## Appendix A  
We used a definition of vector spherical harmonics (VSH) that built upon the definitions of [PERSON] [61], [PERSON] [62], and [PERSON] et al. [63]. The starting point was the solution of the scalar Helmholtz equation \(\
abla^{2}\psi+k^{2}\psi=0\)  
\[\psi_{lm}^{(j)}=-\frac{1}{i\sqrt{l(l+1)}}z_{l}^{(j)}(kr)Y_{lm},\ \psi_{00}^{(j)}=iz_{0}^{(j)}(kr)Y_{00},\]  
where the scalar spherical harmonic is \(Y_{lm}(\hat{\mathbf{r}})=\sqrt{\frac{2l+1}{4\pi}\cdot\frac{(l-m)!}{(l+m)!}}p^{m}( \cos\theta)e^{im\varphi}\).  
The VSH are derived from \(\psi_{lm}^{(j)}\), analogous to the definitions of [PERSON] [61], as  
\[\mathbf{L}_{lm}^{(j)}=\frac{1}{k}\
abla\psi_{lm}^{(j)},\ \mathbf{M}_{lm}^{(j)}=\
abla \times\left(\mathbf{r}\psi_{lm}^{(j)}\right),\ \mathbf{N}_{lm}^{(j)}=\frac{1}{k}\
abla\times\mathbf{M}_{lm}^{(j)}\]  
\[\mathbf{L}_{00}^{(j)}=\frac{1}{k}\
abla\psi_{00}^{(j)},\ \mathbf{M}_{00}^{(j)}=0,\ \mathbf{N}_{00}^{(j)}=0\]  
In spherical coordinates, these VSH can be expressed as  
\[\mathbf{L}_{lm}^{(j)}=\frac{1}{k}\
abla\psi_{lm}^{(j)}=\frac{1}{k}\frac{\partial \psi_{lm}^{(j)}}{\partial r}\hat{\mathbf{r}}+\frac{1}{kr}\frac{\partial\psi_{lm}^ {(j)}}{\partial\theta}\hat{\mathbf{\theta}}+\frac{1}{kr}\frac{im}{\sin\theta} \psi_{lm}^{(j)}\hat{\mathbf{\varphi}}\]  
\[\mathbf{M}_{lm}^{(j)}=\
abla\times\left(\mathbf{r}\psi_{lm}^{(j)}\right)=\frac{im}{ \sin\theta}\psi_{lm}^{(j)}\hat{\mathbf{\theta}}-\frac{\partial\psi_{lm}^{(j)}}{ \partial\theta}\hat{\mathbf{\varphi}}\]  
\[\mathbf{N}_{lm}^{(j)}=\frac{1}{k}\
abla\times\mathbf{M}_{lm}^{(j)}=\frac{l(l+1)}{kr} \psi_{lm}^{(j)}\hat{\mathbf{r}}+\frac{1}{kr}\frac{\partial^{2}}{\partial r\partial \theta}\left[\tau\psi_{lm}^{(j)}\right]\hat{\mathbf{\theta}}+\frac{1}{kr}\frac{im }{\sin\theta}\frac{\partial}{\partial r}\left[\tau\psi_{lm}^{(j)}\right]\hat{ \mathbf{\varphi}}\]  
These harmonics are directly related to the harmonics of [PERSON] [62]  
\[\mathbf{X}_{lm}(\hat{\mathbf{r}})=\frac{1}{\sqrt{l(l+1)}}\mathbf{L}Y_{lm},\ \hat{\mathbf{r}}Y_{lm},\ \hat{\mathbf{r}}\times\mathbf{X}_{lm},\]  
where \(\mathbf{L}=-i(\mathbf{r}\times\
abla)\) is the angular momentum operator  
\[\mathbf{M}_{lm}^{(j)}=\frac{z_{l}^{(j)}(kr)}{i\sqrt{l(l+1)}}\mathbf{r}\times\
abla Y_{ lm}=z_{l}^{(j)}(kr)\mathbf{X}_{lm}(\hat{\mathbf{r}})\]  
\[\mathbf{N}_{lm}^{(j)}=\frac{1}{k}\
abla\times\left(z_{l}^{(j)}(kr)\mathbf{X}_{lm}(\hat {\mathbf{r}})\right)=\frac{i\sqrt{l(l+1)}}{kr}z_{l}^{(j)}(kr)\hat{\mathbf{r}}Y_{lm}+ \frac{1}{kr}\frac{\partial}{\partial r}\left(rz_{l}^{(j)}(kr)\right)\hat{\mathbf{ r}}\times\mathbf{X}_{lm}\]  
The vector spherical harmonics \(\mathbf{L}_{lm}^{(j)},\ \mathbf{N}_{lm}^{(j)},\mathbf{M}_{lm}^{(j)}\) can be expressed using the harmonics \(\mathbf{Y}_{jm}^{l}(\hat{\mathbf{r}})\) of [PERSON] et al. [63].  
\[\begin{pmatrix}\mathbf{L}_{lm}^{(j)}\\ \mathbf{M}_{lm}^{(j)}\\ \mathbf{N}_{lm}^{(j)}\end{pmatrix}=\begin{pmatrix}\frac{iz_{l-1}^{(j)}(kr)}{\sqrt{ (l+1)(2l+1)}}&0&\frac{iz_{l+1}^{(j)}}{\sqrt{l(2l+1)}}\\ 0&z_{l}^{(j)}&0\\ i\sqrt{\frac{l+1}{2l+1}}z_{l-1}^{(j)}&0&-i\sqrt{\frac{l}{2l+1}}z_{l+1}^{(j)} \end{pmatrix}\begin{pmatrix}\mathbf{Y}_{lm}^{l-1}\\ \mathbf{Y}_{lm}^{l+1}\\ \mathbf{Y}_{lm}^{l+1}\end{pmatrix} \tag{11}\]According to the quantum theory of angular momentum, the vector spherical harmonics of [PERSON] et al. [63] can be expressed on a spherical basis by using Clebsch-Gordan coefficients.  
\[\mathbf{Y}_{jm}^{l}(\hat{\mathbf{r}})=\sum\limits_{m^{\prime},\sigma}lm^{\prime}1\sigma| jm\gamma_{lm^{\prime}}(\hat{\mathbf{r}})\mathbf{e}_{\sigma}=\sum\limits_{m^{\prime}, \sigma}C_{lm^{\prime},l\sigma}^{jm}Y_{lm^{\prime}}\mathbf{e}_{\sigma}\]  
where the spherical basis vectors are the eigenstates of the spin operators \(S^{2}\) and \(S_{z}\).  
\[\mathbf{e}_{\pm 1}=\mp\frac{1}{\sqrt{2}}(\hat{\mathbf{x}}\pm i\hat{\mathbf{y}}),\;\mathbf{e}_{0}= \hat{\mathbf{z}}\]