# Wave Diffraction from a Bicone Conjoined with an Open-Ended Conical Cavity
## 9 Numerical Analysis  
All scattered field characteristics are calculated by reducing the ISLAE (73). We compared the solutions with those obtained from Equations (76) and (78) and found almost exact matching. This fact confirms the correctness of the solutions. The expressions (25)-(29) are applied for calculation of the expansion coefficients for the field representation (18). Based on this, we analyse the near- and far-field characteristics of our scatterer for different geometrical and frequency parameters. For the analysis of the radiated far field (\(r\rightarrow\infty\)), we apply the asymptotic representation of Formula (18) for the region \(D_{3}\) as  
\[H_{\varphi}^{t}(\theta)\sim H(\theta)\frac{e^{ikr}}{r}, \tag{99}\]  
where the far-field pattern \(D(\theta)=|H(\theta)|\) is defined by  
\[D(\theta)=Z^{-1}\sqrt{\frac{\pi}{2}}\biggl{|}\sum_{n=1}^{\infty}\frac{x_{n}^{( 3)}\Psi_{\
u_{n}-1/2}(\cos\theta)}{K_{\
u_{n}}(sq_{1})}\biggr{|}. \tag{100}\]  
Let the bicone \(Q\) be placed in a hypothetical environment with single electrical and magnetic parameters and excited by the radial electric dipole of unit amplitude \(A^{(0)}=1\). Let us consider the reduced ISLAE (73) and analyse the corresponding finite \(N\times N\) set of linear algebraic equations, where \(N\) is the truncation order. In Figure 2a, we represent the dependencies of the relative error \(e(N)\) of the near-\(H_{\varphi}(r,\theta)\)-field calculation on the truncation parameter \(N\) for \(kr=ka_{1}+1\) and for three points \(\theta=\gamma_{1},(\gamma_{1}+\gamma_{2})/2\) and \(\gamma_{2}\)\[e(N)=100\%\Big{|}H_{\varphi}^{(N+1)}-H_{\varphi}^{(N)}\Big{|}\,\Big{/}\,\Big{|}H _{\varphi}^{(N)}\Big{|},\]  
where \(H_{\varphi}^{(N)}=H_{\varphi}^{(N)}(r,\theta)\) is the magnetic field, calculated using \(N\times N\) set of linear algebraic equations. We see from this figure that the truncation parameter must be greater than \(ka_{1}\), and we chose it from the condition \(N=[|sa_{1}|]+[q]\) with \(4\leq q\leq 10\), where \([\cdot]\) denotes the entire part. To verify the mode matching, we calculate \(|H_{\varphi}|\) on virtual spherical surfaces, the radii of which are somewhat greater and lower than \(ka_{1}\), and find an excellent adjustment of the field behaviour on the above-mentioned virtual surfaces (see Figure 2b). Figure 3a,b, shows the far field of the scatterers with a wide biconical region \(\gamma_{2}-\gamma_{1}\ \geq\ 80^{\circ}\). Curves 1 and 2 in this figure show the field distributions for different positions of the cavity termination. Curve 3 corresponds to the far field for the scatterer without an internal diaphragm. We observe in Figure 3 the effective far-field radiation along the semi-infinite conical surface \(Q_{1}\). The field oscillations in this figure can be explained by the interference of the multiple waves scattered from the biconical waveguide surfaces, illuminated by the edge waves of the open cavity and the dipole. In order to study the resonance properties of our scatterer, we analyse the dependencies of the modulus of the magnetic far field at the conical surface \(\theta=\gamma_{1}\) on \(kc_{1}\). The value \(|H(\theta=\gamma_{1})|\) is proportional to the \(j_{r}\) component of the far electric current density at the semi-infinite conical surface. Its analysis allows us to understand how the field radiated from the resonant volume penetrates the biconical area. Let us consider a bicone with wide conical (\(\pi-\gamma_{2}=60^{\circ}\)) and narrow biconical (\(\gamma_{2}-\gamma_{1}=31^{\circ}\)) regions formed by a cone \(Q_{1}\) with \(\gamma_{1}=89^{\circ}\) and a truncated one with \(\gamma_{2}=120^{\circ}\) (see Figure 1b). This geometry can be considered as a simple model of a nano-probe with the resonance volume over the plane. Curves 1 and 2 in Figure 4a show the dependencies of the magnetic field module \(|H(\theta=\gamma_{1})|\) for the total field and the module of the TEM mode on the parameter \(kc_{1}\).