We numerically and experimentally investigate collective resonant responses of coherent metasurfaces consisting of planar random-flip asymmetrically split rings. The collective narrow band response becomes broader and eventually disappears with increasing the percentage of flipped split rings. The exploited physical mechanism is the degraded collective response due to random microscopic multipole excitation. Our study provides a method to construct random metasurfaces and is helpful to have an insight into their underlying physics.

Artificial metasurfaces are planar periodic structures comprised of subwavelength resonators.1–3 In recent years, metasurfaces have emerged as a platform for multifunctional spatial light field manipulation, enabling flexible control over amplitude,4,5 phase,6,7 polarization,8,9 and frequency10,11 of electromagnetic waves. They have attracted much attention in various applications, particularly in polarization conversion12–15 and holographic imaging.16–19 The interaction strength between metamolecules plays an important role in engineering electromagnetic responses of metasurfaces. Coherent and incoherent metasurfaces have been experimentally investigated.20 The responses of incoherent metasurfaces are determined by the properties of individual metamolecules, and interactions between the metamolecules are negligible; as a result, the responses are weakly affected by metamolecule disorder. In contrast, coherent metasurfaces are characterized by strong interactions among the metamolecules, and their responses are mainly dominated by collective excitations of metamolecules.20–26 Understanding and harnessing the collective resonance in coherent metasurfaces is crucial for achieving advanced optical functionalities and surpassing the capabilities of incoherent metasurfaces.

In the field of metasurfaces with ordered arrangements, a well-established framework has been developed to explain the electromagnetic response.27,28 Regardless of the geometric shape and composition of the unit cells, the overall response of the metasurface can be attributed to various multipole contributions, further categorized in terms of electric, magnetic, and toroidal multipoles.29–32 In general, electric and magnetic multipole modes originate from the distribution of current and charge in individual unit cells of metasurfaces. Toroidal multipole modes can be visualized as currents flowing on the surface of the torus along its meridians, and an array of magnetic dipole modes is arranged in a head-to-tail configuration along the torus.33–38 

To investigate electromagnetic responses of a random metasurface, one can directly change the spatial distribution or random-flip arrangement of the metasurface units.20,39,40 Random distribution of metasurface elements disturbs the initial interference condition and recognizes coherent or incoherent metasurfaces.20,39 Random-flip operation is another way to design a random metasurface.40 A random-flip metasurface with arrays of gold nanorods has been verified to realize diffuse reflection and reciprocity-protected transmission, offering a promising way for optical spectroscopy and imaging.40 Nevertheless, spatial random distribution20,39 and space inversion (flip) operation40 have introduced challenges to the design and fabrication of metasurfaces. Flexible random metasurfaces with easy fabrication are desirable. It is necessary to develop random metasurfaces by use of planar rotation operation (i.e., planar inversion or flip).

In this work, we propose and experimentally validate a random method to manipulate electromagnetic responses of a coherent metasurface with asymmetrically split rings (ASRs). The random-flip metasurface is achieved by random planar rotation or planar flip. The proposed metasurface utilizes a straightforward structure and accessible random way. By adjusting the proportion of flipped unit cells, resonant properties of the coherent metasurface have been investigated as well as its underlying mechanism. The planar random-flip operation offers an opportunity to study the interaction of metamolecules and understand the mechanism of the coherent metasurface. Importantly, the pathways can be enriched for introducing the concept of “disorder” in the random metasurface. The dependence of collective electromagnetic responses on random planar flip can be further understood and randomly engineered properties of the resonators can be exploited in various applications such as scattering manipulation, antenna directivity control, and sensing.

To study the collective response of a coherent metasurface with ASRs that was termed,20,39 we have designed a planar random-flip coherent metasurface (PRFCM). The PRFCM is composed of copper ASR resonators, which can be easily fabricated by standard lithographic techniques on a 1.6 mm thick FR4 dielectric substrate, as illustrated in Fig. 1(a). The inner and outer radii of the ASRs are 3 and 3.4 mm, and the open angles of the arcs are 160° and 130°, respectively. Each ASR occupies a square translation cell of 7.5 × 7.5 mm2. The random metasurface has two different unit cells, normal and rotated unit cells. These two unit cells are distinguished by yellow and blue colors, referred to as “Unit 1” and “Unit 2,” respectively. The original configuration is designated as “Unit 1,” while “Unit 2” is obtained under planar rotation or flip operation of “Unit 1” (i.e., “Unit 1” is rotated 180° with respect to the z-axis), as presented in Fig. 1(b).

FIG. 1.

(a) Schematic structure of the proposed PRFCM. The yellow square represents “Unit 1,” while the blue square represents “Unit 2” with planar rotation operation of 180° with respect to the z-axis. The inset shows the geometrical parameters of the unit cell. The period of the unit cell is l = 7.5 mm, the inner and outer radii of the ASR are r1 = 3 mm and r2 = 3.4 mm, the thickness of the FR4 substrate and ASR are h0 = 1.6 mm and h1 = 0.035 mm, and the open angles of the arcs correspond to 160° and 130°. (b) Front views of two unit cells. (c)–(e) The percentages of flipped unit cells of three metasurfaces correspond to 0%, 25%, and 50%, respectively.

FIG. 1.

(a) Schematic structure of the proposed PRFCM. The yellow square represents “Unit 1,” while the blue square represents “Unit 2” with planar rotation operation of 180° with respect to the z-axis. The inset shows the geometrical parameters of the unit cell. The period of the unit cell is l = 7.5 mm, the inner and outer radii of the ASR are r1 = 3 mm and r2 = 3.4 mm, the thickness of the FR4 substrate and ASR are h0 = 1.6 mm and h1 = 0.035 mm, and the open angles of the arcs correspond to 160° and 130°. (b) Front views of two unit cells. (c)–(e) The percentages of flipped unit cells of three metasurfaces correspond to 0%, 25%, and 50%, respectively.

Close modal
It is necessary to evaluate how the planar rotation operation affects the performance of the metasurface. As shown in the insets of Figs. 2(a)2(c), the three metasurfaces are composed of Unit 1, parallel arrangement of Unit 1 and Unit 2, and cross arrangement of two Unit 1 and two Unit 2. Three unit cells occupy areas of 7.5 × 7.5, 15 × 7.5, and 15 × 15 mm2. To investigate the dependence of electromagnetic response on the flipped unit cells in the metasurface, transmission spectra and multipole calculation are separately performed for three metasurfaces using finite-element-method commercial software COMSOL Multiphysics. Copper is regarded as a perfect electric conductor (PEC) in the microwave frequency band. The substrate is made of FR4 with a relative permittivity of 4.3 and a loss tangent of 0.025. Figures 2(a)2(c) present their simulated transmission spectra for the normally incident x-polarized wave. In the first metasurface, there is a high transmission peak A2 in the transmittance spectrum at 10.75 GHz, while there are two transmission dips A1 and A3 at 10.05 and 12.75 GHz, as presented in Fig. 2(a). Despite the structural changes, their transmittance spectrums closely resemble that of the first metasurface, as presented in Figs. 2(b) and 2(c). Compared with the first metasurface, all resonant frequencies of two other metasurfaces are slightly shifted due to coupling interaction between metamolecules. In order to understand the contributions of microscopic multipole excitations to the electromagnetic responses, multipole decomposition of the transmission spectra is theoretically analyzed, including electric dipole (ED), magnetic dipole (MD), toroidal dipole (TD), electric quadrupole (EQ), and magnetic quadrupole (MQ) modes, as shown in Figs. 2(d)2(f). In Cartesian coordinates, the scattering cross section under harmonic excitation of exp(iωt) can be expressed as the following formulas:38 
I = 2 ω 4 3 c 3 | P | 2 + 2 ω 4 3 c 3 | M | 2 + 2 ω 6 3 c 5 | T | 2 + ω 6 5 c 5 Q α β Q α β + ω 6 20 c 5 M α β M α β + ,
(1)
where P, M, T, Qαβ, and Mαβ represent the components of ED, MD, TD, EQ, and MQ modes, respectively. These five terms considered here are strong, while higher-order electric and magnetic multipoles are generally negligible at low levels of precision. In this equation, the expressions of modes are given below:
P = 1 i ω j d 3 r ,
(2)
M = 1 2 c ( r × j ) d 3 r ,
(3)
T = 1 10 c [ ( r j ) r 2 r 2 j ] d 3 r ,
(4)
Q α β = 1 i ω [ r α j β + r β j α 2 3 ( r j ) ] d 3 r ,
(5)
M α β = 1 3 c [ ( r × j ) α r β + ( r × j ) β r α ] d 3 r ,
(6)
where j represents the current density, c and ω are the speed and angular frequency of the incident wave, and α, β = x, y, z indicates the axis.
FIG. 2.

Electromagnetic responses of three metasurfaces corresponding to different columns. (a)–(c) Simulated transmission spectra of three metasurfaces for normally incident x-polarized wave. As illustrated in the insets, the unit cells are Unit 1, parallel arrangement of Unit 1 and Unit 2, and cross arrangement of two Unit 1 and two Unit 2, respectively. The transmission peaks and dips are marked by A, B, and C with predefined subscript. (d)–(f) Contributions of the five strongest multipolar excitations of three metasurfaces. (g)–(i) The decomposition components of MD mode in (d), TD mode in (e), and MD mode and TD mode in (f) along three Cartesian coordinates.

FIG. 2.

Electromagnetic responses of three metasurfaces corresponding to different columns. (a)–(c) Simulated transmission spectra of three metasurfaces for normally incident x-polarized wave. As illustrated in the insets, the unit cells are Unit 1, parallel arrangement of Unit 1 and Unit 2, and cross arrangement of two Unit 1 and two Unit 2, respectively. The transmission peaks and dips are marked by A, B, and C with predefined subscript. (d)–(f) Contributions of the five strongest multipolar excitations of three metasurfaces. (g)–(i) The decomposition components of MD mode in (d), TD mode in (e), and MD mode and TD mode in (f) along three Cartesian coordinates.

Close modal

According to the decomposition calculations shown in Fig. 2(d), the transmission peak A2 of the first metasurface is primarily attributed to the excitation of the MD mode, while the two transmission dips A1 and A3 are mainly dominated by the ED mode, which is consistent with the result in Ref. 20. The radiating power of the MD mode is much stronger than others at peak A2. Actually, the anti-phase surface currents along two arcs generate MD mode along the z-axis. As verified in Fig. 2(g), the dominant component of the MD mode at peak A2 is the z-component. The low-frequency transmission dip A1 and high-frequency transmission peak A3 are contributed by the induced surface currents along the long and short metallic arcs, respectively. However, for two other periodic metasurfaces with rotated unit cells, although their transmission spectra are almost kept the same as that of the first metasurface, they undergo different multipole decomposition analyses. For the metasurface with parallel arrangement of Unit 1 and Unit 2, the transmission peak B2 is mostly contributed by the TD mode in Fig. 2(e), where the y-component of the TD mode is predominant as illustrated in Fig. 2(h). The planar rotation operation leads to a head-to-tail configuration of magnetic dipole modes; thus, the TD mode occurs. Notably, the transmission peak C2 lacks any predominant contribution due to the cross configuration of Unit 1 and Unit 2 in Figs. 2(f) and 2(i), where excitations of electromagnetic modes in four adjacent elements may cancel with each other. In these two metasurfaces, all transmission dips (B1, B3, C1, and C3) are mainly influenced by the ED and EQ modes, as depicted in Figs. 2(e), 2(f), 2(h), and 2(i).

To intuitively understand the multipole decomposition analyses, the induced surface current distributions of three metasurfaces are illustrated in Fig. 3 to show orientations of multipole moments at various resonant frequencies. In the first metasurface with Unit 1, two ASRs are almost equally excited at A2 and the surface current flow directions are contrary; thus, the MD mode is generated along the z-axis in Fig. 3(d). At resonances A1 and A3, the long and short arcs are individually excited, and as a result, the ED modes occur as depicted in Figs. 3(a) and 3(g). The results agree well with the theoretical results in Figs. 2(d) and 2(g). In the metasurfaces with parallel and cross arrangements of Unit 1 and Unit 2, it is found that the fundamental excitation features of the resonant surface currents remain unchanged. At resonances B2 and C2, two ASRs are almost equally excited as shown in Figs. 3(e) and 3(f), while the long and short arcs are still individually excited at resonances B1, C1 and B3, C3, as shown in Figs. 3(b), 3(c), 3(h), and 3(i), respectively. At resonances B1, C1 and B3, C3, the magnetic responses are quite weak, while electric responses are much stronger. The electric responses are mainly attributed to ED and EQ excitations. As depicted in Figs. 3(e) and 3(f), the important feature is that orientations of MD modes of Unit 1 and Unit 2 are contrary at resonances B2 and C2. In Fig. 3(e), two contrary MD modes along the z-axis generate a y-oriented TD mode, which is helpful in understanding the calculation results in Figs. 2(e) and 2(h). In Fig. 3(f), the cross arrangement of two Unit 1 and two Unit 2 brings about two y-oriented TD modes, but their directions are contrary; thus, excitations of multipoles totally cancel and a dark mode forms corresponding to the transmission peak C2. The sharp spectral features of three metasurfaces at resonances in Figs. 2(a)2(c) can be explained by their weak coupling with free space, resulting from the collective response of periodic arrays of Unit 1 and Unit 2. This phenomenon is so-called trapped mode resonance with antiphase excitation of two asymmetrical arcs. The scattered electromagnetic fields produced by such current configurations are very weak, which dramatically reduce coupling to free space.41 The planar rotation or flip operation offers a way to modulate microscopic excitations of multipoles, especially producing a TD mode.

FIG. 3.

Simulated surface current distributions of three metasurface at three resonant frequencies aligned in a column. (a), (d), and (g) Surface current distributions of the metasurface with Unit 1 at A1, A2, and A3. (b), (e), and (h) Surface current distributions of the metasurface with the parallel arrangement of Unit 1 and Unit 2 at B1, B2, and B3. (c), (f), and (i) Surface current distributions of the metasurface with the cross arrangement of two Unit 1 and two Unit 2 at C1, C2, and C3. Here, the black arrows indicate the instantaneous directions of the surface currents. The blue, green, red, and cyan arrows denote the instantaneous directions of ED, MD TD, and EQ modes, respectively.

FIG. 3.

Simulated surface current distributions of three metasurface at three resonant frequencies aligned in a column. (a), (d), and (g) Surface current distributions of the metasurface with Unit 1 at A1, A2, and A3. (b), (e), and (h) Surface current distributions of the metasurface with the parallel arrangement of Unit 1 and Unit 2 at B1, B2, and B3. (c), (f), and (i) Surface current distributions of the metasurface with the cross arrangement of two Unit 1 and two Unit 2 at C1, C2, and C3. Here, the black arrows indicate the instantaneous directions of the surface currents. The blue, green, red, and cyan arrows denote the instantaneous directions of ED, MD TD, and EQ modes, respectively.

Close modal

Based on the aforementioned study, planar random-flip coherent metasurface can be constructed via planar rotation operation. The spectral characteristics of the coherent metasurface can be manipulated by adjusting the percentage of Unit 2 (rotated ASRs). Three metasurfaces with 0%, 25%, and 50% are fabricated, and their structures are illustrated in Figs. 1(c)1(e). The total size of each sample corresponds to 20 × 20 unit cells, which is large enough to consider the properties of the metasurface. Next, the electromagnetic properties of the PRFCM are numerically investigated using commercial software CST Microwave Studio. The PRFCM is illuminated by x-polarized electromagnetic waves propagating along the −z direction. Open boundary conditions are employed along the x, y, and z directions.

Figures 4(a), 4(c), and 4(e) correspond to the simulated transmission, reflection, and absorption spectra of three PRFCMs, respectively. The measured transmission, reflection, and absorption spectra of three PRFCMs are in Figs. 4(b), 4(d), and 4(f). Black, blue, and red lines, respectively, correspond to the PRFCM with 0%, 25%, and 50% flipping percentages. For simplicity, all simulations are completed at normal incidence, while the measured results were achieved at oblique incidence of ∼15° due to the limitation of microwave antennas’ size.42 The measured transmission and reflection spectra well agree with the simulated ones. However, there is a discrepancy between the simulated and measured absorption spectra, resulting from the difference between the simulated and measured angles of incidence. As the flipping ratio increases, the transmission peaks decrease and the Q-factor gradually decreases at the trapped-mode frequency of 9.85 GHz. However, the spectral characteristics far away from the trapped-mode frequency remain unaffected.

FIG. 4.

Dependences of properties of the PRFCM on the flipping percentage. Simulated (a) transmission, (c) reflection, and (e) absorption spectra of the PRFCM with the flipping percentages of 0%, 25%, and 50%. Measured (b) transmission, (d) reflection, and (f) absorption spectra of the PRFCM with the flipping percentages of 0%, 25%, and 50%.

FIG. 4.

Dependences of properties of the PRFCM on the flipping percentage. Simulated (a) transmission, (c) reflection, and (e) absorption spectra of the PRFCM with the flipping percentages of 0%, 25%, and 50%. Measured (b) transmission, (d) reflection, and (f) absorption spectra of the PRFCM with the flipping percentages of 0%, 25%, and 50%.

Close modal

In order to fully describe the properties of the PRFCM, it is necessary to show how random metasurfaces are affected by different designs at a fixed flipping percentage. Figure 5 illustrates the simulated transmission, reflection, and absorption spectra of the PRFCMs with 25% and 50% flipping percentages, respectively. For each flipping percentage, three arbitrary random designs are selected, marked by different color lines. Obviously, the spectra of the three random samples with the same flipping percentage are similar. Remarkably, the spectra of the PRFCM definitively depend on the flipping percentage rather than the unit cell selection of random flipping.

FIG. 5.

Dependences of electromagnetic responses of the PRFCMs on the flipped region. (a)–(c) Simulated transmission, reflection, and absorption spectra of the PRFCM with the flipping percentages of 25%. (d)–(f) Simulated transmission, reflection, and absorption spectra of the PRFCM with the flipping percentages of 50%. For a fixed flipping percentage, three arbitrary random cases are considered.

FIG. 5.

Dependences of electromagnetic responses of the PRFCMs on the flipped region. (a)–(c) Simulated transmission, reflection, and absorption spectra of the PRFCM with the flipping percentages of 25%. (d)–(f) Simulated transmission, reflection, and absorption spectra of the PRFCM with the flipping percentages of 50%. For a fixed flipping percentage, three arbitrary random cases are considered.

Close modal

At the trapped-mode resonance, strong magnetic dipoles are generated.41 To investigate impacts of the disorder on the electromagnetic responses, near-field magnetic field maps of the whole PRFCMs are simulated at the trapped-mode resonant frequency, as shown in Figs. 6(a)6(c). The excitations are quantified by assessing the normalized magnetic field intensity I = | H z | 2 / | H z | 2 , where | H z | 2 is the average magnetic field intensity over the entire large metasurface.39 The corresponding probability density functions (PDF) coarse-grained at the unit cell level are presented in Figs. 6(d)6(f). The PDF is estimated through the ratio p ( I ) / d I, where p ( I ) is the probability that the intensity of a metamolecule lies in the interval ( I - d I / 2 , I + d I / 2 ) and d I is a small intensity interval.39 

FIG. 6.

(a)–(c) Normalized near field magnetic intensity maps of the PRFCM at the trapped-mode resonant frequency with the flipping percentages of 0%, 25%, and 50%. Here, the near magnetic field is collected at a distance of 1 mm from the metasurface. (d)–(f) The PDF coarse-grained at the unit cell level around the trapped-mode resonant frequency with the flipping percentages of 0%, 25%, and 50%.

FIG. 6.

(a)–(c) Normalized near field magnetic intensity maps of the PRFCM at the trapped-mode resonant frequency with the flipping percentages of 0%, 25%, and 50%. Here, the near magnetic field is collected at a distance of 1 mm from the metasurface. (d)–(f) The PDF coarse-grained at the unit cell level around the trapped-mode resonant frequency with the flipping percentages of 0%, 25%, and 50%.

Close modal

For the flipping percentage of 0%, the regular array exhibits a narrow band distribution of magnetic field intensity density. A relatively uniform distribution of near-field intensity is formed across the entire plane in Fig. 6(a), verified by the symmetric distribution in Fig. 6(d). Conversely, as the flipping percentage of the PRFCM increases, random-flip disorder results in a significantly wider, strongly skewed distribution with the emergence of high intensities in Figs. 6(e) and 6(f). Obviously, the increasing flip percentage results in the emergence of magnetic hot spot with high intensities while more dark areas (dark spots) appear according to the curves in Figs. 6(e) and 6(f) and magnetic intensity maps in Figs. 6(b) and 6(c). The number of magnetic hot spots in the PRFCM with a flipping percentage of 50% is much less than that in the PRFCM with a flipping percentage of 25%. Therefore, random planar rotation or planar flip operation can degrade the collective response of the coherent metasurface.

In summary, the random metasurface with ASRs has been proposed via planar rotation or planar flip operation. We have theoretically and experimentally investigated the dependence of the collective response of the planar random-flip coherent metasurface on the flipping percentage and further studied the near-field coupling between metamolecules of the metasurface. The random planar rotation or planar flip operation can degrade the collective response of the coherent metasurface. The underlying physics mechanism can be well understood by multipole decomposition and induced surface current distributions. The planar rotation or flip operation offers a way to modulate microscopic excitations of multipoles, especially producing a TD mode as well as hot and dark spots. The experiment results well agree with the simulated ones. The planar rotation or planar flip strategy may be used to develop random metasurfaces and expand the application possibilities of coherent metasurfaces.

This work was supported by the National Natural Science Foundation of China (NNSFC) (Nos. 62275061 and 62175049), the Natural Science Foundation of Heilongjiang Province in China (No. LH2021A008), and the Fundamental Research Funds for the Central Universities (No. 3072022TS2509).

The authors have no conflicts to disclose.

Botian Sun: Conceptualization (lead); Investigation (lead); Writing – original draft (lead). Kaihao Zheng: Conceptualization (equal); Investigation (equal). Zhaoqi Jiang: Investigation (equal). Wenjia Li: Investigation (supporting). Bo Lv: Investigation (supporting). Yuxiang Li: Investigation (supporting). Zheng Zhu: Investigation (supporting). Chunying Guan: Conceptualization (equal); Supervision (supporting); Writing – review & editing (supporting). Jinhui Shi: Conceptualization (equal); Supervision (lead); Writing – review & editing (lead).

The data that support the findings of this study are available from the corresponding authors upon reasonable request.

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