# Axial-Symmetric Diffraction Radiation Antenna with a Very Narrow Funnel-Shaped Directional Diagram
## 3 Frequency Domain Representations  
The solution \(U(g,t)\) to the problem (1) and (4) (we construct it by implementing the standard computational schemes of the finite difference method [41]) and the solution \(\widetilde{U}(g,k)\) to the problem  
\[\left\{ \begin{array}{ll}\big{[}\partial_{z}^{2}+\partial_{\rho}\big{(}\rho^{- 1}\partial_{\rho}\rho\big{)}+\widetilde{\varepsilon}(g)k^{2}\big{]}\widetilde {U}(g,k)=0;&g=\{\rho,z\}\in\Omega_{\rm int}\\ \widetilde{\mathbf{E}}_{\rm tg}(q,k)\big{|}_{q=\{\rho,\rho,z\}\in\Sigma}=0,& \widetilde{U}(0,z,k)=0\ \ \ \text{for}\ \ \ \{0,z\}\in\overline{\Omega_{\rm int}}\\ \widetilde{\mathbf{E}}_{\rm tg}(q,k),&\widetilde{\mathbf{H}}_{\rm tg}(q,k)\ \ \ \text{are continuous when crossing}\ \ \Sigma^{z,\sigma}\\ \text{and boundaries}\ \ \ \text{L}\times[0,2\pi],&\text{L}^{\pm}\times[0,2\pi], \end{array} \right. \tag{13a}\] \[\widetilde{U}(g,k)=\widetilde{V}_{p}(g,k)+\widetilde{U}^{+}(g,k)=\widetilde{V}_ {p}(g,k)+\sum\limits_{n\in\{n\}}\widetilde{U}^{+}_{np}(g,k)=\] (13b) \[=A_{p}(k)\exp\Big{(}i\gamma_{p}^{+}z\Big{)}\mu_{p}^{+}(\rho)+\sum\limits_{n \in\{n\}}B_{np}^{+}(k)\exp(-i\gamma_{n}^{+}z)\mu_{n}^{+}(\rho);\ \ \ g\in \overline{\Lambda},\] \[\widetilde{U}(g,k)=\widetilde{U}^{-}(g,k)=\sum\limits_{n\in\{n\}}\widetilde{U}^{-} _{np}(g,k)=\sum\limits_{n\in\{n\}}B_{np}^{-}(k)\exp\big{(}-i\gamma_{n}^{-}z \big{)}\mu_{n}^{-}(\rho);\ \ \ g\in\overline{\mathbb{B}},\] (13c) \[\widetilde{U}(g,k)=\sum\limits_{m=1}^{\infty}\widetilde{U}_{mp}(g,k)=\frac{1}{ \sqrt{r}}\sum\limits_{m=1}^{\infty}C_{mp}(k)H^{(1)}_{m+1/2}(kr)\widetilde{H}_ {m}(\cos\theta);\ \ \ g\in\overline{\Omega_{\rm ext}}, \tag{13d}\]  
are connected by integral transformation \(\widetilde{f}(k)=\int_{0}^{\infty}f(t)\exp(ikt)dt\)[25; 32; 34] (actually by integral transformation \(\widetilde{f}(k)=\int_{0}^{T}f(t)\exp(ikt)dt\); \(T<\infty\) is an observation time in computational experiments). This transformation matches the characteristics of the time domain \(f(t)\) with the characteristics of the frequency domain \(\widetilde{f}(k)\)--\(\widetilde{f}(k)\leftrightarrow f(t)\). Here, \(H^{(1)}_{m+1/2}(\cdots)\) are cylindrical Hankel functions, \(\widetilde{U}(g,k)=\widetilde{E}_{\phi}(g,k)\) in the case of monochromatic \(TE_{0}\)-waves and \(\widetilde{U}(g,k)=\widetilde{H}_{\phi}(g,k)\) in the case of monochromatic \(TM_{0}\)-waves; \(k\) is a wavenumber (frequency parameter or just frequency, is related with frequency \(f[Hz]\) by the ratio \(f[Hz]=\frac{v[m/s]}{2\pi}k[rad/m]\), where \(v[m/s]\) is a speed of light in vacuum), \(\widetilde{\varepsilon}(g)=\varepsilon(g)+i\eta_{0}\sigma(g)/k\) and  
\[\gamma_{n}^{\pm}=\sqrt{k^{2}-\big{(}\lambda_{n}^{\pm}\big{)}^{2}};\ \ \ \text{Re}\gamma_{n}^{\pm}\geq 0,\ \ \ \text{Im}\gamma_{n}^{\pm}\geq 0 \tag{14}\]  
are longitudinal wavenumbers of \(TE_{0n}\)- or \(TM_{0n}\)- monochromatic eigen waves propagating in waveguides A and B with (if \(\text{Im}\gamma_{n}^{\pm}>0\)) or without (if \(\text{Im}\gamma_{n}^{\pm}=0\)) attenuation. In Formula (13b,c), the term with the complex amplitude \(A_{p}\) corresponds to the monochromatic wave \(\widetilde{V}_{p}(g,k)\ \leftrightarrow\ V_{p}(g,t)\) arriving at the boundary \(\text{L}^{+}\), and the terms with the amplitudes, \(B_{np}^{+}\), \(B_{np}^{-}\), and \(C_{mp}\), to the waves of the secondary field \(\widetilde{U}^{+}_{np}(g,k)\ \leftrightarrow\ U^{+}_{np}(g,t)\), \(\widetilde{U}^{-}_{np}(g,k)\ \leftrightarrow\ U^{-}_{np}(g,t)\), and \(\widetilde{U}_{mp}(g,k)\leftrightarrow U_{mp}(g,t)\), both in the waveguides A, B and in a free space. If we compare representations (13b-d) and (4), it becomes obvious that  
\[A_{p}(k)\ \leftrightarrow\ v_{p}(0,t),\ \ B_{np}^{\pm}(k)\ \leftrightarrow\ u_{ npnp}^{\pm}(0,t), \tag{15}\]  
or, in another notation, \(A_{p}(k)\ =\ \widetilde{v}_{p}(0,k)\), \(B_{np}^{\pm}(k)\ =\ \widetilde{u}_{np}^{\pm}(0,k)\).