Further Details
Units
SI units are employed everywhere.
Duality Relations
In many places the duality relations [1, p. 120]
| electric current | $\rightarrow$ | magnetic current |
|---|---|---|
| $\bm e$ | $\rightarrow$ | $\bm h$ |
| $\bm h$ | $\rightarrow$ | $-\bm e$ |
| $\varepsilon$ | $\rightarrow$ | $\mu$ |
| $\mu$ | $\rightarrow$ | $\varepsilon$ |
are employed in order to compute, e.g., the field of the magnetic current counterparts to the electric currents.
Rotations
Due to the rotational symmetry the fields for different orientations of the excitations can be computed via rotations.
Plane Wave
A plane wave excitation with arbitrary direction $\hat{\bm k}$ and polarization $\hat{\bm p}$ (forming a valid combination) can be related to the case $\hat{\bm k} = \hat{\bm e}_z$ and polarization $\hat{\bm p} = \hat{\bm e}_x$ by a rotation matrix
\[\bm R = \begin{bmatrix} \hat{\bm p} & \hat{\bm k} \times \hat{\bm p} & \hat{\bm k} \end{bmatrix} \,.\]
It constitutes a change from one right-handed orthogonal basis $(\hat{\bm e}_x^\prime , \hat{\bm e}_y^\prime, \hat{\bm e}_z^\prime )$ to another right-handed orthogonal basis $(\hat{\bm e}_x = \hat{\bm p}, \hat{\bm e}_y = \hat{\bm k} \times \hat{\bm p}, \hat{\bm e}_z = \hat{\bm k}$. Hence, $\bm R$ relates vectors (and points) $\bm v'$ in the primed coordinate system $(x',y',z')$ to vectors (and points) $\bm v$ in the unprimed one $(x,y,z)$ by
\[\bm v = \bm R \bm v' \qquad \text{and} \qquad \bm v' = \bm R^\mathrm{T} \bm v\]
leveraging that $\bm R^{-1} = \bm R^\mathrm{T}$.
Ring-Currents and Dipoles
Ring-currents and dipoles are characterized by only one vector $\hat{\bm p}$ (defining the orientation). An arbitrary $\hat{\bm p}$ can be related to an initial one $\hat{\bm p}_0$ (e.g., $\hat{\bm p}_0 = \hat{\bm e}_z^\prime$) by a right-handed rotation around a rotation axis $\hat{\bm a} = (a_x, a_y, a_z)$ by an angle $\alpha$. The rotation axis is given by
\[\hat{\bm a} = \cfrac{\hat{\bm p}_0 \times \hat{\bm p}}{|\hat{\bm p}_0 \times \hat{\bm p}|}\]
and the angle $\alpha$ is encoded in
\[\cos(\alpha) = \hat{\bm p}_0 \cdot \hat{\bm p} \qquad \text{and} \qquad \sin(\alpha) = |\hat{\bm p}_0 \times \hat{\bm p}|\,.\]
The rotation matrix is then found by the Rodrigues' rotation formula as
\[\bm R = \mathbf{I} + \sin(\alpha)\bm{K} + (1 - \cos(\alpha)) \bm{K}^2\]
with
\[\bm K = \begin{bmatrix} 0 & -a_z & a_y \\ a_z & 0 & -a_x \\ -a_y & a_x & 0 \end{bmatrix} \,.\]
Vectors (and points) $\bm v'$ in the initial coordinate system $(x',y',z')$ (with, e.g., $\hat{\bm e}_z^\prime = \hat{\bm p}_0$) are then related to vectors (and points) $\bm v$ in the unprimed one $(x,y,z)$ by
\[\bm v = \bm R \bm v' \qquad \text{and} \qquad \bm v' = \bm R^\mathrm{T} \bm v\]
leveraging that $\bm R^{-1} = \bm R^\mathrm{T}$.