the radiation conditions (\[2-5\]).

Now we address the matching conditions at the sphere surface. By making use of Eqs. (\[3-1\]) we can write, in an explicit form, the radial (r) and tangential (t) components of electric and magnetic fields in the case of spherical symmetry. For TE-modes these equations read $$\begin{aligned}
E^{\text{TE}}_{klm,\text{r}}&=&0\,{,} \label{a-20}\\
E^{\text{TE}}_{klm,\text{t}}&=&a_{lm}(k)\,f_{kl}^{\text{TE}}(r)\,\mathbf{X}_{lm}\,{,}
\label{a-21}\\ H^{\text{TE}}_{klm,\text{r}}&=&\frac{1}{kr}\left (
\frac{\varepsilon}{\mu} \right
)^{1/2}\sqrt{l(l+1)}\,a_{lm}(k)f_{kl}^{\text{TE}}(r)\,Y_{lm} \,{,}
\label{a-22}\\ H^{\text{TE}}_{klm,\text{t}}&=&\frac{i}{kr}\left (
\frac{\varepsilon}{\mu} \right
)^{1/2}a_{lm}(k)\,\frac{d}{dr}\left( rf_{kl}^{\text{TE}}(r)
\right)\,\mathbf{X}_{lm}^\perp\,{,} \label{a-23}\end{aligned}$$ and the same for the TM-modes $$\begin{aligned}
E^{\text{TM}}_{klm,\text{r}}&=&-\frac{1}{kr}\left (
\frac{\mu}{\varepsilon} \right
)^{1/2}\sqrt{l(l+1)}\,b_{lm}(k)f_{kl}^{\text{TM}}(r)\,Y_{lm} \,{,}
\\ E^{\text{TM}}_{klm,\text{t}}&=&-\frac{i}{kr}\left (
\frac{\mu}{\varepsilon} \right
)^{1/2}b_{lm}(k)\,\frac{d}{dr}\left( rf_{kl}^{\text{TM}}(r)
\right)\,\mathbf{X}_{lm}^\perp\,{,} \label{a-25}\\
H^{\text{TM}}_{klm,\text{r}}&=&0\,{,} \label{a-26}\\
H^{\text{TM}}_{klm,\text{t}}&=&b_{lm}(k)\,f_{kl}^{\text{TM}}(r)\,\mathbf{X}_{lm}\,{.}
\label{a-27}\end{aligned}$$ Here $\mathbf{X}_{lm}$ are the vector spherical harmonics [@Jackson] $$\label{vsh}
\mathbf{X}_{lm}(\theta,\phi)=\frac{\mathbf{L}\,Y_{lm}(\theta,\phi)}{\sqrt{l(l+1)}}\,{,}\quad
l\geq1\,{,}$$ where $\mathbf{L}$ is the angular momentum operator $$\mathbf{L}=-i\,(\mathbf{r}\times \bm{\nabla})\,{.}$$ The vector spherical harmonic $\mathbf{X}_{lm}^{\perp}$ is obtained from $\mathbf{X}_{lm}$ after rotation by the angle $\pi/2$ around the normal $\mathbf{n}=\mathbf{r}/r$. From Eqs.(\[a-20\]) – (\[a-27\]) it follows, in particular, that the tangential components of electric field in TE- and TM-modes are orthogonal each other and the same holds for the magnetic field. It implies that the matching conditions on the sphere surface do not couple TE- and TM-modes.

At the sphere surface the tangential components of electric and magnetic fields are continuous (see Eqs. (\[a-20\]) – 