Controlling Dispersion Characteristics of Terahertz Metasurface

Terahertz (THz) metasurfaces have been explored recently due to their properties such as low material loss and ease of fabrication compared to three-dimensional (3D) metamaterials. Although the dispersion properties of the reflection/transmission-type THz metasurface were observed in some published literature, the method to control them at will has been scarcely reported to the best of our knowledge. In this context, flexible dispersion control of the THz metasurface will lead to great opportunities toward unprecedented THz devices. As an example, a THz metasurface with controllable dispersion characteristics has been successfully demonstrated in this article, and the incident waves at different frequencies from a source in front of the metasurface can be projected into different desired anomalous angular positions. Furthermore, this work provides a potential approach to other kinds of novel THz devices that need controllable metasurface dispersion properties.


S.1 Current distributions on the Unit Cell
The current distributions on a unit cell with different parameter L at 250GHz are presented Fig. S1 for references.
Note that the two loops and I-shaped dipole are designed to be proportionally changed with respect to L, for considerations of controlling the reflection phase curve. As L is changed, other parameters are fixed as given in the caption of Fig. S1. It is also clear that as L is increased from 240 to 360μm, the currents are concentrated on the outer loop, the inner loop, and the edges of the I-shaped dipole, respectively. The current distributions in Fig.   S1 indicate that the three components can resonate individually at 250GHz by properly changing their physical sizes.

S.3 Control of Range and Slope of Phase Curve
To control the dispersion of the DCTM, the unit cell is extensively studied to explore its electromagnetic properties.
Fig . S4 shows the control of the slope and range of the reflection phase by combining different physical parameters.
In Fig. S4 (a), there are 14 curves in total which can be divided into three groups. Each group of the phase curves has a common cross point, meaning that different slopes as well as different reflection phase ranges can be obtained as parameter w2 is changed from 10 to 30 μm. Meanwhile, three groups have three horizontally separated cross points which provide more solutions to control the slope and range of the reflection phase. In Fig. S4 (b), several examples are shown to simultaneously control the slope and range by the physical parameter b, and comparatively only the phase range is tuned in Fig. S4 (c). In Fig. S4 (c), a set of parallel and linear phase curves can be obtained as w1 varies. The three figures mentioned above give us clear information that the required phase by the DCTM to control the dispersive properties can be satisfied by the proposed unit cells.
The operating principle of the controllable range and slope of the reflection phase is as follows. There are three resonant components in the unit cell, between which the mutual coupling dominates the range and slope of the reflection phase. Stronger electric coupling between three components, which is determined by the two gaps with a size g3 in Figure 3b, will push the three resonances closer, leading to a linear reflection phase curve. The phase range is controlled by the separation of the three resonances. For a very small g3, strong mutual coupling will make the three resonances indistinguishable, resulting in the reduced reflection phase range. Comparatively, the magnetic coupling between three components is determined by the two gaps with dimensions g1 and g2. Improving the magnetic coupling can also enhance the reflection phase range, but at the cost of reduced reflectivity due to stronger concentration of the electric current distributions on the three components.

S.4 Phase Requirements in Metasurface Grating Design
With the point source placed at position (0, 0, F) and F = Dx = Dy/2 = 25λ0 at the center frequency f0, we assume that the metasurface is discretized into 50 × 100 unit cells along the x-and y-axis directions, which is larger than the

Required phase (deg)
Numerical order of unit cells on Line 1 In Fig. S6 (a), the required phase of the unit cell at different position on Line 2 versus normalized frequency is shown, as the phase curve of the first unit cell is assumed beforehand. This transition point is located at the 33 rd unit cell, and the slope has already become positive from a negative value. This conclusion can also be proved by the data in Fig. S6 (b). For the curve of different frequencies, there is a common cross point at the 33 rd unit cell, across which the slope of the required phase curve versus frequency is actually reversed. For the plane-wave incident case, the required phase by the unit cell is actually independent on the size of the DCTM along the y axis. Therefore, only the desired properties of unit cells along Line 1 are shown in Fig. S7. It can be seen from Fig. S7 (a) that the required phase curve for the unit cell far away from the starting point of Line 1 becomes 6 steeper, but no phase advance versus frequency is observed. Meanwhile, the slope of each curve is much smaller than that in Fig. S5 (a), and the total required reflection phase range is also much smaller. In Fig. S7 (b), the phase curves are linear instead of the nonlinear ones in Fig. S5 (b). It is obvious that design of such a DCTM in the plane-wave incident case is much easier than the general case studied above.

Required phase (deg)
Numerical order of unit cell on Line 1 Fig. S7. For the plane-wave incident case, the required reflection phase of the unit cells on Line 1 versus (a) frequency and (b) numerical order of the unit cell. The phase curve of the first unit cell is the same to that in Fig. S5. The phase curves in Fig. S7 (b) become linear, instead of the nonlinear ones in Fig. S5 (b).

S.5 Reflection Phase Database
A database of the reflection phase of the unit cell is firstly built to map the physical sizes to the reflection phase at 200, 225, 250, 275 and 300GHz, respectively. The five parameters and the discretized steps are shown in Table IV. Here the dimensions of the unit cells are fixed to be Lx = Ly = 350μm, when taking the oblique incidence on the metasurface into account. The first four is discretized by a step of 5μm to reduce the required computational time in the full-wave simulations. After a careful parametric sweeping, an interpolation process is performed to obtain more detailed phase values with a step size of 0.5μm for the first four parameters, and parameter b is discretized by a step of 0.01 within the interpolation process. Then, a six-dimensional database has been established, in which the index of each element indicates the physical sizes of the unit cell and the value of that element corresponds to the reflection phase, as mentioned in the main content.

S.6 Fabricated Prototype
Photographs of the fabricated metasurface prototype are shown in Fig. S8. Its total sizes are 17.5 × 14 mm 2 in the xand y-axis directions. From the zoom-in view in the inset of Fig. S8, it can be seen that there are still some defects in a small portion of unit cells, which are one of the reasons causing differences between simulations and measurements. Fig. S8. Fabricated DCTM prototype and the zoom-in view. A ruler is placed beside the fabricated prototype for reference. The aluminum plate with very small roughness like a mirror can be clearly seen.

S.7 Error Analysis
There are many possible errors causing the discrepancies between measurements and simulations. The most significant are as follows.