Negative refraction in twisted hyperbolic metasurfaces

2.1 Structural design

The sketch of a single unit of the designed metasurfaces is schematically shown in the left panel of Figure 1, where the parameters are: p = 9 mm, a = 4.5 mm, b = 2.6 mm, c = d = 1.6 mm, and e = 0.56 mm. To fulfill the requirement of rotation in the following steps, the parameters must meet the condition: ( b + c + 3 × e ) 2 + ( a + e + d ) 2 < p 2 . In the middle panel, the copper patterns are periodically arranged on a lossless substrate (F₄BM300) with a permittivity of 3.0, and the thicknesses of the dielectric layer and metal patterns are 1 and 0.035 mm, respectively. Surface waves at the topological transition frequencies along orthogonal directions of the first Brillouin zone are rotated along the horizontal and vertical directions when the unit cells rotate by the angle α (0 ≤ α ≤ 45°) counterclockwise, achieving multi-directional SPP propagation. Finally, as depicted in the right panel of Figure 1, an integrated model composed of arrays of units A (α ₁ = 45°) and units B (α ₂ = 135°) is established to realize dual-channel negative refraction and nondiffraction negative refraction when the SPPs reach the interface between the two different arrays of units and then propagate along two paths perpendicular to the original directions.

Figure 1:

Schematic diagram of the proposed twisted HMSs consisting of a dielectric/Cu bilayer, where p = 9 mm, a = 4.5 mm, b = 2.6 mm, c = d = 1.6 mm, and e = 0.56 mm, and the counterclockwise rotation angle of the unit cell is α.

2.2 Basic physical properties

Based on the effective medium theory (EMT), for the electric HMSs, the relative permeability μ ˆ is reduced to a unit tensor, and the relative permittivity tensor is given by [60, 61]:

The subscripts xx and yy denote the components parallel to the x and y axes, respectively. The dispersion relation of the HMSs can be determined by substituting Eq. (1) into the Maxwell’s equations. In particular, for the TM mode, the dispersion relation can be written as:

where k ₀ is the wave-vector in free space, and k _x, k _y are the x and y components of the wave-vector, respectively. Then, the rotation matrix T ˆ ( α ) is adopted to the designed model to better understand the dispersion when the unit cell rotates counterclockwise at an angle α around the z axis:

The relative permittivity tensor in Eq. (1) should be rewritten as:

In this case, the dispersion relation can be derived as:

where A = cos 2 ( α ) ϵ x x + sin 2 ( α ) ϵ y y , B = sin 2 ( α ) ϵ x x + cos 2 ( α ) ϵ y y , and C = sin ( 2 α ) ( ϵ y y − ϵ x x ) . When α = 90°, the dispersion relation Eq. (5) changes to:

We can clearly see that when the structural unit rotates 90°, the IFC rotates 90° accordingly. However, there are nondiagonal terms in the permittivity tensor when α = 0–90°:

Let us take α = 30° as an example:

At this time, the IFCs are no longer ideal ellipses, straight lines or hyperbolae, but have the feature of distortion. The rotation of the unit cell will strongly modify the IFC shape. As a result, the propagation properties of electromagnetic waves will become more diverse by tuning the rotational angle. The twisted models can not only apply to the engineering of SPP transport but also have potential applications in the design of new active devices.

To gain a better understanding of the electromagnetic characteristics of the presented HMS, we numerically calculated its band diagrams using the eigenvalue module of the commercial software Computer Simulation Technology (CST) along the highly symmetrical points in the first Brillouin zone. Based on the equation k = π × θ/180p, the values of k _x and k _y are obtained, and the 3D energy band diagrams/IFCs of the first two energy bands in the first Brillouin zone are presented in Figure 2(a)–(d) for different rotation angles, while the inset in Figure 2(a) indicates the first Brillouin zone and the highly symmetrical points. It is obvious that when the primitive cell rotates 0–45° counterclockwise, the IFCs rotate counterclockwise and the band overlap appears in the second IFC when α = 45°. Additionally, there are varying degrees of distortion. The presence of the extensive overlap means that there are no Lifshitz-like continuous topological transitions in the band diagrams, so the flat band is difficult to be independently excited. Fortunately, there are two ways to break the overlap to obtain a new self-collimation frequency: one is to break the extensive overlap between different energy bands, and the other is to generate a new resonance frequency excited by the incident electromagnetic waves [42]. Therefore, a new self-collimation frequency can be obtained with the appearance of new resonance modes by the counterclockwise rotation around the z-axis, which will simultaneously break the symmetry in the x and y directions. For example, when the unit cells rotate counterclockwise around the z axis, a tilted self-collimation frequency appears when α = 45°. Another appealing feature is that the extreme points (red areas) are no longer at the high symmetry points, which originates from the primitive cell rotation and the lattice no longer having a reflection surface, so the extreme points are not necessary at the boundary of the Brillouin zone. In order to study the underlying mechanism of these changes, we discuss the transmission spectra in the following.

Figure 2:

The 3D energy band diagrams/IFCs of the first two bands in the first Brillouin zone when the rotation angles are (a) 0° (the inset in the band diagram illustrates the first Brillouin zone and the highly symmetrical points), (b) 15°, (c) 30°, and (d) 45°, respectively. The unit of the frequency values in the IFCs is GHz.

As shown in Figure 3(a) and (b), a slight red shift appears in the transmission dip as the model rotates when the incident light is polarized along the x direction, and the coupling in the x-direction also rotates counterclockwise. However, a new dip appears at 12.77 GHz as an octapole when the model rotates 45°. Figure 3(c) and (d) exhibit, respectively, new resonance modes of the hexapole (13.26 GHz for α = 15° and 13.10 GHz for α = 30°) and octapole interactions along the y direction when the incident light is polarized in the y-direction. The higher-order coupling indicates that the metasurface manipulates the interaction between light and matter in new ways, and thus the suppression of radiation loss will be achieved. Besides, because the higher-order multi-dipoles and new types of interactions between the x and y directions are excited with the rotation of the unit cells, not only the energy bands rotate accordingly, but distortions of the band diagrams also appear. Therefore, the rotation operation can be used as an effective solution to obtain flat lines without distortion in the IFC, so as to achieve self-collimation frequencies.

Figure 3:

Transmittance spectra and E_z field distributions. (a and c) Normalized transmittance spectra when the incident waves are polarized in the x and y directions, respectively. (b and d) E _z field distributions on the surface of the unit cells at the resonance valleys corresponding to (a) and (c).

Based on the normalized transmittance spectra discussed above, we make a quantitatively characterization of the permittivity by treating the proposed twisted HMSs as homogeneous anisotropic effective-medium slabs, and the effective permittivity can be obtained by the S-parameter retrieval process [62], [63], [64]. The S matrix can be written as: S = ( S 11 S 12 S 21 S 22 ) , where the first and second subscripts indicate the output and input ports, respectively. Therefore, S ₁₁ and S ₂₁ denote, respectively, the reflection and transmission coefficients, which are related to the effective refractive index n and the impedance z of the dielectric slab by the following expressions:

where h is the thickness of the dielectric slab. The effective permittivity ε can be calculated by ϵ = n / z as follows:

The corresponding calculation results are plotted in Figure 4. When the rotation angle α = 0° and ϵ x y = ϵ y x = 0 , the effective permittivity only has diagonal terms, which successfully predict the effective medium properties. The gray areas (7.21–8.67 GHz and <13.42 GHz) represent the hyperbolic IFCs and topological transition frequency (7.21 GHz) where ɛ _xx  × ɛ _yy  < 0. However, there is obvious deviation in frequencies between the simulated IFCs and the calculated relative permittivity, mainly resulting from the constructed 2D homogeneous dielectric slab. This is due in part to the permittivity of the realistic 3D structure determined by the transcendental equation is more complicated and cannot be solved, and another cause is the small S parameter data in magnitude that have a great influence on the retrieval process [65]. In the case of 0° < α < 45° and ϵ x y = ϵ y x ≠ 0 , the relative permittivity obtained from EMT shows good agreement with Eq. (7). It can be seen that the permittivities ɛ _xx and ɛ _yy change dramatically with the breaking of the symmetry in the x and y directions due to the rotation of the models. Besides, the off-diagonal relative permittivities ɛ _xy and ɛ _yx denote the novel interactions between the x and y directions, and thus, the energy bands twist accordingly and the distortion occurs as shown in Figure 2. Due to the symmetry of the sample, the relative permittivities ɛ _xx  = ɛ _yy , and ɛ _xy  = ɛ _yx  = 0 when α = 45°, which is consistent with the permittivity tensor given in Eq. (7).

Figure 4:

Relative permittivity obtained from EMT when the counterclockwise rotation angles of the sample are (a) α = 0°, (b) α = 15°, (c) α = 30°, and (d) α = 45°, respectively. The gray regions in (a) exhibit the hyperbolic IFCs predicted by EMT.

2.3 Multi-directional SPP propagation

As shown in the right panel in Figure 1, a microwave dipole, resonating in the z direction, can generate surface waves on the surface of the electromagnetic HMSs. The CST time-domain solver was used to simulate the electric field E _z distributions of the rotated model (α = 0–45°) at the respective transition frequencies of the first two energy bands as depicted in Figure 5. As the surface plasmon group velocity is perpendicular to the IFC ( V ⇀ g = ∇ k ⇀ ω ) , self-collimation only occurs at the transition frequencies in two perpendicular directions. With the rotation of the model, the IFCs rotate counterclockwise, so the surface waves at the topological transition frequencies rotate counterclockwise accordingly. However, because of the obvious overlap and perturbation in the IFCs along with the rotation, the rotation angle of the surface waves at the topological transition frequencies is different from the twisted angle of the model when 0° < α < 45°. In this case, we obtain nondiffraction-like SPP transmission by searching the frequency at which the photon density of states reaches a maximum. The mismatching between the measured wave vectors and the calculated IFCs is owing to the distorted and extremely overlapped IFCs [66]. Consistent with the energy bands and IFCs described above, there are tilted lines in of the second energy band IFCs when the model rotates 45°, and therefore, there is nondiffraction transmission in the diagonal direction. Due to the high degeneracy of the self-collimating frequency, the surface waves are better confined than at other frequencies, so we choose the unit cell with α = 45° and α = 135° as the primitive cells for the following model.

Figure 5:

The twisted samples and E _z distributions at the transition frequencies when the counterclockwise rotation angles of the samples are (a) α = 0°, (b) α = 15°, (c) α = 30°, and (d) α = 45°, respectively. β ₁ and β ₂ are the counterclockwise rotation angles of the propagation directions of the SPPs with respect to the x and y axes, respectively.

2.4 Negative refraction

Due to the rotational symmetry, compared with the model whose rotation angle is α, the IFC of the model with a rotation angle of α +90° is relatively rotated by 90°. By combining different twisted meta-resonators, SPP propagation can be redirected at the interface at will. For example, observation of negative refraction phenomenon is presented in Figure 6 when the SPPs traverse an interface between two metasurfaces whose rotation angles are 45° (unit A) and 135° (unit B), respectively. We select two typical negative refraction phenomena and the corresponding IFCs for analyzing the underlying mechanism. As illustrated in Figure 6, the bottom half of the metasurface consists of arrays of units A (29 × 8 unit cells) while the top half array is made up of units B (29 × 8 unit cells). The electric dipole is located at the center of the boundary at the bottom side.

Figure 6:

Calculated E _z distributions in the xy plane at 0.05 mm above the integrated structure and the corresponding IFCs at (a) 8.10 GHz and (b) 8.50 GHz, respectively.

Interestingly, at 8.10 GHz, as the unit A has a hyperbolic IFC, the excited SPPs transmit along two crossing straight lines. Therefore, for units B, the IFC rotates 90° relative to that of units A, and the surface waves rotate 90° as well. The surface waves can reach the interface along two transmission paths, and they travel in the directions perpendicular to the original trajectories after passing through the interface, forming a dual-channel negative refraction phenomenon. On the other hand, at 8.50 GHz, the IFCs of units A are four straight lines with a rotational angle of 45° with respect to the horizontal direction. Therefore, the surface waves transmit along the diagonal directions without diffraction. At this time, the photon density of states reaches a maximum, which is expressed as nondiffraction negative refraction. The proposed HMS thus provides a new method for negative refraction phenomena, anomalous wave propagation, asymmetric transmission, etc. And there will be a wider range of applications in the areas of surface optoelectronic device integration, sensors, hyperlenses, and in-plane beam steering.