# Axial-Symmetric Diffraction Radiation Antenna with a Very Narrow Funnel-Shaped Directional Diagram
## 4 Radial Emitters with a Narrow Funnel-Shaped Radiation Pattern
The structure of the grating period or the value of the parameter \(c\) at the local intervals of the coordinate \(\rho\) variation, which add up to the interval of length \(L\), determining the real dimensions of the aperture of the synthesized device, can be changed and is selected in such a way as to fulfill the known requirements [46], which ensure the radiation pattern of the required quality and are related to the implementation of a certain amplitude distribution of the radiation field. Below, we restrict ourselves to considering prototypes of emitters with an aperture determined by a relatively small size of \(10l<L<30l\) (the structure of the grating period and the parameter \(c\) are constant) and focus on clarifying the question of whether it is possible, in principle, to form a very narrow funnel-shaped radiation pattern by structures with a radial arrangement of the main functional elements: a grating and a dielectric waveguide (Figure 2). In the prototype shown in Figure 2 (the proportions in the image of the details of all analyzed structures are preserved), the circular feeder waveguide A (\(a=0.15\) is its radius) moved into a radial waveguide and then into an open waveguide--a dielectric one (\(\epsilon=2.1\)). A reflective grating with parameters \(l=0.14\), \(d=0.07\), and \(h=0.07\) (width and depth of slots) was placed at a distance \(c=0.03\) from the dielectric waveguide. The period length \(l\) is determined from condition (21) after calculating the propagation constants \(\overline{\chi}(k)\) of a surface wave excited in a radial dielectric waveguide by a \(TE_{01}\)-wave of a circular waveguide A. Specifically, the value \(\overline{\chi}(k)\approx 46.4\) for the frequency \(k=35\) (\(\gamma(k)=\overline{\chi}(k)/k\approx 1.33\)) was taken into account. The values of \(\overline{\chi}(k)\) we obtain by calculating the phase incursion \(\psi(k)=\arg\widetilde{U}(g_{2},k)-\arg\widetilde{U}(g_{1},k)=\overline{\chi} (k)\) of the field \(\widetilde{U}(g,k)\) caused by the displacement of the observation point along the dielectric waveguide axis \(z=const\) from the point \(g_{1}=\{z,\rho_{1}\}\) to the point \(g_{2}=\{z,\rho_{2}\},\rho_{2}-\rho_{1}=1.0\)[17,25]. According to the concepts discussed above, the considered emitter at frequencies close to \(k=35\) should form a diagram, with one of the lobes directed practically along the axis \(z\). This is true in a flat \(\phi=const\) representation. The rotation of this beam with variation of the coordinate \(\phi\) generates a 'funnel' in space, which we would like to see in the field's distribution emitted by the antenna. The results of numerical simulation in the frequency band \(25\leq k\leq 45\) (in the waveguide A, not more than two \(TE_{0n}\)-waves are propagating: \(k_{1}^{+}\approx 17.7\), \(k_{2}^{+}\approx 32.3\),  
Figure 2: Radial diffraction radiation antenna. Radiation pattern \(D(\phi,k,\infty)\) in the frequency band \(25\leq k\leq 45\) and at frequencies \(k=32.1\) and \(k=38.0\). \(L\) is a parameter that determines the size of the emitting aperture. \(k_{3}^{+}\approx 46.8\); hereafter, the frequencies \(k_{n}^{\pm}\) are the cut-off points of \(TE_{0n}\)- and \(TM_{0n}\)-waves in the waveguides A and B, i.e., \(\gamma_{n}^{\pm}(k_{n}^{\pm})=0\)) confirmed the expectations (Figure 2). The main and well-pronounced lobes of the radiation patterns at frequencies \(k\) close to \(k=32\) and \(k=38\) are oriented in the direction \(\theta=1^{\circ}\). However, instead of one arc in the plane \(\{k,\theta\}\) associated with the zeroth spatial harmonic of the grating propagating without attenuation, we see two shifted relative to each other in frequency and intersecting in the region \(k\approx 35\). This allows us to conclude that not one but two competing surface waves with close values \(\overline{\chi}(k)\) are excited in a dielectric waveguide, and in our calculations, we determined their average value.