Metasurface Design on Low-Emissivity Glass via a Physically Constrained Search Method

Abstract

Low-emissivity (Low-E) glass, crucial for thermal insulation, significantly attenuates wireless signals, hindering 5G communication. Metasurface technology offers a solution, but the existing designs often neglect the etching ratio constraint and lack physical interpretability. In this work, we propose a physically constrained search method that incorporates prior knowledge of the capacitive equivalent circuit to guide the design of metasurfaces on Low-E glass. First, the equivalent circuit type of the metasurface is determined as a capacitive structure through transmission line model analysis. Then, a random walk-based search is conducted within the solution space of topological patterns corresponding to capacitive structures, ensuring etching ratio constraints and maintaining structural continuity. Using this method, we design a metasurface pattern optimized for 5G communication, which demonstrates over 30 dB improvement in signal transmission compared with full-coating Low-E glass. A fabricated 300 mm × 300 mm prototype, etched with a ratio of 19.5%, demonstrates a minimum transmission loss of 2.509 dB across the 24–30 GHz band with a 3 dB bandwidth of 4.28 GHz, fully covering the 5G n258 band (24.25–27.5 GHz). Additionally, the prototype maintains a transmission coefficient reduction of no more than 3 dB under oblique incidence angles from 0° to 50°, enabling robust 5G connectivity.

1. Introduction

Low-emissivity (Low-E) glass has gained significant attention for its energy conservation properties. It consists of two layers of dielectric glass with a Low-E coating that is transparent to visible light but blocks solar radiation in the infrared (IR) and ultraviolet (UV) ranges [1]. By minimizing IR and UV radiation, Low-E glass improves thermal insulation and reduces energy needs for heating and cooling. These advantages make it crucial for building energy conservation [2] and automotive glazing applications [3].

However, the Low-E coating also attenuates wireless signals, posing a challenge for modern 5G/6G communications [4]. The attenuation in the 0.8–6 GHz frequency band is 25–30 dB [5], while, in the 26–40 GHz frequency band, it increases to 25–40 dB [6]. Electromagnetic metasurfaces have emerged as a promising solution to mitigate the attenuation of wireless signals caused by the Low-E coating. Metasurfaces have been widely applied in 5G/6G communications for functions such as reflectarrays [7,8,9], beam steering [10,11], frequency-selective surfaces [12,13], and absorbers [14,15]. The metasurface on Low-E glass is realized by etching periodic patterns onto the Low-E coating. However, the etched area must be constrained as the thermal transmittance of Low-E glass is positively correlated with the etching ratio [16].

The traditional metasurface design methods include the parameter sweeping method and optimization algorithms. The parameter sweeping method relies on the designer’s intuition to heuristically identify candidate patterns, followed by parameter scans to enhance transmission performance. However, since the pattern shape remains fixed throughout the process, a significant portion of the potential design space remains unexplored. The commonly investigated patterns include cross-dipole [5,17], Jerusalem-cross [18], rhombus [19], hexagon [20], and square patch structures [3,6,16,21,22]. In traditional methods, the transmission line (TL) model is commonly used to analyze metasurfaces with substrates such as Low-E glass. This model calculates the optimal equivalent circuit parameters of the metasurface, followed by parameter sweeping based on predefined pattern shapes to complete the design. However, studies based on the TL model exhibit certain limitations: the TL model is often employed for theoretical validation after the optimal structure has been determined [23], and the analyzed structures are fixed, resulting in limited design flexibility [24]. Optimization algorithms, such as adjoint-based topological optimization [25], genetic algorithms [26], and particle swarm optimization [27], have improved the efficiency of the search process compared to parameter sweeping. However, they still suffer from limited design flexibility and a lack of physical interpretability.

Unlike traditional methods, artificial intelligence (AI)-based methods offer a high degree of freedom for metasurface design and can theoretically achieve global optimality [28]. Representative examples include the generative adversarial network (GAN) [29,30] and diffusion probabilistic model [31,32,33]. These methods learn complex mappings from large volumes of training data and subsequently generate metasurface patterns that meet transmission requirements. However, two major limitations persist in these AI-based methods. First, the generated patterns tend to be irregular and discontinuous. Second, the resulting designs lack physical interpretability. Although some studies have explored integrating physical principles or domain knowledge with AI-based methods for metasurface design [34,35], such integration has not yet been extended to metasurfaces on Low-E glass.

In this work, we identify through the TL model that the optimal equivalent circuit for enhancing the transmission coefficient on Low-E glass corresponds to a capacitive structure. This capacitive structure imposes topological constraints on the possible shapes of the unit cell pattern. Building on this insight, we propose a random walk-based search method to explore the design space under these topological constraints. This method yields a metasurface pattern whose transmission coefficient closely approximates the theoretical optimum. Compared to traditional methods and AI-based methods, the proposed method offers a physically constrained search within a meaningful design space while simultaneously ensuring design freedom, etching ratio constraints, and pattern continuity. The simulation results demonstrate that our design outperforms the conventional square patch pattern. To validate the theoretical findings, a prototype Low-E glass sample is fabricated, demonstrating a minimum transmission loss of 2.509 dB within the 24–30 GHz frequency band and a 3 dB bandwidth of 4.28 GHz, fully covering the entire 5G n258 operating range (24.25–27.5 GHz) [36]. Furthermore, the etching ratio is 19.5%.

2. Physically Constrained Search Method

Most patterns used in existing Low-E glass structures consist of square patches [6,21,22] and convoluted square-loop patches [16]. A common feature of these patterns is that the etched sections separate the Low-E coating into isolated blocks. From an equivalent circuit perspective, such metasurfaces exhibit a capacitive circuit behavior [37,38]. Here, we theoretically substantiate the necessity of designing capacitive structures on Low-E glass through the TL model and equivalent circuit. The TL model, which is widely used in the field of wireless communications to model electromagnetic wave propagation in layered media [39], is adopted here for system modeling. The structure of the Low-E glass and its corresponding TL model are illustrated in Figure 1. For the ultra-thin Low-E coating layer, which corresponds to the metasurface, its thickness is approximately 260 nm—much smaller than the signal wavelength of about 10 mm (corresponding to 30 GHz). At this scale, the influence of the metasurface layer on electromagnetic wave propagation is modeled as lumped elements, consisting of a series connection of resistance R, capacitance C, and inductance L [37]. Considering the etching ratio constraint, which limits the pattern complexity, higher-order capacitance and inductance effects are neglected in the equivalent circuit. The glass is treated as a transmission line with intrinsic impedance $Z_r$, and air is approximated as a vacuum. Under normal incidence, the transmission coefficient $\tau$ for both TE and TM polarizations can be calculated using the same expression:

$$M_{\mathrm{LowE}}=M_{\mathrm{glass}}{M}_{\mathrm{pattern}}{M}_{\mathrm{air}}{M}_{\mathrm{glass}}=\left(\begin{array}{cc}A& B\\ C& D\end{array}\right)$$

(1)

$$M_{\mathrm{glass}}=\left(\begin{array}{cc}cos\left({\beta }_{r}h\right)& j{Z}_{r}sin\left({\beta }_{r}h\right)\\ {\displaystyle \frac{j}{Z_r}}sin\left({\beta }_{r}h\right)& cos\left({\beta }_{r}h\right)\end{array}\right)$$

(2)

$$M_{\mathrm{pattern}}=\left(\begin{array}{cc}1& 0\\ Y_{\mathrm{pattern}}& 1\end{array}\right)$$

(3)

$$M_{\mathrm{air}}=\left(\begin{array}{cc}cos\left({\beta }_{0}d\right)& j{Z}_{0}sin\left({\beta }_{0}d\right)\\ {\displaystyle \frac{j}{Z_0}}sin\left({\beta }_{0}d\right)& cos\left({\beta }_{0}d\right)\end{array}\right)$$

(4)

$$Y_{\mathrm{pattern}}={\displaystyle \frac{1}{Z_{\mathrm{pattern}}}}={(R+{\displaystyle \frac{-j}{2\pi fC}}+2\pi fLj)}^{-1}$$

(5)

$${\beta }_0={\displaystyle \frac{2\pi f}{c}},{\beta }_r=\sqrt{{\epsilon }_r}{\beta }_0$$

(6)

$$Z_0=\sqrt{{\displaystyle \frac{{\mu }_0}{{\epsilon }_0}}},Z_r=\sqrt{{\displaystyle \frac{{\mu }_0}{{\epsilon }_r{\epsilon }_0}}}={\displaystyle \frac{Z_0}{\sqrt{{\epsilon }_r}}}$$

(7)

$$\tau ={\displaystyle \frac{2}{A+\frac{B}{Z_0}+C{Z}_0+D}}$$

(8)

where c represents the speed of light in a vacuum, $Z_0$ represents the wave impedance in a vacuum, h represents the thickness of glass, d represents the spacing of air gap, $Y_{\mathrm{pattern}}$ represents the admittance of the metasurface, $Z_{\mathrm{pattern}}$ represents the impedance of the metasurface, f represents the electromagnetic wave frequency, ${\mu }_0$ represents the vacuum permeability, ${\epsilon }_0$ represents the vacuum permittivity, ${\epsilon }_r$ represents the relative permittivity of glass, and ${\beta }_0$ as well as ${\beta }_r$ represent the propagation constants in vacuum and glass, respectively. After substituting all known quantities and applying Equations (1)–(8), $\tau$ can then be determined as follows:

$$\tau =\tau (R,C,L;f)$$

(9)

Equation (9) indicates that $\tau$ depends on the equivalent circuit parameters R, C, and L. Thus, the optimal set $(R_{\mathrm{opt}},C_{\mathrm{opt}},L_{\mathrm{opt}})$ that maximizes transmission coefficient $\tau$ over the specified frequency range $(f_1,f_2)$ can be expressed as

$$(R_{\mathrm{opt}},C_{\mathrm{opt}},L_{\mathrm{opt}})=\underset{R,C,L}{argmax}{\int }_{f_1}^{f_2}\tau (R,C,L;f)df$$

(10)

For example, we assume that the transmission band of interest in this optimization is 24.25–27.5 GHz. Numerical calculations are carried out in MATLAB (R2021a) using the parameters provided in

Table 1. TL model parameters.

Name	Parameter	Value
Thickness of Glass	h [mm]	6.0
Spacing of Air Gap	d [mm]	12.0
Relative Permittivity of Glass	${\epsilon }_r$	$7.216\times (1-0.01597j)$
Vacuum Permittivity	${\epsilon }_0$ [F/m]	$8.8542\times {10}^{-12}$
Vacuum Permeability	${\mu }_0$ [H/m]	$1.2566\times {10}^{-6}$

to determine the optimal set, resulting in the following values:

$$(R_{\mathrm{opt}},C_{\mathrm{opt}},L_{\mathrm{opt}})=(0,29.0\mathrm{fF},0)$$

(11)

$${\tau }_{\mathrm{opt}}\left(f\right)=\tau (R=R_{\mathrm{opt}},C=C_{\mathrm{opt}},L=L_{\mathrm{opt}};f)$$

(12)

Here, $R_{\mathrm{opt}}$ = 0 is intuitive as a lossless material can achieve superior transmission performance compared to a lossy material. Figure 2 presents a two-dimensional contour plot of the transmission capability T as a function of $(C,L)$ within the specified frequency band for $R=0$, where T is defined as

$$T={\displaystyle \frac{1}{{\int }_{24.25G}^{27.5G}df}}{\int }_{24.25G}^{27.5G}\tau (0,C,L;f)df$$

(13)

Figure 2 demonstrates the trend of T as a function of $(C,L)$, indicating the existence of a pair $(C_{\mathrm{opt}}=29.0\mathrm{fF},L_{\mathrm{opt}}=0)$ that maximizes T. In the optimal equivalent circuit parameter set, $L_{\mathrm{opt}}=0$ indicates that the corresponding metasurface structure is capacitive. This result theoretically explains why previous studies predominantly selected patch structures for the unit pattern on Low-E glass.

However, the optimal values of the equivalent circuit alone are insufficient as the mapping between these values and specific metasurface patterns remains a complex and implicit relationship. The capacitive structure imposes topological constraints on feasible pattern candidates, requiring the metasurfaces to consist of isolated patches. The diversity of capacitive patterns can be interpreted through variations in the partitioned boundary curves. The simplest example is a square patch, where the boundary is a straight line. This insight reframes the search for suitable capacitive patterns as a task of exploring different configurations of these boundary curves, as illustrated in Figure 3.

To identify patterns that satisfy the topological constraint of capacitive structures and match the desired capacitance, we propose a random walk-based search method. Considering the polarization characteristics of incident electromagnetic waves, symmetry allows the search to be restricted to a one-eighth segment of the unit cell [40], as shown in Figure 4a. To facilitate the representation of curve shapes, we adopt a discrete cell coding scheme [41], in which each unit cell is divided into a $2a\times 2a$ grid, with unit period p. The random walk-based search is then carried out within this one-eighth region, as detailed in the following steps:

(1) To obtain different curves, we adopt a random walk method. Beginning from the initial point $(1,1)$, the walk proceeds by choosing each step’s direction with probabilities $(p_{\mathrm{up}},p_{\mathrm{down}},p_{\mathrm{left}},p_{\mathrm{right}})$, moving either up, down, left, or right to an adjacent grid point. The four direction probabilities are calculated based on the etching ratio constraint, with the specific calculation provided in Supplementary Materials. Due to symmetry, the walk is constrained by the boundary, defined by the three lines $y=0$, $y=x$, and $x=a$, forming a triangular region. A single-step movement is illustrated in Figure 4b. By continuously performing the single-step walk, a curve that satisfies the etching ratio constraint is generated. Repeating this process produces multiple curves with different shapes.

(2) The curve must satisfy two key constraints. First, it must ensure connectivity by linking $(1,1)$ to the line $x=a$, thereby partitioning the cell into sections compatible with a capacitive pattern structure. Second, it must adhere to an etching ratio constraint, which is reinterpreted as a limit on the curve length within the discrete grid description:

$$l\le {\displaystyle \frac{\gamma a^2}{2}}$$

(14)

where l represents the curve length and $\gamma$ represents the maximum permissible etching ratio.

Here, we set the unit size p to 1.2 mm and determine the grid number $a={\displaystyle \frac{p}{2w}}=20$ based on the etching precision $w=30$ μm, and the etching ratio is set to 20%. Through the random walk-based search method, 8484 candidate patterns were generated and their transmission coefficients were simulated using CST Microwave Studio (version 2021). The Low-E coating is modeled as a thin film with a sheet resistance of 1 Ω/sq. The optimal pattern is shown in Figure 5. The comparison among the simulation results of the transmission coefficient $\tau$ for the searched pattern and the square pattern [21] and the optimal ${\tau }_{\mathrm{opt}}$ parameters described by Equation (12) is shown in Figure 6a. It is evident that the pattern obtained from our search is slightly lower than the theoretical optimal case of the TL model. Potential reasons for this discrepancy include (1) our simulation employs actual lossy material; (2) in practice, the metasurface is not purely capacitive and exhibits a small inductance value. However, the simulation results of the designed pattern closely align with the theoretical optimum obtained from the TL model in the considered transmission band, indicating that our search method is both feasible and effective. Compared to fully coated Low-E glass, the Low-E glass with the metasurface pattern obtained through the search method exhibits a transmission coefficient that is at least 30 dB higher across the entire frequency band, as shown in Figure 6b. The proposed random walk-based search method can be viewed as an enumeration approach, with its key advantage lying in the use of free-form curves to represent capacitive metasurfaces. Future improvements may involve integrating this method with heuristic optimization algorithms to enhance search efficiency.

To further demonstrate the effectiveness of the physical constraints applied in our search method, we perform a Monte Carlo-based estimation to quantify the design space under different conditions. In a completely unconstrained scenario where each of the $2a\times 2a=40\times 40$ grid cells is treated as a binary variable, the total number of possible binary patterns reaches an astronomical order of $2^{1600}$, rendering brute-force search infeasible. Even when the search is limited to free-form patterns within one-eighth of the unit cell, the number of candidate patterns remains on the order of $2^{200}\approx {10}^{60}$. By imposing the physical constraint that the pattern must represent a capacitive structure, a continuous and isolated patch topology, effective design space is significantly reduced. Monte Carlo-based estimation shows that the number of feasible connected curves satisfying this capacitive condition is approximately $4\times {10}^{17}$, which is several orders of magnitude smaller than the unconstrained case. Detailed estimation procedures are provided in the Supplementary Materials. This substantial reduction in search space makes the search not only computationally feasible but also physically meaningful. These results highlight the importance and efficacy of introducing physical constraints in the metasurface design process.

3. Fabrication and Measurement

To experimentally validate the performance of Low-E glass and its compatibility with the manufacturing process, a prototype glass with dimensions of $300\mathrm{mm}\times 300\mathrm{mm}$ is fabricated, consisting of $250\times 250$ unit cells. The sample size is chosen to approximate the periodic boundary conditions. The unit pattern is shown in Figure 7a, and the etching pattern is smoothed to facilitate industrial-scale production, achieving an etching ratio of 19.5%. The simulation results indicate that the transmission performance of the smoothed pattern is comparable to that of the originally searched pattern, as shown in Figure 7b. The manufactured prototype glass is displayed in Figure 8.

To verify the transmission properties of the fabricated glass, we establish the setup shown in Figure 9. Two horn antennas are mounted at either end of a guide rail for precise alignment. A vector network analyzer (Agilent N5247A) connects to the antennas, which serve as the transmitter and receiver of RF signals. The antennas are separated by 60 cm to satisfy the far-field condition, with the Low-E glass placed at the midpoint. The entire measurement system is situated in a microwave-resistant chamber composed of wave-absorbing materials to prevent interference from stray waves. The results are normalized to the reference results of the air window.

The CST-simulated and experimentally measured transmission coefficient $\tau$ values of the prototype Low-E glass are shown in Figure 10. The amplitude decrease and frequency drift in the measurement can be attributed to two factors: (1) The fabrication tolerances with regard to glass thickness and spacing. Since the wavelength of the band is in proximity to the glass spacing and glass thickness, an effect analogous to that of a Fabry–Pérot cavity is produced [6]. (2) The fabrication tolerances of etching width. Despite these non-idealities, the prototype glass exhibits highest transmission of −2.509 dB at 24.8 GHz and achieves 3 dB bandwidth of 4.28 GHz.

4. Discussion

4.1. Angular Analysis

In the mmWave band, propagation is overwhelmingly governed by the direct line-of-sight (LoS) component, while multipath contributions are typically weak and can be neglected [42]. Therefore, improving transmission under normal incidence directly strengthens the primary communication link, which is the most critical factor for practical deployments. Figure 11a presents the measured oblique incidence transmission coefficient of the prototype glass. The results show that the prototype maintains stable transmission within 0–20°, retaining 88% of the normal incidence 3 dB bandwidth across 24.25–27.5 GHz. This confirms the prototype’s robustness under realistic angular variations.

To further evaluate the transmission performance of the prototype glass relative to commonly used building glass, a bare glass with identical thickness and air spacing but without the Low-E coating was fabricated. A comparison with Low-E glass (without metasurface) is not included because its transmission under oblique incidence is significantly lower, making such a comparison less meaningful. The reason for choosing bare glass as a reference is that modern building windows are largely composed of glass, and the fact that the prototype glass can achieve transmission performance under oblique incidence comparable to bare glass is of practical significance. To simplify the analysis, the enhancement of TE polarization is used to describe the difference in communication performance of the prototype glass compared to the bare glass [23], as illustrated in Equation (15).

$$Enhancemen{t}^{\mathrm{TE}}={\tau }_{\mathrm{metasurface}}^{\mathrm{TE}}\left(\mathrm{dB}\right)-{\tau }_{\mathrm{bare}}^{\mathrm{TE}}\left(\mathrm{dB}\right)$$

(15)

The enhancement in a wide angular range ($\theta =0^{\circ }-{50}^{\circ }$) is shown in Figure 11b. Figure 11b illustrates that, in a wide angular range, the prototype glass exhibits attenuation of no more than 3 dB compared to bare glass within the 24.25–27.5 GHz frequency band. The corresponding simulation and experimental comparisons at each incidence angle are provided in the Supplementary Materials. Furthermore, in certain frequency ranges, the prototype glass demonstrates better communication performance than bare glass, indicated by an enhancement greater than 0 dB. These results provide experimental evidence that Low-E glass with a metasurface can achieve communication performance comparable to that of bare glass. A preliminary TL model for oblique incidence is provided in the Supplementary Materials, and future work will incorporate oblique incidence transmission performance into the design methodology.

4.2. Optical Transparency and Thermal Insulation

Compared with bare and Low-E glass, the prototype glass maintains a comparable level of visible-light transmittance. The measured results of bare glass, Low-E glass (without metasurface), and the prototype glass are shown in Figure 12 and

Table 2. Measured transmittance of visible light, IR, and UV.

Glass Type	Visible Light	IR	UV	IR Blocking Compared to Bare Glass	UV Blocking Compared to Bare Glass
Bare Glass	79.5%	53.4%	72.4%	0	0
Low-E Glass	62.8%	1.9%	23.5%	51.5%	48.9%
Prototype Glass	64.2%	14.7%	30.9%	38.7%	41.5%

. Compared to bare glass, the prototype glass exhibits only a modest reduction of 15.3% in visible-light transmittance while achieving reductions of 38.7% and 41.5% in IR and UV transmittance, respectively. Furthermore, the prototype glass shows infrared and ultraviolet blocking capabilities comparable to those of Low-E glass (without metasurface). These results demonstrate that the prototype glass not only inherits the thermal-insulation functionality of Low-E glass but also preserves high visible-light transparency, which is of great significance for practical building applications.

4.3. Methodological Comparison and Computational Complexity

Different metasurface designs for Low-E glass exhibit varied transmission coefficients due to differences in transmission bands, Low-E coating materials, and the thickness and permittivity of the glass, as shown in

Table 3. Comparison of different metasurface designs on Low-E glass.

Ref.	Meta Layer	Glass Layer	BW [GHz]	−3 dB BW [GHz]	Highest Transmission Coefficient	Applied Design Method	Design Complexity
[5]	1	1	0.8–6	0.4 *	−9.0 dB @ 1.40 GHz	Parameter Sweeping	$\mathrm{O}\left({10}^3\right)$
[6]	1	2	26–40	2.0 *	−2.0 dB @ 37.5 GHz	Parameter Sweeping	34,944
[21]	1	2	25–31	1.4 *	−3.0 dB @ 25.0 GHz *	Parameter Sweeping	$\mathrm{O}\left({10}^2\right)$
[16]	1	2	0–6	3.2 *	−1.28 dB @ 2.35 GHz	Parameter Sweeping	$\mathrm{O}\left({10}^3\right)$
[4]	3	2	1–6	3.0 *	−2.23 dB @ 3.2 GHz *	Parameter Sweeping	$\mathrm{O}\left({10}^3\right)$
This work	1	2	24–30	4.28	−2.509 dB @ 24.8 GHz	Physically constrained Search	8484

* Approximate value. BW: bandwidth.

. These are comparative works that involve the fabrication of Low-E glass prototypes. Regarding the “design complexity” in the table, it reflects the number of CST simulation runs required as simulation time constitutes the main bottleneck in the design process. If the original work reports specific parameter scanning settings, the complexity is provided as the exact number. Otherwise, it is estimated by multiplying 10 for each design variable dimension. Our prototype demonstrates an excellent −3 dB bandwidth, meeting the high-bandwidth requirements of 5G communications.

To compare the efficiency of different design methodologies, we evaluate the computational complexity in terms of total design time:

(1) Conventional parameter sweeping methods rely solely on CST simulations, with total design time as follows:

$$T_{\mathrm{conv}}=N_{\mathrm{sim},\mathrm{conv}}\times t_{\mathrm{sim}}$$

(16)

where $N_{\mathrm{sim},\mathrm{conv}}$ is the number of simulation runs and $t_{\mathrm{sim}}$ is the average time per simulation.

(2) AI-based inverse design methods involve three stages. First, a dataset is generated via full-wave simulations:

$$T_{\mathrm{data}}=N_{\mathrm{sim},\mathrm{data}}\times t_{\mathrm{sim}}$$

(17)

Second, model training typically requires tens of hours on GPU, depending on network size and dataset scale. For example, a diffusion-based model requires about 92 h of training [31], while a TNN model requires only 24 h [43]. Third, the inverse design (prediction) is fast, taking less than a minute per design: 71.2 s for the diffusion-based model [31] and 0.023 s for the TNN model [43]. Therefore, the total design time is dominated by dataset generation and training:

$$T_{\mathrm{AI}}\approx T_{\mathrm{data}}+T_{\mathrm{train}}$$

(18)

(3) Proposed method combines CST simulations with a random walk-based search implemented in MATLAB:

$$T_{\mathrm{ours}}=N_{\mathrm{sim},\mathrm{ours}}\times t_{\mathrm{sim}}+T_{\mathrm{MATLAB}}$$

(19)

where $T_{\mathrm{MATLAB}}$ is the additional search overhead, approximately 5.6 s, which is negligible compared to CST simulation time.

All three methodologies share the simulation component $N_{\mathrm{sim}}\times t_{\mathrm{sim}}$, with $t_{\mathrm{sim}}\approx$ 35 s per run. Therefore, this component depends solely on the number of simulations, which corresponds to the “design complexity” listed in Table 3. AI-based methods additionally require a large dataset. For example, publicly available datasets used in [31,43] contain 174,483 samples, making dataset construction more time-consuming than our method. The time consumption of each methodology is summarized in

Table 4. Time consumption of each methodology.

Method	Dominant Cost Term	Total Time
Parameter Sweeping	$T_{\mathrm{conv}}$	$N_{\mathrm{sim},\mathrm{conv}}\times$ 35 s *
AI-based (diffusion)	$T_{\mathrm{data}}+T_{\mathrm{train}}$ = 174,483× 35 s + 92 h	1788 h
AI-based (TNN)	$T_{\mathrm{data}}+T_{\mathrm{train}}$ = 174,483 × 35 s + 24 h	1720 h
Proposed Method	$N_{\mathrm{sim},\mathrm{ours}}\times t_{\mathrm{sim}}+T_{\mathrm{MATLAB}}$ = 8484 × 35 s + 5.6 s	82 h

* $N_{\mathrm{sim},\mathrm{conv}}$ corresponds to the “design complexity” listed in Table 3.

. Consequently, the proposed method achieves a total computational complexity comparable to conventional parameter sweeping methods while remaining significantly lower than AI-based approaches.

5. Conclusions

In conclusion, our work proposes a physically constrained design method for wireless signal transparency metasurfaces on Low-E glass. Our method addresses the limitations of previous parameter sweeping methods that rely on fixed patterns and empirical intuition, and it clarifies the common physical characteristics underlying the previously adopted patterns. It provides both a theoretical basis and a practical methodology for designing transmission-optimized metasurfaces on Low-E glass.

We first model the transmission problem using the TL model, representing the metasurface as a series-equivalent circuit. Through numerical calculation, we obtain the optimal equivalent circuit parameters, revealing that the optimal transmission structure corresponds to the capacitive structure. Based on the topological characteristics of capacitive structures—namely isolated block-shaped patterns separated by boundary curves—we introduce a random walk-based search method to generate various boundary curves while ensuring both topological and etching ratio constraints. To demonstrate the scalability of our method, we present design results under rhombic and hexagonal periodic shapes in the Supplementary Materials.

The designed pattern demonstrates excellent communication performance, achieving over a 30 dB transmission improvement compared with full-coating Low-E glass and a 0.45 GHz wider 3 dB bandwidth than the conventional square patch pattern. The fabricated prototype exhibits a peak transmission coefficient of −2.509 dB at 24.8 GHz, fully covering the 5G n258 band with a total 3 dB bandwidth of 4.28 GHz and an etching ratio of 19.5%. It also maintains stable performance under oblique incidence, comparable to glass without Low-E coating, demonstrating the practical applicability of the proposed design.

By addressing the signal attenuation issue of Low-E glass, this work facilitates broader adoption of Low-E glass in both architectural and automotive applications, as well as 5G communications, contributing to energy efficiency and environmental sustainability.