## 2 Theoretical Analysis
### The First Part  
As shown in Figure 2, the first section of this part is a tapered helix inside a uniform hollow dielectric rod covered with 1 \(\lambda\) length of the circular metal waveguide. The rest of 1.7 \(\lambda\) of the first part is linearly tapered at the same rate as the helix inside. The helix has a uniform turn spacing of 0.21 \(\lambda\), so there are a total of 13 turns for the whole length of 2.7 \(\lambda\). The diameter of the helix turns is linearly tapered from 0.493 \(\lambda\) to 0.123 \(\lambda\) with the tapering cone angle of \(\alpha\approx 7.8^{\circ}\). As the tapered helix being primary feeding, the operating band and directivity can be predicted according to [19]. Additionally, the normalized phase velocity p of wave propagation along the helix can also be calculated based on (1) [19].  
\[p=\frac{L_{\lambda}/n}{S_{\lambda}+[(2n+1)/2n]} \tag{1}\]  
where \(n\) is the turn number of the Helix, \(S_{\lambda}\) is the normalized spacing between the adjacent turns, and \(L_{\lambda}\) is the total length of the helix in straighten, and there is \(p=0.79\) for uniform helix. The total length of the tapered helix can be calculated as:  
\[L_{\lambda}=\frac{S_{\lambda}tan\frac{\phi}{2}}{2\pi}\left(\frac{t^{2}}{8}+ \frac{Q}{2}Int-\frac{Q^{2}}{8t^{2}}\right) \tag{2}\]  
where \(t=\sqrt{\varphi_{t}^{2}+Q}+\varphi_{t}\), and \(\varphi_{t}=26\pi\), \(Q=1+\frac{1}{tan^{2}\frac{\phi}{2}}\). For this tapered helix, there is \(p\approx 0.56\).  
The directivity of helix is calculated as [17]  
\[D_{0}(dimensionless)\cong 15 nC_{\lambda}^{2}S_{\lambda} \tag{3}\]  
where \(C_{\lambda}\) is the normalized circumference of Helix, and \(C_{\lambda}\) can be calculated as follows:  
\[C_{\lambda}=\sqrt{\left(\frac{L_{\lambda}}{n}\right)^{2}-S_{\lambda}^{2}} \tag{4}\]  
Thus the theoretical directivity of helix is  
\[D_{0}(dimensionless)\cong\left\{\begin{array}{ll}15.8\;\mathrm{dB}\;for\; uniform\;helix\\ 12.6\;\mathrm{dB}\;for\;tapered\;helix\end{array}\right. \tag{5}\]  
According to [20], the bandwidth of a tapered helix is higher than a uniform helix, although a tapered helix has a lower gain than a uniform helix. The simulated results of uniform and tapered helix are shown in Figure 3. Now the key factor is to improve the gain of a tapered helix. The gain of the tapered helix is increased with a tapered dielectric cover of thickness 0.11 \(\lambda\), as shown in Figure 4a,b.  
Figure 2: Schematic design of Embedded Helix and dielectric rod antenna.  
It can be seen from Figure 5a that the dielectric cover has little influence on the return loss of the tapered helix due to the small thickness of the cover [21]. The front-to-back ratio (FBR) of tapered helix with and without tapered dielectric cover is discussed in Figure 5b. The 8 dB front-to-back ratio is improved with a tapered dielectric cover, although little gain effects with this tapering, as shown in Figure 5c. The diameter of the tapering part of the dielectric cover can be selected by using (6) and (7) [22].  
\[d_{\text{max}}\approx\frac{\lambda_{0}}{\sqrt{\pi(\varepsilon_{r}-1)}} \tag{6}\]  
\[d_{min}\approx\frac{\lambda_{0}}{\sqrt{2.5\pi(\varepsilon_{r}-1)}} \tag{7}\]  
Figure 4: (**a**) Gives the schematic of tapered helix with tapered dielectric cover (**b**) The gain and side lobe level comparison of tapered helix with and without tapered dielectric cover.  
Figure 3: (**a**,**b**) Gives the schematic of tapered helix and uniform helix with metal reflector (**c**) Bandwidth comparison of tapered and uniform helix. (**d**) The gain and side lobe level comparison with tapered and uniform helix.  
where \(d_{\max}\) and \(d_{min}\) are the maximum and minimum diameters of the dielectric rod supporting the lowest and highest critical frequencies of the operating band, respectively, and \(\epsilon_{r}\) is the relative dielectric constant of the rod.