# 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  
Axially symmetric emitters, excited by eigen \(TE_{0p}\)- or \(TM_{0p}\)- pulse wave \(V_{p}(g,t)=v_{p}(z,t)\mu_{p}^{+}(\rho)\), \(g=\{\rho,z\}\in\) A of circular or coaxial waveguide A, coming on the virtual boundary L\({}^{+}\) (see Figure 1: the structures are homogeneous along the axis \(\phi\)) at time moments \(t>0\), are studied by solving the following model problem of the method of exact absorbing conditions [25; 32; 34]:  
\[\left\{ \begin{array}{ll}\left[-\varepsilon(g)\partial_{t}^{2}-\sigma(g) \eta_{0}\partial_{t}+\partial_{p}\big{(}\rho^{-1}\partial_{p}\rho\big{)}+ \partial_{z}^{2}\right]U(g,t)=0;&g=\{\rho,z\}\in\Omega_{\text{int}},\ \ \ t>0\\ \par L(g,0)=0,&\partial_{t}U(g,t)\big{|}_{t=0}=0;&g\in\overline{\Omega_{\text {int}}}\\ \par\mathbf{E}_{\text{tg}}(q,t)\ \ \text{and}\ \mathbf{H}_{\text{tg}}(q,t),\ \ q=\{\rho,\phi,z \}\ \ \text{are continuous when crossing}\ \ \ \Sigma^{\varepsilon,\sigma},\\ \par\mathbf{E}_{\text{tg}}(q,t)\big{|}_{q\in\Sigma}=0,\ \ \ U(0,z)=0\ \ \text{for}\ \ \{0,z\}\in\overline{\Omega_{\text{int}}}\ \ \text{and}\ \ \ D^{-}[U(g,t)]\big{|}_{g\in\text{L}^{-}}=0,\\ \par D^{+}\big{[}U(g,t)-V_{p}(g,t)\big{]}\big{|}_{g\in\text{L}^{+}}=0,\ \ \ D[U(g,t)]\big{|}_{g\in\text{L}}=0;&t\geq 0. \end{array} \right. \tag{1}\]  
Here, \(V_{p}(g,t)\) is the \(E_{\phi}\)- or \(H_{\phi}\)- field component of the exciting \(TE_{0p}\)- or \(TM_{0p}\)- pulse waveguide wave; \(v_{p}(z,t)\) and \(\mu_{p}^{+}(\rho)\) are its space-time amplitude and transverse eigen function respectively; \(\overline{\Theta}\) is the closure of G; \(U(g,t)=E_{\phi}(g,t)\) in the case of \(TE_{0}\)-waves (\(E_{\rho}=E_{z}=H_{\phi}\equiv 0\), \(\partial_{\phi}=0\)) and \(U(g,t)=H_{\phi}(g,t)\) for \(TM_{0}\)-waves (\(H_{\rho}=H_{z}=E_{\phi}\equiv 0\), \(\partial_{\phi}=0\)); \(\mathbf{E}(q,t)=\{E_{\rho},E_{\phi},E_{z}\}\) and \(\mathbf{H}(q,t)=\big{\{}H_{\rho},H_{\phi},H_{z}\}\) are the vectors of the electric and magnetic fields, respectively; \(\{\rho,\phi,z\}\) and \(\{r,\vartheta,\phi\}\) are the cylindrical and spherical coordinates, respectively; piecewise constant functions \(\sigma(g)\geq 0\) and \(\varepsilon(g)>0\) specify a conductivity and relative permittivity of non-magnetic elements of the structure; \(\eta_{0}=(\mu_{0}/\varepsilon_{0})^{1/2}\) is an impedance of free space; \(\varepsilon_{0}\) and \(\mu_{0}\) are electric and magnetic vacuum constants, respectively. The surfaces \(\Sigma=\Sigma_{\phi}\times[0\leq\phi\leq 2\pi]\) of perfectly conducting elements of the structure and surfaces \(\Sigma^{\varepsilon,\sigma}=\Sigma^{\varepsilon,\sigma}_{\phi}\times[0\leq \phi\leq 2\pi]\), where the constitutive parameters \(\sigma(g)\) and \(\varepsilon(g)\) are discontinuous, are assumed to be sufficiently smooth. Nonzero components of the electromagnetic field lying in the half-plane \(\Omega_{\text{total}}=\{q=\{\rho,\phi,z\}:\ \rho>0,\ |z|\geq 0,\ \phi=const\}\) are determined by the expressions  
\[\partial_{t}H_{\rho}(g,t)=\eta_{0}^{-1}\partial_{z}E_{\phi}(g,t),\ \ \partial_{t}H_{z}(g,t)=-\eta_{0}^{-1}\rho^{-1}\partial_{\rho}\big{[}\rho E_{\phi} (g,t)\big{]} \tag{2}\]  
in the case of \(TE_{0}\)-waves, and in the case of \(TM_{0}\)-waves, by relations  
\[ \begin{array}{ll}\Big{[}\varepsilon(g)\eta_{0}^{-1}\partial_{t}+\sigma(g) \Big{]}E_{\rho}(g,t)=-\partial_{z}H_{\phi}(g,t),\\ \Big{[}\varepsilon(g)\eta_{0}^{-1}\partial_{t}+\sigma(g)\Big{]}E_{z}(g,t)=\rho ^{-1}\partial_{\rho}\big{[}\rho H_{\phi}(g,t)\big{]}.\end{array}  \tag{3}\]  
Computational domain \(\Omega_{\text{int}}\) of the problem (1) is a part of the plane \(\Omega_{\text{total}}\), limited by contour \(\Sigma^{\varepsilon,\sigma}_{\phi}\), axis \(z\), virtual boundaries \(\text{L}^{\pm}\) (input and output ports in the cross section of virtual waveguides A and B) and by a'spherical' boundary \(\text{L}=\{g=\{r,\vartheta\}:\ r=L\}\), separating \(\Omega_{\text{int}}\) from free space \(\Omega_{\text{ext}}=\{g=\{r,\vartheta\}:\ r>L\}\) in \(\Omega_{\text{total}}\).