From e6320802a065efd439bd08afb1c82ec67291da18 Mon Sep 17 00:00:00 2001 From: "Steven G. Johnson" Date: Thu, 9 Jan 2025 16:09:36 -0500 Subject: [PATCH] Update doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md --- doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md b/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md index da53432bb..45c3ba211 100644 --- a/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md +++ b/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 *nonaxisymmetric* even if it is on-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) which involves an integration over $\phi$ resulting in a radiation pattern that is axisymmetric.) -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 at $m = \pm 1$ (although it is possible to combine these into a single $m=+1$ simulation at $\pm \omega$ as described below). 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 on-axis (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 at $m = \pm 1$ (although it is possible to combine these into a single $m=+1$ simulation at $\pm \omega$ as described below). 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 $e^{im\phi}$. The Fourier-series expansion involves nonnegative $m$ values because the $-m$ fields can be obtained directly from the $+m$ fields using the relation $\vec{E}_{m}(r, z)e^{im\phi-i\omega t} = \left[\vec{E}_{-m}(r, z)e^{-im\phi+i\omega t} \right]^*$. There is a similar relation for $\vec{H}_{m}$. This involves taking the $\vec{E}_{m}$ fields at $-\omega$ (not $\omega$!) and conjugating them to get the $\vec{E}_{-m}$ fields at $\omega$. Implementing this in a simulation involves two aspects: (1) two sources with frequencies $\omega$ and $-\omega$ and (2) two DFT field monitors at $\omega$ and $-\omega$. (1) is necessary because it ensures the source current $\mathcal{J}$ is purely real and therefore does not change under a complex conjugation. We will compute the radiation pattern for $E_x$ and $E_y$ dipoles at $r = 0.1$ μm.