# A Novel Metasurface Lens Design for Synthesizing Plane Waves in Millimeter-Wave Bands
## 3 Metasurface Lens Design
### Phase-Shift Element  
The FSS structure shown in Figure 2a is the proposed metasurface lens's phase-shift element, for which a periodic metallic array was designed on the top side of the dielectric substrate. The metal layer was assumed to be a perfect electric conductor, and the substrate was modeled as a low-absorption material. The geometry of the phase-shift element is illustrated in Figure 2b, where symmetrical copper Jerusalem-cross elements are printed on the Rogers RO4003C substrate with a relative permittivity of 3.38 in the metasurface-lens design. The Rogers substrate was used because it exhibits a very low absorption loss and small refractive-index variations in millimeter-wave bands. The metal layer for the design was chosen to be 35 um-thick copper and the dielectric substrate was 1.524 mm-thick PCB; the detailed dimensions of the phase-shift element are listed in Table 1.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**a** & **b** & **c** & **d** \\ \hline
3.9 mm & 0.6–3.9 mm & 0.2–0.5 mm & 3.1 mm \\ \hline \hline \end{tabular}
\end{table}
Table 1: Dimensions of the phase-shift element.  
Figure 1: Geometry of the proposed 101 \(\times\) 101 metasurface lens.  
Figure 2: Schematic diagram of the two-layer metasurface lens. **(a)** The copper layer is patterned on a RO4003C PCB substrate. **(b)** Schematics diagram of the unit cell of the designed structure.  
In order to investigate the millimeter transmission response of the designed FSS structure, the finite-difference time-domain (FDTD) method was adopted for a simulation. During the simulation modeling, the dielectricity and permeability of the structure were taken into account, and the perfect matched layer conditions were applied to limit the computation domain. The direction of the incident wave was from the copper Jerusalem-cross elements to the Rogers RO4003C substrate; the electric field from the feeding antenna array made electrons oscillate, then an induced current was formed on the metal surface. Furthermore, for the periodic phase-shift element in the metal layer, a small part of the incident millimeter-wave energy maintained the oscillation of electrons, while the other part of the energy continued to spread through the FSS structure. In other words, in certain frequencies, according to the law of conservation of energy, the energy maintaining the electron oscillation was absorbed as a dielectric loss and the other part of the energy was spread through the FSS structure so that the reflection was zero and the frequency was the resonant point. The phase of the transmission coefficient against the side length of the phase-shift element's length is shown in Figure 3. As b increased from 0.6 to 3.9 mm, the phase delay had a variation range from 0\({}^{\circ}\) to 360\({}^{\circ}\), accompanied by an acceptable stable insertion loss. Due to a spherical wave from the feeding-antenna array, the phase changed sharply on the vertical plane at a reduced distance. Therefore, different length elements were arranged in different places on the PCB substrate to compensate for the phase variation, which formed several circular annular zones.