Aperiodic-metamaterial-based absorber

Recent years have seen the rapid emergence of a new class of perfect absorbers based on metamaterials, which have shown unique characteristics and abilities to control electromagnetic waves in an amazing manner.^1–13 The property of near-perfect absorption of the metamaterial-based absorber thus paves the way for achieving numerous potential applications in sensing,^14,15 cloaking,^16–18 photovoltaics,^19–23 and so on. So far, most studies of the metamaterial-based absorber are implemented by periodically arranging the unit cells with predesigned dimensions.^24–30 Periodic absorbers are generally regarded as homogenized structures, and their absorption behaviors are analyzed by effective constitutive parameter calculations. What if the unit cell in the metamaterial is not periodically arranged, for example, aperiodically? Can we still obtain the desired absorption performance or a vast difference from those in the periodic one will be observed? Indeed, this quest motivates us to construct aperiodic metamaterial absorbers and conduct a systematic research. In addition, it is known that quasicrystal structures or deterministic aperiodic structures such as Fibonacci, Thue-Morse, and Rudin-Shapiro possess unique light localization, multiple bandgap, and multi-frequency characters,^31–36 which have drawn intriguing attention and shown potential applications in surface-enhanced Raman scattering, lasing, and optical mode control.^37–42 Especially, aperiodic structures could have higher filling factors and show lower sensitivities to fabrication precision. These may provide a new avenue to design more efficient and robust absorbers. However, investigations on aperiodic-metamaterial-based absorbers have rarely been reported.

Here, we numerically and experimentally study the absorption performance of aperiodic-metamaterial-based absorbers. As typical representatives of the aperiodic designs, we consider three structures: the 2D Fibonacci (FMA), Thue-Morse (TMA), and quasicrystal (QMA) sequences. The periodic design (periodic-metamaterial-based absorber, PMA) with the same unit cell is also investigated as a comparison. We find similar absorbing behaviors between the aperiodic absorbers and the periodic one, while a pronounced dependence of the absorption on the spacer thickness shows diverse performances between them. Meanwhile, we also observed that the performance of aperiodic-metamaterial-based absorbers is successfully maintained under oblique incidence. Our presented studies offer substantial progress in the understanding of absorbing behaviors of aperiodic structures and thus contribute to a systematic analysis of the metamaterial-based absorber.

Figure 1(a) illustrates the schematic diagram of a metamaterial unit cell excited by a normally incident terahertz wave. The unit cell comprises a cross-shaped aluminum structure at the top layer, a lossy polyimide with a permittivity of ε = 2.93 and a loss tangent of δ = 0.044 at 1.0 THz at the middle layer and an aluminum ground plane of 200 nm on a silicon substrate, respectively. The geometric parameters of the unit cell are also illustrated in Fig. 1(a). Figures 1(b)–1(e) show the 2D metamaterial absorbers with four patterns of periodic, Fibonacci, Thue-Morse, and eight-fold quasicrystal sequences, respectively, and they share the same unit cell. Here, the periodic sequence is a square array with a period P. The Fibonacci sequence can be obtained by combining two different elements A and B according to the inflation rule: A → AB, B → A, while the Thue-Morse sequence is generated by the rule of A → AB, B → BA^36,39 and the minimum interparticle separation in these lattices matches the grating constant of the reference periodic absorber P. The eight-fold quasicrystal sequence is based on two types of rhombuses: a square tile and a thin tile with a vertex angle of 45°, with the equal length of the four rhombus sides defined as P.³⁴ Compared with the periodic patterns, all these three aperiodic structures show some disorder, and disordering of these lattices can be derived from the disorder parameter.^43,44

Numerical simulations on the spectral response of the metamaterial absorbers were performed by using the commercial software package CST Microwave Studio. The entire simulation area of absorbers is 6 mm × 6 mm. Open boundary conditions are applied in both the x and y directions. The S-parameters of absorbers are calculated by the time-domain solver. The absorptance A(ω) of absorbers was then obtained from the S-parameters by $A ( ω ) =1−R ( ω ) −T ( ω ) =1− S 11 2 − S 21 2$⁠.^1,8 Given an incident wave with the electric field polarized in the x direction, the simulated reflection and absorption of the four metamaterial absorbers with the same thickness of the spacer layer are shown in Fig. 2, where h = 15 μm. The reflection is considerably suppressed at the resonant frequency of 0.61 THz for the PMA and the other aperiodic-metamaterial absorbers. For the PMA, the strong coupling between electric and magnetic responses makes it impedance-matched to free space to minimize the reflectivity, which causes the electromagnetic energy to be trapped in the composite structure. In contrast, FMA, TMA, and QMA lack global periodicity and translational invariance, which makes it hard to describe the electromagnetic behaviors by the rule of periodic sequences. In fact, they can be regarded as an inhomogeneous distribution of the most represented spatial periods in the structures manifested by the dominant diffraction peaks.^34,36,39 Due to the fact that the four metamaterial absorbers share the same minimum interparticle separation P, it is obvious that they have the same resonant frequency, which can be described by a simple LC circuit model $f=1/2π L C / 2 ∼1/b$⁠.²⁶ We can then deduce that the resonant frequency is only sensitive to the length of the cross structure, which agrees well with the results of Fig. 2, too. It is interesting to notice that the PMA and QMA have a perfect absorption of 99.8% at the designed frequency, but the performances of the FMA and TMA are limited to 81.7% and 83.3%, respectively.

To validate the simulation, the reflection and transmission of the four sequences were measured by the terahertz time-domain spectroscopy system at normal incidence. All samples with the same unit cell parameters as those used in the simulations were fabricated by conventional photolithography, which are depicted in the insets of Fig. 2. The experimentally measured reflection and absorption spectra are shown in Fig. 2 as dashed lines. It can be seen that the measured spectra showed excellent agreement with simulations over all frequencies characterized. The measured absorption of the PMA and QMA reached 96% and 93.8%, respectively, slightly lower than the simulated values. By contrast, the absorptions of the FMA and TMA were as high as 93% and 91.3%, respectively, showing lower values than the PMA and QMA. The slight deviations between the experiment and the simulation are due to the fact that the processed thickness of the polyimide was somewhat larger than the designed value, and this also could be seen from the following discussions.

Figures 3(a)–3(d) show the simulated electric field distributions at the top layer of the four absorbers. No remarkable coupling between neighboring resonators for all four lattices is observed, and this implies that the aperiodic arrangement does not induce any strong coupling effect here. From this point, the in-plane unit cells could show similar optical responses, although they are arranged in a different way. However, it is known that the metamaterial-based absorber is sensitive to the thickness of the spacer due to the interaction between the top metallic structure and the ground metallic plane. This could also be seen clearly from the calculated electromagnetic surface current distributions in Fig. 3(e). Charges of opposite signs accumulate at the ends of the cross structure, and this electric dipole oscillates in two antiparallel surface currents at the ground plane and top layer resulting in resonant magnetic response. Therefore, the absorption of the metamaterial-based absorber is dominated by both the in-plane electric resonance and the out-of-plane magnetic resonance. In other words, the optical response of the absorber is relevant to the electric resonance induced ε_eff and the magnetic resonance induced μ_eff. When we arrange the unit cell in a different way, ε_eff changes little but μ_eff varies considerably, thus leading to an evident change of the impedance $Z eff = μ e f f / ε e f f$⁠. The effective impedances of the absorbers can be calculated by $Z eff =± 1 + S 11 2 − S 21 2 / 1 − S 11 2 − S 21 2$⁠.^45,46 Figures 3(f)–3(i) show the extracted equivalent impedance for h = 15 μm for the four samples. It is seen that the equivalent impedances of the PMA and QMA have a near-unity real part and minimized imaginary part, indicating that the condition of impedance matching is properly satisfied. However, for the FMA and TMA, although the imaginary part of the impedance is close to zero, the real part is far smaller than one, which gives rise to the reflections of the absorbers.

The aforementioned explanation is also confirmed by the following simulations. The optical responses of the four samples with various thicknesses of the spacer changing from 20 to 30 μm were simulated. Figure 4(a) represents the absorption peak values. For the PMA and QMA, when the polyimide thickness becomes larger, the corresponding absorption peaks deteriorate gradually. However, the Fibonacci and Thue-Morse structures show a different behavior. The absorption peaks are enhanced with the increase of the spacer thickness. When the thickness of the spacer reaches 30 μm, the peak values are over 98%. These characteristics are also consistent with the calculated effective impedance. Figures 4(b) and 4(c) show, respectively, the real and imaginary parts of the calculated effective impedance for the metamaterial absorbers. It is found that the real parts of the PMA and QMA remain in the vicinity of 1.0 in all cases, while the corresponding values of the FMA and TMA keep growing and reach 1.0 when h = 30 μm. However, the imaginary parts of the impedance show the opposite trend as shown in Fig. 4(c). Since that, we could obtain the perfect absorption for the PMA and QMA at h = 15 μm, as well as for the FMA and TMA at h = 30 μm.

Next, as a key feature of the robustness of our design, the resonant absorption behavior with the cross structure under oblique incidence was also experimentally investigated. Figure 5(a) shows the employed angle-resolved terahertz time-domain spectroscopy. Two fiber-coupled terahertz antennas were used as the transmitter and detector, respectively. The antennas were mounted on two guide rails that could rotate concentrically with the rotator. The rotator controlled the incidence angle of the terahertz beam $α$⁠, and the detection angle was equal to the angle of incidence. The terahertz wave emitted from the transmitter was collimated by lens L1 to deliver a nearly parallel beam with a diameter of 4 mm. The samples were positioned at the center of the rotator and kept still. The reflected terahertz beam was collected by lens L2 and received by the detector. To illustrate the experimental setup, an enlarged view of the sample and polarization direction (TM polarization) is shown at the right side of Fig. 5(a). Figures 5(b)–5(d) demonstrate, respectively, the measured absorption spectra of the PMA, FMA, TMA, and QMA under various incident angles. As the incident angle increases, the absorption peaks of the four metamaterial absorbers change slightly, and a blue shift of the absorption peaks occurs. It is interesting that the lattice arrangement has no direct relationship with the angle-independence of the absorption. At an incident angle of 40°, the absorptions of the PMA and QMA were as high as 90%, and the absorptions of the FMA and TMA were still above 80%.

Furthermore, in order to elucidate the polarization dependence of the metamaterial absorbers, we numerically and experimentally examined the reflection and absorption spectra of some absorbers with polarization-dependent structures, as shown in Fig. 6. The unit cell is chosen as a classic split-ring resonator structure with a fixed period P = 135 μm. Because the split-ring resonator has a different resonance for the x and y polarizations, the sample thus shows polarization-dependent absorption features. Figures 6(a), 6(c), 6(e), and 6(g) represent the performance of the absorbers under x polarization, while Figs. 6(b), 6(d), 6(f), and 6(g) are for the case with the y polarization. It can be found that these four absorbers have two distinct absorption peaks under x and y polarizations and these two peaks are independent of each other. Although the absorbers share the same unit cell, their resonant frequencies show an obvious deviation. For the PMA, the frequencies are {0.64, 0.93 THz}, while for the FMA, TMA, and QMA, they are, respectively, {0.66, 0.95 THz}, {0.65, 0.95 THz}, and {0.61, 0.94 THz}. The frequency shift comes from the mutual influence of the resonance between adjacent unit cells. It should be noted that the PMA and QMA show a better performance than the FMA and TMA for both two polarizations, which agrees well with the previous conclusion for the cross structure. The corresponding measured results are indicated by the dashed lines, which reveal a good agreement with the simulated results and prove the absorption of metamaterials. The difference in bandwidth is primarily due to the fabrication errors of the polyimide and resonators.

In summary, we theoretically and experimentally investigate the aperiodic metamaterial based consisting of 2D Fibonacci, Thue-Morse, and eight-fold quasicrystals at both normal and oblique incidences. Aperiodic absorbers exhibit similar optical responses as the periodic one. However, effective impedance matching analyses show different absorption tendencies with the spacer thickness. Angle and polarization dependence of the aperiodic-metamaterial-based absorbers are also presented. Our work elaborates novel aperiodic absorbers beyond pervious works, thus promising a route towards developing unique integrated devices.

This work was supported by the National Key Basic Research Program of China (Grant No. 2014CB339800), the National Science Foundation of China (Grant Nos. 61422509, 61735012, 61420106006, 61427814, and 61605143), and the U.S. National Science Foundation (Grant No. ECCS-1232081).