# Microwave Liquid Crystal Technology
## 6 Dielectric Waveguides  
Dielectric waveguides are predestinated to overcome aforementioned drawbacks of metallic rectangular waveguides. They offer a low-cost fabrication and low-loss propagation. For example, 3D-printing or injection molding fabrication can be used, and no metallic component is required for the waveguide itself. In addition, the design of LC-based dielectric waveguides is comparatively simple. Due to their electrically unshielded design, electrodes can be placed around the dielectric waveguide without disturbing the propagating wave. Their guiding characteristic can be approximated by the geometrics approach, see Figure 22. Assuming a dielectric slab with a core material, \(\epsilon_{1}\) surrounded by a cladding material \(\epsilon_{2}\), with \(\epsilon_{1}>\epsilon_{2}\). Internal reflections at the boundary layer of core and cladding occur according to [PERSON]'s law if the angle \(\theta\) is below the critical angle  
Figure 21: Simulated S-parameter results of the electrically biased W-band metallic waveguide phase shifter. \[\cos\theta_{\rm C}=\frac{\varepsilon_{2}}{\varepsilon_{1}}. \tag{17}\]  
With the internal reflections, the propagating wave is confined in the dielectric core. Two types of dielectric waveguides are the most prominent ones, the step-index fiber and subwavelength fiber, see Figure 22.  
Whereas the subwavelength fiber consists of only one single dielectric material, the step-index fiber comprises of two dielectrics. In both cases, evanescent fields are propagating outside the dielectric core. Since the subwavelength fiber only consists of one single dielectric core, the evanescent field propagates outside the waveguide in air, which leads to a low-loss propagating. On the other hand, these fields are sensitive to discontinuities or external influences, such as metallic structures. Using a cladding as for the step-index fiber, all evanescent fields are already decaying inside the dielectric waveguide. These two characteristics are very important for LC-based tunable dielectric waveguides, which will be discussed in the following. The design of a tunable step-index fiber designed for W-band frequencies which has been recently published in [42] is shown in Figure 23.  
The presented design uses Rogers 3003 as non-tunable core material with a permittivity of \(\varepsilon_{r}=3\) and dissipation factor of \(\tan\delta\leq 0.05\). As cladding material Rexolite 1422 from C-Lec Plastics Inc., Philadelphia, PA, USA is used, with \(\varepsilon_{r}=2.53\) and \(\tan\delta=0.000\,66\). To achieve tunability, LC is used as core material instead of Rogers 3003. The core is untunable and still manufactured from Rogers 3003, see Figure 23. The phase constant of the propagating wave can now be adjusted by means of the LCs permittivity. As the perpendicular permittivity is below the permittivity of the surrounding cladding, RF leakage would occur, as the propagation condition is not fulfilled anymore. To prevent RF leakage, air gaps are included in the cladding to reduce its effective permittivity. With this design  
Figure 23: Cross section of the step-index fiber phase shifter with the LC cavity highlighted in red. Figure 22: **(Left)** Schematic cross-section of a DW used for the geometric approach, with \(\varepsilon_{1}>\varepsilon_{2}>\varepsilon_{0}\). Cross-section of two common dielectric waveguide types: (Center) Step-index fiber with \(\varepsilon_{1}>\varepsilon_{2}\) and (Right) Subwavelength fiber. The principle E-field distributions for both types are depicted. step, the propagation condition is still fulfilled, even for the lowest LC permittivity, and the evanescent fields are still decaying inside the effective cladding. The main advantage of the step-index fiber is that electrodes can be directly attached on the cladding. Therefore, the design of the biasing electrodes can be kept very simple.