Hello mathemagician folks and math enthusiasts,
I am currently working on a project that requires me to simulate electromagnetic fields from dipole antennas. I am working with a set of 3 antennas, all placed at the origin but each rotated around one of the cartesian coordinate system axis. The following link will help you visualize the dipoles' placement:
The field generated by each antenna can be viewed as a set of spherical equations, but to avoid redundancy, let us consider only the electric field in the $(\theta,\phi)$ coordinate.
For the sake of this discussion, let us define the following notations:
$\Lambda^{(\alpha)}_{\beta}$ is the vector/quantity $\Lambda$ belonging to the $\alpha$ polarized (directed) dipole, in the geographical $\beta$ coordinate.
$\Lambda^{(\alpha)}_{\beta_{\alpha}}$ is the vector/quantity $\Lambda$ belonging to the $\alpha$ polarized (directed) dipole, in the local $\beta$ coordinate of to the $\alpha$ polarized (directed) dipole.
$\alpha$ is referring to the unit vector from the cartesian coordinate system. In our discussion x, y or z.
The geographical spherical coordinate system is following the ISO 31-11, Misner et al. (1973, p. 205) convention, i.e, $\theta$ starts at z, $\phi$ starts at x
We define $R(\theta,\phi)$ to be a function matrix that enables us to convert data represented in local coordinates to geographical coordinates. $$ R(\theta^{(\alpha)}, \phi^{(\alpha)})=\begin{pmatrix} f(\theta^{(\alpha)},\phi^{(\alpha)})\cdot\hat{\theta} & g(\theta^{(\alpha)},\phi^{(\alpha)})\cdot\hat{\phi}\\[0.3cm] h(\theta^{(\alpha)},\phi^{(\alpha)})\cdot\hat{\theta} & i(\theta^{(\alpha)},\phi^{(\alpha)})\cdot\hat{\phi} \end{pmatrix}{} \quad $$ The elements inside this matrix are to be found. $$ \begin{pmatrix} E^{(\alpha)}_{\theta}\\[0.3cm] E^{(\alpha)}_{\phi} \end{pmatrix}{} = R(\theta^{(\alpha)}, \phi^{(\alpha)}) \begin{pmatrix} E^{(\alpha)}_{\theta_{\alpha}}\\[0.3cm] E^{(\alpha)}_{\phi_{\alpha}} \end{pmatrix}{} \quad\text{for } \alpha = x,y,z $$
For the $\hat{z}$-polarized dipole, its electric field is naturally in geographical $(\theta, \phi)$ coordinates, we therefore obtain: $$ \begin{cases} \underline{E}^{(z)}_{\theta}=\underline{E}^{(z)}_{\theta_{z}}\\[0.3cm] \underline{E}^{(z)}_{\phi}=\underline{E}^{(z)}_{\phi_{z}} \end{cases}{} $$\par For the $\hat{x}$-polarized dipole we obtain: $$ \begin{pmatrix} E^{(x)}_{\theta}\\ E^{(x)}_{\phi} \end{pmatrix}{} = R(\theta^{(x)}, \phi^{(x)}) \begin{pmatrix} E^{(x)}_{\theta_{x}}\\ E^{(x)}_{\phi_{x}} \end{pmatrix} $$ For the third dipole, $\hat{y}$-polarized we obtain: $$ \begin{pmatrix} E^{(y)}_{\theta}\\ E^{(y)}_{\phi} \end{pmatrix}{} = R(\theta^{(y)},\phi^{(y)}) \begin{pmatrix} E^{(y)}_{\theta_{y}}\\ E^{(y)}_{\phi_{y}} \end{pmatrix}$$
The Rotation matrix derivation is where I have troubles. Any tips on where to start?