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\begin{document}

\preprint{AIP/123-QED}



\begin{equation}
\begin{split}
&E_z^{(4)}(r,\theta)=\sum_{n=1,3,5}^\infty{A_n^{(4)}}\sin(n\theta){K_n}(h_4{r})\\&+\sum_{n=0,2,4}^\infty{B_n^{(4)}}\cos(n\theta){K_n}(h_4{r})+\sum_{n=1,3,5}^\infty{S_n^{(4)}}\sin(n\theta){I_n}(h_4{r})\\&+\sum_{n=0,2,4}^\infty{T_n^{(4)}}\cos(n\theta){I_n}(h_4{r})\\
&B_z^{(4)}(r,\theta)=\sum_{n=2,4,6}^\infty{C_n^{(4)}}\sin(n\theta){K_n}(h_4{r})\\&+\sum_{n=1,3,5}^\infty{D_n^{(4)}}\cos(n\theta){K_n}(h_4{r})+\sum_{n=2,4,6}^\infty{U_n^{(4)}}\sin(n\theta){I_n}(h_4{r})\\&+\sum_{n=1,3,5}^\infty{V_n^{(4)}}\cos(n\theta){I_n}(h_4{r})\\
\end{split}
\end{equation}
Taking into account the  Eqs.(13-14) the continuity of $E_t$  for a certain point $(r_m,\theta_m)$ on the boundaries are written as
\begin{equation}
\begin{split} 
&E_t^{(1)}(r_m,\theta_m)=\\&\frac{ik_z}{h_1^2}\bigg\lbrace\sum_{n=1,3,5}^\infty{A_n^{(1)}}[h_1f(\theta_m)\sin(n\theta_m)J'_n(h_1{r_m})+\\&(\frac{n}{r_m})g(\theta_m)\cos(n\theta_m)J_n(h_1{r_m})]\\&+\sum_{n=0,2,4}^\infty{B_n^{(1)}}[h_1f(\theta_m)\cos(n\theta_m)J'_n(h_1{r_m})\\&-(\frac{n}{r_m})g(\theta_m)\sin(n\theta_m)J_n(h_1{r_m})]\\&+\sum_{n=1,3,5}^\infty{S_n^{(1)}}[h_1f(\theta_m)\sin(n\theta_m)N'_n(h_1{r_m})+\\&(\frac{n}{r_m})g(\theta_m)\cos(n\theta_m)N_n(h_1{r_m})]\\&+\sum_{n=0,2,4}^\infty{T_n^{(1)}}[h_1f(\theta_m)\cos(n\theta_m)N'_n(h_1{r_m})\\&-(\frac{n}{r_m})g(\theta_m)\sin(n\theta_m)N_n(h_1{r_m})]\\&+\sum_{n=2,4,6}^\infty(\frac{\omega}{k_zc}){C_n^{(1)}}[(\frac{n}{r_m})f(\theta_m)\cos(n\theta_m)J_n(h_1{r_m})\\&-h_1g(\theta_m)\sin(n\theta_m)J'_n(h_1{r_m})]\\&+\sum_{n=1,3,5}^\infty(\frac{\omega}{k_zc}){D_n^{(1)}}[(-\frac{n}{r_m})f(\theta_m)\sin(n\theta_m)J_n(h_1{r_m})\\&-h_1g(\theta_m)\cos(n\theta_m)J'_n(h_1{r_m}]\\&+\sum_{n=2,4,6}^\infty(\frac{\omega}{k_zc}){U_n^{(1)}}[(\frac{n}{r_m})f(\theta_m)\cos(n\theta_m)N_n(h_1{r_m})\\&-h_1g(\theta_m)\sin(n\theta_m)N'_n(h_1{r_m})]\\&+\sum_{n=1,3,5}^\infty(\frac{\omega}{k_zc}){V_n^{(1)}}[(-\frac{n}{r_m})f(\theta_m)\sin(n\theta_m)N_n(h_1{r_m})\\&-h_1g(\theta_m)\cos(n\theta_m)N'_n(h_1{r_m}]\bigg\rbrace\\
\end{split} 
\end{equation}
\begin{equation}
\begin{split} 
&B_t^{(1)}(r_m,\theta_m)=\\&\frac{ik_z}{h_1^2}\bigg\lbrace\sum_{n=1,3,5}^\infty(\frac{k_1^2}{\omega{k_z}}){A_n^{(1)}}[-(\frac{n}{r_m})f(\theta_m)\cos(n\theta_m)J_n(h_1{r_m})\\&+h_1g(\theta_m)\sin(n\theta_m)J'_n(h_1{r_m})]\\&+\sum_{n=0,2,4}^\infty{B_n^{(1)}}(\frac{k_1^2}{\omega{k_z}})[(\frac{n}{r_m})f(\theta_m)\sin(n\theta_m)J_n(h_1{r_m})\\&+h_1g(\theta_m)\cos(n\theta_m)J'_n(h_1{r_m})]\\&+\sum_{n=1,3,5}^\infty(\frac{k_1^2}{\omega{k_z}}){S_n^{(1)}}[-(\frac{n}{r_m})f(\theta_m)\cos(n\theta_m)N_n(h_1{r_m})\\&+h_1g(\theta_m)\sin(n\theta_m)N'_n(h_1{r_m})]\\&+\sum_{n=0,2,4}^\infty{T_n^{(1)}}(\frac{k_1^2}{\omega{k_z}})[(\frac{n}{r_m})f(\theta_m)\sin(n\theta_m)N_n(h_1{r_m})\\&+h_1g(\theta_m)\cos(n\theta_m)N'_n(h_1{r_m})]\\&+\sum_{n=2,4,6}^\infty{C_n^{(1)}}[h_1f(\theta_m)\sin(n\theta_m)N'_n(h_1{r_m})\\&+(\frac{n}{r_m})g(\theta_m)\cos(n\theta_m)J_n(h_1{r_m})]\\&+\sum_{n=1,3,5}^\infty{D_n^{(1)}}[h_1f(\theta_m)\cos(n\theta_m)J'_n(h_1{r_m})\\&-(\frac{n}{r_m})g(\theta_m)\sin(n\theta_m)J_n(h_1{r_m})]\\&+\sum_{n=2,4,6}^\infty{U_n^{(1)}}[h_1f(\theta_m)\sin(n\theta_m)N'_n(h_1{r_m})\\&+(\frac{n}{r_m})g(\theta_m)\cos(n\theta_m)N_n(h_1{r_m})]\\&+\sum_{n=1,3,5}^\infty{V_n^{(1)}}[h_1f(\theta_m)\cos(n\theta_m)N'_n(h_1{r_m})\\&-(\frac{n}{r_m})g(\theta_m)\sin(n\theta_m)N_n(h_1{r_m})]\bigg\rbrace
\end{split} 
\end{equation}
\begin{equation}
\begin{split}
&E_t^{(2)}(r_m,\theta_m)=\\&\frac{ik_z}{h_2^2}\bigg\lbrace\sum_{n=1,3,5}^\infty{A_n^{(2)}}[h_2f(\theta_m)\sin(n\theta_m)J'_n(h_2{r_m})\\&+(\frac{n}{r_m})g(\theta_m)\cos(n\theta_m)J_n(h_2{r_m})]\\&+\sum_{n=0,2,4}^\infty{B_n^{(2)}}[h_2f(\theta_m)\cos(n\theta_m)J'_n(h_2{r_m})\\&-(\frac{n}{r_m})g(\theta_m)\sin(n\theta_m)J_n(h_2{r_m})]\\&+\sum_{n=2,4,6}^\infty(\frac{\omega}{k_zc}){C_n^{(2)}}[(\frac{n}{r_m})f(\theta_m)\cos(n\theta_m)J_n(h_2{r_m})\\&-h_2g(\theta_m)\sin(n\theta_m)J'_n(h_2{r_m})]\\&+\sum_{n=1,3,5}^\infty(\frac{\omega}{k_zc}){D_n^{(2)}}[(-\frac{n}{r_m})f(\theta_m)\sin(n\theta_m)J_n(h_2{r_m})\\&-h_2g(\theta_m)\cos(n\theta_m)J'_n(h_2{r_m}]\bigg\rbrace\\
\end{split}
\end{equation}
\begin{equation}
\begin{split}
&B_t^{(2)}(r_m,\theta_m)=\\&\frac{ik_z}{h_2^2}\bigg\lbrace\sum_{n=1,3,5}^\infty(\frac{k_2^2}{\omega{k_z}}){A_n^{(2)}}[-(\frac{n}{r_m})f(\theta_m)\cos(n\theta_m)J_n(h_2{r_m})\\&+h_2g(\theta_m)\sin(n\theta_m)J'_n(h_2{r_m})]\\&+\sum_{n=0,2,4}^\infty{B_n^{(2)}}(\frac{k_2^2}{\omega{k_z}})[(\frac{n}{r_m})f(\theta_m)\sin(n\theta_m)J_n(h_2{r_m})\\&+h_2g(\theta_m)\cos(n\theta_m)J'_n(h_2{r_m})]\\&+\sum_{n=2,4,6}^\infty{C_n^{(2)}}[h_2f(\theta_m)\sin(n\theta_m)J'_n(h_2{r_m})\\&+(\frac{n}{r_m})g(\theta_m)\cos(n\theta_m)J_n(h_2{r_m})]\\&+\sum_{n=1,3,5}^\infty{D_n^{(2)}}[h_2f(\theta_m)\cos(n\theta_m)J'_n(h_2{r_m})\\&-(\frac{n}{r_m})g(\theta_m)\sin(n\theta_m)J_n(h_2{r_m})]\bigg\rbrace
\end{split}
\end{equation}
\begin{equation}
\begin{split}
&E_t^{(3)}(r_m,\theta_m)=\\&\frac{ik_z}{h_3^2}\bigg\lbrace\sum_{n=1,3,5}^\infty{A_n^{(3)}}[-h_3f(\theta_m)\sin(n\theta_m)K'_n(h_3{r_m})\\&-(\frac{n}{r_m})g(\theta_m)\cos(n\theta_m)K_n(h_3{r_m})]\\&+\sum_{n=0,2,4}^\infty{B_n^{(3)}}[-h_3f(\theta_m)\cos(n\theta_m)K'_n(h_3{r_m})\\&+(\frac{n}{r_m})g(\theta_m)\sin(n\theta_m)K_n(h_3{r_m})]\\&+\sum_{n=1,3,5}^\infty{S_n^{(3)}}[-h_3f(\theta_m)\sin(n\theta_m)I'_n(h_3{r_m})\\&-(\frac{n}{r_m})g(\theta_m)\cos(n\theta_m)I_n(h_3{r_m})]\\&+\sum_{n=0,2,4}^\infty{T_n^{(3)})}[-h_3f(\theta_m)\cos(n\theta_m)I'_n(h_3{r_m})\\&+(\frac{n}{r_m})g(\theta_m)\sin(n\theta_m)I_n(h_3{r_m})]\\&+\sum_{n=2,4,6}^\infty(\frac{\omega}{k_zc}){C_n^{(3)}}[(-\frac{n}{r_m})f(\theta_m)\cos(n\theta_m)K_n(h_3{r_m})\\&+h_3g(\theta_m)\sin(n\theta_m)K'_n(h_3{r_m})]\\&+\sum_{n=1,3,5}^\infty(\frac{\omega}{k_zc}){D_n^{(3)}}[(\frac{n}{r_m})f(\theta_m)\sin(n\theta_m)K_n(h_3{r_m})\\&+h_3g(\theta_m)\cos(n\theta_m)K'_n(h_3{r_m})]\\&+\sum_{n=2,4,6}^\infty(\frac{\omega}{k_zc}){U_n^{(3)}}[(-\frac{n}{r_m})f(\theta_m)\cos(n\theta_m)I_n(h_3{r_m})\\&+h_3g(\theta_m)\sin(n\theta_m)I'_n(h_3{r_m})]\\&+\sum_{n=1,3,5}^\infty(\frac{\omega}{k_zc}){V_n^{(3)}}[(\frac{n}{r_m})f(\theta_m)\sin(n\theta_m)I_n(h_3{r_m})\\&+h_3g(\theta_m)\cos(n\theta_m)I'_n(h_3{r_m})]\bigg\rbrace
\end{split}
\end{equation}
\begin{equation}
\begin{split}
&B_t^{(3)}(r_m,\theta_m)\\&=\frac{ik_z}{h_3^2}\bigg\lbrace\sum_{n=1,3,5}^\infty(\frac{\varepsilon_3\omega}{k_zc}){A_n^{(3)}}[(\frac{n}{r_m})f(\theta_m)\cos(n\theta_m)K_n(h_3{r_m})\\&-h_3g(\theta_m)\sin(n\theta_m)K'_n(h_3{r_m})]\\&
+\sum_{n=0,2,4}^\infty(\frac{\varepsilon_3\omega}{k_zc}){B_n^{(3)}}[(-\frac{n}{r_m})f(\theta_m)\sin(n\theta_m)K_n(h_3{r_m})\\&-h_3g(\theta_m)\cos(n\theta_m)K'_n(h_3{r_m})]\\&+\sum_{n=1,3,5}^\infty(\frac{\varepsilon_3\omega}{k_zc}){S_n^{(3)}}[(\frac{n}{r_m})f(\theta_m)\cos(n\theta_m)I_n(h_3{r_m})\\&-h_3g(\theta_m)\sin(n\theta_m)I'_n(h_3{r_m})]\\&+\sum_{n=0,2,4}^\infty(\frac{\varepsilon_3\omega}{k_zc}){T_n^{(3)}}[(-\frac{n}{r_m})f(\theta_m)\sin(n\theta_m)I_n(h_3{r_m})\\&-h_3g(\theta_m)\cos(n\theta_m)I'_n(h_3{r_m})]\\&+\sum_{n=2,4,6}^\infty{C_n^{(3)}}[-h_3f(\theta_m)\sin(n\theta_m)K'_n(h_3{r_m})\\&-(\frac{n}{r_m})g(\theta_m)\cos(n\theta_m)K_n(h_3{r_m})]
\\&+\sum_{n=1,3,5}^\infty{D_n^{(3)}}[-h_3f(\theta_m)\cos(n\theta_m)K'_n(h_3{r_m})\\&+(\frac{n}{r_m})g(\theta_m)\sin(n\theta_m)K_n(h_3{r_m})]\\&+\sum_{n=2,4,6}^\infty({U_n^{(3)}}[-h_3f(\theta_m)\sin(n\theta_m)I'_n(h_3{r_m})\\&-(\frac{n}{r_m})g(\theta_m)\cos(n\theta_m)I_n(h_3{r_m})]\\&+\sum_{n=1,3,5}^\infty{V_n^{(3)}}[-h_3f(\theta_m)\cos(n\theta_m)I'_n(h_3{r_m})\\&+(\frac{n}{r_m})g(\theta_m)\sin(n\theta_m)I_n(h_3{r_m})]\bigg\rbrace
\end{split}
\end{equation}
\begin{equation}
\begin{split}
&E_t^{(4)}(r_m,\theta_m)=\\&\frac{ik_z}{h_4^2}\bigg\lbrace\sum_{n=1,3,5}^\infty{A_n^{(4)}}[h_4f(\theta_m)\sin(n\theta_m)K'_n(h_4{r_m})\\&+(\frac{n}{r_m})g(\theta_m)\cos(n\theta_m)K_n(h_4{r_m})]\\&+\sum_{n=0,2,4}^\infty{B_n^{(4)}}[h_4f(\theta_m)\cos(n\theta_m)K'_n(h_4{r_m})\\&-(\frac{n}{r_m})g(\theta_m)\sin(n\theta_m)K_n(h_4{r_m})]\\&+\sum_{n=1,3,5}^\infty{S_n^{(4)}}[h_4f(\theta_m)\sin(n\theta_m)I'_n(h_4{r_m})\\&+(\frac{n}{r_m})g(\theta_m)\cos(n\theta_m)I_n(h_4{r_m})]\\&+\sum_{n=0,2,4}^\infty{T_n^{(4)}}[h_4f(\theta_m)\cos(n\theta_m)I'_n(h_4{r_m})\\&-(\frac{n}{r_m})g(\theta_m)\sin(n\theta_m)I_n(h_4{r_m})]\\&+\sum_{n=2,4,6}^\infty(\frac{\omega}{k_zc}){C_n^{(4)}}[(\frac{n}{r_m})f(\theta_m)\cos(n\theta_m)K_n(h_4{r_m})\\&-h_4g(\theta_m)\sin(n\theta_m)K'_n(h_4{r_m})]\\&+\sum_{n=1,3,5}^\infty(\frac{\omega}{k_zc}){D_n^{(4)}}[(-\frac{n}{r_m})f(\theta_m)\sin(n\theta_m)K_n(h_4{r_m})\\&-h_4g(\theta_m)\cos(n\theta_m)K'_n(h_4{r_m})]\\&+\sum_{n=2,4,6}^\infty(\frac{\omega}{k_zc}){U_n^{(4)}}[(\frac{n}{r_m})f(\theta_m)\cos(n\theta_m)I_n(h_4{r_m})\\&-h_4g(\theta_m)\sin(n\theta_m)I'_n(h_4{r_m})]\\&+\sum_{n=1,3,5}^\infty(\frac{\omega}{k_zc}){V_n^{(4)}}[(-\frac{n}{r_m})f(\theta_m)\sin(n\theta_m)I_n(h_4{r_m})\\&-h_4g(\theta_m)\cos(n\theta_m)I'_n(h_4{r_m})]\bigg\rbrace
\end{split}
\end{equation}
\begin{equation}
\begin{split}
&B_t^{(4)}(r_m,\theta_m)=\\&\frac{ik_z}{h_4^2}\bigg\lbrace\sum_{n=1,3,5}^\infty(\frac{k_4^2}{\omega{k_z}}){A_n^{(4)}}[(-\frac{n}{r_m})f(\theta_m)\cos(n\theta_m)K_n(h_4{r_m})\\&+h_4g(\theta_m)\sin(n\theta_m)K'_n(h_4{r_m})]\\&+\sum_{n=0,2,4}^\infty(\frac{k_4^2}{\omega{k_z}}){B_n^{(4)}}[(\frac{n}{r_m})f(\theta_m)\sin(n\theta_m)K_n(h_4{r_m})\\&+h_4g(\theta_m)\cos(n\theta_m)K'_n(h_4{r_m})]\\&+\sum_{n=1,3,5}^\infty(\frac{k_4^2}{\omega{k_z}}){S_n^{(4)}}[(-\frac{n}{r_m})f(\theta_m)\cos(n\theta_m)I_n(h_4{r_m})\\&+h_4g(\theta_m)\sin(n\theta_m)I'_n(h_4{r_m})]\\&+\sum_{n=0,2,4}^\infty(\frac{k_4^2}{\omega{k_z}}){T_n^{(4)}}[(\frac{n}{r_m})f(\theta_m)\sin(n\theta_m)I_n(h_4{r_m})\\&+h_4g(\theta_m)\cos(n\theta_m)I'_n(h_4{r_m})]\\&+\sum_{n=2,4,6}^\infty{C_n^{(4)}}[h_4f(\theta_m)\sin(n\theta_m)K'_n(h_4{r_m})\\&+(\frac{n}{r_m})g(\theta_m)\cos(n\theta_m)K_n(h_4{r_m})]
\\&+\sum_{n=1,3,5}^\infty{D_n^{(4)}}[h_4f(\theta_m)\cos(n\theta_m)K'_n(h_4{r_m})\\&-(\frac{n}{r_m})g(\theta_m)\sin(n\theta_m)K_n(h_4{r_m})]\\&+\sum_{n=2,4,6}^\infty{U_n^{(4)}}[h_0f(\theta_m)\sin(n\theta_m)I'_n(h_4{r_m})\\&+(\frac{n}{r_m})g(\theta_m)\cos(n\theta_m)I_n(h_4{r_m})]\\&+\sum_{n=1,3,5}^\infty{V_n^{(4)}}[h_0f(\theta_m)\cos(n\theta_m)I'_n(h_4{r_m})\\&-(\frac{n}{r_m})g(\theta_m)\sin(n\theta_m)I_n(h_4{r_m})]\bigg\rbrace\\
\end{split}
\end{equation}
.


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