# Wave Diffraction from a Bicone Conjoined with an Open-Ended Conical Cavity
## 1 Introduction  
The progress of wave diffraction simulation for determining the coupling between the radiating elements in radiophysics, optics, and acoustics often relies on the availability of solutions to canonical problems. This paper considers the bicone conjoined with an open-ended conical cavity formed by a perfectly conducting semi-infinite cone and a truncated one with an internal spherical termination. Variations in its geometric parameters make it possible to simulate wideband antennas, reference measurement probes, and verification antennas, which are the canonical area of applying the biconical surfaces [1; 2; 3; 4]. Nowadays, an interest in bicones is associated with the design of nanopropes and their application in nanotechnologies [5; 6; 7]. The main reason for their application is TEM wave excitation in the coaxial biconical region and the electromagnetic energy concentration near the conical vertices [6]. If, for example, the semi-infinite biconical shoulder degenerates into the plane, this structure becomes a model of the functional elements of an optic near-field microscope [6; 7]. This structure includes the flat \"preparation\" desk illuminated by an open-ended conical waveguide and the remote sensing TEM wave \"detector\" created by the biconical area. When the truncated semi-infinite cone degenerates into the plane with a semi-spherical cavity, we arrive at the model of the cavity-like defect [8] detected by the conical probe [9]. This structure also has the potential to be applied for modeling of the absorbent properties of scattering surfaces [10; 11; 12]. Since the scatterer under consideration consists of fragments of canonical surfaces in a spherical coordinate system, this study aims to rigorously solve a corresponding wave diffraction problem. Such a solution correctly takes into account the wave interaction between the different scattering elements of a biconical structure and can be used to verify modern software packages and approximate approaches. For our study, we modified the well-known mode-matching technique; the modification involves transformations that allow the fields singularity at the edges to be explicitly taken into account. The canonical results of wave diffraction from finite/truncated biconical scatterers have been obtained using field representations through the series of the normal wavesof subdomains and application of traditional mode matching for seeking the unknown complex amplitudes of the scattered modes. In the literature, studies of wave diffraction from biconical structures are mainly focused on bicones formed by finite conical shoulders, or finite and semi-infinite ones. As a rule, they consider bicones formed by a hollow cone and closed by a spherical cap or partially filled by the dielectric. The general scheme of usage of the mode-matching method and its application for solving various problems of wave diffraction from bicones, as well as an analysis of the earlier works in this direction, was given in [13] and later in [14; 15]. The solution of wave diffraction problems for particular cases of biconical scatterers is given via the Wiener-Hopf technique, which involves a Kontorovich-Lebedev integral transform [16]. Despite the simplicity of mode matching and its physical validity, the mode-matching series is the singular cause of the field singularity at the edges. Therefore, it needs to be taken into account. In our previous study, an explicit inversion of the mode-matching singularity was proposed for finite/truncated conical and biconical scatterers [17; 18; 19; 20; 21; 22; 23; 24; 25]; this technique is generally called an analytical regularization procedure. The main reason for such processing is to guarantee an accurate solution to the problem which satisfies all the necessary conditions, including the edge condition, for any geometrical and frequency parameters except the spectrum points. The basic principles of this powerful tool are presented in [26; 27; 28; 29].