Introduction

Metasurface has attracted a lot of attentions due to its gorgeous performances and ultrathin thickness compared to conventional bulk optical components1,2,3. It is composed of 2D nanostructure arrays with gradient phase change, and can arbitrarily manipulate magnitude, phase, and polarization of electromagnetic waves at a sub-wavelength scale along propagation direction. So far, metasurfaces of V-shaped antennas, C-shaped antennas and rectangular patches have been reported to artificially control the characteristics of waves4,5,6. In specific, when a polarized wave is incident on the metasurface antennas, both antenna modes can be excited but with substantially different amplitude and phase caused by their distinctive resonance conditions, then anomalous cross-polarized scattered light are produced. Based on these exotic properties, various metasurface-based optical devices including flat lens7,8,9,10, beam reflectors11,12, wave plates13,14, vortex generators15,16,17, meta-deflectors18, holograms19,20,21 and other novel, high-performance versatile photonic metadevices22,23 have been studied. Nonetheless, the optical characteristics are usually correlated with the geometry of the designed object, for example, cylinders lead light to a line, spheres result in a point and arbitrarily shaped objects introduce optical aberrations. As in actual operation the geometry is affected by many uncertain factors including deformation, vibration, non-standard surfaces, non-perfectly planar, etc., all of which increase the uncertainty of light manipulation in phase, amplitude and polarization22,23, stabilization of the optical functionalities has been a long-standing design challenges for optical engineering.

In this paper, the stabilization of anomalous refraction and reflection characteristics in a bended V-shaped antenna metasurface are investigated. Theoretical and experimental results show that the anomalous reflection is sensitive to deformation of the metasurface geometry. In contrast, the anomalous refraction is, surprisingly, barely affected by relatively small-angle tilting or bending and able to support perfect beam orienting.

Results

Operation principle and analysis

As is known, a metasurface consists of arrays of plasmonic antennas with linear gradient phase can generate anomalous reflection and refraction which is in agreement with generalized Snell’s laws derived from Fermat’s principle. When a light is incident on the interface of Fig. 1(a) at angle θi, due to the Fermat’s principle, phase difference between the two light paths should be zero and satisfy k0ni sin (θi)dx + (ϕ + dϕ) = k0nt sin (θt)dx + ϕ. In which, θt is angle of refraction, dϕ and ϕ + dϕ are severally the phase discontinuities at locations where the two paths cross the interface, dx is the distance between the crossing points. ni and nt are refractive indices, k0 is vacuum wave vector. Then, from above we can get the generalized Snell’s law of refraction1

$${n}_{t}\,\sin \,({\theta }_{t})-{n}_{i}\,\sin \,({\theta }_{i})=\frac{{\lambda }_{0}}{2\pi }\frac{d\phi }{dx}$$
(1)
Figure 1
figure 1

Light propagation in metasurface. (a) Schematic used to derive the generalized Snell’s law of refraction. (b) Refraction path considering the initial interface of (a) is rotated α angle.

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Similar to the refraction, it has

$$\sin \,({\theta }_{r})-\,\sin \,({\theta }_{i})=\frac{{\lambda }_{0}}{2\pi {n}_{i}}\frac{d\phi }{dx}$$
(2)

where θr is the reflection angle.

In actual operation, uncertain factors such as deformation, vibration, non-standard surfaces, non-perfectly planar, etc., may affect the geometry of an optical object and thus the stabilization of its functionalities. To investigate the stabilization of the metasurface’s optical characteristics, we first analyze the impact on the anomalous refraction and reflection when the metasurface is artificially tilted. Considering the initial interface of Fig. 1(a) is rotated α angle and become tilted interface of Fig. 1(b), where the original refraction path and normal vector for the initial interface are marked by gray lines. In Fig. 1(b), \({\theta ^{\prime} }_{i}\) and \({\theta ^{\prime} }_{t}\) are severally the new angle of incidence and refraction, dϕ′ and ϕ′ + dϕ′ are the phase discontinuities at the locations where the two light paths cross tilted interface, respectively. βt is the angle difference between refraction for the initial and tilted interfaces, and \({\theta ^{\prime} }_{i}={\theta }_{i}-\alpha \), \({\theta ^{\prime} }_{t}={\theta }_{t}+{\beta }_{t}-\alpha \). According to Snell’s law21, it has:

$${n}_{t}\,\sin \,({\theta }_{t}+{\beta }_{t}-\alpha )-{n}_{i}\,\sin \,({\theta }_{i}-\alpha )=\frac{{\lambda }_{0}}{2\pi }\frac{d\phi ^{\prime} }{dx^{\prime} }$$
(3)

We note that when phase gradient along the interface are designed to be constant, \(\frac{d\varphi ^{\prime} }{dx^{\prime} }\) in Eq. (3) will be equal to \(\frac{d\varphi }{dx}\) in Eq. (1). Especially, if the metasurface is thin enough, it can be treated as an interface and nt ≈ ni in Eq. (3) can be provided. Under these conditions, for small α, we can get βt ≈ 0 due to Eqs (1, 3). This implies that a small tilting rotating angle of the metasurface can almost have no impact on the propagation direction of its anomalous refractions.

Similarly, for the reflection, it has \({\theta ^{\prime} }_{i}={\theta }_{i}-\alpha \), \({\theta ^{\prime} }_{r}={\theta }_{t}+\alpha -{\beta }_{r}\), and

$$\sin \,({\theta }_{r}+\alpha -{\beta }_{r})-\,\sin \,({\theta }_{i}-\alpha )=\frac{{\lambda }_{0}}{2\pi {n}_{i}}\frac{d\phi ^{\prime} }{dx^{\prime} }$$
(4)

While βr is the angle difference between reflections for initial and tilted interfaces. From Eqs (2, 4) we can know even α is small, changed angle βr ≈ 2α still can be produced.

Evidently, the above results demonstrate that the anomalous reflection is sensitive to tilting rotations of the object, however, the anomalous refraction direction only be insignificantly affected by the tilting and lead to perfect stabilization of light propagation direction. Such stabilization can be attributed to the peculiar physical structure and mechanism of metasurface for anomalous refraction. That is, in general refraction system, it has nt sin (θt) = ni sin (θi), and the refraction is caused by the difference of nt and ni, a tilting angle α will result in new incident θi − α and refraction θt + βt − α, and the refraction requirement of ni ≠ nt will make direction changed angle βt impossible be zero. In metasurface, the refraction is mainly caused by the \(\frac{{\lambda }_{0}}{2\pi }\frac{d\phi ^{\prime} }{dx^{\prime} }\) (Eq. (3)). Especially when it is thin enough and treated as interface, approximate ni = nt make direction changed angle βt ≈ 0 be possible for small α due to Eqs (1,3). While in system such as grating and mirror, the direction control of the light beam is primarily dominated by reflection, which is sensitive to the tilting α. Comparatively, only the anomalous refraction by metasurface can offer direction changed angle βt ≈ 0 for α tilting. This has important significance to stabilize beam orienting under various conditions such as deformation, vibration, non-standard surfaces, non-perfectly planar in actual operation, which are bound up with the arrangements of various tilting.

Designed procedure of metasurfaces

Specifically, the schematic of the V-shaped antenna metasurface is shown in Fig. 2(a). In which, an array of super-unit-cells arranged periodically is fabricated on flexible substrate polyimide (PI) and the super-cell includes eight anisotropic V-shaped resonating units. Resonant units with gradient phase spacing are arranged in a linear manner, this super-unit-cells can cover a phase of 0 to 2π and can achieve anomalous refraction and reflection with cross-polarization, obeying generalized Snell’s law21. Here we take 4.3 THz (corresponding to wavelength 69.8 μm) TM (electric field along y direction) beam normal incident from bottom of the PI substrate (Fig. 2(a)) for example, the PI substrate dielectric constant is 3.5 and it’s thickness 100 μm, the V-shape copper antenna unit is as with thickness 200 nm and width of arm 1.8 μm. Then, to satisfy the requirement of gradient phase increase from 0 to 2π, the geometric parameters of the antennas are calculated by CST STUDIO SUITE (CST) and present as following. In detail, periodic lengths of units are px = py = 25 μm, the geometric parameters from 1 to 4 units are severally L1 = 12.1 μm, L2 = 10.5 μm, L3 = 9.8 μm, L4 = 7.5 μm, and α1 = 0°, α2 = 30°, α3 = α4 = 90°. The units of 5 to 8 are obtained by making a mirror symmetry for 1 to 4 units.

Figure 2
figure 2

Designed flat V-shaped metasurface and its characteristics. (a) Schematic of proposed V-shaped metasurface. (b) Total scattering pattern when TM wave (electric field along y direction) is incident on the metasurface; scattering pattern of cross-polarized Ex is shown in the inset. (c) Field distribution of cross-polarized Ex along propagation direction (x-z plane).

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For this antennas metasurface shown in Fig. 2(a) with TM incidence, its scattering pattern is illustrated in Fig. 2(b), which shows that three peaks are situated at angle 0, −20 and −160 degrees, respectively. Furthermore, scattering pattern for the cross-polarized Ex is separately shown in Fig. 2(b) inset, in which there are two peaks near −20 and −160 degrees. By comparing Fig. 2(b) with Fig. 2(b) inset, we get the following conclusion: the V-shaped metasurface in Fig. 2(a) can produce co-polarized transmission, cross-polarized refraction and reflection at 0, −20 and −160 degrees, respectively.

In addition, the cross-polarized Ex field distribution along propagation direction (x-z plane) is illustrate in Fig. 2(c). In which, the cross-polarized anomalous refraction and reflection severally appear at −20 and −160 degrees in accordance with the results of Fig. 2(b).

To explore the optical stabilization V-shape metasurface, the anomalous refraction and reflection characteristics when the metasurface object is in bending status are numerically studied. We note that a bending can be approximately regarded as the arrangement of successive tilting. The schematic of bending metasurface is shown in Fig. 3(a). In which, R is the radius of curvature of the metasurface and TM wave is incident from bottom of the metasurface along z axis. When upward bending R are 3 mm, 4 mm, 5 mm, the cross-polarized Ex field distributions are severally illustrated in Fig. 3(b–d). From Fig. 3(b–d), it can be seen that show that both anomalous refraction and reflection are obtained. However, the reflection field forms converging, and for R = 3 mm, 4 mm, 5 mm, the converging points are severally near (−577, 1565), (−746, 1738) and (−811, 2172) of (x, z). On the other hand, the refraction are almost not affected, and for R = 3 mm, 4 mm, 5 mm, the scattering patterns of the anomalous refraction in Fig. 3(b–d) are correspondingly illustrated in Fig. 3(e–g) which show that the anomalous angle are all near −20 degrees. The above results are in good accordance with the former prediction from Eqs (3, 4), which imply that anomalous reflection is sensitive to the tilting rotations and the anomalous refraction only be insignificantly affected by the tilting rotation angle α.

Figure 3
figure 3

Upward bending V-shaped metasurfaces and its characteristics. (a) Schematic of upward bending V-shaped metasurface. (b–d) Field distributions of cross-polarized Ex for R = (b) 3 mm, (c) 4 mm, (d) 5 mm, respectively. (e–g) Scattering patterns of cross-polarized Ex for R = (e) 3 mm, (f) 4 mm, (g) 5 mm corresponding to (b–d).

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For the downward bending, the shematic is shown in Fig. 4(a). When R = −3 mm, −4 mm, −5 mm, the cross-polarized Ex field distributions are severally displayed in Fig. 4(b–d), respectively, and the scattering patterns of anomalous refraction are correspondinglly illustrated in Fig. 4(e–g). Similar to the upward bending, the angle of anomalous refraction are all near −20 degree. Nonetheless, unlike the upward bending, the reflection field for the downword bending is diverging.

Figure 4
figure 4

Downward bending V-shaped metasurfaces and its characteristics. (a) Schematic of downward bending V-shaped metasurface. (b–d) Field distributions of cross-polarized Ex for R = (b) −3 mm, (c) −4 mm, (d) −5 mm, respectively. (e–g) Scattering patterns of cross-polarized Ex for R = (e) −3 mm, (f) −4 mm, (g) −5 mm corresponding to (b–d).

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Sample fabrication and THz measurements

To experimentally confirm our design idea of the bending metasurface, we fabricated V-shaped copper antennas array on flexible PI film by conventional UV lithography. In Fig. 5(a,b), the sample and the microscopic image are shown. The schematic diagram and the measurement setup of experimental instruments for sample, as shown in Fig. 6. We chose a quantum cascade laser (QCL) as the source to generate incident light at a frequency of 4.3 THz. The direction of polarization of the incident light is along the y-direction and is focused by a parabolic mirror. Two terahertz polarizers are placed sequentially in front of the sample, which is caught by two movable holders and different R are provided by tuning the distance of the holders. The polarized direction of the first polarizer is at 45 degree to x axis. The second polarizer’s polarized direction is along y axis to offer TM incidence. We chose the Golay Cell terahertz detector to receive the signal, which is 25 cm from the sample. It is mounted on a rotating table and can be rotated from −30 degrees to 30 degrees on the x-y plane to detect transmission.

Figure 5
figure 5

Sample fabrication. (a) Sample of the V-shaped copper antennas metasurface. (b) Part microscopic image of the sample.

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Figure 6
figure 6

Experimental instruments of THz measurement setup with schematic diagram shown in inset.

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Measurement results

The experimental results of transmissions for bending R = 0 mm, 3 mm, 4 mm, 5 mm, −3 mm, −4 mm, −5 mm, are shown in Fig. 7. The results show that each curve has two peaks severally located at 0 angle and around −20 degree. Since the former numerical calculation has demonstrated that the co-polarized transmission is at angle 0 and the cross-polarized refraction at angle 20, these experimental results agree very well wtih the numerical predictions. Particularly, the refractions are almost all located near −20 degree for R = 0 mm, 3 mm, 4 mm, 5 mm, −3 mm, −4 mm, −5 mm, verifying that the refraction direction is insignificantly affected by the deformation of geometry and lead to perfect stabilization of light propagation direction.

Figure 7
figure 7

Measurement results of transmissions for bending R = 0 mm, 3 mm, 4 mm, 5 mm, −3 mm, −4 mm, −5 mm, respectively.

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Discussion

The influence of the refraction and reflection when the metasurface is artificially tilted are theoretically analyzed, and the influences for upward and downward bending metasurface for R = 0 mm, 3 mm, 4 mm, 5 mm, −3 mm, −4 mm, −5 mm are numerically and experimentally studied. The results show that the anomalous reflection is sensitive to the deformation of the geometry. However, small angle tilt almost has no influence on the propagation direction of anomalous refraction, leading to perfect beam orienting. As in real-world applications, the optical objects are affected by multiple uncertain factors including deformation, vibration, non-standard surfaces, non-perfectly planar, etc., our findings shed new light on the stabilization issue of optical functionality.

Materials and Methods

Sample fabrication

The fabrication procedure of V-shaped copper antennas array metasurface was achieved by typical UV lithography. First, All V-shaped structures were composed of 200 nm thick copper layer on the flexible PI film by conventional electron-beam (E-beam) evaporation. After, positive photoresist SUN-115P was spin coated onto the copper film by a spin coater at a spin speed of 4000 rpm for 30 s. This was followed by hot plate baking it at 100 °C for 1 minute, the feature of V-shape antenna arrays was transferred from mask to the photoresist surface by UV lithography. Subsequently, the film was placed in developer of SUN-238D for 4 s, and then etched the exposed copper by reactive ion etching. Subsequently, the V-shaped metallization of Cu (200 nm) was evaporated and lifed-off. In order to experimentally confirm our design idea, we made several metasurfaces by typical UV lithography, as shown in Fig. 5(a,b).