fix formula in tutorial for extraction efficiency within angular cone

doc/docs/Python_Tutorials/Cylindrical_Coordinates.md

@@ -622,20 +622,19 @@ Results are shown in the table below. At this resolution, the relative error is
 
 The extraction efficiency computed thus far is for *all* angles. To compute the extraction efficiency within an angular cone (i.e., as part of an overall calculation of the [radiation pattern](Near_to_Far_Field_Spectra.md#radiation-pattern-of-an-antenna)), we would need to surround the emitting structure with a closed box of near-field monitors. However, because the LED slab is infinitely extended a non-closed box must be used.  This will introduce [truncation errors](Near_to_Far_Field_Spectra.md#truncation-errors-from-a-non-closed-near-field-surface) which are unavoidable.
 
-In principle, computing extraction efficiency first involves computing the radiation pattern $P(\theta, \phi)$ (the power as a function of [spherical angles](https://en.wikipedia.org/wiki/Spherical_coordinate_system)), and then computing the fraction of this power (integrated over the azimuthal angle $\phi$) that lies within a given angular cone $\theta \in [0,\theta_0]$.   It turns out that there is a simplification because we can compute the azimuthal $P(\theta) = \int P(\theta, \phi) d\phi$ more efficiently without first computing $P(\theta, \phi)$.   However, it is instructive to explain how to compute both $P(\theta, \phi)$ and the extraction efficiency.
+In principle, computing extraction efficiency first involves computing the radiation pattern $P(\theta, \phi)$ (the power as a function of [spherical angles](https://en.wikipedia.org/wiki/Spherical_coordinate_system)), and then computing the fraction of this power (integrated over the azimuthal angle $\phi$) that lies within a given angular cone $\theta \in [0,\theta_0]$.   By convention, $\theta = 0$ is in the $+z$ direction and $\theta = \pi / 2$ is $+r$. It turns out that there is a simplification because we can compute the azimuthal $P(\theta) = \int P(\theta, \phi) d\phi$ more efficiently without first computing $P(\theta, \phi)$.   However, it is instructive to explain how to compute both $P(\theta, \phi)$ and the extraction efficiency.
 
 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 $\pm m$.
 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 $P(\theta) = \int P(\theta, \phi) d\phi$ in order to obtain the extraction efficiency, 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 four 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.  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: $P(\theta) = \int_{0}^{2\pi} \left[ P_{m=0}(\theta) + 2\sum_{m=1}^{M} P_m(\theta)  \right] d\phi =  2\pi \Re\left[ \left[\vec{E}_{m=0}(\theta) \times \vec{H}^*_{m=0}(\theta)\right]\cdot\hat{r} + 2\sum_{m=1}^M \left[\vec{E}_{m}(\theta) \times \vec{H}^*_{m}(\theta)\right]\cdot\hat{r} \right]$.
-3. Compute the fraction of the radial Poynting flux within an angular cone $\left[ \int_0^\theta P(\theta') d\theta' \right] / \left[ \int_0^{\pi/2} P(\theta') d\theta' \right]$ by some discretized integral, e.g. a [trapezoidal rule](https://en.wikipedia.org/wiki/Trapezoidal_rule).
-4. Multiply (3) by the extraction efficiency.
+3. Compute the extraction efficiency within an angular cone $\left[ \int_0^\theta \sin(\theta') P(\theta') d\theta' \right] / \left[ \int_0^{\pi/2} \sin(\theta') P(\theta') d\theta' \right]$ by some discretized integral, e.g. a [trapezoidal rule](https://en.wikipedia.org/wiki/Trapezoidal_rule). See [Tutorial/Cylindrical Coordinates/Radiation Pattern of a Disc in Cylindrical Coordinates](Near_to_Far_Field_Spectra.md#radiation-pattern-of-a-disc-in-cylindrical-coordinates).
 
 The simulation script is in [examples/point_dipole_cyl.py](https://github.com/NanoComp/meep/blob/master/python/examples/point_dipole_cyl.py).