizing the variations of the phase of the reflection coefficient within a given frequency band.

The optimization of RIS structures closely parallels the challenges encountered in antenna optimization. In the next section, we provide the detailed formulation for calculating the phase of the reflection coefficient of a unit cell using array simulation.

2.3.1 Unit cell reflection coefficient phase calculation

The evaluation of the phase of the unit cell starts with the assumption that the unit cell lies in the x y-plane, with the structure illuminated by an incident plane wave, denoted E i n c, arriving with a specified polarization, azimuth, and elevation angle. The solver computes the corresponding scattered electric fields, which inherently satisfy the periodicity conditions. These scattered fields are then decomposed into specular and higher-order Floquet modes, depending on the electrical size and periodicity of the unit cell.

Figure 1 shows the coordinate system and the relationship between the incident and reflected fields, including the definition of incidence angles (θ i n c , φ i n c and reflected (received) angles θ r e c , φ r e c . Please note that θ i n c = θ r e c, and φ is the azimuthal angle in the x y-plane, and φ i n c − φ r e c = 180 ° (Balanis, 2024). Figure 1 shows the coordinate system, the incident and the reflected wave vectors.

The incident and reflected waves can be determined using Equations 1-4: where k i n c is the incident wave vector and k r e c is the reflected wave vector, k = 2 π / λ, is the wave number and λ is the wavelength in free-space. a ^ is a unit vector in the direction of a ϵ x , y , z. E i n c and E r e f are the magnitudes of the incident and reflected electric fields in V/m, respectively. θ = 0 represents the broadside angle and θ = 90 ° represents the plane of RIS (assumed x y plane). The far-field electric field is evaluated at a fixed far-field distance (default of 100 λ and transformed from Cartesian to spherical coordinates. r o b s represents the location of the observation point:

E i n c r = E i n c e − j k i n c · r e ^ i n c (1)

E r e c r = E r e c e − j k r e c · r e ^ r e c (2)

k i n c = k sin ⁡ θ i n c ⁡ cos ⁡ φ i n c x ^ + cos ⁡ θ i n c ⁡ sin ⁡ φ i n c y ^ − cos ⁡ θ i n c z ^ (3)

k r e c = k sin ⁡ θ r e c ⁡ cos ⁡ φ r e c x ^ + cos ⁡ θ r e c ⁡ sin ⁡ φ r e f y ^ + cos ⁡ θ r e c z ^ (4)

r o b s = R sin ⁡ θ o b s ⁡ cos ⁡ φ o b s x ^ + cos ⁡ θ o b s ⁡ sin φ o b s y ^ + cos ⁡ θ o b s z ^ (5)

r o b s = R r ^ o b s = R m u l t λ r ^ o b s (6)

The fields consist of the Transverse Magnetic (TM) and Transverse Electric (TE) polarizations, as depicted in Figure 1. The incident polarization vectors are defined as Equations 7, 8:

e ^ T M = k i n c × z ^ × k i n c k i n c × z ^ × k i n c (7)

e ^ T E = z ^ × k i n c z ^ × k i n c (8)

To calculate the complex scattering coefficient for a unit cell (Equation 9) is used:where 2 π R / k A c e l l is a normalization factor, assuming an array of periodic structure with N x × N y units and the area of A c e l l = d x × d y. 2 π R is the spherical wave spreading factor and k A c e l l is the effective electrical size of the unit cell. This normalization is done to make the coefficient independent of the observation distance and the scale of the RIS size. Finally, the effect of the array is included by adding the array factor (A F):

Γ u n i t = E r e c E i n c · 2 π R k A c e l l (9)

Γ a r r a y = Γ u n i t . A F θ r e c , φ r e c A F θ i n c , φ i n c (10)

A F θ , φ = 1 N x sin ⁡ N x ψ x / 2 sin ⁡ ψ x / 2 1 N y sin ⁡ N y ψ y / 2 sin ⁡ ψ y / 2 (11)

ψ x = k d x ⁡ sin ⁡ θ ⁡ cos ⁡ φ (12)

ψ y = k d y ⁡ sin ⁡ θ ⁡ sin ⁡ φ (13)

The objective of optimization is to minimize the phase of Γ across a range of frequencies and various scan angles. For calculating the cost function expressed as

Minimize ∠ Γ = arctan R e Γ I m Γ for the given frequencies and scan angles (14)

The program first determined the phase for different scan angles and frequencies. The program uses the standard deviations over both frequency and scan angles to calculate the cost value.

2.4 GA optimization

GAs are population-based global optimizers that iteratively evolve a set of candidate solutions through selection, crossover, and mutation. Unlike gradient-based methods, GA makes no assumptions about convexity, smoothness, or even continuity of the objective. That independence from derivatives is critical for full-wave electromagnetic (EM) objectives whose response surfaces are often jagged due to resonances, mode switching, and discrete geometry changes. In pixel-based layouts, toggling a single metallization cell can re-route surface currents, introduce/close slots, or alter effective inductive–capacitive loading, producing abrupt changes in S₁₁, gain, bandwidth, or RIS reflection phase. GA’s stochastic exploration and recombination make it robust to such nonconvex, multi-modal landscapes.

Pixelization maps the design directly into a binary chromosome; each pixel is a gene with state 0/1. This natural genotype is a major practical advantage: it eliminates real-to-binary encodings and simplifies constraint handling. Another reason GA is well-suited to miniaturized patches and RIS unit cells is the frequent need to trade off conflicting metrics, for example, shifting resonance downward (size reduction) tends to narrow bandwidth and may reduce gain/efficiency. GA’s population-based nature supports multi-objective or weighted-objective formulations, maintaining a Pareto-like diversity of solutions that capture different balances between S₁₁ shift, −10 dB bandwidth, broadside/peak gain, and pattern stability. This diversity is operationally valuable: designers can inspect multiple near-optima and select a design consistent with fabrication and system-level constraints. Since the aim is to miniaturize the antenna by creating slots, the cost function should include the resonance frequency of S₁₁. Optionally, the user can choose to combine the cost function with the gain and frequency bandwidth. To normalize these values, the cost function is defined as: B W is the −10 dB frequency bandwidth. G a i n o r i g i n a l is the maximum realized gain of the square patch, and G a i n is the maximum realized gain of the current antenna being evaluated. f r e s _ O r i g i n a l is the frequency of the minimum S₁₁ for the square patch, and similarly, f r e s is the resonance frequency and the frequency of the minimum S₁₁ for the current antenna that is being evaluated. w 1 and w 2 are the weights of the penalty. The user provides the percentages in the GUI. The total weight should be 1, therefore, the third coefficient is calculated by subtracting w 1 and w 2 from 1. In case the program finds the solution does not have any S₁₁ below −10 dB a large value is assigned to the cost value.

C o s t = w 1 B W o r i g i n a l B W + w 2 G a i n o r i g i n a l G a i n + 1 − w 1 − w 2 × f r e s f r e s _ O r i g i n a l (15)

Finally, GA scales seamlessly from single elements to periodic unit cells. For arrays/RIS, a GA loop can evaluate infinite-array behavior by applying the periodic boundary conditions for scan-dependent impedance and phase. When paired with parallel evaluation (each individual evaluated on a separate worker), GA can handle moderate-to-large pixel grids within practical wall-time budgets. It is worth noting that for single-element patches and array unit cells, the overarching goal is to shrink size by shifting the resonance frequency. But for RIS the goal is to have phase stability. The optimization process is similar, but since for RIS optimization we considered stability of the phase angle over the frequency band and scan angles, the cost function is defined by Equation 16:

C o s t f o r R I S = 1 N f N a n g ∑ i = 1 N f ∑ j = 1 N a n g ∠ Γ f i , θ j − a v e r a g e ∠ Γ f i , θ j 2 (16)

Figure 2 illustrates a flowchart of the GA as it was implemented in the code.

2.5 SADEA optimization

SADEA augments a global evolutionary search (differential evolution) with a learned surrogate model that approximates the expensive objective function, significantly reducing the number of high-fidelity simulations needed to reach high-quality designs (Liu et al., 2021). In SADEA, a lightweight model, often radial basis function (RBF) or Kriging/Gaussian process, is fit to the archive of evaluated designs and used to propose new candidates via an infill/merit criterion that balances exploitation (expected improvement in predicted objective) against exploration (space-filling to avoid local traps). This hybrid strategy preserves the global search properties of evolutionary algorithms while slashing wall-clock time on expensive black-box problems such as full-wave EM simulations (MathWorks, 2026a), (MathWorks, 2026b).

Pixel-based antenna and RIS unit-cell optimization is an ideal match for SADEA because it exhibits moderate-to-high dimensionality decision variables after enforcing symmetry/masks, strong nonlinearity, and mixed discrete–continuous constraints, while each evaluation can require long full-wave simulation. By steering the search with a surrogate, updated iteratively from true simulations, SADEA can concentrate simulations on promising regions and avoid wasting evaluations on obviously poor pixel masks, which is precisely where pure evolutionary methods struggle in EM tasks. Empirical studies on SADEA families (using RBF or Kriging) report substantial efficiency gains on expensive engineering problems, including antenna optimization, compared to unguided global search. Figure 3 shows the flowchart of SADEA as it was implemented for the RIS optimization.

2.6 Graphical user interface

A MATLAB-based GUI was developed to streamline the process of parameter selection and simulation. The GUI has the following main panels (Figure 4).

Input Parameter Panel: In this section, users can set key antenna parameters like frequency (range and initial value) and choose the antenna type: single, array, or RIS. Depending on which antenna type is selected, additional parameters can be configured in a corresponding subpanel displayed within three tabs, as shown in Figure 5.

Patch Panel: This panel specifies the physical geometry and material characteristics of the microstrip patch used in simulation and optimization. It allows users to configure ground-plane and substrate properties, as well as define the patch discretization by setting the number of cells (pixels) along the x and y directions.

Optimization Panel: The GUI features dedicated controls for launching optimization routines, including algorithm selection (e.g., GA, or SADEA) and configuration of optimization settings.

Action Panel: The Action Panel offers execution and control functions, enabling users to run simulations and optimization tasks once antenna parameters have been defined. Users can save parameters or load previously saved simulation settings. Additionally, if parallel capabilities are available, they can be included.

The integration of these panels and components within the MATLAB environment ensures a user-friendly workflow for antenna design and optimization, supporting both novice and advanced users in achieving reliable results.

3 Sample results

In this section, several representative optimization scenarios are presented. The first example demonstrates the use of GA to miniaturize a single square patch antenna while simultaneously preserving acceptable bandwidth and gain performance. The second example extends this approach to an infinite-array configuration, applying the same pixel-based GA framework to optimize a periodic unit cell under array boundary conditions. The third example focuses on RIS design, where both GA and SADEA are employed to maximize the stability of the reflection-phase response across a specified bandwidth and over a defined range of scan angles. Together, these examples highlight the flexibility of the optimization framework across single-element antennas, arrays, and RIS structures.

3.1 Single patch antenna miniaturization

For this example, we start with a square patch antenna with an air substrate. The substrate has a thickness of 4.3 mm. The square patch has a side of 51.5 mm, and the ground plane is in a square shape with a side length of 75 mm. The feed location is estimated inside the program, and in this case, is x = − 12 m m and y = − 4.3 m m measured from the center. The center frequency is chosen as 2.6 G H z, and the frequency band is chosen as 1.8–2.8 GHz. The patch was divided into a 10 × 10 pixelized grid. GA was used for the optimization with 10 generations of 10 individuals in each generation. The best solution from each generation was chosen as the elite solution to be passed to the next-generation. Referring to Equation 15, the weights for the BW and Gain are 30% (0.3) each, and the remaining 40% (0.4) is assigned to the resonance frequency of S₁₁.

To ensure continuous current flow, adjacent metallized pixels are designed with a 10% overlap. Figure 6a illustrates the extension applied to a single cell. In Figure 6b, a 6 × 6 grid is presented in one color and connected to three quadrilateral-symmetric counterparts, each represented in a different color. This arrangement forms a larger grid of interconnected cells, where the extensions create overlaps that eliminate discontinuities in the current path.

Cells located along the perimeter of the patch (labeled as Edge Cells in Figure 6b) remain unmodified to preserve current continuity. Additionally, the region surrounding the feed is structurally critical and is intentionally excluded from modification to maintain stable impedance characteristics.

Figure 6c shows the original and the optimized S₁₁. The starting resonance frequency is 2.65 GHz, and the optimization has brought it down to 2.35 GHz with a comparable gain and bandwidth. The original patch has a realized gain of 8.88 dBi and a bandwidth of 120 MHz. The optimized patch antenna has a gain of 8.55 dBi and a bandwidth of 110 MHz. The patch shape is shown in Figure 6d. It is noticeable how the masking of the freed location and mirroring it has created smaller slots around the center. This is to preserve the feed location and maintain a good impedance matching. One can also notice the symmetry enforced in both directions.

3.2 Infinite array

The infinite array example is the expansion of the single-element example that was explained previously. In this case, the single element is extended to an infinite array by adding the periodic boundary conditions. The patch and ground sizes are the same. The substrate is changed to “Foam”, since in MATLAB, the periodic boundaries cannot be applied to the air substrate. The computational complexity in the infinite array is much more than that of a single element. This example was run on a parallel machine with 12 workers. The baseline antenna element has a resonance frequency at 2.66 GHz, with a bandwidth of 20 MHz and an element gain of 6.02 dBi. After optimization using GA and pixelization using a 10 × 10-pixel grid, the optimized patch has a resonance frequency of 2.26 GHz, a bandwidth of 232 MHz, and an element gain of 6.06 dBi. Please note that the array gain should be calculated by multiplying the element gain by the Array Factor gain. Figure 7a shows the optimization S₁₁ results, and Figure 7b shows the optimized patch shape.

3.3 RIS

One example of RIS optimization was done with both GA and SADEA optimizers. In this example, a 20 × 20 unit periodic structure on “Foam” with the square patch size of 5.15 mm and reflector size of 7.5 mm was considered. The substrate thickness was 4.3 mm. The optimization was run for the bandwidth of 1.5 GHz–3 GHz with nine sampling points in the frequency band. The scan angle was changed from θ i n c = 0°–60°, with the steps of 10°. The incident and reflected φ angles were assumed to be 0°. Each square patch was divided into 10 × 10 (100) pixels. TM polarization was assumed, and 12 parallel cores were used. The cost function was defined as Equation 16 and was calculated with “std2” applied to the matrix of reflection coefficient phases.

For GA, the tolerance was set at 0.001, for a population size of 12 and a maximum generation of 13. The probability of crossover is 0.9, and the Elite Count is 1.

Figure 8 shows the shape of the optimized patch. Figure 9 shows the comparison of the baseline and optimized ∠ Γ over the frequency band. Figure 10, on the other hand, shows the variation of the ∠ Γ over the scan angles, for a fixed frequency.

The same problem was set up to be optimized with SADEA. Each row of n x binary pixels is encoded as a decimal in [0 , 2 n x − 1]. Initial population size was 10, the maximum number of function evaluations was 15, and the number of iterations was 15. Figure 11 shows the optimized patch, and Figures 12, 13 show the results of optimizations. Table 1 summarizes the settings of GA and SADEA.

TABLE 1. GA and SADEA settings and number of evaluations.

To compare the results of optimization via GA vs. SADEA, we define a figure of merit (F o M) as:

F o M = max ∠ Γ − min ∠ Γ (17)

The results for different frequencies are summarized in Table 2, and for different scan angles are given in Table 3. It is observed that in some cases, SADEA has shown better results, and in some cases GA. This example is meant to show how the program can work with both optimization methods and does not aim to quantitatively show the advantages of any of the two methods. Depending on the type of problem, one method could be better than the other. The advantage of SADEA is the smaller number of required simulations.

TABLE 2. F o M for different frequencies over the given angles for the results optimized by GA vs. those optimized by SADEA (results are in degrees).

TABLE 3. F o M for different scan angles over the frequencies for the results optimized by GA vs. those optimized by SADEA (results are in degrees).

4 Discussion

The results presented in this work highlight the advantages of combining pixel-based geometric parameterization with evolutionary and surrogate-assisted optimization strategies for the design of compact antennas, periodic unit cells, and RIS elements. Pixelization provides a flexible, topology-free representation that allows the optimizer to explore non-intuitive current paths and miniaturization mechanisms that would be difficult to obtain through traditional parametric tuning. Genetic Algorithms proved effective for navigating the large, discrete, and highly nonconvex design space created by these pixel masks, offering the population diversity and stochastic search needed to uncover viable miniaturized configurations while still maintaining acceptable bandwidth and gain. In contrast, the surrogate-assisted SADEA approach addresses the computational challenges inherent in full-wave evaluations by reducing the number of expensive simulations, enabling efficient exploration, especially in cases such as RIS phase optimization, where the cost of evaluating each candidate is significantly higher. Together, the two methods complement one another in balancing global exploration, convergence reliability, and computational cost.

Beyond algorithmic considerations, the study also demonstrates how a consistent workflow—encompassing geometry construction, constraint enforcement, EM evaluation, and optimization—can be applied uniformly across single-element antennas, infinite arrays, and RIS structures. The ability to incorporate symmetry, feed-region masks, connectivity rules, and periodic boundary conditions into the optimization loop ensures that the generated designs remain physically meaningful and fabrication-compliant. At the same time, the multi-objective formulation allows controlled trade-offs among miniaturization, bandwidth retention, and radiation performance, reflecting the practical requirements of tri-band Wi-Fi and other wideband communication systems. Collectively, these capabilities show that the proposed framework is both general and scalable, offering a foundation for developing more advanced structures and optimization objectives in future work.

5 Conclusion and future work

This work introduced a unified, MATLAB-based framework for pixel-level optimization of planar electromagnetic structures, targeting Wi-Fi band applications where compact size and broadband behavior are essential. By discretizing patches and unit cells into binary pixels and enforcing fabrication-aware constraints (symmetry, feed-area masks, connectivity), we enabled direct GA optimization of miniaturized microstrip antennas that preserve −10 dB bandwidth and peak realized gain within practical limits (8–9 dBi for a single microstrip patch antenna). In parallel, we leveraged SADEA to address the high cost of full-wave evaluations, especially for RIS unit cells requiring stable reflection-phase across wide bandwidths and scan angles. The same methodology extends naturally to infinite-array modeling via periodic boundary conditions, allowing array-level unit-cell optimization without prohibitive simulation budgets. Overall, the framework demonstrates that pixel-based search, guided by evolutionary methods and surrogate modeling, can deliver compact antennas and RIS elements with robust performance while remaining compatible with conventional PCB processes. The method is frequency independent and can be applied to Wi-Fi, 6G, and mmWave systems.

Several directions for future work are envisioned:

• RIS phase-angle targeting across bandwidth and scan: The objective formulation will be extended to direct phase targeting, whereby each unit cell is optimized to realize a prescribed reflection-phase angle (for example, 0°, 90°, 180°, 270°) over a specified bandwidth and range of incidence or scan angles. The cost function will incorporate phase error metrics (mean and worst-case), phase-slope linearity for broadband stability, and amplitude flatness to limit magnitude ripple. Fabrication-aware constraints will enforce minimum feature sizes, conductive continuity, and symmetry consistent with PCB processes.
• Multi-state (2-bit) RIS unit cells: The binary pixel encoding will be generalized to multi-state metallization for 2-bit RIS elements, yielding four discrete phase states per unit cell. This requires expanding the decision vector to integer variables with state space {0,1,2,3}, defining state-specific electromagnetic evaluations, and optimizing inter-state consistency (small amplitude variation between states) while minimizing phase quantization error. Surrogate-assisted strategies will be emphasized to control evaluation counts as the number of states multiplies the simulation workload.
• Feed-location co-optimization for miniaturized patches: for single-element antennas, the decision vector will be augmented to include feed position (x, y) and, optionally, feed geometry parameters. The objective will balance resonance down-shift (miniaturization) with −10 dB bandwidth and realized gain, while constraints maintain impedance stability and pattern symmetry. A two-stage strategy is envisaged: initial topology discovery with a nominal feed, followed by joint refinement of pixel mask and feed parameters under tighter convergence criteria and denser frequency sampling.
• Hybrid and hierarchical search strategies: Hybrid GA/SADEA pipelines will be explored, using GA for early topology exploration and SADEA for accelerated convergence in promising regions via a learned surrogate. In parallel, hierarchical pixelization (coarse-to-fine grids and grouped-pixel encodings) will be employed to manage dimensionality, with progressive ungrouping near the optimum to recover fine current-path features that govern bandwidth and phase stability.
• Parallel and distributed execution at scale: Given the cost of full-wave evaluations, the parallel execution model will be further optimized across local multi-core machines and clusters. This includes adaptive batch sizes for candidate evaluation, license-aware scheduling for electromagnetic solvers, and checkpoint/restart mechanisms for fault tolerance in long runs. For array and RIS studies, angle-batching strategies (evaluating multiple incidence angles per candidate) will be used to amortize geometry construction and precomputation overheads.
• Automation, reporting, and reproducibility: The graphical interface and scripts will be extended with automated report generation (Pareto summaries, phase-stability charts, bandwidth/gain trade-offs), provenance tracking (random seeds, solver versions, mesh settings), and standardized export of stack-up and geometry specifications. These additions aim to streamline design iteration and improve reproducibility across projects and teams.
• Fabrication and measurements: Future investigations will include fabrication and experimental validation of representative optimized antenna and RIS structures as well as detailed electromagnetic analysis of the resulting geometries to better understand the underlying physical mechanisms responsible for their performance and to derive simplified design guidelines.

Generative AI statement

The author(s) declared that generative AI was not used in the creation of this manuscript.

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Supplementary material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fanpr.2026.1830853/full#supplementary-material

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Footnotes

Edited by: Francesca Benassi, University of Bologna, Italy

Reviewed by: Islam Islamov, Baku Engineering University, Azerbaijan

Lorenzo Bastia, University of Bologna, Italy