\bar{\bar{\mathrm{O}}}_{3}\mathbf{b}^E_3
        \end{split}&
        \begin{split}             \mathbf{O}^M&=\frac{\sqrt{105 \pi} E_0}{i \eta k^5} \bar{\bar{\mathrm{O}}}_{3}\mathbf{b}^M_3
        \end{split}
    \end{aligned}$$

\[eqn:multipole\]

where $\mathbf{d}^p = [d^p_x, d^p_y, d^p_z]^\top$, $\mathbf{Q}^p = [Q^p_{xx}, Q^p_{xy}, Q^p_{xz}, Q^p_{yy}, Q^p_{yz}]^\top$, $\mathbf{O}^p = [O^p_{xxx}, O^p_{xxy}, O^p_{xxz}, O^p_{xyy}, O^p_{xyz}, O^p_{yyy}, O^p_{yyz}]^\top$, $\mathbf{b}^{p}_{n} = [b^{p}_{n,-n},b^{p}_{n,-n+1},\cdots,b^{p}_{n,n-1},b^{p}_{n,n}]^\top$ with superscript $p$ = $E$ or $M$ denoting electric or magnetic multipoles, respectively. The basis transformation matrices are introduced as

$$        \bar{\bar{\mathrm{O}}}_1
        =\begin{bmatrix}
        1 & 0 & -1 \\
        -i & 0 & -i \\
        0 & \sqrt{2} & 0
        \end{bmatrix}$$

$$        \bar{\bar{\mathrm{O}}}_2=
        \begin{bmatrix}
        1 & 0 & -\sqrt{2/3} & 0 & 1 \\
        -i & 0 & 0 & 0 & i \\
        0 & 1 & 0 & -1 & 0 \\
        -1 & 0 & -\sqrt{2/3} & 0 & -1 \\
        0 & -i & 0 & -i & 0
        \end{bmatrix}$$ $$        \bar{\bar{\mathrm{O}}}_3=
        \begin{bmatrix}
        1 & 0 & -\sqrt{3/5} & 0 & \sqrt{3/5} & 0 & -1 \\
        -i & 0 & i\sqrt{1/15} & 0 & i\sqrt{1/15} & 0 & -i \\
        0 & \sqrt{2/3} & 0 & -\sqrt{4/5} & 0 & \sqrt{2/3} & 0 \\
        -1 & 0 & -\sqrt{1/15} & 0 & \sqrt{1/15} & 0 & 1 \\
        0 & -i\sqrt{2/3} & 0 & 0 & 0 & 