## 2 Materials and Methods  
### Challenges in the Current State of the Art  
A two-conductor transmission line facilitated the propagation of transverse electromagnetic (TEM) waves [27]. Typically, such a transmission line consists of a thin signal line, a ground plane or two ground planes, and a dielectric material between them [28]. Introducing air cavities into the substrate is widely recognized as an effective strategy to reduce the loss tangent and, consequently, dielectric losses [19; 20; 21; 22; 23; 24; 25; 26; 29].  
The use of large cavities spanning almost the entire dielectric substrate introduces additional challenges for integration processes and mechanical stress, whereas periodic small cavities are more conducive to integration. However, periodic small cavities can lead to impedance mismatches in the two-conductor transmission line. This challenge is analyzed in detail in the subsequent sections, and a methodology to mitigate it is proposed. To demonstrate the problem and its resolution, an SICL is utilized as a representative example. The SICL consists of an inner conductor, an outer conductor, and a dielectric material that separates them, as illustrated in Figure 1. The inner conductor is a metallic line, while the outer conductor is constructed from two metallic layers and two rows of metallic vias.  
Figure 1: Structure of an SICL.  
In Figure 1, \(l\) represents the length of the SICL, \(h\) is half the thickness of the dielectric layer, \(s\) is the spacing between adjacent vias, \(d\) is the diameter of the vias, and \(a\) and \(b\) are the widths of the outer and inner conductors, respectively. The TEM single-mode operating frequency band of the SICL ranges from DC to \(f_{\mathrm{TE10}}\), which can be calculated using Equation (1) [11]:  
\[f_{\mathrm{TE10}}=\frac{c}{2\sqrt{\varepsilon_{\mathrm{r}}}}\bigg{(}a-\frac{d^{ 2}}{0.95s}\bigg{)}^{-1} \tag{1}\]  
where \(c\) is the speed of light in a vacuum, and \(\varepsilon_{\mathrm{r}}\) is the relative permittivity of the dielectric material.  
Inserting air cavities into the substrate, within the constraints of allowable process limits, effectively reduces the equivalent loss tangent and, consequently, the dielectric loss. These air cavities can take various forms, such as rectangular prisms or cylinders. Figure 2 illustrates an SICL with two rows of periodically arranged cylindrical air cavities, where Y0 represents the longitudinal section through the middle of the substrate. To maintain sufficient mechanical strength in the substrate, the air cavities are spaced appropriately, as shown in the longitudinal section Y0 depicted in Figure 3a. The diameter of each air cavity is denoted as \(d_{a}\), the spacing between adjacent cavities as \(s_{a}\), and the distance between the two rows as \(w_{a}\).  
Figure 3: Profile of an SICL with two rows of periodically arranged cylindrical air cavities, viewed from the longitudinal section Y0: (**a**) overall view; (**b**) vertical view.  
Figure 2: An SICL with two rows of periodic cylindrical air cavities.  
However, the introduction of air cavities can result in mismatches of the effective characteristic impedance. The effective characteristic impedance of the transmission line with periodic air cavities at any cross-section can be defined as the characteristic impedance of a uniform transmission line formed by extending that cross-section along the direction of current propagation. As illustrated in Figure 3b, at cross-sections X1, X2, and X3 the ratio of air to the dielectric substrate varies. This variation leads to different effective relative permittivity values, such that \(\varepsilon_{r}(X1)>\varepsilon_{r}(X2)>\varepsilon_{r}(X3)\). Consequently, the effective characteristic impedance at these positions follows the relationship \(Z(X1)<Z(X2)<Z(X3)\).