Update Cylindrical_Coordinates.md (#2965)

* Update Cylindrical_Coordinates.md

* Update doc/docs/Python_Tutorials/Cylindrical_Coordinates.md

* Update doc/docs/Python_Tutorials/Cylindrical_Coordinates.md

doc/docs/Python_Tutorials/Cylindrical_Coordinates.md

@@ -898,11 +898,11 @@ 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 ($m = 0, 1, ..., M$), 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): $\vec{E}_{tot}(\theta, \phi) = \vec{E}_{m=0}(\theta)e^{im\phi} + 2\sum_{m=1}^M \Re\{\vec{E}_m(\theta)e^{im\phi}\}$ and $\vec{H}_{tot}(\theta, \phi) = \vec{H}_{m=0}(\theta)e^{im\phi} + 2\sum_{m=1}^M \Re\{\vec{H}_m(\theta)e^{im\phi}\}$.  Note that $\vec{E}_m$ and $\vec{H}_m$ are generally complex, and are conjugates for $\pm m$.
+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 DFT-monitor 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.  As a result, the procedure for computing the extraction efficiency within an angular cone for a dipole source at $r > 0$ involves three steps:
+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:
 
 1. For each simulation in the Fourier-series expansion ($m = 0, 1, ..., M$), 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.
 2. Obtain the powers $P(\theta)$ from these far fields by summing: $P(\theta) = \int_{0}^{2\pi} \left[ P_{m=0}(\theta) + 2\sum_{m=1}^{M} P_m(\theta)  \right] d\phi =  2\pi \Re\left[ \left[\vec{E}_{m=0}(\theta) \times \vec{H}^*_{m=0}(\theta)\right]\cdot\hat{r} + 2\sum_{m=1}^M \left[\vec{E}_{m}(\theta) \times \vec{H}^*_{m}(\theta)\right]\cdot\hat{r} \right]$.