Fig. 1 depicts the proposed FSS structure in detail. The structure of the C-sandwich model comprises three dielectric layers and two foam layers. To achieve wideband characteristics, the FSS arrays are etched on either side of the core substrate. The top FSS array is composed of a wire-gridded square-loop element, and the bottom FSS array consists of square patch elements with the same periodicity D. The width of the wire grid is w₁, the square loop has a length of l₁ and a width w₂, while the gaps between the bottom square patch are of width s₁.

According to equivalent circuit theory [13], when FSS elements are excited by incident EM waves, the induced current is formed on conductive FSS elements. In Fig. 2, the metallic lines can be modeled as equivalent inductances, and the gaps can be modeled as equivalent capacitances. The equivalent circuit of the top arrays is presented in Fig. 2(a), showing an inductance L₀ connected in parallel with a series LC resonator (L₁ and C₁). Meanwhile, Fig. 2(b) indicates that the bottom arrays can be modeled as capacitance C₂.

As discussed in [9], the frequency response of an FSS at oblique incidence can be improved by bonding a multilayer dielectric composed of low and high permittivity layers to FSS arrays. Therefore, as shown in Fig. 1, a skin substrate (h₁ in thickness) with high permittivity ɛ_skin and a foam substrate (h₂ in thickness) with low permittivity are used to construct the outer dielectric layer. To simplify the model, the foam layers were replaced by air layers.

As for C-sandwich impedance matching, the high-permittivity skin and low-permittivity foam/air layers act as an impedance transformer around the FSS core. This reduces the impedance mismatch of obliquely incident TE/TM waves, thereby mitigating the typically observed angle-induced shift and narrowing of the passband. Equipped with the core FSS substrate (h₃ in thickness), the proposed C-sandwich model can successfully exhibit wideband properties that remain stable at different incident angles and polarizations. According to transmission line theory, the dielectric substrate can be modeled as a short transmission line. Therefore, using the transfer matrix method [13], the transmission characteristics of the proposed model can be calculated using its transmission line model, as shown in Fig. 3, where Z₀ refers to the wave impedance in free space.

To obtain the dimensions of the proposed FSS structure for the desired wideband performance, the following design procedure was adopted:

With regard to Step 1, the equivalent circuit of the proposed FSS model is shown in Fig. 2. As mentioned above, the circuit model of the top FSS array comprises a parallel connection between the inductance L₀ and the series LC resonator (L₁ and C₁), and the bottom array is modeled as capacitance C₂.

In Step 2, drawing on the study of basic FSS elements in [13], we calculated the equivalent impedance of the proposed FSS structure for different incident angles and polarizations. Based on the equivalent circuits presented in Fig. 2, the impedances of the top FSS array ZtopTE,TM and the bottom FSS array ZbotTE,TM were obtained as follows:

where ξ = 1 − sin²θ/2ɛ_eff is the influence factor [15] of incident angle θ. Furthermore, based on the equations in [13], the circuit parameters noted in Fig. 2 were derived as follows:

The geometrical parameters in the above equations were determined based on Fig. 1, with ɛ₀ and μ₀ being the electromagnetic parameters of free space. Notably, the effective permittivity ɛ_eff of the metallic arrays on the dielectric medium can be defined as follows:

where ɛ_core is the permittivity of the core substrate.

In Step 3, based on the transmission line model shown in Fig. 3, the transmission ABCD matrix was constructed as following:

where

for the different polarizations (TE/TM), with

Here, ZiTE,TM is the characteristic impedance, ɛ_i is the relative permittivity, and h_i is the thickness of the different layers in Fig. 1. Furthermore, k_t is the transverse vector with incident angle θ, k₁ is the propagation constant in the substrate, and k₀ is the wave number in free space.

The transmission matrix of the FSS layer in the Fig. 3 under TE and TM polarizations can be represented as:

where

Subsequently, drawing on the ABCD matrix presented above, the transmission characteristics of the proposed FSS were derived. In this context, the analytical transmission coefficients S₂₁ for TE/TM polarization can be expressed as follows [13]:

where Z₀ is the intrinsic wave impedance in free space.

In Step 4, considering the proposed structure’s practical application, the relative permittivity of the skin substrate and core substrate were determined to be 4.4 and 2.65, respectively. Furthermore, to determine the unknown geometrical parameters in Fig. 1—D, w₁, l₁, w₂, s₁, h₁, h₂, and h₃—we applied the GA curve-fitting method to samples of the desired frequency response. Selected sampling frequencies and their corresponding |S₂₁| values are listed in Table 1, demonstrating the bandpass response in the C and X bands. Furthermore, to ensure good transmission performance of the FSS over different incident angles and polarizations, weighting factors were introduced into the adaptive function as follows:

where r indicates the TE and TM polarizations, m refers to the number of frequency points, n denotes the number of angle points, W_TE,TM is the weighting factor for TE/TM polarization, and W_n is the weighting factor for the different incident angles. Furthermore, |S21TE,TM|(m,n) is the magnitude of the transmission coefficients at different frequencies and angles, which were derived from Step 3, and |S_{21 0} |(m) represents the target |S₂₁| values sampled at different frequencies (in Table 1).

To strike an appropriate balance between model accuracy and optimization complexity, the frequency samples listed in Table 1 were selected to span the mid-band region of the passband, as well as its lower and upper edges. A higher sampling density was applied in the transition regions, and an additional 1–2 out-of-band samples were included to ensure adequate suppression constraints. The target |S₂₁| values were set based on the system specification and loss budget: ≥ −2 dB in-band (flat-top), −3 to −5 dB at band edges to control the slope, and ≤ −20 dB out-of-band. Notably, the same targets were used for TE/TM and for θ = 0°, 25°, and 50°. Meanwhile, the weights (W_TE = W_TM = 1) prioritized on-axis performance while also enforcing oblique stability. We kept the sample set compact (13 frequencies × 3 angles × 2 polarization), refining it locally only when the residuals clustered near an edge, thereby limiting GA complexity and improving reproducibility.

For the proposed design, we considered W_TE = W_TM = 1 (assigning equal weight to both TE and TM polarizations to ensure wideband transmission performance), W_n = 1(θ ≤ 50°), and W_n = 0.1(θ > 50°) (to ensure high transmission performance for oblique incidence θ = 0°–50°). The GA was implemented to optimize the structural parameters of the FSS, with the searching ranges set to 4 mm ≤ l₁ < D ≤ 10 mm, 0.1 mm ≤ w_1,2 ≤ 3 mm, 0.1 mm ≤ s₁ ≤ 3 mm, 0.5 mm ≤ h_1,3 < h₂ ≤ 5 mm. The stopping criteria of the GA [16] were as follows: the population size is 100, the maximum number of generations is 100, the number of stall generations is 50, the crossover fraction is 0.75, and the function tolerance is e⁻⁶. Ultimately, using GA optimization to minimize the adaptive function (17), the optimal unknown geometrical parameters were obtained (see Table 2).

In Step 5, the ECM data obtained in the previous step were employed as the initial values to build the FSS model in AN-SYS HFSS. By conducting a parameter sweep in the simulation software, the wideband response of the proposed structure was further optimized. Notably, a comparison of the optimization results and the ECM data, presented in Table 2, showed close agreement, indicating the potential for significant time savings in the initial stages of FSS design. In addition, the effects of the different dimension parameters could be easily determined from the ECM equations, providing significant benefits for the HFSS parameter sweep.

Based on Step 4, the optimization range of the incident angle was 0°–50°. Fig. 4(a) illustrates the computed transmission coefficient |S₂₁| (θ = 0°, 25°, and 50°) for the structural dimensions obtained using ECM based on the analysis presented in Section II with regard to the sampling points in Table 1. The results confirm that the resonant frequencies of the response are close to the desired sampling points, which indicates that the ECM results closely match the expected broadband response.

Fig. 4(b) presents the simulated |S₂₁| (θ = 0°, 25°, and 50°) results obtained in Step 5 by performing HFSS using the optimized dimensions listed in Table 2, along with the results for TE and TM polarizations. The results confirm that the designed FSS guarantees the transmission of EM waves (transmission efficiency is more than 80%) even when the incident angle is 50° in the C and X frequency bands.

Fig. 4 also shows that the response curve for ECM is slightly different from that obtained for HFSS at high frequencies, which may be attributed to attributed to the fact that the coupling capacitance between the top and bottom FSS arrays are not accounted for in ECM. Furthermore, the accuracy of the equivalent circuit formulations is lower than that of the full-wave simulation. We intend to address these problems in our future work.

Nevertheless, the optimal HFSS values are in close agreement with the dimensions calculated by the ECM. The satisfactory promising simulation results indicate that the proposed FSS design is fit for broadband applications.

A prototype of the proposed structure was fabricated and measured. The FSS arrays were fabricated on an F4B-2 core substrate (see Fig. 5) characterized by permittivity of 2.65 and a loss tangent of 0.001. The skin substrate was composed of FR4, which had a permittivity of 4.4 and loss tangent of 0.02. The skin and core substrates were separated by medium washers and screws to construct the air layers, as shown in Fig. 5. For the dimensions of the FSS elements, we adhered to the final optimized data provided by the HFSS, as noted in Table 2.

To validate the functionality of the fabricated miniaturized FSS prototype, a measurement framework (see Fig. 6) was employed. The physical dimensions of the FSS arrays were 225 mm × 225 mm, which means that the number of elements was 30 × 30. The measurement setup comprised two broadband horn antennas for transmitting and receiving EM waves. To measure the transmission coefficient of the FSS, we first calculated the |S₂₁| between the horn antennas in the absence of the FSS to generate the calibration data. Following this, we measured the |S₂₁| between the two horns in the presence of the FSS under test. Finally, the |S₂₁| of the FSS was obtained by normalizing the |S₂₁| obtained in the second measurement to the calibration data.

Limited by the experimental conditions, the measurement spanned the frequency range of 2–16 GHz. In Fig. 7, the measured |S₂₁| results are compared with the simulated results obtained from HFSS by varying the angle and polarization of incidence while maintaining the same structural parameters. It is evident that the measured results agree well with the simulated ones from the HFSS, indicating a passband of 4–12 GHz, with |S₂₁| ≥ −2.0 dB, and a stopband after 12.0 GHz, with |S₂₁| ≤ −20 dB. Moreover, the FSS remains stable even when the angle of incidence increases to 50°. Fig. 7 also shows small ripples across the range of the measured frequency responses, possibly caused by truncation and assembly errors between the transmit and receive antennas. To mitigate this issue and improve the accuracy of the measurement results, we conducted repeated tests and adjustments—optimizing the alignment of the transmit and receive antennas, reducing truncation effects by refining the edge treatment of the FSS prototype, and calibrating the measurement system multiple times—to effectively minimize the generation of these small ripples. Overall, despite these minor residual fluctuations, the overall trend of the measured frequency response remains consistent with the simulated results, and the key performance indicators of the FSS radome are still satisfied, further confirming the feasibility of the proposed design.

Table 3 provides a comparison of the proposed FSS design with those previously reported in the literature [9, 10]. The superior features achieved by the former compared to those of the latter are as follows: