# Axial-Symmetric Diffraction Radiation Antenna with a Very Narrow Funnel-Shaped Directional Diagram
## 2 Models of the Method of Exact Absorbing Conditions Time Domain Representations
The arc L of radius \(L\) covers all scattering inhomogeneities of the domain \(\Omega_{\text{int}}\). Exact absorbing conditions \(\left.D^{+}\big{[}U(g,t)-V_{p}(g,t)\big{]}\right|_{g\in\text{L}^{+}}=0,\ \left.D^{-}[U(g,t)]\right|_{g\in\text{L}^{-}}=0\) and \(\left.D[U(g,t)]\right|_{g\in\text{L}}=0\) for virtual boundaries are the ideal model for waves  
\[U(g,t)-V_{p}(g,t)=U^{+}(g,t)=\sum\limits_{n\in\{n\}}U^{+}_{np}(g,t)=\sum\limits _{n\in\{n\}}u^{+}_{np}(z,t)\mu^{+}_{n}(\rho);\ \ g\in\overline{\text{A}},\ \ \ t\geq 0,\]  
\[U(g,t)=U^{-}(g,t)=\sum\limits_{n\in\{n\}}U^{-}_{np}(g,t)=\sum\limits_{n\in\{n\}} u^{-}_{np}(z,t)\mu^{-}_{n}(\rho);\ \ g\in\overline{\text{B}},\ \ \ t\geq 0,\ \ \text{and} \tag{4}\]  
\[U(g,t)=\sum\limits_{m=1}^{\infty}U_{mp}(g,t)=\sum\limits_{m=1}^{\infty}u_{mp}(r, t)\overline{\mu}_{m}(\vartheta);\ \ \ r\geq L,\ \ 0\leq\vartheta\leq\pi,\ \ \ t\geq 0,\]  
traveling from domain \(\Omega_{\text{int}}\) into waveguides A, B, and domain \(\Omega_{\text{ext}}\)[25; 32; 34]. Here, \(\{n\}=0,1,2,\ldots\) only in the case of coaxial waveguides and only for \(TM_{0}\)-waves; in all other cases \(\{n\}=1,2,3,\ldots\). Explicit analytic representations for transverse functions \(\mu_{n}^{\pm}(\rho)\) and the corresponding eigen values \(\lambda_{n}^{\pm}\) are well known [25; 31; 32; 34]. For example, for \(TE_{0}\)-waves functions \(\mu_{n}^{\pm}(\rho)\) of coaxial waveguide (\(b\) is the radius of its inner conductor, \(a\) is the radius of its outer conductor) have a form  
\[\left\{ \begin{array}{ll}\mu_{n}(\rho)=G_{1}(\lambda_{n},\rho)\sqrt{2}\big{[} a^{2}G_{0}^{2}(\lambda_{n},a)-b^{2}G_{0}^{2}(\lambda_{n},b)\big{]}^{-1/2};&b< \rho<a\\ \lambda_{n}>0\ \ \text{ are the roots of the equation }\ G_{1}(\lambda,a)=0;\\ G_{q}(\lambda,\rho)=J_{q}(\lambda\rho)N_{1}(\lambda b)-N_{q}(\lambda\rho)I_{ 1}(\lambda b);&q=0,1\end{array} \right. \tag{5}\]  
and  
\[\left\{ \begin{array}{ll}\mu_{n}(\rho)=J_{1}(\lambda_{n}\rho)\sqrt{2}\big{[} aJ_{0}(\lambda_{n}a)\big{]}^{-1};&0<\rho<a\\ \lambda_{n}>0\ \ \text{ are the roots of the equation }\ J_{1}(\lambda a)=0\end{array} \right. \tag{6}\]  
for a circular waveguide. Here, \(J_{q}(\dots)\)\(\mathit{\mathit{\mathit{\mathit{\mathit{\mathit{\mathit{\mathit{\mathit{\mathit{\mathit{ \mathit{\mathit{\mathit{\mathit{\mathit{\mathit{\mathit{\mathit{\mathit{   \mathitmathitmathitmathit{     }}}}}}}}}}}}}}N_{q}(\dots)\) are cylindrical Bessel and Neumann functions. The transverse functions  
\[\overline{\mu}_{m}(\theta)=\overline{\mu}_{m}(\cos\theta)=\sqrt{(2m+1)/(2m(m+ 1))}\ P_{m}^{1}(\cos\theta), \tag{7}\]  
(\(P_{m}^{1}(\dots)\) are associated Legendre functions) have relevant eigen values \(\overline{\lambda}_{m}=\sqrt{m(m+1)}\), \(m=1,2,3,\dots\)[32; 34]. The works [25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36; 37] are devoted to the theory and construction of exact absorbing conditions for open initial-boundary value problems in computational electrodynamics. It is possible to find there, in particular, such analytical representations for integro-differential operators \(D^{\pm}[\dots]\) and \(D[\dots]\) of the problem (1) (presume--see Figure 1--that virtual boundaries L\({}^{\pm}\) lie in the plane \(z=0\)):  
Figure 1: Schematic representation of the geometry of an axially symmetric emitter ’radial dielectric waveguide—concentric grating’ in a half-plane \(\Omega_{\text{total}}\).