Wavefront Control with Nanohole Array-Based Out-of-Plane Metasurfaces

Planar metasurfaces provide exceptional wavefront manipulation at the subwavelength scale by controlling the phase of the light. Here, we introduce an out-of-plane nanohole-based metasurface design with the implementation of a unique self-rolling technique. The photoresist-based technique enables the fabrication of the metasurface formed by nanohole arrays on gold (Au) and silicon dioxide (SiO2) rolled-up microtubes. The curved nature of the tube allows the fabrication of an out-of-plane metasurface that can effectively control the wavefront compared to the common planar counterparts. This effect is verified by the spectral measurements of the fabricated samples. In addition, we analytically calculated the dispersion relation to identify the resonance wavelength of the structure and numerically calculate the phase of the transmitted light through the holes with different sizes. Our work forms the basis for the unique platform to introduce a new feature to the metasurfaces, which may find many applications from stacked metasurface layers to optical trapping particles inside the tube.


Dispersion relation calculations
The dispersion relation of multilayered metamaterial helps to determine the existence of SPPs excited through a metal layer sandwiched between two dielectric media. To comprehend the excitation of internal and external SPPs in stacked hole RUTs, the dispersion relation is formulated using the Helmholtz equations to determine the field solutions along the propagation direction as well as the evanescent fields confined in the transverse direction of the multilayered structure. As shown in Figure S1 (a), the layers were assumed to be parallel in the x-axis with a relative permittivity dependent on one spatial coordinate ( = (z)). The y-direction of the multilayered structure is considered to be infinite and homogeneous ( ∂/∂y = 0). The transverse confined field in the z-direction is related to the attenuation coefficients in two metallic media with a field dependence of z > t d /2 + t m , t d /2 < z < (t d /2 + t m ), −t d /2 < z < t d /2, and (−t d /2 + t m ) < z < −t d /2 as shown in Figure S1 (a). Here we considered the transverse magnetic (TM) plasmonic modes in each region by applying the required boundary conditions. We defined the vector potentials of S-1 the electric field and magnetic field as F and A, respectively.
For simplicity, we limit the formulation to the symmetric case whereby both the semiinfinite layers relative to the dielectric constants are equal to 1 = 2 = air while m1 = m2 = Au depicts the Au region. d represents the permittivity of the sandwiched dielectric region.
Based on the aforementioned conditions, the A can be expressed as A zẑ = Ae kzz e iβxẑ , where e kzz describes the electromagnetic field depth dependence with k z = ± β 2 − k 2 0 and k 0 = 2π/λ. k 0 depicts the vacuum wavenumber and propagation wavenumber along x direction shown in Figure S1 (b) is represented as β. Since metals are lossy, the field penetration depth is minimal and essentially the dominant field will be surface waves concentrated at  Figure S1: (a) Geometry of a metal/dielectric interface, with a dielectric sandwiched between two metal layers. (b) Dispersion relation of SPPs excited by the metal-insulator-metal structure that depicts the internal and external SPP modes which propagate along the internal and external metal-insulator interfaces. The light line air and d show the dispersion relation in free space and the dielectric medium, respectively. The SPPs even and odd show the symmetric and anti-symmetric modes excited by a dielectric sandwiched between two metal layers.
Solving equation set S1-S3 and applying the boundary conditions yields the field solutions which are represented below; where , µ, and ε 0 represent the angular frequency, permeability, and relative permittivity of the multilayered media, respectively. From the above equations, the dispersion relation S-3 can be obtained by solving the system of linear equations and applying continuity boundary conditions at every metal-dielectric interface expressed as, and simplified to and describes the SPPs in the multilayered structure. The dispersion relation helps to predict analytically the extraordinary optical transmission (EOT) peaks from the excitation of both internal and external SPP modes. This model formulated to depict the EOT peaks at different spectral wavelengths does not consider the holes stacked in the multilayered structure. Stacked holes can be accounted for by implementing the conservation of momentum and energy between the incident optical field and the periodicity of the rectangular array.
The relation of SPPs and conservation of momentum can be expressed as where k spp ≡ β( ) is the wave vector of the SPP, G x = 2π/a x , and G y = 2π/a y are the reciprocal lattice vectors for the stacked holes, and k 0 sin φ is the in-plane component of the incident wave vector. a x and a y are the lattice periodicity in the x and y-directions.
Notably, the normalized frequency at large Bloch wave vectors tends to approach the SPP frequency, which is expressed as with ε i = ε air or ε d for external or internal SPPs, respectively. However, for a short wave S-4 vectors β k p , the SPP propagation constant is related to the internal SPPs, and is expressed as where k p = 2π/λ p , and the external frequency of the low frequency range can be formulated as E-field response Figure S2 presents the E-field response of multiple unit cells at 640 and 780 nm.

Phase response of uniform hole size
The phase of the E-field response of the supercell with the same hole sizes for planar and curved cases are presented in Figure S3. Figure S4 shows the E-field for a single hole with different tilt angles. S