+r_\gamma r_\alpha J_\beta \\
            &\hspace{10mm}
            -\frac{1}{5}\delta_{\alpha\beta}(r^2J_\gamma+2{\mathbf{(r \cdot J)}}r_\gamma)
            -\frac{1}{5}\delta_{\beta\gamma}(r^2J_\alpha+2{\mathbf{(r \cdot J)}}r_\alpha)
            -\frac{1}{5}\delta_{\gamma\alpha}(r^2J_\beta+2{\mathbf{(r \cdot J)}}r_\beta)\Big]
            \frac{j_2(kr)}{(kr)^2}\\
            &\hspace{5mm}
            +\frac{3k^2}{4}
            \Big[7{\mathbf{(r \cdot J)}}r_\alpha r_\beta r_\gamma
            -r^2(r_\alpha r_\beta J_\gamma +r_\beta r_\gamma J_\alpha +r_\gamma r_\alpha J_\beta)\\
            &\hspace{10mm}
            +\frac{1}{5}\delta_{\alpha\beta}r^2(r^2J_\gamma+5{\mathbf{(r \cdot J)}}r_\gamma)
            +\frac{1}{5}\delta_{\beta\gamma}r^2(r^2J_\alpha+5{\mathbf{(r \cdot J)}}r_\alpha)
            +\frac{1}{5}\delta_{\gamma\alpha}r^2(r^2J_\beta+5{\mathbf{(r \cdot J)}}r_\beta)\Big]
            \frac{j_4(kr)}{(kr)^4}
            {\Bigg\}}
        \end{split}\\
        \begin{split}             O^M_{\alpha\beta\gamma} &=
            \frac{35}{2}
            \int{\mathrm{d}^3\mathbf{r}~}{\Big[}
            r_\alpha r_\beta {\mathbf{(r \times J)}}_\gamma
            +r_\beta r_\gamma {\mathbf{(r \times J)}}_\alpha
            +r_\gamma r_\alpha {\mathbf{(r \times J)}}_\beta\\
            &\hspace{5mm}
            -\frac{1}{5}\delta_{\alpha\beta}r^2{\mathbf{(r \times J)}}_\gamma
            -\frac{1}{5}\delta_{\beta\gamma}r^2{\mathbf{(r \times J)}}_\alpha
            -\frac{1}{5}\delta_{\gamma\alpha}r^2{\mathbf{(r \times J)}}_\beta
            {\Big]}
            \frac{j_3(kr)}{(kr)^3}
        \end{split}
    \end{aligned}$$

where $\alpha, \beta, \gamma = x, y, z$, and the multipoles are symmetrical and traceless [@Gurvitz2019]. The familiar approximate Cartesian multipoles and toroidal multipoles can be readily obtained by taking the long wavelength limit [@Alaee2018]. See Appendix \[appendix:multipoles\] for more details on Cartesian multipoles.

An important result from Ref. [@Alaee2018] is that the exact Cartesian multipoles can be expressed from the spherical multipoles. They have identical physical meaning, but are expressed in different basis, which can be systematically transformed to each other as

$$\begin{aligned}
        \begin{split}             \mathbf{d}^E&=\frac{\sqrt{3 \pi} E_0}{\eta \omega k^2} \bar{\bar{\mathrm{O}}}_1\mathbf{b}^E_1
        \end{split}&
        \begin{split}             \mathbf{d}^M&=\frac{\sqrt{3 \pi} E_0}{i \eta k^3} \bar{\bar{\mathrm{O}}}_1\mathbf{b}^M_1
        \end{split}\\
        \begin{split}             \mathbf{Q}^E&=\frac{\sqrt{20 \pi} E_0}{\eta \omega k^3} \bar{\bar{\mathrm{O}}}_{2}\mathbf{b}^E_2
        \end{split}&
        \begin{split}             \mathbf{Q}^M&=\frac{\sqrt{20 \pi} E_0}{i \eta k^4} \bar{\bar{\mathrm{O}}}_{2}\mathbf{b}^M_2
        \end{split}\\
        \begin{split}             \mathbf{O}^E&=\frac{\sqrt{105 \pi} E_0}{\eta \omega k^4} 