Update doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md

doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md

@@ -1710,7 +1710,7 @@ Earlier we showed how to compute the [radiation pattern of an antenna with linea
 
 In this example, the radiation pattern is computed for $\phi = 0$ (i.e., the $rz$ or $xz$ plane). Note that the radiation pattern for an $\hat{x}$ or $\hat{y}$ polarized dipole is *not* rotationally invariant about the $z$ axis. This means that the radiation pattern depends on the choice of $\phi$. (This is different than the computation of the [extraction efficiency for nonaxisymmetric dipoles](Cylindrical_Coordinates.md#nonaxisymmetric-dipole-sources) for which the radiation pattern is rotationally invariant because the dipoles are arranged along the circumference of a circle.)
 
-For (1), there are two dipole configurations: $E_x$ and $E_z$. An $E_z$ dipole is positioned at $r = 0$ with $m = 0$. This involves a single simulation. An $E_x$ dipole at $r = 0$, however, involves the superposition of left- and right-circularly polarized dipoles ($E_r \pm iE_\phi$) as described in [Tutorial/Scattering Cross Section of a Finite Dielectric Cylinder](Cylindrical_Coordinates.md#scattering-cross-section-of-a-finite-dielectric-cylinder). This requires *two* simulations. The computation of the radiation pattern of an $E_x$ dipole at $r = 0$ is different from the [computation of its extraction efficiency](Local_Density_of_States.md#extraction-efficiency-of-a-light-emitting-diode-led) which involves a *single* $E_r$ source with either $m = +1$ or $m = -1$. This is because the latter calculation involves a circularly polarized source which emits exactly half the power as a linearly polarized source even though their radiation patterns are different: $\frac{1}{2}(1 + \cos^2\theta)$ vs. $\cos^2\theta$.
+For (1), there are two dipole configurations: $E_x$ and $E_z$. An $E_z$ dipole is positioned at $r = 0$ with $m = 0$. This involves a single simulation. An $E_x$ dipole at $r = 0$, however, involves the superposition of left- and right-circularly polarized dipoles ($E_r \pm iE_\phi$) as described in [Tutorial/Scattering Cross Section of a Finite Dielectric Cylinder](Cylindrical_Coordinates.md#scattering-cross-section-of-a-finite-dielectric-cylinder). This requires *two* simulations. The computation of the radiation pattern of an $E_x$ dipole at $r = 0$ is different from the [computation of its extraction efficiency](Local_Density_of_States.md#extraction-efficiency-of-a-light-emitting-diode-led) which involves a *single* $E_r$ source with either $m = +1$ or $m = -1$.  This is because extraction efficiency involves integration over $\phi$, in which case the $m = \pm 1$ cross terms cancel and you simply get identical integrated powers from $m = \pm 1$.
 
 For (2), an $E_x$ (or equivalently an $E_r$) dipole positioned at $r > 0$ requires a [Fourier-series expansion of the fields](Cylindrical_Coordinates.md#nonaxisymmetric-dipole-sources) from an $E_r$ "ring" current source with azimuthal dependence $\exp(im\phi)$. The $m = -1$ fields can be obtained directly from the $m = +1$ fields using the formulas $(E_r, E_\phi, E_z)_m = (E_r, -E_\phi, E_z)_{-m}$ and $(H_r, H_\phi, H_z)_m = (-H_r, H_\phi, -H_z)_{-m}$. These formulas can be used to simplify the expressions for the Fourier-series expansion of the fields at $\phi = 0$: $\vec{E}_{tot}(\theta) = (E_r, 0, E_z)_{m=0} + 2\sum_{m=1}^M (E_r, 0, E_z)_m$ and $\vec{H}_{tot}(\theta) = (0, H_\phi, 0)_{m=0} + 2\sum_{m=1}^M (0, H_\phi, 0)_m$. An $E_y$ (or equivalently an $E_\phi$) dipole involves flipping the sign of the $-m$ fields resulting in: $\vec{E}_{tot}(\theta) = (0, E_\phi, 0)_{m=0} + 2\sum_{m=1}^M (0, E_\phi, 0)_m$ and $\vec{H}_{tot}(\theta) = (H_r, 0, H_z)_{m=0} + 2\sum_{m=1}^M (H_r, 0, H_z)_m$. We will compute the radiation pattern for $E_x$ and $E_y$ dipoles at $r = 0.1$ μm.