Update doc/docs/Python_Tutorials/Cylindrical_Coordinates.md

doc/docs/Python_Tutorials/Cylindrical_Coordinates.md

@@ -899,7 +899,7 @@ In principle, computing extraction efficiency first involves computing the radia
 To compute the radiation pattern $P(\theta, \phi)$ requires three steps:
 
 1. For each simulation in the Fourier-series expansion ($m = -M, \ldots , 0, \ldots, M$), compute the far fields $\vec{E}_m$, $\vec{H}_m$ for the desired $\theta$ points in the $rz$ ($\phi = 0$) plane, at an "infinite" radius (i.e., $R \gg \lambda$) using a [near-to-far field transformation](../Python_User_Interface.md#near-to-far-field-spectra).
-2. Obtain the *total* far fields at these points, for a given $\phi$ by summing the far fields from (1): $\vec{E}_{tot}(\theta, \phi) = \vec{E}_{m=0}(\theta)e^{im\phi} +\sum_{m=-M}^M \vec{E}_m(\theta)e^{im\phi}$ and $\vec{H}_{tot}(\theta, \phi) = \vec{H}_{m=0}(\theta)e^{im\phi} + \sum_{m=-M}^M \vec{H}_m(\theta)e^{im\phi}$.  Note that $\vec{E}_m$ and $\vec{H}_m$ are generally complex.  (The $\pm m$ terms are related by a mirror flip in $\phi$, and also by complex conjugation if you also flip the sign of the frequency, so it is possible to combine their calculation.)
+2. Obtain the *total* far fields at these points, for a given $\phi$ by summing the far fields from (1): $\vec{E}_{tot}(\theta, \phi) = \vec{E}_{m=0}(\theta)e^{im\phi} +\sum_{m=-M}^M \vec{E}_m(\theta)e^{im\phi}$ and $\vec{H}_{tot}(\theta, \phi) = \vec{H}_{m=0}(\theta)e^{im\phi} + \sum_{m=-M}^M \vec{H}_m(\theta)e^{im\phi}$.  Note that $\vec{E}_m$ and $\vec{H}_m$ are generally complex.  (The $\pm m$ terms are related by a mirror flip in $\phi$, and also by complex conjugation if you also flip the sign of the frequency and conjugate the source, so it is possible to combine their calculation.)
 3. Compute the radial Poynting flux $P_i(\theta_i, \phi)$ for each of $N$ points $i = 0, 1, ..., N - 1$ on the circumference using $\Re\left[\left[\vec{E}_{tot}(\theta_i, \phi) \times \vec{H}^*_{tot}(\theta_i, \phi)\right]\cdot\hat{r}\right]$.
 
 However, if you want to compute the extraction efficiency within an angular cone given $P(\theta) = \int P(\theta, \phi) d\phi$, the calculations simplify because the cross terms in $\vec{E}_{tot} \times \vec{H}^*_{tot}$ between different $m$'s integrate to zero when integrated over $\phi$ from $0$ to $2\pi$.  Thus, one can replace step (2) with a direct computation of the powers $P(\theta)$ rather than summing the fields.  Furthermore $P_{-m}(\theta, \phi) = $P_{m}(\theta, -\phi)$ so $P_{-m}(\theta) = $P_{m}(\theta)$.  As a result, the procedure for computing the extraction efficiency within an angular cone for a dipole source at $r > 0$ involves three steps: