# Axial-Symmetric Diffraction Radiation Antenna with a Very Narrow Funnel-Shaped Directional Diagram
## 1 Introduction  
A plane, density-modulated electron beam flying at a constant speed over a one-dimensional periodic grating generates in the surrounding space uniform plane electromagnetic waves; their number, the length, and direction of propagation are determined by the flow's velocity and modulation period, as well as by the length of the grating's period. This is a 'coherent' Smith-Purcell radiation (SPR) [1, 2, 3]. The same flow, passing at a constant speed in a medium, where the speed of light is less than its speed, generates there two homogeneous plane electromagnetic waves, diverging in a wedge from the direction of movement of the flow. The length of these waves and the direction of their propagation are determined by the period of modulation of the flow, the ratio of its speed to the speed of light in the medium, and by the sign of the refractive index of the medium. This is 'coherent' Vavilov-Cherenkov radiation (VChR) [4, 5, 6]. In this electromagnetic scenario, the field of the flow of charged particles generates the field of plane waves, above or below the grating, or in a sufficiently optically dense medium. We observe the same type of radiation and with the same characteristics in the case when we replace the electron flow with an open guiding structure (dielectric waveguide, for example), which supports the propagation of its eigen surface wave with phase velocity coinciding with the electron flow velocity. Here, we have the implementation of wave analogs of SPR and VChR. The analogy here is almost complete, since the field of the electron density-modulated flow is practically identical to the field of the eigen surface wave of the open guide line. Thedifferences may be in the polarization of the fields, but they cannot change the picture of what is happening. In diffraction radiation antennas (or simply in diffraction antennas) [7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24], wave analogs of SPR and VChR are implemented. It is impossible to say that they were very widespread, but several dozen devices of this type, unique in their characteristics, have been constructed at [PERSON]'s Institute for Radiophysics and Electronics of the Academy of Sciences of Ukraine (Kharkiv) for radar and radiometric systems of various purposes. These devices enabled the discovery of the solution to several urgent and practically important problems of remote sensing of the Earth using aerospace carriers, problems of monitoring the airfield and protecting the perimeters of airports, problems of preventing collisions in automobile, river, and sea transport, problems of 'blind' landing of helicopters, and problems of short-range radar and radiometry. The problems that have to be solved for creating efficiently operating diffraction radiation antennas are essentially the same as in the case of phased antenna arrays--it is necessary to ensure, first of all, the required amplitude-phase distribution of the field at sufficiently large (in relation to the size of the period of the used gratings) apertures. These problems have to be solved in different ways: in antenna arrays, each elementary emitter (usually, it is associated with one period of the grating) is excited according to a certain program, and in diffraction, antennas all such emitters are excited by the field of surface wave of an open guiding line; it is necessary to ensure that the phase velocity of this wave is not distorted, and that the amplitude decreases meet the required amplitude distribution of the field at the aperture. The costs of these solutions' implementation differ significantly--with diffraction antennas, they are much lower. In operating systems, mainly linear [7; 9; 12; 16; 18; 19; 20; 21; 22] and planar [7; 10; 11; 13; 14; 15; 16; 17; 24] antennas of diffraction radiation were used. In the radiator of a linear antenna, the grating width and the effective width of the open (dielectric) waveguide are from half to two wavelengths, and the length of the radiating aperture is from several tens to several hundred wavelengths.