# Sub-Diffraction Focusing Using Metamaterial-Based Terahertz Super-Oscillatory Lens
## 2 Materials and Methods
### Design and Fabrication
In the case of using square unit patterns, the focusing is affected by the direction of the incident electromagnetic wave polarization. Figure 4 shows the calculated intensity distribution of linearly polarized light (along the y axis) incident on a square pattern lens, and this distribution is normalized by the maximum intensity of the focal spot. Figure 4a shows the intensity distribution on the plane parallel to the direction of polarization (yz plane), and Figure 4b shows the intensity distribution on the plane perpendicular to the direction of polarization (xz plane). These graphs show that the light is strongly focused on the plane perpendicular to the direction of polarization, generating an elliptical spot at a focal length of 7.9 mm. Figure 4c is a graph comparing the spot size where the blue and red lines show the sizes of the spot on the planes parallel and perpendicular to the direction of polarization, respectively. It can be clearly observed that the spot size on the plane perpendicular to the polarization is about half of the spot size in the parallel direction. This difference in resolution makes the square unit pattern difficult to use for product inspection. Thus, the circular unit pattern is preferred. Figure 3: Calculation results: (**a**) intensity distribution through the square patterns; (**b**) the intensity distribution and phase distribution through the circular patterns; (**c**) spot width of the square patterns (blue dots) and circular patterns (red dots); and (**d**) intensity distribution due to the 3 \(\times\) 3 circular unit patterns at 8-mm distance from the lens. As already mentioned above, there are three important parameters for lens design: the width of the slits, the diameter of the ring \(D\), and the distance between rings \(l\). Figure 5 shows the results of calculations of how these three parameters affect the focusing ability. Figure 5a,b compare the light-gathering ability of three patterns that differ only in ring diameter. Figure 5a shows the intensity distribution after light passed through each lens. The upper graph shows the results using a lens with a ring diameter 20 \(\upmu\)m smaller than the design value of 690 \(\upmu\)m while the lower graph shows the results using a lens pattern with a ring diameter of 710 \(\upmu\)m, which is 20 \(\upmu\)m larger than the design value. As in Figure 3b, a spot appears around 7.9 mm in both results, and the focus repeatedly appears in the direction directly above the lens. Figure 5b compares the spot sizes generated by these lenses. The blue plot represents the spot sizes using a ring diameter of 670 \(\upmu\)m, and the red plot shows the results for a ring diameter of 710 \(\upmu\)m. The spot has a size of 0.63\(\lambda\) at a designed focal length of around 7.9 mm. Comparing Figure 3b with Figure 5a,b, it can be confirmed that the focal length at which the spot appears is the same as that of the designed lens because the lattice constant does not change when the ring diameter is changed, as determined from Equation (3). Figure 4: Calculation results of using square patterns with a linearly polarized beam along the y direction: intensity distribution on (**a**) the plane parallel to the polarization (yz plane) and (**b**) the plane perpendicular to the polarization (xz plane); (**c**) spot widths on the plane parallel to the polarization (blue dots) and the plane perpendicular to the polarization (red dots). Figure 5: _Cont._  
Figure 5c shows the intensity distribution after light passes through each lens along x = 0 mm. The depth of field (DOF) [32], which is generally defined as the range of 80% of the spot's central intensity, is 2.1 mm for the first spot and 2.45 mm for the second spot using either 670 um- or 710 um-diameter lenses, which are consistent with the results using the 690 um-diameter lens. This confirms that the diameter of the ring does not affect the DOF, which is the range of light that can be focused by this SOL. Next, we elucidated the focusing ability by changing only the slit width.