sphere the tangential component of the electric field should vanish. This leads to the following frequency equation for TE-modes $$\label{2-7} h^{(1)}_l\left ( \frac{\omega}{c}\,a \right
)=0{,}\quad l\geq 1$$ and for TM-modes $$\label{2-8} \frac{d}{d r}\left ( r \,h^{(1)}_l\left (
\frac{\omega}{c}\,r \right ) \right )=0, \quad r=a,\quad l\geq
1\,{.}$$ The spherical Hankel function $h^{(1)}_l(z)$ is $e ^{i z}$ multiplied by the polynomial in $1/z$ of a finite order [@AS]. Hence frequency equations (\[2-7\]) and (\[2-8\]) have a finite number of roots which are in the general case complex numbers. For $l=1$ (the lowest oscillations) Eqs.(\[2-7\]) and (\[2-8\]) assume the form $(z=a\,\omega/c)$ $$\begin{aligned}
h^{(1)}_l(z)&=&-\frac{1}{z}\,e^{i z}\left ( 1+\frac{i}{z} \right
)=0\quad (\text{TE modes}),
 \label{2-8a} \\
\frac{d }{d z}\left (z\, h^{(1)}_l(z)\right
)&=&-\frac{i}{z^2}\,e^{i z}\left ( z^2+i z-1 \right )=0\quad
(\text{TM modes})\,{.}
 \label{2-9}\end{aligned}$$ Thus the lowest eigenfrequencies are $$\begin{aligned}
\frac{\omega}{c}&=&-\frac{i}{a}\quad (\mbox{TE modes})\,{,}
\label{2-10}
\\
\frac{\omega}{c}&=&-\frac{1}{2a}(i\pm\sqrt 3)\quad (\text{TM
modes})\,{.} \label{2-11}\end{aligned}$$

The complex eigenfrequencies lead to a specific time and spatial dependence of the respective natural modes and ultimately of the electromagnetic fields. So, with allowance of (\[2-10\]), we obtain $$\label{2-12} e^{-i \omega t}f_{k1}^{\text{TE}}(r)=-i
\,C\,\frac{a}{r}\,e^{(r-ct)/a}\left ( 1-\frac{a}{r} \right ),
\quad r\geq a\,{.}$$ Thus, the eigenfunctions are exponentially going down in time and exponentially going up when $r$ increases. Such a time 