# Axial-Symmetric Diffraction Radiation Antenna with a Very Narrow Funnel-Shaped Directional Diagram
## 4 Radial Emitters with a Narrow Funnel-Shaped Radiation Pattern
The considered model did not allow to determine the radiation efficiency in the directions of interest, since a significant part of the energy, as seen from Figure 2, leaves the open end of the radial dielectric waveguide into the corner sector \(50^{\circ}<\theta<120^{\circ}\). In other models presented in this section, we have corrected this drawback by adding a coaxial waveguide B, removing the remained and not the radiated part of the primary \(TE_{01}\)-wave energy (Figure 3). The propagation constants \(\overline{\chi}(k)\) of the radial dielectric waveguide were calculated for the structure (see Figure 3) with following parameters: the radii \(b^{\pm}\) of the inner conductors of the coaxial waveguides A and B are equal to 0.12 and 15.84, respectively; the radii \(a^{\pm}\) of their outer conductors are 1.0 and 16.72; and the thickness of the radial dielectric (\(\epsilon=2.1\)) waveguide is equal to 0.88. The cutoff points of the first three \(TE_{0n}\)-waves in the waveguide A are equal to \(k_{1}^{+}\approx 2.75\), \(k_{2}^{+}\approx 5.14\), and \(k_{3}^{+}\approx 7.55\), and in the waveguide B, \(k_{1}^{-}\approx 2.46\), \(k_{2}^{-}\approx 4.93\), and \(k_{3}^{-}\approx 7.39\). For the analysis, a frequency band \(3.1\leq k\leq 7.3\), where no more than two \(TE_{0n}\)-waves can propagate in each of these waveguides without attenuation, was chosen. At the frequency \(k=5.2\), the average for this range, \(\overline{\chi}(k)=6.96\), \(\gamma(k)=\overline{\chi}(k)/k\approx 1.34\), and the value \(l\) satisfying condition (21) is 0.9. Let us load an open dielectric waveguide with a reflective grating with defined above period \(l=0.9\) and \(d=h=0.8\) (aiming distance between the waveguide and the grating is \(c=0.1\)) and excite the structure with a pulsed \(TE_{01}\)-wave, such that  
\[ \begin{split}& V_{1}(g,t):\ v_{1}\big{(}z:\ g\in\mathrm{L}^{+},t \big{)}=4\frac{\sin\left[\Delta k(t-\overline{T})\right]}{\big{(}t-\overline {T}\big{)}}\cos\left[\widetilde{k}\big{(}t-\widetilde{T}\big{)}\right]\chi \big{(}\overline{T}-t\big{)}=F_{1}(t);\\ &\widetilde{k}=5.2,\ \ \ \Delta k=2.1,\ \ \ \widetilde{T}=50,\ \ \ \overline{T}=100.\end{split}  \tag{22}\]  
Figure 3: Surface wave propagation constants \(\overline{\chi}(k)\) of a radial dielectric waveguide. Here, \(\chi(\ldots)\) is the Heaviside step function, \(\widetilde{k}\) is the central frequency of the pulse \(F_{1}(t)\), \(\widetilde{T}\) and \(\widetilde{T}\) are its delay time and duration. The pulse \(F_{1}(t)\) occupies the frequency band \(\widetilde{k}-\Delta k\leq k\leq\widetilde{k}+\Delta k\)[31, 34]; in our case, this is a chosen for analysis interval \(3.1\leq k\leq 7.3\). The numerical results are presented in Figure 4. Directional properties of the emitter are much better than in the case considered above--sufficiently 'powerful' side lobes in the radiation pattern are almost completely absent. The minimum deviation of the beam corresponding to the zero spatial harmonic of the grating from the axis \(z\) was, in this case, equal to \(2^{\circ}\). The lower fragments of Figure 4 show the dependences \(D(\theta,k,\infty)\) at frequencies \(k=5.08\) and \(k=6.01\); the radiation efficiency here is, respectively, \(\eta(k)=0.43\) and \(\eta(k)=0.17\). It is clear that, at least with the achieved value of \(\eta(k)=0.43\) and the selected type of periodic structure, one can count on a successful solution of the problem associated with the formation of the required amplitude distribution of the radiation field of the antenna, having an aperture size several times larger than the aperture size \(L\) of the considered prototype [17, 20, 25]. The upper fragment of Figure 4 tells us that a decrease in frequency (transition to the range \(2.9\leq k\leq 4.9\), where only \(TE_{01}\)-waves propagate without attenuation in feeder coaxial waveguides A and B) will allow us to get rid of the 'two arcs' effect in the radiation  
Figure 4: Radial diffraction radiation antenna.