# Potential of Liquid-Crystal Materials for Millimeter-Wave Application
## 3 Millimeter Wave Fresnel Lens Fabricated Using 3D Printer
### Design of Fresnel Lens Substrate  
Figure 5a shows a typical Fresnel lens fabricated using a 3D printer in this study, where a period of the Fresnel zone was divided by eight steps. It is not easy to attain eight or more step divisions if we use the mechanical milling method because the size of the milling cutter is limited [35]. However, the 3D printer has sufficient precision to attain the MMW Fresnel lens, which has a larger number of divided steps. As mentioned above, the lens surface was approximated by the discrete steps, and the radius was suitably designed for MMW applications using Equation(1), and Figure 5b shows an example of a cross-sectional structure for \(P\) = 4 [54]. Here, \(m\) is the zone number of the Fresnel lens, \(P\) is the dividing number of steps for each Fresnel zone period, \(f\) is the focal length, and \(\lambda\) is the wavelength of the MMW. If the dividing number of steps \(P\) increases, a higher diffraction efficiency can be obtained. However, a sufficiently high efficiency can be obtained even with values of \(P\) between 4 and 8. The optimal depth of a step for maximal diffraction efficiency can be designed using Equation (2), which is determined from the refractive index difference between the substrate and the cavity space. This fact means that the step depth must increase if we introduce the LC material to the cavity, because unity in the equation should be changed to the refractive index of the LC material. The processing accuracy of a domestic 3D printer is enough for the fabrication of an MMW Fresnel lens. Although we can see some roughness on the surface caused by the lamination process of the 3D  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \(\lambda_{\mathbf{g}}\) (mm) & \(\mathbf{n}\) & \(\varepsilon_{\mathbf{r}}\) & \(\alpha\) (mm\({}^{-1}\)) \\ \hline Measurement & 2.77 & 2.71 & 7.34 & 0.079 \\ Reference & - & 3.6 & 13 & 0.011 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Evaluation results of the Si substrate.  
printer, as shown in Figure 5c, it is not a serious problem for MMW applications, because excellent MMW focusing properties were confirmed from the measurement results of lens properties.  
\[r_{m}=\sqrt{\frac{2 mf\lambda}{P}+\left(\frac{m\lambda}{P}\right)^{2}}; \tag{1}\]  
\[d=\frac{\lambda}{P(n-1)}. \tag{2}\]  
Figure 6 shows the measurement data of the MMW focusing properties of the fabricated Fresnel lenses, whose focal lengths were 30 mm and 50 mm, with a diameter of about 50 mm. The 70 GHz MMW generated by the Gunn diode oscillator was radiated from a circular horn antenna, and it was incident normally to the Fresnel lens about 350 mm awayfrom the antenna. The spatial distribution of the MMW intensity in a focal plane of the lens was measured by scanning a diode detector set on a linear stage. It was observed that excellent focusing properties could be obtained in comparison with the background MMW intensity level even though there were only two periods of Fresnel structure. The measured diameter of the focusing spot of about 5 mm is close to the diffraction limit, if we consider that the wavelength of the MMW is 4.3 mm. It is also observed that the spot diameter of the lens with a focal length of 30 mm is a little smaller than that of the lens with a focal length of 50 mm. The difference is reasonable if we consider the difference between their numerical apertures.