# Axial-Symmetric Diffraction Radiation Antenna with a Very Narrow Funnel-Shaped Directional Diagram
## 3 Frequency Domain Representations
Similarly, we obtain  
\[\frac{C_{mp}(k)H^{(1)}_{m+1/2}(kr)}{\sqrt{r}}\ \leftrightarrow\ u_{mp}(r,t)\ \ \ \text{or}\ \ \ \frac{C_{mp}(k)H^{(1)}_{m+1/2}(kr)}{\sqrt{r}}=\widetilde{u}_{mp}(r,k). \tag{16}\]  
In the boundary value problem (13), expressions (13b-d) represent the so-called 'partial radiation conditions' [38], corresponding to the physically grounded requirementof the absence in the field \(\widetilde{U}(g,k)\) waves coming from infinity. An exception is possible only for the 'incident' wave. In our case, this wave is determined by the function  
\[\widetilde{V}_{p}(g,k)=A_{p}(k)\exp\Big{(}i\gamma_{p}^{+}z\Big{)}n_{p}^{+}(\rho );\ \ \ \text{Im}\gamma_{p}^{+}=0,\ \ \ g\in\ \text{ A}. \tag{17}\]  
In the frequency domain, the structure under consideration is characterized (in part) by the reflection coefficients \(R_{np}(k)\) (the conversion coefficients of the \(p\)-th incident from the waveguide A wave into \(n\)-th reflected wave) and the transmission coefficients \(T_{np}(k)\) (the conversion coefficients of the \(p\)-th wave of the waveguide A into the \(n\)-th wave of the waveguide B). These coefficients, given by \(R_{np}(k)=B_{np}^{+}/A_{p}\), \(T_{np}(k)=B_{np}^{-}/A_{p}\), and values  
\[W_{\text{abs}}(k)=\frac{k^{2}}{\gamma_{p}^{+}\beta}\int\limits_{\Omega_{\text{ int}}}\text{Im}\overline{\varepsilon}(g)\Big{|}\tilde{\mathbf{E}}(g,k)\Big{|}^{2}dg,\ \ \ W_{np}^{+}(k)=\big{|}R_{np}\big{|}^{2}\frac{\text{Re}\gamma_{n}^{+}}{\gamma_{p }^{+}},\ \ \ W_{np}^{-}(k)=\big{|}T_{np}\big{|}^{2}\frac{\text{Re}\gamma_{n}^{-}}{ \gamma_{p}^{+}} \tag{18}\]  
(\(\beta=1\) for \(TE_{0}\)-waves, and \(\beta=\eta_{0}^{2}\) for \(TM_{0}\)-waves, \(dg=\rho d\,\rho dz\)) determine [34] the relative fraction of the energy absorbed in non-ideal elements of the structure and directed to each of the waves propagating in the waveguides A and B. For any finite value \(k=2\pi/\lambda\) (\(\lambda\) is the wavelength in free space), the number \(N^{+}+N^{-}\), \(N^{\pm}=\sum_{n}(\text{Re}\gamma_{n}^{\pm}/|\gamma_{n}^{\pm}|)\) of such waves taking away the energy supplied to the axially symmetric structure is finite. It follows from (18) that the fraction of the energy emitted by the structure into free space through the virtual boundary L (radiation efficiency) can be calculated by the formula  
\[\eta(k)=1-W_{\text{abs}}(k)-\sum_{n}\Big{(}W_{np}^{+}(k)+W_{np}^{-}(k)\Big{)}. \tag{19}\]  
Normalized radiation pattern on the arc \(r=M\geq L\)  
\[=A_{p}(k)\exp\Big{(}i\gamma_{p}^{+}z\Big{)}\mu_{p}^{+}(\rho)+\sum\limits_{n \in\{n\}}B_{np}^{+}(k)\exp(-i\gamma_{n}^{+}z)\mu_{n}^{+}(\rho);\ \ \ g\in \overline{\Lambda},\]0  
determines the spatial orientation and 'energy capacity' of waves radiated into free space. Here, \(\widetilde{\mathbf{E}}_{\text{tg}}(M,\vartheta,k)\) is a tangential (with respect to the spherical surface \(r=M\)) component of the monochromatic electric field \(\widetilde{\mathbf{E}}(q,k)\). The value \(M\) determines the zone (near, mid, far) where the diagram \(D(\vartheta,k,M)\) is calculated. We presume that the boundary of the near zone is given by equality \(M=L\), and the far zone is determined by such values of \(M\), further increase in which does not lead to significant changes in the function for all considered values \(k\). The main lobe of the radiation pattern is directed at an angle \(\overline{\vartheta}(k)\) such that \(D\big{(}\overline{\vartheta}(k),k,M)=1\). The width of the main lobe \(\vartheta_{0.5}(k)\) is defined as the angle between two (\(\vartheta^{+}\) and \(\vartheta^{-}\)) directions in the main lobe of the diagram for which the radiated power is halved in relation to the maximum, i.e., \(\vartheta_{0.5}(k)=|\vartheta^{+}-\vartheta^{-}|\), where \(D(\vartheta^{+},k,M)=0.5\) and \(D(\vartheta^{-},k,M)=0.5\).