# Axial-Symmetric Diffraction Radiation Antenna with a Very Narrow Funnel-Shaped Directional Diagram
## 4 Radial Emitters with a Narrow Funnel-Shaped Radiation Pattern  
It is possible that the implementation of wave analogs of Smith-Purcell coherent radiation in axially symmetric structures, which is going to be considered below, is the only fairly simple and least expensive way to build antennas with a very narrow funnel-shapedradiation pattern. The principle remains the same as with previously designed linear and planar antennas: the eigen slow wave of an open guiding line'sweeps out' the periodic structure, and spatial propagating harmonics create the radiation field. The peculiarity lies in the fact that the role of an open guiding line is assigned to a planar radial dielectric waveguide, and a grating with concentric guiding lines, determining its period \(l\), is put on a round disc installed on the axis \(z\). The synthesis of an emitter with a very narrow funnel-shaped radiation pattern is reduced to the search for such a periodic structure that could provide the required amplitude and phase distribution of the radiation field at its zero spatial harmonic if the condition  
\[\vartheta(k)=-\alpha_{0}(k)=\arcsin(\Phi_{0}/k)=\arcsin(\gamma(k)-\lambda/l) \approx 0 \tag{21}\]  
holds. Here, \(\gamma(k)=\overline{\chi}(k)/k\) and \(\overline{\chi}(k)=\Phi_{1}(k)\) are deceleration coefficient and propagation constant of the eigen surface wave of a radial dielectric waveguide, in which component \(E_{\phi}\) (in the case of \(TE_{0}\)-waves) or \(H_{\phi}\) (in the case of \(TM_{0}\)-waves) may be presented in the form \(A(k,\rho,z)\exp(i\overline{\chi}(k)\rho)\); \(\Phi_{n}(k)\) are longitudinal (along \(\rho\)) wavenumbers of grating's spatial harmonics of the number \(n\)[42; 43; 44]. Connecting values \(\overline{\chi}(k)\) and \(\Phi_{1}(k)\), we implement in a form suitable for our particular case the well-known fact about the 'identity' of the field of the eigen surface wave of a planar dielectric waveguide and the field of one of the evanescent spatial harmonics of the grating [17; 25; 45]. This identity is manifested in the fact that the secondary fields of periodic structures generated by the fields of these generally different waves are the same. This allows us, when solving problems of analysis and synthesis of diffraction radiation antennas, to proceed from the fact that the characteristic bursts of their radiation pattern functions, both in magnitude and in the corresponding values of the angular coordinate, correlate fairly well with the amplitudes and space orientation of the spatial harmonics of the secondary fields of infinite gratings excited by the corresponding evanescent spatial harmonic. The effectiveness and validity of the approach to the construction of diffraction radiation antennas based on such concepts have already been proven by all the previous experience of researchers working in this direction. Condition (21) formally reflects all that discussed above. If, for some values of \(k=2\pi/\lambda\), \(\gamma(k)\), and \(l\), in the secondary field of an infinite grating excited by the first evanescent harmonic, only the fundamental harmonic (harmonic with a number \(n=0\)) is propagating, then, in the half-plane \(\Omega_{\rm total}\), it will go into free space at an angle \(\vartheta(k)=-\alpha_{0}(k)=\arcsin(\Phi_{0}/k)\)[44]. The equality \(\arcsin(\Phi_{0}/k)=\arcsin(\gamma_{\gamma}(k)-\lambda/l)\) follows from the relations \(\overline{\chi}(k)=\Phi_{1}(k)\) and \(\Phi_{n}=\Phi_{0}+2\pi n/l\)[44]. Condition (21) allows for each given frequency value to determine the period of the grating \(l\approx\lambda/\gamma(k)\), capable of forming, together with a radial dielectric waveguide, supporting the propagation of a surface wave with a deceleration coefficient \(\gamma(k)\), a radiation field with the main lobe of the directional pattern 'pressed' to the axis \(z\).