## 3 Results for SPDR with Dielectric Samples  
Having defined the problem geometry and parameters, the next step was to determine resonant frequencies of the SPDR. Firstly, we have focused on the fundamental mode without any sample to cross verify and compare our FD-FEM and FDTD approaches. The E-field profile of the considered TE01 \(\delta\) measurement mode at 4.95 GHz and its E-field profile is given in Figure 2.  
Figure 1: Visualization of the considered SPDR: (**a**) in 3D; (**b**) materials used and geometry with details; (**c**) 2D representation using axial symmetry. Figure 2: Normalized \(E_{\phi}\) component of the fundamental mode of the considered SPDR without any sample (air) is obtained from FD-FEM solver. In addition to the no sample case, we have also obtained several resonant frequencies for different sample thicknesses and different sample permittivities, including analysis of real-life material samples, namely quartz (\(\varepsilon_{sample}=3.82\)) and sapphire (\(\varepsilon_{sample}=9.40\)). Additionally, a non-planar (cone-shaped) sample (to represent realistic deformation), which is not perfectly horizontal in the sample slot but bent upwards in the middle, is considered. The results of performed calculations are given in the Table 1, and the discrepancies in the resonant frequency extraction between the two simulation methods do not exceed 4.2 MHz. The comparison of simulated and measured frequency values reveals the differences of ca. 200 MHz. The observed higher difference is understandable, as the physical device may differ from its design due to being a subject to manufacturing modifications and tolerances, making each unit unique in terms of geometry and resulting resonant frequency, which is confirmed with two different resonant frequencies obtained for the two adopted commercial units (Table 1). The extraction method for permittivity is based on measuring the shifts in the resonant frequency of an empty and sample-loaded SPDR; therefore, the frequency shift (and not the frequency itself) directly determines the permittivity value. These frequency shifts have been calculated for the considered simulation scenarios and experiments, and results have been visualized in Figure 3. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Thickness \(t_{sample}\) [mm]** & **Permititivity \(\epsilon_{sample}\)** & **FD-FEM** & **FDTD** & **Measurement 1** & **Measurement 2** \\ \hline
**No sample** & **1.0** & **4.9502** & **4.9466** & **5.1588** & **5.1117** \\
0.5 & 2.0 & 4.9384 & 4.9347 & X & X \\
0.5 & 5.0 & 4.9027 & 4.8990 & X & X \\
0.5 & 10.0 & 4.8434 & 4.8396 & X & X \\
1.0 & 2.0 & 4.9268 & 4.9232 & X & X \\
1.0 & 5.0 & 4.8567 & 4.8529 & X & X \\
1.0 & 10.0 & 4.7411 & 4.7371 & X & X \\
1.5 & 2.0 & 4.9151 & 4.9114 & X & X \\
1.5 & 5.0 & 4.8108 & 4.8069 & X & X \\
1.5 & 10.0 & 4.6415 & 4.6373 & X & X \\
0.542-deformed & 3.82 & 4.9148 & 4.9111 & X & X \\
0.542- & 3.82 & 4.9140 & 4.9103 & 5.1223 & 5.0770 \\
0.475 & 9.40 & 4.8554 & 4.8516 & 5.0629 & 5.0200 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Resonant frequency of the fundamental mode of SPDR loaded with dielectric samples varying in thickness and permittivity: simulations and measurements performed by the authors. Figure 3: Resonant frequency shifts with respect to the case without inserting a sample for FEM and FDTD simulations for the samples (**a**) with 0.5 mm thickness, (**b**) with 1.0 mm thickness, and (**c**) with 1.5 mm thickness. (**d**) Resonant frequency shifts for the quartz sample; (**e**): resonant frequency shifts for the sapphire sample. The obtained values of the frequency shifts are consistent between FDTD and FEM simulations performed for arbitrary samples' thicknesses and permittivities, not exceeding 0.5 MHz reported for thick high-permittivity samples.