Implement proper integration weights in cylindrical coordinates

[smartalecH]

As discussed in #2452, the integration weights in cylindrical coordinates are implemented identically as those in cartesian coordinates (which is incorrect). The procedure to fix this is outlined in the code itself:

meep/src/loop_in_chunks.cpp

Lines 112 to 113 in 1fe3899

	Integration Weights in Cylindrical Coordinates
	FIXME: implement this below?

but it's not trivial to implement, particularly with cases at or crossing r=0 (which happens a lot...)

Anecdotally, this doesn't seem to be an issue if R is rather large. However, for smaller cells, the error compounds substantially (and doesn't "easily" disappear with resolution).

[smartalecH]

@oskooi, i've got a draft for a fix to this. You mentioned you have a test problem. Can you link me to it?

[oskooi]

The test that I had in mind is to demonstrate that the normalized radiated flux $P(r) / r^2$ is independent of the location of the point source for $r \approx 0$. Currently, $P(r) / r^2 \rightarrow C$ for a constant $C$ only for $r \rightarrow \infty$ as shown in the plots of #2108 (comment).

To set this up, we can simply modify the tutorial script python/examples/point_dipole_cyl.py. The function led_flux returns the radiated flux (flux_air) and the total emitted flux (flux_src) separately as a 2-tuple:

meep/python/examples/point_dipole_cyl.py

Lines 141 to 149 in b415cbf

	for m in ms:
	flux_air, flux_src = led_flux(
	layer_thickness,
	dipole_height,
	rp,
	m,
	)
	flux_air_tot += flux_air if m == 0 else 2 * flux_air
	flux_src_tot += flux_src if m == 0 else 2 * flux_src

The key metric to check would be flux_air_tot / rpos**2 vs. rpos for several values of rpos near 0.

Note that this test does not use the LDOS feature.

[oskooi]

Also note that in Tutorial/Nonaxisymmetric Dipole Sources, I intentionally used "large" values of the source position (rpos > 3.5) to ensure the extraction-efficiency results were reasonably accurate:

meep/python/examples/point_dipole_cyl.py

Lines 132 to 133 in b415cbf

	rpos = [3.5, 6.7, 9.5]
	for rp in rpos:

Hopefully once this bug is fixed, we can modify this tutorial to include point-source locations closer to $r = 0$.