## 2 Materials and Methods  
### Basic Mechanism of RF Devices Using LCs and Consideration of the Cell Gap  
The tunability of LCs is the main reason to use them in RF applications. Dielectric anisotropy of LCs permits control of their orientation by an electric field. The dielectric constants of devices can be regulated by changing the angle between the RF field and the LC's directors. Typically, a dielectric constant can be considered using [PERSON]'s law  
\[\overrightarrow{\
abla}\cdot\overrightarrow{E}=\frac{\rho}{\varepsilon_{0} \varepsilon_{r}}=\frac{\rho}{\varepsilon} \tag{1}\]  
where \(\overrightarrow{E}\) [V m\({}^{-1}\)] is the electric field, \(\rho\) [C m\({}^{-3}\)] is charge density, \(\varepsilon_{0}\) = 8.85 \(\times\) 10\({}^{-12}\) [F m\({}^{-1}\)] is the electric constant of the vacuum, and \(\varepsilon_{r}\) is a relative dielectric constant. The wave-occupied volume ratio \(WoVR\) is helpful to analyze the dielectric constant of RF devices that use LCs [26]. Using this concept, it is easy to explain the E flux on the tunable region and tunability. \(WoVR\) can be induced by [PERSON]'s law as  
\[\Phi_{\text{E}}=\frac{Q_{enc}}{\varepsilon_{0}}=\oint_{s}\overrightarrow{E} \cdot dA\]  
\[\varepsilon_{eff}=\sum_{k=1}^{n}\varepsilon_{k}*\frac{flux\;in\;region\;k}{ total\;flux}=\sum_{k=1}^{n}(\varepsilon_{k}*WoVR_{k}), \tag{2}\]where \(\Phi_{\rm E}\) [V m] is electric flux, \(Q_{enc}\) [C] is total enclosed charges, \(\varepsilon_{eff}\) [F m\({}^{-1}\)] is the effective dielectric constant of the structure, and \(WoVR_{k}\) is the wave-occupied volume ratio in region \(k\). _WoVR_\(k\) depends on the dielectric constant of the region \(\varepsilon_{k}\) and the conductor's geometry. Usually, when we design tunable RF devices using LCs, we have to likely maximize \(\Phi_{\rm E}\) of the LC's tunable region, which is directly related to the tunability of the RF devices using LCs. The maximum obtainable differential phase shift is  
\[\Delta\Phi=\frac{2\pi fL}{c_{0}}\Big{(}\sqrt{\varepsilon_{eff(max.bias)}}- \sqrt{\varepsilon_{eff(ref.bias)}}\ \Big{)}, \tag{3}\]  
where \(f\) [Hz] is the operating frequency, \(L\) [m] is the physical length of the phase shifter, \(c_{0}\) = 3 \(\times\) 10\({}^{8}\) [m s\({}^{-1}\)] is the velocity of the light in a vacuum, \(\varepsilon_{eff(max.bias)}\) is the effective relative dielectric constant with LCs biased by a saturated voltage, and \(\varepsilon_{eff(ref.bias)}\) is the effective relative dielectric constant of a phase shifter while biased by a reference voltage. Finally, tunability is evaluated with the tuning range TR [%] of the RF structure as  
\[\text{TR}(\%)=\frac{\Big{(}\varepsilon_{ref,\|}-\varepsilon_{ref,\bot}\Big{)} }{\varepsilon_{ref,\|}}\ast 100(\%), \tag{4}\]  
where \(\varepsilon_{ref,\|}\) is effective relative dielectric constant when the RF field is aligned with the LC molecular long axis, and \(\varepsilon_{ref,\bot}\) is an effective relative dielectric constant when the RF field is aligned with the LC molecular short axis. Equation (4) demonstrates that the structure should be designed to maximize \((\varepsilon_{ref,\|}-\varepsilon_{ref,\bot})\). The main consideration of this paper, the cell gap (=LC layer thickness), is also related to the \(WoVR\)[27]. The cell gap is important when designing RF devices with LCs, especially for liquid crystal phase shifters (LCPS). Two main types of transmission line topologies are used for passive LCPSs: the microstrip line (MSL) and the coplanar waveguide (CPW) [28; 29; 30; 31]. The currently-used LC MSL phase shifter has a limitation in reducing the cell gap to \(\leq\)100 \(\upmu\)m while matching the impedance, because the narrowness of the MSL inevitably causes a large amount of conduction loss [32]. The grounded-CPW LCPS type has the same limitation because of the conductor-backed design. The floating-electrode-free CPW (FE-free CPW) can be implemented with a cell gap \(\leq\) 20 \(\upmu\)m, but to achieve sufficient tuning range, it requires a cell gap \(\geq\) 100 \(\upmu\)m [25].