Controlling and manipulating electromagnetic radiation by its interaction with materials is a cornerstone of optics. The study of metamaterials that allow selecting their permittivity and permeability freely from positive to negative values has led to new visions including sub-wavelength-resolution imaging [1–3], invisibility cloaking [4–6], and mimicking celestial mechanics [7]. Obviously, seeing sharper or becoming invisible are strong driving forces. However, the development of practical metamaterials for the THz range up to the visible is lagging behind theoretical studies and envisaged applications. A practical metamaterial would be available in copious quantities and enable customized solutions, much like a foil, whereas most of present-day metamaterials are made by time-consuming primary pattern generation and involve dielectric substrates or matrices, which might substantially restrict their usefulness and applicability due to electric, mechanical, and thermal properties of dielectrics as well as their sensitivity to humidity and radiation degradation [8–14].

In earlier work, we mitigated such restrictions by micromanufacturing of free-standing metamaterials from metal-string arrays suspended in free space by plastic window-frames [15]. Strings are a preferred architecture of metamaterials as they extend continuously along one dimension. Made of gold, strings were S-shaped longitudinally. Aligned assembly of two chips of strings created a bi-layer chip in which two layers of S-strings, typically 1 to 10 μm apart, were opposed such as to form the well-known S-string resonator loops [16]. However, window-frames were still rigid and restricted the useful range of incidence angles.

Here, we demonstrate an all-metal approach where the conducting metal is the structural material simultaneously and no rigid window-frame is needed anymore. Individual S-strings are connected by transverse rods creating a space-grid that is self-supporting, locally stiff, but globally flexible. Connections are made between oscillation nodes of the current in the strings to minimize any influence on resonances. Introducing such bonds between the “atoms” of the metamaterial, we form a “crystal lattice”, overcoming conventional “frozen-in solutions” like matrix embedding or thin films on substrates. For their foil-like appearance, we dub such space-grids “meta-foil”. Meta-foils can be tailor-made to virtually any shape, bent, and wrapped around objects to hide and shield them from electromagnetic radiation, thus becoming true metamaterials on curved surfaces. Owing to the metallic interconnects, they are much more robust than earlier bi-layer chips [15].

Figure 1(a) shows a 3D schematic of the meta-foil and its geometric parameters. The plane of the meta-foil corresponds to the y-z plane. S-strings stand upright with leg “b” extending along the x-axis normal to that plane. Adjacent strings are shifted along z by (a-h)/2 with respect to each other to form the resonance loops. Made from the same metal as the S-strings, interconnecting rods or lines run along the y axis, are centered on leg “b”, and repeated with 1S or 2S periods. A normally incident wave propagates in x-direction along leg “b”, with electric and magnetic field vectors pointing along z and y axes, respectively, thus maximizing magnetic coupling for normal incidence in contrast to bi-layer chips. Inductive loops are formed by half an S in one row and the oppositely oriented half S in the adjacent row. Opposite legs “b” form capacitances, loop areas determine the inductance. The time-varying magnetic field normal to the loops induces an electromotive force that drives a loop current, eventually into resonance. In the 1S case, the unit cell consists of one central capacitor formed by two opposite non-connected “b” legs connected to two inductive loops extending in +z- and –z-directions up to and including half of the next interconnecting rod. The 2S case has three identical capacitors in the unit cell with two inner loops involving a series connection of two capacitors and two inductors formed by “a” and “b” legs, and two end loops that are identical to those in the 1S case. In an equivalent circuit diagram, loops are alternatingly twisted to ensure likewise charging of common capacitors.

 

Fig. 1 (a) 3D schematic of a 1SE meta-foil showing lithography layers, coordinate frame, definition of geometric parameters, and their measured values (unit μm). (b) Photo of flat and rolled Au meta-foils 8×7×0.015 mm³ (L×W×H) in size. (c) Scanning electron microscope (SEM) image of a 2SP Au meta-foil, scale bar 100 μm. (d) SEM image of a warped 1SP Au meta-foil, scale bar 50 μm.

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Meta-foils are manufactured by means of 3-level lithography using either UV or X-rays depending on structure geometry. Figure 1(a) illustrates three layers coloured in dark, middle, and light red. Middle and top layers need precise alignment of mask and substrate as well as re-deposition of a gold plating base. Bottom and top layers contain all conductors parallel to the y-z plane, the middle layer the “vias” and interconnecting rods that run along x- and y-directions, respectively. Top and bottom layers are related by a translation of (a-h)/2 along z. Hence, only two masks are needed, in principle, one for bottom and top layers, and one for the middle layer. However, for practical reasons, three masks are used. To keep the resonance frequency within 10% of the nominal value, alignment accuracy must be <20% of the gap between S–strings which controls capacitance and resonance frequency. Omitting details, we note that this 3-level process is also suitable for plastic moulding, enabling cost-effective large-volume manufacturing of meta-foils [17]. Meta-foils presented here feature S-strings that are either equidistant and interconnected after one or two periods of “a” (1SE or 2SE), or grouped in pairs that are further apart than gap d (1SP or 2SP). Measured dimensions are given in Fig. 1(a) while Fig. 1(b)-(d) show a photo and SEM images of meta-foils as manufactured.

The spectral behavior of samples was characterized by Fourier transform interferometry in the far infrared between 2 and 14 THz. Beam spot size on sample was 1.5 mm at normal incidence, total beam divergence 60°. Figure 2(a) displays measured transmission spectra of a 1SE foil versus frequency with the incidence angle α around the z axis running from 0°(9°)81° in comparison with simulated spectra (Fig. 2(b)). Peak positions and widths agree fairly well. MWS commercial software was used for full-wave simulation [18]. Two dominant peaks appear at 3.2 THz and 6.8 THz. From parameter and index retrieval calculations (Fig. 2(c) and (d)) [19], we see that both Re(ε) and Re(μ) are negative around 3.2 THz, so the peak at 3.2 THz (λ = 94 μm) is assigned the well-known left-handed resonance of the fig-8 loop in S-strings [16]. Its wavelength-structure-size ratio of λ/b = 94/15 = 6.26 indicates a reasonable effective-medium approximation. The figure-of-merit FOM = abs(Re(n)/Im(n)) is 3 at 3.27 THz and 5.6 at 3.47 THz from simulation. The peak at 6.8 THz (λ = 44 μm) is a right-handed electrical resonance of one half S acting as an antenna between coupling capacitors or interconnecting rods as nodes. Here, the length of the shortest such resonator is a/2+h = (15.5+5) μm = 20.5 µm, in 7%-agreement with λ/2 = 22 μm. Note that in the shaded frequency ranges <3 THz and >7.4 THz, the refractive index reaches the edge of the Brillouin zone π/(k⋅b), as shown in Fig. 2(d). Therefore, a resonance of permeability (permittivity) and anti-resonance of permittivity (permeability) behavior accompanied by a negative imaginary part of the permittivity (permeability) can be observed. This resonance and anti-resonance behavior is very similar to that observed in the left-handed metamaterial composed of rod and split-ring resonators, which was shown to be a result of the periodic medium model with resonant structures [20]. Reaching the edge of the Brillouin zone in the periodic medium deforms the Drude-like (permittivity) and Lorentz-like (permeability) resonance behavior of the left-handed metamaterial [20]. Below 3 THz, the retrieved refractive index reaches the edge of the Brillouin zone, and the wavelength inside the metamaterial is comparable or smaller than the structure period ruling out the use of an effective permittivity and permeability to characterize the sample. Note as well that the transmission peak at 6.8 THz is due to an electric plasma frequency around 6 THz accompanied by a higher electric resonant frequency around 7.6 THz. From 6 THz to 7.6 THz, both permittivity and permeability are positive corresponding to a pass-band. Above the electrical resonance of 7.6 THz, the negative value of the permittivity and positive value of the permeability lead to a stop-band for which reason we see an isolated transmission peak at 6.8 THz instead of a continuous pass-band.

 

Fig. 2 (a) Measured and (b) simulated transmission spectra of a 1SE sample with the incidence angle α as a parameter varying from 0°(9°)81°, the electric field pointing along the z axis. Spectra are vertically shifted with respect to each other for clarity. (c) and (d) Retrieval calculation of the relative complex permittivity ε, permeability μ, and refractive index n of the 1SE sample. In the shaded frequency ranges <3 THz and >7.4 THz Re(n) reaches the edge of the Brillouin zone.

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When the incidence angle α, defined as the angle between the normal to the meta-foil plane y-z and the wave vector k keeping the latter perpendicular to the z-axis, is varied, the magnetic field component normal to the induction loops changes as H_n=H₀cosα while the electric field points along the S-strings independently of α. H₀ is the magnetic field amplitude of the incident wave. Thus, the magnetically excited left-handed peak at 3.2 THz should vary as cosα which is the case over the wide range from 0° to 81° implying a large latitude for choosing α which may practically lie between ±30° or more (Fig. 3 ). The relative peak width of about 25% also facilitates operation. The cos²α-dependency of the electrical resonance peak is due to the relative phase shift of the electric field between adjacent strings and the short-circuit current generated. Simulated data also fit well.

 

Fig. 3 Transmission peak areas versus incidence angle with a cosα-fit to the 3.2 THz left-handed magnetic resonance and a cos²α–fit to the 6.8 THz right-handed electric resonance (1SE sample).

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Meta-foils can be strongly bent without compromising their function as shown by spectra of a 2SE sample with bending radii varying from infinity to 1 cm (Fig. 4(a) ). The absence of any significant change of the spectra confirms the local rigidity and global flexibility of the meta-foil. Bending to a cylinder around the z-axis deforms cells from their original parallel to a wedge shape. With 1 cm bending radius, the wedge angle between two adjacent strings is 1 mrad and the change of capacitor gap ±15 nm. The relative change being only 0.003, the gap expansion in the upper half of the capacitor and the contraction in the lower cancel each other leaving no measurable effect. The slight amplitude reductions result from the decrease of the normal magnetic field component towards the edges.

 

Fig. 4 (a) Spectral response under bending from flat to 1 cm radius around z axis (2SE) under normal incidence. (b) Peak shift and damping upon filling with PMMA (1SP) under normal incidence. Spectra are vertically shifted with respect to each other for clarity.

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From the 60%-transmission of the left-handed peak at 4 THz (Fig. 4(a)) we infer that the double spacing of interconnecting lines in structure 2SE as compared to 1SE (Fig. 2) favors larger resonance peaks. Moreover, the frequency shift from 3.2 to 4 THz is explained essentially by the actual resonance loops. In the 1SE case, loops are formed by one capacitor short-circuited over the interconnecting line. In the 2SE case, half of the loops are like 1SE, but the other half consists of the canonical S-string loop that has two capacitors in series. Therefore, the frequency is 1.41 times higher, i.e., 4.5 THz. Superposition of peaks at 3.2 THz $3.2{(1+\sqrt{2}/2)}^{1/2}$and 4.5 THz gives one peak at 3.85 THz, close to the measured 4 THz. A more accurate analytical calculation yields a frequency of THz = 4.18 THz, in fair agreement as well.

The resonance frequency of all-metal meta-foils can be shifted by adding dielectrics which enables sensing and tuning (Fig. 4(b)). Peak shift and broadening caused by PMMA reflect quantitatively the relative permittivity of PMMA. The ratio of the three peaks without and with PMMA is 1.54, 1.52, and 1.5 for the 3.5, 6.3, and 7.7 THz peaks, respectively, proportional to the square root of the permittivity and, hence, to the refractive index. In fair agreement, the latter is found as 1.57 in the THz region [21]. The observed damping indicates a PMMA-induced loss and supports the initial statement that matrix-embedded metamaterials may be significantly restricted in their function. Finally, we note without showing graphs that the meta-foil was operated from room temperature to >120 °C without a detectable change of results.

The meta-foil is a new photonic material implementing a “crystalline state” of the “atoms” of the meta-material, i.e., the S-string resonant loops. Its easy handling enables a wide range of THz frequency applications. Optical elements like filters, polarizers, reflectors, and absorbers may be made from meta-foils in various geometrical shapes benefiting from the relative insensitivity of meta-foils to incidence angle, deformation, spatial or angular misalignment, and temperature. Meta-foils can be filled with dielectrics for sensing and tuning. Forthcoming work will focus on stacking to provide extended three-dimensional arrangements and enhance optical properties including resonance widths. Stacks of meta-foils may also serve as polarizing devices. Ultimately, meta-foils will allow introducing an almost arbitrary spatial distribution of refractive index by corresponding variations of geometric parameters, thus enabling index-gradient optics for infrared microscopy and THz imaging as well as radically new optical components.