i\sqrt{2/3} & 0 \\
        i & 0 & i\sqrt{3/5} & 0 & i\sqrt{3/5} & 0 & i \\
        0 & -\sqrt{2/3} & 0 & -\sqrt{4/5} & 0 & -\sqrt{2/3} & 0
        \end{bmatrix}$$

The transformation relations for ED, MD, EQ cases (Eqns. \[eqn:multipole\]a–c) can be found in previous work [@Muhlig2011; @Grahn2012], although the expression for exact Cartesian multipoles has only been published recently [@Alaee2018; @Fernandez-Corbaton2015].

Finally, the farfield radiated fields from the multipoles are

$$\begin{aligned}
        \begin{split}             \mathbf{E} &= 
            \frac{k^2}{4\pi\epsilon} \frac{e^{ikr}}{r}
            {\Bigg[}
            \mathbf{n} \times (\mathbf{d}^E \times \mathbf{n})
            -\frac{ik}{2} \mathbf{n} \times(\hat{Q}^E \times \mathbf{n})
            -\frac{k^2}{4} \mathbf{n} \times(\hat{O}^E \times \mathbf{n})
            {\Big]}\\
            &\hspace{5mm}
            +\frac{\eta k^2}{4\pi} \frac{e^{ikr}}{r}
            {\Bigg[}
            \mathbf{d}^M \times \mathbf{n}
            -\frac{ik}{2} (\hat{Q}^M \times \mathbf{n})
            -\frac{k^2}{4} (\hat{O}^M \times \mathbf{n})
            {\Big]}
        \end{split}\\
        \begin{split}             \mathbf{H} &= 
            -\frac{\omega k}{4\pi} \frac{e^{ikr}}{r}
            {\Bigg[}
            \mathbf{d}^E \times \mathbf{n}
            -\frac{ik}{2} (\hat{Q}^E \times \mathbf{n})
            -\frac{k^2}{4} (\hat{O}^E \times \mathbf{n})
            {\Big]}\\
            &\hspace{5mm}
            +\frac{k^2}{4\pi} \frac{e^{ikr}}{r}
            {\Bigg[}
            \mathbf{n} \times (\mathbf{d}^M \times \mathbf{n})
            -\frac{ik}{2} \mathbf{n} \times (\hat{Q}^M \times \mathbf{n})
            -\frac{k^2}{4} \mathbf{n} \times (\hat{O}^M \times \mathbf{n})
            {\Big]}
        \end{split}
    \end{aligned}$$

\[eqn:radiationfield\]

where $\hat{Q}^p_\alpha = Q^p_{\alpha\beta}n_\beta$ and $\hat{O}^p_\alpha = O^p_{\alpha\beta\gamma}n_\beta n_\gamma$. Total radiation power by multipoles is $$    \begin{split}
        P &= 
        \frac{\omega k^3}{12 \pi \epsilon} \sum_{\alpha}{|d^E_\alpha|^2}
        + \frac{\omega k^5}{160 \pi \epsilon} \sum_{\alpha\beta}{|Q^E_{\alpha\beta}|^2}
        + \frac{\omega k^7}{1680 \pi \epsilon} \sum_{\alpha\beta\gamma}{|O^E_{\alpha\beta\gamma}|^2}\\
        &\hspace{5mm}
        + \frac{\eta k^4}{12 \pi} \sum_{\alpha}{|d^M_\alpha|^2}
        