Clarify that the radiation pattern from a ring-current source with ci…

…rcular polarization has no azimuthal dependence (#2978)

* clarify that ring current source with circular polarization produces a radiation pattern with no azimuthal dependence

* Update doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md

* Update doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md

* Update doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md

---------

Co-authored-by: Steven G. Johnson <stevenj@mit.edu>

doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md

@@ -381,15 +381,15 @@ if __name__ == '__main__':
 Radiation Pattern of a Disc in Cylindrical Coordinates
 ------------------------------------------------------
 
-The near-to-far field transformation feature can also be used in [cylindrical coordinates](Cylindrical_Coordinates.md). As a demonstration, we compute the radiation pattern of a dielectric disc and verify Poynting's theorem: the total radiated flux computed from the far fields is equivalent to using the near fields via `add_flux`. (The same result is demonstrated in [Tutorial/Radiation Pattern of an Antenna](#radiation-pattern-of-an-antenna) for 2D Cartesian coordinates.)
+The near-to-far field transformation feature can also be used in [cylindrical coordinates](Cylindrical_Coordinates.md). As a demonstration, we compute the radiation pattern of a dielectric disc and verify Poynting's theorem: the total radiated flux computed from the far fields is equivalent to using the near fields via `add_flux`. This example involves a ring-current source generating a circularly polarized field. (The radiation pattern of an off-axis *point* dipole with *linear* polarization is demonstrated in [Tutorial/Radiation Pattern of an Antenna in Cylindrical Coordinates](#radiation-pattern-of-an-antenna-in-cylindrical-coordinates).)
 
-The simulation consists of an $E_r$ point-dipole source ($\lambda$ = 1.0 μm) at $r$ = 0.6 μm embedded within a disc (radius of 1.2 μm) of index $n$ = 2.4 above a perfect-metallic ground plane. This is similar to the configuration in [Tutorial/Extraction Efficiency of a Light-Emitting Diode (LED)](Local_Density_of_States.md#extraction-efficiency-of-a-light-emitting-diode-led). Unlike the infinitely extended slab of the LED, a *finite* structure such as the disc ensures that all the power from the dipole emitter is radiated. The LED contains waveguide modes which are more challenging to disentagle from the radiated power.
+The simulation consists of an $E_r$ "ring"-current source with $\lambda$ = 1.0 μm and radius of 0.6 μm embedded within a disc of radius 1.2 μm, refractive index $n$ = 2.4, and thickness $0.7 \lambda / n$. The disc is positioned above a perfect-metallic ground plane. The source has azimuthal dependence $e^{im\phi}$ with $m = -1$, which leads to circularly polarized fields (unlike $|m| \ne 1$). Unlike the [infinitely extended slab of the LED](Local_Density_of_States.md#extraction-efficiency-of-a-light-emitting-diode-led), a *finite* structure such as the disc ensures that all the power from the source is radiated away. The LED contains waveguide modes which are more challenging to disentagle from the radiated power.
 
 A schematic of the simulation layout is shown below. The flux and near-field monitors (shown in blue) are overlapping.
 
 ![](../images/disc_radiation_layout.png#center)
 
-Obtaining the radiation pattern $P(\theta)$ of the disc involves computing the radial (or "outgoing") flux from the far fields along the circumference of a quarter circle (i.e. angular range of $[0, \pi/2]$). The radius $r$ of the circle needs to be sufficiently large ($\gg \lambda$) to ensure accurate results but is otherwise arbitrary. The total flux is then computed by integrating $P(\theta)$ over the surface of a hemisphere with radius $r$ in [spherical coordinates](https://en.wikipedia.org/wiki/Spherical_coordinate_system):
+Obtaining the radiation pattern $P(\theta)$ of the disc involves computing the radial (or "outgoing") flux from the far fields along the circumference of a quarter circle with angular range of $[0, \pi/2]$. Note that the radiation pattern $P(\theta)$ for fields $\sim e^{im\phi}$ (for any $m$) is independent of $\phi$. The radius $r$ of the far-field circle needs to be sufficiently large ($\gg \lambda$) to ensure accurate results but is otherwise arbitrary. The total flux is then computed by integrating $P(\theta)$ over the surface of a hemisphere with radius $r$ in [spherical coordinates](https://en.wikipedia.org/wiki/Spherical_coordinate_system):
 
 $$P_{total} = \int_0^{2\pi} \int_0^{\frac{\pi}{2}} P(\theta) r^2 \sin(\theta) d\theta d\phi = 2 \pi r^2 \sum_{n=0}^{N-1} P(\theta_n) \sin(\theta_n) \Delta \theta$$