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An ultra-fast method for designing holographic phase shifting surfaces



In this section, we conduct an error analysis of our semi-numerical methodology, and in turn, we validate it with the design of a 1-D beam-steerable antenna system.

Error analysis

To validate our design approach a thorough study is conducted by designing holographic PSSs utilizing traditional metallic resonators (e.g., complementary square ring, square patch, square ring), and we compare our results with the results observed when the conventional design methodology (see “Conventional PSS design methodology” section) is employed.

We start our analysis with the three-layer PSS unit-cell of Fig. 2a that consists of three complementary square ring (CSR) resonators. Notably, the frequency of operation is chosen at 30 GHz. Also, we arbitrarily choose to design a PSS that steers a broadside beam excited by a feed antenna at angle \(\theta _t=33.7^{\circ }\). Based on this \(\theta _t\) value, using (1), the phase progression, \(\phi _p\), is equal to \(60^{\circ }\). For this \(\phi _p\) value, to achieve a \(360^{\circ }\) phase range coverage, \(n=6\) unit-cells need to be used. Following the sub-wavelength principle of operation of metasurfaces, the unit-cell periodicity is chosen at \(S=3\) mm (this is \(\lambda _0/3.33\) at 30 GHz). Also, the width w of the slot, etched in the complementary square ring (see Fig. 2a), is kept constant at 0.1 mm for this analysis. Finally, two Duroid/Rogers 5880 (\(\epsilon _r=2.2\) and \(tan\delta =0.0009\)) substrates are used with a height of \(h=2\) mm (this height is equal to \(\lambda _0/5\) at 30 GHz) each. For this example, the sizes of the complementary rings change by varying \(a_n\) from 0.2 to 2.5 mm with an increment of 0.025 mm (\(M=93\) variations for each layer). Notably, the total number of different combinations of rings for this example is equal to 804,357 (that is \(M^N\) = \(93^3\)), therefore, 804,357 full-wave simulations need to be run with the conventional approach to fully characterize the performance of this PSS. These results are used as reference data to evaluate the accuracy of our approach. For our method, as explained above, only \(3 \times 93 = 279\) full-wave simulations are needed. Notably, this number can be even reduced to \(2 \times 93 = 186\) full-wave simulations since the top and bottom layers, in this example, are identical. Therefore, the electromagnetic properties of the bottom (top) layer, for example, can be obtained by simply inverting the ABCD matrix of the top (bottom) layer. Then, we use the multiplication property of ABCD parameters for cascaded networks to calculate the performance of all the 804,357 different designs for this PSS. Figure 2c shows all the amplitude and phases evaluated for insertion loss less than \(-\,2\) dB (each point corresponds to a different unit-cell with different combinations of CSRs and different sizes). Data with insertion loss greater than \(-\,2\) dB are omitted since they have no practical significance as discussed above. Notably, this CSR unit-cell covers the entire \(360^{\circ }\) phase range, and we can identify different unit-cell combinations that satisfy the desired \(\phi _p=60^{\circ }\) phase progression (see in Fig. 2c, for example, the areas at \(\pm \,180^{\circ }\), \(\pm \,120^{\circ }\), \(\pm \,60^{\circ }\), and \(0^{\circ }\)). The accuracy of our approach is determined by comparing the results of our method to the full-wave simulation reference data. Figure 2d, indicatively, compares the amplitude and phase responses of our method to the ones observed by the conventional approach, for the \(n=6\) unit-cells that offer the required \(60^{\circ }\) phase progression computed above. Using (3) and (4):

$$\begin{aligned} Error_{phase}= \left| \frac{\phi _{ref}-\phi }{2\pi }\right| \times 100\%, \end{aligned}$$


$$\begin{aligned} Error_{amplitude}= \left| T_{ref}-T\right| \times 100\%. \end{aligned}$$


We evaluate the percentage of the absolute error between the two approaches, showing a maximum amplitude and phase error of 1.5% and 1.4%, respectively at \(h = 0.2\lambda \).

A similar analysis is conducted for both the square patch (SP) and square ring (SR)-based unit-cells, showing a maximum error of 2.9% and 5.3%, respectively at \(h = 0.2\lambda \). Figure S4a–d of our Supplementary Material present the equivalent analyses showing the corresponding responses for both SP and SR unit-cells. To further investigate the accuracy of our proposed methodology, we study the phase and amplitude percentage errors as the total height of a three-layer PSS unit-cell varies for all the three different unit-cell designs modeled here (CSR, SP, and SR). Figure 3 shows the corresponding results where the worst-case scenario (e.g., the maximum error) among all cell variations has been used. To evaluate the percentage errors we use as reference phase, \(\phi _{ref}\), and reference amplitude, \(T_{ref}\), the phase and amplitude responses of the full-wave simulations. Notably, \(\phi \) and T represent the phase and amplitude responses obtained by our proposed method. The blue line corresponds to the phase percentage error and the red line corresponds to the amplitude percentage error. As we can see from these results, the square ring introduces a higher overall error compared to the other two resonators, while the square patch exhibits minimal error at \(h = 0.2\lambda \), which is our point of interest here. Nevertheless, what is important to note here, is that for all three resonators, both the phase and amplitude percentage errors increase significantly for heights smaller than \(0.175\lambda \). This behavior is expected and is attributed to the strong mutual coupling between the resonant patches, when they are brought close to each other. Even though this seems to limit the applicability of our proposed methodology to designs that are thicker than \(\lambda /6\), in reality, the majority of antenna designs that operate at millimeter wave frequencies have thicknesses that are greater than \(\lambda /5\). Therefore, our proposed method should be applicable to most practical designs. Also, Olk et al.35 recently introduced a method that takes into account the mutual coupling between the different layers of a unit-cell design, similar to our ABCD approach. Thus, by utilizing Olk’s approach we expect to enhance the accuracy of our proposed methodology, leading to a significant reduction in both amplitude and phase errors. However, such an analysis is not conducted in this study and is left as a future work.

Figure 3
figure 3

Phase and amplitude percentage error as the total height of a three-layer PSS unit-cell varies. Three different unit-cell designs are shown here (CSR, SP, and SR) which have been commonly used in the literature. Here, \(\phi _{ref}\) and \(T_{ref}\) correspond to phase and amplitude responses based on full-wave simulations, and \(\phi \) and T represent phase and amplitude responses obtained from our proposed method.

Figure 4
figure 4

(a) Isometric and side view of our designed holographic metasurface antenna (HMA). In the inset, the monopole feed antenna, and the HMA unit-cell are shown. Comparison of simulated and measured responses; (b) 2-D normalized radiation pattern of the HMA at \(\phi =0^{\circ }\) cut-plane, and (c) axial ratio. (d) Simulated 3-D radiation pattern contour plot.

Figure 5
figure 5

(a) Hybrid PSS super-unit-cell. (b) E-field (\(|E_y|\)) cross-section distribution of an infinite 1-D hybrid PSS. (c) Comparison of the transmission amplitude and phase, between proposed (semi-numerical) and conventional (full-wave) methodology, for the case of properly chosen \(n=3+3\) CSR and SP unit-cells, that when are combined offer the required \(\phi _p=60^{\circ }\) phase progression. The geometrical dimensions for the data points of these unit-cells are tabulated in Table ST1 in the Supplementary Material. (d) Transmission and reflection coefficient of the CSR only and hybrid super-unit-cells.

Validation: design of a 1-D beam-steerable antenna

To validate the accuracy of our proposed methodology, we design a \(10\lambda _0 \times 10\lambda _0\) 1-D beam-steerable antenna system that operates at \(f=30\) GHz consisting of a right-hand circularly polarized (RHCP) HMA with a broadside beam, and a properly designed holographic PSS that steers the broadside beam at \(\theta _t=33.7^{\circ }\). Notably, our proposed method of designing PSS does not limit the use of a base antenna to HMA only, and other antenna types (e.g., antenna arrays, radial-line slot arrays, etc.) can be used. In what follows, we, briefly present the design of the HMA, and then, we design our PSS and simulate and measure the entire antenna system.

Holographic metasurface antenna (HMA)

A \(10 \lambda _0 \times 10\lambda _0\) square spiral holographic metasurface antenna (HMA) is properly designed to feed our 1-D beam-steerable antenna system. Figure 4a shows our HMA which consists of properly designed square metallic patches over the ground plane, and a monopole antenna that excites the HMA at its center. The design process of the HMA is available in Ref.45, and is omitted here for reasons of brevity. A Duroid/Rogers 5880 substrate is used with a thickness of 1.57 mm, a relative permittivity of 2.2, and a loss tangent of 0.0009. Figure S6 in the Supplementary Material shows our prototype. Notably, a 2.4 mm connector is used as our monopole feed. Figure 4b indicates excellent matching between simulation and measured normalized gain. The HMA achieves a maximum broadside gain of 18.3 dBi. Also, as we can see from Fig. 4b,c, the HMA is right-hand circularly polarized (RHCP) with an axial ratio close to 1 dB at the broadside direction for both simulated and measured responses. Figure 4d shows the simulated \(3-D\) radiation pattern contour plot.

Figure 6
figure 6

(a) Isometric view of the 1-D beam-steerable antenna system with the PSS placed above the HMA in a distance \(d=\lambda _0\). (b) Fabricated prototype of 1-D beam-steerable antenna system with measurement setup. The PSS (top) and HMA (bottom) are shown in the inset. 2-D normalized co-pol and cross-pol radiation pattern at different \(\phi _{rot}\) rotation angles of (c) \(0^{\circ }\), (d) \(90^{\circ }\), (e) \(180^{\circ }\), and (f) \(270^{\circ }\), respectively.

Beam-steerable antenna system

As discussed earlier, our goal is to design a 1-D beam-steerable antenna system that operates at \(f=30\) GHz and points its beam at \(\theta _t=33.7^{\circ }\). To achieve this goal, we need to properly design our PSS. First, we start by choosing the appropriate unit-cell design that can (a) cover the entire \(360^{\circ }\) phase range, and (b) achieve a transmission coefficient above \(-\,2\) dB, as discussed above. Notably, in “Theory” section we investigated the design of three different PSSs based on three different three-layer unit-cell designs (CSR, SP, and SR). Figure 2c and Fig. S4a,c show the corresponding amplitude and phase responses, where, it is seen that only the CSR-based unit-cell designs satisfy the two above-mentioned requirements of \(360^{\circ }\) phase range and transmission coefficient of \(S_{12}\) >\(-2\) dB. The next step is to choose \(n=6\) unit-cell designs (we proved this in “Theory” section) to steer the broadside beam at \(33.7^{\circ }\). Figure 2d shows the amplitude and phase for a set of candidate unit-cell designs for our CSR resonator. Next, following Step 7 of our design methodology, we create a super-unit-cell comprised of the 6 CSR unit-cells of Fig. 2d, and we conduct full-wave infinite array simulations to evaluate its electromagnetic performance. Figure 2e,f show the corresponding responses. As we can see from these responses, even though the field distribution is directed towards the expected direction of \(33.7^{\circ }\), the CSR-based super-unit-cell has high reflection and low transmission. This is despite the fact that all the individual unit-cells have very good transmission coefficients, and it is attributed to the mutual coupling between the different unit-cells that has not been taken into account when we conduct infinite array full-wave simulations of the corresponding unit-cells. Notably, this is not a drawback of our methodology, but it is a common behavior that also appears in the conventional approach, where also infinite array simulations are conducted for characterizing the corresponding unit-cells. Only very recently a methodology was introduced in Ref.46, where the transverse coupling between unit-cells has been considered. Nevertheless, to address this challenge, hybrid PSSs have been introduced that use combinations of different unit-cells. Therefore, moving forward with our design, we create a super-unit-cell that is comprised of both CSR and SP unit-cell designs. Figure 5a shows this design. Notably, as we can see from Fig. 5c, we use 3 CSR unit-cell designs to cover the \((-180^{\circ },-60^{\circ })\) phase range, and 3 SP unit-cell designs to cover the \((0^{\circ },120^{\circ })\) phase range. To properly select the unit-cell designs that we need to retain and replace, an intermediate electromagnetic analysis is required to examine the mutual coupling between the different unit-cells. In this analysis, we construct our super-unit-cell by progressively adding different unit-cells and monitoring the resulting transmission and reflection coefficients. When undesirable transmission or reflection coefficients occur, signifying high mutual coupling, we replace the corresponding unit-cell design with one that meets our desired criteria. In our specific case, after conducting this analysis, we identified the 4th unit-cell design within the super-unit-cell shown in Fig. 2e (inset) as responsible for the excessive reflection. Notably, although the 3rd and 5th unit-cell designs in Fig. 2e (inset) did not introduce significant reflection, we replaced them to maintain symmetry in the super-unit-cell. Figure 5b shows the field distribution of the hybrid PSS that is directed towards the expected direction of \(33.7^{\circ }\). Moreover, Fig. 5d shows the response of the hybrid super-unit-cell where both its reflection and transmission coefficients are at \(-\,18.04\) dB and \(-\,0.94\) dB, respectively. These responses are acceptable, and, therefore, we use this as our final super-unit-cell based on which we build our PSS.

Notably, our PSS has a total aperture of \(100 \times 100\) mm\(^2\), which corresponds to \(10\lambda _0\times 10\lambda _0\) at the operating frequency of 30 GHz, and a total thickness of 4 mm, consisting of two dielectric and three metallic layers. To model and build the PSS, we use Duroid/Rogers 5880 as the substrate with a relative permittivity of 2.2 and a loss tangent of 0.0009. Due to the unavailability of a 2 mm custom thickness dielectric substrate in the market, to fabricate our prototype, we combined two single-sided copper laminate 1.57 mm and 0.38 mm Duroid/Rogers 5880 substrates with a prepreg (Rogers 6700 of 0.038 mm thickness) to construct a 2 mm dielectric substrate. Finally, two of these 2 mm layers were joined with the prepreg (Rogers 6700 of 0.038 mm thickness). A detailed view of the PSS and its layers is provided in Fig. S5 of our Supplementary Material. Next, we place our hybrid PSS (see Fig. S2b in our Supplementary Material) above the HMA we designed in “Holographic metasurface antenna (HMA)” section in a distance of \(\lambda _0\) (10 mm at 30 GHz) and characterize the entire beam-steerable antenna system. To hold our PSS in place, we used a 3D-printed plastic mold. Figure 6a shows an isometric view of the entire antenna system and Fig. 6b shows the fabricated prototype placed inside our MVG MicroLab anechoic chamber47. In the inset of Fig. 6b, the fabricated PSS and HMA are shown, respectively.

To examine the beam-steering capability of our hybrid PSS, we conducted a series of tests. Namely, we rotated the PSS at four different azimuthal angles (\(\phi _{rot}=0^{\circ }\), \(90^{\circ }\), \(180^{\circ }\), and \(270^{\circ }\)), while keeping the HMA at the same position. Figure 6c–f compare the simulated with the measured radiation patterns. Our results clearly demonstrate that the PSS preserves the RHCP radiation pattern of the HMA, while tilting the beam to \(\theta _t=33^{\circ }\). Only a \(0.3^{\circ }{-}0.7^{\circ }\) beam angle error is observed, which is of no practical significance. Table 1 tabulates the beam direction for all four cases as well as the corresponding realized gains for both simulated and measured responses. As we can see from this table, a very stable gain of \(16 \pm 0.4\) dBi is achieved. Notably, this gain is 2.7 to 1.9 dB lower compared to the 18.3 dBi gain of the HMA when no PSS is used. This gain reduction is totally expected and is attributed to (a) the insertion loss of 0.94 dB introduced by the PSS (see Fig. 5d), and (b) the cosine roll-off as we steer the beam from broadside to \(33.7^{\circ }\). The slight asymmetry we observe with respect to the different rotation angles is attributed to the fact that the HMA is not entirely symmetric in the azimuthal plane. Also, the measured gains in some cases are slightly higher compared to the simulated ones. This is not surprising and it is attributed to the limitations of our computational resources to evaluate with high accuracy the entire structure. Figure 7a–d present the contour plots of the corresponding \(3-D\) radiation patterns for all the rotation angles. Finally, Fig. 7e–h show, the axial ratio of our design, which is always maintained below 3 dB for all the \(\phi _{rot}\) rotation cases at \(\theta =33^{\circ }\pm 0.7^{\circ }\).

Table 1 Antenna system realized gain and axial ratio for different PSS rotation angles (\(\phi _{rot}\)).

Figure 7
figure 7

(a–d) Simulated 3-D radiation pattern contour plot, and (e–h) simulated and measured axial ratio (AR) for \(0^{\circ }\), \(90^{\circ }\), \(180^{\circ }\), and \(270^{\circ }\) rotation angles (\(\phi _{rot}\)), respectively.

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