Frequency-selective surfaces (FSSs) comprising periodic metallic patches or hole elements have been extensively studied for distinct electromagnetic applications, such as bandpass filters that selectively transmit, reflect, or absorb electromagnetic waves [1–3]. Metallic elements can determine the resonant response to electromagnetic radiation, preventing or allowing the transmission of selective frequencies. Because FSSs have been intensively applied to numerous applications such as radomes [4–6], wave polarizers [7], efficient communication systems [8,9], and reflectors [10,11], design advancement is essential for using periodic metallic arrays with variously shaped elements.

FSS-integrated optical windows have recently attracted significant interest in developing advanced military radar and communication systems. In particular, FSSs operating at GHz frequencies can be used for smart weapons, defense systems, and stealth aircraft, where an appropriate GHz band can be selected for each purpose. FSSs have been mostly studied for flat substrates where periodic structures are integrated. However, curved FSSs are suitable for various practical applications, such as conformal radomes and antennas [2,12–14]. Indeed, the design of curved FSSs is much more complicated than that of flat FSSs owing to the unique geometry of the curved lattice and the increasing number of computational tasks. Therefore, developing a design methodology for investigating the electromagnetic properties of curved FSSs is required.

In this study,we designed various dome-shaped FSSs and theoretically analyzed their bandpass/bandstop characteristics depending on the lattice design and element parameters. Furthermore, the tunability of the operating frequencies was confirmed by controlling the periodicity and diameter of the elements. Moreover, electromagnetic analysis was conducted to determine the high selectivity at the desired frequencies to transmit or reflect by varying the period, diameter, and radius of curvature.

FSS structures comprise circular disk-shaped metal patches or a metal film with hole arrays on curved surfaces (hemispherical dome substrates) with a curvature radius (R) on the millimeter scale. The calculated arrays have square and triangular lattices with patch and hole elements. First, the scripts for modeling the structure on the curved surfaces were first contrived to consider the periodically arranged elements (patches or holes). The periodicity of the array was defined as the length of the arc of the hemispherical dome. Moreover, the positions of the elements were determined using the following parameters: x, y, z, and angles to be located perpendicular to the curved surface (θ, ϕ). Because the dispersive metallic properties were less relevant at lower frequencies than UV-Vis-IR regimes, herein, the materials of the patch and film were set by a perfect electrical conductor that showed the ideal metal nature of zero absorption and 100 % reflection. Note that three-dimensional FSSs have been studied using the finite-difference time-domain (FDTD) software (Lumerical FDTD solution), generally used to model electromagnetic responses with timedependent propagating waves.

3.1. Design of curved FSS

Curved FSSs comprising circular hole elements and a triangular lattice array on hemispherical dome substrates are shown in Fig. 1(a). The modeling of curved FSSs is explained in detail in the experimental section. Moreover, Fig. 1(b) shows the information regarding the configurations of the square and triangular lattices with the patch and hole elements in the top panels, presenting the metallic area (yellow parts) and substrate (blue shaded area and white parts). In two-dimensional FSSs, the design of the arrays to be calculated considers the position with the distance between the elements (x, y; periodicity) and the size of the elements. However, the design of three-dimensional FSSs is complicated because the angles (θ, ϕ) and heights (z) of elements located perpendicular to the curved surfaces must be considered. The bottom panel of Fig. 1(b) shows the parameters that must be considered. Based on widely used lattice design and elements, we have focused on the four types of configurations, including (i) triangular lattice of patch array, (ii) triangular lattice of hole array, (iii) square lattice of patch array, and iv) square lattice of hole array, where the curvature radius (R) was 225 mm. As shown in Figs. 1(c) and 1(d), each optical response with the same element parameters (p = 8 mm and d = 4 mm) extracted through electromagnetic analysis can be engineered for the transmission and shielding properties. A bandpass feature was observed in the hole arrays, whereas a bandstop feature was observed in the patch arrays. Figure 1(c) shows the dependence of the element type (patch or hole) on the triangular lattice. Spectral shapes between the patch and hole element arrays, which were almost inverted, were observed. Babinet’s theory can explain this observation [1,15–18] in that the theory states that patch and hole arrays with similar-sized elements complement each other. Subsequently, the reflection coefficient for one array equals the transmission coefficient of the other, and their optical responses are opposite. Figure 1(d) shows the selectivity between particular frequencies by engineering the lattice structures because they determine the width and intensity of the transmittance band. In both lattice array structures, low transmission of the X-band (~10 GHz) and high transmission of the Ku-band (~15 GHz) were observed; nevertheless, the triangular lattice array exhibited a higher selectivity [T(15 GHz)/T(10 GHz)] than that of the square lattice array. Thus, subsequent electromagnetic simulations were performed using triangular lattice structures for applications in optical filters with high selectivity.

Figure 1. (a) Illustrations of hemispherical FSS with the triangular lattice of circular hole elements with a diameter of d. R is the curvature radius for the hemispherical dome. (b) (top panels) Various configurations showing the square lattice of the patch array and the triangular lattice of the hole array; the yellow area indicates the metallic material. (bottom panel) Definition of the position and period for the array elements in a three-dimensional coordinate. (c) Transmittance spectra for patch (black) and hole (red) arrays on the hemispherical dome substrate with the same triangular lattice structure (p = 8 mm, d = 4 mm). (d) Transmittance spectra for triangular (navy) and square (magenta) lattices (p = 8 mm, d = 4 mm). For (c) and (d), The curvature radius (R) is 225 mm.

3.2. Frequency responses for varying periods and diameters

In Fig. 2, we present the calculations of the transmittance characteristics by changing the period and diameter of the hole and patch elements arranged in a triangular lattice with a curvature radius (R) of 225 mm. First, for the triangular hole lattice arrays, the period- and diameter-dependent transmittances were calculated for a fixed hole diameter of 6 mm and a fixed period of 10 mm. When the period of the array is decreased from 12 to 8 mm, as shown in Fig. 2(a), the resonance peak shifts to higher frequencies, and the overall transmittance and bandwidth increase in the bandpass frequency region. As the diameter of the hole element decreased from 8 to 4 mm, the resonance peak also shifted to higher frequencies, and the transmittance and bandwidth decreased, as shown in Fig. 2(b). The changes in the transmittance and bandwidth can be explained by the proportion of the metallic area to the hole area. The decrease in period decreases the metallic area over the total surface of the structures, whereas the smaller diameter of the hole element increases the metallic area. Moreover, the result confirmed that the decrease in feature sizes (period and diameter of the element) resulted in a blue-shift of the resonance peaks. For the triangular lattice of the patch array, the period- and diameter-dependent transmittances were also investigated for a fixed hole diameter of 4 mm and a fixed period of 8 mm. As the length of the period decreased from 12 to 6 mm, the resonance frequency and bandwidth of the transmission dips increased; nevertheless, the overall transmission decreased, as shown in Fig. 2(c). When the diameter of the circular disk-shaped patches decreased from 4 to 1 mm, the transmission increased, as shown in Fig. 2(d). However, for a 1-mm-sized patch array, the transmittance exhibited a featureless spectrum in the frequency range of interest, indicating a meaningless size for the FSSs. In contrast to the hole arrays, a decrease in the period of the patch array reduces the open area, resulting in an overall low transmission. Furthermore, the smaller size of the elements and periods shifts the bandstop frequencies toward higher frequencies, as shown in Figs. 2(c) and 2(d). Although the spectral shape of the transmittance between the hole and patch arrays for the period- and diameter-dependence is inverted, the resonance frequency shift depends on the size of the periods and elements regardless of the type of hole or patch. Because these characteristics are also observed in the square lattice (data not shown), this observation demonstrates that changes in the period and diameter of the elements strongly affected the control of operating frequencies and transmittance for bandpass and bandstop. However, it is not significantly related to the lattice structure, whether it is a square or triangular lattice.

Figure 2. Calculated transmittance characteristics for triangular lattice arrays with the curvature radius (R) of 225 mm. (a) Period-dependent spectra of hole arrays with d = 6 mm. (b) Diameter-dependent spectra of hole arrays with p = 10 mm. (c) Period-dependent spectra of patch arrays with d = 4 mm. (d) Diameter-dependent spectra of patch arrays with p = 8 mm.

3.3. Engineering bandpass characteristics

To investigate bandpass characteristics, such as bandwidth and selectivity, we focused on the triangular lattice of the hole array structures. The selectivity of particular frequencies can be determined by the bandwidth and transmittance. If the selected frequencies are located in close frequency ranges, a narrow and steep bandwidth is required to produce a high transmission contrast between the selected frequencies. First, we studied the bandpass characteristics depending on the ratio of the period to diameter (p/d). As the p/d ratio increased from 1.2 to 2.0, the bandwidth became narrow, but the transmittance decreased, as shown in Fig. 3. Therefore, the transmittance and bandwidth have a trade-off relationship and can be tuned using the p/d ratio. Second, the effect of the curvature radius (R) of the curved FSSs on the bandpass properties was investigated because the curved structural design could induce a large difference depending on the curvature radius. In Fig. 3, the curvature radius varies from 80 to 160 mm steps of 20 mm under the normal incidence of the Gaussian beam. With an increase in the curvature radius, the transmittance increases, and the resonance peak slightly blue shifts without changing spectral features. Note that the p/d ratio significantly affected the bandwidth of curved FSSs, while the overall transmittance is determined by both the p/d ratio and the curvature radius of the curved FSS structures. Therefore, an appropriate p/d ratio should be considered to design curved FSSs with high selectivity in the radio frequency range. From our results, the highest selectivity of the Ku-band (~15 GHz) to the X-band (~10 GHz) was obtained for a p/d ratio of 1.5, as shown in Fig. 3(b).

Figure 3. Calculated transmittance for the different p/d ratios of (a) 1.2, (b) 1.5, (c) 1.6, and (d) 2.0 through the triangular lattice of the hole array for different radii of curvatures. The selectivity for bandpass characteristics can be tuned by varying the p/d ratio and radius of curvature.

We designed millimeter-scale three-dimensional curved FSS structures on hemispherical dome substrates and theoretically analyzed the optical filter characteristics for two elements (patch and hole) and lattice types (square and triangular) using FDTD methods. The transmission band could be controlled by varying the length of the period and diameter of the elements. When the period of the hole (patch) array decreased, the transmittance increased (decreased) in the frequency range of interest, and the bandwidth for both bandpass and bandstop characteristics broadened. As the diameter of the hole (patch) decreased, the overall transmittance decreased (increased). However, the operating frequencies were blue-shifted by reducing the size of holes and patches regardless of the element type. As the p/d ratio increased, the spectral features changed significantly, indicating that the bandwidth became narrow and the transmittance decreased. When the curvature radius increased, only the transmittance monotonically increased with negligible changes in the spectral features. Therefore, curved FSSs can be designed to exhibit proper spectral shapes with high selectivity for specific frequencies by considering the p/d ratio and curvature radius. These results provide insights into possibilities for improving the performance of optical filters with high selectivity at desired frequencies for practical applications, such as aircraft radomes and antenna systems.

This study was supported by the Agency for Defense Development of the Korean Government (UD200044GD).

The authors declare no conflicts of interest.