# Measurement of Dielectric Properties of Thin Materials for Radomes Using Waveguide Cavities
## 2 Theory and Materials  
As mentioned in the Introduction section, one of the most common methods used for determining dielectric properties is the resonant cavity method. This involves placing the material under test within a waveguide cavity with shims (with irises in their center) on each end, identifying the resonant frequency, and then using that to calculate permittivity and loss tangent [9]. The relevant resonant frequencies, the dimensions of the waveguide cavity, and the mode numbers are used in Equation (2) to determine both TE and TM modes (noting that TM modes have m values greater than zero):  
\[f=\frac{c}{2}\sqrt{\left(\frac{m}{a}\right)^{2}+\left(\frac{n}{b}\right)^{2}+ \left(\frac{p}{l}\right)^{2}}, \tag{2}\]  
where \(f\) is the resonant frequency (Hz); \(c\) is the speed of light; \(m\), \(n\), and \(p\) are given by the chosen mode TE\({}_{\text{mnp}}\) or TM\({}_{\text{mnp}}\); and \(a\), \(b\), and \(l\) are the width, height, and length of the waveguide cavity, as shown in Figures 3 and 4[15]. By determining multiple cutoff frequencies, various mode resonances can be observed within the cavity. The first eight propagating modes of a rectangular waveguide that are excited at the iris of the shim are TE\({}_{10}\), TE\({}_{30}\), TE\({}_{12}\), TM\({}_{12}\), TE\({}_{50}\), TE\({}_{32}\), TM\({}_{32}\), and TE\({}_{52}\)[22; 23]. These modes are excited by a symmetrical iris when the electric field in the y direction has values of \(m\) odd and \(n\) even. A benefit of this measurement method is the ability to look at properties at higher frequencies that are outside the waveguide's recommended frequency range. The orientation of the material under test within the waveguide cavity is also an important factor. Generally, test materials are placed within the x-y plane of the cavity, with the largest side of the sample facing each of the irises, as shown in Figure 5b. However, for the research discussed in this article, we placed our samples in the y-z plane of the cavity, as shown in Figure 5a. This was done so that the sample is exposed to the maximum integrated electric field for the TE\({}_{\text{10p}}\) modes and thus has the highest filling factor (Equation (9)). For \(p>1\), the filling factor for the x-y case was zero for \(p\), even as the sample sat in a null. This can be understood by observing the two modes in Figure 5[21]. As we focused on the  
Figure 4: Rectangular waveguide cavity with the variations in electric field for resonant modes TE\({}_{101}\) (\(p=1\)) and TE\({}_{102}\) (\(p=2\))[21]. Figure 3: Coordinate system and dimensions of a general rectangular waveguide [15]. fundamental \(\mathrm{TE}_{10}\) mode, we observed it travels along the length of the cavity (z-axis). For the first cavity mode of \(\mathrm{TE}_{101}\), there was no null present in the electric field. However, for the second (\(\mathrm{TE}_{102}\)) and higher cavity modes, we observed the presence of a null where the electric field crossed the center of the z-axis. If the material under test was placed within this null, it would not be affected by the electric field, resulting in no useful measurements. Therefore, to ensure that we had multiple modes available for measurement, we avoided placing the sample within a null by placing it in the y-z plane. To calculate the permittivity and loss tangent of a sample material using cavity resonance, specific equations are required. For simulations, different equations were used, as the foam holding the sample in place within the cavity was not simulated. These equations used for calculating real and imaginary permittivity and the loss tangent are shown in Equations (3)-(5) [20].